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# Regularized Matrix Regression
Hua Zhou
Department of Statistics
North Carolina State University
Raleigh, NC 27695-8203
hau_zhou@ncsu.edu Lexin Li
Department of Statistics
North Carolina State University
Raleigh, NC 27695-8203
li@stat.ncsu.edu
###### Abstract
Modern technologies are producing a wealth of data with complex structures.
For instance, in two-dimensional digital imaging, flow cytometry, and
electroencephalography, matrix type covariates frequently arise when
measurements are obtained for each combination of two underlying variables. To
address scientific questions arising from those data, new regression methods
that take matrices as covariates are needed, and sparsity or other forms of
regularization are crucial due to the ultrahigh dimensionality and complex
structure of the matrix data. The popular lasso and related regularization
methods hinge upon the sparsity of the true signal in terms of the number of
its nonzero coefficients. However, for the matrix data, the true signal is
often of, or can be well approximated by, a low rank structure. As such, the
sparsity is frequently in the form of low rank of the matrix parameters, which
may seriously violate the assumption of the classical lasso. In this article,
we propose a class of regularized matrix regression methods based on spectral
regularization. Highly efficient and scalable estimation algorithm is
developed, and a degrees of freedom formula is derived to facilitate model
selection along the regularization path. Superior performance of the proposed
method is demonstrated on both synthetic and real examples.
Key Words: Electroencephalography; multidimensional array; Nesterov method;
nuclear norm; spectral regularization; tensor regression.
## 1 Introduction
Modern scientific applications are frequently producing data sets where the
sampling unit is not in the form of a vector but instead a matrix. Examples
include two-dimensional digital imaging data, which contain the quantized
brightness value of a given color at a number of rows and columns of pixels;
and flow cytometric data, which consist of the fluorescence intensity of
multiple cells at multiple channels. Our motivating example is a study of an
electroencephalography (EEG) data of alcoholism
(http://kdd.ics.uci.edu/datasets/eeg/eeg.data.html). The study consists of 122
subjects with two groups: an alcoholic group and a normal control group, and
each subject was exposed to a stimulus. Voltage values were measured from 64
channels of electrodes placed on the subject’s scalp for 256 time points, so
each sampling unit is a $256\times 64$ matrix. It is of scientific interest to
study the association between alcoholism and the pattern of voltage over times
and channels (Li et al.,, 2010). The generalized linear model (GLM) (McCullagh
and Nelder,, 1983) offers a useful tool for that purpose, where the response
$Y$ is the binary indicator of alcoholic or control, and the predictors
include the matrix-valued EEG data ${\bm{X}}$ and possible covariate vector
${\bm{Z}}$ such as age and gender. However, the classical GLM deals with a
vector of covariates, and the presence of matrix type covariates poses fresh
challenges to statistical analysis. First, naïvely turning a matrix into a
vector results in an exceedingly large dimensionality; for instance, for the
EEG data, the dimension is $p=256\times 64=16,384$, whereas the sample size is
only $n=122$. Second, vectorization destroys the wealth of structural
information inherently possessed in the matrix data; e.g., it is commonly
expected that the voltage values of the adjacent time points and channels are
highly correlated. Given the ultrahigh dimensionality and the complex
structure, _regularization_ becomes crucial for the analysis of such data. In
this article, we propose a novel regularization solution for regression with
matrix covariates that efficiently tackles the ultrahigh dimensionality while
preserving the matrix structure.
A large variety of regularization methods have been developed in recent years.
Among them, penalization has been playing a powerful role in stabilizing the
estimates, improving the risk property, and increasing the generalization
power in classical regressions. Popular penalization techniques include lasso
(Tibshirani,, 1996; Donoho and Johnstone,, 1994), SCAD (Fan and Li,, 2001),
fused lasso (Tibshirani et al.,, 2005), elastic net (Zou and Hastie,, 2005),
and many others. For regressions with matrix covariates, a direct approach is
to first vectorize the covariates then apply the classical penalization
techniques. Regularization helps alleviate the problem that the dimensionality
far exceeds the sample size. However, this is unsatisfactory, since it does
not incorporate the matrix structural information. More importantly, the
solution is based upon a fundamental assumption that the true underlying
signal is sparse in terms of the $\ell_{0}$ norm of the regression parameters.
In matrix regressions, however, it is often the case that the true signal is
of a low rank structure, or can be well approximated by a low rank structure.
As such, sparsity is in terms of the _rank_ of the matrix parameters, which is
intrinsically different from sparsity in the number of nonzero entries.
Figure 1: Comparison of the nuclear norm regularized estimation to the
classical lasso. The matrix covariate is $64\times 64$, and the sample size is
$500$. Top panel: True signal (left), nuclear norm regularized estimate with
minimal BIC (center), and the classical lasso estimate with minimal BIC
(right). Bottom panel: BIC along solution paths for the nuclear norm
regularization (left) and the classical lasso regularization (right).
To see how such a difference affects signal estimation in matrix regressions,
we consider the following illustrative example. We generated a normal response
$Y$ with mean, $\mu=\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{Z}}+\langle{\bm{B}},{\bm{X}}\rangle$, and variance one.
${\bm{Z}}\in\mathrm{I\\!R}\mathit{{}^{5}}$ denotes a usual vector of
covariates with standard normal entries, and
$\mbox{\boldmath$\gamma$}=(1,\ldots,1)^{\mbox{\tiny{\sf T}}}$.
${\bm{X}}\in\mathrm{I\\!R}\mathit{{}^{64\times 64}}$ denotes the matrix
covariates, of which all entries are standard normal, and ${\bm{B}}$ is the
coefficient matrix of the same size. ${\bm{B}}$ is binary, with the true
signal region, which is a cross shape in our example, equal to one and the
rest zero. The inner product between two matrices is defined as
$\langle{\bm{B}},{\bm{X}}\rangle=\langle\mathrm{vec}{\bm{B}},\mathrm{vec}{\bm{X}}\rangle=\sum_{i_{1},i_{2}}\beta_{i_{1}i_{2}}x_{i_{1}i_{2}}$,
where $\mathrm{vec}(\cdot)$ is the vectorization operator that stacks the
columns of a matrix into a vector. We sampled $n=500$ instances
$\\{(y_{i},{\bm{x}}_{i},{\bm{z}}_{i}),i=1,\ldots,500\\}$, and our goal is to
identify ${\bm{B}}$ through a regression of $y_{i}$ on
$({\bm{x}}_{i},{\bm{z}}_{i})$. We note that our problem differs from the usual
edge detection or object recognition in imaging processing (Qiu,, 2005, 2007).
In our setup, all elements of the image ${\bm{X}}$ follow the same
distribution. The signal region is defined through the coefficient image
${\bm{B}}$ and needs to be inferred from the association between $Y$ and
${\bm{X}}$ after adjusting for ${\bm{Z}}$. We also note that the total number
of entries in ${\bm{B}}$ is $4096=64^{2}$, and the number of nonzero ones is
240 (about $5.8\%$). We applied two approaches. The first is lasso to the
vectorized ${\bm{X}}$. That is, we solve the optimization problem
$\displaystyle\min_{{\bm{B}}}\,\frac{1}{2}\sum_{i=1}^{n}(y_{i}-\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{z}}_{i}-\langle{\bm{B}},{\bm{x}}_{i}\rangle)^{2}+\lambda\|\mathrm{vec}{\bm{B}}\|_{1},$
where $\|\mathrm{vec}{\bm{B}}\|_{1}$ is the $\ell_{1}$ norm of the vectorized
${\bm{B}}$, and $\lambda$ is the regularization parameter. The lower right
panel of Figure 1 displays the Bayesian information criterion (BIC) along the
lasso solution path, which suggests a model with the maximum number of
predictors (500) that is allowed given the sample size. The parameter estimate
under this model is shown in the upper right panel, which appears far away
from the truth. The second solution we consider is penalizing the nuclear norm
of ${\bm{B}}$, i.e., we solve
$\displaystyle\min_{{\bm{B}}}\,\frac{1}{2}\sum_{i=1}^{n}(y_{i}-\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{z}}_{i}-\langle{\bm{B}},{\bm{x}}_{i}\rangle)^{2}+\lambda\|{\bm{B}}\|_{*},$
where the nuclear norm $\|{\bm{B}}\|_{*}=\sum_{j}\sigma_{j}({\bm{B}})$, and
$\sigma_{j}({\bm{B}})$’s are the singular values of the matrix ${\bm{B}}$. The
nuclear norm $\|{\bm{B}}\|_{*}$ is a suitable measure of the “size” of a
matrix parameter, and is a convex relaxation of
$\text{rank}({\bm{B}})=\|\sigma({\bm{B}})\|_{0}$. This is analogous to the
$\ell_{1}$ norm for a vector (Recht et al.,, 2010). The lower left graph of
Figure 1 displays the BIC along the solution path of the nuclear norm
penalized matrix regression, and the upper middle panel shows the
corresponding estimate with minimal BIC. It is clearly seen that the nuclear
norm estimate achieves a substantially better recovery than the lasso
estimate. One might argue that fused lasso (Tibshirani et al.,, 2005) might
give a better recovery of such piecewise constant signals. However, there are
numerous low rank signals, e.g., $(01\ldots 01)^{\mbox{\tiny{\sf T}}}(10\ldots
10)$, which are extremely non-smooth and would fail fused lasso.
More generally, in this article, we propose a family regularized regression
models with matrix covariates based on spectral regularization. Our
contributions are multifold. First, we employ a spectral regularization
formulation and integrate within a generalized linear model (GLM) framework.
The resulting model works for a variety of penalization functions, including
lasso, elastic net, SCAD, and many others, as well as different types of
response variables, including normal, binary and count outcomes. Second, we
develop a highly efficient and scalable Nesterov algorithm for model
estimation with explicit, non-asymptotic convergence rate. We emphasize that
such a highly scalable algorithm is critical for analyzing large-scale and
ultrahigh-dimensional matrix data. Third, we derive the effective degrees of
freedom of selected models, which is crucial for tuning of the regularization
parameter. The result can be viewed as an extension of the degrees of freedom
development from the classical lasso model (Zou et al.,, 2007) and the group
lasso model (Yuan and Lin,, 2006) to the generalized linear matrix model. On
the other hand, our proposal is _not_ simply another variant of lasso and
alike. We aim at _matrix regression_ problems, which are important in imaging
and other scientific applications but have received relatively little
attention.
Our proposal is related to but also distinct from two recent developments
involving matrix data. The first is a recent proposal of a family of
generalized linear models with matrix or tensor (multidimensional arrays)
covariates (Zhou et al.,, 2012). The basic idea is to impose a particular low
rank structure (CP decomposition) on ${\bm{B}}$ and then introduce a sparse
penalty on the coefficients of ${\bm{B}}$. That solution fits the model at a
_fixed_ rank of the matrix/tensor regression parameters, and thus corresponds
to the hard thresholding in the classical vector covariate case. In contrast,
our solution does not fix the rank of the parameters and is a soft
thresholding procedure. Moreover, even when using a convex penalty function
such as the lasso penalty, the approach of Zhou et al., (2012) involves a
challenging non-convex optimization task, whereas the solution in this paper
remains a convex problem. We also note that Hung and Wang, (2011) considered
matrix logistic regression, which is a special case of Zhou et al., (2012),
and they did not investigate any sparsity regularization. The second related
work is the line of research in matrix completion, where nuclear norm type
regularization has been widely employed (Candès and Recht,, 2009; Mazumder et
al.,, 2010; Cai et al.,, 2010). However, the two approaches are different in
that, the matrix completion problem aims to recover a low rank matrix when
only a small portion of its entries are observed., whereas our approach
concerns about regressions with matrix covariates.
The rest of the article is organized as follows. We formulate the spectral
regularization for matrix regression in Section 2, and develop a highly
scalable algorithm for the associated optimization in Section 3. We derive the
degrees of freedom formula in Section 4, and investigate the numerical
performance of the proposed method in Section 5. We conclude the paper with a
discussion of potential future research in Section 6. We delegate all the
technical proofs to the Appendix.
## 2 Spectral Regularization
We first fix the notations. For any matrix
${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times p_{2}}}$,
$\sigma({\bm{B}})=(\sigma_{1}({\bm{B}}),\ldots,\sigma_{q}({\bm{B}}))$,
$q=\min\\{p_{1},p_{2}\\}$, denotes the vector of decreasingly ordered singular
value mapping of ${\bm{B}}$. That is
$\sigma_{1}({\bm{B}})\geq\sigma_{2}({\bm{B}})\geq\ldots\geq\sigma_{r}({\bm{B}})>\sigma_{r+1}({\bm{B}})=\ldots=\sigma_{q}({\bm{B}})=0$,
where $r=\mbox{rank}({\bm{B}})$. Let $Y$ denote the response variable,
${\bm{Z}}\in\mathrm{I\\!R}\mathit{{}^{p_{0}}}$ the vector covariate,
${\bm{X}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times p_{2}}}$ the 2D matrix
covariate, and $(y,{\bm{x}},{\bm{z}})$ their sample instances.
We consider the generalized linear model setup, where $Y$ belongs to an
exponential family with probability mass function or density
$\displaystyle
p(y|{\bm{x}},{\bm{z}})=\exp\left\\{\frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)\right\\},$
and the first conditional moment is
$\mathrm{E}(Y|{\bm{X}},{\bm{Z}})=\mu=b^{\prime}(\theta)$, and $\mu$ is of the
form
$\displaystyle g(\mu)=\eta=\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{Z}}+\langle{\bm{B}},{\bm{X}}\rangle,$ (1)
where $g$ is a known link function,
$\mbox{\boldmath$\gamma$}\in\mathrm{I\\!R}\mathit{{}^{p_{0}}}$, and
${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times p_{2}}}$. For simplicity, in
our subsequent development of the regularized matrix model estimation, we drop
the vector covariate ${\bm{Z}}$ and its associated parameter $\gamma$.
However, the results can be extended straightforwardly to incorporate
${\bm{Z}}$ and $\gamma$. Also, we only consider GLM with a univariate
response, whereas extensions to more complex models such as quasi-likelihood
models and multivariate responses are straightforward.
For generality in terms of penalty, we consider the spectral regularization
problem
$\displaystyle\min_{{\bm{B}}}\,h({\bm{B}})=\ell({\bm{B}})+J({\bm{B}}),$ (2)
where $\ell({\bm{B}})$ is a loss function; for the GLM, we use the negative
log-likelihood as the loss. $J({\bm{B}})=f\circ\sigma({\bm{B}})$, where
$f:\mathrm{I\\!R}\mathit{{}^{q}}\to\mathrm{I\\!R}\mathit{}$ is a function of
the singular values of ${\bm{B}}$. The choice
$f({\bm{w}})=\lambda\sum_{j=1}^{q}|w_{j}|$ and the least squares loss
corresponds to the special case of the nuclear norm regularization problem we
considered in the Introduction. In general, for sparsity of the spectrum, $f$
takes the general form
$\displaystyle f({\bm{w}})=\sum_{j=1}^{q}P_{\eta}(|w_{j}|,\lambda),$
where $P$ is a scalar penalty function, $\eta$ is the parameter indexing the
penalty family, and $\lambda$ is the tuning constant. We list some commonly
used penalty functions below.
* •
Power family (Frank and Friedman,, 1993)
$\displaystyle P_{\eta}(|w|,\lambda)$ $\displaystyle=\lambda|w|^{\eta},\hskip
14.45377pt\eta\in(0,2].$
Two important special cases of this family are the lasso penalty when $\eta=1$
(Tibshirani,, 1996; Chen et al.,, 2001) and the ridge penalty when $\eta=2$
(Hoerl and Kennard,, 1970).
* •
Elastic net (Zou and Hastie,, 2005)
$\displaystyle P_{\eta}(|w|,\lambda)$
$\displaystyle=\lambda\left\\{(\eta-1)w^{2}/2+(2-\eta)|w|\right\\},\hskip
14.45377pt\eta\in[1,2].$
Varying $\eta$ from 1 to 2 bridges the lasso to the ridge penalty.
* •
Log penalty (Candès et al.,, 2008; Armagan et al.,, 2011)
$\displaystyle P_{\eta}(|w|,\lambda)$
$\displaystyle=\lambda\ln(\eta+|w|),\hskip 14.45377pt\eta>0.$
* •
SCAD (Fan and Li,, 2001), in which the penalty is defined via its partial
derivative
$\displaystyle\frac{\partial}{\partial|w|}P_{\eta}(|w|,\lambda)$
$\displaystyle=\lambda\left\\{1_{\\{|w|\leq\lambda\\}}+\frac{(\eta\lambda-|w|)_{+}}{(\eta-1)\lambda}1_{\\{|w|>\lambda\\}}\right\\},\hskip
14.45377pt\eta>2.$
Integration shows SCAD as a natural quadratic spline with knots at $\lambda$
and $\eta\lambda$
$\displaystyle P_{\eta}(|w|,\lambda)$
$\displaystyle=\begin{cases}\lambda|w|&|w|<\lambda\\\
\lambda^{2}+\frac{\eta\lambda(|w|-\lambda)}{\eta-1}-\frac{w^{2}-\lambda^{2}}{2(\eta-1)}&|w|\in[\lambda,\eta\lambda]\\\
\lambda^{2}(\eta+1)/2&|w|>\eta\lambda\end{cases}.$
For small signals $|w|<\lambda$, it acts as lasso; for large signals
$|w|>\eta\lambda$, the penalty flattens and leads to the unbiasedness of the
regularized estimate.
* •
MC+ penalty (Zhang,, 2010), which is similar to SCAD and is defined by the
partial derivative
$\displaystyle\frac{\partial}{\partial|w|}P_{\eta}(|w|,\lambda)$
$\displaystyle=\lambda\left(1-\frac{|w|}{\lambda\eta}\right)_{+}.$
Integration shows that the penalty function
$\displaystyle P_{\eta}(|w|,\lambda)$
$\displaystyle=\left(\lambda|w|-\frac{w^{2}}{2\eta}\right)1_{\\{|w|<\lambda\eta\\}}+\frac{\lambda^{2}\eta}{2}1_{\\{|w|\geq\lambda\eta\\}},\hskip
14.45377pt\eta>0,$
is quadratic on $[0,\lambda\eta]$ and flattens beyond $\lambda\eta$. Varying
$\eta$ from 0 to $\infty$ bridges hard thresholding ($\ell_{0}$ regression) to
lasso ($\ell_{1}$) shrinkage.
We also comment that, besides the above sparsity penalties, other forms of
regularization can be useful, depending on the scientific question of
interest. For instance, the choice
$f({\bm{w}})=\lambda\sum_{j=1}^{q-1}|w_{j}-w_{j+1}|=\lambda(w_{1}-w_{r})$
produces the regularization for the “spiked” matrix model, i.e., matrices with
clustered eigen-/singular values (Johnstone,, 2001).
Convexity is essential for studying convergence properties of optimization
problems. We first state the necessary and sufficient condition for the
convexity of the regularizer $J$. Its proof follows from the theory of
spectral function (Borwein and Lewis,, 2006) and is given in the Appendix.
###### Lemma 1.
The functional $J({\bm{B}})=f\circ\sigma({\bm{B}})$ is convex and lower
semicontinuous if and only if $f$ is convex and lower semicontinuous.
Furthermore, for a convex $f$, the subdifferential of $J$ at ${\bm{B}}$, which
admits singular value decomposition
${\bm{U}}\mathrm{diag}({\bm{b}}){\bm{V}}^{\mbox{\tiny{\sf T}}}$, is
$\displaystyle\partial
J({\bm{B}})=\partial(f\circ\sigma)({\bm{B}})={\bm{U}}\mathrm{diag}[\partial
f({\bm{b}})]{\bm{V}}^{\mbox{\tiny{\sf T}}}.$
Lemma 1 immediately leads to the optimality condition when both loss and
regularizer are convex.
###### Theorem 1.
When both the loss $\ell$ and $f$ are convex, all local minima of the
regularized program (2) are global minimum and is unique if $\ell$ is strictly
convex. A matrix
${\bm{B}}={\bm{U}}\mathrm{diag}({\bm{b}}){\bm{V}}^{\mbox{\tiny{\sf T}}}$ is a
global minimum if and only if
$\displaystyle{\mathbf{0}}_{p_{1}\times
p_{2}}\in\nabla\ell({\bm{B}})+{\bm{U}}\mathrm{diag}[\partial
f({\bm{b}})]{\bm{V}}^{\mbox{\tiny{\sf T}}}.$
When either the loss $\ell$ or $f$ is non-convex, the regularized objective
function (2) may be non-convex and there lacks an easy-to-check optimality
condition.
## 3 Estimation Algorithm
We utilize the powerful Nesterov optimal gradient method (Nesterov,, 1983,
2004) for minimizing the non-smooth and possibly non-convex objective function
(2). We first state a matrix thresholding formula for spectral regularization,
which forms the building blocks of the Nesterov algorithm.
###### Proposition 1.
For a given matrix ${\bm{A}}$ with singular value decomposition
${\bm{A}}={\bm{U}}\mathrm{diag}({\bm{a}}){\bm{V}}^{\mbox{\tiny{\sf T}}}$, the
optimal solution to
$\displaystyle\min_{{\bm{B}}}\frac{1}{2}\|{\bm{B}}-{\bm{A}}\|_{\mathrm{F}}^{2}+f\circ\sigma({\bm{B}})$
shares the same singular vectors as ${\bm{A}}$ and its ordered singular values
are the solution to
$\displaystyle\min_{{\bm{b}}}\frac{1}{2}\|{\bm{b}}-{\bm{a}}\|_{2}^{2}+f({\bm{b}}).$
An immediate consequence of Proposition 1 is the following well-known singular
value thresholding formula for nuclear norm regularization (Cai et al.,,
2010).
###### Corollary 1.
For a given matrix ${\bm{A}}$ with singular value decomposition
${\bm{A}}={\bm{U}}\mathrm{diag}({\bm{a}}){\bm{V}}^{\mbox{\tiny{\sf T}}}$, the
optimal solution to
$\displaystyle\min_{{\bm{B}}}\frac{1}{2}\|{\bm{B}}-{\bm{A}}\|_{\mathrm{F}}^{2}+\lambda\|{\bm{B}}\|_{*}$
shares the same singular vectors as ${\bm{A}}$ and its singular values are
$b_{i}=(a_{i}-\lambda)_{+}$.
Given the above matrix thresholding formula, we are ready to present the
Nesterov algorithm for minimizing (2). The Nesterov method has attracted
increasing attention in recent years due to its efficiency in solving
regularization problems (Beck and Teboulle, 2009b, ). It resembles the
classical gradient descent algorithm in that only the first order gradients of
the objective function are utilized to produce next algorithmic iterate from
current search point, and as such is simple to implement. It differs from the
gradient descent algorithm by extrapolating the previous two algorithmic
iterates to generate the next search point. This extrapolation step incurs
trivial computational cost but improves the convergence rate dramatically. It
has been shown to be optimal among a wide class of convex smooth optimization
problems (Nemirovski,, 1994; Nesterov,, 2004).
We summarize our Nesterov method for solving spectral regularization problem
(2) in Algorithm 1. Each iteration consists of three steps: (i) predict a
search point ${\bm{S}}$ by a linear extrapolation from previous two iterates
(line 1 of Algorithm 1), (ii) perform gradient descent from the search point
${\bm{S}}$ possibly with Armijo type line search (lines 1-1), and (iii) force
the descent property of the next iterate (lines 1-1).
1 Initialize ${\bm{B}}^{(0)}={\bm{B}}^{(1)}$, $\delta>0$, $\alpha^{(0)}=0$,
$\alpha^{(1)}=1$ ;
2 repeat
3
${\bm{S}}^{(t)}\leftarrow{\bm{B}}^{(t)}+\left(\frac{\alpha^{(t-1)}-1}{\alpha^{(t)}}\right)({\bm{B}}^{(t)}-{\bm{B}}^{(t-1)})$
;
4 repeat
5
${\bm{A}}_{\mathrm{temp}}\leftarrow{\bm{B}}^{(t)}-\delta\nabla\ell({\bm{S}}^{(t)})$
;
6 Compute SVD
${\bm{A}}_{\mathrm{temp}}={\bm{U}}\text{diag}({\bm{a}}){\bm{V}}^{\mbox{\tiny{\sf
T}}}$ ;
7
${\bm{b}}\leftarrow\text{argmin}_{{\bm{x}}}(2\delta)^{-1}\|{\bm{x}}-{\bm{a}}\|_{2}^{2}+f({\bm{x}})$
;
8
${\bm{B}}_{\mathrm{temp}}\leftarrow{\bm{U}}\text{diag}({\bm{b}}){\bm{V}}^{\mbox{\tiny{\sf
T}}}$ ;
9 $\delta\leftarrow\delta/2$ ;
10
11 until _$h({\bm{B}}_{\mathrm{temp}})\leq
g({\bm{B}}_{\mathrm{temp}}\mid{\bm{S}}^{(t)},\delta)$ _;
12 if _$h({\bm{B}}_{\mathrm{temp}})\leq h({\bm{B}}^{(t)})$ _ then
13 ${\bm{B}}^{(t+1)}\leftarrow{\bm{B}}_{\mathrm{temp}}$ ;
14
15 else
16 ${\bm{B}}^{(t+1)}\leftarrow{\bm{B}}^{(t)}$ ;
17
18 end if
19 $\alpha^{(t+1)}\leftarrow(1+\sqrt{1+(2\alpha^{(t)})^{2}})/2$
20until _objective value converges_ ;
Algorithm 1 Nesterov method for spectral regularized matrix regression (2).
In step (i), $\alpha^{(t)}$ is a scalar sequence that plays a critical role in
the extrapolation. We update this sequence as in the original Nesterov method
(line 1 of Algorithm 1), whereas other sequences, for instance
$\alpha^{(t)}=(t-1)/(t+2)$, can also be used. In step (ii), the gradient
descent is based on the first order approximation to the loss function at the
current search point ${\bm{S}}^{(t)}$,
$\displaystyle g({\bm{B}}|{\bm{S}}^{(t)},\delta)$ $\displaystyle=$
$\displaystyle\ell({\bm{S}}^{(t)})+\langle\nabla\ell({\bm{S}}^{(t)}),{\bm{B}}-{\bm{S}}^{(t)}\rangle+\frac{1}{2\delta}\|{\bm{B}}-{\bm{S}}^{(t)}\|_{\text{F}}^{2}+J({\bm{B}})$
$\displaystyle=$
$\displaystyle\frac{1}{2\delta}\|{\bm{B}}-[{\bm{S}}^{(t)}-\delta\nabla\ell({\bm{S}}^{(t)})]\|_{\text{F}}^{2}+J({\bm{B}})+c^{(t)},$
where the constant $\delta$ is determined during the line search and the
constant $c^{(t)}$ collects terms irrelevant to the optimization. The “ridge”
term $(2\delta)^{-1}\|{\bm{B}}-{\bm{S}}^{(t)}\|_{\text{F}}^{2}$ acts as a
trust region and shrinks the next iterate towards ${\bm{S}}^{(t)}$. If the
loss function $\ell\in\mathcal{C}_{1,1}$, which denotes the class of functions
that are convex, continuously differentiable and the gradient satisfies
$\|\nabla\ell({\bm{u}})-\nabla\ell({\bm{v}})\|\leq\mathcal{L}(\ell)\|{\bm{u}}-{\bm{v}}\|$
with a known gradient Lipschitz constant $\mathcal{L}(\ell)$ for all
${\bm{u}},{\bm{v}}$, then $\delta$ is fixed at $\mathcal{L}(\ell)^{-1}$. In
practice, the gradient Lipschitz constant is often unknown. Then $\delta$ is
updated dynamically to capture the unknown $\mathcal{L}(\ell)$ using the
classical Armijo line search rule (Nocedal and Wright,, 2006; Lange,, 2004).
Solution to the surrogate function $g({\bm{B}}|{\bm{S}}^{(t)},\delta)$ is
obtained by Proposition 1. Singular value decomposition is performed on the
intermediate matrix
${\bm{A}}_{\mathrm{temp}}={\bm{B}}^{(t)}-\delta\nabla\ell({\bm{S}}^{(t)})$.
The next iterate ${\bm{B}}^{(t+1)}$ shares the same singular vectors as
${\bm{A}}$ and its singular values ${\bm{b}}^{(t+1)}$ are determined by
minimizing $(2\delta)^{-1}\|{\bm{b}}-{\bm{a}}\|_{2}^{2}+f({\bm{b}})$, where
${\bm{a}}=\sigma({\bm{A}}_{\mathrm{temp}})$. For a nuclear norm regularization
$f({\bm{w}})=\lambda\sum_{j}|w_{i}|$, the solution is given by soft
thresholding the singular values $b_{i}^{(t+1)}=(a_{i}-\lambda\delta)_{+}$. In
this special case, only the top singular values/vectors need to be retrieved.
The Lanczos method (Golub and Van Loan,, 1996) is extremely efficient for this
purpose. For a linear regularization function
$f({\bm{w}})=\lambda\|{\bm{D}}{\bm{w}}\|_{1}$ where ${\bm{D}}$ has full column
rank, reparameterization ${\bm{c}}={\bm{D}}{\bm{b}}$ turns the problem to:
$\min_{{\bm{c}}}\frac{1}{2\delta}\|({\bm{D}}^{\mbox{\tiny{\sf
T}}}{\bm{D}})^{-1}{\bm{D}}^{\mbox{\tiny{\sf
T}}}{\bm{c}}-{\bm{a}}\|_{2}^{2}+\lambda\|{\bm{c}}\|_{1},$ which is a standard
lasso problem with many efficient solvers available. When ${\bm{D}}$ does not
have a full column rank, we append extra rows such that the expanded matrix,
denoted by $\tilde{\bm{D}}$, has full column rank and then solve the above
lasso problem with $\tilde{\bm{D}}$ and only part of ${\bm{c}}$ penalized.
For minimization of a smooth convex function $\ell$ in $\mathcal{C}_{1,1}$, it
is well-known that the Nesterov method is optimal with the convergence rate at
order $O(t^{-2})$, where $t$ indicates the iterate number. In contrast, the
gradient descent has a slower convergence rate of $O(t^{-1})$. Our nuclear
norm regularization problem (2) is non-smooth, but the same convergence result
can be established, which is summarized in Theorem 2. Its proof is omitted for
brevity, while readers are referred to Beck and Teboulle, 2009b .
###### Theorem 2.
Suppose $\ell$ is continuously differentiable with a gradient Lipschitz
constant $\mathcal{L}(\ell)$. Let ${\bm{B}}^{(t)}$ be the iterates generated
by the Nesterov method described in Algorithm 1. Then the objective value
$h({\bm{B}}^{(t)})$ monotonically converges. Furthermore, if $J$ is convex,
then
$\displaystyle
h({\bm{B}}^{(t)})-h({\bm{B}}^{*})\leq\frac{4\mathcal{L}(\ell)\|{\bm{B}}^{(0)}-{\bm{B}}^{*}\|_{\mathrm{F}}^{2}}{(t+1)^{2}}$
(3)
for all $t\geq 0$ and any minimum point ${\bm{B}}^{*}$.
We make a few important remarks here. The first remark regards the
monotonicity of the objective function during iterations. Because of the
extrapolation step, the objective values of algorithmic iterates
$f({\bm{B}}^{(t)})$ are not guaranteed to be monotonically decreasing. When
the loss $\ell\in{\cal C}_{1,1}$ and the regularizer $J$ is convex,
convergence of the objective values is guaranteed with the explicit
convergence rate (3). Because of potential use of a non-convex $J$, we enforce
monotonicity of algorithmic iterates (lines 1-1 in Algorithm 1), which is
essential for the convergence of at least the objective values. After each
gradient descent step, if the new iterate fails to decrease the objective
value, then the current iterate is same as the previous one. In other words,
the next gradient descent is initiated from the previous iterate. Fortunately,
the fast convergence rate (3) still holds under the assumptions $\ell\in{\cal
C}_{1,1}$ and $J$ is convex. See Beck and Teboulle, 2009a for the argument.
The second remark is about the non-convex loss function. The Nesterov method
and its convergence properties hinge upon convexity of the loss $\ell$. It
covers many commonly used statistical models, including linear model and GLMs
with canonical links. For GLM with non-canonical link, the loss function may
be non-convex, which could cause trouble in the Nesterov method. The
iteratively reweighted least squares strategy can be applied in this scenario.
At each IWLS step, the Nesterov method is used to solve the penalized weighted
least squares problem, which is convex.
The final remark is about an efficient way for estimating the Lipschitz
constant $L$ for the GLM loss. Each step halving in the line search part of
Algorithm 1 involves an expensive singular value decomposition. Therefore even
a rough initial estimate of $L$ potentially cuts the computational cost
significantly. Recall that a twice differentiable function $f$ is
$L$-Lipschitz continuously differentiable if and only if
${\bm{v}}^{\mbox{\tiny{\sf T}}}d^{2}f({\bm{u}}){\bm{v}}\leq
L\|{\bm{v}}\|_{2}^{2}$ for all ${\bm{v}}$. The Fisher information matrix of a
GLM model with systematic part (1) is:
${\bm{I}}({\bm{B}})=\mathrm{E}[d^{2}\ell({\bm{B}})]=\sum_{i=1}^{n}\omega_{i}(\mathrm{vec}{\bm{x}}_{i})(\mathrm{vec}{\bm{x}}_{i})^{\mbox{\tiny{\sf
T}}}$, where $\omega_{i}=\\{\mu_{i}^{\prime}(\eta_{i})/\sigma_{i}\\}^{2}$,
$\eta_{i}$ is the systematic part, $\mu_{i}$ is the mean, and $\sigma_{i}^{2}$
is the variance corresponding to the $i$th observation. Then in light of the
Cauchy-Schwartz inequality
$\displaystyle{\bm{v}}^{\mbox{\tiny{\sf
T}}}{\bm{I}}({\bm{B}}){\bm{v}}=\sum_{i}\omega_{i}({\bm{v}}^{\mbox{\tiny{\sf
T}}}\mathrm{vec}{\bm{x}}_{i})({\bm{v}}^{\mbox{\tiny{\sf
T}}}\mathrm{vec}{\bm{x}}_{i})^{\mbox{\tiny{\sf
T}}}\leq\sum_{i}\omega_{i}\|\mathrm{vec}{\bm{x}}_{i}\|_{2}^{2}\|{\bm{v}}\|_{2}^{2}=\|{\bm{v}}\|_{2}^{2}\left(\sum_{i}\omega_{i}\|{\bm{x}}_{i}\|_{\mathrm{F}}^{2}\right),$
and thus an initial estimate of $L$ is given by
$L\approx\sum_{i}\\{\mu_{i}^{\prime}(\eta_{i})\\}^{2}/\sigma_{i}^{2}\;\|{\bm{X}}_{i}\|_{\mathrm{F}}^{2}$.
## 4 Degrees of Freedom
In this section, we address the problem of choosing the tuning parameter
$\lambda$ that yields the best model along the regularization path according
to certain criteria. Cross validation is commonly used for parameter tuning in
practice. However, for large data, it may incur considerable computation
burden. There exist computationally attractive alternatives, such as Akaike
information criterion (AIC) (Akaike,, 1974) and Bayesian information criterion
(BIC) (Schwarz,, 1978), which often yield performance comparable to cross
validation in practice.
Consider a normal model under the GLM (1). For simplicity, we again drop the
covariate vector ${\bm{Z}}$:
$\displaystyle Y=\langle{\bm{X}},{\bm{B}}\rangle+\epsilon$ (4)
where $\epsilon$ is a normal error with mean zero and variance $\sigma^{2}$.
Let $y_{i}$ denote the $i$th observation of $Y$, and $\hat{y}_{i}(\lambda)$
denote the estimated response under a given tuning parameter $\lambda$ from
the minimization of (2). Then for this normal model, AIC and BIC are defined
by
$\displaystyle\mathrm{AIC}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}\\{y_{i}-\hat{y}_{i}(\lambda)\\}^{2}}{\sigma^{2}}+2\mathrm{df}(\lambda)$
$\displaystyle\mathrm{BIC}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}\\{y_{i}-\hat{y}_{i}(\lambda)\\}^{2}}{\sigma^{2}}+\ln(n)\mathrm{df}(\lambda).$
In applications, the variance $\sigma^{2}$ is often unknown but can be
estimated from the fitted value by least squares estimation. An essential
element in the above model selection criteria is the effective degrees of
freedom $\mathrm{df}(\lambda)$ of the selected model. Using Stein’s theory of
unbiased risk estimation (Stein,, 1981), Efron, (2004) showed that
$\displaystyle\mathrm{df}(\lambda)=\mathrm{E}\left\\{\mathrm{tr}\left(\frac{\partial\hat{\bm{y}}}{\partial{\bm{y}}}\right)\right\\}=\frac{1}{\sigma^{2}}\sum_{i=1}^{n}\mathrm{cov}(\hat{y}_{i}(\lambda),y_{i})$
with expectation taken with respect to $Y$,
${\bm{y}}=(y_{1},\ldots,y_{n})^{\mbox{\tiny{\sf T}}}$, and
$\hat{\bm{y}}=(\hat{y}_{1}(\lambda),\ldots,\hat{y}_{n}(\lambda))^{\mbox{\tiny{\sf
T}}}$. This formulation has been productively used to derive the degrees of
freedom estimate in least angle regression (Efron et al.,, 2004), lasso (Zou
et al.,, 2007), group penalized regression (Yuan and Lin,, 2006), and sign-
coherent group penalized regression (Chiquet et al.,, 2011).
We derive a degrees of freedom estimate for the nuclear norm regularized
estimate under normal model. For an orthonormal design, i.e.,
$\mbox{\boldmath$\Xi$}^{\mbox{\tiny{\sf
T}}}\mbox{\boldmath$\Xi$}={\bm{I}}_{p_{1}p_{2}}$ with the matrix $\Xi$ having
rows $(\mathrm{vec}{\bm{x}}_{i})^{\mbox{\tiny{\sf T}}}$, the derived estimate
is unbiased for the true degrees of freedom. In practice, it yields results
comparable to cross validation even for non-orthogonal designs. The technical
proof is relegated to the Appendix.
###### Theorem 3.
Assume that the data is generated from model (4) with
$\mathrm{vec}{\bm{x}}_{i}$ orthonormal. Consider the nuclear norm regularized
estimate
$\displaystyle\widehat{\bm{B}}_{\lambda}=\mathrm{argmin}_{{\bm{B}}}\frac{1}{2}\sum_{i}(y_{i}-\langle{\bm{X}}_{i},{\bm{B}}\rangle)^{2}+\lambda\|{\bm{B}}\|_{*}$
with singular values
$\sigma(\widehat{\bm{B}}_{\lambda})=(b_{1}(\lambda),\ldots,b_{q}(\lambda))$
where $q=\min\\{p_{1},p_{2}\\}$. Let $\widehat{\bm{B}}_{\mathrm{LS}}$ be the
usual least squares estimate and assume that it has distinct positive singular
values $\sigma_{1}>\cdots>\sigma_{q}>0$. With the convention $\sigma_{i}=0$
for $i>q$, the following expression is an unbiased estimate of the degree of
freedom of the regularized fit
$\displaystyle\widehat{\mathrm{df}}(\lambda)=\sum_{i=1}^{q}1_{\\{b_{i}(\lambda)>0\\}}\left(1+\sum_{1\leq
j\leq p_{1},j\neq
i}\frac{\sigma_{i}(\sigma_{i}-\lambda)}{\sigma_{i}^{2}-\sigma_{j}^{2}}+\sum_{1\leq
j\leq p_{2},j\neq
i}\frac{\sigma_{i}(\sigma_{i}-\lambda)}{\sigma_{i}^{2}-\sigma_{j}^{2}}\right).$
This formula for the degrees of freedom is interesting in several aspects.
First it does not involve any information on the singular vectors of the least
squares estimate, but only requires singular values. Second,
$\widehat{\mathrm{df}}(\lambda)$ is continuous in $\lambda$, in contrast to
the piecewise constant degrees of freedom estimate for the classical lasso
(Zou et al.,, 2007). Third, at $\lambda=0$,
$\widehat{\bm{B}}_{0}=\widehat{\bm{B}}_{\mathrm{LS}}$ almost surely has a full
rank and the degrees of freedom is
$\displaystyle\sum_{i=1}^{q}\left(1+\sum_{1\leq j\leq p_{1},j\neq
i}\frac{\sigma_{i}^{2}}{\sigma_{i}^{2}-\sigma_{j}^{2}}+\sum_{1\leq j\leq
p_{2},j\neq i}\frac{\sigma_{i}^{2}}{\sigma_{i}^{2}-\sigma_{j}^{2}}\right)$
$\displaystyle=$ $\displaystyle q+q(q-1)+q(p_{1}+p_{2}-2q)=p_{1}p_{2},$
which is exactly the number of parameters without any regularization.
Figure 2 plots the estimated degrees of freedom
$\widehat{\mathrm{df}}(\lambda)$ for a
$\widehat{\bm{B}}_{\mathrm{LS}}\in\mathrm{I\\!R}\mathit{{}^{64\times 64}}$, as
well as the naïve count of the number of free parameters in the fitted matrix
parameter $\widehat{\bm{B}}_{\lambda}$ of rank $r(\lambda)$, which equals
$r(\lambda)(p_{1}+p_{2})-r^{2}(\lambda)$. The estimated degrees of freedom
appears smaller than the naïve count, reflecting the overwhelming shrinkage
effect over model searching. At $\lambda=0$, it coincides with the number of
matrix elements $64^{2}=4096$, reflecting the effect of no shrinkage.
Figure 2: Degrees of freedom estimate $\widehat{\mathrm{df}}(\lambda)$ versus
the number of parameters in the estimated model $\widehat{\bm{B}}_{\lambda}$
with a $\widehat{\bm{B}}_{\mathrm{LS}}\in\mathrm{I\\!R}\mathit{{}^{64\times
64}}$.
Finally we notice that the degrees of freedom estimate in Theorem 3 is
limiting as it requires existence of the least squares estimate
$\widehat{\bm{B}}_{\mathrm{LS}}$ which is not true when $n<p_{1}p_{2}$. In
this case we may use a ridge estimate
$\widehat{\bm{B}}_{\mathrm{ridge}(\tau)}$ where
$\displaystyle\mathrm{vec}\widehat{\bm{B}}_{\mathrm{ridge}(\tau)}=(\mbox{\boldmath$\Xi$}^{\mbox{\tiny{\sf
T}}}\mbox{\boldmath$\Xi$}+\tau{\bm{I}}_{p_{1}p_{2}})^{-1}\mbox{\boldmath$\Xi$}^{\mbox{\tiny{\sf
T}}}{\bm{y}},$
which always exists and is unique. Assume that
$\widehat{\bm{B}}_{\mathrm{ridge}(\tau)}$ admits a singular value
decomposition
$\widehat{\bm{B}}_{\mathrm{ridge}(\tau)}={\bm{U}}\mathrm{diag}(\mbox{\boldmath$\sigma$}){\bm{V}}^{\mbox{\tiny{\sf
T}}}$. The following degree of freedom formula
$\displaystyle\widehat{\mathrm{df}}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{q}1_{\\{b_{i}(\tau)>0\\}}\left[1+\frac{1}{1+\tau}\sum_{1\leq
j\leq p_{1},j\neq
i}\frac{\sigma_{i}\\{(1+\tau)\sigma_{i}-\lambda\\}}{\sigma_{i}^{2}-\sigma_{j}^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+\frac{1}{1+\tau}\sum_{1\leq j\leq
p_{2},j\neq
i}\frac{\sigma_{i}\\{(1+\tau)\sigma_{i}-\lambda\\}}{\sigma_{i}^{2}-\sigma_{j}^{2}}\right]$
generalizes Theorem 3 and is unbiased for the true degree of freedom under the
same assumptions as Theorem 3. Its proof is given in the Appendix.
## 5 Numerical Examples
We have conducted intensive numerical studies with two aims: first, we
investigate the empirical performance of the proposed spectral regularized
regression with matrix covariates, and second, we compare with the
corresponding classical regularization solutions. Four methods are under
comparison: a matrix regression with nuclear norm regularization (since it
takes the form $f({\bm{w}})=\lambda\sum_{j=1}^{q}|w_{j}|$ in the
regularization problem (2), we call this solution matrix lasso, or simply,
m.lasso), a usual vector regression after vectorizing the matrix covariate
with a lasso penalty (lasso), a matrix regression with power spectral
regularization (matrix power, or simply, m.power), and the corresponding
vector regression with a power penalty (power). For the power penalty, we
fixed the coefficient $\eta=0.5$. Our goal here is not to best tune $\eta$.
Instead, we examine this penalty since it yields a nearly unbiased estimate.
We also examined another unbiased penalty, SCAD, which yields very similar
results as power $\eta=0.5$, and thus its results are not reported here for
brevity. We also note that the lasso penalty is a convex penalty, while the
power $\eta=0.5$ is non-convex. We summarize our findings in three examples.
First, we elaborated on the illustrative example by examining a number of
different geometric and natural shapes. Second, we generated some synthetic
data and compared different regularization solutions under varying ranks and
sparsity. Lastly, we revisited the motivating electroencephalography (EEG)
data analysis mentioned in the Introduction.
Example 1: 2D Shapes. We elaborated on the illustrative example in the
Introduction, by employing the same model setup, but examining a variety of
signal shapes. We presented the true signal followed by the estimates from the
four aforementioned regularization methods, where the regularization parameter
$\lambda$ was tuned by BIC. The signals of square, cross, T-shape are given in
Figure 3, where the true signal is of a low rank structure (rank 1 or 2). The
signals of triangle, disk and butterfly are given in Figure 4, where the true
signal is not of an exact low rank structure, however, can be well
approximated by so (Zhou et al.,, 2012). It is seen that the matrix version of
regularized estimators clearly outperform their vector version counterparts,
for both lasso and power penalties. Comparing matrix lasso with matrix power,
the two yield comparable results, while the former is better for the high rank
signals, and the latter is better for the low rank signals. We will further
verify this observation in the next simulation example.
---
Figure 3: Comparison of the matrix and vector version of regularized
estimators for low rank signals.
---
Figure 4: Comparison of the matrix and vector version of regularized
estimators for high rank signals. Table 1: Parameter estimation of a normal
model. Reported are the mean and standard deviation (in the parenthesis) of
the RMSE for ${\bm{B}}$ out of 100 data replications.
Sparsity Rank $s$ Method $R=1$ $R=5$ $R=10$ $R=20$ $1\%$ m.lasso 0.031 (0.006)
0.074 (0.016) 0.086 (0.015) 0.090 (0.013) lasso 0.022 (0.017) 0.022 (0.010)
0.023 (0.013) 0.022 (0.006) m.power 0.010 (0.001) 0.063 (0.021) 0.097 (0.021)
0.107 (0.017) power 0.009 (0.025) 0.005 (0.001) 0.007 (0.017) 0.006 (0.001)
$5\%$ m.lasso 0.044 (0.007) 0.172 (0.021) 0.199 (0.022) 0.212 (0.020) lasso
0.212 (0.049) 0.214 (0.035) 0.215 (0.032) 0.217 (0.026) m.power 0.010 (0.001)
0.171 (0.029) 0.236 (0.029) 0.254 (0.024) power 0.270 (0.060) 0.271 (0.042)
0.272 (0.039) 0.275 (0.032) $10\%$ m.lasso 0.051 (0.008) 0.250 (0.028) 0.288
(0.023) 0.309 (0.023) lasso 0.320 (0.046) 0.343 (0.040) 0.343 (0.034) 0.345
(0.030) m.power 0.009 (0.001) 0.258 (0.041) 0.351 (0.033) 0.375 (0.032) power
0.399 (0.056) 0.427 (0.050) 0.428 (0.044) 0.426 (0.036) $20\%$ m.lasso 0.062
(0.011) 0.350 (0.031) 0.418 (0.031) 0.450 (0.031) lasso 0.475 (0.050) 0.523
(0.050) 0.539 (0.049) 0.534 (0.044) m.power 0.009 (0.001) 0.354 (0.041) 0.511
(0.039) 0.562 (0.044) power 0.585 (0.062) 0.645 (0.060) 0.667 (0.060) 0.663
(0.055) $50\%$ m.lasso 0.087 (0.015) 0.556 (0.030) 0.700 (0.044) 0.792 (0.040)
lasso 0.775 (0.044) 1.054 (0.081) 1.112 (0.089) 1.130 (0.069) m.power 0.010
(0.001) 0.493 (0.042) 0.760 (0.055) 0.903 (0.062) power 0.952 (0.057) 1.300
(0.107) 1.365 (0.113) 1.388 (0.082)
Table 2: Prediction of a normal model. Reported are the mean and standard
deviation (in the parenthesis) of the RMSE for $y$ out of 100 data
replications.
Sparsity Rank $s$ Method $R=1$ $R=5$ $R=10$ $R=20$ $1\%$ m.lasso 1.707 (0.195)
4.597 (1.091) 5.525 (1.009) 5.865 (0.826) lasso 1.610 (0.978) 1.544 (0.487)
1.622 (0.762) 1.530 (0.233) m.power 1.175 (0.042) 3.996 (1.164) 5.534 (1.047)
5.992 (0.874) power 1.236 (1.240) 1.133 (0.742) 1.168 (0.962) 1.066 (0.039)
$5\%$ m.lasso 2.278 (0.309) 11.031 (1.418) 12.837 (1.578) 13.754 (1.352) lasso
13.488 (2.923) 13.625 (2.115) 13.688 (2.047) 13.855 (1.609) m.power 1.177
(0.043) 10.253 (1.479) 13.013 (1.593) 14.118 (1.407) power 14.713 (3.384)
14.877 (2.109) 14.933 (2.018) 15.113 (1.597) $10\%$ m.lasso 2.542 (0.420)
16.101 (1.940) 18.590 (1.568) 19.810 (1.514) lasso 19.601 (2.661) 21.257
(2.488) 21.165 (2.052) 21.187 (1.742) m.power 1.179 (0.045) 15.386 (2.161)
18.840 (1.588) 20.239 (1.597) power 21.138 (2.879) 22.796 (2.782) 22.708
(2.323) 22.546 (1.776) $20\%$ m.lasso 3.188 (0.655) 22.723 (2.157) 26.783
(2.264) 28.874 (2.256) lasso 28.687 (3.017) 31.862 (3.219) 32.588 (3.182)
32.449 (2.767) m.power 1.169 (0.042) 21.401 (2.309) 27.103 (2.171) 29.271
(2.189) power 30.879 (3.284) 34.350 (3.495) 35.013 (3.600) 34.998 (2.994)
$50\%$ m.lasso 4.566 (1.026) 35.651 (2.153) 45.132 (2.989) 50.916 (3.054)
lasso 45.815 (2.749) 62.834 (4.896) 66.490 (5.523) 67.478 (4.444) m.power
1.180 (0.047) 29.638 (2.079) 42.104 (2.381) 48.892 (2.945) power 49.798
(3.091) 68.073 (5.719) 71.626 (6.063) 72.729 (4.986)
Example 2: Synthetic Data. We consider a class of synthetic signals to compare
various regularization methods under different ranks and sparsity levels.
Specifically, we generated the matrix covariates ${\bm{X}}$ of size $64\times
64$ and the $5$-dimensional vector covariates ${\bm{Z}}$, both of which
consist of independent standard normal entries. We set the sample size
$n=500$, whereas the number of parameters is $64\times 64+5=4,101$. We set
$\mbox{\boldmath$\gamma$}=(1,\ldots,1)^{\mbox{\tiny{\sf T}}}$, and generated
the true array signal as ${\bm{B}}={\bm{B}}_{1}{\bm{B}}_{2}^{\mbox{\tiny{\sf
T}}}$, where ${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times R}}$,
$d=1,2$. $R$ controls the rank of the generated signal. Moreover, each entry
of ${\bm{B}}$ is $0/1$, and the percentage of non-zero entries is controlled
by a sparsity level constant $s$. That is, each entry of ${\bm{B}}_{d}$ is a
Bernoulli with probability of one equal to $\sqrt{1-(1-s)^{1/R}}$. We varied
the rank $R=1,5,10$ and $20$, and the level of (non)sparsity
$s=0.01,0.05,0.1,0.2$ and $0.5$. (So $s=0.05$ means about $5\%$ of entries of
${\bm{B}}$ are ones and the rest are zeros.) We generated both a normal and a
binomial response $Y$ with the systematic part as in (1), $\mu=\eta$ for the
normal model, and $\mu=1/\\{1+\exp(-\eta)\\}$ for the binomial.
We evaluated the performance of each method from two aspects: parameter
estimation and prediction. For the former, we employed BIC for parameter
tuning and the root mean squared error (RMSE) as the evaluation criterion. For
the latter, we used an independent validation data set to tune the
regularization parameter and a testing data set to evaluate the prediction
error, which is measured by the RMSE of the response for the normal case, and
the mis-classification error rate for the binomial case. Those are all common
practices in the literature. We replicated the experiment 100 times, and
report the mean and standard deviation of the corresponding criterion out of
100 replications in Tables 1 – 4. The best outcomes are highlighted in bold
face. Since the RMSE for the vector-valued coefficient $\gamma$ shows the same
qualitative pattern as that for the array coefficient ${\bm{B}}$, we only
present the results for ${\bm{B}}$ for brevity.
Table 3: Parameter estimation of a binomial model. Reported are the mean and
standard deviation (in the parenthesis) of the RMSE for ${\bm{B}}$ out of 100
data replications.
Sparsity Rank $s$ Method $R=1$ $R=5$ $R=10$ $R=20$ $1\%$ m.lasso 0.087 (0.023)
0.094 (0.021) 0.096 (0.015) 0.094 0.014) lasso 0.095 (0.024) 0.097 (0.022)
0.097 (0.016) 0.095 0.014) m.power 0.077 (0.020) 0.091 (0.020) 0.094 (0.015)
0.093 (0.013) power 0.095 (0.024) 0.097 (0.022) 0.098 (0.016) 0.095 0.014)
$5\%$ m.lasso 0.210 (0.042) 0.222 (0.032) 0.231 (0.027) 0.231 (0.021) lasso
0.220 (0.042) 0.225 (0.032) 0.233 (0.027) 0.233 (0.021) m.power 0.193 (0.042)
0.216 (0.032) 0.226 (0.027) 0.228 (0.021) power 0.221 (0.042) 0.225 (0.032)
0.233 (0.027) 0.233 (0.021) $10\%$ m.lasso 0.304 (0.043) 0.333 (0.035) 0.339
(0.031) 0.331 (0.028) lasso 0.315 (0.043) 0.336 (0.035) 0.342 (0.031) 0.333
(0.028) m.power 0.287 (0.043) 0.326 (0.035) 0.334 (0.031) 0.327 (0.028) power
0.315 (0.043) 0.337 (0.035) 0.342 (0.031) 0.333 (0.028) $20\%$ m.lasso 0.438
(0.044) 0.502 (0.047) 0.510 (0.038) 0.515 (0.039) lasso 0.449 (0.044) 0.506
(0.047) 0.513 (0.038) 0.517 (0.039) m.power 0.419 (0.044) 0.494 (0.047) 0.504
(0.038) 0.510 (0.038) power 0.449 (0.044) 0.507 (0.047) 0.513 (0.038) 0.518
(0.039) $50\%$ m.lasso 0.695 (0.038) 0.990 (0.072) 1.044 (0.076) 1.056 (0.057)
lasso 0.706 (0.038) 0.997 (0.073) 1.049 (0.077) 1.061 (0.058) m.power 0.676
(0.039) 0.979 (0.072) 1.034 (0.076) 1.048 (0.057) power 0.706 (0.038) 0.997
(0.073) 1.049 (0.077) 1.061 (0.058)
Table 4: Prediction of a binomial model. Reported are the mean and standard
deviation (in the parenthesis) of the misclassification error for $y$ out of
100 data replications.
Sparsity Rank $s$ Method $R=1$ $R=5$ $R=10$ $R=20$ $1\%$ m.lasso 0.231 (0.029)
0.337 (0.030) 0.362 (0.024) 0.372 (0.024) lasso 0.411 (0.038) 0.416 (0.037)
0.413 (0.035) 0.415 (0.031) m.power 0.252 (0.028) 0.346 (0.027) 0.370 (0.025)
0.382 (0.029) power 0.763 (0.075) 0.764 (0.071) 0.777 (0.082) 0.771 (0.068)
$5\%$ m.lasso 0.199 (0.021) 0.363 (0.023) 0.393 (0.022) 0.409 (0.026) lasso
0.459 (0.022) 0.460 (0.025) 0.460 (0.030) 0.461 (0.023) m.power 0.220 (0.025)
0.368 (0.023) 0.398 (0.024) 0.416 (0.025) power 0.853 (0.091) 0.854 (0.088)
0.824 (0.080) 0.850 (0.080) $10\%$ m.lasso 0.194 (0.023) 0.359 (0.027) 0.396
(0.024) 0.404 (0.027) lasso 0.469 (0.022) 0.466 (0.022) 0.467 (0.023) 0.462
(0.020) m.power 0.217 (0.027) 0.368 (0.027) 0.404 (0.024) 0.413 (0.024) power
0.864 (0.079) 0.865 (0.095) 0.860 (0.086) 0.874 (0.076) $20\%$ m.lasso 0.193
(0.030) 0.343 (0.027) 0.375 (0.027) 0.396 (0.026) lasso 0.469 (0.022) 0.466
(0.025) 0.470 (0.025) 0.465 (0.023) m.power 0.217 (0.030) 0.355 (0.024) 0.384
(0.027) 0.403 (0.023) power 0.886 (0.094) 0.870 (0.091) 0.859 (0.089) 0.858
(0.089) $50\%$ m.lasso 0.187 (0.023) 0.282 (0.033) 0.312 (0.032) 0.344 (0.027)
lasso 0.472 (0.019) 0.471 (0.025) 0.470 (0.023) 0.465 (0.023) m.power 0.214
(0.023) 0.308 (0.030) 0.336 (0.028) 0.359 (0.024) power 0.872 (0.091) 0.860
(0.089) 0.869 (0.083) 0.873 (0.092)
We make the following observations. First, for the normal response, the
proposed matrix version of estimators almost always outperform the
corresponding vector version in terms of both parameter estimation and
prediction, and this holds true for both the lasso and power penalties. The
only exception is that when the signal is extremely sparse ($s=1\%$ of non-
zero elements), where the usual vector version of regularized estimators
perform slightly better. Second, for the binomial response, the matrix version
is superior than the vector version for all ranks and sparsity levels, and
such a superiority is more striking in terms of predicting the binary outcome.
Third, as for the effects of the signal rank and sparsity, the regular version
of regularized estimators perform better when the signal is more sparse (a
smaller $s$), while is relatively insensitive to the rank ($R$). By contrast,
the proposed matrix version of regularized estimators perform better when the
rank is smaller, and is insensitive to the coefficient sparsity. Those
patterns agree with our expectations since the former penalizes directly on
the coefficient sparsity, whereas the latter penalizes on the rank sparsity.
Finally, comparing the lasso penalty with the power penalty, the two yield
similar results, while the lasso usually performs better when the rank is
large, and the power penalty is better when the rank is small. Such patterns
agree with what we observed in Example 1, and can provide some useful
guidelines when choosing a penalty function given the data.
Figure 5: EEG signals of an alcoholic subject (left) and a control subject
(right). Each curve tracks signals from one electrode.
Example 3: EEG Data Analysis. We analyzed the motivating EEG data in the
Introduction and compared a number of regularization estimators. The data set
consists of 77 individuals with alcoholism and 44 individuals as the control.
For each subject, 64 channels of electrodes were placed at different locations
of scalp and the voltage values are recorded at 256 time points. Besides, each
subject performed 120 trials under three types of stimuli: single stimulus,
two matched stimuli and two unmatched stimuli. One primary interest was to
study the association between alcoholism and the pattern of voltage values
over times and channels. Figure 5 displays the EEG signals from a randomly
chosen alcoholic subject and a control subject, where the horizontal axis is
the time, the vertical axis is the voltage, and each curve represents an
electrode channel. It is clearly seen that the alcoholic and control subjects
demonstrate distinct characteristics.
Li et al., (2010) and Hung and Wang, (2011) both analyzed this same data set.
Following their practice, we focused on the data under the single stimulus
condition only, and averaged for all the trials under that condition for each
subject. The resulting covariates ${\bm{x}}_{i}$ are $256\times 64$ matrices,
and the response $y_{i}$ is a binary variable indicating whether the $i$th
subject is alcoholic ($y_{i}=1$) or not ($y_{i}=0$), $i=1,\ldots,122$. We
applied matrix lasso, lasso, matrix power and power to the data, and evaluated
each solution via cross-validation based misclassification error. More
specifically, we divided the full data into a training and a testing sample
using $k$-fold cross-validation. When $k$ equals the sample size, it is the
leave-one-out cross-validation. Then for the training data, we further
employed a 5-fold cross-validation to tune the shrinkage parameter $\lambda$.
We then applied the tuned model that is fully based on the training data now
to the testing data and evaluated the misclassification error rate for
testing. Table 5 reports the results for leave-one-out, 5, 10, and 20-fold
cross-validation results. It is clearly seen that the matrix version of the
regularized estimators achieve smaller misclassification errors compared to
the vector version counterparts. Comparing the lasso and the power penalties,
the lasso achieves a slightly better classification accuracy.
Table 5: Misclassification error rate for EEG data.
Method leave-one-out 5-fold 10-fold 20-fold m.lasso 0.230 0.214 0.222 0.181
lasso 0.246 0.287 0.271 0.264 m.power 0.246 0.222 0.228 0.213 power 0.254
0.329 0.244 0.276
We also make a few remarks regarding to this data analysis. First, Hung and
Wang, (2011) analyzed the same data using a matrix logistic regression, which
is a rank-1 special case of the tensor GLM of Zhou et al., (2012). Their
method does not work for $n<p$, and thus a ridge type regularization was
employed. The ridge tuning parameter was chosen so that the leave-one-out
classification accuracy is maximized, and the corresponding best
classification result was reported, which is slightly better than the
classification accuracy reported in Table 5. We believe, however, a fair
evaluation of prediction should have the parameter tuning solely based on the
training data, and then report the corresponding testing error. If one tunes
the parameter based on the reported testing error, the result would be over
optimistic. Second, both Li et al., (2010) and Hung and Wang, (2011)
preprocessed the EEG data by reducing the dimensionality of the matrices to a
smaller scale. Both used principal components analysis (PCA) for reduction,
but they employed different variants of PCA for matrices. Part of reason for
the dimension reduction preprocessing was that their proposed numerical
methods can not directly handle the size of $256\times 64$ of the EEG data. By
contrast, our proposal is not limited by the matrix size and were directly
applied to the original data, since our Nesterov algorithm is highly efficient
and scalable. On the other hand, we agree that such preprocessing could
potentially improve the overall classification accuracy by removing noisy
irrelevant information. However, we note that, PCA is known as an unsupervised
dimension reduction approach that can not guarantee full preservation of all
relevant information. Moreover, it introduces another layer of tuning,
including choosing the right version of PCA for matrices, and determining the
optimal number of principal components. Our goal of this data analysis has
primarily been the comparison of the regularized matrix regression estimators
with the vector counterparts, so we have chosen not to include any
preprocessing that requires a significant amount of additional tuning.
## 6 Discussion
Motivated by modern scientific data arising in areas such as imaging and
cytometry, we study in this article the problem of regressions with matrix
covariates. Regularization is bound to play a crucial role in such regressions
due to the ultrahigh dimensionality and complex structure of the matrix data.
We have proposed a class of regularized matrix regression models by penalizing
the spectrum of the matrix parameters. It is based upon the observation that
the matrix signals are often of, or can be well approximated by, a low rank
structure. Consequently, the new method focuses on the sparsity in terms of
the rank of the matrix parameters rather than the number of nonzero entries,
and is intrinsically different from the classical lasso and related
penalization approaches.
In many applications, sparsity is sought in certain pre-specified _basis_
systems rather than the original coordinates. Specifically, the systematic
component takes the form
$\displaystyle\eta=\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{Z}}+\langle{\bm{B}},{\bm{S}}_{1}^{\mbox{\tiny{\sf
T}}}{\bm{X}}{\bm{S}}_{2}\rangle,$
where ${\bm{S}}_{j}\in\mathrm{I\\!R}\mathit{{}^{p_{j}\times q_{j}}}$ contains
$q_{j}$ basis vectors for $j=1,2$. For 2D images, over-complete wavelet basis
($q_{j}>p_{j}$) are often used in both dimensions. For the EEG data that are
channels by time points, wavelet or Fourier basis can be applied to the time
dimension. It is of direct interest to seek a sparse low rank representation
of the signal ${\bm{B}}$ in the basis system by solving the regularized
regression. In that sense, our proposed regularized matrix regression can be
considered as the matrix analog of the classical basis pursuit problem (Chen
et al.,, 2001).
We have concentrated on problems with matrix covariates throughout this
article. In applications such as anatomical magnetic resonance imaging (MRI)
and functional magnetic resonance imaging (fMRI), the covariates are in the
form of multidimensional arrays, a.k.a. tensors ($D>2$). It is natural to
extend the work here to regularized tensor regressions. However, the problem
formulation requires an appropriate norm for tensors that is in analogous to
the nuclear norm for matrices, and the involved regularization and
optimization are to be fundamentally different from the methods for matrices.
We are currently pursuing this line of extension, and will report the results
elsewhere.
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|
arxiv-papers
| 2012-04-15T22:58:36 |
2024-09-04T02:49:29.716309
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hua Zhou and Lexin Li",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1204.3331"
}
|
1204.3336
|
# spectrums of equivalent Schauder operators
Luo Yi Shi Department of Mathematics, Tianjin Polytechnic University,
Tianjin, 300160,CHINA sluoyi@yahoo.cn , Yang Cao Institution of
Mathematics, Jilin University, Changchun, 130012, China caoyang@jlu.edu.cn
and Geng Tian Institution of Mathematics, Jilin University, Changchun,
130012, China tiangeng09@mails.jlu.edu.cn
# spectrums of equivalent Schauder operators
Luo Yi Shi Department of Mathematics, Tianjin Polytechnic University,
Tianjin, 300160,CHINA sluoyi@yahoo.cn , Yang Cao Institution of
Mathematics, Jilin University, Changchun, 130012, China caoyang@jlu.edu.cn
and Geng Tian Institution of Mathematics, Jilin University, Changchun,
130012, China tiangeng09@mails.jlu.edu.cn
###### Abstract.
Assume that $T_{1},T_{2}$ are equivalent Schauder operators. In this paper, we
show that even in this case their Schauder spectrum may be very different in
the view of operator theory. In fact, we get that if a self-adjoint Schauder
operator $A$ has more than one points in its essential spectrum
$\sigma_{e}(A)$, then there exists a unitary spread operator $U$ such that the
Schauder spectrum $\sigma_{S}(UA)$ contains a ring which is depended by the
essential spectrum; if there is only one point in $\sigma_{e}(A)$ and
satisfies some conditions then there exists a unitary spread operator $U$ such
that the Schauder spectrum $\sigma_{S}(UA)$ contains the circumference which
is depended by the essential spectrum.
###### 2000 Mathematics Subject Classification:
Primary 47B37, 47B99; Secondary 54H20, 37B99
## 1\. Introduction
In their paper [3], Cao give an operator theory description of bases on a
separable Hilbert space $\mathcal{H}$. To study operators on $\mathcal{H}$
from a basis theory viewpoint, it is naturel to consider the behavior of
operators related by equivalent bases. For examples, they show that there
always be some strongly irreducible operators in the orbit of equivalent
Schauder matrices([4]). However, in the usual way a spectral method
consideration of operators in the equivalent orbit is also important to the
joint research both on operator theory and Schauder bases. Cao introduces the
conception Schauder spectrum to do this work. The main purpose of this paper
is to show that the Schauder spectrum of Schauder operators in a given orbit
can be very different.
Recall that a sequence of vectors $\\{f_{n}\\}_{n=1}^{\infty}$ in
$\mathcal{H}$ is said to be a Schauder basis [13, 9] for $\mathcal{H}$ if
every element $f\in\mathcal{H}$ has a unique series expansion $f=\sum
c_{n}f_{n}$ which converges in the norm of $\mathcal{H}$. If $\\{f_{n}\\}$ is
Schauder basic for $\mathcal{H}$, the sequence space associated with
$\\{f_{n}\\}$ is defined to be the linear space of all sequences $\\{c_{n}\\}$
for which $f=\sum c_{n}f_{n}$ is convergent. Two Schauder bases
$\\{f_{n}\\}_{n=1}^{\infty}$ and $\\{g_{n}\\}_{n=1}^{\infty}$ are equivalent
to each other if they have the same sequence space(cf, [13], definition 12.1,
p131, [5], p163). Denote by $\omega$ the countable infinite cardinal. In paper
[2], Cao.e.t considered the $\omega\times\omega$ matrix whose column vectors
comprise a Schauder basis and call them the Schauder matrix. An operator has a
Schauder matrix representation under some ONB is called a Schauder operator.
Given an orthonormal basis(ONB in short) $\varphi=\\{e_{n}\\}_{n=1}^{\infty}$,
the vector $f_{n}$ in a Schauder basis sequence
$\psi=\\{f_{n}\\}_{n=1}^{\infty}$ corresponds an $l^{2}$ sequence
$\\{f_{mn}\\}_{m=1}^{\infty}$ defined uniquely by the series
$f_{n}=\sum_{m=1}^{\infty}f_{mn}e_{m}$. The matrix
$F_{\psi}=(f_{mn})_{\omega\times\omega}$ is called the Schauder matrix of
basis $\psi$ under the ONB $\varphi$.
Assume that $\psi_{1},\psi_{2}$ are equivalent Schauder bases and
$T_{\psi_{1}},T_{\psi_{2}}$ are the operators defined by Schauder matrices
$F_{\psi_{1}}$ and $F_{\psi_{2}}$ respectively under the same ONB. Then there
are no difference between $\psi_{1}$ and $\psi_{2}$ from the view of bases of
the Hilbert space. Are there some notable differences between the operators
$T_{\psi_{1}}$ and $T_{\psi_{2}}$ from the view of operator theory? From the
Arsove’s theorem([1], or theorem 2.12 in [2]), there is some invertible
operator $X\in L(\mathcal{H})$ such that $XT_{\psi_{1}}=T_{\psi_{2}}$ holds.
Hence for a Schauder basis $\psi=\\{f_{n}\\}_{n=1}^{\infty}$, the set defined
as
$\mathcal{O}_{gl}(\psi)=\\{X\psi;X\in gl(\mathcal{H})\\}$
in which $X\psi=\\{Xf_{n}\\}_{n=1}^{\infty}$ and $gl(\mathcal{H})$ consists of
all invertible operators in $L(\mathcal{H})$ contains exactly all equivalent
bases to $\psi$. Moreover, the set
$\mathcal{O}_{gl}(F_{\psi})=\\{M_{X}F_{\psi};M_{X}\hbox{ is the matrix of some
operator }X\in gl(\mathcal{H})\\}$
consists of all Schauder matrix equivalent to $F_{\psi}$. In the operator
level, we define
$\mathcal{O}_{gl}(T_{\psi})=\\{XT_{\psi};X\in gl(\mathcal{H})\\}.$
Then the set $\mathcal{O}_{gl}(T_{\psi})$ consists of operators related to
bases equivalent to $\psi$. Similarly, we consider following sets:
$\begin{array}[]{c}\mathcal{O}_{u}(\psi)=\\{U\psi;U\in U(\mathcal{H})\\},\\\
\mathcal{O}_{u}(F_{\psi})=\\{M_{U}F_{\psi};M_{U}\hbox{ is the matrix of some
unitary operator }U\\},\\\ \mathcal{O}_{u}(T_{\psi})=\\{UT_{\psi};U\in
U(\mathcal{H})\\},\end{array}$
where $U(\mathcal{H})$ consists of all unitary operators in $L(\mathcal{H})$.
Roughly speaking, by these set we bind operators related to equivalent bases
of the basis $\psi$ with the same basis const. It is easy to check that a
Schauder operator $T_{\psi}$ must be injective and having a dense range.
Denote by $T_{\psi}=UA_{\psi}$ the polar decomposition of $T_{\psi}$, then the
partial isometry $U$ must be a unitary operator. Then the orbit
$\mathcal{O}_{u}(T_{\psi})$ is just the orbit $\mathcal{O}_{u}(A_{\psi})$ in
which $A_{\psi}$ is the self-adjoint operator defined by the polar
decomposition of $T_{\psi}$. In this paper we focus on unitary operators with
a nice basis theory understanding, that is, a slight generalization of spread
form defined by W. T. Gowers and B. Maurey([6], [7]).
For a complex number $\lambda,\lambda$ will be called in the Schauder spectrum
of $T$ denoted by $\sigma_{S}(T)$ if and only if there is no ONB such that
$\lambda I-T$ has a matrix representation as a Schauder matrix. It is
obviously, $\sigma(T)\supset\sigma_{S}(T)=\sigma_{p}(T)\cup\sigma_{r}(T)$ in
which $\sigma_{r}(T)=\\{\lambda\in\mathbb{C},\overline{\textup{Ran}(\lambda
I-T)}\neq\mathcal{H}\\}$.
Now we state our main theorem:
###### Theorem 1.1.
Assume that $A$ is a self-adjoint Schauder operator.
(i) If $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}],\lambda_{1}>0$ and
$\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$, then there exists a unitary spread
operator $U$ such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for
any rings $R$ in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;
(ii) If $\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$ and
$0<\lambda_{1}<\lambda_{2}$, then there exists a unitary spread operator $U$
such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for any rings $R$
in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;
(iii) If there exists only one point $\lambda_{1}\in\sigma_{e}(A)$,
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{1}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then there exists a unitary spread operator $U$ such that the Schauder
spectrum $\sigma_{S}(UA)\supseteq\\{\lambda,|\lambda|=\lambda_{1}\\}$.
That is, if $T$ is a Schauder operator, then there exist operator
$T_{1}\in\mathcal{O}_{u}(T)$ such that $\sigma_{S}(T_{1})$ has a certain
thickness. Related concept will be clear in later section.
We organize our paper as follows. In section 2, we introduce some notations
and lemmas which will be used in the main theorem; in section 3, we research
the case that the spectrum of self-adjoint Schauder operator has only two
points; In section 4 we research the case that the essential spectrum of self-
adjoint Schauder operator has only two points; In section 5, we research the
case that there is no point spectrum in the spectrum of self-adjoint Schauder
operator. At last, we get that if $A$ is a self-adjoint Schauder operator with
at least two essential spectrum, then exists $UA\in\mathcal{O}_{u}(A)$ such
that $\sigma_{S}(A)$ is thin and $\sigma_{S}(UA)$ has a certain thickness.
###### Remark 1.2.
In the seminar held at Jilin university, Cao shows that for a Schauder
operator $T$ there must be some unitary spread $U$ such that the Schauder
operator $UT$ has an empty Schauder spectrum. In this sense, our result in
this paper show that the Schauder spectrum of $UT$ may be very bad.
## 2\. Notation and auxiliary results
In this section we will introduce some notation for convenience, and some
lemmas which will be used in the main theorem.
Throughout this paper, let
$R_{\lambda_{1},\lambda_{2}}=\\{\lambda,\lambda_{1}\leq|\lambda|\leq\lambda_{2}\\}$,
$R_{\lambda_{1}}=\\{\lambda,|\lambda|=\lambda_{1}\\}$,
$R_{\lambda_{2}}=\\{\lambda,|\lambda|=\lambda_{2}\\}$ and
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,\lambda_{1}<|\lambda|<\lambda_{2}\\}$
for $0<\lambda_{1}<\lambda_{2}$. If $E$ is a subset of complex plane
$\mathbb{C}$ and $0\notin E$, let
$E^{-1}=\\{\lambda,\frac{1}{\lambda}\in\textup{E}\\}$, $\textup{Card}\\{E\\}$
denote the cardinal number of $E$.
Recall the definition of the spread from $A$ to $B$ given by W. T. Gowers and
B. Maurey.
###### Definition 2.1.
([7], p549) Given an ONB $\\{e_{n}\\}_{n=1}^{\infty}$ and two infinite subsets
$A,B$ of $\mathbb{N}$. Let $c_{00}$ be the vector space of all sequences of
finite support. Let the elements of $A$ and $B$ be written in increasing order
respectively as $\\{a_{1},a_{2},\cdots\\}$ and $\\{b_{1},b_{2},\cdots\\}$.
Then $e_{n}$ maps to $0$ if $n\notin A$, and $e_{a_{k}}$ maps to $e_{b_{k}}$
for every $k\in N$. Denote this map by $S_{A,B}$ and call it the spread from
$A$ to $B$.
Using spread forms, we can write some unitary operator into their linear
combination. See the Example 4.13 in [2].
###### Definition 2.2.
([2], Definition 4.14) A unitary operator $U$ on $\mathcal{H}$ is said to be a
unitary spread if there is a sequence $\\{S_{A_{n},B_{n}}\\}_{n=1}^{\infty}$
of spreads such that the series $\sum_{n=1}^{\infty}S_{A_{n},B_{n}}$ converges
to $U$ in strongly operator topology (SOT). Moreover, $U$ will be called a
finite unitary spread if $U$ can be written as a finite linear combination.
In the paper [3], Cao.e.t proved that for each bijection $\sigma$ on the set
$\mathbb{N}$, the unitary operator $U_{\sigma}$ is a unitary spread.
###### Lemma 2.3.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$, and there
exists $x\neq 0$ such that $||Ax||=\lambda_{i}||x||$. Then
$\lambda_{i}\in\sigma_{p}(A)$, and $x\in\textup{Ker}(\lambda_{i}I-A)$,
$i=1,2$.
###### Proof.
Indeed we only need to prove the case of $i=1$. The proof of the case of
$i=2$, is minor modifications of the proof of the analogous statements in the
case of $i=1$ by consider $A^{-1}$ and will be omitted.
Since $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$, we know that
$(Ax,x)\geq\lambda_{1}||x||$ for any $x\neq 0$. Hence,
$||(\lambda_{1}I-A)x||^{2}=\lambda_{1}^{2}||x||+||Ax||^{2}-2\lambda_{1}(Ax,x)\leq
0$, it follows that $||(\lambda_{1}I-A)x||=0$. That is to say
$x\in\textup{Ker}(\lambda_{1}I-A)$ and $\lambda_{1}\in\sigma_{p}(A)$. ∎
###### Lemma 2.4.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$. Then, for any
unitary operator $U$,
(i) $\sigma(UA)\subseteq R_{\lambda_{1},\lambda_{2}}$;
(ii) If $\lambda_{1},\lambda_{2}\notin\sigma_{p}(A)$, then $\sigma_{p}(UA)\cap
R_{\lambda_{i}}=\emptyset$;if $\lambda_{1},\lambda_{2}\in\sigma_{p}(A)$, then
$\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{i}}\\}\leq\textup{dim
Ker}(\lambda_{i}I-A),i=1,2$.
###### Proof.
(i) It is well known that if $T$ is an invertible operator, then
$\sigma(T^{-1})=\\{\lambda,\lambda^{-1}\in\sigma(T)\\}$, and $r(T)\leq||T||$
for any $T\in B(\mathcal{H})$. Thus $\sigma(UA)=\frac{1}{\sigma((UA)^{-1})}$
and $||UA||=||A||=\lambda_{2},||(UA)^{-1}||=||A^{-1}||=\frac{1}{\lambda_{1}}$.
It follows that $\lambda\notin\sigma(UA)$ when $|\lambda|>\lambda_{2}$ and
$\lambda\notin\sigma((UA)^{-1})$ when $|\lambda|>\frac{1}{\lambda_{1}}$.
Hence, $\sigma(UA)\subseteq R_{\lambda_{1},\lambda_{2}}$, for any unitary
operator $U$.
(ii)Indeed we only need to prove the case of $i=1$. The proof of the case of
$i=2$, is minor modifications of the proof of the analogous statements in the
case of $i=1$ by consider $A^{-1}$ and will be omitted.
Assume $U$ is a unitary operator and $\lambda\in\sigma_{p}(UA)\cap
R_{\lambda_{1}}$. Then there exists $x\neq 0$ such that $UAx=\lambda x$ and
$||Ax||=||UAx||=\lambda_{1}||x||$. By Lemma 2.3, $\lambda_{1}\in\sigma_{p}(A)$
and $x\in\textup{Ker}(\lambda_{1}I-A)$. Hence, $A$ and $U$ have the matrix
forms
$A=\left[\begin{array}[]{cc}\lambda_{1}I&\\\
&A_{1}\end{array}\right]\begin{matrix}\mbox{$\textup{Ker}(\lambda I-UA)$}\\\
\mbox{$\textup{Ker}(\lambda I-UA)^{\perp}$}\\\
\end{matrix},U=\left[\begin{array}[]{cc}U_{11}&U_{12}\\\
U_{21}&U_{22}\end{array}\right]\begin{matrix}\mbox{$\textup{Ker}(\lambda
I-UA)$}\\\ \mbox{$\textup{Ker}(\lambda I-UA)^{\perp}$}\\\ \end{matrix},$
and $\textup{Ker}(\lambda I-UA)\subseteq\textup{Ker}(\lambda_{1}I-A)$.
For any $x\in\textup{Ker}(\lambda I-UA)$, $UAx=\lambda x$. Since $U$ is a
unitary operator, it is easy to check that $U_{12}=U_{21}=0$. Hence, $UA$ has
the matrix form
$UA=\left[\begin{array}[]{cc}\lambda_{1}U_{11}&\\\
&U_{22}A_{1}\end{array}\right]\begin{matrix}\mbox{$\textup{Ker}(\lambda
I-UA)$}\\\ \mbox{$\textup{Ker}(\lambda I-UA)^{\perp}$}\\\ \end{matrix},$
in which $U_{11}$ and $U_{22}$ are unitary operators and
$\sigma_{P}(UA)=\sigma_{P}(\lambda_{1}U_{11})\cup\sigma_{P}(U_{22}A_{1}),$
$\textup{Card}\\{\sigma_{P}(\lambda_{1}U_{11})\\}\leq\textup{dim Ker}(\lambda
I-UA)\leq\textup{dim Ker}(\lambda_{1}I-A).$
If there exists another $\delta\in\sigma_{p}(U_{22}A_{1})\cap
R_{\lambda_{1}}$, repeating the above process, we can get that
$\textup{Ker}(\delta I-U_{22}A_{1})\subseteq\textup{Ker}(\lambda_{1}I-A)$ and
$\textup{Ker}(\delta I-U_{22}A_{1})\bot\textup{Ker}(\lambda I-UA)$.
Repeating the above process, we can obtain that
$\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq\textup{dim
Ker}(\lambda_{1}I-A).$
∎
###### Remark 2.5.
By the above lemma, we know that if the spectrum $\sigma(A)$ of a self-adjoint
Schauder operator is contained in an interval, then the Schauder spectrum
$\sigma_{S}(UA)$ must be contained in the ring which is depended by the
interval.
## 3\. Only two points in $\sigma(A)$
In this section, we will research the case that the spectrum of self-adjoint
Schauder operator $A$ has only two points $\lambda_{1},\lambda_{2}$ and
$0<\lambda_{1}<\lambda_{2}$.
According to Lemma 2.4, we know that for any unitary operator $U$, there
exists at most denumerable subsets $\sigma_{1}$ in $R_{\lambda_{1}}$ and
$\sigma_{2}$ in $R_{\lambda_{2}}$ such that
$\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup
R_{\lambda_{1},\lambda_{2}}^{o}$. In this section, we will show that if
ker$(\lambda_{i}-A)=\infty,i=1,2$, then for any at most denumerable subsets
$\sigma_{1}$ in $R_{\lambda_{1}}$, $\sigma_{2}$ in $R_{\lambda_{2}}$ and a
ring $R$ in $R_{\lambda_{1},\lambda_{2}}^{o}$, there exists a unity operator
$U$ such that $\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup R$. Hence,
there exists $UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness.
###### Lemma 3.1.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $0<\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R_{\lambda_{1},\lambda_{2}}^{o}$.
###### Proof.
By the classical spectral theory of normal operator, we have following
orthogonal decomposition of $A$,
$A=\oplus_{n\in\mathbb{Z}}A_{n},$
in which $A_{0}=\lambda_{2}I,A_{-1}=\lambda_{1}I,$ $A_{n}=\lambda_{1}I$ for
all $n\geq 1$ and $A_{n}=\lambda_{2}I$ for all $n\leq-2$.
Now we choose an ONB $\\{e_{k}^{(n)}\\}_{k=1}^{\infty}$, for each
$n\in\mathbb{Z}$. And let $U$ be the unitary spread operator defined as
$Ue_{k}^{(n)}=e_{k}^{(n+1)},n\in\mathbb{Z},k\in\mathbb{N}.$
For a vector $x\in\mathcal{H}$ now under the ONB constructed it has a
$l_{2}$-sequence coordinate in the form
$x=\sum_{n\in\mathbb{Z}}x^{(n)}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}x_{k}^{(n)}e_{k}^{(n)}.$
Now simply we have
$UAx=UA(\sum_{n\in\mathbb{Z}}x^{(n)})=\sum_{n\in\mathbb{Z}}UAx^{(n)}=\sum_{n\in\mathbb{Z}}A_{n-1}x^{(n-1)}.$
Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have
$\lambda x^{(n)}=A_{n-1}x^{(n-1)}.$
Therefore, following equations hold:
$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$
$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$
That is to say $x^{(n)}=\lambda^{-n}\lambda_{1}^{n-1}\lambda_{2}x^{(0)},n\geq
1$ and $x^{(n)}=\lambda^{-n}\lambda_{1}^{-1}\lambda_{2}^{n+1}x^{(0)},n\leq-1.$
Since $0<\lambda_{1}<\lambda_{2}$ it is easy to see that if
$\lambda_{1}<|\lambda|<\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if $\lambda_{1}\leq|\lambda|$
then $||x^{(n)}||>1$ for $n\geq 1$, if $|\lambda|\geq\lambda_{2}$ then
$||x^{(-n)}||>1$ for $n\geq 1$, i.e. if $\lambda_{1}\leq|\lambda|$ or
$|\lambda|\geq\lambda_{2}$ then $\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$.
Hence, $\sigma_{p}(UA)=\\{\lambda,\lambda_{1}<|\lambda|<\lambda_{2}\\}$.
∎
###### Proposition 3.2.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $0<\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$.
###### Proof.
Firstly, we prove that if
$R=\\{\lambda,(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}<|\lambda|<(\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}})^{\frac{1}{m_{1}+m_{2}}}\\}$
is a ring in $R_{\lambda_{1},\lambda_{2}}^{o}$ for some integers
$n_{1},n_{2},m_{1},m_{2}$, then there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R$.
We assign the same notations used in the proof of Lemma 3.1.
$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$
$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$
Let
$A_{0}=A_{1}=\cdots A_{n_{1}}=\lambda_{1}I,$ $A_{n_{1}+1}=A_{n_{1}+2}=\cdots
A_{n_{1}+n_{2}}=\lambda_{2}I,$ $\vdots$
$A_{k_{1}n_{1}+k_{2}n_{2}+k_{3}}=\lambda_{1}I,$
for any $k_{1},k_{2}$ and $1\leq k_{3}\leq n_{1}$,
$A_{k_{1}n_{1}+k_{2}n_{2}+k_{3}}=\lambda_{2}I,$
for any $k_{1},k_{2}$ and $n_{1}+1\leq k_{3}\leq(n_{1}+n_{2})$.
And let
$A_{-1}=\cdots A_{-m_{2}}=\lambda_{2}I,$ $A_{-m_{2}-1}=A_{-m_{2}-2}=\cdots
A_{-m_{2}-m_{1}}=\lambda_{1}I,$ $\vdots$
$A_{-k_{1}m_{2}-k_{2}m_{1}-k_{3}}=\lambda_{2}I,$
for any $k_{1},k_{2}$ and $1\leq k_{3}\leq m_{1}$,
$A_{-k_{1}m_{2}-k_{2}m_{1}-k_{3}}=\lambda_{1}I,$
for any $k_{1},k_{2}$ and $m_{2}+1\leq k_{3}\leq(m_{1}+m_{2})$.
Then we have
$x^{(k(n_{1}+n_{2}))}=\lambda^{-k(n_{1}+n_{2})}\lambda_{1}^{kn_{1}}\lambda_{2}^{kn_{2}}x^{(0)},k\geq
1,$
and
$x^{-(k(m_{1}+m_{2}))}=\lambda^{-k(m_{1}+m_{2})}\lambda_{1}^{-km_{1}}\lambda_{2}^{-km_{2}}x^{(0)},k\geq
1.$
It is easy to see that if
$(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}<|\lambda|<(\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}})^{\frac{1}{m_{1}+m_{2}}}$
then $\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if
$(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}\geq|\lambda|$
there exists $N$ such that $||x^{(n)}||>1$ for $n>N$, if
$|\lambda|\geq(\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}})^{\frac{1}{m_{1}+m_{2}}}$
there exists $N$ such that $||x^{(-n)}||>1$ for $n>N$, i.e. if
$\lambda_{1}\leq|\lambda|$ or $|\lambda|\geq\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$. Hence, $\sigma_{p}(UA)=R$.
Now we turn to the more general situation.
Since
$\lim_{n_{1}\rightarrow\infty}(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}=\lambda_{1},\lim_{n_{2}\rightarrow\infty}(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}=\lambda_{2}$.
We can get that there exists a unitary spread operator $U$ such that
$\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$.
∎
###### Lemma 3.3.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}$ for any at most
denumerable subsets $\sigma_{1}$ in $\\{\lambda,|\lambda|=\lambda_{1}\\}$ and
$\sigma_{2}$ in $\\{\lambda,|\lambda|=\lambda_{2}\\}$.
###### Proof.
Since $A$ is a self-adjoint operator, by the classical spectral theory of
normal operator, we have following orthogonal decomposition of $A$
$\begin{matrix}\begin{bmatrix}\lambda_{1}I&\\\
&\lambda_{2}I\end{bmatrix}&\end{matrix}.$
Let $U=U_{1}\oplus U_{2}$, in which $U_{1},U_{2}$ are unity operators such
that
$\sigma_{p}(\lambda_{1}U_{1})=\sigma_{1},\sigma_{p}(\lambda_{2}U_{2})=\sigma_{2}$.
Then $U$ is a unity operator and $\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}$. ∎
According to the Lemmas 2.4, 3.1, 3.3 and the Proposition 3.2, we can get the
following theorem:
###### Theorem 3.4.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $0<\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}\cup R$ for any at most
denumerable subsets $\sigma_{1}$ in $\\{\lambda,|\lambda|=\lambda_{1}\\}$,
$\sigma_{2}$ in $\\{\lambda,|\lambda|=\lambda_{2}\\}$ and $R$ is a ring in the
ring $R_{\lambda_{1},\lambda_{2}}^{o}$. Moreover, for any unitary operator
$U$, there exists at most denumerable subsets $\sigma_{1}$ in
$R_{\lambda_{1}}$ and $\sigma_{2}$ in $R_{\lambda_{2}}$ such that
$\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup
R_{\lambda_{1},\lambda_{2}}^{o}$.
###### Proof.
Since $A$ is a self-adjoint operator, by the classical spectral theory of
normal operator, we have following orthogonal decomposition of $A=A_{1}\oplus
A_{1}$ where $A_{1}$ is a self-adjoint operator satisfying that
$\sigma(A_{1})=\\{\lambda_{1},\lambda_{2}\\}$ and dim
ker$(\lambda_{i}-A)=\infty$. By Lemmas 3.1, 3.3 and the Proposition 3.2, we
get that there exists a unitary operator $U$ such that
$\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}\cup R$ for any at most denumerable
subsets $\sigma_{1}$ in $\\{\lambda,|\lambda|=\lambda_{1}\\}$, $\sigma_{2}$ in
$\\{\lambda,|\lambda|=\lambda_{2}\\}$ and $R$ is a ring in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$. The last part of this theorem is obvious by
the Lemma 2.4. ∎
###### Remark 3.5.
By the above theorem, we know that if the spectrum $\sigma(A)$ of a self-
adjoint Schauder operator has only two points $\lambda_{1},\lambda_{2}$ and
ker$(\lambda_{i}-A)=\infty,i=1,2$, then for any ring $R$ in
$R_{\lambda_{1},\lambda_{2}}^{o}$ and at most denumerable subsets $\sigma_{1}$
in $\\{\lambda,|\lambda|=\lambda_{1}\\}$, $\sigma_{2}$ in
$\\{\lambda,|\lambda|=\lambda_{2}\\}$, there exists $UA\in\mathcal{O}_{u}(A)$
such that $\sigma_{S}(UA)$ contains $\sigma_{1}\cup\sigma_{2}\cup R$. i.e.
$\sigma_{S}(UA)$ has a certain thickness, $\sigma_{S}(A)$ is thin. In other
words, there is no ONB such that $\lambda I-UA$ has a matrix representation as
a Schauder matrix for $\lambda\in\sigma_{1}\cup\sigma_{2}\cup R$.
## 4\. Only two points in $\sigma_{e}(A)$
In this section, we will research the case that the essential spectrum of
self-adjoint operator $A$ has only two points $\lambda_{1},\lambda_{2}$ and
$0<\lambda_{1}<\lambda_{2}$. We will show that for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$,
there exists a unitary spread operator $U$ such that
$R_{\lambda_{1}\lambda_{2}}\supseteq\sigma_{p}(UA)\supseteq R$. i.e. there
exists $UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness.
###### Theorem 4.1.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
(i) $\sigma(A)=\sigma_{p}(A)\cup\\{\lambda_{1},\lambda_{2}\\}$,
$0<\lambda_{1}<\lambda_{2}$ and $\lambda_{1},\lambda_{2}$ are the unique
accumulation points of $\sigma(A)$;
(ii) For each $t\in\sigma_{p}(A)$, dim $\ker(A-tI)=1$.
Then there exists a unitary spread operator $U$ such that
$R_{\lambda_{1}\lambda_{2}}\supseteq\sigma_{p}(UA)\supseteq R$ for any rings
$R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.
Moreover, if $t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$, then there exists a
unitary spread operator $U$ such that $\sigma_{p}(UA)=R$ for any rings $R$ in
the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$;
for any unitary operator $U$, $\sigma_{p}(UA)\subset
R_{\lambda_{1},\lambda_{2}}=\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$
and $\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq
1,\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq 1$.
###### Proof.
We only prove the case that $R=R_{\lambda_{1}\lambda_{2}}$, the proof of the
more general situation is similar to the Proposition 3.2 and we omit it.
The self-adjoint operator satisfying the conditions appearing in the
proposition has a spectrum in the following form:
$\sigma(A)=\\{t_{1},t_{2},\cdots,t_{k},\cdots\\}\cup\\{r_{1},r_{2},\cdots,r_{k},\cdots\\}\cup\\{\lambda_{1},\lambda_{2}\\},$
in which $\lambda_{1}$ is the accumulation point of the sequence
$\\{t_{k}\\}$, $\lambda_{2}$ is the accumulation point of the sequence
$\\{r_{k}\\}$.
Choose the subsequences $\\{t_{nk}\\}_{k=1}^{\infty}$, $n\geq 0$ of
$\\{t_{k}\\}$ and $\\{r_{nk}\\}_{k=1}^{\infty}$, $n\geq 1$ of $\\{r_{k}\\}$
satisfying the following properties:
(i)
$\lim_{k\rightarrow\infty}t_{nk}=\lim_{n\rightarrow\infty}t_{nk}=\lambda_{1}$,
$\lim_{k\rightarrow\infty}r_{nk}=\lim_{n\rightarrow\infty}r_{nk}=\lambda_{2}$;
(ii) There exist $t_{nk}$ and $r_{nk}$ such that $t_{nk}=t_{k_{0}}$,
$r_{nk}=r_{k_{1}}$ for any $t_{k_{0}}\in\\{t_{k}\\},r_{k_{1}}\in\\{r_{k}\\}$;
(iii) $t_{n_{1}k_{1}}\neq t_{n_{2}k_{2}},r_{n_{1}k_{1}}\neq r_{n_{2}k_{2}}$
when $n_{1}\neq n_{2}$ or $k_{1}\neq k_{2}$.
Let $J_{n}=\\{t_{nk}\\},n\geq 0,J_{n}=\\{r_{-nk}\\},n\leq-1.$ We rearrange
these intervals as follows: $I_{0}=J_{0},I_{n}=J_{n+1}$ for $n\geq 1$,
$I_{-1}=J_{1},I_{n}=J_{n+1}$ for $n\leq-2$.
Denote $E_{n}=E_{I_{n}}$ the spectral projection on the interval $I_{n}$ and
by $H_{n}=$Ran$(E_{n})$ for $n\in\mathbb{Z}$. Now we choose an ONB
$\\{e_{k}^{(n)}\\}_{k=1}^{\infty}$, for each $n\in\mathbb{Z}$. Since each
$H_{n}$ is a reducing subspace of $A$, we can write $A$ into the direct sum:
$A=\oplus_{n=-\infty}^{+\infty}A_{n}.$
Let
$\sup_{k}\\{t_{nk}\\}=\alpha_{n}^{(1)},\inf_{k}\\{t_{nk}\\}=\alpha_{n}^{(2)}$
for $n\geq 0$,
$\sup_{k}\\{r_{nk}\\}=\beta_{n}^{(1)},\inf_{k}\\{r_{nk}\\}=\beta_{n}^{(2)}$
for $n\geq 1$. Then
$\lim_{n\rightarrow\infty}\alpha_{n}^{(1)}=\lim_{n\rightarrow\infty}\alpha_{n}^{(2)}=\lambda_{1},\lim_{n\rightarrow\infty}\beta_{n}^{(1)}=\lim_{n\rightarrow\infty}\beta_{n}^{(2)}=\lambda_{2}$.
Now let $U$ be the unitary spread operator defined as
$Ue_{k}^{(n)}=e_{k}^{(n+1)},n\in\mathbb{Z},k\in\mathbb{N}.$
For a vector $x\in\mathcal{H}$ now under the ONB constructed it has a
$l_{2}$-sequence coordinate in the form
$x=\sum_{n\in\mathbb{Z}}x^{(n)}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}x_{k}^{(n)}e_{k}^{(n)}.$
Now simply we have
$UAx=UA(\sum_{n\in\mathbb{Z}}x^{(n)})=\sum_{n\in\mathbb{Z}}UAx^{(n)}=\sum_{n\in\mathbb{Z}}A_{n-1}x^{(n-1)}.$
Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have
$\lambda x^{(n)}=A_{n-1}x^{(n-1)}.$
Therefore, following equations hold:
$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$
$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$
Hence,
$\lambda^{-n}\beta_{0}^{(2)}\alpha_{2}^{(2)}\alpha_{3}^{(2)}\cdots\alpha_{n}^{(2)}\leq||x^{(n)}||\leq\lambda^{-n}\beta_{0}^{(1)}\alpha_{2}^{(1)}\alpha_{3}^{(1)}\cdots\alpha_{n}^{(1)},n\geq
1;$
$\frac{\lambda^{-n}}{\alpha_{1}^{(1)}\beta_{1}^{(1)}\beta_{2}^{(1)}\cdots\beta_{-n-1}^{(1)}}\leq||x^{(n)}||\leq\frac{\lambda^{-n}}{\alpha_{1}^{(2)}\beta_{1}^{(2)}\beta_{2}^{(2)}\cdots\beta_{-n-1}^{(2)}},n\leq-1.$
Since
$\lim_{n\rightarrow\infty}\alpha_{n}^{(1)}=\lim_{n\rightarrow\infty}\alpha_{n}^{(2)}=\lambda_{1},\lim_{n\rightarrow\infty}\beta_{n}^{(1)}=\lim_{n\rightarrow\infty}\beta_{n}^{(2)}=\lambda_{2}$,
it is easy to see that if $\lambda_{1}<|\lambda|<\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if $\lambda_{1}<|\lambda|$
there exists $N$ such that $||x^{(n)}||>1$ for $n>N$, if
$|\lambda|>\lambda_{2}$ there exists $N$ such that $||x^{(-n)}||>1$ for $n>N$,
i.e. if $\lambda_{1}<|\lambda|$ or $|\lambda|>\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$. That is to say
$\\{\lambda,\lambda_{1}\leq|\lambda|\leq\lambda_{2}\\}\supseteq\sigma_{p}(UA)\supseteq\\{\lambda,\lambda_{1}<|\lambda|<\lambda_{2}\\}$.
Moreover, if $t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$ then
$t_{k}>\lambda_{1},r_{k}<\lambda_{2}$. So
$\alpha_{n}^{(i)}\geq\lambda_{1},\beta_{n}^{(i)}\leq\lambda_{2}$ for all $n$
and $i=1,2$. It is easy to see that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.
Furthermore, by Lemma 2.4, we get that for any unitary operator $U$,
$\sigma_{p}(UA)\subset\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$
and $\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq
1,\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq 1$. ∎
###### Remark 4.2.
(i) Trivial modifications adapt the proof of Theorem 4.1, we can weaken the
condition dim $\ker(A-tI)=1$ to dim $\ker(A-tI)<\infty$.
(ii) In the Theorem 4.1, we obtained that there exists a unitary operator $U$
such that $\sigma_{p}(UA)\supseteq R$ for any rings $R$ in the ring
$R_{\lambda_{1}\lambda_{2}}$. Moreover, we got
$\sigma_{p}(UA)=R_{\lambda_{1}\lambda_{2}}$ if adding the condition that
$t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$. The following examples illustrate
that this condition is necessary.
###### Example 4.3.
We assign the same notations used in the Theorem 4.1.
(1) Let $\lambda_{1}=1$, $\lambda_{2}>1$, and $t_{n1}=1-\frac{1}{n}$,
$t_{nk}=\frac{k+n-1}{k+n}+\frac{\frac{k+n}{k+n+1}-\frac{k+n-1}{k+n}}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$ and $r_{k}<r_{k+1}$ for all $k\geq 1$. Then according
to the proof of Theorem 4.1 and let $A_{n}=\oplus_{k=1}^{\infty}t_{nk}$,
$x^{(0)}=e_{0}^{(0)}$ in Theorem 4.1, we obtain that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|\leq|\lambda|<\lambda_{2}\\}$.
(2) Let $\lambda_{2}=1$, $\lambda_{1}<1$, and $r_{n1}=1+\frac{1}{n}$,
$r_{nk}=\frac{k+n+1}{k+n}-\frac{\frac{k+n+1}{k+n}-\frac{k+n+2}{k+n+1}}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$ and $t_{k}>t_{k+1}$ for all $k\geq 1$. Then according
to the proof of Theorem 4.1 and let $A_{n}=\oplus_{k=1}^{\infty}t_{nk}$,
$x^{(0)}=e_{0}^{(0)}$ in Theorem 4.1, we obtain that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|<|\lambda|\leq\lambda_{2}\\}$.
(3) Let $\lambda_{1}=1$, $\lambda_{2}=2$, and $t_{n1}=1-\frac{1}{n}$,
$r_{n1}=2+\frac{2}{n}$,
$t_{nk}=\frac{k+n-1}{k+n}+\frac{\frac{k+n}{k+n+1}-\frac{k+n-1}{k+n}}{k+n-1}\cdot
n$,
$r_{nk}=(2+\frac{2}{k+n-1})-\frac{(2+\frac{2}{k+n-1})-(2+\frac{2}{k+n})}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$. Then according to the proof of Theorem 4.1 and let
$A_{n}=\oplus_{k=1}^{\infty}t_{nk}$, $x^{(0)}=e_{0}^{(0)}$ in Theorem 4.1, we
obtain that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$.
Trivial modifications adapt the proof of the Theorem 4.1, we can get the
following Proposition.
###### Corollary 4.4.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
(i) $\sigma(A)=\sigma_{p}(A)\cup\\{\lambda_{1}\\}$, $0<\lambda_{1}$ and
$\lambda_{1}$ is the unique accumulation point of $\sigma(A)$;
(ii) For each $t\in\sigma_{p}(A)$, dim $\ker(A-tI)<\infty$;
(iii)
$\sigma_{p}(A)=\\{t_{1},t_{2},\cdots,t_{k},\cdots\\}\cup\\{r_{1},r_{2},\cdots,r_{k},\cdots\\}$,
$t_{k}<t_{k+1},r_{k}>r_{k+1}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then, there exists a unitary spread operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$.
###### Example 4.5.
Let $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\sigma_{p}(A)\cup\\{1\\}$,
$\sigma_{p}(A)=\\{t_{nk},r_{nk}\\}_{k,n=1}^{\infty}$, in which
$t_{n1}=1-\frac{1}{n}$,
$t_{nk}=\frac{k+n-1}{k+n}+\frac{\frac{k+n}{k+n+1}-\frac{k+n-1}{k+n}}{k+n-1}\cdot
n$, $r_{n1}=1+\frac{1}{n}$,
$r_{nk}=\frac{k+n+1}{k+n}-\frac{\frac{k+n+1}{k+n}-\frac{k+n+2}{k+n+1}}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$, for each $t\in\sigma_{p}(A)$, dim $\ker(A-tI)=1$. By
Corollary 4.4, and (1), (2) of Example 4.3, we can get that there exists a
unitary spread operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=1\\}$.
###### Remark 4.6.
By the Theorem 4.1, we know that if the essential spectrum of self-adjoint
operator $A$ has only two points $\lambda_{1},\lambda_{2}$ and
$0<\lambda_{1}<\lambda_{2}$ and for each $t\in\sigma_{p}(A)$, dim
$\ker(A-tI)<\infty$, then for any ring $R$ in
$R_{\lambda_{1},\lambda_{2}}^{o}$, there exists $UA\in\mathcal{O}_{u}(A)$ such
that $\sigma_{S}(UA)$ contains $R$. i.e. $\sigma_{S}(UA)$ has a certain
thickness, $\sigma_{S}(A)$ is thin. In other words, there is no ONB such that
$\lambda I-UA$ has a matrix representation as a Schauder matrix for
$\lambda\in R$.
## 5\. No points spectrum in $\sigma(A)$
In this section, we will research the case that there is no point spectrum in
$\sigma(A)$. i.e. $\sigma(A)=[\lambda_{1},\lambda_{2}],0<\lambda_{1}$.
According to Lemma 2.4, we know that for any unitary operator $U$, there
exists at most denumerable subsets $\sigma_{1}$ in $R_{\lambda_{1}}$ and
$\sigma_{2}$ in $R_{\lambda_{2}}$ such that
$\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup
R_{\lambda_{1},\lambda_{2}}^{o}$. In this section, we will show that if
ker$(\lambda_{i}-A)=\infty,i=1,2$, then for any at most denumerable subsets
$\sigma_{1}$ in $R_{\lambda_{1}}$, $\sigma_{2}$ in $R_{\lambda_{2}}$ and a
ring $R$ in $R_{\lambda_{1},\lambda_{2}}^{o}$, there exists a unity operator
$U$ such that $\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup R$. i.e.
there exists $UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness.
###### Theorem 5.1.
Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$ and
$\sigma_{p}(A)=\varnothing$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.
###### Proof.
There is a sequence $\alpha_{n}\longrightarrow\lambda_{2}$ such that
$\alpha_{n+1}>\alpha_{n}$ for each $n\geq 1$. Moreover, the range of spectral
projection $E_{[\alpha_{n},\alpha_{n+1}]}$ is an infinite subspace; and a
sequence $\beta_{n}\longrightarrow\lambda_{1}$ such that
$\beta_{n}>\beta_{n+1}$ for each $n\geq 1$. Moreover, the range of spectral
projection $E_{[\beta_{n+1},\beta_{n}]}$ is an infinite subspace.
Now we rearrange these intervals as follows.
$J_{n}=[\alpha_{n},\alpha_{n+1}),n\geq 0,$
$J_{n}=[\beta_{-n+1},\beta_{-n}),n\leq-1.$
Let $I_{0}=J_{0},I_{n}=J_{-n+1}$ for $n\geq 1$, $I_{-1}=J_{-1},I_{n}=J_{-n}$
for $n\leq-1$.
Denote $E_{n}=E_{I_{n}}$ the spectral projection on the interval $I_{n}$ and
by $H_{n}=$Ran$(E_{n})$ for $n\in\mathbb{Z}$. Now we choose an ONB
$\\{e_{k}^{(n)}\\}_{k=1}^{\infty}$, for each $n\in\mathbb{Z}$. Since each
$H_{n}$ is a reducing subspace of $A$, we can write $A$ into the direct sum:
$A=\oplus_{n=-\infty}^{+\infty}A_{n}.$
And $\alpha_{0}||x||\leq||A_{0}x||\leq\alpha_{1}$ for $x\in H_{0}$,
$\beta_{0}|x||\leq||A_{-1}x||\leq\beta_{-1}||x||$ for $x\in H_{-1}$,
$\beta_{-n}||x||\leq||A_{n}x||\leq\beta_{-n-1}$ for $x\in H_{n}$, $n\geq 1$,
$\alpha_{n}||x||\leq||A_{0}x||\leq\alpha_{n+1}$ for $x\in H_{n}$ $n\leq-1$.
Now let $U$ be the unitary spread operator defined as
$Ue_{k}^{(n)}=e_{k}^{(n+1)},n\in\mathbb{Z},k\in\mathbb{N}.$
For a vector $x\in\mathcal{H}$ now under the ONB constructed it has a
$l_{2}$-sequence coordinate in the form
$x=\sum_{n\in\mathbb{Z}}x^{(n)}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}x_{k}^{(n)}e_{k}^{(n)}.$
Now simply we have
$UAx=UA(\sum_{n\in\mathbb{Z}}x^{(n)})=\sum_{n\in\mathbb{Z}}UAx^{(n)}=\sum_{n\in\mathbb{Z}}A_{n-1}x^{(n-1)}.$
Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have
$\lambda x^{(n)}=A_{n-1}x^{(n-1)}.$
Therefore, following equations hold:
$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$
$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$
Since $\alpha_{0}||x||\leq||A_{0}x||\leq\alpha_{1}$ for $x\in H_{0}$,
$\beta_{0}|x||\leq||A_{-1}x||\leq\beta_{-1}||x||$ for $x\in H_{-1}$,
$\beta_{-n}||x||\leq||A_{n}x||\leq\beta_{-n-1}$ for $x\in H_{n}$, $n\geq 1$,
$\alpha_{n}||x||\leq||A_{0}x||\leq\alpha_{n+1}$ for $x\in H_{n}$ $n\leq-1$ and
$\beta_{n}\longrightarrow\lambda_{1},\alpha_{n}\longrightarrow\lambda_{2}$, it
is easy to see that if $\lambda_{1}<|\lambda|<\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if $\lambda_{1}\leq|\lambda|$
there exists $N$ such that $||x^{(n)}||>1$ for $n>N$, if
$|\lambda|\geq\lambda_{2}$ there exists $N$ such that $||x^{(-n)}||>1$ for
$n>N$, i.e. if $\lambda_{1}<|\lambda|$ or $|\lambda|>\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$. Hence,
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.
The proof of the more general situation is similar to the Proposition 3.2. ∎
###### Remark 5.2.
By the Theorem 5.1, we know that if $A$ is a self-adjoint operator satisfying
that $\sigma(A)=[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$ and
$\sigma_{p}(A)=\varnothing$, then for any ring $R$ in
$R_{\lambda_{1},\lambda_{2}}^{o}$, there exists $UA\in\mathcal{O}_{u}(A)$ such
that $\sigma_{S}(UA)$ contains $R$. i.e. $\sigma_{S}(UA)$ has a certain
thickness, $\sigma_{S}(A)$ is thin. In other words, there is no ONB such that
$\lambda I-UA$ has a matrix representation as a Schauder matrix for
$\lambda\in R$.
Trivial modifications adapt the proof of the Theorems of 3.4, 4.1 and 5.1, we
can get the following proposition:
###### Proposition 5.3.
Assume that $A$ is a self-adjoint operator.
(i) If $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}],\lambda_{1}>0$ and
$\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$, then there exists a unitary spread
operator $U$ such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$;
(ii) If $\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$ and
$0<\lambda_{1}<\lambda_{2}$, then there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)\supseteq R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$. Moreover, if there exist sequence
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$, then there exists a unitary spread
operator $U$ such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1}\lambda_{2}}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$;
for any unitary operator $U$,
$\sigma_{p}(UA)\subset\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$
and $\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{i}}\\}\leq\textup{dim
Ker}(\lambda_{i}I-A),i=1,2$;
(iii) If there exists only one point $\lambda_{1}\in\sigma_{e}(A)$,
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{2}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then there exists a unitary spread operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$.
As we know,
$\sigma(T)\supset\sigma_{S}(T)=\sigma_{p}(T)\cup\\{\lambda\in\mathbb{C},\overline{\textup{Ran}(\lambda
I-T)}\neq\mathcal{H}\\}$ for every $T\in B(\mathcal{H})$. Hence, by the
Proposition 5.3, we obtain the main theorem:
###### Theorem 5.4.
Assume that $A$ is a self-adjoint Schauder operator.
(i) If $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}],\lambda_{1}>0$ and
$\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$, then there exists a unitary spread
operator $U$ such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for
any rings $R$ in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;
(ii) If $\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$ and
$0<\lambda_{1}<\lambda_{2}$, then there exists a unitary spread operator $U$
such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for any rings $R$
in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;
(iii) If there exists only one point $\lambda_{1}\in\sigma_{e}(A)$,
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{1}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then there exists a unitary spread operator $U$ such that the Schauder
spectrum $\sigma_{S}(UA)\supseteq\\{\lambda,|\lambda|=\lambda_{1}\\}$.
According to the Proposition 5.3 and Theorem 5.4, we know that if a self-
adjoint operator $A$ has more than one points in its essential spectrum, then
there exists a unitary spread operator $U$ such that $\sigma_{p}(UA)$ contains
a ring which is depended by the essential spectrum, i.e. there exists
$UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness; if there is only one point in the
essential spectrum and satisfies some conditions, then there exists a unitary
spread operator $U$ such that $\sigma_{p}(UA)$ contains the circumference
which is depended by the essential spectrum, i.e. there exists
$UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is at most denumerable and
$\sigma_{S}(UA)$ is uncountable. Furthermore, by Lemma 2.4, we know that if
$\sigma_{e}(A)$ has only one point $\lambda_{1}$ and $\\{t_{k}\\}$ (or
$\\{r_{k}\\}$) contained in $\sigma(A)$ and satisfy that $t_{k}<t_{k+1}$ (or
$r_{k}>r_{k+1}$), $t_{k}\rightarrow\lambda_{1}$(or
$r_{k}\rightarrow\lambda_{1}$), then for any unity operator $U$,
$\sigma_{p}(UA)\neq R_{\lambda_{1}}$. However, we don’t know if there exist
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{2}$, does there exist a
unitary operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$. It is easy to know that
if $A=\lambda I$, then the point spectrum of $UA$ is at most denumerable for
any unitary operator. We call a normal operator $A$ is non-trivial, if
$A\neq\lambda I$ for any $\lambda\in\mathbb{C}$. Hence, we have the following
question:
###### Question 5.5.
Assume that $A$ is a non-trivial invertible self-adjoint operator, and there
exists only one point $\lambda_{1}\in\sigma_{e}(A)$, $\\{t_{k}\\}$ and
$\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{2}$. Whether there must
be a unity operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$?
## References
* [1] Arsove, Maynard G. Similar bases and isomorphisms in Fr chet spaces. Math. Ann. 135, 1958, 283-293.
* [2] Y. Cao G. Tian and B. Z. Hou, Schauder Bases and Operator Theory, preprint. Avaliable at http://arxiv.org/abs/1203.3603.
* [3] Y. Cao B. Z. Hou and G. Tian, On unitary operators in spread form(in Chinese), accepted.
* [4] Y. Q. Ji G. Tian and Y. Cao, Strongly Irreducible Schauder Operators, preprint.
* [5] Garling, D. J. H., Symmetric bases of locally convex spaces, Studia Math. 30, 1968, 163-181.
* [6] W. T. Gowers and B. Maurey, The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), no. 4, 851-874.
* [7] W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997) no. 4, 543–568.
* [8] S. Jaffard and R. M. Young, A representation theorem for Schauder bases in hilbert space, Proc. Ame. Math. soc. 126 (1998) 553–560.
* [9] C. W. McArthur, developments in schauder basis theory, Bulletin of American Mathematical Society, 78 (1972) no. 6, 877–901.
* [10] Robert E. Megginson, An introuduction to Banach Space Theory, GTM183, Springe-Verlag, 1998.
* [11] A. M. Olevskii, On operators generating conditional bases in a Hilbert space, Translated from Matematicheskie Zametki, Vol(12), No.1, pp. 73-84, July, 1972.
* [12] Allen L. Shields, Weighted shift operators and analytic function theory , in: Topics in Operator Theory, Math. Surveys No. 13, 49-128, Amer. Math. Soc., Providence (1974).
* [13] I. Singer, Bases in Banach Space I, Springer-verlag, 1970.
|
arxiv-papers
| 2012-04-15T23:45:48 |
2024-09-04T02:49:29.723391
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luoyi Shi, Yang Cao, Geng Tian",
"submitter": "Cao Yang",
"url": "https://arxiv.org/abs/1204.3336"
}
|
1204.3361
|
# $B_{s}\to\phi l^{+}l^{-}$ decays in the topcolor-assisted technicolor model
Lin-Xia Lü1111E-mail: lvlinxia@sina.com, Xing-qiang Yang1, and Zong-chang
Wang1
1 Physics and electronic engineering college, Nanyang Normal University,
Nanyang, Henan 473061, P.R. China
###### Abstract
Using the updated form factors within the light-cone QCD sum rule approach, we
calculate the new physics contributions to rare semileptonic
$\bar{B}_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ decays from the new
particles appearing in the topcolor-assisted technicolor (TC2) model. In our
evaluations, we find that: (i) the branching ratio, normalized forward-
backward asymmetry and lepton polarization asymmetries show highly sensitivity
to charged top-pions contributions and little sensitivity to $Z^{\prime}$
contributions. The TC2 enhancements to the branching ratios of these decays
can reach a factor of $\sim 2$; (ii) the NP enhancement to the forward-
backward asymmetry of the decay $B_{s}\to\phi\mu^{+}\mu^{-}$ is in the range
$-13\%$ to $3\%$, but $-9\%$ to $-6\%$ for decay
$B_{s}\to\phi\tau^{+}\tau^{-}$ compared to the SM predictions; (iii) the TC2
model provide an enhancement of about $12\%$ to the longitudinal polarization
asymmetry $P_{L}$ for decay $B_{s}\to\phi\mu^{+}\mu^{-}$, but a decrease of
about $10\%$ to the transverse polarization asymmetry $P_{T}$ for the decay
$B_{s}\to\phi\tau^{+}\tau^{-}$.
###### pacs:
13.20.He, 12.15.Ji, 12.60.Nz, 14.40.Nd
## I Introduction
High energy physics experiments are designed to resolve the yet-unanswered
questions in the Standard Model (SM) through searches of new physics (NP)
using two approaches: high energy or high luminosity approach. The first
approach is to use high energy collider to produce and discover new particles
directly. The second one is to measure flavor physics observables at for
example B factory experiments and search for the signal or evidence of a
deviation from the SM prediction. A natural place is to investigate the
flavor-changing neutral current (FCNC) processes in B meson rare decays. In
the SM, the rare B decays are all induced by the so-called box and/or penguin
diagrams. Since these rare decay modes are highly suppressed in the SM, they
may serve as a good hunting ground for testing the SM and probing possible NP
effects.
At the quark level, the decays $B_{s}\to\phi l^{+}l^{-}$ proceed via FCNC
transition $b\to sl^{+}l^{-}$. The decays $B_{s}\to\phi l^{+}l^{-}$ will be
one of the most important rare decays to be studied at the LHCb experiment and
other B physics experiments. The theoretically predicted branching ratio has
the value of $\sim 1.65\times 10^{-6}$ for the $B_{s}\to\phi\mu^{+}\mu^{-}$
mode Geng2003-sm . The experimental observation was first done by CDF
collaboration CDFPhill1 and the updated branching fraction is CDFPhill2
${\cal B}(\bar{B}_{s}\to\phi\mu^{+}\mu^{-})=[1.47\pm 0.24({\rm stat.})\pm
0.46({\rm syst.})]\times 10^{-6}.$ (1)
which is consistent with the SM prediction. However, the experimental errors
are still large and this decay allow for sizable NP contributions. We will
evaluate the effects of possible NP in these decays in the topcolor-assisted
model.
The decays $B_{s}\to\phi l^{+}l^{-}$ have been studied by employing the low-
energy effective Hamiltonian and nonperturbative approaches to compute the
decay form factors in the framework of the SM Geng2003-sm ; Deandrea . Many
studies about possible new physics contributions to these decays induced by
loop diagrams involving various new particles have been published, for
example, in the two Higgs doublet model Erkol , the universal extra dimension
scenario Mohanta , the family non-universal $Z^{\prime}$ model qinchang and
other new physics scenarios Bsphill .
The topcolor-assisted (TC2) Hill model is one of the important candidates for
a mechanism of natural electroweak symmetry breaking. In the TC2 model, the
non-universal gauge boson $Z^{\prime}$, top-pions $\pi_{t}^{0,\pm}$, top-Higgs
boson $h_{t}^{0}$ and other bound states may provide potentially large loop
effects on low energy observables. In this paper, we will investigate the new
contributions from the new particles predicted by the TC2 model to the
branching ratios, the forward-backward asymmetry, and double lepton
polarization of the decays $B_{s}\to\phi l^{+}l^{-}$.
The paper is arranged as follows. In Section II, we give a brief review of the
topcolor-assisted technicolor model. In Section III, we present the
theoretical framework for $B_{s}\to\phi l^{+}l^{-}$ decays within the SM and
the TC2 model, then give the definitions and the derivations of the form
factors in the decays $B_{s}\to\phi l^{+}l^{-}$ using the updated form factors
within the light-cone QCD sum rule. In Section IV, we introduce the basic
formula for experimental observables, including dilepton invariant mass
spectrum, forward-backward asymmetry (FBA), and lepton polarization. In
Section V, we present our numerical results for these decays in the SM and the
TC2 model. The conclusions are presented in the final section.
## II Outline of the TC2 model
To completely avoid the problems arising from the elementary Higgs field in
the SM, various kinds of dynamical electroweak symmetry breaking (EWSB) models
have been proposed, among which the topcolor scenario is attractive because it
can explain the large top quark mass and provide a possible EWSB mechanism
Hill . Almost all of these kinds of models propose that the underlying
interactions, topcolor interactions, should be flavor nonuniversal. These non-
universal interactions in the mass eigenstate basis generate tree-level flavor
changing (FC) couplings and result in a rich phenomenology.
A key feature of the TC2 model is the presence of top-
pions($\pi_{t}^{0,\pm}$), the non-universal gauge boson ($Z^{\prime}$) and the
top-Higgs ($h_{t}^{0}$). These new particles treat the third-generation
fermions differently from those in the first and second generations and thus
can lead to tree-level flavor changing (FC) couplings. When one writes the
non-universal interactions in the quark mass eigen-basis, the FC couplings of
top-pions to quarks can be written as tc201 ; tc204 :
$\displaystyle\frac{m_{t}^{*}}{\sqrt{2}F_{\pi}}\frac{\sqrt{\nu_{w}^{2}-F_{\pi}^{2}}}{\nu_{w}}\left[iK_{UR}^{tc}K_{UL}^{tt^{*}}\bar{t}_{L}c_{R}\pi_{t}^{0}+\sqrt{2}K_{UR}^{tc^{*}}K_{DL}^{bb}\bar{c}_{R}b_{L}\pi_{t}^{+}+\sqrt{2}K_{UR}^{tc}K_{DL}^{bb^{*}}\bar{b}_{L}c_{R}\pi_{t}^{-}\right.$
$\displaystyle\hskip
42.67912pt\left.+\sqrt{2}K_{UR}^{tc^{*}}K_{DL}^{ss}\bar{t}_{R}s_{L}\pi_{t}^{+}+\sqrt{2}K_{UR}^{tc}K_{DL}^{ss^{*}}\bar{s}_{L}t_{R}\pi_{t}^{-}\right].$
(2)
Here $\nu_{w}=\nu/\sqrt{2}=174\rm~{}GeV$, $F_{\pi}\approx 50{\rm GeV}$ is the
top-pion decay constant, $K_{UL(R)}$ and $K_{DL(R)}$ are rotation matrices and
satisfy the equations $K_{UL}^{+}M_{U}K_{UR}=M_{U}^{dia}$ and
$K_{DL}^{+}M_{D}K_{DR}=M_{D}^{dia}$, in which $M_{U}$ and $M_{D}$ are up-quark
and down-quark mass matrices, respectively. The values of the coupling
parameters can be taken as tc204 :
$K_{UL}^{tt}\approx K_{DL}^{bb}\approx K_{DL}^{ss}\approx 1,\hskip
28.45274ptK_{UR}^{tc}\leq\sqrt{2\varepsilon-\varepsilon^{2}}.$ (3)
In our numerical analysis, we will take
$K_{UR}^{tc}=\sqrt{2\varepsilon-\varepsilon^{2}}$. The predicted top-Higgs
$h_{t}^{0}$ is a $t\bar{t}$ bound analogous to the $\sigma$ particle in low
energy QCD, and its Feynman rules are similar to the SM Higgs boson.
The Flavor diagonal (FD) couplings of top-pions to fermions take the form Hill
; tc203 :
$\displaystyle\frac{m_{t}^{*}}{\sqrt{2}F_{\pi}}\frac{\sqrt{\nu_{w}^{2}-F_{\pi}^{2}}}{\nu_{w}}\left[i\bar{t}\gamma^{5}t\pi_{t}^{0}+\sqrt{2}\bar{t}_{R}b_{L}\pi_{t}^{+}+\sqrt{2}\bar{b}_{L}t_{R}\pi_{t}^{-}\right]$
$\displaystyle+\frac{m_{b}^{*}}{\sqrt{2}F_{\pi}}\left[i\bar{b}\gamma^{5}b\pi_{t}^{0}+\sqrt{2}\bar{t}_{L}b_{R}\pi_{t}^{+}+\sqrt{2}\bar{b}_{R}t_{L}\pi_{t}^{-}\right]+\frac{m_{l}}{\nu}\bar{l}\gamma^{5}l\pi^{0}_{t}.$
(4)
The TC2 model postulates that topcolor interactions mainly couple to the third
generation fermions, and give rise to the main part of the quark mass
$m_{t}^{*}=m_{t}(1-\varepsilon)$, while the masses of the ordinary fermions
are induced by ETC (extended technicolor) interactions with
$m_{b}^{*}=m_{b}-0.1\varepsilon m_{t}$.
The FC couplings of the non-universal gauge boson $Z^{\prime}$ to fermions,
which may provide significant contributions to some FCNC processes, can be
written as tc210 :
$\displaystyle{\cal
L}^{FC}_{Z^{\prime}}=-\frac{g_{1}}{2}\cot{\theta^{\prime}}Z^{\prime\mu}\left\\{\frac{1}{3}D_{L}^{bb}D_{L}^{bs*}\bar{s}_{L}\gamma_{\mu}b_{L}-\frac{2}{3}D_{R}^{bb}D_{R}^{bs*}\bar{s}_{R}\gamma_{\mu}b_{R}+{\rm
h.c.}\right\\},$ (5)
Here $g_{1}$ is the ordinary hypercharge gauge coupling constant,
$D_{L},D_{R}$ are matrices which rotate the weak eigen-basis to the mass
eigen-basis for the down-type left and right hand quarks. The FD couplings of
$Z^{\prime}$ to fermions can be written as Hill ; tc201 ; tc203 :
$\displaystyle{\cal L}^{FD}_{Z^{\prime}}$ $\displaystyle=$
$\displaystyle-\sqrt{4\pi
K_{1}}\left\\{Z^{\prime}_{\mu}\left[\frac{1}{2}\bar{\tau}_{L}\gamma^{\mu}\tau_{L}-\bar{\tau}_{R}\gamma^{\mu}\tau_{R}+\frac{1}{6}\bar{t}_{L}\gamma^{\mu}t_{L}+\frac{1}{6}\bar{b}_{L}\gamma^{\mu}b_{L}+\frac{2}{3}\bar{t}_{R}\gamma^{\mu}t_{R}\right.\right.$
(6) $\displaystyle-$
$\displaystyle\left.\left.\frac{1}{3}\bar{b}_{R}\gamma^{\mu}b_{R}\right]-\tan^{2}\theta^{\prime}Z^{\prime}_{\mu}\left[\frac{1}{6}\bar{s}_{L}\gamma^{\mu}s_{L}-\frac{1}{3}\bar{s}_{R}\gamma^{\mu}s_{R}-\frac{1}{2}\bar{\mu}_{L}\gamma^{\mu}\mu_{L}-\bar{\mu}_{R}\gamma^{\mu}\mu_{R}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.\frac{1}{2}\bar{e}_{L}\gamma^{\mu}e_{L}-\bar{e}_{R}\gamma^{\mu}e_{R}\right]\right\\}.$
Here $\theta^{\prime}$ is the mixing angle, and $K_{1}$ is the coupling
constant with $\tan\theta^{\prime}=\frac{g_{1}}{\sqrt{4\pi K_{1}}}$.
## III Effective Hamiltonian and form factors
In the TC2 model, after neglecting the doubly Cabibbo-suppressed
contributions, the effective hamiltonian for the transition $b\rightarrow
sl^{+}l^{-}$ has the following structure dai ; wenjunli :
${\cal
H}=-\frac{4G_{F}}{\sqrt{2}}V^{*}_{ts}V_{tb}\sum\limits_{i=1}^{10}[C_{i}(\mu){\cal
O}_{i}(\mu)+C_{Q_{i}}(\mu)Q_{i}(\mu)]$ (7)
where $V^{*}_{ts}V_{tb}$ is the CKM factor, and $G_{F}$ is the Fermi coupling
constant. $C_{i}$ and $C_{Q_{i}}$ are the Wilson coefficients at the
renormalization point $\mu=m_{W}$, ${\cal O}_{i}$’s ($i=1,\cdots,10$) are the
operators in the SM and the explicit expressions can be found in Ref. buras52
, and $Q_{i}$’s come from the diagrams exchanging the neutral particles in TC2
and are dai ; wenjunli
$\displaystyle Q_{1}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}b^{\alpha}_{R})(\bar{l}l)\,,$
$\displaystyle Q_{2}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}b^{\alpha}_{R})(\bar{l}\gamma_{5}l)\,,$
$\displaystyle Q_{3}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}b^{\alpha}_{R})\left(\sum_{q}\bar{q}^{\beta}_{L}q^{\beta}_{R}\right)\,,$
$\displaystyle Q_{4}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}b^{\alpha}_{R})\left(\sum_{q}\bar{q}^{\beta}_{R}q^{\beta}_{L}\right)\,,$
$\displaystyle Q_{5}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}b^{\beta}_{R})\left(\sum_{q}\bar{q}^{\beta}_{L}q^{\alpha}_{R}\right)\,,$
$\displaystyle Q_{6}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}b^{\beta}_{R})\left(\sum_{q}\bar{q}^{\beta}_{R}q^{\alpha}_{L}\right)\,,$
$\displaystyle Q_{7}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}\sigma^{\mu\nu}b^{\alpha}_{R})\left(\sum_{q}\bar{q}^{\beta}_{L}\sigma_{\mu\nu}q^{\beta}_{R}\right)\,,$
$\displaystyle Q_{8}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}\sigma^{\mu\nu}b^{\alpha}_{R})\left(\sum_{q}\bar{q}^{\beta}_{R}\sigma_{\mu\nu}q^{\beta}_{L}\right)\,,$
$\displaystyle Q_{9}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}\sigma^{\mu\nu}b^{\beta}_{R})\left(\sum_{q}\bar{q}^{\beta}_{L}\sigma_{\mu\nu}q^{\alpha}_{R}\right)\,,$
$\displaystyle Q_{10}$
$\displaystyle=\frac{g_{s}^{2}}{16\pi^{2}}(\bar{s}^{\alpha}_{L}\sigma^{\mu\nu}b^{\beta}_{R})\left(\sum_{q}\bar{q}^{\beta}_{R}\sigma_{\mu\nu}q^{\alpha}_{L}\right)\,.$
(8)
where $\alpha$ and $\beta$ denote color indices. The subscripts $L$ and $R$
refer to left- and right- handed components of the fermion fields. $e$ and
$g_{s}$ are the electromagnetic and strong coupling constants respectively.
In terms of the above effective Hamiltonian (7), the decay amplitude of
$b\rightarrow sl^{+}l^{-}$ can be written as wenjunli :
$\displaystyle{\cal M}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha_{em}}{2\sqrt{2}\pi}V_{tb}V_{ts}^{*}\Bigg{\\{}-2\widetilde{C}_{7}^{eff}\hat{m}_{b}\bar{s}i\sigma_{\mu\nu}\frac{\hat{q}^{\nu}}{\hat{s}}(1+\gamma_{5})b\bar{l}\gamma^{\mu}l+\widetilde{C}_{9}^{eff}\bar{s}\gamma_{\mu}(1-\gamma_{5})b\bar{l}\gamma^{\mu}l$
(9)
$\displaystyle+\widetilde{C}_{10}^{eff}\bar{s}\gamma_{\mu}(1-\gamma_{5})b\bar{l}\gamma^{\mu}\gamma_{5}l+C_{Q_{1}}\bar{s}(1+\gamma_{5})b\bar{l}l+C_{Q_{2}}\bar{s}(1+\gamma_{5})b\bar{l}\gamma_{5}l\Bigg{\\}}.$
In the SM, the effective Wilson coefficients which enter the decay
distributions are written as buras52
$C_{9}^{\rm eff}(\hat{s})=C_{9}+Y(\hat{s})\,,$ (10)
in which $Y(\hat{s})$ stands for the matrix element of four-quark operators
and given by
$\displaystyle Y(\hat{s})$ $\displaystyle=$ $\displaystyle
h(z,\hat{s})\big{(}3C_{1}+C_{2}+3C_{3}+C_{4}+3C_{5}+C_{6}\big{)}-\frac{1}{2}h(1,\hat{s})\big{(}4C_{3}+4C_{4}+3C_{5}+C_{6}\big{)}\,$
(11)
$\displaystyle-\frac{1}{2}h(0,\hat{s})\big{(}C_{3}+3C_{4}\big{)}+\frac{2}{9}\big{(}3C_{3}+C_{4}+3C_{5}+C_{6}\big{)}\,.$
Here the long-distance contributions from the resonant states have been
neglected because they could be excluded by experimental analysis CDFPhill1 .
The detailed discussion of the resonance effects can be found in Ref. resEff .
In the TC2 model, After the breaking of the extended gauge group to their
diagonal subgroups, the non-universal massive gauge boson $Z^{\prime}$ is
produced. It generally couples to the third-generation fermions and have large
tree-level flavor changing couplings. The non-universal gauge boson
$Z^{\prime}$ can give a correction to the function $C_{0}(x)$ of the SM smf .
For $l=e,\mu$, the $C_{01}^{TC2}(x)$ is chongxingyue99
$\displaystyle C_{01}^{TC2}(y_{t})$ $\displaystyle=$
$\displaystyle\frac{-tan^{2}\theta^{\prime}M_{Z}^{2}}{M_{Z^{\prime}}^{2}}\left[K_{ab}(y_{t})+K_{c}(y_{t})+K_{d}(y_{t})\right],$
(12)
with $y_{t}=m_{t}^{*2}/M_{W}^{2}$. For the decay process
$B_{s}\to\phi\tau^{+}\tau^{-}$, the factor $-tan^{2}\theta^{\prime}$ should be
replaced by 1. For the convenience of the reader, we present the functions
$K_{ab}(y_{t})$, $K_{c}(y_{t})$ and $K_{d}(y_{t})$ in the Appendix A.
The charged top-pions $\pi_{t}^{\pm}$ can give contributions to the
corresponding SM functions $C_{0}(x)$, $D_{0}(x)$, $E_{0}(x)$ and
$E^{\prime}_{0}(x)$. The explicit expressions of these functions are xiao2001
:
$\displaystyle C_{02}^{TC2}(z_{t})$ $\displaystyle=$
$\displaystyle\frac{m_{\pi}^{2}}{4\sqrt{2}G_{F}M_{W}^{2}F_{\pi}^{2}}\left[-\frac{z_{t}^{2}}{8(1-z_{t})}-\frac{z_{t}^{2}}{8(1-z_{t})^{2}}{\rm
log}[z_{t}]\right],$ (13) $\displaystyle D_{0}^{TC2}(z_{t})$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}G_{F}F_{\pi}^{2}}\left[\frac{47-79z_{t}+38z_{t}^{2}}{108(1-z_{t})^{3}}+\frac{3-6z_{t}^{2}+4z_{t}^{3}}{18(1-z_{t})^{4}}{\rm
log}[z_{t}]\right],$ (14) $\displaystyle E_{0}^{TC2}(z_{t})$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}G_{F}F_{\pi}^{2}}\left[\frac{7-29z_{t}+16z_{t}^{2}}{36(1-z_{t})^{3}}-\frac{3z_{t}^{2}-2z_{t}^{3}}{6(1-z_{t})^{4}}{\rm
log}[z_{t}]\right],$ (15) $\displaystyle E_{0}^{{}^{\prime}TC2}(z_{t})$
$\displaystyle=$
$\displaystyle\frac{1}{8\sqrt{2}G_{F}F_{\pi}^{2}}\left[-\frac{5-19z_{t}+20z_{t}^{2}}{6(1-z_{t})^{3}}+\frac{z_{t}^{2}-2z_{t}^{3}}{(1-z_{t})^{4}}{\rm
log}[z_{t}]\right].$ (16)
Here $z_{t}={m^{*}_{t}}^{2}/m_{\pi_{t}^{\pm}}^{2}$.
The neutral top-pion $\pi_{t}^{0}$ and top-Higgs $h_{t}^{0}$ can also give
contributions to the rare decays $B_{s}\to\phi l^{+}l^{-}$ through the new
operators given in Eq. (8) chongxingyue99 . The corresponding Wilson
coefficients are written as:
$C_{Q_{1}}=\frac{\sqrt{\nu_{w}^{2}-F_{\pi}^{2}}}{\nu_{w}}\left[\frac{m_{b}^{*}m_{l}\nu}{2\sqrt{2}sin^{2}\theta_{w}F_{\pi}m_{\pi_{t}^{0}}^{2}}C_{0}(x_{t})+\frac{V_{ts}m_{l}m_{t}^{*}m_{b}^{*2}M_{W}^{2}}{4\sqrt{2}\nu
g_{2}^{4}F_{\pi}^{3}m_{\pi_{t}^{0}}^{2}}C(x_{s})\right].$ (17)
Here $x_{s}={m^{*}_{t}}^{2}/m_{\pi_{t}^{0}}^{2}$, $g_{2}$ is the $SU(2)$
coupling constant, and $C_{0}(x_{t})$ is the Inami-Lim function in the SM smf
. The expression of $C_{Q_{2}}$ is same as that of $C_{Q_{1}}$ except for the
masses of the scalar particles.
Exclusive decays are described in terms of matrix elements of the quark
operators in Eq. (9) over meson states, which are described by several
independent form factors. For $B_{s}\to\phi l^{+}l^{-}$, the related
transition matrix elements are defined as $(q=p-k)$ Ball:2004rg
$\displaystyle\langle\phi(k)|(V-A)_{\mu}|B(p)\rangle$ $\displaystyle=$
$\displaystyle-i\epsilon^{*}_{\mu}(m_{B_{s}}+m_{\phi})A_{1}(s)+i(p+k)_{\mu}(\epsilon^{*}p)\,\frac{A_{2}(s)}{m_{B_{s}}+m_{\phi}}$
$\displaystyle+iq_{\mu}(\epsilon^{*}p)\,\frac{2m_{\phi}}{s}\,\left(A_{3}(s)-A_{0}(s)\right)+\epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}p^{\rho}k^{\sigma}\,\frac{2V(s)}{m_{B_{s}}+m_{\phi}}\,.$
(18)
with
$A_{3}(s)=\frac{m_{B_{s}}+m_{\phi}}{2m_{\phi}}\,A_{1}(s)-\frac{m_{B_{s}}-m_{\phi}}{2m_{\phi}}\,A_{2}(s)$
and $A_{0}(0)=A_{3}(0)$,
$\displaystyle\langle{\phi(k)}|\bar{s}\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b|B(p)\rangle$
$\displaystyle=$ $\displaystyle
i\epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}p^{\rho}k^{\sigma}\,2T_{1}(s)$ (19)
$\displaystyle{}+T_{2}(s)\left\\{\epsilon^{*}_{\mu}(m_{B_{s}}^{2}-m_{\phi}^{2})-(\epsilon^{*}k)\,(p+k)_{\mu}\right\\}$
$\displaystyle{}+T_{3}(s)(\epsilon^{*}p)\left\\{q_{\mu}-\frac{s}{m_{B_{s}}^{2}-m_{\phi}^{2}}\,(p+k)_{\mu}\right\\}.$
with $T_{1}(0)=T_{2}(0)$. $\epsilon_{\mu}$ is the polarization vector of the
$\phi$ meson. The physical range in $s=q^{2}$ extends from $s_{\rm min}=0$ to
$s_{\rm max}=(m_{B_{s}}-m_{\phi})^{2}$.
Table 1: Form factors for $B_{s}\to\phi$ transition within the light-cone QCD sum rule. | $F(0)$ | $r_{1}$ | $m_{R}^{2}$ | $r_{2}$ | $m^{2}_{\rm fit}$
---|---|---|---|---|---
$V^{B_{s}\to\phi}$ | $0.434$ | $1.484$ | $5.32^{2}$ | $-1.049$ | $39.52$
$A_{0}^{B_{s}\to\phi}$ | $0.474$ | $3.310$ | $5.28^{2}$ | $-2.835$ | $31.57$
$A_{1}^{B_{s}\to\phi}$ | $0.311$ | — | — | $0.308$ | $36.54$
$A_{2}^{B_{s}\to\phi}$ | $0.234$ | $-0.054$ | — | $0.288$ | $48.94$
$T_{1}^{B_{s}\to\phi}$ | $0.349$ | $1.303$ | $5.32^{2}$ | $-0.954$ | $38.28$
$T_{2}^{B_{s}\to\phi}$ | $0.349$ | — | — | $0.349$ | $37.21$
$\tilde{T}_{3}^{B_{s}\to\phi}$ | $0.349$ | $0.027$ | — | $0.321$ | $45.56$
Form factors for $B_{s}\to\phi$ transition have been updated recently in the
light-cone QCD sum rule approach Ball:2004rg . For the $q^{2}$ dependence of
the form factors, they have been parameterized by a simple formulae with two
or three parameters. The form factors $V$, $A_{0}$ and $T_{1}$ are
parameterized by
$\displaystyle F(s)=\frac{r_{1}}{1-s/m^{2}_{R}}+\frac{r_{2}}{1-s/m^{2}_{\rm
fit}},$ (20)
For the form factors $A_{2}$ and $\tilde{T}_{3}$, it can be expanded to the
second order around the pole, giving
$\displaystyle F(s)=\frac{r_{1}}{1-s/m^{2}}+\frac{r_{2}}{(1-s/m)^{2}}\,,$ (21)
where $m=m_{\rm fit}$ for $A_{2}$ and $\tilde{T}_{3}$. The fit formula for
$A_{1}$ and $T_{2}$ is
$\displaystyle F(s)=\frac{r_{2}}{1-s/m^{2}_{\rm fit}}.$ (22)
The form factor $T_{3}$ can be obtained by
$T_{3}(s)=\frac{m_{B_{s}}^{2}-m_{\phi}^{2}}{s}\big{[}\tilde{T}_{3}(s)-T_{2}(s)\big{]}$.
All of the form factors are collected Table 1.
## IV Basic Formula for Observables
In this section, we give formula for experimental observables including
dilepton invariant mass spectrum, forward-backward asymmetry (FBA), and lepton
polarization.
From Eqs. (9-19), the decay matrix element of $B_{s}\to\phi l^{+}l^{-}$ can be
written in the form
${\cal
M}=-\frac{G_{F}\alpha_{em}}{2\sqrt{2}\pi}V_{tb}V^{*}_{ts}m_{B_{s}}\left[{\cal
T}^{1}_{\mu}(\overline{l}\gamma^{\mu}l)+{\cal
T}^{2}_{\mu}(\overline{l}\gamma^{\mu}\gamma_{5}l)+{\cal
S}(\overline{l}l)\right\\}$ (23)
with
$\displaystyle{\cal T}^{1}_{\mu}$ $\displaystyle=$ $\displaystyle
A(\hat{s})\epsilon_{\mu\rho\alpha\beta}\epsilon^{*\rho}\hat{p}^{\alpha}\hat{k}^{\beta}-iB(\hat{s})\epsilon^{*}_{\mu}+iC(\hat{s})(\epsilon^{*}\cdot\hat{p})(\hat{p}+\hat{k})_{\mu},$
(24) $\displaystyle{\cal T}^{2}_{\mu}$ $\displaystyle=$ $\displaystyle
E(\hat{s})\epsilon_{\mu\rho\alpha\beta}\epsilon^{*\rho}\hat{p}^{\alpha}\hat{k}^{\beta}-iF(\hat{s})\epsilon^{*}_{\mu}+iG(\hat{s})(\epsilon^{*}\cdot\hat{p})(\hat{p}+\hat{k})_{\mu}+iH(\hat{s})(\epsilon^{*}\cdot\hat{p})\hat{q}_{\mu},$
(25) $\displaystyle{\cal S}$ $\displaystyle=$ $\displaystyle
i2\hat{m}_{\phi}(\epsilon^{*}\cdot\hat{p}){\cal S}_{2}(\hat{s})$ (26)
where $\hat{m}=\frac{m}{m_{B_{s}}}$, $\hat{p}=\frac{p}{m_{B_{s}}}$, and the
auxiliary functions are then given by:
$\displaystyle A(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{2}{1+\hat{m}_{\phi}}\widetilde{C}_{9}^{eff}(\hat{s})V(\hat{s})+\frac{4\hat{m}_{b}}{\hat{s}}\widetilde{C}_{7}^{eff}T_{1}(\hat{s}),$
(27) $\displaystyle B(\hat{s})$ $\displaystyle=$
$\displaystyle(1+\hat{m}_{\phi})\widetilde{C}_{9}^{eff}(\hat{s})A_{1}(\hat{s})+\frac{2\hat{m}_{b}}{\hat{s}}(1-\hat{m}^{2}_{\phi})\widetilde{C}_{7}^{eff}T_{2}(\hat{s}),$
(28) $\displaystyle C(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{1+\hat{m}_{\phi}}\widetilde{C}_{9}^{eff}(\hat{s})A_{2}(\hat{s})+\frac{2\hat{m}_{b}}{1-\hat{m}^{2}_{\phi}}\widetilde{C}_{7}^{eff}\left(T_{3}(\hat{s})+\frac{1-\hat{m}^{2}_{\phi}}{\hat{s}}T_{2}(\hat{s})\right),$
(29) $\displaystyle E(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{2}{1+\hat{m}_{\phi}}\widetilde{C}_{10}^{eff}V(\hat{s}),$
(30) $\displaystyle F(\hat{s})$ $\displaystyle=$
$\displaystyle(1+\hat{m}_{\phi})\widetilde{C}_{10}^{eff}A_{1}(\hat{s}),$ (31)
$\displaystyle G(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{1+\hat{m}_{\phi}}\widetilde{C}_{10}^{eff}A_{2}(\hat{s}),$
(32) $\displaystyle H(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{2\hat{m}_{\phi}}{\hat{s}}\widetilde{C}_{10}^{eff}\left(A_{3}(\hat{s})-A_{0}(\hat{s})\right)+\frac{\hat{m}_{\phi}}{\hat{m}_{l}(\hat{m}_{b}+\hat{m}_{s})}C_{Q_{2}}A_{0}(\hat{s}),$
(33) $\displaystyle{\cal S}_{2}(\hat{s})$ $\displaystyle=$
$\displaystyle-\frac{1}{(\hat{m}_{b}+\hat{m}_{s})}A_{0}(\hat{s})C_{Q_{1}}.$
(34)
The contributions of $Z^{\prime}$ and charged top-pions are translated through
the RGE step into modifications of the effective Wilson coefficients
$\widetilde{C}_{7}^{eff}$, $\widetilde{C}_{9}^{eff}$ and
$\widetilde{C}_{10}^{eff}$, while the contributions of neutral top-pion and
top-Higgs are incorporated in the terms of $H(\hat{s})$ and ${\cal
S}_{2}(\hat{s})$.
The two kinematic variables $\hat{s}$ and $\hat{u}$ are chosen to be
$\displaystyle\hat{s}$ $\displaystyle=$
$\displaystyle\hat{q}^{2}=(\hat{p}_{+}+\hat{p}_{-})^{2},$ (35)
$\displaystyle\hat{u}$ $\displaystyle=$
$\displaystyle(\hat{p}-\hat{p}_{-})^{2}-(\hat{p}-\hat{p}_{+})^{2},$ (36)
which are bounded as
$\displaystyle(2\hat{m}_{l})^{2}\leq$ $\displaystyle\hat{s}$
$\displaystyle\leq(1-\hat{m}_{\phi})^{2},$ (37)
$\displaystyle-\hat{u}(\hat{s})\leq$ $\displaystyle\hat{u}$
$\displaystyle\leq\hat{u}(\hat{s}),$ (38)
with $\hat{m}_{l}=m_{l}/m_{B}$. Here the variable $\hat{u}$ is related to the
angle $\theta$ between the momentum of the $B$-meson and that of $l^{+}$ in
the center of mass frame of the dileptons $l^{+}l^{-}$ through the relation
$\hat{u}=-\hat{u}(\hat{s})\cos\theta$. $\hat{u}(\hat{s})$ can be written as
follows
$\hat{u}(\hat{s})=\sqrt{\lambda\big{(}1-4\frac{\hat{m}^{2}_{l}}{\hat{s}}\big{)}},$
(39)
with
$\displaystyle\lambda$ $\displaystyle\equiv$
$\displaystyle\lambda(1,\hat{m}^{2}_{\phi},\hat{s})$ (40) $\displaystyle=$
$\displaystyle
1+\hat{m}^{4}_{\phi}+\hat{s}^{2}-2\hat{s}-2\hat{m}^{2}_{\phi}(1+\hat{s}).$
Keeping the lepton mass and integrating over $\hat{u}$ in the kinematic region
given in Eq. (38), we can obtain the differential decay rates for the decays
$B_{s}\to\phi l^{+}l^{-}$:
$\displaystyle\frac{dBr}{d\hat{s}}$ $\displaystyle=$
$\displaystyle\tau_{B_{s}}\frac{G^{2}_{F}\alpha_{em}^{2}m^{5}_{B_{s}}}{2^{10}\pi^{5}}|V_{tb}V^{*}_{ts}|^{2}\hat{u}(\hat{s})D^{\phi},$
(41) $\displaystyle D^{\phi}$ $\displaystyle=$
$\displaystyle\frac{|A|^{2}}{3}\hat{s}\lambda(1+2\frac{\hat{m}^{2}_{l}}{\hat{s}})+\frac{|E|^{2}}{3}\hat{s}\hat{u}(\hat{s})^{2}+|{\cal
S}_{2}|^{2}(\hat{s}-4\hat{m}^{2}_{l})\lambda$ (42)
$\displaystyle+\frac{1}{4\hat{m}^{2}_{\phi}}\left[|B|^{2}(\lambda-\frac{\hat{u}(\hat{s})^{2}}{3}+8\hat{m}^{2}_{\phi}(\hat{s}+2\hat{m}^{2}_{l}))+|F|^{2}(\lambda-\frac{\hat{u}(\hat{s})^{2}}{3}+8\hat{m}^{2}_{\phi}(\hat{s}-4\hat{m}^{2}_{l}))\right]$
$\displaystyle+\frac{\lambda}{4\hat{m}^{2}_{\phi}}\left[|C|^{2}(\lambda-\frac{\hat{u}(\hat{s})^{2}}{3})+|G|^{2}\left(\lambda-\frac{\hat{u}(\hat{s})^{2}}{3}+4\hat{m}^{2}_{l}(2+2\hat{m}^{2}_{\phi}-\hat{s})\right)\right]$
$\displaystyle-\frac{1}{2\hat{m}^{2}_{\phi}}\left[Re(BC^{*})(1-\hat{m}^{2}_{\phi}-\hat{s})(\lambda-\frac{\hat{u}(\hat{s})^{2}}{3})\right.$
$\displaystyle\left.+Re(FG^{*})\left((1-\hat{m}^{2}_{\phi}-\hat{s})(\lambda-\frac{\hat{u}(\hat{s})^{2}}{3})+4\hat{m}^{2}_{l}\lambda\right)\right]$
$\displaystyle-2\frac{\hat{m}^{2}_{l}}{\hat{m}^{2}_{\phi}}\lambda\left[Re(FH^{*})-Re(GH^{*})(1-\hat{m}^{2}_{\phi})\right]+|H|^{2}\frac{\hat{m}^{2}_{l}}{\hat{m}^{2}_{\phi}}\hat{s}\lambda$
The normalized forward-backward asymmetries (FBA) is defined as
$\displaystyle{\cal A}_{FB}(\hat{s})=\int
d\hat{s}~{}\frac{\int^{+1}_{-1}dcos\theta\frac{d^{2}Br}{d\hat{s}dcos\theta}{\rm
Sign}(cos\theta)}{\int^{+1}_{-1}dcos\theta\frac{d^{2}Br}{d\hat{s}dcos\theta}}.$
(43)
According to this definition, the explicit expressions of FBA for the
exclusive decays is:
$\displaystyle\frac{d{\cal A}_{FB}}{d\hat{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{D^{\phi}}\hat{u}(\hat{s})\Bigg{\\{}\hat{s}[Re(BE^{*})+Re(AF^{*})]$
(44) $\displaystyle+\frac{\hat{m}_{l}}{\hat{m}_{\phi}}[Re({\cal
S}_{2}B^{*})(1-\hat{s}-\hat{m}^{2}_{\phi})-Re({\cal
S}_{2}C^{*})\lambda]\Bigg{\\}}.$
Now we are ready to present the analytical expressions of lepton polarization
by defining:
$\frac{d\Gamma(\hat{n})}{d\hat{s}}=\frac{1}{2}\big{(}\frac{d\Gamma}{d\hat{s}}\big{)}_{0}[1+(P_{L}\hat{e}_{L}+P_{N}\hat{e}_{N}+P_{T}\hat{e}_{T})\cdot\hat{n}]$
(45)
where the subscript $"0"$ corresponds to the unpolarized decay case, $P_{L}$
and $P_{T}$ are the longitudinal and transverse polarization asymmetries in
the decay plane respectively, and $P_{N}$ is the normal polarization asymmetry
in the direction perpendicular to the decay plane.
The lepton polarization asymmetry $P_{i}$ can be derived by
$P_{i}(\hat{s})=\frac{d\Gamma(\hat{n}=\hat{e}_{i})/d\hat{s}-d\Gamma(\hat{n}=-\hat{e}_{i})/d\hat{s}}{d\Gamma(\hat{n}=\hat{e}_{i})/d\hat{s}+d\Gamma(\hat{n}=-\hat{e}_{i})/d\hat{s}}\;$
(46)
the results are
$\displaystyle P_{L}$ $\displaystyle=$ $\displaystyle\frac{1}{D^{\phi}}{\cal
D}\Bigg{\\{}\frac{2\hat{s}\lambda}{3}Re(AE^{*})+\frac{(\lambda+12\hat{s}\hat{m}^{2}_{\phi})}{3\hat{m}^{2}_{\phi}}Re(BF^{*})\Bigg{.}$
(47)
$\displaystyle\Bigg{.}-\frac{\lambda(1-\hat{m}^{2}_{\phi}-\hat{s})}{3\hat{m}^{2}_{\phi}}Re(BG^{*}+CF^{*})+\frac{\lambda^{2}}{3\hat{m}_{\phi}}Re(CG^{*})\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{2\hat{m}_{l}\lambda}{\hat{m}_{\phi}}[Re(F{\cal
S}^{*}_{2})-\hat{s}Re(H{\cal S}^{*}_{2})-(1-\hat{m}^{2}_{\phi})Re(G{\cal
S}^{*}_{2})]\Bigg{\\}},$ $\displaystyle P_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{D^{\phi}}\frac{-\pi\sqrt{\hat{s}}\hat{u}(\hat{s})}{4\hat{m}_{\phi}}\Bigg{\\{}\frac{\hat{m}_{l}}{\hat{m}_{\phi}}\left[Im(FG^{*})(1+3\hat{m}^{2}_{\phi}-\hat{s})\right.\Bigg{.}$
(48)
$\displaystyle\Bigg{.}\left.+Im(FH^{*})(1-\hat{m}^{2}_{\phi}-\hat{s})-Im(GH^{*})\lambda\right]\Bigg{.}$
$\displaystyle\Bigg{.}+2\hat{m}_{\phi}\hat{m}_{l}[Im(BE^{*})+Im(AF^{*})]\Bigg{.}$
$\displaystyle\Bigg{.}-(1-\hat{m}^{2}_{\phi}-\hat{s})Im(B{\cal
S}^{*}_{2})+\lambda Im(C{\cal S}_{2}^{*})\Bigg{\\}},$ $\displaystyle P_{T}$
$\displaystyle=$
$\displaystyle\frac{1}{D^{\phi}}\frac{\pi\sqrt{\lambda}\hat{m}_{l}}{4\sqrt{\hat{s}}}\Bigg{\\{}4\hat{s}Re(AB^{*})\Bigg{.}$
(49)
$\displaystyle\Bigg{.}+\frac{(1-\hat{m}^{2}_{\phi}-\hat{s})}{\hat{m}^{2}_{\phi}}\left[-Re(BF^{*})+(1-\hat{m}^{2}_{\phi})Re(BG^{*})+\hat{s}Re(BH^{*})\right]\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{\lambda}{\hat{m}^{2}_{\phi}}[Re(CF^{*})-(1-\hat{m}^{2}_{\phi})Re(CG^{*})-\hat{s}Re(CH^{*})]\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{(\hat{s}-4\hat{m}^{2}_{l})}{\hat{m}_{\phi}\hat{m}_{l}}[(1-\hat{m}^{2}_{\phi}-\hat{s})Re(F{\cal
S}^{*}_{2})-\lambda Re(G{\cal S}^{*}_{2})]\Bigg{\\}}.$
where ${\cal D}=\sqrt{1-4\frac{\hat{m}^{2}_{l}}{\hat{s}}}$, $D^{\phi}$ are
given in Eq. (42).
## V Numerical result
In the numerical calculations, we fix the SM parameters as follows UTfitCKM ;
PDG10 ; PMass .
$\displaystyle A=0.8095,\ \lambda=0.22545,\ \overline{\rho}=0.132\pm 0.02,\
\overline{\eta}=0.367\pm 0.013.$ $\displaystyle\ m_{c}=1.4\mbox{ GeV},\
m_{b}=4.8\mbox{ GeV},\ m_{t}=172.4\mbox{ GeV},\ $ $\displaystyle
m_{\mu}=0.1057\mbox{ GeV},\ m_{\tau}=1.7769\mbox{ GeV}\ m_{W}=80.4\mbox{
GeV},\ $ $\displaystyle m_{Z}=91.18\mbox{GeV},\ m_{B_{s}}=5.36\mbox{ GeV},\
m_{\phi}=1.02\mbox{ GeV},\ $ $\displaystyle\alpha_{em}=\frac{1}{137},\
\alpha_{s}(m_{Z})=0.118,\ sin^{2}\theta_{W}=0.23,\ \tau_{B_{s}}=1.46\times
10^{-12}\mbox{s}.$ (50)
In the TC2 model, the new physics contributions depend on new parameters which
have been constrained by theory arguments and by experimental results.
$\varepsilon$ denotes the portion of the top quark mass generated by the
extended technicolor. The experimental constraints on $\varepsilon$ from the
data of radiative decay $b\to s\gamma$ are weak bsgam . However, from the
theoretical point of view, $\varepsilon$ is favored in the range of
$[0.03,0.1]$ Hill .
On the theoretical side, Ref. Hill estimated that the mass of top-pions
should be a few hundred GeV using quark loop approximation, and Refs. Hill ;
tc204 evaluated the mass of top-Higgs to be about $2m_{t}$. On the
experimental side, the neutral top-pion and the top-Higgs are weakly
restricted. Meanwhile, the mass of the charged top-pion have been strongly
constrained. For example, the absence of $t\to\pi_{t}^{+}b$ indicates that
$m_{\pi_{t}^{+}}>165{\rm GeV}$ Balaji , and the analysis of $R_{b}$ reveals
that $m_{\pi_{t}^{+}}>220{\rm GeV}$ Burdman ; Hill95 .
As for the bounds on the mass of $Z^{\prime}$, precision electroweak data show
that $m_{Z^{\prime}}$ must be larger than $1{\rm TeV}$ Chivukula . The vacuum
tilting, the confinement from Z-pole physics, and U(1) triviality need
$K_{1}\leq 1$ Popovic . When considering experimentally much better measured
modes such as $B\to\mu^{+}\mu^{-}$ and $B\to Kl^{+}l^{-}$, we can easily
obtain the constraints on the free parameters $m_{Z^{\prime}}$ and $K_{1}$.
For example, for $K_{1}=0.4$, we must have $1290{\rm GeV}<m_{Z^{\prime}}<{\rm
1787GeV}$ chongxingyue99 .
The differential branching fraction of exclusive decay $B\to
K^{*}\mu^{+}\mu^{-}$ has been already measured by BaBar, Belle, CDF and LHCb.
The latest LHCb result which corresponds to an integrated luminosity of 1
${\rm fb}^{-1}$ in the low $q^{2}$ region is LHCb-ksmumu
$\left[\frac{dBr}{dq^{2}}(B\to K^{*}\mu^{+}\mu^{-})\right]_{[1,6]}=(0.42\pm
0.04\pm 0.04)\times 10^{-7}c^{4}/{\rm GeV^{2}}.$ (51)
With the above precise measurement, we give the plots of differential
branching fraction $dBr/dq^{2}(B\to K^{*}\mu^{+}\mu^{-})$ at low $q^{2}$ in
function of the mass $M_{Z^{\prime}}$ (left panel) and of the mass
$m_{\pi_{t}^{+}}$ (right panel) in Fig. 1. The solid lines denote the LHCb
central value, while the dotted lines show the $3\sigma$ bound including the
experimental errors with theoretical ones given in Table 5 of Ref. ksmumu-
theory (added in quadrature). The dashed and dash-dotted curve corresponds to
the TC2 prediction for $\varepsilon=0.04$ and $\varepsilon=0.08$,
respectively. It is easy to see that the whole parameter space of
$M_{Z^{\prime}}$ is excluded for $\varepsilon=0.04$, but allowed for
$\varepsilon=0.08$ by this differential branching fraction. The mass of top-
pion $\pi_{t}^{+}$ below 450 GeV for $\varepsilon=0.04$ and below 400 GeV for
$\varepsilon=0.08$ are also excluded by the LHCb data.
Figure 1: Plots of differential branching fractions $dBr/dq^{2}(B\to
K^{*}\mu^{+}\mu^{-})$ at low $q^{2}$ in function of the mass $M_{Z^{\prime}}$
(left panel) and of the mass $m_{\pi_{t}^{+}}$ (right panel).
After taking into account the new constraints from $B\to K^{*}\mu^{+}\mu^{-}$
decay, we will make numerical calculations by using the input parameters in
the following ranges:
$\displaystyle m_{\pi_{t}^{+}}$ $\displaystyle=$ $\displaystyle(350-600){\rm
GeV},\quad m_{\pi_{t}^{0}}=m_{h_{t}^{0}}=(200-500){\rm GeV},\quad
m_{Z^{\prime}}=(1200-1800){\rm GeV},$ $\displaystyle\varepsilon$
$\displaystyle=$ $\displaystyle(0.06-0.1),\quad K_{1}=(0.3-1),\quad
F_{\pi}=50{\rm GeV}.$ (52)
Using the above input parameters, we will calculate the physics observables as
defined in previous sections and study the sensitivity to the new physics
corrections appeared in the TC2 model. The invariant mass spectra and
branching ratios are almost the same for electron and muon modes because the
mass of electron and muon are small. Meanwhile, the electron polarization is
very difficult to measure, so we only consider
$B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ decays in this work.
Using Eq. (41) and the input parameters as given above, it is easy to
calculate the branching ratio
$Br(B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-})$. In the SM, the
numerical results are
$\displaystyle Br(B_{s}\to\phi\mu^{+}\mu^{-})$ $\displaystyle=$ $\displaystyle
1.54^{+0.28}_{-0.25}\times 10^{-6},$ $\displaystyle
Br(B_{s}\to\phi\tau^{+}\tau^{-})$ $\displaystyle=$ $\displaystyle
1.65^{+0.30}_{-0.28}\times 10^{-7}.$ (53)
where the error corresponds to the uncertainty of input parameters of form
factors.
In the TC2 model, both the new penguin and tree level diagrams contribute
through constructive interference with their SM counterparts and consequently
provide large enhancements with respect to the SM predictions. For the typical
values of $F_{\pi}=50GeV$, $\varepsilon=0.08$, $K_{1}=0.4$,
$m_{\pi^{+}_{t}}=450GeV$, $m_{\pi^{0}_{t}}=m_{h^{0}_{t}}=300GeV$ and
$M_{Z^{\prime}}=1500GeV$, one has
$\displaystyle Br(B_{s}\to\phi\mu^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1.55\times 10^{-6}&{\rm only}\ \
Z^{\prime}\ \ {\rm considered},\\\ 3.07\times 10^{-6}&{\rm only}\ \
\pi_{t}^{+}\ \ {\rm considered},\\\ 3.76\times 10^{-6}&{\rm both}\ \
Z^{\prime}\ {\rm and}\ \ \pi_{t}^{+}\ \ {\rm considered}.\end{array}\right.$
(57) $\displaystyle Br(B_{s}\to\phi\tau^{+}\tau^{-})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1.80\times 10^{-7}&{\rm only}\ \
Z^{\prime}\ \ {\rm considered},\\\ 2.92\times 10^{-7}&{\rm only}\ \
\pi_{t}^{+}\ \ {\rm considered},\\\ 3.14\times 10^{-7}&{\rm both}\ \
Z^{\prime}\ {\rm and}\ \ \pi_{t}^{+}\ \ {\rm considered}.\end{array}\right.$
(61)
Figure 2: Plots of branching ratios of
$Br(B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-})$ decays versus
$M_{Z^{\prime}}$ in the SM and TC2 model. The dashed, short-dashed lines and
solid curve correspond to the TC2 and SM results, respectively. The dotted
lines denote the CDF data with $1\sigma$ error:
$Br(B_{s}\to\phi\mu^{+}\mu^{-})=(1.47\pm 0.52)\times 10^{-6}$.
In Fig. 2, we show the branching ratios of decays
$B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ as a function of the mass
parameter $M_{Z^{\prime}}$ in the SM and TC2 model. The solid line refers to
the SM prediction, while the dashed, short-dashed curves correspond to
theoretical prediction with the inclusion of the new physics effects of the
TC2 model for $\varepsilon=0.08$ and $\varepsilon=0.1$, respectively. The
dotted lines denote the CDF data with $1\sigma$ error. From this figure, we
can see that the new physics enhancements can be significant in size. For
$B_{s}\to\phi\mu^{+}\mu^{-}$ decay mode, the values of branching ratio
basically remain unchanged within the range of $M_{Z^{\prime}}=1200\sim
1800GeV$. The theoretical predictions of $Br(B_{s}\to\phi\tau^{+}\tau^{-})$
have some sensitivity to the parameter $M_{Z^{\prime}}$ because the
nonuniversal gauge boson $Z^{\prime}$ has large couplings to the third
generation fermion with respect to the first two generations.
Figure 3: Plots of branching ratios of
$Br(B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-})$ decays versus
$m_{\pi_{t}^{+}}$ in the SM and TC2 model. The dashed, short-dashed lines and
solid curve represent the TC2 and SM results, respectively.
In Fig. 3, we show the branching ratios of decays
$B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ as a function of the mass
parameter $m_{\pi_{t}^{+}}$ in the SM and TC2 model. One can see from Fig. 3
that the new physics enhancements to the two studied decays are still large in
size when the parameter $m_{\pi_{t}^{+}}$ varies. The branching ratios
$Br(B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-})$ are not very sensitive
to the variations of the input parameter $\varepsilon$. For $\varepsilon=0.08$
and $\varepsilon=0.1$, the enhancement to the
$Br(B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-})$ can reach a factor of
$\sim 2$. The uncertainty of the data is still large. Further improvement of
the data will be very helpful to test or constrain the parameter
$m_{\pi_{t}^{+}}$ in the TC2 model from these decays.
Figure 4: Plots of forward-backward asymmetries of
$A_{FB}(B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-})$ decays versus
$m_{\pi_{t}^{+}}$ in the SM and TC2 model. The dashed, dash-dotted lines and
solid curve stand for the TC2 and SM results, respectively.
In Fig. 4, we show the forward-backward asymmetries of decays
$B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ as a function of the mass
parameter $m_{\pi_{t}^{+}}$ in the SM and TC2 model. The solid line denotes
the SM prediction, while the dashed, dash-dotted curves correspond to the
theoretical prediction of TC2 model for $\varepsilon=0.08$ and
$\varepsilon=0.1$, respectively. For $B_{s}\to\phi\mu^{+}\mu^{-}$ decay, the
theoretical prediction of the forward-backward asymmetry in the SM is:
$A_{FB}(B_{s}\to\phi\mu^{+}\mu^{-})=-0.149\pm 0.001$. In the TC2 model, when
the $\pi_{t}^{+}$ mass is in the range of $350GeV\sim 600GeV$, the value of
$A_{FB}(B_{s}\to\phi\mu^{+}\mu^{-})$ is in the range of $-0.168\sim-0.146$ for
$\varepsilon=0.08$. For $B_{s}\to\phi\tau^{+}\tau^{-}$ decay, the forward-
backward asymmetry amounts to $-0.038\sim-0.037$ for $\varepsilon=0.08$, which
is comparable with the SM result of $-0.035\pm 0.0001$.
Figure 5: Plots of $P_{L}$ and $P_{T}$ for decay $B_{s}\to\phi\mu^{+}\mu^{-}$
in the SM and TC2 model. The dashed, dash-dotted lines and solid curve display
the central values of the TC2 and SM predictions, respectively. The two dotted
lines show the uncertainties of form factors induced by F(0) in the SM.
In Figs. 5 and 6, we present the longitudinal and transverse polarization for
decays $B_{s}\to\phi\mu^{+}\mu^{-}$ and $B_{s}\to\phi\tau^{+}\tau^{-}$. The
solid line is the SM prediction, while the dashed, dash-dotted curves are the
theoretical prediction of TC2 model for $\varepsilon=0.08$ and
$\varepsilon=0.1$, respectively. From these figures, it easy to see that the
variations of the input parameter $\varepsilon$ can only provide a few percent
change of the lepton polarization for
$B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ decays. For the decay
$B_{s}\to\phi\mu^{+}\mu^{-}$, the $P_{L}$ is suppressed by about $110\%$ at
most at the small momentum transfer, while at $\hat{s}>0.07$, it will become
larger than that of the SM, and the new physics contribution in TC2 model
provide an enhancement of $\sim 12\%$. For the $P_{T}$ part, the new physics
contribution result in a $(8\sim 18)\%$ decrease. As for
$B_{s}\to\phi\tau^{+}\tau^{-}$, the deviation from the SM prediction appears
when $\hat{s}>0.5$ for $P_{L}$. $P_{T}$ is decreased with respect to the SM
prediction by about $10\%$ in all the di-lepton invariant mass range. Thus,
the measurement of $P_{L}$ for $B_{s}\to\phi\mu^{+}\mu^{-}$ and $P_{T}$ for
$B_{s}\to\phi\tau^{+}\tau^{-}$ will distinguish between the SM and the TC2
model.
Figure 6: Plots of $P_{L}$ and $P_{T}$ for decay
$B_{s}\to\phi\tau^{+}\tau^{-}$ in the SM and TC2 model. Other captions are
same as Fig. 5.
## VI Summary
In this paper, we carried out a study of the new physics contributions to the
branching ratios, forward-backward asymmetries and lepton polarization for the
decays $B_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ in the TC2 model by
using form factors calculated within the light-cone QCD sum rule approach.
In Section II, a brief review about the topcolor-assisted technicolor model
was given. In Section III, we presented the theoretical framework for
$B_{s}\to\phi l^{+}l^{-}$ decays within the TC2 model, then give the
definitions and the derivations of the form factors in the decays
$B_{s}\to\phi l^{+}l^{-}$ using the updated form factors within the light-cone
QCD sum rule. In Section IV, we introduced the basic formula for experimental
observables. In Section V, we calculated the branching ratio, forward-backward
asymmetry, and lepton polarization of $B_{s}\to\phi l^{+}l^{-}$ and made
phenomenological analysis for these decays in the SM and the TC2 model. From
the numerical results, we found the following features about the new physics
effects:
* •
The branching ratios of
$\bar{B}_{s}\to\phi\mu^{+}\mu^{-},\phi\tau^{+}\tau^{-}$ decays are essentially
unaffected by the $Z^{\prime}$ contributions, while charged top-pions
interaction can lead to striking effects in these decay distributions. For
$\varepsilon=0.08$ and $\varepsilon=0.1$, the enhancement can reach a factor
of $\sim 2$.
* •
For the forward-backward asymmetry of the decay $B_{s}\to\phi\mu^{+}\mu^{-}$,
the NP enhancement is in the range $-13\%$ to $3\%$. For
$B_{s}\to\phi\tau^{+}\tau^{-}$ decay, the NP effects is about $-9\%$ to $-6\%$
compared to the SM predictions.
* •
For the lepton polarization, $P_{L}(B_{s}\to\phi\mu^{+}\mu^{-})$ is increased
by about $12\%$. However, $P_{T}(B_{s}\to\phi\mu^{+}\mu^{-})$ is decreased by
$(8\sim 18)\%$. As for $B_{s}\to\phi\tau^{+}\tau^{-}$, the deviation from the
SM prediction appears when $\hat{s}>0.5$ for $P_{L}$. In the $P_{T}$ part, the
SM prediction will be decreased by about $10\%$.
An improved measurement of $Br(\bar{B}_{s}\to\phi\mu^{+}\mu^{-})$ and first
measurements of the longitudinal polarization asymmetry, $P_{L}$, in
$B_{s}\to\phi\mu^{+}\mu^{-}$ and of the transverse polarization asymmetry,
$P_{T}$, in $B_{s}\to\phi\tau^{+}\tau^{-}$ at LHCb and super-flavor factories
(BellII and the proposed Super-B ) will allow to distinguish between the SM
and the TC2 model.
## Acknowledgments
The authors would like to thank Prof. Zhen-jun Xiao for helpful comments and
suggestions on the manuscript. The work is supported by the National Science
Foundation under contract No. 10947020, and Natural Science Foundation of
Henan Province under Grant No. 112300410188.
## Appendix A Relevant functions in the TC2 model
In this Appendix, we give the explicit expressions of functions that related
to the rare B decays studied here in the framework of the TC2 model.
$\displaystyle K_{ab}(x)$ $\displaystyle=$
$\displaystyle-\frac{2g^{2}c_{w}^{2}I_{1}(x)}{3g_{2}^{2}(v_{d}+a_{d})},$ (62)
$\displaystyle K_{c}(x)$ $\displaystyle=$
$\displaystyle\frac{2f^{2}c_{w}^{2}}{g_{2}^{2}}\left[\frac{2I_{2}(x)}{3(v_{u}+a_{u})}+\frac{I_{3}(x)}{6(v_{u}-a_{u})}\right],$
(63) $\displaystyle K_{d}(x)$ $\displaystyle=$
$\displaystyle\frac{2f^{2}c_{w}^{2}}{g_{2}^{2}}\left[\frac{2I_{4}(x)}{3(v_{u}+a_{u})}+\frac{I_{5}(x)}{6(v_{u}-a_{u})}\right],$
(64) $\displaystyle C(x)$ $\displaystyle=$
$\displaystyle\frac{I_{1}(x)}{-[0.5(Q-1)s_{w}^{2}+0.25]}.$ (65)
Here $g=\sqrt{4\pi K_{1}}$, $s_{w}=\sin\theta_{w}$, $a_{u,d}=I_{3}$,
$v_{u,d}=I_{3}-2Q_{u,d}s_{w}^{2}$, and $u,d$ stand for the up and down type
quarks, respectively.
$\displaystyle I_{1}(x)$ $\displaystyle=$
$\displaystyle-(0.5(Q-1)s_{w}^{2}+0.25)(x^{2}ln(x)/(x-1)^{2}-x/(x-1)-x(0.5(-0.5772$
(66)
$\displaystyle+ln(4\pi)-ln(M_{W}^{2}))+0.75-0.5(x^{2}ln(x)/(x-1)^{2}-1/(x-1)))),$
$\displaystyle I_{2}(x)$ $\displaystyle=$
$\displaystyle(0.5Qs_{w}^{2}-0.25)(x^{2}ln(x)/(x-1)^{2}-2xln(x)/(x-1)^{2}+x/(x-1)),$
(67) $\displaystyle I_{3}(x)$ $\displaystyle=$ $\displaystyle-
Qs_{w}^{2}(x/(x-1)-xln(x)/(x-1)^{2}),$ (68) $\displaystyle I_{4}(x)$
$\displaystyle=$ $\displaystyle
0.25(4s_{w}^{2}/3-1)(x^{2}ln(x)/(x-1)^{2}-x-x/(x-1)),$ (69) $\displaystyle
I_{5}(x)$ $\displaystyle=$
$\displaystyle-0.25Qs_{w}^{2}x(-0.5772+ln(4\pi)-ln(M_{W}^{2})+1-xln(x)/(x-1))$
(70) $\displaystyle-s_{w}^{2}/6(x^{2}ln(x)/(x-1)^{2}-x-x/(x-1)).$
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|
arxiv-papers
| 2012-04-16T04:31:35 |
2024-09-04T02:49:29.729419
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lin-Xia L\\\"u, Xing-Qiang Yang, and Zong-Chang Wang",
"submitter": "Linxia Lu",
"url": "https://arxiv.org/abs/1204.3361"
}
|
1204.3474
|
# Phase Diagrams, Distinct Conformal Anomalies and Thermodynamics of Spin-1
Bond-Alternating Heisenberg Antiferromagnetic Chain in Magnetic Fields
Xin Yan†, Wei Li†, Yang Zhao, Shi-Ju Ran, and Gang Su Email: gsu@gucas.ac.cn
Theoretical Condensed Matter Physics and Computational Materials Physics
Laboratory, College of Physical Sciences, Graduate University of Chinese
Academy of Sciences, P. O. Box 4588, Beijing 100049, China
###### Abstract
The ground state phase diagrams and thermodynamic properties of a spin-1 bond-
alternating Heisenberg antiferromagnetic chain with a single-ion anisotropy in
longitudinal and transverse magnetic fields are investigated jointly by means
of the infinite time evolving block decimation, the linearized tensor
renormalization group and the density matrix renormalization group methods. It
is found that in the magnetic field-bond alternating ratio plane six phases
such as singlet dimer, Haldane, two Tomonaga-Luttinger liquid, 1/2
magnetization plateau and spin polarized phases are identified in a
longitudinal field, while in a transverse field there are fives phases
including singlet dimer, Haldane, Z2 symmetry breaking, quasi-1/2
magnetization plateau and quasi spin polarized phases. A reentrant behavior of
the staggered magnetization in a transverse field is observed. The quantum
critical behaviors in longitudinal and transverse magnetic fields are
disclosed to fall into different universality classes with corresponding
conformal field central charge $c=1$ and $1/2$, respectively. The experimental
data of the compound NTENP under both longitudinal and transverse magnetic
fields are nicely fitted, and the Luttinger liquid behavior of low temperature
specific heat experimentally observed is also confirmed.
###### pacs:
75.10.Jm, 75.40.Mg, 05.30.-d, 02.70.-c
## I Introduction
Low dimensional quantum spin systems have been active subjects in quantum
many-body physics for many years. Owing to strong quantum fluctuations and
competitions between various interactions in these systems, a number of exotic
and fascinating quantum emergent phenomena are expected to occur, which thus
arouses persistently considerable interest not only in condense matter physics
but also in other fields such as quantum information and quantum computation
andrew .
Among others, the spin-1 bond alternating Heisenberg antiferromagnetic chain
(BAHAFC) in longitudinal and transverse magnetic fields is of particular
interest, for a series of materials, such as NENP S.Ma , NMOAP Narumi , NDOAP
Narumi2 , and NTENP escuer can be well described by the spin S=1 BAHAFC, and
the ideal model compound Ni(C9H24N4)(NO2)ClO4 (NTENP) has been extensively
studied both experimentally and theoretically in recent years [Narumi, ;
Narumi3, ; Narumi2, ; Zheludev, ; Hagiwara, ; Suzuki, ; regnault, ; Hagiwara2,
; glazkov, ], where the effects of magnetic field and the bond alternating
ratio on low-lying magnetic excitations, spin correlations and low-temperature
specific heat of the compound NTENP have been investigated. It is known that
in a magnetic field the one-dimensional (1D) Heisenberg quantum spin chains
with periodic ground states would exhibit a topological quantization of
magnetization according to the necessary condition $n(S-m)=integer$ Oshikawa ,
where $n$ is the period of the system, $S$ is the spin and $m$ is the
magnetization per site. Such magnetic plateau states have been addressed in
some polymerized Heisenberg antiferromagnetic or ferrimagnetic spin chains
(e.g. Refs. [su, ]). High-field magnetization measurements on NTENP also
revealed that a $m=1/2$ magnetization plateau appears around 700 kOe Narumi ;
Narumi2 ; the neutron scattering experiments indicated that the spin dynamics
of NTENP quite differs from that of a Haldane spin chain Zheludev ; Hagiwara ;
glazkov ; and the low-temperature specific heat measurement showed a Tomonaga-
Luttinger liquid (TLL) behavior in NTENP in a longitudinal magnetic field
Hagiwara2 . Although the experimental studies on NTENP gain obvious advances,
some ambiguities still remain on further understanding its physical
properties. For instance, the compound NTENP was experimentally studied under
both longitudinal and transverse magnetic fields, but only the data in a
longitudinal field have been fitted by quantum Monte Carlo simulations; no
direct numerical evidence of low-temperature specific heat of this material
was reported to verify the TLL behavior; the sharp peaks of the specific heat
in the NTENP were observed experimentally in both longitudinal and transverse
magnetic fields, where the peak positions move to high temperature side with
the increase of the transverse field, while retain almost intact with
increasing the longitudinal field; and so on. The remaining issues are very
worthy to address.
Apart from that the spin-1 BAHAFC with a single-ion anisotropy [Eq. (1) below]
is believed to be a pertinent model in describing the physical characters of
NTENP, which has been studied previously Zheludev ; Gomez ; Tsvelik ; Kitazawa
; Singh ; Kohno , this model itself has also fascinating properties that
deserve to investigate. When the bond alternating ratio $\alpha=0$, it reduces
to the singlet dimers; when $\alpha=1$, it becomes a spin-1 uniform chain
(Haldane chain). In these two special cases, the system shows an excitation
gap from the singlet dimer or Haldane ground state to the triplet excited
state. The latter uniform chain under both longitudinal ($h_{z}$) and
transverse ($h_{x}$) magnetic fields has been extensively discussed (e.g.
Refs. [Affleck, ; X.G.Wen, ; Xing, ]). However, the overall phase diagrams in
$\alpha-h_{z,x}$ plane for the S=1 BAHAFC with a single-ion anisotropy are
still absent. Therefore, it is really necessary to tackle these above
questions.
In this paper, by means of jointly the infinite time evolving block decimation
(iTEBD) G. Vidal , the linearized tensor renormalization group (LTRG) Li , and
the density matric renormalization group (DMRG) methods White , we shall study
systematically the ground state phase diagrams, magnetic and thermodynamic
properties of the S=1 BAHAFC with a single-ion anisotropy in longitudinal and
transverse magnetic fields. The iTEBD, LTRG and DMRG methods are the numerical
algorithms with very good accuracy and high efficiency that were recently
developed for low-dimensional quantum lattice systems, which allow for
calculating with nice precision the critical properties and the extremely low
temperature behaviors of quantum many-body lattice systems. These methods can
assist us to fit well the experimental data on the NTENP with the accurately
calculated results, thereby capable of reasonably determining the material
parameters and better understanding the fundamental features of the NTENP in
both longitudinal and transverse magnetic fields.
## II Model and Ground State Phase Diagrams
### II.1 Model Hamiltonian and Calculational Method
Let us start with the model Hamiltonian given by
$\displaystyle H$ $\displaystyle=$ $\displaystyle
J\sum_{i}^{L/2}(\mathbb{S}_{2i-1}\cdot\mathbb{S}_{2i}+\alpha\mathbb{S}_{2i}\cdot\mathbb{S}_{2i+1})$
(1) $\displaystyle+$
$\displaystyle\sum_{i}^{L}[\Delta(S_{i}^{z})^{2}-g_{\|}\mu_{B}h_{z}S_{i}^{z}-g_{\bot}\mu_{B}h_{x}S_{i}^{x}],$
where $J$ is the coupling constant, $L$ (even) is the number of lattice sites,
$\mathbb{S}_{i}$ is the spin operator at $i$th site, $\Delta$ is the single-
ion anisotropy, $g_{\|}$, $g_{\bot}$ are the Landé g-factors, $\mu_{B}$ is the
Bohr magneton, $h_{z}$ and $h_{x}$ are the longitudinal and transverse
magnetic fields, respectively.
In the following calculations, $g_{\|}\mu_{B}=g_{\bot}\mu_{B}=1$ is assumed
unless the fittings to experimental data are concerned, and $J$ is taken as
the energy scale. The ground state properties were studied by utilizing the
iTEBD G. Vidal imaginary time projection scheme along with DMRG method White
, and the thermodynamic properties and fittings to the experimental data of
NTENP were performed by invoking the LTRG approach Li . In iTEBD calculations
we take the smallest Trotter step $\tau=10^{-7}$.
### II.2 Magnetization and Phase Diagram in a Longitudinal Field
Figure 1: (Color online) (a) The magnetic curves of spin-1 bond alternating
Heisenberg antiferromagnetic chain (BAHAFC) with a single-ion anisotropy in
longitudinal magnetic fields $h_{z}$, where three magnetization plateaux
($m=0$, 1/2, and 1) are observed. The inset shows the singularities of the
susceptibility as a function of magnetic field, which signal the occurrence of
quantum phase transitions. (b) The ground state phase diagram in the plane of
the bond alternating ratio ($\alpha$) _vs._ longitudinal magnetic field
($h_{z}$), where the Haldane, singlet-dimer (SD), m=1/2 magnetic plateau,
Tomonaga-Luttinger liquid (TLL), and spin polarized phases are identified. The
single-ion anisotropy $\Delta=0.25$ is taken.
Figure 1(a) presents the field dependence of magnetization per site $m_{z}$
for the system defined by Eq. (1) in the longitudinal magnetic field for
$\alpha=0.45$ and $0.8$, where the bond dimension $D_{c}=120$ of a matrix
product state is presumed. When $\alpha=0.45$, it is clear to see that there
are three magnetization plateaux occurring at $m$=0, 1/2 and 1 in some regions
of $h_{z}$. The $m$=0 plateau suggests that a gap remains in the absence of
magnetic field, implying that the system is in the singlet-dimer phase, as it
should be in the same phase as $\alpha=0$. Based on the valence-bond solid
picture, the $m=1/2$ plateau corresponds to that one of the two bonds in each
coupled spin-1 dimer is broken su . The $m=1$ plateau is in the spin fully
polarized state. Near the critical fields, the magnetization per site $m_{z}$
depends on the magnetic field in a square root behavior, similar to the case
of spin-1 HAFC in a magnetic field. In the ranges between the plateaus, the
spin correlations should be in a power law decay, indicating a quasi long-
range order. The critical magnetic fields for closure of the gap can be
accurately determined by the field dependence of the susceptibility, where the
sharp peaks appear, as shown in the inset of Fig. 1(a). When $\alpha=0.8$, the
three magnetic plateaux still remain, but the width of $1/2$ plateau
decreases, indicating that the corresponding gap becomes narrower. For the
spin-1 BAHAFC, as the ground state period is $n=2$, the topological
quantization condition $n(S-m)=integer$ is satisfied by $m$=0, 1/2 and 1,
i.e., there should exist at most three plateaux in the longitudinal field for
$0\leq\alpha<1$, which is well confirmed in Fig. 1(a). At the special case of
$\alpha=1$, $n$ becomes 1, and there should be two plateaus at $m$=0 and 1.
For $0\leq\alpha\leq 1$, we have calculated the magnetization curves and the
corresponding susceptibilities in the ground state, and collected all critical
values of magnetic fields at which the susceptibility exhibits obvious
singularities, which allows us for readily obtaining a global ground state
phase diagram in $\alpha-h_{z}$ plane for the spin-1 BAHAFC with a single-ion
anisotropy $\Delta=0.25$, as shown in Fig. 1(b). There are six phases,
including the singlet-dimer (SD) phase, Haldane phase, two TLL phases
separated by the $m=1/2$ magnetic plateau phase, and the spin fully polarized
phase. At a critical bond alternating ratio $\alpha_{c}\approx 0.6$, there
exists a quantum phase transition from the gapped SD state into the gapped
Haldane phase. The $m$=1/2 plateau state persists for all values of $\alpha$
until it disappears at $\alpha=1$. At all six phase boundaries, the gap
closes. In these phases, the spin-spin correlation function $\langle
S_{i}^{z}S_{j}^{z}\rangle$ reveals different spatial behaviors, which is
short-range ordered except that in the TLL phase it decays in an algebraic
way.
### II.3 Magnetization and Phase Diagram in a Transverse Field
Figure 2: (Color online) (a) The magnetic curves of S=1 BAHAFC with a single-
ion anisotropy in transverse magnetic fields $h_{x}$. The up-left inset shows
the staggered magnetization along the $y$ axis, and the down-right inset
presents the corresponding susceptibility as a function of magnetic field. For
$\alpha=0.1$, three plateau-like steps are observed. (b) The ground state
phase diagram in the plane of the bond alternating ratio ($\alpha$) _vs._
transverse magnetic field ($h_{x}$), where the Haldane, singlet-dimer (SD), Z2
symmetry breaking Néel ordered, quasi 1/2 magnetic plateau, and quasi spin
polarized phases are identified. The single-ion anisotropy $\Delta=0.25$ is
taken.
In a transverse magnetic field $h_{x}$, the field dependence of magnetization
per site $m_{x}$ and $m_{y}$ and the corresponding susceptibility of the model
under interest are presented in Fig. 2(a) for $\alpha=0.1$ and 0.45. Three
steps around $m_{x}$=0, 1/2 and 1 in magnetic curves ($m_{x}$ _vs._ $h_{x}$)
are observed for $\alpha=0.1$. The corresponding susceptibility $\chi$ is
illustrated in the lower inset of Fig. 2(a), where one may notice that, due to
the non-conservation of total $S_{x}$, at these three steps $\chi$ does not
exactly vanish although it is negligibly small. Thus, we call these steps as
quasi magnetic plateaus. In the regions between the quasi plateaus, $m_{y}$
has large values as shown in the upper inset of Fig. 1(a). The reason is that
the existence of the single-ion anisotropy makes the spins tend to arrange in
$xy$ plane, and when the magnetic field is applied along the $x$ direction,
the staggered magnetization $m_{y}$ is induced to form a canted Ising order.
In addition, for $\alpha=0.1$ we observed from the upper inset of Fig. 2(a)
that with increasing $h_{x}$ the staggered magnetization $m_{y}$ first is
negligible small, and then exhibits a sharp valley structure, which
illustrates a reentrant behavior of the staggered magnetization Diep ; Hieida
. For $\alpha=0.45$, only two steps appear in the magnetic curve, and the
quasi $m=1/2$ step disappears. In this case, $m_{y}$ has even larger values in
the region between the two steps. For other larger $\alpha$, the magnetic
curves in the transverse field $h_{x}$ have the behaviors similar to that of
$\alpha=0.45$.
Utilizing the method similar to Fig. 1(b) and sweeping various values of
$\alpha$, we can collect all critical magnetic fields by finding the singular
positions of susceptibility, and then obtain the whole ground state phase
diagram in $\alpha-h_{x}$ plane for the system under investigation, as
depicted in Fig. 2(b). It can be seen that there are five phases, namely, the
SD, Haldane, quasi $m=1/2$ magnetic plateau, Z2 symmetry breaking (or spin
canted) and the quasi spin polarized phases. The quantum critical point
$\alpha_{c}\approx 0.6$ is recovered here. In contrast to the case under a
longitudinal field where the $m=1/2$ plateau phase persists into the whole
region of $\alpha<1$ and separates two TLL phases, the quasi $m=1/2$ plateau
phase sets only in a small region ($\alpha\leq 0.24$). Starting from the SD
and Haldane phases, when we increase the field $h_{x}$, the gap is gradually
closed at the phase boundaries. If one continues to increase the magnetic
field $h_{x}$, another gap opens because of the breaking of the discrete
symmetry, giving rise to a Z2 symmetry breaking phase, which displays a long
range behavior.
## III Entanglement Entropy and Conformal Anomalies
Figure 3: (Color online) The normalized Schmidt coefficient $\lambda_{\nu}$ as
a function of the number of kept states $\nu$ for different phases, where
$D_{c}=80$.
This present system demonstrates quite different behaviors in longitudinal and
transverse magnetic fields. To gain further insight into the underlying
physics behind this character, we have studied the conformal anomalies of this
system at the critical regimes. The conformal field theory (CFT) tells us that
the conformal invariance at the critical point sets useful constraints on the
critical behaviors of two-dimensional classical or 1D quantum systems
conformal , and the universality class can be characterized by the conformal
anomaly or central charge $c$ of the Virasoro algebra. For a system with a
continuous symmetry $G$, if G=SU(N), the possible conformal central charge is
given by $c=(N^{2}-1)k/(N+k)$, $k=1,2,3,...$ Affleck0 . For the spin-S uniform
antiferromagnetic quantum chains, G=SU(2), $c=3S/(1+S)$. For S=1/2, $c=1$;
S=1, $c=3/2$. For the present spin-1 BAHAFC with a single-ion anisotropy, the
SU(2) symmetry is not satisfied, and the conformal central charge should be
calculated via other ways. The von Neumann entropy $S$, defined by
$S=-Tr(\hat{\rho}_{sys}\log_{2}\hat{\rho}_{sys})=-Tr(\hat{\rho}_{env}\log_{2}\hat{\rho}_{env}),$
(2)
offers a possible way to get the central charge of quantum spin chains, where
$\hat{\rho}_{sys(env)}$ is the reduced density matrix (DM) of system
(environment). In critical and noncritical regimes, the entanglement entropy
has different asymptotic behaviors arealaw ; Vidal2 . In critical regimes, the
CFT predicts wilczek
$S\approx\frac{c}{3}\log_{2}(L)+k,$ (3)
where $L$ is the number of spins for a block embedding in an infinite chain,
$c$ is the central charge and $k$ is a non-universal constant. In noncritical
regimes, $S$ vanishes for all L or grows monotonically with L until
saturation. In the following, we shall invoke the iTEBD method to study the
entanglement entropy for the case in a transverse magnetic field. Because of
the nearly unitary evolution, the iTEBD algorithm gives the canonical form of
infinite matrix product states (iMPS). We can make a Schmidt decomposition on
the ground state wave function such that
$|\psi\rangle=\sum_{\nu=1}^{D_{c}}\lambda_{\nu}|\psi^{A}_{\nu}\rangle|\psi^{B}_{\nu}\rangle$
(4)
on each bond, where $|\psi_{\nu}^{A,B}\rangle$ is the orthonormalized basis
states (Schmidt bases) for two parts $A$ and $B$ of the infinite chain, and
$\lambda_{\nu}$ is the Schmidt coefficient (SC). Fig. 3 presents
$\lambda_{\nu}$ as a function of $\nu$ for different phases. It is seen that
in the TLL phase, the SC attenuates much slower than those in
Figure 4: (Color online) The entanglement entropy $S$ as a function of chain
length $L$ calculated by the iTEBD in a transverse field for $\alpha=0.45$ and
$D_{c}=100$, and by the DMRG in a longitudinal magnetic field for $\alpha=0.6$
where the optimal states were kept as 400. In both cases, the single-ion
anisotropy $\Delta=0.25$. The solid line is the fitting curve to Eq. (3),
giving the central charge $c=1/2$ at a critical transverse field
$h_{x}^{c}=0.349$; the inset shows that the entanglement entropy as a function
of $h_{x}$ for a semi-infinite chain length is singular at $h_{x}^{c}=0.349$.
The dashed line is the fitting curve for the case (TLL phase) at a
longitudinal magnetic field $h_{z}=0.3$, giving the central charge $c=1$.
other gapped phases. The double degeneracy of SC only appears in the Haldane
phase, indicating the existence of the topological string order
entang_spectrum . Given a canonical form of iMPS, if the system in Eq. (2) is
chosen as the semi-infinite chain, it is easy to directly calculate the von
Neumann entropy,
$S=-\sum_{\nu=1}^{D_{c}}\lambda_{\nu}^{2}\log_{2}(\lambda_{\nu}^{2})$. In
addition, we can also calculate the entanglement entropy when the system is
successive L spins embedding in an infinite chain, where we may set $L=2$ for
an example. In particular, we can write
$\displaystyle|\psi\rangle=\sum_{\alpha,\beta,\gamma=1}^{D_{c}}\sum_{i,j=1}^{d}\lambda_{\alpha}\Gamma^{i}_{\alpha,\beta}\lambda_{\beta}\Gamma^{j}_{\beta,\gamma}\lambda_{\gamma}|\Phi_{\alpha}\rangle|i\rangle|j\rangle|\Phi_{\gamma}\rangle,$
(5)
where $\Gamma^{i}_{\alpha,\beta}$ is the tensor on the $i$-th site,
$|i\rangle(|j\rangle)$ is the spin basis states on the $i(j)$-th site,
$|\Phi_{\alpha}\rangle$ and $|\Phi_{\gamma}\rangle$ are the left Schmidt basis
of the $i$-th site and the right Schmidt basis of the $j$-th site,
respectively. By contracting two tensors, we will obtain the reduced DM
$\displaystyle\hat{\rho}$ $\displaystyle=$ $\displaystyle
Tr_{ij}|\psi\rangle\langle\psi|$ (6) $\displaystyle=$
$\displaystyle\sum_{\alpha,\gamma=1}^{D_{c}}\sum_{\alpha^{\prime},\gamma^{\prime}=1}^{D_{c}}[\sum_{i,j=1}^{d}\sum_{\beta,\beta^{\prime}=1}^{D_{c}}\lambda_{\alpha}\Gamma^{i}_{\alpha,\beta}\lambda_{\beta}\Gamma^{j}_{\beta,\gamma}\lambda_{\gamma}$
$\displaystyle\lambda_{\alpha^{\prime}}(\Gamma^{i}_{\alpha^{\prime},\beta^{\prime}})^{*}\lambda_{\beta^{\prime}}(\Gamma^{j}_{\beta^{\prime},\gamma^{\prime}})^{*}\lambda_{\gamma^{\prime}}]|\alpha\rangle|\gamma\rangle\langle\alpha^{\prime}|\langle\gamma^{\prime}|.$
One can diagonalize this matrix and calculate the entanglement entropy. For a
larger $L$, we need to contract $L$ tensors and select Schmidt basis to
construct the reduced DM. For all $L$, the dimension of the reduced DM is
$D_{c}^{2}\times{D_{c}^{2}}$. By calculating the von Neumann entropy for a
semi-infinite chain in a transverse magnetic field, we observe that the cusp
position of the entanglement entropy just gives the phase transition point
$h_{x}^{c}=0.349(2)$. By fitting our calculated results to Eq. (3), we find
$c=1/2$ for $h_{x}^{c}=0.349$, as shown in Fig. 4. In the TLL phase of the
case in a longitudinal magnetic field, the iTEBD algorithm gives the ground
state with a staggered magnetization perpendicular to $z$ direction so that it
cannot yield a correct wave function. The reason is that there does not have
an excitation gap and the iTEBD algorithm cannot project the right ground
state wave function. In order to calculate the central charge in the TLL
phase, we utilize the DMRG algorithm with open boundary condition at
$h_{z}=0.3$, $\alpha=0.6$, and $\Delta=0.25$. The result is also included in
Fig. 4, where the fitting result gives the central charge $c=1$ in this TLL
phase note . Consequently, this implies that the critical behaviors of the
spin-1 BAHAFC with a single-ion anisotropy in transverse fields quite differ
from those in longitudinal fields, and the universality falls into two
distinct classes with conformal central charges $c=1/2$ and $c=1$,
respectively.
## IV Thermodynamic Properties and Comparison to Experiments
### IV.1 Susceptibility, Magnetization and Comparison to Experiments
Figure 5: (Color online) The temperature dependence of susceptibility of NTENP
measured experimentally is well fitted to the LTRG calculated data for both
longitudinal and transverse magnetic fields. The experimental data are taken
from Ref. [Narumi2, ].
The thermodynamic properties of this system were explored by using recently
proposed LTRG algorithm Li that allows for accurately calculating the
thermodynamic quantities at very low temperature. In the following LTRG
calculations we keep the Trotter step $\tau=0.1$.
Fig. 5 gives the temperature dependence of susceptibilities of the spin-1
BAHAFC in longitudinal and transverse magnetic fields, where the fittings to
the experimental data of NTENP under both fields are also included. One may
see that the theoretical results are nicely fitted to the experimental data,
generating a set of material parameters of NTENP: $\alpha=0.45$,
$\Delta/J=0.25$, $J=54.2K$, $g_{\|}=2.14$, and $g_{\bot}=2.27$. It should be
noted that in a longitudinal field, these parameters are in agreement with the
previous results from the quantum Monte Carlo calculations Narumi2 , while in
a transverse field the fitting to the experimental data of NTENP is for the
first time done. The magnetization curves of NTENP were measured at T=1.3 K up
to 700 kOe in Ref. [Narumi2, ], which are well fitted to our LTRG calculated
results, giving the same set of fitting parameters as those obtained from the
data of susceptibility except that $g_{\bot}=2.24$ here, as shown in Fig. 6.
Considering that the two sets of experimental data from susceptibility and
magnetic curves are independent, a slight difference on $g_{\bot}$ is
reasonable. Thus, $g_{\bot}$ of NTENP should be around 2.24 $\sim$ 2.27. The
high-field magnetization curve is also well fitted with our LTRG results
(inset of Fig. (6)) using the same parameters.
Figure 6: (Color online) The magnetic curves of NTENP are well fitted to the
LTRG calculated data for both longitudinal and transverse fields, where the
fitting parameters are consistent with those from the susceptibilities. Inset
is the high-field magnetic curve up to 700 kOe fitted with the corresponding
LTRG calculated data. The experimental data are taken from Ref. [Narumi2, ].
### IV.2 Specific Heat in Magnetic Fields
The temperature dependence of the specific heat $C(T)$ of the S=1 BAHAFC in
longitudinal and transverse magnetic fields is obtained by the LTRG method
down to very low temperatures, as shown in Fig. 7 for $h_{z}=0.5$ and
$h_{x}=0.5$. In a transverse field, there are one round peak and a low
temperature shoulder in $C(T)$, while in a longitudinal field the specific
heat exhibits only one broad peak. It appears that at low temperature the
specific heat displays distinct behaviors. To show this point clearly, we have
carefully calculated the specific heat of this model at extremely low
temperatures by the LTRG algorithm that is very powerful and efficient for
calculating the low-temperature properties than other numerical methods gubo .
Shown in the lower inset of Fig. 7, two different behaviors in both fields at
low temperature are clearly demonstrated, where in a longitudinal field, the
specific heat displays a linear T-dependent character, showing a TLL behavior,
while in a transverse field, $C(T)$ exhibits an exponential decay (the fitting
curve is given in the upper inset of Fig. 7), that can be ascribed to the
$Z_{2}$ symmetry breaking with an open of an Ising gap.
Figure 7: (Color online) The temperature dependence of the specific heat of
S=1 BAHAFC under both longitudinal and transverse magnetic fields. Although
the whole profiles look similar for both cases, the extremely low temperature
behaviors shown in the down-left inset quite differ, where the linear-T
dependence of the specific heat in a longitudinal field is clearly seen at low
temperature, suggesting a Tomonaga-Luttinger liquid behavior, while an
exponential decay for the T-dependence of the specific heat is observed in a
transverse field owing to the appearance of an Ising gap, as seen in the up-
right inset. The single-ion anisotropy $\Delta=0.25$ and the bond alternating
ratio $\alpha=0.45$ are taken.
The specific heat $C(T)$ of NTENP was also measured experimentally Hagiwara2 ,
where the linear T-dependence of $C(T)$ at low temperature in a longitudinal
field above the critical field $h{{}_{z}}^{c}=9.3$ T was observed, which was
identified as a TLL behavior. Our low temperature LTRG calculations at
$h_{z}=0.5$ ($>h{{}_{z}}^{c}$) on the S=1 BAHAFC model strongly supports this
experimental observation. In a transverse field, the experiment on NTENP gives
a distinct nonlinear low-temperature behavior of $C(T)$ from that in a
longitudinal field, which is also backed up by our LTRG results. We should
remark here that the sharp peaks in the temperature dependent specific heat of
NTENP were observed in both fields, which cannot be explained by using this
spin-1 BAHAFC model, as revealed by our present studies, because we do not
find any field-dependent sharp peaks of $C(T)$ in this 1D model. Those sharp
peaks may signal the field-induced long-range orders from the 3D effect of
inter-chain interactions in NTENP. It is this reason that makes us not
directly fit our LTRG results with the experimental data of specific heat of
NTENP. Nevertheless, our present studies on the spin-1 BAHAFC model may give a
possible clue to understand the experimentally observed low-temperature sharp
peaks of $C(T)$ Hagiwara2 . In a longitudinal field, as long as the magnetic
field is higher than the critical field ($h{{}_{z}}^{c}$), the system will
enter the TLL phase that has a quasi LRO, and the true LRO can be established
only by interchain couplings. Thus the peak position is almost determined by
the strength of interchain interactions and hardly moves with the increase of
the magnetic field. In a transverse field, in the range that the experiment
was performed, the staggered magnetization that could enhance the interchain
couplings increases monotonously with increasing the magnetic field, so the
low-temperature sharp peak moves to the high temperature side with the
increase of the transverse field.
## V Summary and Conclusion
In summary, we have investigated the spin-1 BAHAFC model with a single-ion
anisotropy in longitudinal and transverse magnetic fields by employing the
iTEBD and LTRG methods. The ground state phase diagrams in the plane of field
versus the bond alternating ratio under both fields are obtained, where
various phases are identified for two cases. From the entanglement entropy,
the conformal central charges in both critical longitudinal and transverse
fields are determined to be $c=1$ and $1/2$, respectively, suggesting that the
universality in critical regimes falls into different classes for both fields.
The TLL behavior at low $T$ observed experimentally in NTENP is verified via
our accurate calculations, while an exponential decay of low-temperature
specific heat is uncovered in a transverse field. The experimental data of the
model material NTENP are well fitted with our LTRG results, and the parameters
for characterizing NTENP are determined.
###### Acknowledgements.
We are indebted to Bo Gu, Fei Ye, Shou-Shu Gong, Bin Xi, J. Sirker, and Qing-
Rong Zheng for stimulating discussions. This work is supported in part by the
NSFC (Grants No. 10934008, No. 90922033), the MOST (Grant No. 2012CB932900)
and the Chinese Academy of Sciences.
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|
arxiv-papers
| 2012-04-16T13:17:36 |
2024-09-04T02:49:29.740559
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Yan, Wei Li, Yang Zhao, Shi-Ju Ran, Gang Su",
"submitter": "Xin Yan",
"url": "https://arxiv.org/abs/1204.3474"
}
|
1204.3538
|
# Muon Collider: Plans, Progress and Challenges
Ronald Lipton Fermi National Accelerator Laboratory, Batavia, Il, USA
###### Abstract
We in the physics community expect the LHC to uncover new physics in the next
few years. The character and energy scale of the new physics remain unclear,
but it is likely that data from the LHC will need to be complemented by
information from a lepton collider which can provide for precise examination
of new phenomena. We describe the concept, accelerator design, and detector
R&D for a high energy Muon Collider as well as the challenges associated with
the machine and its detector environment.
## I Muon Colliders - Who Ordered That?
The use of muons in a high energy collider appears to be a desperate measure.
After all muons have a lifetime of $2.2\mu s$ and will only survive for about
2000 turns in a 1.5 TeV storage ring. But muons also have distinct advantages
as projectiles in a colliding beams accelerator. They are pointlike, so one
can adjust the center of mass energy of the collision precisely and study
resonance structures and threshold effects in great detail. Secondly, they are
207 times more massive than the electron, meaning that muons radiate
$(m_{\mu}/m_{e})^{4}$ times less than an electron traveling with he same
radius of curvature and energy. This means that a muon collider is a circular,
rather than a linear, machine. It also means that beamsthralung effects,
radiation due to beam-beam interactions, would be much smaller in a muon
collider than an $e^{+}e^{-}$ machine, allowing for precise beam constraints
and energy measurements. Finally the mass-dependent coupling of the Higgs to
the ($\mu^{+}\mu^{-}$) system is 40,000 times larger than the coupling to
$e^{+}e^{-}$, making a muon collider an ideal candidate for direct study of s
channel Higgs.
Muons in a storage ring are reused, each muon has $\approx$2000 chances to
collide with the opposing beam before it decays. This relaxes requirements on
emittance and makes constraints on beam dimensions much more forgiving than a
linear collider with its extremely tight beam focussing requirements. Such a
machine would also be compact, with machines up to 5 TeV fitting on the
Fermilab site. The machine is also likely to use significantly less power than
a comparable electron collider. Initial calculations indicate the wall power
needed for a 3 TeV Muon Collider would be 1/3 of that for a 3 TeV CLIC and 2/3
of that for a 0.5 TeV ILC (Table 1). A muon collider could scale to multi-TeV
energies without excessive penalties in power or cost.
### I.1 Accelerator Challenges
The short muon lifetime means that everything must be done quickly. Muons must
be produced and collected, cooled, and re-accelerated rapidly. This forces
many of the components to provide multiple functions, combining cooling,
acceleration, and focussing. The beam transport system must handle the
radiation and heat load associated with electrons from muon decay. Detectors
must be well-shielded from the bulk of the decay backgrounds. The Muon
Accelerator Program (MAP) MAP has been organize to study the many technical
challenges associated with a Muon Collider.
Figure 1 shows a schematic of a possible Muon Collider Complex. The front-end
muon source would use protons from a 4 MW accelerator (e.g. Project X)
impinging on a mercury jet target. This is followed by a decay channel, beam
bunching and bunch rotation, and an initial cooling stage. This first stage
could be identical to the front end of a Neutrino Factory. The initial section
is followed by a cooling section that would further cool the beam in both
momentum and position space. Finally the muons are accelerated in a multi-
racetrack section before being injected into the storage ring. The Muon
Collider accelerator is described in more detail in Mike Zisman’s contribution
to this conferenceZisman and in publications of the Muon Accelerator Program
Mucol Palmer:2007zzc .
Figure 1: Schematic of the muon collider complex..
#### I.1.1 Ionization Cooling
Previously used beam cooling techniques, such as stochastic or electron
cooling will not cool a muon beam quickly enough for the muons to be used in a
collider. Instead, muons will be cooled utilizing ionization energy loss,
combining a low Z absorber ’with a high field solenoid and normally conducting
RF cavity. Muons lose both transverse and longtitudinal momentum in the
absorber. The RF restores longtudinal momentum, reducing $p_{x,y}/p_{z}$.
Multiple scattering acts to reheat the muons, resulting in an equilibrium
emittance that is proportional to $(1/(X_{0}\times dE_{\mu}/ds)$. The optimum
absorber material has maximum energy loss per radiation length. Liquid
hydrogen is the best candidate. Muon cooling ideas are being tested in the
MICE experiment at RAL. This experiment will measure the properties of single
muons before and after passing through an ionization cooling
sectionKaradzhov:2010zz .
#### I.1.2 RF Breakdown
After the absorber longitudinal momentum of the muon is restored by
accelerating the beam through an rf cavity. The overall cooling effect is more
efficient when a solenoid focuses the beam while still in the cavity.
Unfortunately, the breakdown voltage of an RF cavity is significantly degraded
in a magnetic field Stratakis:2009zz . Electrons are emitted from the cavity
surface in regions which are not perfectly smooth. These electrons are guided
in the magnetic field and impact on localized regions on the wall of the
cavity. In the absence of a field the electrons tend to be more dispersed and
the resulting impact damage damage is not localized. The MAP program is
studying how these effects can be mitigated by improving the surface material
properties (beryllium coatings) or by interrupting the free flow of the field-
emitted electrons by filing the cavity with gas Freemire:2011zz .
#### I.1.3 Neutrino Radiation
Muons in the collider ring decay at the rate of $1.28\times
10^{10}decays/m/s$. At this rate the radiation due to interacting neutrinos at
the site boundary is a concern. The neutrino interaction rate increases as
$energy^{3}$. Local hot spots will occur which correspond to straight sections
in the storage ring. Off site radiation can be minimized by limiting the
length of straight sections, increasing the depth of the collider ring, and
managing the operating parameters to maximize luminosity/dose. However,
neutrino-induced radiation may be the practical limit to the maximum energy of
a machine located on the Fermilab site.
Table 1: Comparison of the machine parameters for a 1.5 and 3 TeV Muon Collider with designs for CLIC at 3 TeV. | $\mu^{+}\mu^{-}$ | $\mu^{+}\mu^{-}$ | $e^{+}e^{-}$ CLIC |
---|---|---|---|---
CM Energy | 1.5 | 3.0 | 3.0 | TeV
Luminosity | 1 | 2-4 | 2 | $10^{34}cm^{2}sec^{-1}$
rms bunch height | 6 | 4 | 0.001 | $\mu m$
Diameter/length | 2 | 4 | 48 | km
Wall power | 147 | 159 | 560 | MW
## II Physics and Detector Studies
Although there is a revived muon accelerator effort, there has been little
corresponding detector effort which could study the large beam-related
backgrounds with modern simulation tools and detector technologies. This
effort was commissioned at the Muon Collider 2011 meeting in Telluride at the
end of June 2011. Initial studies were performed using ILCROOT, LCSIM, and
GEANT frameworks, with the goal of understanding the characteristics of the
backgrounds and what sorts of tools would be needed to build a detector for
the Muon Collider environment.
Both CLIC and the Muon Collider would need to study new phenomena with a level
of precision that can not be achieved at LHC. The experimental conditions in
CLIC and the Muon Collider are quite different. CLIC has a bunch train with
0.5 ns bunch spacing. The Muon Collider will have a single bunch each of
$\mu^{+}$ and $\mu^{-}$, colliding at $10\mu s$ intervals. Energy spread of
the $e^{+}e^{-}$ colliding beams scales as $\delta E/E\propto\gamma^{2}$. The
effect of this on the center of mass energy resolution is shown in figure 2.
The excellent Muon Collider energy resolution allows for precise measurements
of s-channel resonances and precise turn-on threshold scans. The higher muon
mass means that there is a significant rate for s-channel Higgs-like particle
production which is not available in a $e^{+}e^{-}$ machine. Finally, although
the muon beams are born polarized, the large phase space acceptance at low
pion momentum needed for efficient muon collection means that the colliding
beam polarization is likely to be below 20%. Additional polarization can be
obtained at the expense of luminosity by limiting the initial phase space
acceptance.
Figure 2: (left) Beam energy distribution in an $e^{+}e^{-}$ and muon collider
. The tail in the $e^{+}e^{-}$ collider is due to beam-beam (beamstrahlung)
effects and will occur in any $e^{+}e^{-}$ collider with a tight final focus.
(right)Cross sections as a function on center-of-mass energy for various
reactions in a muon collider.
### II.1 Machine-Detector Interface and Backgrounds
The physics environment of the Muon Collider is dominated by very large flux
of high energy electrons from muon decay. These electrons interact in the beam
transport system as well as in the shielding around the interaction point.
Carefully designed shielding, both for the experiment and the beam transport
system is necessary to keep backgrounds manageable. A feature that
distinguishes the Muon Collider from other experiments is a tungsten/borated
polyethylene “nose” that extends at a $10^{\circ}$ angle 6 cm from the
interaction point. The cone is designed to absorb the intense electromagnetic
radiation due to muon decay that accompanies the muon beams and reduces the
overall background level by three orders of magnitude. Over the past year
there has been a substantial effort to fully model the backgrounds in a model
detector at the Muon Collider. There are now both MARS and G4beamline models
of the interaction region with detailed simulations of the particle flux.
Figure 3 shows the background flux entering the detector region in a typical
Muon Collider interaction. Total non-ionizing background is about 10% that of
the LHC, but the crossing interval is 400 times longer, resulting in high
instantaneous flux. The background is very different in character than that of
either the LHC or CLIC. It is dominated by soft photons and low energy
neutrons emerging from the shielding surrounding the detector. A typical
background event has 164 TeV of photons, 172 TeV of neutrons, and 184 TeV of
muons. With the exception of muons and charged hadrons the background spectrum
is dominated by low energy particles. Only a small fraction of the background
originates from the vicinity of the interaction region. This means that most
of the decay background is out of time with respect to particles originating
from the $\mu^{+}\mu^{-}$ collision.
Figure 3: Energy distributions of particles entering the detector region from
a MARS simulation of Muon Collider beam backgroundsStriganov .
### II.2 Background Rejection Techniques
The fact that much of the background is soft and out of time gives us two
handles on the design of an experiment that can cope with the high levels of
background. Timing is especially powerful. Figure 3 shows the fraction of
background and signal hits in the tracker preserved as a function of the width
of the timing gate. An important feature is that the local gate t=0 is defined
as the time when relativistic particle emerging from the interaction point
arrives at the detector. Therefore a very tight cut can be made, still
preserving the bulk of the tracks of interest. A 1 ns cut rejects two orders
of magnitude of the overall background and about 4 orders of magnitude of
neutron background.
A detailed tracking study was performed using the ILCROOT framework which
includes a all silicon detector similar to the one proposed for SiDMazzacane .
The tracker is fully pixelated with $50\mu m$ pixels in the tracker and $20\mu
m$ pixels in the vertex section. Hits are required to have a threshold of 3000
electrons, which eliminates much of the soft photon and neutron background.
Reconstructed tracks are required to have an impact parameter less then 3 mm.
With no timing cut the reconstruction program fails due to the large number of
hits. However if the $\Delta t$ cut is 3 ns only 11 background tracks are
found and a 1 ns cut further reduces this to 3 tracks, all with low momentum.
Replacing the timing cut by a time stamp would allow the hit arrival time to
be used in the track fit, providing both background rejection and some level
of particle identification.
Soft, uncorrelated hits can also be eliminated by exploiting the correlation
between two closely spaced tracking layers. This was first suggested by Steve
Geer in early Muon Collider studies and a similar concept is being explored
for a track trigger for CMSLipton:2011zz . In that design two silicon sensors
are spaced by a 1 mm thick interposer and only hits which are correlated
between the top and bottom layers are used for the trigger. Initial studies
indicate that such an arrangement would also be extremely effective at
reducing Muon Collider backgrounds and might reduce dependence on the timing
cut.
Figure 4: Fraction of hits accepted as a function of the timing gate
widthTerentev . The gate start is defined by the time of flight for a
relativistic particle emerging from the interaction point.
Timing is also crucial for calorimetry. Our initial ILCROOT simulation studied
a dual readout “Adriano” heavy glass/scintillator calorimeter with $4\times 4$
cm cells with 7.5 interaction lengths Vito . There are two longitudinal
sections 20 and 160 long, each with front and rear readout through SIPMs.
Different timing cuts are used for the scintillation and Cerenkov light in the
front section (15 and 6 ns, respectively) with a 22 ns cut for the rear
section. In the central barrel region backgrounds deposit an average energy
per tower of 5.33 GeV per event with RMS fluctuations of 540 MeV in the front
section. The rear section sees an average energy of 630 MeV with fluctuations
of 430 MeV. Further analysis will be needed to optimize cuts and determine jet
energy resolution.
An optimal calorimeter design might combine fast timing with the
reconstruction ability of pixelated calorimeters being studied for particle
flow. A pixelated imaging sampling calorimeter with 200 $\mu m$ square cells
was proposed by R. Raja Raja . In this design a 2 ns “traveling trigger” gate
referenced to the time of flight with respect to the beam crossing is used to
reject out-of-time hits. In this case background rejection was found to be
$3\times 10^{-2}$ to $4\times 10^{-4}$. This sort of calorimeter can implement
compensation by recognizing hadronic interaction vertices and using the number
of such vertices to correct the energy. Initial estimates of the resolution of
such a compensated calorimeter is $60\%/\sqrt{E}$. In contrast to relativistic
tracks and electromagnetic showers, hadronic showers can take significant time
to developsimon . Further study is needed to understand the tradeoff between
background rejection provided by a short time gate and the loss of energy
resolution caused by the slow time development of hadronic showers .
We have learned that tracking seems possible in a Muon Collider detector.
Calorimetery is more challenging, but progress is being made on imaging
calorimeter concepts that appear to meet the physics needs. Precise timing and
pixelated detectors will be crucial to a successful Muon Collider detector.
Both come at a cost. For example the time resolution, $\sigma(t)\approx
risetime\times(noise/signal)$. The signal/noise and gain-bandwidth of typical
electronic front ends are proportional to the transductance of the front end
transistor - which in turn is proportional to front-end current. This means
that electronics will necessarily dissipate significant power and, in contrast
to planned ILC detectors, detectors for the Muon Collider will have to be
water cooled with associated increase in mass. The large background of non-
ionizing radiation means that silicon detector will have to be kept cold,
around -10 C, again increasing the detector mass. Such a detector will
resemble an LHC experiment more closely than those planned for ILC or CLIC.
## III Conclusions
Both accelerator and detector aspects of a muon collider are extremely
challenging. The Muon Accelerator Program has been formed to study and
evaluate the accelerator challenges. A complementary effort in now beginning
to study physics and detector aspects. Initial results of these studies
indicate that detectors can be designed that withstand the fierce backgrounds.
Such a detector is likely to be more massive than a corresponding ILC
detector, but could have unique capabilities. Fast timing and fine
segmentation appear to be crucial.
## References
* (1) http://map.fnal.gov/
* (2) ICFA Beam Dynamics Newsletter No. 55, August 2011. http://www-bd.fnal.gov/icfabd/Newsletter55.pdf
* (3) Mike Zisman, submitted to Proc. of the DPF-2011 Conference, Providence, RI, August 8-13, 2011 arXiv:1109.3086v1 [physics.acc-ph]
* (4) R. B. Palmer, J. S. Berg, R. C. Fernow, J. C. Gallardo, H. G. Kirk, Y. Alexahin, D. Neuffer, S. A. Kahn et al., [arXiv:0711.4275 [physics.acc-ph]].
* (5) Y. Karadzhov, “Status of MICE, the international Muon Ionisation Cooling Experiment,” PoS ICHEP2010, 323 (2010).
* (6) D. Stratakis, J. C. Gallardo, R. B. Palmer, AIP Conf. Proc. 1222 (2010) 303-307.
* (7) B. Freemire et al., “High Pressure RF Cavity Test at Fermilab,” PAC-2011-MOP032.
* (8) Corrado Gatto, TIPP 2011
* (9) Nikolai Mokhov, Muon Collider 2011, https://indico.fnal.gov/getFile.py/access?contribId=5&sessionId=2&resId=0&materialId=slides&confId=4146
* (10) Sergei Striganov, Muon Collider 2011, https://indico.fnal.gov/getFile.py/access?contribId=17&sessionId=5&resId=0&materialId=slides&confId=4146
* (11) Nikolai Terentiev, Muon Collider 2011, https://indico.fnal.gov/getFile.py/access?contribId=20&sessionId=5&resId=0&materialId=slides&confId=4146
* (12) Anna Mazzacane, Muon Collider 2011, https://indico.fnal.gov/getFile.py/access?contribId=19&sessionId=5&resId=0&materialId=slides&confId=4146
* (13) R. Lipton, Nucl. Instrum. Meth. A636, S160-S163 (2011).
* (14) Vito diBenedetto, Muon Collider 2011, https://indico.fnal.gov/conferenceOtherViews.py?view=standard&confId=4146.
* (15) Rajendran Raja, LCWS 2011, http://ilcagenda.linearcollider.org/contributionDisplay.py?sessionId=33&contribId=26&confId=5134
* (16) Frank Simon for the CALICE Collaboration, submitted to TIPP 2011 arXiv:1109.3143v1 [physics.ins-det]
|
arxiv-papers
| 2012-04-16T15:48:37 |
2024-09-04T02:49:29.746495
|
{
"license": "Public Domain",
"authors": "Ronald Lipton",
"submitter": "Ronald Lipton",
"url": "https://arxiv.org/abs/1204.3538"
}
|
1204.3547
|
Computer Model Calibration Using the Ensemble Kalman Filter
Dave Higdon, Statistical Sciences Group, Los Alamos National Laboratory
Matt Pratola, Statistical Sciences Group, Los Alamos National Laboratory
Jim Gattiker, Statistical Sciences Group, Los Alamos National Laboratory
Earl Lawrence, Statistical Sciences Group, Los Alamos National Laboratory
Charles Jackson, University of Texas Institute for Geophysics
Michael Tobis, University of Texas Institute for Geophysics
Salman Habib, High Energy Physics Division, Argonne National Laboratory
Katrin Heitmann, High Energy Physics Division, Argonne National Laboratory
Steve Price, Fluid Dynamics Group, Los Alamos National Laboratory
The ensemble Kalman filter (EnKF) (Evensen, , 2009a) has proven effective in
quantifying uncertainty in a number of challenging dynamic, state estimation,
or data assimilation, problems such as weather forecasting and ocean modeling.
In these problems a high-dimensional state parameter is successively updated
based on recurring physical observations, with the aid of a computationally
demanding forward model that propagates the state from one time step to the
next. More recently, the EnKF has proven effective in history matching in the
petroleum engineering community (Evensen, , 2009b; Oliver and Chen, , 2010).
Such applications typically involve estimating large numbers of parameters,
describing an oil reservoir, using data from production history that
accumulate over time. Such history matching problems are especially
challenging examples of computer model calibration since they involve a large
number of model parameters as well as a computationally demanding forward
model. More generally, computer model calibration combines physical
observations with a computational model – a computer model – to estimate
unknown parameters in the computer model. This paper explores how the EnKF can
be used in computer model calibration problems, comparing it to other more
common approaches, considering applications in climate and cosmology.
Keywords: computer experiments; model validation; data assimilation;
uncertainty quantification; Gaussian process; parameter estimation; Bayesian
statistics
## 1 Introduction
The ensemble Kalman filter (EnKF) has proven effective in quantifying
uncertainty in a number of challenging dynamic, state estimation, or data
assimilation, problems. Applications include weather forecasting (Houtekamer
et al., , 2005), ocean modeling (Evensen, , 2003), storm tracking (Aksoy et
al., , 2009), hydrology (Moradkhani et al., , 2005) and wildfire modeling
(Mandel et al., , 2004), just to name a few. In these data assimilation
problems, a high-dimensional state parameter is successively updated based on
recurring physical observations, with the aid of a computationally demanding
forward model that propagates the state from one time step to the next. The
EnKF iteratively updates an ensemble of state vectors, using a scheme
motivated by the standard Kalman filter (Meinhold and Singpurwalla, , 1983;
West and Harrison, , 1997), producing an updated ensemble of states that is
affected by both the forward model and the physical observations. More
recently, the EnKF has proven effective in history matching in the petroleum
engineering community (Evensen, , 2009b; Oliver and Chen, , 2010). Such
applications typically involve estimating large numbers of parameters,
describing an oil reservoir, using data from production history that
accumulate over time. Such history matching problems are especially
challenging examples of computer model calibration since they involve a large
number of model parameters as well as a computationally demanding forward
model. Unlike standard data assimilation problems, here focus is on estimation
of a static model parameter vector, rather than an evolving state vector.
This paper explores how the EnKF can be used in computer model calibration
problems that typically have a collection of model parameters to be
constrained using physical observations. We first use a simple 1-d inverse
problem to describe standard Bayesian approaches to produce a posterior
distribution for the unknown model parameter vector, as well as the resulting
model prediction. We then go on to describe how the EnKF can be used to
address this basic problem, with examples taken from the literature in climate
and cosmology. We end with conclusions summarizing the strengths and
weaknesses of using the EnKF for computer model calibration.
### 1.1 A simple inverse problem
A simple inverse problem in which $\eta(\cdot)$, given by the black line,
denotes the forward model, mapping the unknown parameter $\theta$ into an
observable $\eta(\theta)$.
The physical observation $y$ is a noisy version of $\eta(\theta)$:
$y=\eta(\theta)+\epsilon$
where $\epsilon\sim N(0,\sigma_{y}^{2}=.1^{2})$. The horizontal gray line and
band denote the measurement ($y=.8$) and its uncertainty $y\pm 2\sigma_{y}$.
The model parameter $\theta$ is given a $N(0,1)$ prior. The resulting
posterior for $\theta$ is given by the shaded density at the bottom of the
figure.
Figure 1: A simple 1-dimensional inverse problem and resulting posterior
density.
In order to describe the basic approaches to inverse problems, we first
describe a simple, 1-dimensional inverse problem shown in Figure 1. We take
$\eta(\cdot)$ to denote the forward model. It requires a single model
parameter $\theta$, producing a univariate output $\eta(\theta)$ that is
comparable to a physical measurement $y$. Here we take the sampling model for
$y$ to be normally distributed about the forward model’s output when the true
value of the model parameter is input
$L(y|\eta(\theta))\propto\exp\\{-\mbox{\small$\frac{1}{2}$}\sigma^{-2}_{y}(y-\eta(\theta))^{2}\\}$
where the observation error is assumed known to be $\sigma_{y}=.1$.
After specifying a standard normal prior for the model parameter, with
$\pi(\theta)$ denoting the prior density, the posterior density is given by
$\displaystyle\pi(\theta|y)$ $\displaystyle\propto$ $\displaystyle
L(y|\eta(\theta))\times\pi(\theta)$ $\displaystyle\propto$
$\displaystyle\exp\\{-\mbox{\small$\frac{1}{2}$}\sigma^{-2}_{y}(y-\eta(\theta))^{2}\\}\times\exp\\{-\mbox{\small$\frac{1}{2}$}\theta^{2}\\}.$
Thus an evaluation of the posterior requires a run of the forward model. While
this simple, 1-dimensional density is trivial to evaluate, many inverse
problems have to deal with a large model parameter vector (dimensions ranging
from $10$ to $10^{8}$) as well as a computationally demanding forward model
that may take a long time to evaluate (our experience ranges from seconds to
weeks).
#### 1.1.1 Using a Gaussian process emulator
While Markov chain Monte Carlo (MCMC) remains a popular approach for exploring
the resulting posterior distribution (Kaipio and Somersalo, , 2004; Tarantola,
, 2005), the demands required by the size of the model parameter vector and
the computational demands of the forward model have inspired recent research
focused on overcoming these hurdles. These research efforts range from
response surface approximation of the forward model $\eta(\cdot)$ (Kennedy and
O’Hagan, , 2001; Higdon et al., , 2005), to constructing reduced, or
simplified forward models (Galbally et al., , 2010; Lieberman et al., , 2010),
to polynomial chaos approximations of the prior model (Ghanem and Doostan, ,
2006; Marzouk and Najm, , 2009), to exploiting multiple model fidelities
(Christen and Fox, , 2005; Efendiev et al., , 2009).
The forward model $\eta(\cdot)$ is run at 4 values of the input parameter,
producing model observations $\eta^{\circ}$ (black dots). A GP prior
$\pi(\eta(\cdot))$ is specified for the forward model.
likelihood (measurement) $\displaystyle y\sim N(\eta(\theta),\sigma_{y}^{2})$
prior for $\eta(\cdot)$ $\displaystyle\pi(\eta(\cdot))$ likelihood
(simulations) $\displaystyle\eta^{\circ}\sim\pi(\eta(\cdot))$ prior for
$\theta$ $\displaystyle\pi(\theta)$ $\displaystyle\Rightarrow\mbox{posterior
}$ $\displaystyle\pi(\theta,\eta(\cdot)|y,\eta^{\circ})$
Conditioning on both the physical observation (light, horizontal band) and the
four model runs, the posterior distribution for both $\eta(\cdot)$ and
$\theta$ is produced (shaded density).
Figure 2: Using a Gaussian process prior for the forward model to reduce the
number of model runs necessary for posterior exploration for the simple
inverse problem.
Gaussian processes (GPs) are commonly used to emulate the computer model
response, producing a probabilistic description of the response at untried
parameter settings – see Kennedy and O’Hagan, (2001) or Bayarri et al.,
(2007) for just a couple of examples. This basic approach is depicted in
Figure 2. In this case, a GP prior is used to model the unknown function
$\eta(\cdot)$ and a collection of forward model runs
$\eta^{\circ}=(\eta(\theta_{1}^{\circ}),\ldots,\eta(\theta_{m}^{\circ}))^{\prime}$,
over a collection of input parameter settings
$\theta^{\circ}=(\theta_{1}^{\circ},\ldots,\theta_{m}^{\circ})^{\prime}$, are
used to infer the forward model response at untried input parameter settings.
Thus the basic fomulation (1.1) is augmented to incorporate this GP prior for
the forward model
$\eta(\cdot)\sim GP(m(\cdot),C(\cdot,\cdot)),$
where the mean function $m(\cdot)$ may be a constant (Sacks et al., , 1989;
Kennedy and O’Hagan, , 2001), or a more complicated regression function (Craig
et al., , 2001; Vernon et al., , 2010), and the covariance function
$C(\cdot,\cdot)$ is typically of product form, requiring just a single
additional parameter for each dimension of the input parameter. For this
simple problem, we take the mean and covariance functions as fixed, leading to
the posterior form
$\pi(\theta|y,\eta^{\circ})\propto\exp\\{-\mbox{\small$\frac{1}{2}$}(v_{\theta}+\sigma^{2}_{y})^{-1}(y-\mu_{\theta})^{2}\\}\times\exp\\{-\mbox{\small$\frac{1}{2}$}\theta^{2}\\}.$
Here $\mu_{\theta}$ and $v_{\theta}$ are the mean and variance given by the GP
model after conditioning on the forward model runs $\eta^{\circ}$
$\displaystyle\mu_{\theta}$ $\displaystyle=$ $\displaystyle
C(\theta,\theta^{\circ})C(\theta^{\circ},\theta^{\circ})^{-1}(\eta^{\circ}-m(\theta^{\circ}))+m(\theta)$
$\displaystyle v_{\theta}$ $\displaystyle=$ $\displaystyle
C(\theta,\theta)-C(\theta,\theta^{\circ})C(\theta^{\circ},\theta^{\circ})^{-1}C(\theta^{\circ},\theta),$
where, for example, $C(\theta^{\circ},\theta^{\circ})$ produces a $m\times m$
matrix whose $ij$ entry is $C(\theta^{\circ}_{i},\theta^{\circ}_{j})$ and
$C(\theta,\theta^{\circ})$ produces a $m$-vector whose $j$th element is
$C(\theta,\theta^{\circ}_{j})$. See Higdon et al., (2005) for details
regarding the posterior specification when the GP (and other) parameters are
not taken as fixed, and the input parameter is multivariate.
#### 1.1.2 Using the ensemble Kalman filter
Below we briefly describe two basic variants of the the EnKF for computer
model calibration, differing in how they use the ensemble of model runs to
approximate, and represent, the resulting posterior distribution. In both
cases, an ensemble of draws $\theta^{\circ}$ from the prior distribution of
the model parameter are paired with the resulting simulation output to produce
an ensemble of $(\theta^{\circ},\eta(\theta^{\circ}))$ pairs, from which the
sample covariance is used to produce an approximation to the posterior
distribution. Hence we treat the input parameter settings
$\theta^{\circ}_{1},\ldots,\theta^{\circ}_{m}$ as $m$ draws from the prior
distribution $\pi(\theta)$. Note that even though the distribution of the
simulator response $\eta(\theta)$ is completely determined by the distribution
for $\theta$, the EnKF uses a joint normal model for $(\theta,\eta(\theta))$
to motivate its calculations.
Next we describe two variants of the EnKF algorithm for computer model
calibration. One uses a Gaussian representation of the posterior distribution,
the other uses an ensemble representation.
#### Gaussian representation
The first approach fits a multivariate normal distribution to the ensemble for
$(\theta^{\circ},\eta(\theta^{\circ}))$. The algorithm is depicted in the left
frame of Figure 3 and described below.
1. 1.
For each of the $m$ simulations form the ensemble of joint vectors
$\begin{pmatrix}\theta_{k}^{\circ}\\\
\eta(\theta^{\circ}_{k})\end{pmatrix},\,k=1,\ldots,m.$ (2)
With these $m$ vectors, compute the sample mean vector $\mu_{\rm pr}$ and
sample covariance matrix $\Sigma_{\rm pr}$. For the simple inverse problem
here, $\mu_{\rm pr}$ is a $2$-vector and $\Sigma_{\rm pr}$ is $2\times 2$, but
this recipe is quite general.
2. 2.
In this simple inverse problem, the physical observation $y$ corresponds to
the 2nd element of the joint $(\theta,\eta(\theta))$ vector. Take ${H}$ be
$(0,1)^{\prime}$ to be the observation matrix. The likelihood can be written
$L(y|\eta(\theta))\propto\exp\left\\{-\frac{1}{2}\left(y-H\begin{pmatrix}\theta\\\
\eta(\theta)\end{pmatrix}\right)^{\prime}\Sigma_{y}^{-1}\left(y-H\begin{pmatrix}\theta\\\
\eta(\theta)\end{pmatrix}\right)\right\\}.$ (3)
More generally the observation operator $H$ can select elements of
$\eta(\theta)$ that are observed, or can be specified to interpolate between
values of the simulator output.
3. 3.
Combining the normal approximation to the prior with the normal likelihood
results in an updated, or posterior, distribution for $(\theta,\eta)$ for
which
$\begin{pmatrix}\theta\\\ \eta\end{pmatrix}|y\sim N(\mu_{\rm post},\Sigma_{\rm
post}),$ (4)
where
$\Sigma_{\rm post}^{-1}=\Sigma_{\rm pr}^{-1}+H^{\prime}\Sigma_{y}^{-1}H$ (5)
and
$\mu_{\rm post}=\Sigma_{\rm post}\left(\Sigma_{\rm pr}^{-1}\mu_{\rm
pr}+H^{\prime}\Sigma_{y}^{-1}y\right).$ (6)
Note that the posterior mean can be rewritten in a form more commonly used in
Kalman filtering
$\mu_{\rm post}=\mu_{\rm pr}+\Sigma_{\rm pr}H^{\prime}(H\Sigma_{\rm
pr}H^{\prime}+\Sigma_{y})^{-1}(y-H\mu_{\rm pr})$
where $\Sigma_{\rm pr}H^{\prime}(H\Sigma_{\rm pr}H^{\prime}+\Sigma_{y})^{-1}$
is the Kalman gain matrix.
The joint normal computations used here effectively assume a linear plus
Gaussian noise relationship between $\theta$ and $\eta(\theta)$, inducing a
normal posterior for $\theta$.
Figure 3: Left: Gaussian representation of the posterior distribution for
$(\theta,\eta(\theta))$ resulting from the ensemble Kalman filter (EnKF). The
approximate normal prior distribution for $(\theta,\eta(\theta))$ is depicted
by the black ellipse, estimated from the ensemble (circle plotting symbols).
The resulting posterior distribution is approximated as normal, depicted by
the gray ellipse. The marginal posterior for $\theta$ is given by the shaded
density. Right: the ensemble representation of the posterior distribution for
$(\theta,\eta(\theta))$ resulting from the EnKF. Here the updated sample (gray
dots) are approximate draws from the posterior distribution.
#### Ensemble representation
The second approach is basically the usual EnKF as applied to time evolving
systems, but here, only for a single time step. The goal is to perturb each
member of the ensemble $(\theta^{\circ}_{k},\eta(\theta^{\circ}_{k}))$, in
order to produce an updated member $(\eta_{k}^{(1)},\theta^{(1)}_{k})$ whose
mean and variance match the posterior produced by the Gaussian representation
EnKF described above. This updated member is not produced with the simulator
so that $\eta^{(1)}_{k}$ will not be equal to the simulator evaluated at
updated parameter value $\eta(\theta^{(1)}_{k})$. Here we describe the
perturbed data version of the EnKF described in Evensen, (2009b). A number of
variants of this basic approach exist; see Anderson, (2001) and Szunyogh et
al., (2008), for example. The algorithm is given below.
1. 1.
Construct the sample covariance matrix $\Sigma_{\rm pr}$ as in Step 1 of the
previous algorithm.
2. 2.
For $k=1,\dots,m$ do:
1. (a)
Draw a perturbed data value $y_{k}\sim N(y,\Sigma_{y})$.
2. (b)
Produce the perturbed ensemble member
$\begin{pmatrix}\theta^{(1)}_{k}\\\ \eta_{k}^{(1)}\end{pmatrix}=\Sigma_{\rm
post}\left(\Sigma_{\rm
pr}^{-1}\begin{pmatrix}\theta^{\circ}_{k}\cr\eta(\theta^{\circ}_{k})\end{pmatrix}+H^{\prime}\Sigma_{y}^{-1}y_{k}\right).$
(7)
where $\Sigma_{\rm pr}$ and $\Sigma_{\rm post}$ are defined in the previous
algorithm. Note this perturbation of the ensemble member can be equivalently
written using the more standard Kalman gain update:
$\begin{pmatrix}\theta^{(1)}_{k}\\\
\eta_{k}^{(1)}\end{pmatrix}=\begin{pmatrix}\theta^{\circ}_{k}\cr\eta(\theta^{\circ}_{k})\end{pmatrix}+\Sigma_{\rm
pr}H^{\prime}(H\Sigma_{\rm
pr}H^{\prime}+\Sigma_{y})^{-1}(y_{k}-\eta(\theta^{\circ}_{k}))$ (8)
3. 3.
Treat this updated, $m$ member ensemble
$\begin{pmatrix}\theta^{(1)}_{k}\\\
\eta_{k}^{(1)}\end{pmatrix},\,k=1,\ldots,m.$
as draws from the updated, posterior distribution for $(\theta,\eta)$ given
the initial ensemble $(\theta^{\circ},\eta(\theta^{\circ}))$ and the physical
observation $y$.
This approach uses a Bayesian update of two normal forms, with each ensemble
member updated separately. Here the normal prior is centered at the ensemble
member, and the normal likelihood is centered at the perturbed data value,
rather than at the ensemble mean and the actual data value. This update sets
the new ensemble value $(\theta^{(1)}_{k},\eta^{(1)}_{k})$ to the mean of this
resulting combination of normal distributions.
This produces a posterior ensemble for the joint distribution of
$(\theta,\eta)$, given by the gray dots in the right hand frame of Figure 3.
Hence the difference between these two representations can be seen Figure 3 –
compare the gray ellipse in the left frame, representing the updated normal
posterior, to the gray dots in the right frame, representing draws using this
ensemble representation.
Note that if we take $(\theta^{\circ}_{k},\eta^{\circ}_{k})$ to be a draw from
a distribution with mean $\mu_{\rm pr}$ and variance $\Sigma_{\rm pr}$,
applying (7) – or equivalently (8) – produces a random variable
$(\theta^{(1)}_{k},\eta^{(1)}_{k})$ with mean and variance given in (6) and
(5). Hence the mean and variance of the ensemble members
$(\theta^{(1)}_{k},\eta^{(1)}_{k})$ matches that of the Gaussian
representation of the EnKF (4). Even though the first and second moments of
the two EnKF representations match in distribution, the ensemble
representation appears to better capture the true, right skewed posterior
(compare to Figure 1).
#### A two-stage approach
Figure 4 shows how one can repeatedly apply the EnKF to improve the accuracy
of the of the normal representation of $\eta(\theta)$ where the posterior mass
for $\theta$ is concentrated. This iterative strategy is closer to the
original use of the EnKF for state-space estimation in non-linear, dynamic
systems. Also, this two stage approach easily generalizes to additional
stages.
For this two-stage EnKF, we artificially break the information from the
likelihood into two even pieces
$L(y|\eta(\theta))\propto\exp\left\\{-\frac{1}{2}\frac{1}{(2\sigma_{y}^{2})}(y-\eta(\theta))^{2}\right\\}\times\exp\left\\{-\frac{1}{2}\frac{1}{(2\sigma_{y}^{2})}(y-\eta(\theta))^{2}\right\\}$
as if $y$ were observed twice, with twice the error variance. Then the EnKF is
first applied to one of these $y$ values, with twice the error varriance,
producing an ensemble representation
$\theta^{(1)}_{1},\ldots,\theta^{(1)}_{m}$ of the posterior distribution for
$\theta$ given this partial piece of information. Next, the forward model is
run again at each of these new parameter settings, producing the ensemble
$(\theta^{(1)}_{k},\eta(\theta^{(1)}_{k})),\;k=1,\ldots,m$. This new ensemble
is now the starting point for a second EnKF update, again using $y$ with twice
the error variance.
This second update can produce a Gaussian representation (the gray ellipse in
the right frame of Figure 4), or an ensemble representation
$(\theta^{(2)}_{k},\eta^{(2)}_{k}),\;k=1,\ldots,m$ (the gray dots in the right
frame of Figure 4). As can be seen in the right frame of Figure 4, the second
Gaussian representation of the relationship between $\theta$ and
$\eta(\theta)$ is more accurate because the
$(\theta^{(1)},\eta(\theta^{(1)}))$ ensemble covers a narrower range, over
which $\eta(\theta)$ is more nearly linear.
Clearly, the choice of using two even splits of the likelihood information is
somewhat arbitrary – both the number of splits and the partitioning of
information to each split could be made in many ways. The cost of additional
forward model evaluations has to be weighed against the benefits of a slightly
more accurate Gaussian representation of $\eta(\theta)$ over a restricted
range of values for $\theta$.
Figure 4: A two-stage EnKF solution to the simple inverse problem. Here the
EnKF is applied twice, using the same observation $y$, but assuming it is
observed with twice the variance. Left: at the first stage, the ensemble
representation is used, assuming the observation has twice the variance,
giving a data uncertainty that is a factor of $\sqrt{2}$ larger than in the
previous figures. Right: the second stage starts with the updated ensemble,
evaluatingng the forward model at each $\theta^{(1)}_{k}$, producing a new
ensemble
$(\theta^{(1)}_{1},\eta(\theta^{(1)}_{1})),\ldots,(\theta^{(1)}_{m},\eta(\theta^{(1)}_{m}))$.
This new ensemble is updated once more, again using the data with twice the
variance. The resulting uncertainty can be represented with a Gaussian
distribution (shaded density) or an ensemble
$(\theta^{(2)}_{1},\eta^{(2)}_{1}),\ldots,(\theta^{(2)}_{m},\eta^{(2)}_{m})$
(shaded dots).
#### 1.1.3 Embedding the EnKF into a Bayesian formulation
As noted in a number of references (Anderson and Anderson, , 1999; Shumway and
Stoffer, , 2010; Stroud et al., , 2010), the EnKF can be embedded in a
likelihood or Bayesian formulation. For this simple inverse problem lends
itself to the Bayesian formulation below,
$\displaystyle\mbox{sampling model: }y|\eta(\theta)$ $\displaystyle\sim$
$\displaystyle N(\eta(\theta),\sigma_{y}^{2})$ $\displaystyle\mbox{prior
model: }(\theta,\eta)$ $\displaystyle\sim$ $\displaystyle N(\mu_{\rm
pr},\Sigma_{\rm pr}),$
or, equivalently
$\displaystyle y|\eta(\theta)$ $\displaystyle\sim$ $\displaystyle
N(\eta(\theta),\sigma_{y}^{2})$ $\displaystyle\eta|\theta$ $\displaystyle\sim$
$\displaystyle N\left(\mu_{{\rm pr}2}+\Sigma_{{\rm pr}22}^{-1}\Sigma_{{\rm
pr}21}(\theta-\mu_{{\rm pr}1}),\Sigma_{{\rm pr}22}-\Sigma_{{\rm
pr}21}\Sigma_{{\rm pr}11}^{-1}\Sigma_{{\rm pr}12}\right)$
$\displaystyle\theta$ $\displaystyle\sim$ $\displaystyle N(\mu_{{\rm
pr}1},\Sigma_{{\rm pr}11}).$
In looking at the prior specification of $\eta|\theta$ above, it’s apparent
that the mean is just the linear regression estimate of $\eta$ given $\theta$.
Hence where a GP model is used in the formulation described in Figure 2, the
EnKF implicitly uses a linear regression-based emulator. While this simple
form can only account for linear effects and no interactions, the tradeoff is
that this emulator can be estimated quickly, can handle large ensemble sizes
$m$, and can handle moderately high-dimensional input parameter, and output
spaces.
The EnKF uses the initial sample of model runs
$(\theta^{\circ}_{1},\eta(\theta^{\circ}_{1})),\ldots,(\theta^{\circ}_{m},\eta(\theta^{\circ}_{m}))$
to produce the standard plug-in estimates for $\mu_{\rm pr}$ and $\Sigma_{\rm
pr}$ – the sample mean and covariance. In static inverse problems, where quick
turn-around of results isn’t crucial, one could specify priors for these
parameters, producing a more fully Bayesian solution. An obvious choice might
take vague, normal prior for $\mu_{\rm pr}$, and an inverse wishart for
$\Sigma_{\rm pr}$ (West and Harrison, , 1997), if $m$ is sufficiently large
relative to the dimensionality of $y$ and $\theta$.
In cases where the dimensionality of $\mu_{\rm pr}$ and $\Sigma_{\rm pr}$ is
large (much larger than the ensemble size $m$) covariance tapering, or some
other form of localization is used to deal with spurious correlations produced
in the standard sample covariance estimate (Furrer and Bengtsson, , 2007;
Evensen, , 2009b; Stroud et al., , 2010). The above specification suggests the
use of variable selection (Wasserman, , 2000; Tibshirani, , 1996), or
compressed sensing (Baraniuk, , 2007) could make a viable alternative for
estimating the regression function for $\eta$ given the ensemble draws for
$\theta$, producing the updated ensemble. Finally, we note that a bootstrap
could be a useful tool for accounting for the uncertainty in the ensemble-
based estimates for $\mu_{\rm pr}$ and $\Sigma_{\rm pr}$ since it does not
require any additional model runs be carried out.
## 2 Applications
This section describes three applications in the statistical analysis of
computer models that make use of the EnKF. The first two are calibration
examples, one taken from cosmology, the second from climate. The last explores
how this EnKF representation can be used to for experimental design,
determining optimal spatial locations at which to take ice sheet measurements.
The goal of these examples are to suggest possible uses of EnKF ideas, rather
than providing definitive analyses in problems involving inference with the
aid of computationally demanding computer models.
### 2.1 Calibration of cosmological parameters
Perhaps the simplest cosmological model in agreement with available physical
observations (e.g. the large scale structure of the universe, the cosmic
microwave background) is the $\Lambda$-cold dark matter ($\Lambda$CDM) model.
This model, controlled by a small number of parameters, determines the
composition, expansion and fluctuations of the universe.
This example focuses on model calibration, combining observations from the
Sloan Digital Sky Survey (Adelman-McCarthy et al., , 2006), giving a local
spatial map of large galaxies, with large-scale $N$-body simulations,
controlled by five $\Lambda$CDM model parameters, evolving matter over a
history that begins with the big bang, and ends at our current time, about 14
billion years later. An example of the physical observations produced by the
SDSS are shown in the left frame of Figure 5. It shows a slice of the 3-d
spatial map of large galaxies. Along with spatial position, the estimated mass
for each of galaxy is also recorded.
The computational model predicts the current spatial distribution of matter in
the universe, given the parameters of the $\Lambda$CDM model, requiring
substantial computing effort. For a given parameter setting, a very large-
scale $N$-body simulation is carried out. The simulation initializes dark
matter tracer particles according to the cosmic microwave background and then
propagates them according to gravity and other forces up to the present time.
The result of one such simulation is shown in the middle frame of Figure 5.
Different cosmologies (i.e. cosmological parameter settings) yield simulations
with different spatial structure. We would like to determine which cosmologies
are consistent with physical observations of the SDSS given in the left frame
of Figure 5.
Figure 5: Left: Physical observations from the Sloan Digital Sky Survey
(Credit: Sloan Digital Sky Survey). Middle: Simulation results from an
$N$-body simulation. Right: Power spectra for the Matter density fields. The
gray lines are from 128 simulations; the black lines give spectrum estimates
derived from the physical observations.
Direct comparison between the simulation output and the SDSS data is not
possible since the simulations evolve an idealized, periodic cube of particles
corresponding to clusters of galaxies, while the SDSS data give a censored,
local snapshot of the large scale structure of the universe. Instead, we
summarize the simulation output and physical observations by their dark matter
power spectra which describe the spatial distribution of matter density at a
wide range of length scales.
Computing the matter power spectrum is trivial for the simulation output since
it is defined on a periodic, cubic lattice. In contrast, determining matter
power spectrum from the SDSS data is a far more challenging task since one
must account for the many difficulties that accompany observational data:
nonstandard survey geometry, redshift space distortions, luminosity bias and
noise, just to name a few. Because of these challenges, we use the published
data and likelihood of Tegmark et al., (2004) which is summarized by the
black lines in the right hand frame of Figure 5. The resulting data correspond
to 22 pairs $(y_{i},k_{i})$ where $y_{i}$ is a binned estimate of the log of
the power, and $k_{i}$ denotes the wavenumber corresponding to the estimate.
The data vector $y=(y_{1},\ldots,y_{22})^{\prime}$ has a diagonal covariance
$\Sigma_{y}$. Two standard deviation error bars are shown in the right frame
of Figure 5 for each observation.
We take the ensemble produced in Heitmann et al., (2006) – a $m=128$ run
orthogonal array-based latin hypercube sample (LHS) over the 5-d rectangular
parameter space detailed in Table 1. Since this sample was originally
generated to produce a multivariate GP emulator – predicting the simulated
matter power spectrum as a function of the 5-d parameter inputs – it is not a
draw from a normal prior as is standard for EnKF applications. Nevertheless,
this sample can be used to estimate $\mu_{\rm pr}$ and $\Sigma_{\rm pr}$ from
Section 1.1.2. The restricted ranges of the parameters will need to be
reconciled with the eventual normal description of the parameter posterior, or
resulting EnKF sample.
Table 1: $\Lambda$CDM parameters and their lower and upper bounds.
parameter description lower upper $n$ spectral index 0.8 1.4 $h$ Hubble
constant 0.5 1.1 $\sigma_{8}$ galaxy fluctuation amplitude 0.6 1.6
$\Omega_{\rm CDM}$ dark matter density 0.0 0.6 $\Omega_{\rm B}$ baryonic
matter density 0.02 0.12
For each of the $m=128$ parameter settings prescribed in the LHS, the
simulation produces a 55-vector of log power spectrum outputs, given by the
gray lines in the right hand frame of Figure 5. Of the 55 elements in the
simulation output vector, 22 of the elements are at the wavenumber $k$
corresponding to the physical observations. Concatenating the parameter
settings with the with the simulation output produces $m=128$ vectors of
length $5+55$
$\begin{pmatrix}\theta_{k}^{\circ}\\\
\eta(\theta^{\circ}_{k})\end{pmatrix},\,k=1,\ldots,m.$
We take $H$ to be the $22\times 60$ incidence matrix, selecting the elements
of the vector $(\theta^{\circ},\eta(\theta^{\circ}))$ that correspond to the
physical observations. This, along with the physical observations $y$ and
corresponding measurement covariance $\Sigma_{y}$ are the necessary inputs to
carry out the Gaussian and ensemble representations of the EnKF described in
Sections 1.1.2.
The estimates of the posterior distribution for the 5-dimensional parameter
vector is shown in the left frame of Figure 6. The presence of some slight
skewness is noticible in the estimate produced by the ensemble representation.
Also produced in these two estimation schemes is an estimate of the fitted log
power spectrum, along with uncertainties, given in the right frame of Figure
6.
Figure 6: The estimated posterior distribution for the model parameters
(left) and the log power (right). Left: One- and two-dimensional marginals for
the estimated posterior for the cosmological parameters. The upper triangle
shows the 128 updated ensemble members (light circle plotting symbols) and a
sample from the Gaussian representation (black dots). The lower triangle shows
estimated 90% hpd contours for both the ensemble (light lines) and Gaussian
representations (black lines). Right: posterior mean and pointwise 90%
credible bands for log power spectrum for the matter density of the universe.
Light lines give the estimate produced by the ensemble representation; black
lines give the estimate produced by the Gaussian representation.
These results, produced by the EnKF, can be compared to the posterior in
Higdon et al., (2010), which was produced using a multivariate GP emulator,
with a far more elaborate statistical formulation. The resulting posteriors
are similar, but both EnKF estimates seem to “chop off” tails in the posterior
for the cosmological parameters that are present in the GP emulator-based
analysis.
### 2.2 Optimal location of ice sheet measurements
This second application comes from an ongoing effort to use the community ice
sheet model (CISM) (Rutt et al., , 2009; Price et al., , 2011) along with
physical measurements to better understand ice sheet behavior and its impact
on climate. This study considers a model of an idealized ice sheet over a
rectangular region which is flowing out to sea on one side, while accumulating
ice from prescribed precipitation over a time of 1000 years. This
implementation of the CISM depends on two parameters – $\theta_{1}$ a constant
in the Glen-Nye flow law (Greve and Blatter, , 2009), controlling the
deformation of the ice sheet, and $\theta_{2}$ which controls the heat
conductivity in the ice sheet. A few time snapshots of the model output are
shown in Figure 7 for a particular choice of model parameters $\theta_{1}$ and
$\theta_{2}$.
Figure 7: Output from the idealized ice sheet model. The idealized ice sheet
is described by height over a 36m $\times$ 30m base, bounded on three sides by
ledges. The fourth side is open to the ocean. Over the span of 1000 years, the
ice flows into the ocean, while being replenished by a prescribed
precipitation. Of interest in this application is the thickness of the ice
sheet at 1000 years.
While this configuration does not realistically represent important ice sheets
in Greenland or Antarctica, it is a testbed where methodology can be evaluated
for model calibration and/or planing measurement campaigns. After 1000 years,
the thickness of the ice sheet could be measured to inform about the model
parameters $\theta_{1}$ and $\theta_{2}$. The goal of this application is to
use an ensemble of $m=20$ model runs at different
$\theta=(\theta_{1},\theta_{2})$ input settings to find a best set of 5 or 10
locations at which to measure the ice sheet thickness.
The parameter settings and resulting ice sheet thickness (after 1000 years)
for the $m=20$ model runs are shown in Figure 8. Thickness is produced on a
$36\times 30$ rectangular lattice of spatial locations. From this figure it’s
clear that the modeled ice sheet thickness is larger for smaller values of
$\theta_{1}$, and larger values of $\theta_{2}$.
Figure 8: Output from the ensemble of 20 model runs, showing thickness of the
ice sheet after 1000 years, plotted in the $(\theta_{1},\theta_{2})$-parameter
space. $\theta_{1}$ controls deformation of the ice sheet; $\theta_{2}$
controls heat conductivity in the ice sheet. Each image shows thickness as a
function of spatial location; the center of the image marks the
$(\theta_{1},\theta_{2})$ input setting at which the model was run. The
grayscale indicates log thickness.
The model runs produce an ensemble of $2+36\cdot 30=p$-vectors
$(\theta_{k},\eta(\theta_{k}))$, $k=1,\ldots,m$. We consider $n$ ice thickness
measurements taken at $n$ of the $36\cdot 30$ spatial grid locations given by
the model. A given set of $n$ measurement locations, is indexed by the
$n\times p$ incidence matrix $H$, which will contain a single 1 in each of its
$n$ rows. Thus there are $\binom{n}{p-2}$ possible measurement designs under
consideration, each determined by which of the last $p-2$ columns of $H$
contain a 1.
We use the Gaussian representation of the EnKF to describe the resulting
uncertainty in $\theta$, giving a simple means to compare designs which are
determined by $H$. Assuming the $n$ thickness measurements have independent
measurement errors, with a standard deviation of one meter, means $\Sigma_{y}$
is the $n\times n$ identity matrix. Then the resulting posterior variance for
the joint parameter-output vector is given by (5)
$\Sigma_{\rm post}^{-1}=\Sigma_{\rm pr}^{-1}+H^{\prime}\Sigma_{y}^{-1}H,$
with the upper $2\times 2$ submatrix of $\Sigma_{\rm post}$ describing the
posterior variance for the parameter vector $\theta$.
The sample covariance estimate for $\Sigma_{\rm pr}$, estimated from only
$m=20$ model runs, gives some spurious estimates for the elements of the
covariance matrix, leading to aberrant behavior in estimates for conditional
mean and variance for $\theta$ given $\eta$. If we define
$\mu_{\rm pr}=\begin{pmatrix}\mu_{\theta}\\\ \mu_{\eta}\end{pmatrix}\mbox{ and
}\Sigma_{\rm pr}=\begin{pmatrix}\Sigma_{\theta\theta}&\Sigma_{\theta\eta}\\\
\Sigma_{\eta\theta}&\Sigma_{\eta\eta}\end{pmatrix},$ (9)
corresponding to the 2-vector $\theta$ and the $p-2$-vector $\eta$, a spatial
tapering covariance matrix $R(r)$ can be used to help stabilize these
estimates (Kaufman et al., , 2008; Furrer and Bengtsson, , 2007) Hence we can
produce an improved estimate
$\Sigma_{\eta\eta}(r)=S\circ R(r)$
where $S$ denotes the sample covariance matrix from the samples
$\eta_{1},\ldots,\eta_{m}$, and $\circ$ denotes the elementwise product of the
matrix elements. Here we take $R(r)$ to be the spatial correlation matrix
induced by the isotropic exponential correlation function, with a correlation
distance of $r$. The value for $r$ is taken to be the maximizer of the
likelihood of prior samples $\eta_{1},\ldots,\eta_{m}$.
$L(r)\propto\prod_{k=1}^{m}|\Sigma_{\eta\eta}(r)|^{-\frac{1}{2}}\exp\left\\{-\frac{1}{2}\left(\eta_{k}-\mu_{\eta}\right)^{\prime}\Sigma_{\eta\eta}^{-1}\left(\eta_{k}-\mu_{\eta}\right)\right\\}$
Using this plug-in estimate for $r$, and treating $\Sigma_{\rm pr}$ as known,
the Bayesian $D$-optimal design that maximizes the prior-posterior gain
Shannon information is simply the $H$ that minimizes the determinant of the
$\Sigma_{\rm post}^{\theta}$ – the upper $2\times 2$ submatrix of $\Sigma_{\rm
post}$ (Chaloner and Verdinelli, , 1995).
Of course, since only a small number $n$ of observations are likely to be
taken, one need not compute using the full $p\times p$ matrix $\Sigma_{\rm
post}$. If we define $H_{\eta}$ to be the $n\times(p-2)$ restriction of $H$,
removing the first two columns of $H$, then the posterior covariance matrix
for $\theta$ can be written
$\Sigma^{\theta}_{\rm
post}=\Sigma_{\theta\theta}-\Sigma_{\theta\eta}H^{\prime}_{\eta}(H_{\eta}\Sigma_{\eta\eta}(r)H^{\prime}_{\eta}+\Sigma_{y})^{-1}H_{\eta}\Sigma_{\eta\theta}.$
Here the computations require only the solve of a relatively small $n\times n$
system.
Figure 9: Estimates of the Bayesian $D$-optimal designs for locations at
which to measure the ice sheet depth for $n=3$, 5, and 10. The estimates use a
plug-in estimate for the spatial covariance distance of the covariance taper
matrix, and Federov’s exchange algorithm to carry out the optimization.
We use the exchange algorithm of Fedorov, (1972) to search for the design
$H_{\eta}$ that approximately minimizes the determinant of
$\Sigma^{\theta}_{\rm post}$. The estimated optimal sampling locations for
depth measurements with $n=3$, 5, and 10, are shown in Figure 9. While the
optimization algorithm only guarantees a local maximum, we tried a large
number of restarts, with the configurations giving the minimal determinant of
$\Sigma_{\rm post}^{\theta}$ shown in Figure 9.
### 2.3 Calibration of parameters in the Community Atmosphere Model
This final application is an adaptation of the application described in
Jackson et al., (2008), in which multiple very fast simulated annealing
(MVFSA) was used to approximate the posterior distribution of climate model
parameters. Here we use the EnKF to carry out model calibration, considering a
more recent ensemble of 1,400 model runs, using the community atmosphere model
CAM 3.1, as described in Jackson et al., (2008). In this application 15 model
parameters are sampled uniformly over a 15-dimensional rectangle whose ranges
are apparent in Figure 11. The model parameters, output fields, and
corresponding physical observation fields are listed in Table 2.
Table 2: Climate model inputs, outputs and physical data
inputs outputs and physical data description description $\theta_{1}$
effective radius of liquid cloud droplets over sea ice $\eta_{1},y_{1}$
shortwave cloud forcing $\theta_{2}$ cloud particle number density over ocean
& land $\eta_{2},y_{2}$ precipitation over ocean $\theta_{3}$ effective radius
of liquid cloud droplets over land $\eta_{3},y_{3}$ two meter air temperature
$\theta_{4}$ time scale for consumption rate of deep CAPE $\eta_{4},y_{4}$
zonal winds at 300mb $\theta_{5}$ cloud particle number density over warm land
$\eta_{5},y_{5}$ vertically averaged relative humidity $\theta_{6}$ threshold
for autoconversion of warm ice $\eta_{6},y_{6}$ air temperature $\theta_{7}$
threshold for autoconversion of cold ice $\eta_{7},y_{7}$ latent heat flux
over ocean $\theta_{8}$ effective radius of liquid cloud droplets over ocean
$\theta_{9}$ environmental air entrainment rate $\theta_{10}$ initial cloud
downdraft mass flux $\theta_{11}$ low cloud relative humidity $\theta_{12}$
ice fall velocities $\theta_{13}$ low cloud relative humidity $\theta_{14}$
deep convection precipitation efficiency $\theta_{15}$ cloud particle number
density over sea ice
The computational model, described in detail in Jackson et al., (2008),
produces a large number of outputs that could be compared to physical
observations. We focus on a subset of the outputs (listed in Table 2) explored
in the original investigation. Each of these outputs is recorded as a field
over the globe, averaged over 11 years (from 1990 to 2001), separately for
each season (December – February, DJF; March – May, MAM; June – August, JJA;
September – November, SON) . The images in Figure 10 show the two-meter air
temperature observations.
Rather that work directly with the model output and observed fields, we
project these fields onto a precomputed empirical orthogonal function (EOF)
basis, producing a small vector of weights – one for each basis function – to
represent each field (von Storch and Zwiers, , 1999). As in Jackson et al.,
(2008), the EOF bases are computed from a long pilot run, separately from any
of the model runs used to make the ensemble. The resulting weights for the
two-meter air temperature are shown by the light/green dashes (model) and the
black dots (observation) in Figure 10. We use 5 EOF basis elements for each
output-season combination. Thus the model output $\eta$ and observation fields
$y$ are each summarized by a $7\times 4\times 5$ vector of weights,
corresponding to output, season, and EOF basis respectively.
The long pilot run is also used to estimate the variation in the outputs
expected just due to variation in climate. Thus for each output, season, and
EOF, a variance $\sigma^{2}_{\rm clim}$ is also estimated. We scale the EOF
bases so that each $\sigma^{2}_{\rm clim}$ is estimated to be 1. Thus, the
error bars in Figure 10 are $\pm 2$ because of this scaling. This scaling also
makes the actual values of the $y$-axis in the figure essentially meaningless.
Figure 10: Physical observations and uncertainty (black), prior simulations
(light/green), and posterior predictions (dark/blue) for seasonal 2-meter air
temperature, averaged from 1990 to 2001. The averages are computed for each
season (DJF = Dec, Jan, Feb, and so on). The observed and simulated
temperature fields are projected onto five EOF basis functions for each
season, producing five EOF weights for each field. The basis was estimated
using a single, long pilot run. The black dot shows EOF weights corresponding
to the physical observations, the solid black line shows a 2-$\sigma_{\rm
clim}$ bound for climate variation computed from the pilot run; the dashed
black lines show additional uncertainty due to the estimated discrepancy
error. The light/green lines are a sample of outputs from the ensemble of
model runs. The dark/blue lines give the corresponding ensemble representation
for the updated (i.e. posterior) model predictions. The scale of the $y$-axis
has been standardized so that the estimated climate variance is one for each
of the basis weights. The images above the plots show the physically observed
two-meter air temperature fields.
Even with this variation estimated from the pilot run, it is expected that
there will still be a discrepancy between the physical observations and the
model output, even at the best parameter setting $\theta$, for at least some
of the outputs. Hence we specify $\Sigma_{y}$ to be the sum of the variance
due to climate variation $I_{140}$ and a diagonal covariance matrix that
accounts for this additional discrepancy $\Sigma_{\delta}$. For each output
$i$ we allow a different precision $\lambda_{i}$ for the discrepancy that is
common across seasons and EOF bases. This gives
$\Sigma_{\delta}=\mbox{diag}\left(\lambda_{1}^{-1},\ldots,\lambda_{7}^{-1}\right)\otimes
I_{20}$
so that
$\Sigma_{y}=I_{140}+\Sigma_{\delta}.$
The black dotted lines in Figure 10 show this additional uncertainty due to
model discrepancy for the 2 meter air temperature, governed by $\lambda_{3}$.
We specify independent $\Gamma(a=1,b=.001)$ priors for each $\lambda_{i}$,
$i=1,\ldots,7$.
In order to estimate these precision parameters, we note that the full
140-dimensional observation vector $y$ is modeled as the sum of normal terms
$y=\eta+\epsilon_{\rm clim}+\epsilon_{\rm discrep}$
where $\eta\sim N(\mu_{\eta},\Sigma_{\eta\eta})$, with $\mu_{\eta}$ and
$\Sigma_{\eta\eta}$ estimated from the prior ensemble as defined in (9),
$\epsilon_{\rm clim}\sim N(0,I_{140})$, and $\epsilon_{\rm discrep}\sim
N(0,\Sigma_{\delta})$. If we define
$V(\lambda)=\Sigma_{\eta\eta}+I_{140}+\Sigma_{\delta},$
we get the posterior distribution for the 7-vector $\lambda$
$\displaystyle\pi(\lambda|y)$ $\displaystyle\propto$
$\displaystyle|V(\lambda)|^{-\frac{1}{2}}\exp\left\\{-\mbox{\small$\frac{1}{2}$}(y-\mu_{\eta})^{\prime}V(\lambda)^{-1}(y-\mu\eta)\right\\}$
$\displaystyle\times\prod_{i=1}^{7}\lambda_{i}^{a-1}e^{-b\lambda_{i}}.$
We use the posterior mean as plug-in estimates for $\lambda$, determining
$\Sigma_{y}$.
Now, given the dimension reduction from using the EOF bases estimated from the
pilot run, the estimate for $\Sigma_{y}$, and the 1400 member ensemble of
$15+140$-vectors $(\theta,\eta(\theta))$, and the ensemble-based estimates
$\mu_{\rm pr}$ and $\Sigma_{\rm pr}$, the updated posterior distribution for
$\eta$ and $\theta$ is computed using the ensemble representation. The
dark/blue dashes in Figure 10 show the posterior ensemble for the model
outputs in the EOF weight space for the two meter air temperature. Figure 11
shows the posterior ensemble of parameter values $\theta$. The prior ensemble
was sampled uniformly over the 15-dimensional rectangle depicted in the
figure.
Figure 11: Ensemble representation for the posterior distribution of the 15
model parameters after conditioning on 9 data fields. The parameter settings
for the initial ensemble are uniform over the 15-dimensional rectangle
depicted here.
While the formulation presented here is very similar to that of Jackson et
al., (2008), we used an additive discrepancy covariance matrix, with
different precisions for each output type; theirs used a $\Sigma_{y}$ that is
proportional to the estimated climate variation. Also, this analysis used
fewer types of physical observations. The resulting posterior distribution for
$\theta$ is similar, with a bit more posterior spread in this analysis. Also,
the EnKF analysis requires only an ensemble of model runs, with no need for
the sequential sampling required for MVFSA.
Finally, we point out that Annan et al., (2005) also use the EnKF to carry
out parameter estimation on a climate model. That example uses a multi-stage
estimation approach, collecting observations over ten successive years. That
paper also uses synthetic observations so that $\Sigma_{y}$ can be specified
without the need for estimation.
## 3 Discussion
This paper highlights a number of features of the EnKF from the perspective of
model calibration and shows examples of how it can be used in a variety of
applications. Implicitly, the EnKF uses a multiple linear regression emulator
to model the mapping between model parameters and outputs. This makes it easy
for this approach to handle large ensembles – often a challenge for approaches
that use GP-based emulators – as well as model outputs that are noisy or
random. This also suggests regression-based approaches for dealing with high-
dimensional input and output spaces may be helpful in EnKF applications.
While the EnKF nominally starts with an initial ensemble from the prior
distribution for $\theta$, it’s clear this prior will have little impact on
the final results if the physical observations are fairly constraining, as in
the examples presented here. The uniform designs used in the applications here
have little impact on the posterior results.
The results depend far more on specifications for covariance matrices, and how
a large covariance matrix is estimated from a relatively small ensemble of
model runs. We used likelihood and Bayesian approaches for estimation of
covariance parameters; a variety of alternative approaches exist in the
literature (Tippett et al., , 2003; Stroud and Bengtsson, , 2007; Evensen, ,
2009a; Stroud et al., , 2010; Kaufman et al., , 2008).
The resulting posterior distribution for $\theta$ tends to chop off tails that
would be present in a more exact formulation. This is clear from comparing the
analyses of the simple inverse problem laid out in Section 1.1. This is
largely due to the linearity of the regression based emulator implicitly used
in the EnKF. We have also seen this phenomena in the ice sheet and cosmology
applications when comparing to calibration analyses based on more exacting GP
emulators.
Finally we note that the ability of the EnKF to quickly provide “rough and
ready” results makes it ideal for more computationally demanding tasks such
experimental design or other optimization problems that require many
iterations of the estimation process. The ice sheet application of Section 2.2
is one such example.
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|
arxiv-papers
| 2012-04-16T16:03:23 |
2024-09-04T02:49:29.751920
|
{
"license": "Public Domain",
"authors": "Dave Higdon, Matt Pratola, James Gattiker, Earl Lawrence, Salman\n Habib, Katrin Heitmann, Steve Price, Charles Jackson, Michael Tobis",
"submitter": "Dave Higdon",
"url": "https://arxiv.org/abs/1204.3547"
}
|
1204.3731
|
# Towards Real-Time Summarization of Scheduled Events from Twitter Streams
Arkaitz Zubiaga
Damiano Spina
Queens College City University of New York New York, NY, USA
arkaitz.zubiaga@qc.cuny.edu NLP&IR Group ETSI Informática UNED Madrid, Spain
damiano@lsi.uned.es Enrique Amigó
Julio Gonzalo
NLP&IR Group ETSI Informática UNED Madrid, Spain enrique@lsi.uned.es NLP&IR
Group ETSI Informática UNED Madrid, Spain julio@lsi.uned.es
###### Abstract
This paper explores the real-time summarization of scheduled events such as
soccer games from torrential flows of Twitter streams. We propose and evaluate
an approach that substantially shrinks the stream of tweets in real-time, and
consists of two steps: (i) sub-event detection, which determines if something
new has occurred, and (ii) tweet selection, which picks a representative tweet
to describe each sub-event. We compare the summaries generated in three
languages for all the soccer games in _Copa America 2011_ to reference live
reports offered by Yahoo! Sports journalists. We show that simple text
analysis methods which do not involve external knowledge lead to summaries
that cover 84% of the sub-events on average, and 100% of key types of sub-
events (such as goals in soccer). Our approach should be straightforwardly
applicable to other kinds of scheduled events such as other sports, award
ceremonies, keynote talks, TV shows, etc.
###### category:
H.3.3 Information Storage and Retrieval Information Search and Retrieval
###### category:
H.1.2 Models and Principles User/Machine Systems
###### keywords:
Human information processing
###### keywords:
twitter, real-time, events, summarization
††titlenote: The present paper gives more technical and experimental details
about the work published as a poster at HT’2012 [8].††terms: Experimentation
## 1 Introduction
Twitter111http://twitter.com/ has gained widespread popularity as a
microblogging site where users share short messages (_tweets_). Twitter users
not only tweet about their personal issues or nearby events, but also about
news and events of interest to some community [5]. Twitter has become a
powerful tool to stay tuned to current affairs. It is known that, in
particular, Twitter users exhaustively share messages about (all kinds of)
events they are following live, occasionally giving rise to related trending
topics [9].
The community of users live _tweeting_ about a given event generates rich
contents describing sub-events that occur during an event (e.g., goals, red
cards or penalties in a soccer game). All those users share valuable
information providing live coverage of events [1]. However, this overwhelming
amount of information makes difficult for the user: (i) to follow the full
stream while finding out about new sub-events, and (ii) to retrieve from
Twitter the main, summarized information about which are the key things
happening at the event. In the context of exploring the potential of Twitter
as a means to follow an event, we address the (yet largely unexplored) task of
summarizing Twitter contents by providing the user with a summed up stream
that describes the key sub-events. We propose a two-step process for the real-
time summarization of events –sub-event detection and tweet selection–, and
analyze and evaluate different approaches for each of these two steps. We find
that Twitter provides an outstanding means for detailed tracking of events,
and present an approach that accurately summarizes streams to help the user
find out what is happening throughout an event. We perform experiments on
scheduled events, where the start time is known. By comparing different
summarization approaches, we find that learning from the information seen
before throughout the event is really helpful both to determine if a sub-event
occurred, and to select a tweet that represents it.
To the best of our knowledge, our work is the first to provide an approach to
generate real-time summaries of events from Twitter streams without making use
of external knowledge. Thus, our approach might be straightforwardly applied
to other kinds of scheduled events without requiring additional knowledge.
## 2 Dataset
We study the case of tweets sent during the games of a soccer competition.
Sports events are a good choice to explore for summarization purposes, because
they are usually reported live by journalists, providing a reference to
compare with. We set out to explore the _Copa America 2011_ championship,
which took place from July $1^{\text{st}}$ to $24^{\text{th}}$, 2011, in
Argentina, where 26 soccer games were played. Choosing an international
competition with a wide reach enables to gather and summarize tweets in
different languages. The official start times for the games were announced in
advance by the organization.
During the period of the _Copa America_ , we gathered all the tweets that
contained any of #ca2011, #copaamerica, and #copaamerica2011, which were set
to be the official Twitter hashtags for the competition. For the 24 days of
collection, we retrieved 1,425,858 unique tweets sent by 290,716 different
users. These tweets are written in 30 different languages, with a majority of
76.2% in Spanish, 7.8% in Portuguese, and 6.2% in English. The tweeting
activity of the games considerably varies, from 11k tweets for the least-
active game, to 74k for the most-active one, with an average of 32k tweets per
game.
In order to define a reference for evaluation, we collected the live reports
for all the games given by Yahoo! Sports222
http://uk.eurosport.yahoo.com/football/ copa-america/fixtures-results/. These
reports include the annotations of the most relevant sub-events throughout a
game. 7 types of annotations are included: goals (54 were found for the 26
games), penalties (2), red cards (12), disallowed goals (10), game starts
(26), ends (26), and stops and resumptions (63). On average, each game
comprises 7.42 annotations. Each of these annotations includes the minute when
it happened. We manually annotated the beginning of each game in the Twitter
streams, so that we could infer the timestamp of each annotation from those
minutes. The annotations do not provide specific times with seconds, and the
actual timestamp may vary slightly. We have considered these differences for
the evaluation process.
## 3 Real-Time Event Summarization
We define real-time event summarization as the task that provides new
information about an event every time a relevant sub-event occurs. To tackle
the summarization task, we define a two-step process that enables to report
information about new sub-events in different languages. The first step is to
identify at all times whether or not a specific sub-event occurred in the last
few seconds. The output will be a boolean value determining if something
relevant occurred; if so, the second step is to choose a representative tweet
that describes the sub-event in the language preferred by the user. The
aggregation of these two processes will in turn provide a set of tweets as a
summary of the game (see Figure 1).
Figure 1: Two-step process for real-time event summarization.
### 3.1 First Step: Sub-Event Detection
The first part of the event summarization system corresponds to the sub-event
detection. Note that, being a real-time sub-event detection, the system has to
determine at all times whether or not a relevant sub-event has occurred,
clueless of how the stream will continue to evolve. Before the beginning of an
event, the system is provided with the time that it starts, as scheduled in
advance, so the system knows when to start looking for new sub-events. With
the goal of developing a real-time sub-event detection method, we rely on the
fact that relevant sub-events trigger a massive tweeting activity of the
community. We assume that the more important a sub-event is, the more users
will tweet about it almost immediately. This is reflected as peaks in the
histogram of tweeting rates (see Figure 2 for an example of a game in our
dataset). In the process of detecting sub-events, we aim to compare 2
different ideas: (i) considering only sudden increase with respect to the
recent tweeting activity, and (ii) considering also all the previous activity
seen during a game, so that the system learns from the evolution of the
audience. We compare the following two methods that rely on these 2 ideas:
1. 1.
Increase: this approach was introduced by Zhao et al. [7]. It considers that
an important sub-event will be reflected as a sudden increase in the tweeting
rate. For time periods defined at 10, 20, 30 and 60 seconds, this method
checks if the tweeting rate increases by at least 1.7 from the previous time
frame for any of those periods. If the increase actually occurred, it is
considered that a sub-event occurred. A potential drawback of this method is
that not only outstanding tweeting rates would be reported as sub-events, but
also low rates that are preceded by even lower rates.
2. 2.
Outliers: we introduce an outlier-based approach that relies on whether the
tweeting rate for a given time frame stands out from the regular tweeting rate
seen so far during the event (not only from the previous time frame). We set
the time period at 60 seconds for this approach. 15 minutes before the game
starts, the system begins to learn from the tweeting rates, to find out what
is the approximate audience of the event. When the start time approaches, the
system begins with the sub-event detection process. The system considers that
a sub-event occurred when the tweeting rate represents an outlier as compared
to the activity seen before. Specifically, if the tweeting rate is above 90%
of all the previously seen tweeting rates, the current time frame will be
reported as a sub-event. This threshold has been set a priori and without
optimization. The outlier-based method incrementally learns while the game
advances, comparing the current tweeting rate to all the rates seen
previously. Different from the increase-based approach, our method presents
the advantages that it considers the specific audience of an event, and that
consecutive sub-events can also be detected if the tweeting rate remains
constant without increase. Accordingly, this method will not consider that a
sub-event occurred for low tweeting rates preceded by even lower rates, as
opposed to the increase-based approach.
Figure 2: Sample histogram of tweeting rates for a soccer game (Argentina vs
Uruguay), where several peaks can be seen.
Since the annotations on the reference are limited to minutes, we round down
the outputs of the systems to match the reference. Also, the timestamps
annotated for the reference are not entirely precise, and therefore we accept
as a correct guess an automatic sub-event detection that differs by at most
one minute from the reference.
This evaluation method enables us to compare the two systems to infer which of
them performs best. Table 1 shows the precision (P), recall (R) and F-measure
(F1) of the automatically detected sub-events with respect to the reference,
as well as the average number of sub-events detected per game (#). Our
outliers approach clearly outperforms the baseline, improving both precision
(75.8% improvement) and recall (3.7%) for an overall 40% gain in F1. At the
same time, the compression rate for the outliers approach almost doubles that
of the baseline (56.4%). From the average of 32k tweets sent per game, the
summarization to 25.6 tweets represents a drastic reduction to only 0.079% of
the total. Keeping the number of sub-events small while effectiveness improves
is important for a summarization system in order to provide a concise and
accurate summary. The outperformance of the outlier-based approach shows the
importance of taking into account the audience of a specific game, as well as
the helpfulness of learning from previous activity throughout a game.
| P | R | F1 | #
---|---|---|---|---
Increase | 0.29 | 0.81 | 0.41 | 45.4
Outliers | 0.51 | 0.84 | 0.63 | 25.6
Table 1: Evaluation of sub-event detection approaches.
### 3.2 Second Step: Tweet Selection
The second and final part of the summarization system is the tweet selection.
This second step is only activated when the first step reports that a new sub-
event occurred. Once the system has determined that a sub-event occurred, the
tweet selector is provided with the tweets corresponding to the minute of the
sub-event. From those tweets, the system has to choose one as a representative
tweet that describes what occurred. This tweet must provide the main
information about the sub-event, so the user understands what occurred and can
follow the event. Here we compare two tweet selection methods, one relying
only on information contained within the minute of the sub-event, and another
considering the knowledge acquired during the game. We test them on the output
of the outlier-based sub-event detection approach described above, as the
approach with best performance for the first step.
To select a representative tweet, we get a ranking of all the tweets. To do
so, we score each tweet with the sum of the values of the terms that it
contains. The more representative are the terms contained in a tweet, the more
representative will be the tweet itself. To define the values of the terms, we
compare two methods: (i) considering only the tweets within the sub-event (to
give highest values to terms that are used frequently within the sub-event),
and (ii) taking into account also the tweets sent before throughout the game,
so that the system can make a difference from what has been the common
vocabulary during the event (to give highest values to terms that are
especially used within the minute and not so frequently earlier during the
event). We use the following well-known approaches to implement these two
ideas:
1. 1.
TF: each term is given the value of its frequency as the number of occurrences
within the minute, regardless of its prior use.
2. 2.
KLD: we use the Kullback-Leibler divergence [4] (see Equation 1) to measure
how frequent is a term $t$ within the sub-event ($H$), but also considering
how frequent it has been during the game until the previous minute ($G$).
Thus, KLD will give a higher weight to terms frequent within the minute that
were less frequent during the game. This may allow to get rid of the common
vocabulary all along the game, and rather provide higher rates to specific
terms within the sub-event.
$D_{\mathrm{KL}}(H\|G)=H(t)\log\frac{H(t)}{G(t)}$ (1)
With these two approaches, the sum of values for terms contained in each tweet
results in a weight for each tweet. With weights given to all tweets, we
create a ranking of tweets sent during the sub-event, where the tweet with
highest weight ranks first. We create these rankings for each of the languages
we are working on. The tweet that maximizes this score for a given language is
returned as the candidate tweet to show in the summary in that language. The
two term weighting methods were applied to create summaries in three different
languages: Spanish, English, and Portuguese. We test them on the output of the
outlier-based sub-event detection approach described above, as the approach
with best performance for the first step. Thus, we got six summaries for each
game, i.e., TF and KLD-based summaries for the three languages. These six
summaries were manually evaluated by comparing them to the reference. Table 2
shows some tweets included in the KLD-based summary in English.
Sub-event | Selected Tweet | Narrator’s Comment
---|---|---
Game start | RT @user: Uruguay-Argentina. The Río de la Plata classic. The 4th vs the 5th in the last WC. History doesn’t matter. Argentina must win. #ca2011 | The referee gets the game under way
Goal | Gol! Gol! Gol! de Perez Uruguay 1 vs Argentina 0 Such a quick strike and Uruguay is already on top. #copaamerica | GOAL!! Forlan’s free kick is hit deep into the box and is flicked on by Caceres. Romero gets a hand on it but can only push it into the path of Perez who calmly strokes the ball into the net.
Goal | Gooooooooooooooooal Argentina ! Amazing pass from Messi, Great positioning & finish from Higuain !! Arg 1 - 1 Uru #CopaAmerica | GOAL!! Fantastic response from Argentina. Messi picks the ball up on the right wing and cuts in past Caceres. The Barca man clips a ball over the top of the defence towards Higuain who heads into the bottom corner.
Red card | Red card for Diego Pérez, his second yellow card, Uruguay is down to 10, I don’t know if I would have given it. #CopaAmérica2011 | You could see it coming. How stupid. Another needless free kick conceded by Perez and this time he is given his marching order. He purposely blocks off Gago. Uruguay have really got it all to do now.
Red card | #ca2011 Yellow for Mascherano! Double yellow! Adios! 10 vs 10! Mascherano surrenders his captain armband! | It’s ten against ten. Macherano comes across and fouls Suarez. He’s given his second yellow and his subsequent red.
Game stop (full time) | Batista didn’t look too happy at the game going to penalties as the TV cut to hit at FT, didn’t appear confident #CA2011 | The second half is brought to an end. We will have extra time.
Game end | Uruguay beats Argentina! 1-1 (5-4 penalty shoot out)! Uruguay now takes on Peru in Semis. #copaamerica | ARGENTINA 4-5 - URUGUAY WIN. Caceres buries the final penalty into the top right-hand corner.
Table 2: Example of some tweets selected by the (outliers+KLD) summarization
system, compared with the respective comments narrated on Yahoo! Sports.
In the manual evaluation process, each tweet in a system summary is classified
as correct if it can be associated to a sub-event in the reference and is
descriptive enough (note that there might be more than one correct tweet
associated to the same sub-event). Alternatively, tweets are classified as
novel (they contain relevant information for the summary which is not in the
reference) or noisy. From these annotations, we computed the following values
for analysis and evaluation: (i) recall, given by the ratio of sub-events in
the reference which are covered by a correct tweet in the summary; and (ii)
precision, given by the ratio of correct + novel tweets from a whole summary
(note that redundancy is not penalized by any of these measures).
| es | en | pt
---|---|---|---
Goals (54) | TF | 0.98 | 0.98 | 0.98
KLD | 1.00 | 1.00 | 1.00
Penalties (2) | TF | 1.00 | 0.50 | 1.00
KLD | 1.00 | 0.50 | 1.00
Red cards (12) | TF | 0.75 | 0.75 | 0.83
KLD | 0.92 | 0.92 | 1.00
Disallowed | TF | 0.40 | 0.50 | 0.40
goals (10) | KLD | 0.40 | 0.50 | 0.30
Game starts (26) | TF | 0.73 | 0.74 | 0.79
KLD | 0.84 | 0.79 | 0.83
Game ends (26) | TF | 1.00 | 1.00 | 1.00
KLD | 1.00 | 1.00 | 1.00
Game stops & | TF | 0.62 | 0.60 | 0.57
resumptions (63) | KLD | 0.68 | 0.60 | 0.59
Overall | TF | 0.79 | 0.74 | 0.78
KLD | 0.84 | 0.77 | 0.82
Table 3: Recall of reported sub-events for summaries in Spanish (es), English
(en), and Portuguese (pt).
Table 3 shows recall values as the coverage of the two approaches over each
type of sub-event, as well as the macro-averaged overall values. These results
corroborate that simple state-of-the-art approaches like TF and KLD score
outstanding recall values. Nevertheless, KLD shows to be slightly superior
than TF for recall. Regarding the averages of all kinds of sub-events, recall
values are near or above 80% for all the languages. It can also be seen that
some sub-events are much easier to detect than others. It is important that
summaries do not miss the fundamental sub-events. For instance, all the
summaries successfully reported all the goals and all the game ends, which are
probably the most emotional moments, when users extremely coincide sharing.
However, other sub-events like game stops and resumptions, or disallowed
goals, were sometimes missed by the summaries, with recall values near 50%.
This shows that some of these sub-events may not be that shocking sometimes,
depending on the game, so fewer users share about them, and therefore are
harder to find by the summarization system. For instance, one could expect
that users would not express high emotion when a boring game with no goals
stops for half time. Likewise, this shows that those sub-events are less
relevant for the community. In fact, from these summaries, users would
perfectly know when a goal is scored, when it finished, and what is the final
result.
| es | en | pt
---|---|---|---
TF | 0.79 | 0.74 | 0.79
KLD | 0.84 | 0.79 | 0.83
Table 4: Precision of summaries in Spanish (es), English (en), and Portuguese
(pt).
Table 4 shows precision values as the ratio of useful tweets for the three
summaries generated in Spanish, English and Portuguese. The results show that
a simple TF approach is relatively good for the selection of a representative
tweet, with precision values above 70% for all three languages. As for recall
values, KLD does better than TF, with precision values near or above 80%. This
shows that taking advantage of the differences between the current sub-event
and tweets shared before considerably helps in the tweet selection. Note also
that English summaries reach 0.79 precision even if the tweet stream is, in
that case, an order of magnitude smaller than their Spanish counterpart,
suggesting that the method works well at very different tweeting rates.
## 4 Related Work
Automatic summarization of events from tweets is still in its infancy as a
research field. Some have tackled the task in an offline mode, after the
events were finished. For instance, Hannon et al. [3] present an approach for
the automatic generation of video highlights for soccer games after they
finished. They set a fixed number of sub-events that want to be included in
the highlights, and select that many video fragments with the highest tweeting
activity. Others, such as Petrović et al. [6], have shown the potential of
Twitter for the detection and discovery of events from tweets. While some have
studied events after they happened, there is very little research dealing with
the real-time study of events to provide near-immediate information. Zhao et
al. [7] detect sub-events occurred during NFL games, using an approach based
on the increase of the tweeting activity. We set this approach as the baseline
in our sub-event detection process. Afterward, they apply a specific lexicon
provided as input to identify the type of sub-event. Different from this, our
approach aims to be independent of the event, providing a summarized stream
instead of categorizing sub-events. Chakrabarti and Punera [2] were the first
to present an approach –which is based on Hidden Markov Models– for
constructing real-time summaries of events from tweets. However, their
approach requires prior knowledge of similar events, and so it is not easily
applicable to previously unseen types of events.
## 5 Conclusions
We have presented a two-step summarization approach that, without making use
of external knowledge, identifies relevant sub-events in soccer games and
selects a representative tweet for each of them. Using simple text analysis
methods such as KLD, our system generates real-time summaries with precision
and recall values above 80% when compared to manually built reports. The fact
that users tweet at the same time, with overlapping vocabulary, helps not only
detecting that a sub-event occurs, but also selecting a representative tweet
to describe it. Our study also shows that considering all previous information
seen during the event is really helpful to this end, yielding superior results
than taking into account just the most recent activity. The activity for the
soccer games studied in this work varies from 11k to 74k tweets sent, showing
that regardless of the audience tweeting about an event, our method
effectively reports the key sub-events occurred during a game. Finally, all of
the most relevant types of sub-events, such as goals and game ends, are
reported almost perfectly.
Note that our method does not rely on any external knowledge about soccer
events (except for the schedule time to begin), so it can be straightforwardly
applied to other kinds of events. As future work, we intend to evaluate the
performance of the method on other kinds of scheduled events such as award
ceremonies, keynote talks, other types of sport events, product presentations,
TV shows, etc.
## 6 Acknowledgments
This work has been part-funded by the Education Council of the Regional
Government of Madrid, MA2VICMR (S-2009/TIC-1542), the Innovation project
Holopedia (TIN2010-21128-C02-01), the European Community’s Seventh Framework
Programme (FP7/ 2007-2013) under grant agreement nr. 288024 (LiMoSINe project)
and the Spanish Ministry of Education for a doctoral grant (AP2009-0507).
## References
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* [2] D. Chakrabarti and K. Punera. Event summarization using tweets. In Proceedings of the fifth International AAAI Conference on Weblogs and Social Media (ICWSM ’11), pages 66–73. AAAI, 2011.
* [3] J. Hannon, K. McCarthy, J. Lynch, and B. Smyth. Personalized and automatic social summarization of events in video. In Proceedings of the 16th international conference on Intelligent user interfaces (IUI ’11), pages 335–338. ACM, 2011.
* [4] S. Kullback and R. Leibler. On information and sufficiency. The Annals of Mathematical Statistics, 22(1):79–86, 1951.
* [5] E. Mishaud. Twitter: Expressions of the whole self. Master’s thesis, Department of Media and Communications, University of London, 2007.
* [6] S. Petrović, M. Osborne, and V. Lavrenko. Streaming first story detection with application to twitter. In Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics (HLT-NAACL ’10), pages 181–189. ACL, 2010.
* [7] S. Zhao, L. Zhong, J. Wickramasuriya, and V. Vasudevan. Human as real-time sensors of social and physical events: A case study of twitter and sports games. Arxiv preprint arXiv:1106.4300, 2011.
* [8] A. Zubiaga, D. Spina, E. Amigó, and J. Gonzalo. Towards real-time summarization of scheduled events from twitter streams. In Proceedings of the 23nd ACM conference on Hypertext and Social Media (HT ’12), HT ’12. ACM, ACM, 2012.
* [9] A. Zubiaga, D. Spina, V. Fresno, and R. Martínez. Classifying trending topics: A typology of conversation triggers on twitter. In Proceedings of the 20th ACM international conference on Information and knowledge management (CIKM ’11), pages 2461–2464, 2011.
|
arxiv-papers
| 2012-04-17T08:58:39 |
2024-09-04T02:49:29.763220
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arkaitz Zubiaga, Damiano Spina, Enrique Amig\\'o and Julio Gonzalo",
"submitter": "Damiano Spina",
"url": "https://arxiv.org/abs/1204.3731"
}
|
1204.3933
|
SLAC-PUB-14957
BABAR-PUB-12/002
The BABAR Collaboration
# Measurement of Branching Fractions and Rate Asymmetries in the Rare Decays
$B\rightarrow K^{(*)}\ell^{+}\ell^{-}$
J. P. Lees V. Poireau V. Tisserand Laboratoire d’Annecy-le-Vieux de
Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3, F-74941
Annecy-Le-Vieux, France J. Garra Tico E. Grauges Universitat de Barcelona,
Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain A. Palanoab
INFN Sezione di Baria; Dipartimento di Fisica, Università di Barib, I-70126
Bari, Italy G. Eigen B. Stugu University of Bergen, Institute of Physics,
N-5007 Bergen, Norway D. N. Brown L. T. Kerth Yu. G. Kolomensky G. Lynch
Lawrence Berkeley National Laboratory and University of California, Berkeley,
California 94720, USA H. Koch T. Schroeder Ruhr Universität Bochum,
Institut für Experimentalphysik 1, D-44780 Bochum, Germany D. J. Asgeirsson
C. Hearty T. S. Mattison J. A. McKenna University of British Columbia,
Vancouver, British Columbia, Canada V6T 1Z1 A. Khan Brunel University,
Uxbridge, Middlesex UB8 3PH, United Kingdom V. E. Blinov A. R. Buzykaev V.
P. Druzhinin V. B. Golubev E. A. Kravchenko A. P. Onuchin S. I.
Serednyakov Yu. I. Skovpen E. P. Solodov K. Yu. Todyshev A. N. Yushkov
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia M. Bondioli
D. Kirkby A. J. Lankford M. Mandelkern University of California at Irvine,
Irvine, California 92697, USA H. Atmacan J. W. Gary F. Liu O. Long G. M.
Vitug University of California at Riverside, Riverside, California 92521, USA
C. Campagnari T. M. Hong D. Kovalskyi J. D. Richman C. A. West University
of California at Santa Barbara, Santa Barbara, California 93106, USA A. M.
Eisner J. Kroseberg W. S. Lockman A. J. Martinez B. A. Schumm A. Seiden
University of California at Santa Cruz, Institute for Particle Physics, Santa
Cruz, California 95064, USA D. S. Chao C. H. Cheng B. Echenard K. T. Flood
D. G. Hitlin P. Ongmongkolkul F. C. Porter A. Y. Rakitin California
Institute of Technology, Pasadena, California 91125, USA R. Andreassen Z.
Huard B. T. Meadows M. D. Sokoloff L. Sun University of Cincinnati,
Cincinnati, Ohio 45221, USA P. C. Bloom W. T. Ford A. Gaz U. Nauenberg J.
G. Smith S. R. Wagner University of Colorado, Boulder, Colorado 80309, USA
R. Ayad Now at the University of Tabuk, Tabuk 71491, Saudi Arabia W. H. Toki
Colorado State University, Fort Collins, Colorado 80523, USA B. Spaan
Technische Universität Dortmund, Fakultät Physik, D-44221 Dortmund, Germany
K. R. Schubert R. Schwierz Technische Universität Dresden, Institut für
Kern- und Teilchenphysik, D-01062 Dresden, Germany D. Bernard M. Verderi
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128
Palaiseau, France P. J. Clark S. Playfer University of Edinburgh, Edinburgh
EH9 3JZ, United Kingdom D. Bettonia C. Bozzia R. Calabreseab G. Cibinettoab
E. Fioravantiab I. Garziaab E. Luppiab M. Muneratoab M. Negriniab L.
Piemontesea V. Santoroa INFN Sezione di Ferraraa; Dipartimento di Fisica,
Università di Ferrarab, I-44100 Ferrara, Italy R. Baldini-Ferroli A.
Calcaterra R. de Sangro G. Finocchiaro P. Patteri I. M. Peruzzi Also with
Università di Perugia, Dipartimento di Fisica, Perugia, Italy M. Piccolo M.
Rama A. Zallo INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
R. Contriab E. Guidoab M. Lo Vetereab M. R. Mongeab S. Passaggioa C.
Patrignaniab E. Robuttia INFN Sezione di Genovaa; Dipartimento di Fisica,
Università di Genovab, I-16146 Genova, Italy B. Bhuyan V. Prasad Indian
Institute of Technology Guwahati, Guwahati, Assam, 781 039, India C. L. Lee
M. Morii Harvard University, Cambridge, Massachusetts 02138, USA A. J.
Edwards Harvey Mudd College, Claremont, California 91711 A. Adametz U. Uwer
Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120
Heidelberg, Germany H. M. Lacker T. Lueck Humboldt-Universität zu Berlin,
Institut für Physik, Newtonstr. 15, D-12489 Berlin, Germany P. D. Dauncey
Imperial College London, London, SW7 2AZ, United Kingdom P. K. Behera U.
Mallik University of Iowa, Iowa City, Iowa 52242, USA C. Chen J. Cochran
W. T. Meyer S. Prell A. E. Rubin Iowa State University, Ames, Iowa
50011-3160, USA A. V. Gritsan Z. J. Guo Johns Hopkins University,
Baltimore, Maryland 21218, USA N. Arnaud M. Davier D. Derkach G.
Grosdidier F. Le Diberder A. M. Lutz B. Malaescu P. Roudeau M. H. Schune
A. Stocchi G. Wormser Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et
Université Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898 Orsay
Cedex, France D. J. Lange D. M. Wright Lawrence Livermore National
Laboratory, Livermore, California 94550, USA C. A. Chavez J. P. Coleman J.
R. Fry E. Gabathuler D. E. Hutchcroft D. J. Payne C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom A. J. Bevan F. Di
Lodovico R. Sacco M. Sigamani Queen Mary, University of London, London, E1
4NS, United Kingdom G. Cowan University of London, Royal Holloway and
Bedford New College, Egham, Surrey TW20 0EX, United Kingdom D. N. Brown C.
L. Davis University of Louisville, Louisville, Kentucky 40292, USA A. G.
Denig M. Fritsch W. Gradl K. Griessinger A. Hafner E. Prencipe Johannes
Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
R. J. Barlow Now at the University of Huddersfield, Huddersfield HD1 3DH, UK
G. Jackson G. D. Lafferty University of Manchester, Manchester M13 9PL,
United Kingdom E. Behn R. Cenci B. Hamilton A. Jawahery D. A. Roberts
University of Maryland, College Park, Maryland 20742, USA C. Dallapiccola
University of Massachusetts, Amherst, Massachusetts 01003, USA R. Cowan D.
Dujmic G. Sciolla Massachusetts Institute of Technology, Laboratory for
Nuclear Science, Cambridge, Massachusetts 02139, USA R. Cheaib D. Lindemann
P. M. Patel S. H. Robertson McGill University, Montréal, Québec, Canada H3A
2T8 P. Biassoniab N. Neria F. Palomboab S. Strackaab INFN Sezione di Milanoa;
Dipartimento di Fisica, Università di Milanob, I-20133 Milano, Italy L.
Cremaldi R. Godang Now at University of South Alabama, Mobile, Alabama
36688, USA R. Kroeger P. Sonnek D. J. Summers University of Mississippi,
University, Mississippi 38677, USA X. Nguyen M. Simard P. Taras Université
de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 G. De
Nardoab D. Monorchioab G. Onoratoab C. Sciaccaab INFN Sezione di Napolia;
Dipartimento di Scienze Fisiche, Università di Napoli Federico IIb, I-80126
Napoli, Italy M. Martinelli G. Raven NIKHEF, National Institute for Nuclear
Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands C. P.
Jessop J. M. LoSecco W. F. Wang University of Notre Dame, Notre Dame,
Indiana 46556, USA K. Honscheid R. Kass Ohio State University, Columbus,
Ohio 43210, USA J. Brau R. Frey N. B. Sinev D. Strom E. Torrence
University of Oregon, Eugene, Oregon 97403, USA E. Feltresiab N. Gagliardiab
M. Margoniab M. Morandina M. Posoccoa M. Rotondoa G. Simia F. Simonettoab R.
Stroiliab INFN Sezione di Padovaa; Dipartimento di Fisica, Università di
Padovab, I-35131 Padova, Italy S. Akar E. Ben-Haim M. Bomben G. R.
Bonneaud H. Briand G. Calderini J. Chauveau O. Hamon Ph. Leruste G.
Marchiori J. Ocariz S. Sitt Laboratoire de Physique Nucléaire et de Hautes
Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université
Denis Diderot-Paris7, F-75252 Paris, France M. Biasiniab E. Manoniab S.
Pacettiab A. Rossiab INFN Sezione di Perugiaa; Dipartimento di Fisica,
Università di Perugiab, I-06100 Perugia, Italy C. Angeliniab G. Batignaniab
S. Bettariniab M. Carpinelliab Also with Università di Sassari, Sassari, Italy
G. Casarosaab A. Cervelliab F. Fortiab M. A. Giorgiab A. Lusianiac B.
Oberhofab E. Paoloniab A. Pereza G. Rizzoab J. J. Walsha INFN Sezione di
Pisaa; Dipartimento di Fisica, Università di Pisab; Scuola Normale Superiore
di Pisac, I-56127 Pisa, Italy D. Lopes Pegna J. Olsen A. J. S. Smith A. V.
Telnov Princeton University, Princeton, New Jersey 08544, USA F. Anullia R.
Facciniab F. Ferrarottoa F. Ferroniab M. Gasperoab L. Li Gioia M. A. Mazzonia
G. Pireddaa INFN Sezione di Romaa; Dipartimento di Fisica, Università di Roma
La Sapienzab, I-00185 Roma, Italy C. Bünger O. Grünberg T. Hartmann T.
Leddig H. Schröder C. Voss R. Waldi Universität Rostock, D-18051 Rostock,
Germany T. Adye E. O. Olaiya F. F. Wilson Rutherford Appleton Laboratory,
Chilton, Didcot, Oxon, OX11 0QX, United Kingdom S. Emery G. Hamel de
Monchenault G. Vasseur Ch. Yèche CEA, Irfu, SPP, Centre de Saclay, F-91191
Gif-sur-Yvette, France D. Aston D. J. Bard R. Bartoldus J. F. Benitez C.
Cartaro M. R. Convery J. Dorfan G. P. Dubois-Felsmann W. Dunwoodie M.
Ebert R. C. Field M. Franco Sevilla B. G. Fulsom A. M. Gabareen M. T.
Graham P. Grenier C. Hast W. R. Innes M. H. Kelsey P. Kim M. L. Kocian
D. W. G. S. Leith P. Lewis B. Lindquist S. Luitz V. Luth H. L. Lynch D.
B. MacFarlane D. R. Muller H. Neal S. Nelson M. Perl T. Pulliam B. N.
Ratcliff A. Roodman A. A. Salnikov R. H. Schindler A. Snyder D. Su M. K.
Sullivan J. Va’vra A. P. Wagner W. J. Wisniewski M. Wittgen D. H. Wright
H. W. Wulsin C. C. Young V. Ziegler SLAC National Accelerator Laboratory,
Stanford, California 94309 USA W. Park M. V. Purohit R. M. White J. R.
Wilson University of South Carolina, Columbia, South Carolina 29208, USA A.
Randle-Conde S. J. Sekula Southern Methodist University, Dallas, Texas
75275, USA M. Bellis P. R. Burchat T. S. Miyashita Stanford University,
Stanford, California 94305-4060, USA M. S. Alam J. A. Ernst State
University of New York, Albany, New York 12222, USA R. Gorodeisky N. Guttman
D. R. Peimer A. Soffer Tel Aviv University, School of Physics and Astronomy,
Tel Aviv, 69978, Israel P. Lund S. M. Spanier University of Tennessee,
Knoxville, Tennessee 37996, USA J. L. Ritchie A. M. Ruland R. F. Schwitters
B. C. Wray University of Texas at Austin, Austin, Texas 78712, USA J. M.
Izen X. C. Lou University of Texas at Dallas, Richardson, Texas 75083, USA
F. Bianchiab D. Gambaab INFN Sezione di Torinoa; Dipartimento di Fisica
Sperimentale, Università di Torinob, I-10125 Torino, Italy L. Lanceriab L.
Vitaleab INFN Sezione di Triestea; Dipartimento di Fisica, Università di
Triesteb, I-34127 Trieste, Italy F. Martinez-Vidal A. Oyanguren IFIC,
Universitat de Valencia-CSIC, E-46071 Valencia, Spain H. Ahmed J. Albert
Sw. Banerjee F. U. Bernlochner H. H. F. Choi G. J. King R. Kowalewski M.
J. Lewczuk I. M. Nugent J. M. Roney R. J. Sobie N. Tasneem University of
Victoria, Victoria, British Columbia, Canada V8W 3P6 T. J. Gershon P. F.
Harrison T. E. Latham E. M. T. Puccio Department of Physics, University of
Warwick, Coventry CV4 7AL, United Kingdom H. R. Band S. Dasu Y. Pan R.
Prepost S. L. Wu University of Wisconsin, Madison, Wisconsin 53706, USA
###### Abstract
In a sample of 471 million $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$
events collected with the BABAR detector at the PEP-II $e^{+}e^{-}$ collider
we study the rare decays $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$, where
$\ell^{+}\ell^{-}$ is either $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$. We report
results on partial branching fractions and isospin asymmetries in seven bins
of di-lepton mass-squared. We further present $C\\!P$ and lepton-flavor
asymmetries for di-lepton masses below and above the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance. We find no evidence
for $C\\!P$ or lepton-flavor violation. The partial branching fractions and
isospin asymmetries are consistent with the Standard Model predictions and
with results from other experiments.
###### pacs:
13.20.He
## I Introduction
The decays $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ arise from flavor-changing
neutral-current processes that are forbidden at tree level in the Standard
Model (SM). The lowest-order SM processes contributing to these decays are the
photon penguin, the $Z$ penguin and the $W^{+}W^{-}$ box diagrams shown in
Fig. 1. Their amplitudes are expressed in terms of hadronic form factors and
perturbatively-calculable effective Wilson coefficients, $C^{\rm eff}_{7}$,
$C^{\rm eff}_{9}$ and $C^{\rm eff}_{10}$, which represent the electromagnetic
penguin diagram, and the vector part and the axial-vector part of the linear
combination of the $Z$ penguin and $W^{+}W^{-}$ box diagrams, respectively
Buchalla . In next-to-next-to-leading order (NNLO) at a renormalization scale
$\mu=4.8$ $\mathrm{\,Ge\kern-1.00006ptV}$, the effective Wilson coefficients
are $C^{\rm eff}_{7}=-0.304$, $C^{\rm eff}_{9}=4.211$, and $C^{\rm
eff}_{10}=-4.103$ Altmannshofer:2008dz .
Non-SM physics may add new penguin and box diagrams, which can contribute at
the same order as the SM diagrams NewPhysics ; isospin ; Ali:2002jg . Examples
of new physics loop processes are depicted in Fig. 2. These contributions
might modify the Wilson coefficients from their SM expectations Ali:2002jg ;
Zhong:2002nu ; Ali01 ; Khodjamirian:2010vf . In addition, new contributions
from scalar, pseudoscalar, and tensor currents may arise that can modify, in
particular, the lepton-flavor ratios Alok:2010zd ; Yan:2000dc .
Figure 1: Lowest-order Feynman diagrams for $b\rightarrow s\ell^{+}\ell^{-}$.
Figure 2: Examples of new physics loop contributions to $b\rightarrow
s\ell^{+}\ell^{-}$: (a) charged Higgs ($H^{-}$); (b) squark
($\tilde{t},\tilde{c},\tilde{u}$) and chargino ($\chi^{-}$); (c) squark
($\tilde{b},\tilde{s},\tilde{d}$) and gluino ($\tilde{g}$)/neutralino
($\chi^{0}$).
## II Observables
Table 1: The definition of seven $s$ bins used in the analysis. Here $m_{B}$ and $m_{K^{(*)}}$ are the invariant masses of $B$ and $K^{(*)}$, respectively. The low $s$ region is given by $0.10<s<8.12$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, while the high $s$ region is given by $s>10.11$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. | $s$ bin | $s$ min | $s$ max
---|---|---|---
| | (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$)
Low | $s_{1}$ | 0.10 | 2.00
| $s_{2}$ | 2.00 | 4.30
| $s_{3}$ | 4.30 | 8.12
High | $s_{4}$ | 10.11 | 12.89
| $s_{5}$ | 14.21 | 16.00
| $s_{6}$ | 16.00 | $(m_{B}-m_{K^{(*)}})^{2}$
| $s_{0}$ | 1.00 | 6.00
We report herein results on exclusive partial branching fractions and isospin
asymmetries in six bins of $s\equiv m^{2}_{\ell\ell}$, defined in Table 1. We
further present results in the $s$ bin $s_{0}=1.0-6.0~{}{\rm GeV}^{2}/c^{4}$
chosen for calculations inspired by soft-collinear effective theory (SCET)
scet . In addition, we report on direct $C\\!P$ asymmetries and the ratio of
rates to dimuon and dielectron final states in the low $s$ and high $s$
regions separated by the $J/\psi$ resonance. We remove regions of the long-
distance contributions around the $J/\psi$ and $\psi{(2S)}$ resonances. New
BABAR results on angular observables using the same dataset and similar event
selection will be reported shortly.
The $B\rightarrow K\ell^{+}\ell^{-}$ and $B\rightarrow K^{*}\ell^{+}\ell^{-}$
total branching fractions are predicted to be $(0.35\pm 0.12)\times 10^{-6}$
and $(1.19\pm 0.39)\times 10^{-6}$ (for $s>0.1~{}{\rm GeV}^{2}/c^{4}$),
respectively Ali:2002jg . The $\sim 30\%$ uncertainties are due to a lack of
knowledge about the form factors that model the hadronic effects in the
$B\rightarrow K$ and $B\rightarrow K^{*}$ transitions. Thus, measurements of
decay rates to exclusive final states are less suited to searches for new
physics than rate asymmetries, where many theory uncertainties cancel.
For charged $B$ decays and neutral $B$ decays flavor-tagged through
$K^{*}\rightarrow K^{+}\pi^{-}$ chargeconj , the direct $C\\!P$ asymmetry is
defined as
$\displaystyle{\cal A}_{C\\!P}^{K^{(*)}}\equiv\frac{{\cal
B}(\overline{B}\rightarrow\overline{K}^{(*)}\ell^{+}\ell^{-})-{\cal
B}(B\rightarrow K^{(*)}\ell^{+}\ell^{-})}{{\cal
B}(\overline{B}\rightarrow\overline{K}^{(*)}\ell^{+}\ell^{-})+{\cal
B}(B\rightarrow K^{(*)}\ell^{+}\ell^{-})}\,,$ (1)
and is expected to be ${\cal O}(10^{-3})$ in the SM. However ${\cal
A}_{C\\!P}^{K^{(*)}}$ may receive a significant enhancement from new physics
contributions at the electro-weak scale CPnp .
For $s>0.1$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, the ratio of rates
to dimuon and dielectron final states is defined as
$\displaystyle{\cal R}_{K^{(*)}}\equiv\frac{{\cal B}(B\rightarrow
K^{(*)}\mu^{+}\mu^{-})}{{\cal B}(B\rightarrow K^{(*)}e^{+}e^{-})}\,.$ (2)
In the SM, ${\cal R}_{K^{(*)}}$ is expected to be unity to within a few
percent Hiller:2003js for dilepton invariant masses above the dimuon
kinematic threshold. In two-Higgs-doublet models, including supersymmetry,
these ratios are sensitive to the presence of a neutral Higgs boson. When the
ratio of neutral Higgs field vacuum expectation values $\tan\beta$ is large,
${\cal R}_{K^{(*)}}$ might be increased by up to 10% Yan:2000dc .
The $C\\!P$-averaged isospin asymmetry is defined as
${\cal A}^{K^{(*)}}_{I}\equiv\frac{{\cal B}(B^{0}\rightarrow
K^{(*)0}\ell^{+}\ell^{-})-r_{\tau}{\cal B}(B^{+}\rightarrow
K^{(*)+}\ell^{+}\ell^{-})}{{\cal B}(B^{0}\rightarrow
K^{(*)0}\ell^{+}\ell^{-})+r_{\tau}{\cal B}(B^{+}\rightarrow
K^{(*)+}\ell^{+}\ell^{-})}$,
(3)
where $r_{\tau}\equiv\tau_{B^{0}}/\tau_{B^{+}}=1/(1.071\pm 0.009)$ is the
ratio of $B^{0}$ and $B^{+}$ lifetimes PDG . ${\cal A}^{K^{*}}_{I}$ has a SM
expectation of $+6$% to $+13$% as $s\rightarrow 0$ isospin . This is
consistent with the measured asymmetry $3\pm 3\%$ in $B\rightarrow
K^{*}\gamma$ kstg . A calculation of the predicted $K^{*+}$ and $K^{*0}$ rates
integrated over the low $s$ region yields ${{\cal A}}^{K^{*}}_{I}=-0.005\pm
0.020$ Beneke ; Feldmann:ckm2008 . In the high $s$ region, we may expect
contributions from charmonium states as an additional source of isospin
asymmetry. However the measured asymmetries in the $J/\psi K^{(*)}$ and
$\psi{(2S)}K^{(*)}$ modes are all below 5% PDG .
## III BABAR Experiment and Data Sample
We use a data sample of 471 million $B\bar{B}$ pairs collected at the
$\mathchar 28935\relax(4S)$ resonance with the BABAR detector BaBarDetector
at the PEP-II asymmetric-energy $e^{+}e^{-}$ collider at the SLAC National
Accelerator Laboratory. Charged particle tracking is provided by a five-layer
silicon vertex tracker and a 40-layer drift chamber in a 1.5 T solenoidal
magnetic field. We identify electrons with a CsI(Tl) electromagnetic
calorimeter, and muons using an instrumented magnetic flux return. Electron
and muon candidates are required to have momenta
$p>0.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the laboratory frame. We
combine up to three photons with electrons when they are consistent with
bremsstrahlung, and do not use electrons that are associated with photon
conversions to low-mass $e^{+}e^{-}$ pairs. We identify charged kaons using a
detector of internally reflected Cherenkov light, as well as
$\mathrm{d}E/\mathrm{d}x$ information from the drift chamber. Charged tracks
other than identified $e$, $\mu$ and $K$ candidates are treated as pions.
Neutral $K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ candidates are
required to have an invariant mass consistent with the nominal $K^{0}$ mass,
and a flight distance from the $e^{+}e^{-}$ interaction point that is more
than three times its uncertainty.
## IV Event Selection
We reconstruct $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ signal events in the
following eight final states:
* •
$B^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\mu^{+}\mu^{-}$,
* •
$B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$,
* •
$B^{0}\rightarrow K^{0}_{\scriptscriptstyle S}e^{+}e^{-}$,
* •
$B^{+}\rightarrow K^{+}e^{+}e^{-}$,
* •
$B^{+}\rightarrow K^{*+}(\rightarrow K^{0}_{\scriptscriptstyle
S}\pi^{+})\mu^{+}\mu^{-}$,
* •
$B^{0}\rightarrow K^{*0}(\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$,
* •
$B^{+}\rightarrow K^{*+}(\rightarrow K^{0}_{\scriptscriptstyle
S}\pi^{+})e^{+}e^{-}$,
* •
$B^{0}\rightarrow K^{*0}(\rightarrow K^{+}\pi^{-})e^{+}e^{-}$.
We reconstruct $K^{0}_{\scriptscriptstyle S}$ candidates in the
$\pi^{+}\pi^{-}$ final state. We also study the $K^{(*)}h^{\pm}\mu^{\mp}$
final states, where $h$ is a charged track with no particle identification
requirement applied, to characterize backgrounds from hadrons misidentified as
muons. We use a $K^{*}e^{\pm}\mu^{\mp}$ sample to model the combinatorial
background from two random leptons. In each mode, we utilize the kinematic
variables $\mbox{$m_{\rm ES}$}=\sqrt{E^{2}_{\rm CM}/4-p^{*2}_{B}}$ and $\Delta
E=E_{B}^{*}-E_{\rm CM}/2$, where $p^{*}_{B}$ and $E_{B}^{*}$ are the $B$
momentum and energy in the $\mathchar 28935\relax(4S)$ center-of-mass (CM)
frame, and $E_{\rm CM}$ is the total CM energy.
For masses $\mbox{$m_{\rm ES}$}>5.2$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ we perform one-dimensional fits of
the $m_{\rm ES}$ distribution for $K\ell^{+}\ell^{-}$ modes. For
$K^{*}\ell^{+}\ell^{-}$ modes, we include in addition the $K\pi$ mass region
$0.72<m_{K\pi}<1.10$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ in the fit. We
use the sideband $5.20<\mbox{$m_{\rm ES}$}<5.27$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to characterize combinatorial
background shapes and normalizations. For both the $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ modes, we veto the $J/\psi(2.85<m_{\ell\ell}<3.18~{}{\rm
GeV}/c^{2})$ and $\psi{(2S)}(3.59<m_{\ell\ell}<3.77~{}{\rm GeV}/c^{2})$ mass
regions. The vetoed events provide high-statistics control samples that we use
to validate the fit methodology.
The main backgrounds arise from random combinations of leptons from
semileptonic $B$ and $D$ decays. These combinatorial backgrounds from either
$B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ events (referred to as
“$B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ backgrounds”) or continuum
$q\overline{q}$ events ($e^{+}e^{-}\rightarrow q\overline{q},\ q=u,d,s,c$,
referred to as “$q\overline{q}$ backgrounds”) are suppressed using bagged
decision trees (BDTs) bdts . We train eight separate BDTs as follows:
* •
Suppression of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ backgrounds for
$e^{+}e^{-}$ modes in the low $s$ region;
* •
Suppression of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ backgrounds for
$e^{+}e^{-}$ modes in the high $s$ region;
* •
Suppression of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ backgrounds for
$\mu^{+}\mu^{-}$ modes in the low $s$ region;
* •
Suppression of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ backgrounds for
$\mu^{+}\mu^{-}$ modes in the high $s$ region;
* •
Suppression of $q\overline{q}$ backgrounds for $e^{+}e^{-}$ modes in the low
$s$ region;
* •
Suppression of $q\overline{q}$ backgrounds for $e^{+}e^{-}$ modes in the high
$s$ region;
* •
Suppression of $q\overline{q}$ backgrounds for $\mu^{+}\mu^{-}$ modes in the
low $s$ region;
* •
Suppression of $q\overline{q}$ backgrounds for $\mu^{+}\mu^{-}$ modes in the
high $s$ region.
The BDT input parameters include the following observables:
* •
$\Delta E$ of the $B$ candidate;
* •
The ratio of Fox-Wolfram moments $R_{2}$ Foxwolfram and the ratio of the
second-to-zeroth angular moments of the energy flow $L_{2}/L_{0}$ LegMom ,
both event shape parameters calculated using charged and neutral particles in
the CM frame;
* •
The mass and $\Delta E$ of the other $B$ meson in the event (referred to as
the “rest of the event”) computed in the laboratory frame by summing the
momenta and energies of all charged particles and photons that are not used to
reconstruct the signal candidate;
* •
The magnitude of the total transverse momentum of the event in the laboratory
frame;
* •
The probabilities that the $B$ candidate and the dilepton candidate,
respectively, originate from a single point in space;
* •
The cosine values of four angles: the angle between the $B$ candidate momentum
and the beam axis, the angle between the event thrust axis and the beam axis,
the angle between the thrust axis of the rest of the event and the beam axis,
and the angle between the event thrust axis and the thrust axis of the rest of
the event, all defined in the CM frame.
Figure 3 shows the output distributions of the BDTs for Monte Carlo (MC)
simulated signal and combinatorial background for the $e^{+}e^{-}$ sample
below the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance. The
distributions are histograms normalized to unit area. The selections on BDT
outputs are further optimized to maximize the statistical significance of the
signal events, as shown later.
Figure 3: The (a) $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ and (b)
$q\overline{q}$ $e^{+}e^{-}$ BDT outputs for simulated events in the low $s$
region. Shown are the distributions for $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ background (red dashed line),
$q\overline{q}$ background (red dotted line), and signal (blue solid line)
event samples, normalized to unit area.
Another source of background arises from $B\rightarrow D(\rightarrow
K^{(*)}\pi)\pi$ decays if both pions are misidentified as leptons. Determined
from data control samples with high purity BaBarDetector , the
misidentification rates for muons and electrons are $\sim 3\%$ and $\lesssim
0.1\%$ per candidate, respectively. Thus, this background is only significant
for $\mu^{+}\mu^{-}$ final states. We veto these events by requiring the
invariant mass of the $K^{(*)}\pi$ system to be outside the range $1.84-1.90$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ after assigning the pion mass
hypothesis to the muon candidates. Any remaining residual backgrounds from
this type of contribution are parameterized using control samples obtained
from data.
After applying all selection criteria about 85% of signal events contain more
than one $B$ candidate. These candidates differ typically in one charged or
neutral hadron. The average number of candidates per signal event is about
six. To choose the best candidate, we define the ratio
${\lambda}\equiv\frac{{\cal P}_{sig}^{B\bar{B}}+{\cal
P}_{sig}^{q\overline{q}}}{{\cal P}_{sig}^{B\bar{B}}+{\cal
P}_{sig}^{q\overline{q}}+{\cal P}_{bkg}^{B\bar{B}}+{\cal
P}_{bkg}^{q\overline{q}}},$ (4)
where ${\cal P}_{sig}$ and ${\cal P}_{bkg}$ are probabilities calculated from
the corresponding $B\bar{B}$ and $q\overline{q}$ BDT output distributions for
signal and background, respectively. We select the candidate with the largest
$\lambda$ as the best candidate. The probability for a correctly-reconstructed
signal event to be selected as the best candidate is mode-dependent and varies
between about $80\%$ and $95\%$ for $s$ bins below the $J/\psi$ mass, while
for $s$ bins above the $\psi{(2S)}$ mass it varies between about $60\%$ and
$90\%$.
## V Selection Optimization
To optimize the $\Delta E$ selection, we simultaneously vary the upper and
lower bounds of the $\Delta E$ interval to find the values that maximize the
ratio $S/\sqrt{S+B}$ in the signal region ($\mbox{$m_{\rm ES}$}>5.27~{}{\rm
GeV}/c^{2}$, and for $K^{*}$ modes in addition $0.78<m_{K\pi}<0.97~{}{\rm
GeV}/c^{2}$), where $S$ and $B$ are the expected numbers PDG of signal and
combinatorial background events, respectively. We perform separate
optimizations for dilepton masses below and above the $J/\psi$ mass. For some
modes, the optimization tends to select very narrow intervals, which leads to
small signal efficiency. To prevent this, we require the magnitudes of the
$\Delta E$ upper and lower bounds to be 0.04 $\mathrm{\,Ge\kern-1.00006ptV}$
or larger. (Note that the lower bound is always negative and the upper bound
always positive.)
We also optimize the lower bounds on the BDT $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ and $q\overline{q}$ intervals (the
upper bounds on these intervals are always 1.0). We perform fits to extract
signal yields using the fit model described in Sec. VI. For each mode, the
lower bound on the BDT interval is optimized by maximizing the expected signal
significance defined as the fitted signal yield divided by its associated
uncertainty. We determine these from 500 pseudo-experiments using branching
fraction averages PDG . The optimized BDT lower bounds are listed in Tables 2
and 3 for $K\ell^{+}\ell^{-}$ and $K^{*}\ell^{+}\ell^{-}$, respectively.
Figure 4 shows the expected experimental significance in the $B\bar{B}$ BDT
versus the $q\overline{q}$ BDT plane for $B^{0}\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ in bin $s_{2}$. The signal selection efficiency
and the cross-feed fraction (defined in Sec. VI) in each mode and $s$ bin
after the final event selection are also listed in Tables 2 and 3. The
selection efficiencies determined in simulations vary from $11.4\pm 0.2$% for
$K^{0}_{S}\pi^{+}e^{+}e^{-}$ in $s_{6}$ to $33.3\pm 0.3$% for
$K^{+}\mu^{+}\mu^{-}$ in $s_{5}$, where the uncertainties are statistical.
Figure 4: Expected statistical significance of the number of fitted signal events as a function of BDT interval lower bounds for $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ in bin $s_{2}$. The star marks the optimized pair of lower bounds. Table 2: Optimized lower bounds on the BDT intervals, signal reconstruction efficiency, and cross-feed fraction, by $K\ell^{+}\ell^{-}$ mode and $s$ bin. The uncertainties are statistical only. Mode | $s$ bin | $B\bar{B}$ | $q\overline{q}$ | Efficiency | Cross-feed
---|---|---|---|---|---
| | BDT | BDT | $[\%]$ | fraction $[\%]$
$B^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\mu^{+}\mu^{-}$ | $s_{1}$ | 0.20 | 0.80 | $19.9\pm 0.2$ | $8.9\pm 0.3$
| $s_{2}$ | 0.70 | 0.85 | $22.2\pm 0.2$ | $8.6\pm 0.2$
| $s_{3}$ | 0.20 | 0.85 | $25.2\pm 0.1$ | $8.9\pm 0.2$
| $s_{4}$ | 0.70 | 0.70 | $24.3\pm 0.2$ | $9.4\pm 0.2$
| $s_{5}$ | 0.70 | 0.80 | $22.2\pm 0.2$ | $12.0\pm 0.5$
| $s_{6}$ | 0.75 | 0.80 | $16.6\pm 0.1$ | $21.7\pm 0.7$
| $s_{0}$ | 0.50 | 0.85 | $22.7\pm 0.1$ | $8.8\pm 0.1$
$B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ | $s_{1}$ | 0.30 | 0.85 | $21.3\pm 0.2$ | $0.3\pm 0.0$
| $s_{2}$ | 0.15 | 0.85 | $27.0\pm 0.2$ | $0.3\pm 0.0$
| $s_{3}$ | 0.15 | 0.85 | $30.9\pm 0.1$ | $0.3\pm 0.0$
| $s_{4}$ | 0.80 | 0.85 | $31.0\pm 0.2$ | $0.4\pm 0.0$
| $s_{5}$ | 0.65 | 0.85 | $33.3\pm 0.3$ | $2.1\pm 0.1$
| $s_{6}$ | 0.05 | 0.85 | $30.5\pm 0.2$ | $10.4\pm 0.2$
| $s_{0}$ | 0.05 | 0.85 | $13.6\pm 0.1$ | $0.3\pm 0.0$
$B^{0}\rightarrow K^{0}_{\scriptscriptstyle S}e^{+}e^{-}$ | $s_{1}$ | 0.25 | 0.80 | $22.1\pm 0.2$ | $8.3\pm 0.3$
| $s_{2}$ | 0.25 | 0.80 | $25.2\pm 0.2$ | $9.4\pm 0.3$
| $s_{3}$ | 0.65 | 0.80 | $24.3\pm 0.1$ | $9.4\pm 0.2$
| $s_{4}$ | 0.50 | 0.85 | $24.1\pm 0.2$ | $10.9\pm 0.4$
| $s_{5}$ | 0.05 | 0.65 | $23.0\pm 0.2$ | $18.5\pm 0.9$
| $s_{6}$ | 0.25 | 0.70 | $16.5\pm 0.1$ | $35.0\pm 1.1$
| $s_{0}$ | 0.85 | 0.85 | $21.3\pm 0.1$ | $9.2\pm 0.2$
$B^{+}\rightarrow K^{+}e^{+}e^{-}$ | $s_{1}$ | 0.35 | 0.85 | $22.8\pm 0.2$ | $0.4\pm 0.1$
| $s_{2}$ | 0.10 | 0.85 | $28.8\pm 0.2$ | $0.4\pm 0.0$
| $s_{3}$ | 0.10 | 0.85 | $30.8\pm 0.1$ | $0.5\pm 0.0$
| $s_{4}$ | 0.30 | 0.80 | $32.7\pm 0.2$ | $1.1\pm 0.1$
| $s_{5}$ | 0.25 | 0.80 | $31.7\pm 0.3$ | $4.3\pm 0.2$
| $s_{6}$ | 0.50 | 0.85 | $25.1\pm 0.2$ | $12.0\pm 0.3$
| $s_{0}$ | 0.40 | 0.85 | $29.6\pm 0.1$ | $0.5\pm 0.0$
Table 3: Optimized lower bounds on the BDT intervals, signal reconstruction efficiency, and cross-feed fraction, by $K^{*}\ell^{+}\ell^{-}$ mode and $s$ bin. The uncertainties are statistical only. Mode | $s$ bin | $B\bar{B}$ | $q\overline{q}$ | Efficiency | Cross-feed
---|---|---|---|---|---
| | BDT | BDT | $[\%]$ | fraction $[\%]$
$B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{+}\mu^{+}\mu^{-}$ | $s_{1}$ | 0.55 | 0.85 | $13.6\pm 0.1$ | $14.0\pm 0.5$
| $s_{2}$ | 0.80 | 0.85 | $14.6\pm 0.2$ | $19.2\pm 0.7$
| $s_{3}$ | 0.85 | 0.80 | $14.9\pm 0.1$ | $20.7\pm 0.5$
| $s_{4}$ | 0.85 | 0.85 | $14.7\pm 0.1$ | $28.0\pm 0.7$
| $s_{5}$ | 0.15 | 0.85 | $16.4\pm 0.2$ | $59.3\pm 1.3$
| $s_{6}$ | 0.10 | 0.85 | $14.3\pm 0.1$ | $110.8\pm 1.9$
| $s_{0}$ | 0.80 | 0.85 | $14.5\pm 0.1$ | $18.9\pm 0.5$
$B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ | $s_{1}$ | 0.80 | 0.85 | $16.2\pm 0.1$ | $4.9\pm 0.2$
| $s_{2}$ | 0.80 | 0.85 | $19.6\pm 0.2$ | $7.8\pm 0.3$
| $s_{3}$ | 0.75 | 0.85 | $21.3\pm 0.1$ | $10.1\pm 0.2$
| $s_{4}$ | 0.85 | 0.85 | $20.9\pm 0.1$ | $13.8\pm 0.3$
| $s_{5}$ | 0.75 | 0.85 | $22.8\pm 0.2$ | $31.7\pm 0.6$
| $s_{6}$ | 0.80 | 0.80 | $19.5\pm 0.2$ | $61.0\pm 0.9$
| $s_{0}$ | 0.60 | 0.85 | $20.4\pm 0.1$ | $8.9\pm 0.2$
$B^{+}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{+}e^{+}e^{-}$ | $s_{1}$ | 0.45 | 0.70 | $16.6\pm 0.2$ | $17.8\pm 0.6$
| $s_{2}$ | 0.85 | 0.85 | $13.7\pm 0.2$ | $20.7\pm 0.8$
| $s_{3}$ | 0.55 | 0.85 | $16.0\pm 0.1$ | $27.5\pm 0.7$
| $s_{4}$ | 0.40 | 0.85 | $15.4\pm 0.1$ | $41.6\pm 0.9$
| $s_{5}$ | 0.80 | 0.45 | $13.1\pm 0.2$ | $68.6\pm 1.8$
| $s_{6}$ | 0.60 | 0.85 | $11.4\pm 0.2$ | $133.4\pm 2.9$
| $s_{0}$ | 0.70 | 0.85 | $16.0\pm 0.1$ | $23.1\pm 0.5$
$B^{0}\rightarrow K^{+}\pi^{-}e^{+}e^{-}$ | $s_{1}$ | 0.80 | 0.85 | $16.5\pm 0.2$ | $6.8\pm 0.2$
| $s_{2}$ | 0.85 | 0.85 | $18.6\pm 0.2$ | $10.9\pm 0.3$
| $s_{3}$ | 0.80 | 0.80 | $18.5\pm 0.1$ | $11.2\pm 0.3$
| $s_{4}$ | 0.55 | 0.65 | $21.9\pm 0.2$ | $25.6\pm 0.4$
| $s_{5}$ | 0.75 | 0.80 | $19.0\pm 0.2$ | $50.4\pm 0.9$
| $s_{6}$ | 0.05 | 0.80 | $15.1\pm 0.2$ | $110.9\pm 1.8$
| $s_{0}$ | 0.80 | 0.85 | $19.7\pm 0.1$ | $10.8\pm 0.2$
## VI Fit Methodology
We perform one-dimensional fits in $m_{\rm ES}$ for $K\ell^{+}\ell^{-}$ modes
and two-dimensional fits in $m_{\rm ES}$ and $m_{K\pi}$ for
$K^{*}\ell^{+}\ell^{-}$ modes to extract the signal yields. The probability
density function (PDF) for signal $m_{\rm ES}$ is parametrized by a Gaussian
function with mean and width fixed to values obtained from fits to the vetoed
$J/\psi$ events in the data control samples. For $m_{K\pi}$, the PDF is a
relativistic Breit-Wigner line shape Aubert:2008gm . True signal events are
those where all generator-level final-state daughter particles are correctly
reconstructed and are selected to form a $B$ candidate.
For the combinatorial background, the $m_{\rm ES}$ PDF is modeled with a
kinematic threshold function whose shape is a free parameter in the fits argus
, while the $m_{K\pi}$ PDF shape is characterized with the
$K^{*}e^{\pm}\mu^{\mp}$ sample mentioned in Sec. IV. We parameterize the
combinatorial $m_{K\pi}$ distributions with non-parametric Gaussian kernel
density estimator shapes keys (referred to as the “KEYS PDFs”) drawn from the
$K^{*}e^{\pm}\mu^{\mp}$ sample in the full $m_{\rm ES}$ fit region. Since the
correlation between $m_{K\pi}$ and $\Delta E$ is weak, we accept all
$K^{*}e^{\pm}\mu^{\mp}$ events within $|\Delta E|<0.3$
$\mathrm{\,Ge\kern-1.00006ptV}$, rather than imposing a stringent $\Delta E$
selection, in order to enhance sample sizes.
Signal cross-feed consists of mis-reconstructed signal events, in which
typically a low-momentum $\pi^{\pm}$ or $\pi^{0}$ is swapped, added, or
removed in the $B$ candidate reconstruction. We distinguish among different
categories of cross-feed: “self-cross-feed” is when a particle is swapped
within one mode, “feed-across” is when a particle is swapped between two
signal modes with the same final-state multiplicity, and “feed-up (down)” is
when a particle is added (removed) from a lower (higher) multiplicity
$b\rightarrow s\ell^{+}\ell^{-}$ mode. We use both exclusive and inclusive
$b\rightarrow s\ell^{+}\ell^{-}$ MC samples to evaluate the contributions of
the different categories. The cross-feed $m_{\rm ES}$ distribution is
typically broadened compared to correctly reconstructed signal decays. We
combine the cross-feed contributions from all sources into a single fit
component that is modeled as a sum of weighted histograms with a single
overall normalization, which is allowed to scale as a fixed fraction of the
observed correctly reconstructed signal yield. This fixed fraction is
presented as the “cross-feed fraction” in Tables 2 and 3. The modeling of
cross-feed contributions is validated using fits to the vetoed $J/\psi
K^{(*)}$ and $\psi{(2S)}K^{(*)}$ events, in which the cross-feed contributions
are relatively large compared to all other backgrounds.
Exclusive $B$ hadronic decays may be mis-reconstructed as $B\rightarrow
K^{(*)}\ell^{+}\ell^{-}$, since hadrons can be misidentified as muons.
Following a procedure similar to that described in Ref. Aubert:2006vb , we
determine this background by selecting a sample of $K^{(*)}\mu^{\pm}h^{\mp}$
events, in which the muon is identified as a muon and the hadron is
inconsistent with an electron. Requiring identified kaons and pions, we select
subsamples of $K^{(*)}\pi^{+}\pi^{-},K^{(*)}K^{+}\pi^{-},K^{(*)}\pi^{+}K^{-}$,
and $K^{(*)}K^{+}K^{-}$. We obtain weights from data control samples where a
charged particle’s species can be identified with high precision and accuracy
without using particle identification information. The weights are then
applied to this dataset to characterize the contribution expected in our fits
due to misidentified muon candidates. We characterize the misidentification
backgrounds using the KEYS PDFs, with normalizations obtained by construction
directly from the weighted data.
Some charmonium events may escape the charmonium vetoes and appear in our fit
region. Typically, this occurs when electrons radiate a photon or a muon
candidate is a misidentified hadron and the missing energy is accounted for by
a low-energy $\pi^{\pm}$ or $\pi^{0}$. The largest background contributions
from this source are expected in the $K^{*}\mu^{+}\mu^{-}$ and
$K^{*}e^{+}e^{-}$ channels. We model this background using the charmonium MC
samples and determine the leakage into $s$ bins on either side of the $J/\psi$
and $\psi{(2S)}$ resonances. We see a notable charmonium contribution (about
five events) for $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ in bin $s_{3}$.
This leakage is typically caused by a swap between the $\mu^{+}$ and $\pi^{+}$
in a single $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K\pi^{+}$ candidate, where both the $\mu^{+}$
and $\pi^{+}$ are misidentified.
Hadronic peaking background from $B\rightarrow K^{*}\pi^{0}$ and $B\rightarrow
K^{*}\eta$ in which the $\pi^{0}$ or $\eta$ decays via Dalitz pairs shows a
small peaking component in $m_{\rm ES}$ in bin $s_{1}$. Due to the requirement
$s>0.1~{}\rm GeV^{2}/c^{4}$, contributions of $\gamma$ conversions from
$B\rightarrow K^{*}\gamma$ events beyond the photon pole region are found to
be negligible.
### Fit Model for Rate Asymmetries
Using the PDFs described above, we perform simultaneous fits across different
$K^{(*)}\ell^{+}\ell^{-}$ modes. Since efficiency-corrected signal yields are
shared across various decay modes, we can extract rate asymmetries directly
from the fits. The fitted signal yields in $B^{+}$ modes are corrected by the
lifetime ratio $\tau_{B^{0}}/\tau_{B^{+}}$. We also correct the signal yields
for ${\cal B}(K^{*}\rightarrow K\pi)$ in $K^{*}$ modes and ${\cal
B}(K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-})$ in the modes with a
$K^{0}_{\scriptscriptstyle S}$. In the fits for ${\cal A}_{C\\!P}$, we share
the efficiency-corrected signal yield $N_{B}$ as a floating variable for $B\
(q\bar{b},q=u,d)$ events across different flavor-tagging
$K^{(*)}\ell^{+}\ell^{-}$ modes by assuming lepton-flavor and isospin
symmetries. The efficiency-corrected signal yield $N_{\bar{B}}$ for $\bar{B}\
(\bar{q}b)$ events is then defined by $N_{\bar{B}}=N_{B}\cdot(1+{\cal
A}_{C\\!P)})/(1-{\cal A}_{C\\!P})$ and is also shared across corresponding
modes. For the lepton-flavor ratios ${\cal R}_{K^{(*)}}$, we share the
efficiency-corrected signal yield $N_{ee}$ as a floating variable for the two
$B\rightarrow Ke^{+}e^{-}$ or $B\rightarrow K^{*}e^{+}e^{-}$ modes by assuming
isospin symmetry. The efficiency-corrected signal yield $N_{\mu\mu}$ shared
across the corresponding $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ modes is then
defined by $N_{\mu\mu}=N_{ee}\cdot R_{K^{(*)}}$. For the isospin asymmetry
${\cal A}^{K^{(*)}}_{I}$, we share the efficiency-corrected signal yield
$N_{B^{+}}$ as a floating variable for the two $B^{+}\rightarrow
K^{+}\ell^{+}\ell^{-}$ or $B^{+}\rightarrow K^{*+}\ell^{+}\ell^{-}$ modes by
assuming lepton-flavor symmetry. The efficiency-corrected signal yield
$N_{B^{0}}$ shared across the corresponding $B^{0}\rightarrow
K^{*0}\ell^{+}\ell^{-}$ modes is then defined by
$N_{B^{0}}=N_{B^{+}}\cdot(1+{\cal A}^{K{(*)}}_{I})/(1-{\cal A}^{K{(*)}}_{I})$.
## VII Fit Validation
We validate the fit methodology with charmonium control samples obtained from
the dilepton mass regions around the $J/\psi$ and $\psi{(2S)}$ resonances that
are vetoed in the $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ analysis. We measure
the $J/\psi K^{(*)}$ and $\psi{(2S)}K^{(*)}$ branching fractions in each final
state with the optimized BDT selections in bins $s_{3}$ and $s_{4}$,
respectively. Our measurements agree well with the world averages PDG for all
final states. Typical deviations, based on statistical uncertainties only, are
less than one standard deviation ($\sigma$). The largest deviation, in the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ mode, is $1.7\sigma$. For $J/\psi K^{(*)}$ modes,
the statistical uncertainties are considerably smaller than those of the world
averages. We float the Gaussian means and widths of the signal PDFs in the
fits for the $J/\psi K^{(*)}$ modes. The associated uncertainties obtained
from the fits are then used as a source of systematic variation for the signal
PDFs. The typical signal width in $m_{\rm ES}$ is $2.5~{}{\rm MeV}/c^{2}$.
We further validate our fitting procedure by applying it to charmonium events
to extract the rate asymmetries. The measured $C\\!P$ asymmetries ${\cal
A}_{CP}$, lepton-flavor ratios ${\cal R}_{K^{(*)}}$ and isospin asymmetries
${\cal A}_{I}$ are in good agreement with Standard Model expectations or world
averages for ${\cal A}_{I}$.
We also test the methodology with fits to ensembles of datasets where signal
and background events are generated from appropriately normalized PDFs (“pure
pseudo-experiments”). We perform fits to these pseudo-experiments in each mode
and $s$ bin using the full fit model described previously. For ensembles of
1000 pure pseudo-experiments, the pull distributions for the signal yields
show negligible biases. We further fit ensembles of pseudo-experiments in
which the signal events are drawn from properly normalized exclusive MC
samples (“embedded pseudo-experiments”). The pull distributions also show the
expected performance.
We perform fits to ensembles of pure pseudo-experiments in order to estimate
the statistical sensitivity of, and biases related to, the various rate
asymmetry measurements. The pull distributions for ${\cal A}_{C\\!P}$ and
${\cal R}_{K^{(*)}}$ for the low and high $s$ regions show minimal biases. For
${\cal A}_{I}$, we test a series of ${\cal A}_{I}$ input values ($-0.6,\
-0.3,\ 0.0,\ 0.3,\ 0.6$) in each $s$ bin using pure pseudo-experiments to
ensure we obtain unbiased fits under different assumptions of isospin
asymmetry. The ${\cal A}^{K}_{I}$ pulls are generally well-behaved. In the
worst case, the test fits for ${\cal A}^{K}_{I}$ are slightly biased due to
very low signal yield expectations in the $K^{0}_{\scriptscriptstyle
S}\ell^{+}\ell^{-}$ final states.
## VIII Systematic Uncertainties
Since some systematic uncertainties largely cancel in ratios, it is useful to
separate the discussion of systematic uncertainties on partial branching
fractions from that on rate asymmetries.
### VIII.1 Branching Fraction Uncertainties
Systematic uncertainties for branching fractions arise from multiplicative
systematic uncertainties involving the determination of the signal efficiency,
and from additive systematic uncertainties arising from the extraction of
signal yields in the data fits. The multiplicative systematic errors include
contributions from the
* •
Number of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs: This
uncertainty is $0.6\%$.
* •
Tracking efficiency for charged particles: We assign a correlated uncertainty
of 0.3% for each lepton, and 0.4% for each charged hadron including daughter
pions from $K^{0}_{\scriptscriptstyle S}$ decay Allmendinger:2012ch .
* •
Charged particle identification (PID) efficiencies: We employ a data-driven
method to correct PID efficiencies in simulated events. We estimate the
systematic uncertainties from the change in signal efficiency for simulated
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ events after turning off
the PID corrections. The systematic uncertainties are mode dependent and vary
between $0.3\%$ and $1.6\%$.
* •
$K^{0}_{\scriptscriptstyle S}$ identification efficiency: This is determined
as a function of flight distance after applying $K^{0}_{\scriptscriptstyle S}$
efficiency corrections. An uncertainty of $0.9\%$ is obtained by varying the
$K^{0}_{\scriptscriptstyle S}$ selection algorithm.
* •
Event selection efficiency: We measure the efficiency of the BDT selection in
charmonium data control samples and compare with results obtained for
exclusive charmonium samples from simulation. We take the magnitude of the
deviation for any particular final state and $s$ bin as the uncertainty
associated with the BDT lower bounds. If the data and simulation are
consistent within the uncertainty, we then take the uncertainty as the
systematic uncertainty. The systematic uncertainty is found to vary between
0.3% and 9.1% depending on both the mode and the $s$ bin. Due to a strong
correlation between the $\Delta E$ and BDT outputs, uncertainties due to
$\Delta E$ are fully accounted for by this procedure.
* •
Monte Carlo sample size: We find the uncertainty related to the finite size of
the MC sample to be of the order of 1% or less for all modes.
The additive systematic uncertainties involve contributions from the
* •
Signal PDF shapes: We characterize them by varying the PDF shape parameters
(signal mean, signal width, and combinatorial background shape and
normalization) by the statistical uncertainties obtained in the fits to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ data control samples for
$m_{\rm ES}$ and signal MC events for $m_{K\pi}$.
* •
Hadronic backgrounds: We characterize them by varying both the normalization
by the associated statistical uncertainties and by performing fits with
different choices of smoothing parameters for the KEYS PDF shapes.
* •
Peaking backgrounds from charmonium events and $\pi^{0}/\eta$ Dalitz decays:
We vary the normalization for these contributions by $\pm$25%.
* •
Modeling of $m_{K\pi}$ line shapes of the combinatorial background: We
characterize the uncertainties by analyzing data samples selected from the
$\mbox{$m_{\rm ES}$}<5.27$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
sideband, and simulated events.
Table 4 summarizes all sources of systematic uncertainties considered in the
total branching fraction measurements for individual modes. The total
systematic uncertainty for the branching fractions is obtained by summing in
quadrature the above-described uncertainties from different categories.
Table 4: Individual systematic uncertainties [%] for measurements of the total branching fractions in $K^{(*)}\ell^{+}\ell^{-}$ decays. Mode | $K^{0}_{\scriptscriptstyle S}\mu^{+}\mu^{-}$ | $K^{+}\mu^{+}\mu^{-}$ | $K^{0}_{\scriptscriptstyle S}e^{+}e^{-}$ | $K^{+}e^{+}e^{-}$ | $K^{0}_{\scriptscriptstyle S}\pi^{+}\mu^{+}\mu^{-}$ | $K^{+}\pi^{-}\mu^{+}\mu^{-}$ | $K^{0}_{\scriptscriptstyle S}\pi^{+}e^{+}e^{-}$ | $K^{+}\pi^{-}e^{+}e^{-}$
---|---|---|---|---|---|---|---|---
$B\kern 1.43994pt\overline{\kern-1.43994ptB}{}$ counting | $\pm 0.6$ | $\pm 0.6$ | $\pm 0.6$ | $\pm 0.6$ | $\pm 0.6$ | $\pm 0.6$ | $\pm 0.6$ | $\pm 0.6$
Tracking | $\pm 1.4$ | $\pm 1.0$ | $\pm 1.4$ | $\pm 1.0$ | $\pm 1.8$ | $\pm 1.4$ | $\pm 1.8$ | $\pm 1.4$
PID | $\pm 1.6$ | $\pm 0.3$ | $\pm 0.7$ | $\pm 0.4$ | $\pm 1.5$ | $\pm 0.3$ | $\pm 0.5$ | $\pm 1.2$
$K^{0}_{s}$ ID | $\pm 0.9$ | — | $\pm 0.9$ | — | $\pm 0.9$ | — | $\pm 0.9$ | —
BDT selections | $\pm 2.2$ | $\pm 1.7$ | $\pm 4.7$ | $\pm 1.5$ | $\pm 8.3$ | $\pm 2.5$ | $\pm 9.1$ | $\pm 2.7$
MC sample size | $\pm 0.3$ | $\pm 0.3$ | $\pm 0.3$ | $\pm 0.3$ | $\pm 0.4$ | $\pm 0.3$ | $\pm 0.4$ | $\pm 0.4$
Sig. Shape | $\pm 0.5$ | $\pm 0.4$ | $\pm 1.5$ | $\pm 0.4$ | $\pm 1.5$ | $\pm 0.7$ | $\pm 1.5$ | $\pm 0.7$
Hadronic | $\pm 3.3$ | $\pm 5.8$ | — | — | $\pm 2.3$ | $\pm 1.6$ | — | —
Peaking | $\pm 0.3$ | $\pm 0.8$ | $\pm 1.2$ | $\pm 0.8$ | $\pm 0.7$ | $\pm 1.7$ | $\pm 0.8$ | $\pm 1.2$
Comb. $m_{K\pi}$ shape | — | — | — | — | $\pm 1.2$ | $\pm 0.6$ | $\pm 0.6$ | $\pm 1.6$
Total | $\pm 4.7$ | $\pm 6.3$ | $\pm 5.4$ | $\pm 2.2$ | $\pm 9.3$ | $\pm 3.9$ | $\pm 9.5$ | $\pm 4.0$
### VIII.2 Systematic uncertainties for the rate asymmetries
For ${\cal A}_{C\\!P}$, a large portion of the uncertainties associated with
the signal efficiency cancel. We find that the only efficiency-related term
discussed in Sec. VIII.1 that is not negligible for ${\cal A}_{C\\!P}$ is the
one associated with the PID selection. Amongst the efficiency-related
systematics, we therefore only consider this term. We also consider the
additive systematic uncertainties listed in Sec. VIII.1. Our measured ${\cal
A}_{C\\!P}$ central values for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K$
and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*}$ are both well below 1%
and show minimal detector efficiency effects. Potential, additional ${\cal
A}_{C\\!P}$ systematic effects from the assumptions of lepton-flavor and
isospin symmetry are tested by removing these assumptions.
The systematic uncertainties for the lepton-flavor ratios ${\cal R}_{K^{(*)}}$
are calculated by summing in quadrature the systematic errors in the muon and
electron modes. Common systematic effects, such as tracking,
$K^{0}_{\scriptscriptstyle S}$ efficiency, and $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ counting, yield negligible
uncertainties in the ratios. Potential, additional ${\cal R}_{K^{(*)}}$
systematic effects are tested by removing the assumption of isospin symmetry.
For the systematic uncertainties of ${\cal A}_{I}$, we sum in quadrature the
systematic errors in charged and neutral $B$ modes. Common systematic effects,
which include $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ counting and a
large portion of the uncertainties associated with PID and tracking
efficiencies, are negligible. Again, additional tests on ${\cal A}_{I}$
systematics are performed by relaxing the assumption of lepton-flavor
symmetry. Furthermore, as the cross-feed fractions in Tables 2 and 3 are
estimated under the assumption of isospin symmetry, we test this systematic
effect using cross-feed fractions estimated with different ${\cal A}_{I}$
input values.
Our checks on symmetry assumptions described above for ${\cal A}_{C\\!P}$,
${\cal R}_{K^{(*)}}$ and ${\cal A}_{I}$ generally show deviations from the
original measured values below 20% of the associated statistical
uncertainties, and so we do not assign additional uncertainties.
## IX Results
We perform fits for each $K^{(*)}\ell^{+}\ell^{-}$ final state in each $s$ bin
listed in Tables 2 and 3 to obtain signal and background yields, $N_{\rm sig}$
and $N_{\rm bkg}$, respectively. We model the different background components
by the PDFs described in Sec. VI. We allow the shape parameter of the $m_{\rm
ES}$ kinematic threshold function of the combinatorial background to float in
the fits. For the signal, we use a fixed Gaussian shape unique to each final
state, as described previously. We leave the shapes of the other background
PDFs fixed. For the peaking background, we fix the absolute normalization. For
the cross-feed, we fix the normalization relative to the signal yields.
Figure 5 shows as an example the $m_{\rm ES}$ distribution for the combined
$K\ell^{+}\ell^{-}$ modes in bin $s_{4}$, while Fig. 6 shows the $m_{\rm ES}$
and $m_{K\pi}$ mass spectra for the combined $K^{*}\ell^{+}\ell^{-}$ modes in
bin $s_{1}$. The cross-feed contributions and the peaking backgrounds are
negligible for this fit. The combinatorial background dominates and for
$\mu^{+}\mu^{-}$ modes misidentified hadrons are the second largest
background. From the yields in each $s$ bin we determine the partial branching
fractions summarized in Table 5. Figure 7 shows our results for the partial
branching fractions of the $K\ell^{+}\ell^{-}$ and $K^{*}\ell^{+}\ell^{-}$
modes in comparison to results from the Belle and CDF Collaborations belle09 ;
cdf10 and to the prediction of the Ali et al. model Ali:2002jg . Our results
are seen to agree with those of Belle and CDF. Our results are also in
agreement with the most recent partial branching fraction measurements of
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ from LHCb lhcb11 .
The total branching fractions are measured to be
$\displaystyle{\cal B}(B\rightarrow K\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle(4.7\pm 0.6\pm 0.2)\times 10^{-7},$ $\displaystyle{\cal
B}(B\rightarrow K^{*}\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle(10.2_{-1.3}^{+1.4}\pm 0.5)\times 10^{-7}.$
Here, the first uncertainties are statistical, and the second are systematic.
The total branching fractions are shown in Fig. 8 in comparison to
measurements from Belle belle09 and CDF cdf10 and predictions from Ali et
al. Ali:2002jg and Zhong et al. Zhong:2002nu .
Figure 5: The $m_{\rm ES}$ spectrum in bin $s_{4}$ for all $K\ell^{+}\ell^{-}$ modes combined showing data (points with error bars), the total fit (blue solid line), signal component (black short-dashed line), combinatorial background (magenta long-dashed line), hadrons misidentified as muons (green dash-dotted line), and the sum of cross-feed and peaking components (red dotted line). Figure 6: The (a) $m_{\rm ES}$ and (b) $m_{K\pi}$ mass spectra in bin $s_{1}$ for all four $K^{*}\ell^{+}\ell^{-}$ modes combined showing data (points with error bars), the total fit (blue solid lines), signal component (black short-dashed lines), combinatorial background (magenta long-dashed lines), hadrons misidentified as muons (green dash-dotted lines), and the sum of cross-feed and peaking components (red dotted lines). Figure 7: Partial branching fractions for the (a) $K\ell^{+}\ell^{-}$ and (b) $K^{*}\ell^{+}\ell^{-}$ modes as a function of $s$ showing BABAR measurements (red triangles), Belle measurements belle09 (open squares), CDF measurements cdf10 (blue solid squares), and the SM prediction from the Ali et al. model Ali:2002jg with $B\rightarrow K^{(*)}$ form factors ffmodels (magenta dashed lines). The magenta solid lines show the theory uncertainties. The vertical yellow shaded bands show the vetoed $s$ regions around the $J/\psi$ and $\psi{(2S)}$. Figure 8: Total branching fractions for the $K\ell^{+}\ell^{-}$ and $K^{*}\ell^{+}\ell^{-}$ modes (red triangles) compared with Belle belle09 (open squares) and CDF cdf10 (blue solid squares) measurements and with predictions from the Ali et al. Ali:2002jg (light grey bands), and Zhong et al. Zhong:2002nu (dark grey bands) models. Table 5: Measured branching fractions [$10^{-7}$] by mode and $s$ bin. The first and second uncertainties are statistical and systematic, respectively. | $B\rightarrow K\ell^{+}\ell^{-}$ | $B\rightarrow K^{*}\ell^{+}\ell^{-}$
---|---|---
$s$ (${\mathrm{\,Ge\kern-0.80005ptV^{2}\\!/}c^{4}}$) | $N_{\rm sig}$ | ${\cal B}[10^{-7}]$ | $N_{\rm sig}$ | ${\cal B}[10^{-7}]$
0.10–2.00 | $20.6_{-5.4}^{+5.9}$ | $0.71_{-0.18}^{+0.20}\pm 0.02$ | $26.0_{-6.4}^{+7.1}$ | $1.89_{-0.46}^{+0.52}\pm 0.06$
2.00–4.30 | $17.4_{-4.8}^{+5.4}$ | $0.49_{-0.13}^{+0.15}\pm 0.01$ | $14.5_{-4.6}^{+5.3}$ | $0.95_{-0.30}^{+0.35}\pm 0.04$
4.30–8.12 | $37.1_{-7.5}^{+8.0}$ | $0.94_{-0.19}^{+0.20}\pm 0.02$ | $29.3_{-8.3}^{+9.1}$ | $1.82_{-0.52}^{+0.56}\pm 0.09$
10.11–12.89 | $36.0_{-7.6}^{+8.2}$ | $0.90_{-0.19}^{+0.20}\pm 0.04$ | $31.6_{-8.1}^{+8.8}$ | $1.86_{-0.48}^{+0.52}\pm 0.10$
14.21–16.00 | $19.7_{-5.6}^{+6.2}$ | $0.49_{-0.14}^{+0.15}\pm 0.02$ | $24.1_{-6.0}^{+6.7}$ | $1.46_{-0.36}^{+0.41}\pm 0.06$
$>$16.00 | $22.3_{-6.9}^{+7.7}$ | $0.67_{-0.21}^{+0.23}\pm 0.05$ | $14.1_{-5.9}^{+6.6}$ | $1.02_{-0.42}^{+0.47}\pm 0.06$
1.00–6.00 | $39.4_{-7.1}^{+7.7}$ | $1.36_{-0.24}^{+0.27}\pm 0.03$ | $33.1_{-7.8}^{+8.6}$ | $2.05_{-0.48}^{+0.53}\pm 0.07$
To measure direct ${\cal A}_{C\\!P}$, we fit the $B$ and $\bar{B}$ samples in
the two $K^{+}\ell^{+}\ell^{-}$ modes and four $K^{*}\ell^{+}\ell^{-}$ modes
listed in Sec. IV. We perform the measurements in the full $s$ region, as well
as in the low $s$ and high $s$ regions separately. The $B$ and $\bar{B}$ data
sets share the same background shape parameter for the kinematic threshold
function. Figure 9 shows an example fit for the combined $B\rightarrow
K^{*}\ell^{+}\ell^{-}$ modes in the low $s$ region. Table 6 summarizes the
results. Figure 10 shows ${\cal A}_{C\\!P}$ as a function of $s$. Our results
are consistent with the SM expectation of negligible direct ${\cal
A}_{C\\!P}$.
Figure 9: (a)&(c) $m_{\rm ES}$ and (b)&(d) $m_{K\pi}$ fits for ${\cal A}_{C\\!P}$ in the (a)&(b) $\bar{B}$ and (c)&(d) $B$ low $s$ region for all four $K^{*}\ell^{+}\ell^{-}$ modes combined. Data (points with error bars) are shown together with total fit (blue solid lines), combinatorial background (magenta long-dashed lines), signal (black short-dashed lines), hadronic background (green dash-dotted lines), and the sum of cross-feed and peaking background (red dotted lines). Table 6: Measured ${\cal A}_{C\\!P}$ by mode and $s$ region. The first and second uncertainties are statistical and systematic, respectively. “All” refers to the union of $0.10<s<8.12$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and $s>10.11$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. $s$ (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | ${\cal}A_{C\\!P}(B^{+}\rightarrow K^{+}\ell^{+}\ell^{-})$ | $A_{C\\!P}(B\rightarrow K^{*}\ell^{+}\ell^{-})$
---|---|---
All | $-0.03\pm 0.14\pm 0.01$ | $0.03\pm 0.13\pm 0.01$
0.10–8.12 | $0.02\pm 0.18\pm 0.01$ | $-0.13_{-0.19}^{+0.18}\pm 0.01$
$>$10.11 | $-0.06_{-0.21}^{+0.22}\pm 0.01$ | $0.16_{-0.19}^{+0.18}\pm 0.01$
Figure 10: $C\\!P$ asymmetries ${\cal A}_{CP}$ for $K\ell^{+}\ell^{-}$ modes
(red solid triangles) and $K^{*}\ell^{+}\ell^{-}$ modes (red open circles) as
a function of $s$. The vertical yellow shaded bands show the vetoed $s$
regions around the $J/\psi$ and $\psi{(2S)}$.
We fit the $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ samples in the four
$K\ell^{+}\ell^{-}$ modes and four $K^{*}\ell^{+}\ell^{-}$ modes in the low
$s$ and high $s$ regions separately to measure the lepton-flavor ratios.
Figure 11 shows an example fit for the combined $K\mu^{+}\mu^{-}$ and
$Ke^{+}e^{-}$ modes in the high $s$ region. Table 7 and Fig. 12 show ${\cal
R}_{K}$ and ${\cal R}_{K^{*}}$ for
$s>0.1{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. Our results are
consistent with unity as expected in the SM.
Figure 11: $m_{\rm ES}$ fits for ${\cal R}_{K}$ in the (a) $Ke^{+}e^{-}$ and (b) $K\mu^{+}\mu^{-}$ modes in the high $s$ region. Data (points with error bars) are shown together with total fit (blue solid lines), combinatorial background (magenta long-dashed lines), signal (black short-dashed lines), hadronic background (green dash-dotted lines), and the sum of cross-feed and peaking background (red dotted lines). Table 7: Measured ${\cal R}_{K^{(*)}}$ by mode and $s$ region. The first and second uncertainties are statistical and systematic, respectively. “All” refers to the union of $0.10<s<8.12$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and $s>10.11$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. $s$ (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | ${\cal R}_{K}$ | $R_{K^{*}}$
---|---|---
All | $1.00_{-0.25}^{+0.31}\pm 0.07$ | $1.13_{-0.26}^{+0.34}\pm 0.10$
0.10–8.12 | $0.74_{-0.31}^{+0.40}\pm 0.06$ | $1.06_{-0.33}^{+0.48}\pm 0.08$
$>$10.11 | $1.43_{-0.44}^{+0.65}\pm 0.12$ | $1.18_{-0.37}^{+0.55}\pm 0.11$
Figure 12: Lepton flavor ratios ${\cal R}_{K^{(*)}}$ for the
$K\ell^{+}\ell^{-}$ (red solid triangles) and $K^{*}\ell^{+}\ell^{-}$ modes
(red open circles) as a function of $s$. The vertical yellow shaded bands show
the vetoed $s$ regions around the $J/\psi$ and $\psi{(2S)}$.
We fit the data in each $s$ bin separately to determine ${\cal A}_{I}$ for the
four combined $K\ell^{+}\ell^{-}$ and four combined $K^{*}\ell^{+}\ell^{-}$
modes. Figure 13 shows an example fit for bin $s_{2}$. The results are
summarized in Table 8. Figure 14 shows our measurements as a function of $s$
in comparison with those of Belle belle09 . The two sets of results are seen
to agree within the uncertainties. Our results are also consistent with the SM
prediction that ${\cal A}_{I}$ is slightly negative ($\sim-1\%$) except in bin
$s_{1}$, where it is predicted to have a value around $+5\%$ isospin .
Figure 13: The $m_{\rm ES}$ and $m_{K\pi}$ fit projections for the (a)&(b) $K^{*+}\ell^{+}\ell^{-}$ and (c)&(d) $K^{*0}\ell^{+}\ell^{-}$ modes in bin $s_{2}$. Data (points with error bars) are shown together with total fit (blue solid lines), combinatorial background (magenta long-dashed lines), signal (black short-dashed lines), hadronic background (green dash-dotted lines), and the sum of cross-feed and peaking background (red dotted lines). Table 8: Measured ${\cal A}_{I}$ by mode and $s$ bin. The first and second uncertainties are statistical and systematic, respectively. | ${\cal A}_{I}$
---|---
$s$ (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | $B\rightarrow K\ell^{+}\ell^{-}$ | $B\rightarrow K^{*}\ell^{+}\ell^{-}$
0.10–2.00 | $-0.51_{-0.95}^{+0.49}\pm 0.04$ | $-0.17_{-0.24}^{+0.29}\pm 0.03$
2.00–4.30 | $-0.73_{-0.55}^{+0.48}\pm 0.03$ | $-0.06_{-0.36}^{+0.56}\pm 0.05$
4.30–8.12 | $-0.32_{-0.30}^{+0.27}\pm 0.01$ | $0.03_{-0.32}^{+0.43}\pm 0.04$
10.11–12.89 | $-0.05_{-0.29}^{+0.25}\pm 0.03$ | $-0.48_{-0.18}^{+0.22}\pm 0.05$
14.21–16.00 | $0.05_{-0.43}^{+0.31}\pm 0.03$ | $0.24_{-0.39}^{+0.61}\pm 0.04$
$>$16.00 | $-0.93_{-4.99}^{+0.83}\pm 0.04$ | $1.07_{-0.95}^{+4.27}\pm 0.35$
1.00–6.00 | $-0.41\pm 0.25\pm 0.01$ | $-0.20_{-0.23}^{+0.30}\pm 0.03$
Figure 14: Isospin asymmetry ${\cal A}_{I}$ for the (a) $K\ell^{+}\ell^{-}$
and (b) $K^{*}\ell^{+}\ell^{-}$ modes as a function of $s$ (red triangles), in
comparison to results from Belle belle09 (open squares). The vertical yellow
shaded bands show the vetoed $s$ regions around the $J/\psi$ and $\psi{(2S)}$.
Our ${\cal A}_{I}$ measurements in the low $s$ region ($0.10<s<8.12$
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) yield
$\displaystyle{\cal A}^{\rm low}_{I}(B\rightarrow K\ell^{+}\ell^{-})$
$\displaystyle=-0.58_{-0.37}^{+0.29}\pm 0.02$ $\displaystyle[2.1\sigma],$
$\displaystyle{\cal A}^{\rm low}_{I}(B\rightarrow K^{*}\ell^{+}\ell^{-})$
$\displaystyle=-0.25_{-0.17}^{+0.20}\pm 0.03$ $\displaystyle[1.2\sigma],$
where the first uncertainty is statistical and the second is systematic. The
${\cal A}_{I}$ significances shown in the square brackets include all
systematic uncertainties. We estimate the significance by refitting the data
with ${\cal A}_{I}$ fixed to zero and compute the change in log likelihood
$\sqrt{2\Delta\ln{\cal L}}$ between the nominal fit and the null hypothesis
fit.
## X Conclusion
In summary, we have measured total and partial branching fractions, direct
$C\\!P$ asymmetries, lepton-flavor ratios, and isospin asymmetries in the rare
decays $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ using 471 million $B\bar{B}$
pairs. These results provide an update to our previous measurements on
branching fractions and rate asymmetries excluding the $s<0.1$
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ region babarrun5 . The total
branching fractions, ${\cal B}(B\rightarrow K\ell^{+}\ell^{-})=(4.7\pm 0.6\pm
0.2)\times 10^{-7}$ and ${\cal B}(B\rightarrow
K^{*}\ell^{+}\ell^{-})=(10.2_{-1.3}^{+1.4}\pm 0.5)\times 10^{-7}$, are
measured with precisions of $13\%$ and $14\%$, respectively. The partial
branching fractions as a function of $s$ agree well with the SM prediction.
For $0.10<s<8.12$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, our partial
branching fraction results also allow comparisons with SCET based predictions.
$C\\!P$ asymmetries for both $B\rightarrow K\ell^{+}\ell^{-}$ and
$B\rightarrow K^{*}\ell^{+}\ell^{-}$ are consistent with zero and the lepton-
flavor ratios are consistent with one, both as expected in the SM. The isospin
asymmetries at low $s$ values are negative. For $0.10<s<8.12$
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ we measure ${\cal
A}_{I}(B\rightarrow K\ell^{+}\ell^{-})=-0.58^{+0.29}_{-0.37}\pm 0.02$ and
${\cal A}_{I}(B\rightarrow K^{*}\ell^{+}\ell^{-})=-0.25^{+0.20}_{-0.17}\pm
0.03$. The isospin asymmetries are all consistent with the SM predictions. All
results are in good agreement with those of the Belle, CDF, and LHCb
experiments.
## XI ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues
in achieving the excellent luminosity and machine conditions that have made
this work possible. The success of this project also relies critically on the
expertise and dedication of the computing organizations that support BABAR.
The collaborating institutions wish to thank SLAC for its support and the kind
hospitality extended to them. This work is supported by the US Department of
Energy and National Science Foundation, the Natural Sciences and Engineering
Research Council (Canada), the Commissariat à l’Energie Atomique and Institut
National de Physique Nucléaire et de Physique des Particules (France), the
Bundesministerium für Bildung und Forschung and Deutsche
Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare
(Italy), the Foundation for Fundamental Research on Matter (The Netherlands),
the Research Council of Norway, the Ministry of Education and Science of the
Russian Federation, Ministerio de Ciencia e Innovación (Spain), and the
Science and Technology Facilities Council (United Kingdom). Individuals have
received support from the Marie-Curie IEF program (European Union) and the A.
P. Sloan Foundation (USA).
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|
arxiv-papers
| 2012-04-17T22:17:43 |
2024-09-04T02:49:29.773727
|
{
"license": "Public Domain",
"authors": "The BABAR Collaboration",
"submitter": "Liang Sun",
"url": "https://arxiv.org/abs/1204.3933"
}
|
1204.3945
|
# Axially symmetric pseudo-Newtonian hydrodynamics code
Jinho Kim,1 Hee Il Kim,1,211footnotemark: 1 Matthew William
Choptuik,3,411footnotemark: 1 and Hyung Mok Lee111footnotemark: 1
1Department of Physics and Astronomy, FPRD, Seoul National University, Seoul,
151-742, Korea
2Korea Institute of Science and Technology Information, 245 Daehak-ro,
Yuseong-gu, Daejeon, 305-806, Korea
3Department of Physics and Astronomy, University of British Columbia,
Vancouver, Canada
4Canadian Institute for Advanced Research Cosmology & Gravity Program E-mail:
jinho@astro.snu.ac.kr (JK); khi@astro.snu.ac.kr (HIK); choptuik@phas.ubc.ca
(MWC); hmlee@snu.ac.kr (HML)
(Released 2012 Xxxxx XX)
###### Abstract
We develop a numerical hydrodynamics code using a pseudo-Newtonian formulation
that uses the weak field approximation for the geometry, and a generalized
source term for the Poisson equation that takes into account relativistic
effects. The code was designed to treat moderately relativistic systems such
as rapidly rotating neutron stars. The hydrodynamic equations are solved using
a finite volume method with High Resolution Shock Capturing (HRSC) techniques.
We implement several different slope limiters for second order reconstruction
schemes and also investigate higher order reconstructions such as PPM, ENO and
WENO. We use the method of lines (MoL) to convert the mixed spatial-time
partial differential equations into ordinary differential equations (ODEs)
that depend only on time. These ODEs are solved using second and third order
Runge-Kutta methods. The Poisson equation for the gravitational potential is
solved with a multigrid method, and to simplify the boundary condition, we use
compactified coordinates which map spatial infinity to a finite computational
coordinate using a tangent function. In order to confirm the validity of our
code, we carry out four different tests including one and two dimensional
shock tube tests, stationary star tests of both non-rotating and rotating
models and radial oscillation mode tests for spherical stars. In the shock
tube tests, the code shows good agreement with analytic solutions which
include shocks, rarefaction waves and contact discontinuities. The code is
found to be stable and accurate: for example, when solving a stationary
stellar model the fractional changes in the maximum density, total mass, and
total angular momentum per dynamical time are found to be $3\times 10^{-6}$,
$5\times 10^{-7}$ and $2\times 10^{-6}$, respectively. We also find that the
frequencies of the radial modes obtained by the numerical simulation of the
steady state star agree very well with those obtained by linear analysis.
###### keywords:
relativistic processes - gravitation - hydrodynamics hydrodynamics- methods:
numerical
††pagerange: Axially symmetric pseudo-Newtonian hydrodynamics code–A††pubyear:
2012
## 1 Introduction
It is necessary to take into account both special- and general relativistic
effects in the studies of the dynamics of compact astrophysical object such as
neutron stars and black holes. Some pulsars produce pulses of up to 1 KHz,
corresponding to rotation speeds at the surface of around $0.2c$. Their
typical sizes and masses are known to be around $10\rm{km}$ and $1.4\sim
2\mbox{$\rm{M}_{\odot}$}$, respectively, giving compactness,
$GM/Rc^{2}=0.2\sim 0.3$. Therefore, a Newtonian approach cannot properly
describe neutron stars, even for the non-rotating case.
In general relativity, the dynamics of gravity (or spacetime) can be studied
by solving the Einstein equations. The equations of motion for the matter are
given, in part, by the conservation law of the energy-momentum tensor which
itself sources the gravitational field. Computational approaches for solving
general relativistic field equations constitute the field of numerical
relativity.
Over the past few decades, many general relativistic hydrodynamic codes have
been developed, starting with Wilson (1972) who proposed a 3+1 Eulerian
formulation (see also Wilson & Mathew, 2003). Although Wilson’s numerical
approach was widely used to study problems such as core collapse and accretion
disks, it produced large errors when fluid flows became ultra-relativistic
(Centrella & Wilson, 1984; Norman & Winkler, 1986). In order to avoid these
excessive errors, a new formulation was proposed by Marti et al. (1991). This
formulation makes it possible to use existing numerical techniques based on
characteristic approaches for Newtonian hydrodynamics. In particular, these
include High Resolution Shock Capturing (hereafter HRSC) methods that reduce
the order of accuracy near shocks, but minimize the amount of numerical
dissipation. This dissipation is very unnatural and can result in non-physical
effects in the numerical results. Marti’s formulation was extended to the
general relativistic case by the Valencia group (Font et al., 2000), and this
last work forms the basis for most recent general relativistic hydrodynamical
codes. Recent reviews of the formulation and numerical methods can be found in
Martí & Müller (2003) and Font (2008).
However, when working in multiple spatial-dimensions, it still requires a lot
of computational resources to treat fluid dynamics in concert with the
evolution of the general relativistic gravitational field. In addition,
numerical relativity simulations have frequently encountered instabilities
which are often associated with violations of the Hamiltonian and momentum
constraints. (However, with the development of new formulations which cast the
Einstein equations in appropriate hyperbolic forms, as well as the use of
constraint-damping techniques, significant progress has been made on this
front: see Sarbach & Tiglio, 2012 for a very recent review of this subject).
For these reasons, simulations using Newtonian gravity are still used even
though they are not applicable to very compact objects.
The aims of this paper are 1) to introduce a new formulation which applies a
pseudo-Newtonian approach (Kim et al., 2009) to the study of moderately
relativistic objects and 2) to describe a numerical implementation of this
method. In our pseudo-Newtonian approach, which was introduced by Kim et al.
(2009) for steady state models, the gravitational field is treated by a weak
field approximation, but special relativistic effects are correctly taken into
account. Specifically, the Newtonian gravitational potential that appears in
the weak field metric satisfies a Poisson equation, but the mass density that
appears as a source term for that equation is modified to include relativistic
effects. Of course this method cannot be applied to highly relativistic
systems, but Kim et al. (2009) showed that the pseudo-Newtonian formulation is
valid for the modeling of mildly compact objects, such as rotating neutron
stars having surface rotation velocity up to $\sim 0.2c$ and compactness $\sim
0.2$ (Kim et al., 2009). In this paper, we extend the pseudo-Newtonian
approach to hydrodynamical systems where the flows can be ultra-relativistic
and gravity can be moderately strong.
The remainder of the paper is structured as follows: In section 2, we present
the formulation and governing equations for our system, while the numerical
techniques employed in our study are given in section 3. We discuss various
numerical tests of our code’s treatment of hydrodynamics for the case of shock
tubes in 4, and for stationary stars in 5. A test which compares radial
pulsation mode frequencies for polytropic stars determined through dynamical
evolution to those computed in linear theory is detailed in section 6. We
conclude with a summary and discussion in section 7.
Throughout this paper we use units in which $c=G=\mbox{$\rm{M}_{\odot}$}=1$:
these correspond a to unit time $=4.92\times 10^{-3}\rm{ms}$, unit length
$=1.47\,\rm{km}$ and unit mass $=1.99\times 10^{33}\rm{g}$.
## 2 Formulation
Our pseudo-Newtonian method was first discussed in the steady state context by
Kim et al. (2009). We assume the weak field metric,
$\begin{split}ds^{2}&=g_{\mu\nu}dx^{\mu}dx^{\nu}\\\
&=-(1+2\Phi)dt^{2}+(1+2\Phi)^{-1}\delta_{ij}dx^{i}dx^{j},\end{split}$ (1)
where $g_{\mu\nu}$ is the spacetime metric and $\Phi$ is the Newtonian
gravitational potential. With this metric, we neglect all higher order effects
such as frame dragging and describe gravity using only a single gravitational
potential, just as in the Newtonian case. The gravitational potential
satisfies a Poisson equation with the active mass density,
$\rho_{\rm{active}}$, providing the source:
$\nabla^{2}\Phi=4\pi\rho_{\rm{active}}.$ (2)
The active mass density is computed from the relativistic definition of the
energy. For a perfect fluid, the energy momentum tensor can be expressed as
$T^{\mu\nu}=\rho_{0}hu^{\mu}u^{\nu}+Pg^{\mu\nu},$ (3)
where the specific enthalpy is defined by
$h=1+\epsilon+\frac{P}{\rho_{0}}.$ (4)
The active mass density is then given by
$\rho_{\rm{active}}=T-2T^{0}_{0}=T^{i}_{i}-T^{0}_{0}=\rho_{0}h\frac{1+v^{2}}{1-v^{2}}+2P.$
(5)
In Eqs. (3), (4) and (5), $\rho_{0}$ is the rest mass density which is
proportional to the number density of baryons of the fluid, $P$ is the
pressure, $u^{\mu}$ is the four velocity of a fluid element with respect to an
Eulerian observer, $\epsilon$ is the specific internal energy, $T$ is the
trace of the energy-momentum tensor ($T=g_{\mu\nu}T^{\mu\nu}$), and $v$ is the
three dimensional fluid velocity. Unlike the Newtonian case,
$\rho_{\rm{active}}$ includes all sources of energy.
The equations governing the motion of the fluid matter can be derived from the
conservations laws for the energy-momentum tensor and the fluid’s matter
current, i.e., $\nabla_{\mu}T^{\mu\nu}=0$ and $\nabla_{\mu}J^{\mu}=0$. In the
ADM decomposition of spacetime (Arnowitt et al., 1962), the metric
($g_{\mu\nu}$) can be expressed in the following form by considering the
foliation of spacetime using 3-dimensional hypersurfaces defined by $t={\rm
const.}$:
$ds^{2}=-\alpha^{2}dt^{2}+\gamma_{ij}\left(dx^{i}+\beta^{i}dt\right)\left(dx^{j}+\beta^{j}dt\right),$
(6)
Here $\gamma_{ij}$ is the spatial metric, defined on each hypersurface, while
$\alpha$ and $\beta^{i}$ are known as the lapse, and shift vector,
respectively, and encode the 4-fold coordinate freedom of general relativity.
Flux-conservative formulations of hydrodynamics have been applied very
successfully in computational fluid dynamics. To cast the fluid equations in
flux-conservative form we first define so-called conservative variables ($q$)
in terms of the original hydrodynamic variables (so-called primitive
variables, $w$),
$q=\left(\begin{array}[]{c}D\\\ S_{i}\\\
\tau\end{array}\right)=\left(\begin{array}[]{c}\rho_{0}W\\\
\rho_{0}hW^{2}v_{i}\\\
\rho_{0}hW^{2}-P-D\end{array}\right),\,w=\left(\begin{array}[]{c}\rho_{0}\\\
v^{i}\\\ P\end{array}\right),$ (7)
where $W=1/\sqrt{1-\gamma_{ij}v^{i}v^{j}}$. With these definitions, and with
the metric (Eq. (6)), we can then write the Euler equation as (Font et al.,
2000)
$\frac{\partial\left(\sqrt{\gamma}q\right)}{\partial
t}+\frac{\partial\left(\sqrt{-g}f^{i}\right)}{\partial
x^{i}}=\sqrt{-g}\Sigma,$ (8)
where the fluxes $f^{i}$ and the sources $\Sigma$ are given by
$\displaystyle f^{i}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}D\left(v^{i}-\frac{\beta^{i}}{\alpha}\right)\\\
S_{j}\left(v^{i}-\frac{\beta^{i}}{\alpha}\right)+P\delta^{i}_{j}\\\
\tau\left(v^{i}-\frac{\beta^{i}}{\alpha}\right)+Pv^{i}\end{array}\right],$
(12) $\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}0\\\
T^{\mu\nu}\left(\partial_{\mu}g_{\mu j}-\Gamma^{\lambda}_{\mu\nu}g_{\lambda
j}\right)\\\ \alpha\left(T^{\mu
0}\partial_{\mu}\left(\ln\alpha\right)-\Gamma^{0}_{\mu\nu}T^{\mu\nu}\right)\end{array}\right].$
(16)
Here $\sqrt{\gamma}$ and $\sqrt{-g}$ are the determinants of $\gamma_{ij}$ and
$g_{\mu\nu}$, respectively, and are related by
$\sqrt{-g}=\alpha\sqrt{\gamma}$. It is well known that for a perfect fluid,
the system of equations derived from the conservation laws is not closed: the
number of dynamical equations is always less than the number of unknowns.
As is also well known, the equation of state (hereafter EOS) for the fluid
provides an additional equation, but in the general case it also introduces
other unknowns. In order to completely close the hydrodynamical equations, an
energy balance equation is often used. However, under certain circumstances,
we can adopt rather simple EOSs that do not introduce any further variables:
adiabatic and isothermal EOSs provide specific examples.
Realistic EOSs are usually determined by theoretical calculations and
experimental measurements. However, there are physical regimes where our
understanding of the nature of the matter is quite incomplete. Specifically,
in the case where the matter density is significantly above nucleon density,
there remain large uncertainties in the correct EOS. Thus, for example, the
EOS at the core of neutron stars is still not very well understood. Here we
ignore these difficulties, and for the purpose of testing our code, use two
types of very simple EOS. The first is the ideal gas EOS which can be written
in the following form:
$P=\left(\Gamma-1\right)\rho_{0}\epsilon,$ (17)
and corresponds to the isothermal EOS. We use this EOS in the shock tube tests
described in (section 4). The second EOS results from the isentropic
assumption, whereby Eq. (17) becomes the polytropic EOS:
$P=K\rho_{0}^{1+\frac{1}{N}}.$ (18)
Here $K$ and $N$ are the polytropic constant and index respectively. The
polytropic EOS of state is the generalized form of the adiabatic one; a fluid
which is governed by it does not generate entropy, and shock formation is thus
generically prohibited. We use this EOS in the pulsation mode test (section 6
and Appendix A).
Using the above formulation, we are now ready to describe in detail the
pseudo-Newtonian hydrodynamical equations used in our code. We limit our study
here to axisymmetric systems, and adopt cylindrical coordinates ($R,Z,\phi$)
such that
$ds^{2}=-(1+2\Phi)dt^{2}+\frac{1}{1+2\Phi}\left(dR^{2}+dZ^{2}+R^{2}d\phi^{2}\right).$
(19)
The lapse function and shift vector are thus given by $\alpha=\sqrt{1+2\Phi}$
and $\beta^{i}=0$. In addition, we enforce the equatorial symmetry at $z=0$
since the phenomena involving $l=\rm{odd}$ modes are not dominant in most
cases of neutron star dynamics, where $l$ is from the spherical harmonics. In
this coordinate system, the conservative and primitive variables are
$q=\left(\begin{array}[]{c}D\\\ S_{R}\\\ S_{Z}\\\ S_{\phi}\\\
\tau\end{array}\right)=\left(\begin{array}[]{c}\rho_{0}W\\\
\rho_{0}hW^{2}v_{R}\\\ \rho_{0}hW^{2}v_{Z}\\\ \rho_{0}hW^{2}v_{\phi}\\\
\rho_{0}hW^{2}-P-D\end{array}\right),\,w=\left(\begin{array}[]{c}\rho_{0}\\\
v^{R}\\\ v^{Z}\\\ v^{\phi}\\\ P\end{array}\right).$ (20)
The final form of the hydrodynamical equations then becomes
$\frac{\partial\left(\sqrt{\gamma}q\right)}{\partial
t}+\frac{\partial\left(\sqrt{-g}f^{R}\right)}{\partial
R}+\frac{\partial\left(\sqrt{-g}f^{Z}\right)}{\partial Z}=\sqrt{-g}\Sigma$
(21)
where
$\displaystyle f^{R}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}Dv^{R}\\\ S_{R}v^{R}+P\\\
S_{Z}v^{R}\\\ S_{\phi}v^{R}\\\ \tau v^{R}+Pv^{R}\end{array}\right],$ (27)
$\displaystyle f^{Z}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}Dv^{Z}\\\ S_{R}v^{Z}\\\
S_{Z}v^{Z}+P\\\ S_{\phi}v^{Z}\\\ \tau v^{Z}+Pv^{Z}\end{array}\right],$ (33)
$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}0\\\
-\frac{\rho_{\rm{active}}}{1+2\Phi}\frac{\partial\Phi}{\partial
R}+\frac{S_{\phi}v^{\phi}}{R}+\frac{P}{R}\\\
-\frac{\rho_{\rm{active}}}{1+2\Phi}\frac{\partial\Phi}{\partial Z}\\\ 0\\\
-\left(S_{R}\frac{\partial\Phi}{\partial R}+S_{Z}\frac{\partial\Phi}{\partial
Z}\right)\end{array}\right].$ (39)
Using Eq. (19) we have $\sqrt{\gamma}=R\left(1+2\Phi\right)^{-3/2}$, and
$\sqrt{g}=R\left(1+2\Phi\right)^{-1}$. In obtaining the expressions in Eq.
(39) we have used the assumption of slow changes of the potential relative to
the gradients ($\frac{\partial\Phi}{\partial t}\ll\frac{\partial\Phi}{\partial
R}$ or $\frac{\partial\Phi}{\partial Z}$). Recently, Nagakura et al. (2011)
used a similar method in the context of jet propagation in a uniform medium,
but adopted a slightly different linear momentum equation than ours. (See Eq.
(39) and compare with Eqs. (2) and (3) in Nagakura et al., 2011).
Finally, the gravitational Poisson equation in our coordinate system is
$\frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial\Phi}{\partial
R}\right)+\frac{\partial^{2}\Phi}{\partial Z^{2}}=4\pi\rho_{\rm{active}}.$
(40)
Note that the second component of $\Sigma$ contains terms which, individually,
become singular on the axis of symmetry ($R=0$). In addition, there are other
terms in the equations of motion that need to be treated carefully as $R\to
0$. This is done by demanding regularity at the axis, and by considering the
parity of each function, with respect to $R$, in that limit. In particular,
$\rho_{0}$, $v^{Z}$, $v^{\phi}$, $P$, $D$, $S_{Z}$, $S_{\phi}$, and $\tau$ are
all even functions of $R$ as $R\to 0$, while $v^{R}$ and $S_{R}$ are odd.
Taking this into account, Eq. (21) and $\Sigma$ in Eq. (39) become
$\frac{\partial\left(\sqrt{\gamma^{\prime}}q\right)}{\partial
t}+2\frac{\partial\left(\sqrt{-g^{\prime}}f^{R}\right)}{\partial
R}+\frac{\partial\left(\sqrt{-g^{\prime}}f^{Z}\right)}{\partial
Z}=\sqrt{-g^{\prime}}\Sigma$ (41)
and
$\Sigma=\left[\begin{array}[]{ccccc}0\\\ 0\\\
-\frac{\rho_{\rm{active}}}{1+2\Phi}\frac{\partial\Phi}{\partial Z}\\\ 0\\\
-S_{Z}\frac{\partial\Phi}{\partial Z}\end{array}\right],$ (42)
where $\sqrt{\gamma^{\prime}}=\left(1+2\Phi\right)^{-3/2}$ and
$\sqrt{-g^{\prime}}=\left(1+2\Phi\right)^{-1}$. The coefficient of
$\frac{\partial\left(\sqrt{-g}f^{R}\right)}{\partial R}$ in Eq. (41) becomes
$2$ instead of $1$, while the other variables, such as $q$, $f^{R}$ and
$f^{Z}$, are unchanged from Eq. (39).
Finally, using L’Hopital’s theorem at $R=0$, the singular term
$R^{-1}\partial\Phi/\partial R$ in the Poisson equation (40) is replaced by
$\partial^{2}\Phi/\partial R^{2}$.
## 3 Numerical Methods
In this section, we describe our numerical methods for solving the coupled
hydrodynamical and Poisson equations. We mainly use the finite volume methods
for the hydrodynamical equations and the a finite difference approach for the
Poisson equation. In the finite volume method, each grid cell represents
volume averaged hydrodynamic quantities i.e., $\bar{q}=\frac{1}{\Delta V}\int
qdV$. After applying the finite volume method our hydro equations can be
reduced to Riemann problems which consider the time evolution of initial
conditions given by two distinct states that adjoin at some interface (so that
there are, in general, discontinuities across one or more physical quantities
at the interface). A very important property of the finite volume method is
that it maintains the local conservation properties of the flow in the
computational grid.
In the dynamics of compressible fluids, we inevitably encounter discontinuous
behaviors such as shocks, rarefactions or contact discontinuities. To treat
such discontinuities without introducing numerical instabilities or spurious
oscillations, we use High Resolution Shock Capturing (HRSC) techniques that
generically reduce the order of accuracy of the numerical scheme near
discontinuities or when one or more of the fluid variables are at a local
maximum. A key ingredient to the success of the HRSC methods is the
calculation of fluxes through cell boundaries. To compute these fluxes we need
approximate values for the primitive variables at the cell boundaries. We have
implemented second order slope limiters such as minmod (van Leer, 1979),
monotonized central difference (MC hereafter, van Leer, 1977) and superbee
(Roe, 1985), as well as a third order slope limiter proposed by Shibata (2003)
and which is based on the minmod function (3minmod hereafter). Other
reconstruction methods such as the third order Piecewise Parabolic Method (PPM
hereafter, Colella & Woodward, 1984), Essentially Non-Oscillatory method (ENO,
Harten et al., 1987) and Weighted ENO (WENO, Liu et al., 1994; Jiang & Shu,
1996), which has an arbitrary order of accuracy, were also implemented.
In the implementation of HRSC schemes it is not efficient to exactly solve the
Riemann problems which arise since an excessively large amount of
computational resources per cell are then needed to calculate the fluxes.
Thus, an approximate calculation of fluxes is performed. We implemented the
following three schemes: Roe (Roe, 1981), Marquina (Donat & Marquina, 1996;
Donat, 1998), and HLLE (Harten et al., 1983; Einfeldt, 1988; Einfeldt et al.,
1991) approximations. The Roe approximation is based on the Rankine-Hugoniot
jump condition and Marquina’s approach generalizes Roe’s scheme. The HLLE
algorithm comes from a very simple two wave approximation and produces the
most dissipative and stable results. We have mainly used HLLE for reducing
computational cost but found that our results were not significantly
influenced by the type of flux approximation used.
In order to solve the Poisson equation (which is elliptic) for the
gravitational potential we use the multigrid method, which can quickly reduce
low frequency error components in the solution by adopting hierarchical grid
levels (Brandt, 1977). One of the difficulties we often encounter with the
Poisson equation is in the proper implementation of the boundary conditions.
For example, one of the natural boundary conditions is $\Phi=0$ at $\infty$,
but in the coordinates adopted in the previous section the computational
domain cannot reach spatial infinity. We thus now refer to our previous
coordinates as $(r,z)$ and introduce new coordinates $(R,Z)$ which compactify
the spatial domain, mapping the infinities in each spatial direction to finite
coordinate values. Specifically, we choose the same type of compactification
for both $r$ and $z$ coordinates, namely a tangent function, but allow a
certain portion of the domain to remain ”’uncompactified”:
$\displaystyle r=\left\\{\begin{array}[]{ll}R&\mbox{if $R\leq r_{0}$}\\\
r_{0}+r_{1}\tan\left(\frac{R-r_{0}}{r_{1}}\right)&\mbox{if
$R>r_{0}$}\end{array}\right.,$ (45) $\displaystyle
z=\left\\{\begin{array}[]{ll}Z&\mbox{if $Z\leq z_{0}$}\\\
z_{0}+z_{1}\tan\left(\frac{Z-z_{0}}{z_{1}}\right)&\mbox{if
$Z>z_{0}$}\end{array}\right..$ (48)
Here, the four parameters $z_{0}$, $z_{1}$, $r_{0}$ and $r_{1}$ control the
compactification, and we chose this specific form for the coordinate
transformation since it guarantees that the compactified coordinates smoothly
transition to the original ones near the origin. We note that we solve the
hydrodynamical and gravitational equations on separate spatial domains:
$[0:r_{0},0:z_{0}]$ for the hydrodynamic calculations and
$[0:r_{0}+\frac{2}{\pi}r_{1},0:z_{0}+\frac{2}{\pi}z_{1}]$ for the computation
of the gravitational potential. There ranges correspond to $[0:r_{0},0:z_{0}]$
and $[0:\infty,0:\infty]$, respectively, in the original cylindrical
coordinates $(r,z)$. In the compactified coordinates, the Poisson equation is
written as
$\displaystyle\frac{1}{rf(R)}\frac{\partial}{\partial
R}\left(\frac{r}{f(R)}\frac{\partial\Phi(R,Z)}{\partial R}\right)$
$\displaystyle+\frac{1}{g(Z)}\frac{\partial}{\partial
Z}\left(\frac{1}{g(Z)}\frac{\partial\Phi(R,Z)}{\partial
Z}\right)=4\pi\rho_{\rm{active}},$ (49)
where $f(R)$ and $g(Z)$ are given by
$\displaystyle f(R)=\frac{dr}{dR}=\left\\{\begin{array}[]{ll}1&\mbox{if $R\leq
r_{0}$}\\\ \sec^{2}\left(\frac{R-r_{0}}{r_{1}}\right)&\mbox{if
$R>r_{0}$},\end{array}\right.$ (52) $\displaystyle
g(Z)=\frac{dz}{dZ}=\left\\{\begin{array}[]{ll}1&\mbox{if $Z\leq z_{0}$}\\\
\sec^{2}\left(\frac{Z-z_{0}}{z_{1}}\right)&\mbox{if
$Z>z_{0}$}\end{array}\right..$ (55)
As just noted, the domain for the hydrodynamical calculation is finite, i.e.
we do not solve the hydrodynamical equations on the full compactified domain,
and we thus must be careful to choose values of $r_{0}$ and $z_{0}$ large
enough so that there is no outflux of matter through the $r=r_{0}$ and/or
$z=z_{0}$ boundaries. In our code we set $r_{0}=z_{0}=\eta r_{e}$ where $\eta$
is a free parameter and $r_{e}$ is the equatorial radius of the rotating star
as obtained from the procedure we use to calculate the initial stellar model.
For the pulsation mode test described in section 6, a typical choice is
$\eta=2$. This means that the hydrodynamical computational domain extends
twice the distance of the stellar radius in both the $R$ and $Z$ directions:
this choice is found to be sufficient for our study. The values of $r_{1}$ and
$z_{1}$ are automatically determined by requiring the multigrid domain to be 2
times larger than the size of hydrodynamic domain in compactified coordinates,
i.e., $r_{1}=\frac{\pi}{2}r_{0}$ and $z_{1}=\frac{\pi}{2}z_{0}$.
In our multigrid algorithm, we use line relaxation for our basic smoother,
whereby all grid point values given by $R={\rm const.}$ or $Z={\rm const.}$
are updated simultaneously (constant-$R$ and constant-$Z$ sweeps are
alternated). We cannot use point-wise relaxation since, as is well known, such
a technique is not a good smoother when there is significant anisotropy in the
coefficients of the second derivative terms in the elliptic operator being
treated. This is the case in our compactified coordinate system, particularly
near the domain boundaries. We have used second- and fourth-order finite-
difference approximations to the Poisson equation, and these lead to tri- and
pentadiagonal linear systems, respectively, that must be solved to implement
the line relaxations. We use the routines DGTSV (tridiagonal) and DGBSV
(banded/pentadiagonal) routines from LAPACK to perform these solutions.
In order to integrate the discretized hydrodynamical equations, we use the
method of lines (MOL), transforming our partial differential equations in time
and space to ordinary differential equations (ODEs) with respect to the time.
To solve these ODEs, we then employ second and third order Runge-Kutta
methods, which are known to have the Total Variation Diminishing (TVD)
property.
## 4 Shock tube Tests
In order to verify the accuracy and the convergence of our numerical code, we
first carry out rigorous tests using initial configurations having analytic
solutions. In this section, we present the results of such tests for the case
where there is no self-gravity (i.e. pure hydrodynamics). Another test of the
entire code—including our treatment of the gravitational field—is described in
the next section.
Shock tube tests are Riemann problems where the initial configuration of the
fluid is given by two states having, in general, different densities,
pressures and velocities, on the left and right halves of the tube. Three
possible distinct features emerge from the subsequent evolution: a shock, a
rarefaction fan, and a contact discontinuity. We carried out 1D and 2D
numerical simulations with 3 different parameter sets previously used by Zhang
& MacFadyen (2006). These parameters are listed in Table 1, where superscripts
$R$ and $L$ represent the fluid states in the right and left halves,
respectively, of the tube.
Table 1: Initial values of physical quantities for Shock tube tests (Riemann problem). Problem | $\Gamma$ | $\rho_{0}^{L}$ | $\rho_{0}^{R}$ | $v^{L}$ | $v^{R}$ | $P^{L}$ | $P^{R}$
---|---|---|---|---|---|---|---
1 | 5/3 | $10.0$ | $1.0$ | $0.0$ | $0.0$ | $13.33$ | $10^{-8}$
2 | 5/3 | $1.0$ | $1.0$ | $0.0$ | $0.0$ | $1000.0$ | $10^{-2}$
3 | 4/3 | $1.0$ | $1.0$ | $0.9$ | $0.0$ | $1.0$ | $10.0$
### 4.1 1D test in Cartesian Coordinates
We first we present the results of our 1D tests. In these tests, we carefully
examine how the distinct features predicted by the analytic solutions are
reproduced by different methods, and measure the accuracy and convergence rate
of the various solutions obtained.
In problem 1, the initial discontinuity gives 3 different types of solutions
(shock, rarefaction, contact discontinuity). Figure 1 shows the results at
$t=0.4$ obtained using four different methods of reconstruction: minmod (top
left), MC (top right), 3minmod (bottom left) and PPM (bottom right). We
observe that the minmod method is quite dissipative, yielding rather smooth
solutions that cannot accurately describe the shockwave. We also find that at
low resolution the height of the shock is not well reproduced if we use the
minmod method. MC and 3minmod give almost similar results, while PPM shows the
best behaviour near the shock.
00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/15$$v$ | 00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/15$$v$
---|---
00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/15$$v$ | 00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/15$$v$
Figure 1: One dimensional shock tube test of problem 1 at $t=0.4$ with
different reconstruction methods: minmod(top-left), MC (top right), 3minmod
(bottom left), and PPM (bottom right). The initial discontinuity is at
$x=0.5$. We use 512 uniform grid points. The numerical results are shown in 3
different colours: rest mass density (pink), pressure (red) and velocity
(blue). The solid lines show the analytic solutions.
The second test problem (Problem 2) is the so-called blast wave test which
produces a very sharp and thin shell in density between the shock and contact
discontinuity. Generally, numerical codes are not able to perfectly resolve
this very thin shell because it can span only a few grid cells, even in very
high resolution calculations. Nonetheless, this test provides insight as to
how well a code can handle such a feature. As can see in Figure 2, PPM again
gives the best results. although it still shows large errors at the shock.
00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/15$$P/1000$$v$ | 00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/15$$P/1000$$v$
---|---
00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/15$$P/1000$$v$ | 00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/15$$P/1000$$v$
Figure 2: Same as Fig. 1 for problem 2.
The third problem generates a strong reverse shock but numerical solution has
oscillatory features near the shock front. Generally speaking, the oscillation
can be easily damped out if the numerical scheme is significantly dissipative.
Numerical dissipation also tends to weaken the sharpness of the discontinuity.
In Figure 3, one can see that the minmod methods, which, as already noted, is
the most dissipative of the techniques we use, gives relatively small
amplitude oscillations, except near the discontinuity. The more non-
dissipative methods describes the shock features well, but produce rather
large amplitude oscillatory behavior.
00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/25$$v$ | 00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/25$$v$
---|---
00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/25$$v$ | 00.20.40.60.8100.20.40.60.81 $\rho_{0}$, $P$, $v$ $x$-axis$\rho_{0}/10$$P/25$$v$
Figure 3: Same as Figs. 1 and 2 for problem 3.
To quantify the deviation of our numerical results from the analytic
solutions, we use the $L_{1}$ norm of the errors, defined by
$L_{1}=\sum_{i=1}^{N}\Delta x_{i}|q_{i}-q(x_{i})|$, where $q(x_{i})$ is the
value of the analytic solution at point $x_{i}$. We summarize the results in
Table 2. The convergence rate ($\log_{2}\left[L_{1}^{2h}/L_{1}^{h}\right]$) in
the table should be close to 1, which corresponds to the $1^{\textrm{st}}$
order nature of the HRSC scheme near the shock where the most of the
$L_{1}$-norm error occurs. However, it can deviate from that value due to the
oscillatory features near the shock.
Table 2: The $L_{1}$ norm of the error and its convergence rate for each of the test problems using different resolutions and different reconstruction schemes. | N
---|---
| 64 | 128 | 256 | 512 | 1024 | 2048
Problem 1 | minmod | $L_{1}$ norm ($\times 10^{-2}$) | $25.7$ | $16.5$ | $9.48$ | $5.02$ | $2.66$ | $1.49$
convergence rate | - | $0.64$ | $0.80$ | $0.92$ | $0.91$ | $0.84$
MC | - | $15.3$ | $9.43$ | $5.25$ | $2.79$ | $1.46$ | $0.830$
- | - | $0.70$ | $0.85$ | $0.91$ | $0.93$ | $0.82$
3minmod | - | $17.8$ | $11.0$ | $5.82$ | $2.99$ | $1.51$ | $0.816$
- | - | $0.69$ | $0.92$ | $0.96$ | $0.98$ | $0.89$
PPM | - | $12.3$ | $6.55$ | $3.43$ | $1.74$ | $0.877$ | $0.431$
- | - | $0.91$ | $0.93$ | $0.98$ | $0.99$ | $1.0$
Problem 2 | - | - | $30.1$ | $21.0$ | $20.1$ | $15.8$ | $10.9$ | $6.93$
- | - | $0.52$ | $0.061$ | $0.34$ | $0.54$ | $0.65$
- | - | $27.8$ | $18.2$ | $14.9$ | $10.4$ | $6.28$ | $3.77$
- | - | $0.61$ | $0.29$ | $0.52$ | $0.73$ | $0.74$
- | - | $28.3$ | $17.9$ | $13.7$ | $8.84$ | $5.05$ | $2.72$
- | - | $0.66$ | $0.39$ | $0.63$ | $0.81$ | $0.89$
- | - | $29.5$ | $17.9$ | $12.7$ | $7.79$ | $3.73$ | $2.13$
- | - | $0.73$ | $0.49$ | $0.71$ | $1.1$ | $0.81$
Problem 3 | - | - | $15.5$ | $10.0$ | $6.19$ | $3.65$ | $2.37$ | $1.58$
- | - | $0.63$ | $0.69$ | $0.76$ | $0.63$ | $0.59$
- | - | $14.9$ | $7.73$ | $5.40$ | $2.72$ | $1.64$ | $1.04$
- | - | $0.95$ | $0.52$ | $0.99$ | $0.73$ | $0.66$
- | - | $13.0$ | $6.97$ | $4.40$ | $2.25$ | $1.35$ | $0.867$
- | - | $0.90$ | $0.66$ | $0.97$ | $0.74$ | $0.63$
- | - | $7.26$ | $3.93$ | $2.41$ | $1.08$ | $0.547$ | $0.393$
- | - | $0.89$ | $0.70$ | $1.16$ | $0.98$ | $0.47$
0.010.11PPM3-minmodMCminmod $\rm{L}_{1}\rm{errornorm}$ Problem 1Problem
2Problem 3
---
0.40.60.811.21.4PPM3-minmodMCminmod Convergence Rate Problem 1Problem 2Problem
3
Figure 4: The $L_{1}$ norm (top) and convergence rate (bottom) when the number
of grid points is 512 with different reconstruction methods. Three different
shock tube problems are shown with different colors (problem 1: blue, problem
2: red and problem 3: sky blue).
Figure 4 shows the $L_{1}$ norms and convergence rates for each problem when
the grid resolution is $\rm{N}=512$. Although no single method stands out in
our 1D shock tube tests, we conclude from from the values of the $L_{1}$ error
norms and convergence rates that PPM gives the most promising results.
### 4.2 2D test in Cylindrical Coordinates
Since the cylindrical coordinate system we have adopted is curvilinear,
1-dimensional shock tube tests are not sufficient for assessing our code’s
accuracy and convergence. In Cartesian coordinates, fluxes between cells which
have the same state cancel out. For example, if we carry out the shock tube
test in the $x$-direction, then the fluxes in the $y$ and $z$ directions are
identical in every grid cell, meaning that the net flux is 0. Therefore, 1D
shock tube tests performed with codes that use 2- or 3D Cartesian coordinates
produce exactly the same results as a 1D code. However, in cylindrical
coordinates, fluxes do not cancel in this way, but rather are balanced by
source terms. This difference may give additional non-physical effects,
especially near discontinuities.
Therefore, we carried out the first of the shock tube tests listed in Table 1
in cylindrical coordinates, where we placed the discontinuity on the Z=0
plane. Figure 5 shows the solution resulting solution on the $Z$-axis. If we
use the minmod method, the 2D results are similar to the 1 dimensional ones.
In addition, although PPM produces better results minmod, it cannot produce
the sharp features of the shock seen in the 1D test: this is due to the
dissipation caused by the imbalance between the net flux and the source term.
We can also see that 3minmod and PPM yield quite similar results. We checked
the differences in solutions at different $z={\rm const.}$ planes and found
that they are negligibly small ($\sim 10^{-13}$) compared to the truncation
errors. Overall, however, although the 2D results show more dissipation than
the 1D ones, the relative differences in the solutions are not significant
(for our purposes). In particular, both agree acceptably with the analytic
forms.
00.20.40.60.8100.20.40.60.81
$\rho_{0}/10$
$x$-axisminmod3minmodPPM
Figure 5: The solution on the axis at t=0.4 in problem 1 in section 4.1. The 3
different reconstruction methods (minmod: blue, 3-minmod: sky blue and PPM:
magenta) are shown.
## 5 Stationary Star Test
The tests just reported did not involve the effects of the gravitational
field. In this section, we test our treatment of the Poisson equation for the
gravitational potential as well as the hydrodynamics.
With an ideal code, the evolution of a stationary star should also be
stationary. However, in practice, all codes that dynamically evolve stationary
states show some level of fluctuation due to finite grid resolution and
intrinsic errors in the numerical scheme used. In this section, we show the
time evolution of the physical quantities of non-rotating and the rotating
stars, and investigate the dependence of this time behaviour by changing the
resolution of the simulations. Specifically, we use 3 different grid
resolutions : $65\times 65$, $129\times 129$ and $257\times 257$, where $1/2$
of the grid points span the star at the equator.
Our initial models of rotating stars are generated using Hachisu’s Self-
Consistent Field (HSCF: Hachisu, 1986a, b) method—details of the procedure are
described in Kim et al. (2009). In order to generate equilibrium models, we
choose 1) the maximum rest mass density, $\rho_{0}^{\rm{max}}$) 2) the
rotation parameter, $A$, which describes the differential rotation and 3) the
axis ratio which determines how fast the star is rotating. We must also
specify the equation of state (EOS) in our construction of the initial model.
Here, we used the polytropic EOS Eq. (18) with $K=100$ and $N=1$. We choose a
maximum density value of $\rho_{0}^{\rm{max}}=1.28\times 10^{-3}$, which, with
this EOS, produces a 1.4$\rm{M}_{\odot}$star in the non-rotating case. For the
rotating models, we only consider rigid body rotation, which is obtained when
we chose a very large value of $A$. The axis ratio is specified to be $0.75$
resulting in an orbital frequency of $611\rm{Hz}$.
Even with our use of the multigrid technique—which is generally an efficient
method for solving elliptic equations—we still find solution of the Poisson
equation for the gravitational potential to be computationally expensive. We
thus calculate $\Phi$ only every $40$ time steps to reduce the time spent in
the Poisson solver, and find that this produces results which are nearly
equivalent to those obtained when the Poisson equation is solved at each time
step. However, we use time-extrapolated values for the gravitational potential
at the time steps between solves of the Poisson equation in order to avoid
discontinuities in the primitive variables, when abrupt changes of the
gravitational potential occur. We find that these discontinuities give rise to
very unnatural dissipative effects in the simulation resulting, for example,
in a rapid decay in the amplitude of radial oscillations, even when radial
perturbations are explicitly introduced.
-0.01-0.00500.0050.010246810 $\left[\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(0)\right]/\rho_{0}^{\rm{max}}(0)$ time(ms)$\mathcal{R}=3.05\times 10^{-7}$$65\times 65$$129\times 129$$257\times 257$ | -0.01-0.00500.0050.010246810 $\left[\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(0)\right]/\rho_{0}^{\rm{max}}(0)$ time(ms)$\mathcal{R}=3.95\times 10^{-6}$$65\times 65$$129\times 129$$257\times 257$
---|---
-0.01-0.00500.0050.010246810 $\left[\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(0)\right]/\rho_{0}^{\rm{max}}(0)$ time(ms)$\mathcal{R}=1.94\times 10^{-7}$$65\times 65$$129\times 129$$257\times 257$ | -0.01-0.00500.0050.010246810 $\left[\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(0)\right]/\rho_{0}^{\rm{max}}(0)$ time(ms)$\mathcal{R}=3.42\times 10^{-6}$$65\times 65$$129\times 129$$257\times 257$
Figure 6: The time evolution of the maximum rest mass density
changes($\left[\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(t=0)\right]/\rho_{0}^{\rm{max}}(t=0)$)
with different resolutions ($65\times 65$: red, $129\times 129$: dark blue and
$257\times 257$: sky blue). Top figure shows results when we fix the metric
(Cowling approximation) while the bottom one shows the case where we consider
the fully coupled dynamics. In the left panel, we show the figures for a
spherical (non-rotating) star while the right panel shows the corresponding
figures for a rigidly rotating star with axis ratio$=0.75$, which give a
rotational frequency of $611\rm{Hz}$
Figure 6 shows the time evolution of the relative changes of the maximum
density
($\left[\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(0)\right]/\rho_{0}^{\rm{max}}(0)$)
for non-rotating (left panel) and rigidly rotating (right panel) stars. For
the stationary stars, we use the Cowling approximation, which assumes the
gravitational potential is fixed. This gives efficient evolution of the stars,
and can also be used as a testbed for fully coupled evolutions. The results
computed using the Cowling approximation are shown in the top figures. The
maximum density slowly increases with time for the rotating star while it
decreases for the non-rotating star. For grid resolutions greater than
$65\times 65$ the rate of change is almost independent of resolution for the
spherical star, but a slow decrease with resolution is seen for the rotating
star. We define the following dimensionless rate of change:
$\mathcal{R}=\left|t_{\rm{dyn}}\frac{d\ln\rho_{0}^{\rm{max}}}{dt}\right|,$
(56)
where we use $t_{\rm{dyn}}=1/\sqrt{\rho_{0}^{\rm{max}}}$ for simplicity. We
use this quantity—as computed from the highest resolution simulations—as a
label in the figures. The values of $\mathcal{R}$ are within $3\times 10^{-7}$
for non-rotating star and are about 10 times larger for the rotating star,
again with a maximum resolution of $257\times 257$. The inverse of
$\mathcal{R}$ can be interpreted as the time (in units of the dynamical time)
that the simulation could be carried out until the results deviate from the
true solution by $O(1)$. Our results indicate that the error would become
$\sim$ 1% in 30,000 and 3,000 dynamical times for non-rotating and rotating
stars, respectively. We also carried out very long time simulations and found
that $\mathcal{R}$ becomes smaller even though it appears to be almost
constant in the figures. From these results, we conclude that we can use the
code to evolve stellar configurations for several thousand or more dynamical
times.
It is also very important to check the constancy of the conserved quantities
with respect to simulation time. In our formulation we have 2 conserved
quantities: the total rest mass, $M_{0}$, and the total angular momentum, $J$,
which are computing using
$\displaystyle M_{0}=\int
DdV^{(3)}=2\pi\int\frac{\rho_{0}W}{\left(1+2\Phi\right)^{3/2}}RdRdZ,$ (57)
$\displaystyle J=\int
S_{\phi}dV^{(3)}=2\pi\int\frac{\rho_{0}hW^{2}v^{\phi}}{\left(1+2\Phi\right)^{5/2}}R^{3}dRdZ,$
(58)
respectively, and where $dV^{(3)}$ denotes the 3-dimensional volume element.
-0.0001-5e-0505e-050.00010246810 $\left[M_{0}(t)-M_{0}(0)\right]/M_{0}(0)$ time(ms)$\mathcal{R}_{M}=4.79\times 10^{-7}$$65\times 65$$129\times 129$$257\times 257$
---
-0.001-0.0008-0.0006-0.0004-0.000200246810 $\left[J(t)-J(0)\right]/J(0)$ time(ms)$\mathcal{R}_{J}=1.67\times 10^{-6}$$65\times 65$$129\times 129$$257\times 257$
Figure 7: The deviation of total rest mass (upper panel) and angular momentum
(lower panel), which should remain constant, from their initial values with
time, for the same models shown in Fig. 6. Results computed with three
different resolutions ($65\times 65$:red, $129\times 129$:dark blue and
$257\times 257$:sky blue) are shown.
Figure 7 shows the time evolution of these two conserved quantities: total
rest mass (upper panel) and total angular momentum (lower panel). We show the
results only from the rotating star since there is, of course, no angular
momentum for non-rotating stars. The deviation of the total rest mass from the
initial value has two features: short-term fluctuations and long term average
behavior. The shot-term fluctuations depend on the grid resolution, but the
average slopes are almost independent of the resolution. We label the graphs
with $\mathcal{R}_{M}$ and $\mathcal{R}_{J}$ in a manner analogous to Eq. (56)
and Fig. 6, and use these quantities to measure the long-term stability of the
code. Their measured values are consistent with the ones for the central
density ($\mathcal{R}$). The behaviour of $\mathcal{R}_{M}$ is quite similar
for the 3 different grid resolutions, but $R_{J}$ shows considerable
dependence on the grid resolution. We have seen above (see Figure 6) that the
central density fluctuation is significantly dependent on grid resolution only
for the rotating models. We conclude that the main reason for this resolution-
sensitive behavior is the fact that angular momentum conservation is sensitive
to grid resolution. Therefore, simulations for rotating stars require high
grid resolution, otherwise angular momentum conservation will fail, and other
stationary properties of the start (such as central density) will also show
substantial, and non-physical, time evolution.
## 6 Radial Pulsation Frequency Test
Even without any explicitly-added perturbations, it is natural for our
numerical simulation of stationary stars to give rise to normal mode
oscillations due to intrinsic numerical errors. These errors occur for a
variety of reasons, including 1) truncation error due to the discretization
scheme, 2) the artificial atmosphere (floor) whereby the primitive variables
(pressure, density) are restricted from falling below minimum values to avoid
code crashes (the sound velocity becomes unbounded when vacuum is encountered
in the numerical calculations), and 3) the numerical limitation in describing
the stellar surface. Furthermore, the artificial atmosphere is known to excite
higher overtone modes.
The frequencies of various modes depend only on the structure of a given star,
and can be calculated by various methods. As explained above, our stationary
models oscillate even when we do not explicitly introduce external or internal
perturbations. We attempted to compare the frequencies of the modes excited in
our models with those obtained by normal mode analysis. The fundamental mode
(F-mode hereafter) frequency is very closely related to the dynamical time
($\sim 1/\sqrt{\rho}$) and the associated overtones have frequencies of
similar order.
Although using calculations based on cylindrical coordinates is not an
efficient way to compute radial pulsations, our code should still be able to
approximately compute the correct pulsation frequencies. The detailed
perturbation formulations and numerical methods we use for investigating the
radial pulsations are described in Appendix A. For initial conditions we use a
non-rotating equilibrium star with a baryon mass $1.4\mbox{$\rm{M}_{\odot}$}$.
we performed the test with and without the Cowling approximation, and In order
to obtain the mode frequency from the simulations, we analyzed the fluctuation
of the maximum density with time.
Specifically, we carried out Fourier transformation on the maximum density
using the FFTW package (Frigo & Johnson, 2005). To obtain better resolution in
the frequency domain, we use the zero-padding method which adds additional
zeros at the end of the time series data, effectively using interpolation
between points following the basic Fourier transformations. During the process
of obtaining a frequency having a maximum sinusoidal amplitude, leakage may
also cause additional errors. To reduce the effects of this leakage, we
multiply the time series by a window function. Here we used the Hamming window
function defined by
$w_{j}=0.54+0.46\cos\left(\frac{2\pi j}{N}\right),$ (59)
where $j$ is the index of the grid points and $N$ is the total number of
points, prior to zero-padding (Harris, 1978).
Although, as described above, some modes are excited simply due to numerical
error, their amplitudes are too small to be accurately extracted from the
simulation. We therefore introduce an explicit perturbation which can more
strongly excite the radial modes. The perturbation that we used is
$\delta\rho_{0}=B_{s}\sin\left(\pi\frac{r}{r_{s}}\right),$ (60)
where $B_{s}$ is the perturbation amplitude which we set to $B_{s}=0.001$.
00.20.40.60.8100.20.40.60.811.21.41.6 $A/A_{\rm{F-mode}}$
$f/\sqrt{\rho_{0}^{\rm{max}}}$$F$$H_{1}$$H_{2}$$H_{3}$$H_{4}$$H_{5}$
---
00.20.40.60.8100.20.40.60.811.21.41.6 $A/A_{\rm{F-mode}}$
$f/\sqrt{\rho_{0}^{\rm{max}}}$$F$$H_{1}$$H_{2}$$H_{3}$$H_{4}$$H_{5}$
Figure 8: The mode amplitudes of maximum density as a function of frequency of
the star with a baryon mass $1.4\mbox{$\rm{M}_{\odot}$}$. The vertical red
dotted lines show the frequency of the radial pulsation modes computed using
the perturbation method. The top panel shows the result when we use the
Cowling approximation, where the gravitational potential is assumed to be
fixed. In the bottom figure, we obtain the gravitational potential every few
time steps. In the both panels the 3 curves show results obtained using 3
different grid resolutions (sky blue: $257\times 257$, dark blue: $129\times
129$, and red: $65\times 65$)
Figure 8 shows the result after Fourier transformation of the time series data
given by the differences in maximum density relative to the initial time
($\rho_{0}^{\rm{max}}(t)-\rho_{0}^{\rm{max}}(t=0)$), and using calculations at
different resolutions. For comparison purposes, the vertical red lines show
the results computed from linear analysis. The mode labeled as $F$ is the
fundamental mode, while $H_{n}$ denote the $n$-th overtone radial modes. The
results shown in the figure can be summarized as follows:
1. 1.
The most excited mode with the perturbation given by Eq. (60) is the $F$-mode.
By changing the nature of the perturbation we could make one of the overtones
the most highly excited.
2. 2.
At low resolution, the code cannot identify high frequency modes. The reason
for this is the lack of spatial, not temporal, resolution. The eigenfunctions
describing higher overtones have large gradients near the surface which cannot
be accurately represented in the low resolution calculations.
3. 3.
The frequency increases when we use the Cowling approximation. This is a well-
known phenomena irrespective of whether Newtonian or general relativistic
gravitation is used. This issue is discussed in more detail in the appendix.
Table 3 shows the mode frequencies computed from linear analysis as well as
the numerical simulations. Again, the stellar model is a non-rotating
spherical star of mass 1.4 $\rm{M}_{\odot}$. The relative difference between
the linear and full numerical results is listed in the last row. Here the
numerical simulations have been carried out using the highest resolution
($257\times 257$), and we list results computed with and without the Cowling
approximation.
Table 3: Comparison of mode frequencies obtained by numerical simulation ($2\pi f$) and by linear analysis ($\sigma$) Mode | $F$ | $H_{1}$ | $H_{2}$ | $H_{3}$
---|---|---|---|---
$f/\sqrt{\rho_{0}^{\rm{max}}}$ | $0.190$ | $0.482$ | $0.734$ | $0.974$
$\sigma/2\pi/\sqrt{\rho_{0}^{\rm{max}}}$ | $0.190$ | $0.482$ | $0.733$ | $0.974$
Error(%) | $0.000$ | $0.000$ | $0.136$ | $0.000$
$|2\pi f-\sigma|/\sigma$ | | | |
The frequencies we obtained from the numerical simulation with $257\times 257$
grid resolution have relative differences from those computed from linear
analysis of at most $0.1\%$. We thus conclude that the radial mode frequencies
computed from our code agree very well with the ones calculated from linear
theory. The largest difference of 0.1% was found in the second overtone (mode
$H_{2}$), while for other modes we did not find any measurable difference.
## 7 Summary and Discussion
We have developed a new hydrodynamical code which adopts a pseudo-Newtonian
treatment of the gravitational field. This code uses the so called “Valencia
formulation” for the hydrodynamical equations. From the computational
perspective, the code is modular and includes many reconstruction schemes such
as slope limiting techniques (minmod, MC, 3rd order minmod, etc.), PPM and ENO
(WENO). In 1D shock tube tests, we assessed code accuracy relative to analytic
solutions and computed convergence rates of the errors. We found that the
minmod method gives the most diffusive results, smoothing out complex features
near discontinuities. As a result it cannot be used to accurately describe
stellar surfaces, which are characterized by stiff density changes. The MC
method gives the most promising result in the shock tube test and has second
order accuracy. It can capture discontinuities very well in the pulsation mode
test, but also yields additional non-physical effects such as the excitation
of the higher order overtones near the stellar boundary. The 3minmod and PPM
methods can provide higher order accuracy and we have found that they can also
describe the stellar surface well.
In the code we also implemented 3 different flux approximation schemes: Roe,
Marquina, and HLLE. Although the results in this paper were all computed using
the HLLE approach—which is the most dissipative of the three—we have also
found that for the simulations we have considered all produce very similar
results.
In the multigrid module for computing the gravitational potential we have
implemented both second and fourth order finite-difference discretizations.
The actual value of the gravitational potential is slightly different if we
change the order of accuracy. However, the changes of maximum density in time
show very little sensitivity to the order of approximation, and we consider
the difference between the use of the second or fourth order method to be
insignificant.
In the stationary star test which is described in section 5, we evolve
equilibrium solutions describing both non-rotating and rotating stars using
our code. Our code shows stable long-time behavior of the maximum density and
conserved quantities. Based on the rates of change in the maximum density,
total mass and total angular momentum, we estimate that our code can be used
to study evolution in excess of 3,000 dynamical times with 1% error.
In the radial mode test described in section 6, modes are obtained from the
Fourier transformation of the maximum density fluctuations. We also computed
normal modes by linear analysis (see Appendix A) and found that the mode
frequencies generated by our code agree with the results from linear analysis
almost perfectly (less than 0.14%).
This code can be applied to the following astrophysical scenarios:
1. 1.
Phenomena associated with isolated rotating neutron stars, such as
axisymmetric pulsations. Since our approach can be applied to mildly compact
stars, it is very useful to determine the amplitudes and frequencies of the
radial and non-radial modes.
2. 2.
Accretion disks around a neutron star or black hole. It is not sufficient to
treat a disk around a compact object using Newtonian gravity, since the
gravitational field is not weak there. In addition, because the rotational
velocity of the disk is a significant fraction of $c$, we should also take
into account special relativity in our treatment of the hydrodynamics. Our
code can be a very good tool for accretion disk studies.
This work was supported by the NRF grant 2006-341-C00018, by NSERC, and by the
CIFAR Cosmology and Gravity Program.
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## Appendix A Perturbation equation
The eigenfrequencies and eigenfunctions of the radial pulsation of stars are
well-known in Newtonian hydrodynamics as well as in the general relativistic
case. However, the corresponding formulation has not been previously presented
for our pseudo-Newtonian approach. Here, we describe the linearized equations
that can be used to obtain eigenfrequencies and eigenfunctions of the normal
modes of spherical stars using this approximation, and following the general
relativistic framework described in Misner et al., 1973 (MTW hereafter).
First, to describe stellar oscillations—such as those occurring on the
surface—it is much more practical to use a Lagrangian description rather than
the Eulerian one adopted in section 2. The relation between the the Eulerian
and Lagrangian perturbation is (see, e.g., Cox 1980),
$\Delta f(t,r)=\delta f+f_{0}^{\prime}\zeta,$ (61)
where $\zeta$ is a Lagrangian variation in space. The law of baryon number
conservation($\nabla_{\mu}(nu^{\mu})=0$) gives
$\Delta
n=-n_{0}[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\zeta)^{\prime}-3\alpha_{0}^{-1}\delta\alpha],$
(62)
where $\alpha=\sqrt{1+2\Phi}$, $n$ is the baryon number density and ′ denotes
differentiation with respect to $r$ (See MTW Eq.(26.7)). The relation between
$n$ in Eq. (62) and $\rho_{0}$ is $\rho_{0}=m_{b}n$, where $m_{b}$ is baryon
mass and the subscript $0$ denotes the unperturbed state.
Another perturbation equation comes from the adiabatic equation of state which
offers a much easier way to find the pressure variation:
$\Gamma=\frac{n}{P}\frac{dP}{dn}.$ (63)
Since the Lagrangian variations commute with total differentiation (denoted by
$d$), Eq. (63) becomes
$\Gamma=\frac{n}{P}\frac{\Delta P}{\Delta n}.$ (64)
In addition, Eqs. (61), (62) and (64) give the following pressure variation
equation:
$\delta P=-\Gamma
P_{0}[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\zeta)^{\prime}-3\alpha_{0}^{-1}\delta\alpha]-\zeta
P_{0}^{\prime}.$ (65)
The energy conservation equation ($u_{\mu}\nabla_{\nu}T^{\mu\nu}$) gives
$\Delta\rho=\frac{\rho_{0}+P_{0}}{n_{0}}\Delta n.$ (66)
Note that $\rho_{0}$ is the energy density in the unperturbed state, rather
than the rest mass density used in the main text. Combining this with Eq.
(62), we obtain the equation for the energy density variation
$\delta\rho=-(\rho_{0}+P_{0})[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\zeta)^{\prime}-3\alpha_{0}^{-1}\delta\alpha]-\zeta\rho_{0}^{\prime}.$
(67)
The main difference here relative to the general relativistic case arises in
the computation of the perturbation of the gravitational potential. The
Poisson equation gives
$\frac{2}{r}(\alpha_{0}\delta\alpha)^{\prime}+(\alpha_{0}\delta\alpha)^{\prime\prime}=4\pi(\delta\rho+3\delta
P).$ (68)
Note that we should use only the Eulerian variation in this equation since
Lagrangian variation does not commute with partial differentiation. Eq.
(26.16) in MTW involves only first order differential equations—i.e. the
second order differentiations are rewritten in terms of first order ones. On
the other hand, in our case we cannot find any equations which can be used to
eliminate the second order differentiation. That means that we need to find
one more boundary condition to solve this equation.
Finally, the equation of motion of the fluid is obtained from the
4-acceleration ($a_{\mu}=u^{\nu}\nabla_{\nu}u_{\mu}$),
$(\rho_{0}+P_{0})\alpha_{0}^{-4}\ddot{\zeta}=-\delta
P^{\prime}-(\delta\rho+\delta P)\alpha_{0}^{-1}\alpha_{0}^{\prime}\\\
-(\rho_{0}+P_{0})(\alpha_{0}^{-1}\delta\alpha^{\prime}-\alpha_{0}^{-2}\alpha_{0}^{\prime}\delta\alpha).$
(69)
Under the assumption of the adiabatic nature of the oscillation, normal modes
are standing waves, and thus space and time variables can be separated as
follows:
$\zeta(r,t)=\xi(r)e^{i\sigma t}.$ (70)
Then, we can rewrite the equations using $\xi$ and $\sigma$,
$\displaystyle\delta P=-\Gamma
P_{0}[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\xi)^{\prime}-3\alpha_{0}^{-1}\delta\alpha]-\xi
P_{0}^{\prime}$ (71)
$\displaystyle\delta\rho=-(\rho_{0}+P_{0})[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\xi)^{\prime}-3\alpha_{0}^{-1}\delta\alpha]-\xi\rho_{0}^{\prime}$
(72) $\displaystyle(\rho_{0}+P_{0})\alpha_{0}^{-4}\sigma^{2}\zeta=\delta
P^{\prime}+(\delta\rho+\delta P)\alpha_{0}^{-1}\alpha_{0}^{\prime}$
$\displaystyle\qquad\qquad\qquad\qquad+(\rho_{0}+P_{0})(\alpha_{0}^{-1}\delta\alpha^{\prime}-\alpha_{0}^{-2}\alpha_{0}^{\prime}\delta\alpha)$
(73)
To solve Eqs. (71)–(73), we need to impose appropriate boundary conditions.
The first condition is that $\xi/r$ should be regular at the origin, and the
second one is that the pressure variation at the surface must vanish, i.e.,
$\displaystyle\frac{\xi}{r}={\rm finite~{}at~{}}r=0,$ (74)
$\displaystyle\Delta P(r=r_{s})=0.$ (75)
Unlike the general relativistic case, we cannot substitute $\delta\alpha$ and
$\delta\alpha^{\prime}$ in terms of other variations such as $\delta\rho$ and
$\delta P$. Therefore, we need an additional boundary condition for Eq. (68).
We use the properties of the gravitational potential to obtain extra
conditions. First, from the condition that the gravitational potential should
be regular at the center we obtain
$\delta\alpha^{\prime}=0{\rm~{}at~{}}r=0,.$ (76)
Second because the gravitational potential should fall off as $1/r$ beyond the
stellar surface, we have
$\delta\Phi^{\prime}+\frac{\delta\Phi}{r}=0.$ (77)
When we apply the above equation at the stellar boundary ($r=r_{s}$), we get
$\delta\alpha^{\prime}=-\frac{\delta\alpha^{2}-1}{2r\delta\alpha}{\rm~{}at~{}}r=r_{s}.$
(78)
Since Eqs. (71)–(73) and (68) are coupled, we use an iterative method to solve
them.
For the case of the Cowling approximation, which assumes that the
gravitational potential is fixed ($\delta\alpha=0$), the equations simplify
considerably:
$\displaystyle\delta P=-\Gamma
P_{0}[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\xi)^{\prime}]-\xi
P_{0}^{\prime}$ (79)
$\displaystyle\delta\rho=-(\rho_{0}+P_{0})[r^{-2}\alpha_{0}^{3}(r^{2}\alpha_{0}^{-3}\xi)^{\prime}]-\xi\rho_{0}^{\prime}$
(80) $\displaystyle(\rho_{0}+P_{0})\alpha_{0}^{-4}\sigma^{2}\zeta=\delta
P^{\prime}+(\delta\rho+\delta P)\alpha_{0}^{-1}\alpha_{0}^{\prime}$ (81)
If we compare the above equations with Eqs. (71)–(73), we observe that every
coefficient of $\delta\alpha$ is negative: therefore, as mentioned in the main
text, $\sigma$ increases when we apply the Cowling approximation.
We show the solution for $\xi/r$ for the $1.4\mbox{$\rm{M}_{\odot}$}$ star
with $K=100$ and $N=1$ with and without the Cowling approximation in Figure 9.
The $\sigma$ values corresponding to each mode are summarized in Table 3 which
appears in the main text.
00.20.40.60.811.21.41.61.800.20.40.60.81 $\tilde{\xi}/\tilde{\xi}_{s}$ $r/r_{s}$ | -0.8-0.6-0.4-0.200.20.40.60.8100.20.40.60.81 $\tilde{\xi}/\tilde{\xi}_{s}$ $r/r_{s}$
---|---
-0.200.20.40.60.8100.20.40.60.81 $\tilde{\xi}/\tilde{\xi}_{s}$ $r/r_{s}$ | -0.6-0.4-0.200.20.40.60.8100.20.40.60.81 $\tilde{\xi}/\tilde{\xi}_{s}$ $r/r_{s}$
-0.200.20.40.60.8100.20.40.60.81 $\tilde{\xi}/\tilde{\xi}_{s}$ $r/r_{s}$ | -0.6-0.4-0.200.20.40.60.8100.20.40.60.81 $\tilde{\xi}/\tilde{\xi}_{s}$ $r/r_{s}$
Figure 9: Radial pulsation eigenfunction of a 1.4$\rm{M}_{\odot}$star. The
equation of state that we use is the polytropic one with $K=100$ and $N=1$. In
this figure, $\tilde{\xi}=\xi/r$ and $\tilde{\xi}_{s}=\tilde{\xi}(r=r_{s})$
where $r_{s}$ is the surface radius. The dashed and solid lines represent the
results with and without the Cowling approximation, respectively. Each panel
shows different modes (top-left ($F$), top-right ($H_{1}$), middle-left
($H_{2}$), middle- right ($H_{3}$), bottom-left ($H_{4}$) and bottom-right
($H_{5}$)) which have different oscillation frequencies.
|
arxiv-papers
| 2012-04-17T23:54:02 |
2024-09-04T02:49:29.782004
|
{
"license": "Public Domain",
"authors": "Jinho Kim, Hee Il Kim, Matthew William Choptuik and Hyung Mok Lee",
"submitter": "Jinho Kim",
"url": "https://arxiv.org/abs/1204.3945"
}
|
1204.3966
|
# The $\sigma$ law of evolutionary dynamics in community-structured
populations
$Changbing~{}Tang^{a},~{}Xiang~{}Li^{a},~{}Lang~{}Cao^{b},~{}Jingyuan~{}Zhan^{a}$111
_Email address_ :lix@fudan.edu.cn aAdaptive Networks and Control Lab,
Department of Electronic Engineering Fudan University, Shanghai 200433, China
bDepartment of Mathematical Engineering and Information Physics, University of
Tokyo, Tokyo 153-8505, Japan
###### Abstract
Evolutionary game dynamics in finite populations provides a new framework to
understand the selection of traits with frequency-dependent fitness. Recently,
a simple but fundamental law of evolutionary dynamics, which we call $\sigma$
law, describes how to determine the selection between two competing
strategies: in most evolutionary processes with two strategies, $A$ and $B$,
strategy $A$ is favored over $B$ in weak selection if and only if $\sigma
R+S>T+\sigma P$. This relationship holds for a wide variety of structured
populations with mutation rate and weak selection under certain assumptions.
In this paper, we propose a model of games based on a community-structured
population and revisit this law under the Moran process. By calculating the
average payoffs of $A$ and $B$ individuals with the method of effective
sojourn time, we find that $\sigma$ features not only the structured
population characteristics but also the reaction rate between individuals.
That’s to say, an interaction between two individuals are not uniform, and we
can take $\sigma$ as a reaction rate between any two individuals with the same
strategy. We verify this viewpoint by the modified replicator equation with
non-uniform interaction rates in a simplified version of the prisoner’s
dilemma game (PDG).
###### keywords:
Evolutionary game theory, Sojourn time, Non-uniform interaction rate,
Simplified PDG.
††journal: Theoretical Biology
## 1 Introduction
Evolutionary game theory was originally introduced as a tool for studying
animal behavior (Maynard and Price,, 1973; Maynard,, 1982) but has become a
general approach that transcends almost every aspect of evolutionary biology
(Nowak and Sigmund, 2004a, ), and provides a framework to understand the
dynamics of frequency-dependent selection (Hofbauer and Sigmund,, 1998;
Gintis,, 2000; Cressman,, 2003; Nowak and Sigmund, 2004a, ; Nowak,, 2006;
Antal et al., 2009a, ; Wu et al.,, 2010; Gokhale and Traulsen,, 2011; Tarnita
et al.,, 2011).
The traditional approach of evolutionary game theory uses deterministic
dynamics to describe infinitely large, well-mixed populations (Hofbauer and
Sigmund,, 1988, 2003). To understand evolutionary game dynamics in finite-
sized populations, a stochastic approach is developed (Schaffer,, 1988; Fogel
et al.,, 1998; Ficici et al.,, 2000; Alos-Ferrer,, 2003; Perc,, 2007; Traulsen
and Nowak,, 2007). A crucial quantity in the stochastic approach is the
fixation probability of strategies which relies on the individual’s
reproduction ability (Nowak et al., 2004b, ; Taylor et al.,, 2004; Imhof and
Nowak,, 2006; Nowak,, 2006; Traulsen et al., 2006a, ; Lessard and Ladret,,
2007; Bomze and Pawlowitsch,, 2008). In the limit of weak selection, for a
neutral mutant, probability $\rho_{A}$, that an individual using strategy $A$
will invade and take over the whole population of individuals using strategy
$B$, is equal to the reciprocal of the population size, i.e., $1/N$.
Therefore, the selection favors the fixation of invading strategy $A$ if
$\rho_{A}>1/N$ under the weak selection.
Evolutionary game dynamics are also affected by the structure of population.
Lots of early works focused on spatial games with regular lattices (Nowak and
May, 1992; Nakamaru et al., 1997; Hauert and Doebeli, 2004; Szabó and Fath,
1998; Szabó et al., 2000), which recently, have been expanded to general
structured populations, such as graphs (Lieberman et al.,, 2005; Ohtsuki et
al., 2006a, ; Ohtsuki and Nowak, 2006b, ; Ohtsuki et al., 2007a, ; Taylor et
al.,, 2007; Cao and Li,, 2008; Fu et al.,, 2009; Li and Cao,, 2009; Perc and
Wang,, 2010; Wang et al.,, 2011), phenotype space (Antal et al., 2009b, ) and
set structured populations (Tarnita et al., 2009a, ). In a word, a population
structure specifies who interacts with whom, and it greatly affects the
outcome of an evolution. If the fitness of an individual is determined by its
interactions with others, then we are in the world of evolutionary game theory
(Nathanson et al.,, 2009).
Consider a game of two strategies, $A$ and $B$, in a population of fixed size
$N$. Mutual cooperation is rewarded by $R$ for each player, whereas mutual
defection pays each player a punishment $P$. If a cooperator plays against a
defector, the former gets the sucker s payoff $S$ and the latter gets the
temptation to defect $T$. Thus, the interactions are given by the payoff
matrix
$\begin{array}[]{ccc}{}\hfil&A&B\\\ A&R&S\\\ B&T&P\\\ \end{array}$ (1)
Each individual obtains a payoff by interacting with others according to the
population structure, and reproduces either genetically or culturally with a
rate proportional to its payoff. Of course, reproduction is subject to
mutation. Whenever an individual reproduces, the offspring adopts the parent’s
strategy with probability $1-\nu$ and a random strategy with probability
$\nu$. To make the evolutionary dynamics of the process well approximated by
an embedded Markov chain on the pure states (Wu et al.,, 2011), it’s
considered that mutation rate $\nu<(NlnN)^{-1}\ll 1$.
It is said that strategy $A$ is selected over strategy $B$ if it is more
abundant in the stationary distribution of the mutation-selection process.
Tarnita et al., 2009b showed that with the weak selection, the condition that
strategy $A$ is more abundant than strategy $B$ in the stationary distribution
of the mutation-selection process is
$\begin{array}[]{l}\sigma R+S>T+\sigma P\end{array}$ (2)
From this inequality, we know that the condition specifying which strategy is
more abundant is a linear inequality of the payoff values, $R$, $S$, $T$, $P$.
It’s pointed out that parameter $\sigma$, reflects how the structured
populations influence the evolutionary dynamics (Nowak et al.,, 2010), and
this inequality holds for a wide variety of population structures, including
well mixed populations, graphs, phenotype space and set structured populations
(Nathanson et al.,, 2009). Conveniently, we call this simple but fundamental
law as the $\sigma$ law.
Recently, attention are focused on the population structures allowing
selection at multiple levels (Girvan and Newman,, 2002; Newman,, 2006;
Traulsen et al., 2005a, ; Traulsen et al., 2006b, ; Traulsen et al.,, 2008;
Wang et al.,, 2011; Hauert and Imhof,, 2011; Wang et al.,, 2012). In this
case, a population is divided into groups (or communities), and the selection
between individuals can be either in a community or among several communities.
Following this line, we propose a model of games on a community-structured
population based on the graph theory with a finite size. We verify that
evolutionary games on the community-structured population obeys the $\sigma$
law from the viewpoint of interaction under the Moran process with sojourn
time. Along the path of an invasion attempt, we find that the probability of
an interaction between any two individuals with the same strategy is $\sigma$
times of that between different strategies. In other words, $\sigma$
represents the reaction rate between two individuals with the same strategy,
and quantifies the degree that individuals using the same strategy are more
($\sigma>1$) or less ($\sigma<1$) likely to interact than individuals using
different strategies.
## 2 $\sigma$ law in community-structured populations
### 2.1 Model of community structures
Consider a game played in a finite population of the fixed size $H$
distributed over $M$ communities (each community has the same size $N$, and
any two of which have none common member) with strategies $A$ and $B$. Each
individual obtains a payoff according to the payoff matrix $(1)$. At each time
step, a single individual is selected proportional to its fitness for
reproduction, and the offspring either replaces a randomly chosen individual
in this community or moves to another community at any time step to replaces a
randomly chosen individual. In Hauert and Imhof, (2011), the authors discussed
the evolutionary dynamics in a deme structured population, and assumed that
the migration rate is proportional to deme’s fitness. In this paper, we
suppose that there exists a graph in the structured population connecting
communities, and the migration occurs between two communities if there is an
edge between them (See Fig. 1).
Label all communities in the structured populations with $l=1,2,\cdots,M$. The
probability that a selected individual belongs to community $l$ migrates to
community $m$ is given by $\lambda_{lm}$. Hence the migration process is
determined by an $M\times M$ matrix, $\Lambda=[\lambda_{lm}]$, $\lambda_{ll}$
is the probability that an individual stays in community $l$. Note that
$\lambda_{ll}+\sum_{m\in\Omega_{l}}^{M}\lambda_{lm}=1$, where $\Omega_{l}$
denotes the set of all the communities that individuals in community $l$ can
migrate to. Thus $\Lambda$ is a stochastic matrix with an eigenvalue $1$, and
no eigenvalue has an absolute value greater than $1$ (Godsil and Royle,,
2001).
Assume that a mutant $A$-individual (individual using strategy $A$) is
produced from a community $l$ with pure strategy $B$. This mutant replaces a
randomly chosen individual in community $l$ with probability $\lambda_{ll}$,
or migrates to community $m$ and replaces a randomly chosen individual in
community $m$ with probability $\lambda_{lm},m\in\Omega_{l}$. Since the graph
connects all the communities as its nodes, there exists a path along which a
mutant beginning in community $l,l\in\\{1,2,\cdots,M\\}$ can be fixed in the
whole population (note that the path is not unique). An example that the
fixation process of a single mutant in the whole community-structured
population is shown in Fig. 2, where the fixation of a mutant from community
$1$ to take over the whole population corresponds to a path.
### 2.2 $\sigma$ law in community-structured population with sojourn time
Assume that a migration occurs very rarely and much less than mutation
($\lambda_{lm}\ll\nu<(NlnN)^{-1}\ll 1,m\in\Omega_{l})$. In the limit of
$\lambda_{lm}\rightarrow 0$ and $\nu\rightarrow 0$, the population spends
almost all the time in the homogeneous states with all individuals of type $A$
or type $B$. Since mutations occur much more frequently than migrations, a
migrant has either invaded the whole community or been eliminated before the
next migration takes place (Traulsen et al., 2006b, ). Due to this special
structure, the interactions within a community is more frequent than those
between communities. This allows us to use the method of sojourn time in the
Moran process which was applied to explain the one-third law of evolutionary
dynamics in Ohtsuki et al., 2007b .
Suppose that a single individual from $l$-community is selected proportional
to its fitness for reproduction. This newborn has to stay or migrate to
another community and replaces a randomly selected individual, and the total
number of the population keeps unchanged. Based on the migration matrix
$\Lambda$ constructed previously, we know that the newborn stays in community
$l$ with probability $\lambda_{ll}$, and migrate to community $m$ with
probability $\lambda_{lm}$, where $\lambda_{ll}\gg\lambda_{lm},m\neq l$.
First, we discuss the Moran process that happens in $l$-community with
probability $\lambda_{ll}$. Among the total $N$ individuals in community $l$,
$j$ of them follow strategy $A$, and $N-j$ follow strategy $B$. Introduce a
parameter $\omega$ to measure the intensity of selection. The corresponding
fitness of A and B individuals are given by
$\begin{array}[]{l}f_{j}=1-\omega+\omega F_{j}\\\ g_{j}=1-\omega+\omega
G_{j}\end{array}$
where $F_{j}$ and $G_{j}$ are the expected payoffs for $A$ and $B$,
respectively. In this paper, we consider the weak selection, i.e.,
$\omega\rightarrow 0$. In the state space $\\{0,\cdots,N\\}$, there are two
absorbing states, $j=0$ (which means $A$ has become extinct) and $j=N$ (which
means $A$ has reached fixation). Then the transition probability $p_{j,j\pm
1}$ is given by
$\begin{array}[]{l}p_{j,j+1}=(N-j)/N\cdot jf_{j}/[jf_{j}+(N-j)g_{j}]\\\
p_{j,j-1}=j/N\cdot[(N-j)g_{j}]/[jf_{j}+(N-j)g_{j}]\\\
p_{j,j}=1-p_{j,j+1}-p_{j,j-1}\end{array}$ (3)
Consider a path along which a mutant invades a resident population from state
$j$ $(1\leq j\leq N-1)$. Since the mutant eventually gets absorbed into either
state $0$ or state $N$, we are interested in how much time is spent at state
$j$ along the path. In Ohtsuki et al., 2007b , the authors called it the
sojourn time. Given all paths that start at state $i$, let $\bar{t}_{ij}$ be
the mean sojourn time at state $j$ before the absorption into either state $0$
or state $N$, which is described as (see, for example, Ewens, (2004)).
$\bar{t}_{ij}=\sum_{k=0}^{k^{l}}p_{ik}\bar{t}_{kj}+\delta_{ij},~{}~{}~{}\bar{t}_{0j}=\bar{t}_{Nj}=0$
(4)
In particular, if we start from a single mutant with no loss of generality,
i.e., $i=1$, then Eq. (4) can be simplified as
$\overline{t}_{1j}=N/j.$ (5)
This means that the stochastic process spends most of the time around the
absorbing state, $j=0$. Besides, the mean effective sojourn time
$\overline{\tau}_{ij}$ at state $j$ for paths that start at state $i$ is given
by $\overline{\tau}_{ij}=(p_{j,j+1}+p_{j,j-1})\overline{t_{ij}}$. Now, we only
concern the mean effective sojourn time $\overline{\tau}_{1j}$ which starts
from a single mutant under the neutral drift. Since the fitness $f_{j}$ and
$g_{j}$ are very close to $1$ under weak selection, with Eqs. (3) and (5), we
obtain
$\begin{array}[]{l}\overline{\tau}_{1j}=(p_{j,j+1}+p_{j,j-1})\overline{t_{1j}}\\\
=\\{(N-j)/N\cdot
jf_{j}/[jf_{j}+(N-j)g_{j}]+j/N\cdot[(N-j)g_{j}]/[jf_{j}+(N-j)g_{j}]\\}N/j\\\
\approx 2(N-j)/N\end{array}$ (6)
We now can count the average number of games played by $A$-individual and
$B$-individual along a path of invasion. At state $j$, an $A$-individual meets
$(j-1)$ $A$-individuals and $(N-j)$ $B$-individuals, while a $B$-individual
meets $j$ $A$-individuals and $(N-j-1)$ $B$-individuals. The average number of
games played by $A$-individual and $B$-individual at state $j$ can be
summarized as (Ohtsuki et al., 2007b, )
$\left(\begin{array}[]{cc}A\rightarrow A&A\rightarrow B\\\ B\rightarrow
A&B\rightarrow B\\\ \end{array}\right)_{j}=\left(\begin{array}[]{cc}j-1&N-j\\\
j&N-j-1\\\ \end{array}\right)$ (7)
Therefore, the effective number of encounters are
$\sum_{j=1}^{N-1}\overline{\tau}_{1j}\left(\begin{array}[]{cc}j-1&N-j\\\
j&N-j-1\\\ \end{array}\right)=$
$2\left(\begin{array}[]{cc}\sum_{j=1}^{N-1}\frac{1}{N}(j-1)(N-j)&\sum_{j=1}^{N-1}\frac{1}{N}(N-j)(N-j)\\\
\sum_{j=1}^{N-1}\frac{1}{N}j(N-j)&\sum_{j=1}^{N-1}\frac{1}{N}(N-j-1)(N-j)\\\
\end{array}\right)$
$=2\cdot\left(\begin{array}[]{cc}\sum_{j=1}^{N-1}\frac{1}{N}(j-1)N_{B}^{l}&\sum_{j=1}^{N-1}\frac{1}{N}(N-j)N_{B}^{l}\\\
\sum_{j=1}^{N-1}\frac{1}{N}(N-j)N_{A}^{l}&\sum_{j=1}^{N-1}\frac{1}{N}(j-1)N_{A}^{l}\\\
\end{array}\right)$ (8)
where $N_{A}^{l}$ is the number of $A$-individuals in community $l$ at state
$j$, $N_{B}^{l}=N-N_{A}^{l}$ is the number of $B$-individuals in community $l$
at state $j$. Furthermore, denote $I_{XX}^{l}$ the total number of
interactions that an $X$-individual interacts with other $X$-individuals, and
$I_{XY}^{l}(X\neq Y)$ is the total number of interactions that an
$X$-individual interacts with $Y$-individuals, where $X,Y\in\\{A,B\\}$.
Symmetrically, we have the following equations $\langle
I_{AA}^{l}N_{B}^{l}\rangle_{0}=\langle I_{BB}^{l}N_{A}^{l}\rangle_{0}$,
$\langle I_{AB}^{l}N_{B}^{l}\rangle_{0}=\langle
I_{BA}^{l}N_{A}^{l}\rangle_{0}$ at neutrality (Nathanson et al.,, 2009). The
notation $\langle\cdot\rangle_{0}$ denotes the quantity averaged over all
states of the stochastic process under neutral drift, $w=0$. Thus, Eq. (8) can
be simplified as
$2\left(\begin{array}[]{cc}\langle I_{AA}^{l}N_{B}^{l}\rangle_{0}&\langle
I_{AB}^{l}N_{B}^{l}\rangle_{0}\\\ \langle
I_{BA}^{l}N_{A}^{l}\rangle_{0}&\langle I_{BB}^{l}N_{A}^{l}\rangle_{0}\\\
\end{array}\right)$
$~{}~{}~{}\rightarrow~{}\left(\begin{array}[]{cc}\sigma^{l}&1\\\
1&\sigma^{l}\\\ \end{array}\right)$ (9)
where $\sigma^{l}=\langle I_{AA}^{l}N_{B}^{l}\rangle_{0}/\langle
I_{AB}^{l}N_{B}^{l}\rangle_{0}$. On the other hand, the newborn reproduced in
community $l$ migrates to community $m$ with probability $\lambda_{lm}$.
Suppose that in community $m$ there are $h$ $A$-individuals and $N-h$
$B$-individuals, where $N$ is the size of community $m$. Similarly, we can get
the effective sojourn time $\overline{\tau}_{1h}\approx 2(N-h)/N$. And the
effective number of encounters can be written as
$\left(\begin{array}[]{cc}\sigma^{m}&1\\\ 1&\sigma^{m}\\\ \end{array}\right)$
(10)
where $\sigma^{m}=\langle I_{AA}^{m}N_{B}^{m}\rangle_{0}/\langle
I_{AB}^{m}N_{B}^{m}\rangle_{0}$.
Since we only discuss the number of interactions along a path that the mutant
invades the whole population, the total number of interactions is a linear
composition of the number of interactions in every community. So the total
effective number of interactions can be finally written as
$\left(\begin{array}[]{cc}\sigma&1\\\ 1&\sigma\\\ \end{array}\right)$ (11)
where $\sigma=c_{l}\sigma^{l}+\sum_{m\in\Omega_{l}}^{M}c_{m}\sigma^{m}$. Here,
the coefficient $c_{n}(n\in l\cup\Omega_{l})$, reflects the strength of effect
from effective number of interactions in community $n$ to total effective
number of interactions.
Eq. (11) suggests that both types of individuals with the same strategy
interact $\sigma$ times as often as they interact with the other individuals.
Therefore the average payoffs of $A$ and $B$ individuals along an invasion-
path in a community-structured population are $\sigma R+S$ and $T+\sigma P$,
respectively. From this result we can get the $\sigma$ law under the weak
selection in a community-structured population, i.e., the condition that
strategy $A$ is more abundant than strategy $B$ in the stationary distribution
is $\sigma R+S>T+\sigma P$.
## 3 $\sigma$ law in the case of non-uniform interaction
The $\sigma$ law provides a fundamental criterion to specify which strategy is
more abundant in a structured population. But, what’s the $\sigma$? In Nowak
et al., (2010), the authors said that $\sigma$ is the ‘structure coefficient’,
and it reflects how the structured population influences the evolutionary
dynamics. In this section, we point out that $\sigma$ also features the
reaction rate between two individuals with the same strategy.
To verify this point, we will use the evolutionary game theory with non-
uniform interaction rates discussed in Taylor and Nowak, (2006). They assumed
that the probability of interaction between two individuals is dependent of
their strategies, and described the interaction as
$\begin{array}[]{l}A+A^{~{}\underrightarrow{~{}~{}r_{1}~{}~{}}~{}}AA\\\
A+B^{~{}\underrightarrow{~{}~{}r_{2}~{}~{}}~{}}AB\\\
B+B^{~{}\underrightarrow{~{}~{}r_{3}~{}~{}}~{}}BB\end{array}$
Let us consider a simplified version of the prisoner’s dilemma game (PDG) with
the payoff matrix as below
$\begin{array}[]{ccc}{}\hfil&C&D\\\ C&b-c&-c\\\ D&b&0\\\ \end{array}$ (12)
If we discuss the game (12) in a population without community-structure, i.e.
$M=1$, strategy $D$ dominates strategy $C$ with uniform interaction rates
$r_{1}=r_{2}=r_{3}$. Eventually, the entire population consists of defectors.
However, if individuals only interact with opponents of the same strategy,
cooperators cannot be exploited by defectors. In this case, when $r_{2}=0$ and
$r_{1},r_{3}>0$, cooperation is the dominant strategy due to $b-c>0$.
Therefore, $r_{2}\neq 0$ means that cooperators and defectors do interact.
Without loss of generality, we assume that $r_{1}=r_{3}=r>0$, $r_{2}=1$ in a
population with community-structure. Let $x^{l}$ and $y^{l}$ be the
frequencies of individuals adopting strategy $C$ and $D$ in community $l$,
respectively, and $x^{l}+y^{l}=1$. The fitness of individuals are determined
by the average payoffs over a large number of interactions. Therefore, the
fitness of $A$ and $B$ individuals in community $l$ are
$\begin{array}[]{c}f_{C}^{l}=[r(b-c)x^{l}-cy^{l}]/(rx^{l}+y^{l}),~{}~{}~{}~{}f_{D}^{l}=bx^{l}/(x^{l}+ry^{l}),\end{array}$
(13)
In our model, an individual is randomly selected at each time step as a new
vacancy, which is replaced either with the offspring of an individual from the
same community or with an individual migrated from other communities. In
community $l$, the probability of an $X$-individual ($X=C$, or $D$) filling a
new vacancy due to local reproduction is proportional to the product of the
number of $X$-individuals and their fitness, i.e., $N_{X}^{l}f_{X}^{l}$. And,
the probability of an $X$-individual filling in a new vacancy due to global
migration is proportional to the product of the averaged number of
$X$-individuals and the migration rate $\lambda_{l}$, i.e.,
$\lambda_{l}\langle N_{X}\rangle$. Here, we suppose
$\lambda_{lm}=\lambda_{l}=(1-\lambda_{ll})/N_{\Omega_{l}},m\in\Omega_{l}$,
where $N_{\Omega_{l}}$ is the number of set $\Omega_{l}$. Thus, we obtain the
transition probabilities as
$\begin{array}[]{l}T_{C\rightarrow
D}^{l}=(N_{C}^{l}/N)\cdot(N_{D}^{l}f_{D}^{l}+\lambda_{l}\langle
N_{D}\rangle)/[\sum_{C,D}(N_{D}^{l}f_{D}^{l}+\lambda_{l}\langle
N_{D}\rangle)]\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=(N_{C}^{l}/N)\cdot[\overline{f}^{l}/(\overline{f}^{l}+\lambda_{l})\cdot(N_{D}^{l}f_{D}^{l})/(N\overline{f}^{l})+\lambda_{l}/(\overline{f}^{l}+\lambda_{l})\cdot\langle
N_{D}\rangle/N]\end{array}$ $\begin{array}[]{l}T_{D\rightarrow
C}^{l}=(N_{C}^{l}/N)\cdot(N_{C}^{l}f_{C}^{l}+\lambda_{l}\langle
N_{C}\rangle)/[\sum_{C,D}(N_{C}^{l}f_{C}^{l}+\lambda_{l}\langle
N_{C}\rangle)]\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=(N_{D}^{l}/N)\cdot[\overline{f}^{l}/(\overline{f}^{l}+\lambda_{l})\cdot(N_{C}^{l}f_{C}^{l})/(N\overline{f}^{l})+\lambda_{l}/(\overline{f}^{l}+\lambda_{l})\cdot\langle
N_{C}\rangle/N]\end{array}$ (14)
where
$\overline{f}^{l}=f_{C}^{l}\cdot(N_{C}^{l}/N)+f_{D}^{l}\cdot(N_{D}^{l}/N)$,
and $\langle N_{X}\rangle$=$\sum_{l=1}^{M}N_{X}^{l}/M$. From Eq. (14), we know
that after a vacancy appears, either local reproduction occurs with
probability $\overline{f}^{l}/(\overline{f}^{l}+\lambda_{l})$, or global
migration occurs with probability
$\lambda_{l}/(\overline{f}^{l}+\lambda_{l})$.
For sufficiently large but finite populations, Traulsen et al. have shown that
the stochastic process can be well approximated by a set of stochastic
differential equations combining deterministic dynamics and diffusion referred
to as Langevin dynamics (Traulsen et al., 2005b, ; Traulsen et al., 2006c, ).
Hence from Eq. (14), we may derive the Langevin equation to describe the
evolutionary dynamics as
$\begin{array}[]{l}\dot{x}^{l}=a^{l}(x^{l})+b^{l}(x^{l})\xi\end{array}$ (15)
where $a^{l}=T_{D\rightarrow C}^{l}-T_{C\rightarrow D}^{l}$, $b^{l}$ is the
effective terms, and $\xi$ is the uncorrelated Gaussian noise. As
$N\rightarrow\infty$, the diffusion term tends to zero and a deterministic
equation is obtained
$\begin{array}[]{l}\dot{x}^{l}=a^{l}(x^{l})=(N_{D}^{l}/N)\cdot[\overline{f}^{l}/(\overline{f}^{l}+\lambda_{l})\cdot(N_{C}^{l}f_{C}^{l})/(N\overline{f}^{l})+\lambda_{l}/(\overline{f}^{l}+\lambda_{l})\cdot\langle
N_{C}\rangle/N]\\\
~{}~{}~{}~{}~{}~{}-(N_{C}^{l}/N)\cdot[\overline{f}^{l}/(\overline{f}^{l}+\lambda_{l})\cdot(N_{D}^{l}f_{D}^{l})/(N\overline{f}^{l})+\lambda_{l}/(\overline{f}^{l}+\lambda_{l})\cdot\langle
N_{D}\rangle/N]\end{array}$ (16)
Therefore, the equilibria of Eq. (16) which satisfy $T_{D\rightarrow
C}^{l}=T_{C\rightarrow D}^{l}$ are
$\begin{array}[]{l}[-(\alpha-2\beta)(x^{l})^{2}+(\alpha-2\beta)x^{l}+\beta]/[-(r-1)^{2}(x^{l})^{2}+(r-1)^{2}x^{l}+r]=\gamma/N\end{array}$
(17)
where $\alpha=r^{2}(b-c)-(b+c)$, $\beta=-rc$, and
$\gamma=4(N_{C}^{l}-\overline{N}_{C})M\lambda_{l}$ with
$\overline{N}_{C}=\sum_{i=1}^{M}N_{C}^{i}$.
Denote $\Delta N_{C}=N_{C}^{l}-\overline{N}_{C}$. When $\Delta N_{C}=0$, i.e.,
each community has the same numbers of $C$-individuals, Eq. (17) can be
simplified as
$\begin{array}[]{l}-(\alpha-2\beta)(x^{l})^{2}+(\alpha-2\beta)x^{l}+\beta=0\end{array}$
(18)
When $\alpha>-2\beta$, i.e., $r>(b+c)/(b-c)$, there are two interior
equilibria. Specifically, the stable interior equilibrium is
$\begin{array}[]{c}x^{l*}_{s}=1/2+\sqrt{\alpha^{2}-4\beta^{2}}/(2\alpha-4\beta)\end{array}$
and the unstable interior equilibrium is
$\begin{array}[]{c}x^{l*}_{u}=1/2-\sqrt{\alpha^{2}-4\beta^{2}}/(2\alpha-4\beta)\end{array}$
As $r$ increases, the two interior equilibria move symmetrically away from the
bifurcation point $x^{l*}=1/2$, the interior stable equilibrium moves toward
1, and the unstable equilibrium moves toward 0. Therefore, the proportion of
cooperators tends to increase monotonically with $r$ increases, and
$(b+c)/(b-c)$ is the critical value which specifies whether cooperators are
more abundant than defectors or not. That’s to say, cooperators are more
abundant than defectors when
$r(b-c)-c>b$ (19)
This condition coincides with Eq. (2), for $R=b-c,S=-c,T=b,P=0$, and the
interaction rate $r$ plays the same role as $\sigma$ in section 2.
Since the frequencies of individuals adopting strategy $C$ in each community
become stable at the steady state of the evolutionary dynamics, $\Delta
N_{C}\rightarrow 0$. Hence, when $\Delta N_{C}\neq 0$, we write $\Delta
N_{C}=\Delta N_{C0}\cdot\varepsilon$ for some fixed $\Delta N_{C0}>0$, and Eq.
(17) can be simplified as
$\begin{array}[]{l}[\gamma(r-1)^{2}-N(\alpha-2\beta)](x^{l})^{2}-[\gamma(r-1)^{2}-N(\alpha-2\beta)]x^{l}+(N\beta-\gamma
r)=0\end{array}$ (20)
When $N\alpha+2N\beta>\gamma(r+1)^{2}/N$, i.e.,
$r>(b+c+\gamma/N)/(b-c-\gamma/N)$, we obtain the stable interior equilibrium
$\begin{array}[]{c}x^{l*}_{s}=1/2+1/2\sqrt{[\gamma(r+1)^{2}-N(\alpha+2\beta)]/[\gamma(r-1)^{2}-N(\alpha-2\beta)]}\end{array}$
and the unstable interior equilibrium is
$\begin{array}[]{c}x^{l*}_{u}=1/2-1/2\sqrt{[\gamma(r+1)^{2}-N(\alpha+2\beta)]/[\gamma(r-1)^{2}-N(\alpha-2\beta)]}\end{array}$
This conclusion is similar to the case of $\Delta N_{C}=0$, that is, the two
interior equilibria move symmetrically away from the bifurcation point
$x^{l*}=1/2$ and the proportion of cooperators tends to increase monotonically
with $r$ increases. Thus, $(b+c+\gamma/N)/(b-c-\gamma/N)$ is the critical
value which specifies whether cooperators are more abundant than defectors or
not. With simplification, we have the condition that cooperators are more
abundant than defectors
$r(b-c)-c>b+(\gamma/N)(r+1)$ (21)
When $r>1$, a small $\varepsilon>0$ leads to $\gamma\rightarrow 0$. Eq. (21)
is discriminated from Eq. (19) only with the infinitesimal term, and the
second term in the r.h.s of Eq. (21) is negligible in the meaning of
mathematic. Therefore, Eq. (21) also coincides with Eq. (2), for
$R=b-c,S=-c,T=b,P=0$.
To sum up, Eq. (19) and Eq. (21) reflect that the interaction rate $r$ plays
the similar role as $\sigma$ in section 2, and we verify that $\sigma$
features the reaction rate between two individuals with the same strategy.
## 4 Conclusion and Discussion
In this paper, we have proposed a model of games on a community-structured
population to understand the $\sigma$ law with the Moran process and the
ergodic theory. By calculating the average payoffs of $A$ and $B$ individuals
with the effective sojourn time method, we find that, $\sigma$ features not
only the structured populations characteristics, but also the reaction rate
between individuals. Parameter $\sigma$, reflects that individuals using the
same strategy may interact with themselves $\sigma$ times than the
interactions with individuals of the other strategy. That’s to say, an
interaction between two individuals are not uniform with the verification in a
simplified PDG.
In practice, an overlap of communities usually exists in social networks for
example, yet the case with overlap doesn’t be discussed in this paper. Here we
simplify an overlapped community-structured population to a none-overlap
community-structured population. For a simple example without loss of
generality, suppose that a game is played in a finite population with the
fixed size $H$ distributed over $2$ communities, which have one common
individual, as shown in Fig. 3(a). In this case, the overlap individual can be
regarded as two parts, itself and a virtual reproduction which exist in the
two corresponding separated communities, respectively. Therefore, the
overlapped community-structured population with size $H$ becomes a none-
overlap one with size $H+1$ (see Fig. 3(b)), which is discussed in our paper.
Hence, we provide a simplified yet alternative approach to study the
evolutionary games in a community-structured population with overlap.
## Acknowledgment
We were grateful to Lin Wang and Yu Wang for their helpful discussions and
valuable suggestions, and the anonymous reviewers for their constructive
comments to help improve this paper. This work is supported partly by NSFC
(Grant No. 60874089), the National Key Basic Research and Development Program
(Grant No. 2010CB731403), and the NCET program of China (Grant No. 09-0317).
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---
Figure 1.
An illustrative community-structured population with strategies $A$ and $B$.
At each time step, a single individual from the entire population is selected
proportional to its fitness for reproduction, and the offspring either stays
in this community or migrates to another community. There exists a graph
connecting all the communities in the structured population. The migration
occurs between two communities if there is an edge connecting them both.
Figure 2.
An illustrative fixation process of a single mutant in the whole community-
structured population. (a) A single mutant is produced in community $1$ (white
node); (b) This mutant successfully takes over community $1$ with probability
$\lambda_{11}$, and then migrates to community $2$ to replace a randomly
chosen individual with probability $\lambda_{12}$; (c) The mutant takes over
community $2$ with probability $\lambda_{22}$, and then migrates to community
$3$ to replace a randomly chosen individual with probability $\lambda_{23}$;
(d) The mutant takes over community $4$ with probability $\lambda_{44}$, and
then migrates to community $5$ to replace a randomly chosen individual with
probability $\lambda_{45}$.
---
Figure 3.
(a) An overlapped community-structured population with one common individual;
(b) A population with two none-common communities.
|
arxiv-papers
| 2012-04-18T03:17:37 |
2024-09-04T02:49:29.794735
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Changbing Tang, Xiang Li, Lang Cao, Jingyuan Zhan",
"submitter": "Flion Tang",
"url": "https://arxiv.org/abs/1204.3966"
}
|
1204.4083
|
# a note on the depth formula and vanishing of cohomology
Arash Sadeghi Faculty of Mathematical Sciences and Computer, Tarbiat Moallem
University, Tehran, Iran. School of Mathematics, Institute for Research in
Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
sadeghiarash61@gmail.com
###### Abstract.
It is proved that if one of the finite modules $M$ and $N$, over a local ring
$R$, has reducible complexity and has finite Gorenstein dimension then the
depth formula holds, provided $\mbox{Tor}\,^{R}_{i}(M,N)=0$ for $i\gg 0$. We
also study the vanishing of cohomology of a module of finite complete
intersection dimension.
###### Key words and phrases:
Complete intersection dimension, Depth formula, Gorenstein dimension,
Vanishing of cohomology
###### 2000 Mathematics Subject Classification:
13C15, 13D07, 13D02, 13H10
## 1\. introduction
Let $R$ be a local ring. Two $R$–modules $M$ and $N$ satisfy the depth formula
if
$0pt_{R}(M)+0pt_{R}(N)=0ptR+0pt_{R}(M\otimes_{R}N).$
The depth formula was first studied by Auslander [3]. Suppose that
$\mbox{Pd}\,_{R}(M)<\infty$ and that $q$ is the largest integer such that
$\mbox{Tor}\,_{q}^{R}(M,N)$ is nonzero. Auslander proved that if or
$0pt_{R}(\mbox{Tor}\,_{q}^{R}(M,N))\leq 1$ either $q=0$, then the formula
(1.1) $0pt_{R}(M)+0pt_{R}(N)=0ptR+0pt_{R}(\mbox{Tor}\,_{q}^{R}(M,N))-q$
holds [3, Theorem 1.2].
In [12], Huneke and Wiegand showed that two $R$–modules $M$ and $N$ over
complete intersection rings satisfy the depth formula provided
$\mbox{Tor}\,_{i}^{R}(M,N)=0$ for $i>0$. In [13], Iyengar showed that the
depth formula holds for two $R$–modules $M$ and $N$, provided one of the
modules has finite complete intersection dimension and
$\mbox{Tor}\,_{i}^{R}(M,N)=0$ for all $i>0$. In [6], Araya and Yoshino
generalized Auslander’s original result. More precisely, they proved that the
formula (1.1) holds provided one of the modules has finite complete
intersection dimension and $\mbox{Tor}\,^{R}_{i}(M,N)=0$ for $i\gg 0$. In [9],
Bergh and Jorgensen proved that the depth formula holds in certain cases over
Cohen-Macaulay rings, provided one of the modules has reducible complexity and
$\mbox{Tor}\,_{i}^{R}(M,N)=0$ for $i>0$.
In this paper, we generalize the Auslander’s original result for a module of
finite Gorenstein dimension and with reducible complexity.
In section 1, we prove that the formula (1.1) holds provided one of the
modules has reducible complexity and has finite Gorenstein dimension and
$\mbox{Tor}\,_{i}^{R}(M,N)=0$ for $i\gg 0$, which is a generalization of [6,
Theorem 2.5]. Also it can be viewed as a generalization of [9, Corollary 3.4].
In section 2, we study the vanishing of cohomology of a module of finite
complete intersection dimension over a local ring. For an $R$–module $M$ of
finite weak Gorenstein dimension and an $R$–module $N$ of finite complete
intersection dimension and complexity $c$, it is shown that if there exist an
odd number $q\geq 1$, and a number $n>\mbox{w.g.d}\,_{R}(M)$ such that
$\mbox{Ext}\,^{i}_{R}(M,N)=0$ for $i\in\\{n,n+q,\cdots,n+cq\\}$, then
$\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>\mbox{w.g.d}\,_{R}(M)$(see Theorem
4.2). As a consequence, for two $R$–modules $M$ and $N$ of finite complete
intersection dimensions, it is shown that if there exist an odd number $q\geq
1$, and $i>\mbox{CI-dim}\,_{R}(M)$ such that $\mbox{Ext}\,^{j}_{R}(M,N)=0$,
for $j\in\\{i,i+q,\cdots,i+cq\\}$, where
$c=\min\\{\mbox{cx}\,_{R}(M),\mbox{cx}\,_{R}(N)\\}$, then
$\mbox{Ext}\,^{j}_{R}(M,N)=0$ for all $j>\mbox{CI-dim}\,_{R}(M)$, (see
Corollary 4.3). In Theorem 4.4, it is shown that if
$\mbox{w.g.d}\,_{R}(M)<\infty$ and $N$ has reducible complexity such that
$\mbox{Ext}\,^{i}_{R}(M,N)=0$ for $i\gg 0$ then
$\mbox{w.g.d}\,_{R}(M)=\sup\\{i\mid\mbox{Ext}\,^{i}_{R}(M,N)\neq 0\\}$.
## 2\. Preliminaries
Throughout the paper, $R$ is a commutative Noetherian local ring and all
modules are finite (i.e. finitely generated) $R$–modules. Let
$\cdots\rightarrow F_{n+1}\rightarrow F_{n}\rightarrow
F_{n-1}\rightarrow\cdots\rightarrow F_{0}\rightarrow M\rightarrow 0$
be the minimal free resolution of $M$. Recall that the $n^{\text{th}}$ syzygy
of an $R$–module $M$ is the cokernel of the $F_{n+1}\rightarrow F_{n}$ and
denoted by $\Omega^{n}_{R}(M)$, and it is unique up to isomorphism. The
$n^{\text{th}}$ Betti number, denoted $\beta_{n}^{R}(M)$, is the rank of the
free $R$–module $F_{n}$. The complexity of $M$ is defined as follows;
$\mbox{cx}\,_{R}(M)=\inf\\{i\in\mathbb{N}\cup
0\mid\exists\gamma\in\mathbb{R}\text{ such that }\beta_{n}^{R}(M)\leq\gamma
n^{i-1}\text{ for }n\gg 0\\}.$
Note that $\mbox{cx}\,_{R}(M)=\mbox{cx}\,_{R}(\Omega^{i}_{R}(M))$ for every
$i\geq 0$. It follows from the definition that $\mbox{cx}\,_{R}(M)=0$ if and
only if $\mbox{Pd}\,_{R}(M)<\infty$. The complete intersection dimension was
introduced by Avramov, Gasharov and Peeva [5]. A module of finite complete
intersection dimension behaves homologically like a module over a complete
intersection. Recall that a quasi-deformation of $R$ is a diagram
$R\rightarrow A\twoheadleftarrow Q$ of local homomorphisms, in which
$R\rightarrow A$ is faithfully flat, and $A\twoheadleftarrow Q$ is surjective
with kernel generated by a regular sequence. The module $M$ has finite
complete intersection dimension if there exists such a quasi-deformation for
which $\mbox{Pd}\,_{Q}(M\otimes_{R}A)$ is finite. The complete intersection
dimension of $M$, denoted $\mbox{CI-dim}\,_{R}(M)$, is defined as follows;
$\mbox{CI-
dim}\,_{R}(M)=\inf\\{\mbox{Pd}\,_{Q}(M\otimes_{R}A)-\mbox{Pd}\,_{Q}(A)\mid
R\rightarrow A\twoheadleftarrow Q\text{ is a quasi-deformation }\\}.$
By [5, Theorem 5.3], every module of finite complete intersection dimension
has finite complexity.
The concept of modules with reducible complexity was introduced by Bergh [8].
Let $M$ and $N$ be $R$–modules and consider a homogeneous element $\eta$ in
the graded $R$–module
$\mbox{Ext}\,^{*}_{R}(M,N)=\bigoplus^{\infty}_{i=0}\mbox{Ext}\,^{i}_{R}(M,N)$.
Choose a map $f_{\eta}:\Omega^{|\eta|}_{R}(M)\rightarrow N$ representing
$\eta$, and denote by $K_{\eta}$ the pushout of this map and the inclusion
$\Omega^{|\eta|}_{R}(M)\hookrightarrow F_{|\eta|-1}$. Therefore we obtain a
commutative diagram
$\setcounter{MaxMatrixCols}{14}\begin{CD}&&&&&&&&\\\ \ \
&&&&0@>{}>{}>\Omega^{|\eta|}_{R}(M)@>{}>{}>F_{|\eta|-1}@>{}>{}>\Omega^{|\eta|-1}_{R}(M)@>{}>{}>0&\\\
&&&&&&@V{}V{f_{\eta}}V@V{}V{}V@V{}V{{\parallel}}V\\\ \ \
&&&&0@>{}>{}>N@>{}>{}>K_{\eta}@>{}>{}>\Omega^{|\eta|-1}_{R}(M)@>{}>{}>0.&\\\
\end{CD}$
with exact rows. Note that the module $K_{\eta}$ is independent, up to
isomorphism, of the map $f_{\eta}$ chosen to represent ${\eta}$.
###### Definition 2.1.
_The full subcategory of $R$-modules consisting of the modules having
reducible complexity is defined inductively as follows:_
* (i)
_Every $R$-module of finite projective dimension has reducible complexity._
* (ii)
_An $R$-module $M$ of finite positive complexity has reducible complexity if
there exists a homogeneous element $\eta\in\mbox{Ext}\,^{*}_{R}(M,M)$, of
positive degree, such that $\mbox{cx}\,_{R}(K_{\eta})<\mbox{cx}\,_{R}(M)$,
$0pt_{R}(M)=0pt_{R}(K_{\eta})$ and $K_{\eta}$ has reducible complexity._
By [8, Proposition 2.2(i)], every module of finite complete intersection
dimension has reducible complexity. On the other hand, there are modules
having reducible complexity but whose complete intersection dimension is
infinite (see for example, [9, Corollarry 4.7]).
The notion of the Gorenstein(or G-) dimension was introduced by Auslander [2],
and developed by Auslander and Bridger in [4].
###### Definition 2.2.
An $R$–module $M$ is said to be of $G$-dimension zero whenever
* (i)
_the biduality map $M\rightarrow M^{**}$ is an isomorphism;_
* (ii)
_$\mbox{Ext}\,^{i}_{R}(M,R)=0$ for all $i>0$;_
* (iii)
_$\mbox{Ext}\,^{i}_{R}(M^{*},R)=0$ for all $i>0$._
The Gorenstein dimension of $M$, denoted $\mbox{G-dim}\,_{R}(M)$, is defined
to be the infimum of all nonnegative integers $n$, such that there exists an
exact sequence
$0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{0}\rightarrow M\rightarrow
0$
in which all the $G_{i}$ have $G$-dimension zero. By [4, Theorem 4.13], if $M$
has finite Gorenstein dimension then $\mbox{G-dim}\,_{R}(M)=0ptR-0pt_{R}(M)$.
By [5, Theorem 1.4], $\mbox{G-dim}\,_{R}(M)$ is bounded above by the complete
intersection dimension, $\mbox{CI-dim}\,_{R}(M)$, of $M$ and if $\mbox{CI-
dim}\,_{R}(M)<\infty$ then the equality holds.
The notion of the weak Gorenstein dimension was introduced in [11]. An
$R$–module $M$ is said to be of weak Gorenstein dimension zero, written
$\mbox{w.g.d}\,_{R}(M)=0$, if $\mbox{Ext}\,^{i}_{R}(M,R)=0$ for all $i>0$. If
for some integer $t\geq 1$ we have $\mbox{Ext}\,^{t}_{R}(M,R)\neq 0$ and
$\mbox{Ext}\,^{i}_{R}(M,R)=0$ for all $i>t$ then $\mbox{w.g.d}\,_{R}(M)=t$. In
all other cases, i.e. if $\mbox{Ext}\,^{i}_{R}(M,R)\neq 0$ for infinitely many
integer $i>0$, then $\mbox{w.g.d}\,_{R}(M)=\infty$.
Note that, by [4, Theorem 4.13], every module of finite Gorenstein dimension
has finite weak Gorenstein dimension and
$\mbox{G-dim}\,_{R}(M)=\mbox{w.g.d}\,_{R}(M)$. On the other hand, there are
modules having finite weak Gorenstein dimension but whose Gorenstein dimension
is infinite (see [15]).
Let $P_{1}\overset{f}{\rightarrow}P_{0}\rightarrow M\rightarrow 0$ be a finite
projective presentation of $M$. The transpose of $M$, $\mbox{Tr}\,M$, is
defined to be $\mbox{Coker}\,f^{*}$, where $(-)^{*}:=\mbox{Hom}\,_{R}(-,R)$,
which satisfies in the exact sequence
(2.1) $0\rightarrow M^{*}\rightarrow P_{0}^{*}\rightarrow
P_{1}^{*}\rightarrow\mbox{Tr}\,M\rightarrow 0$
and is unique up to projective equivalence. Thus the minimal projective
presentations of $M$ represent isomorphic transposes of $M$. Two modules $M$
and $N$ are called _stably isomorphic_ and write
$\underline{M}\cong\underline{N}$ if $M\oplus P\cong N\oplus Q$ for some
projective modules $P$ and $Q$.
The composed functors $\mathcal{T}_{k}:=\mbox{Tr}\,\Omega^{k-1}$ for $k>0$
introduced by Auslander and Bridger in [4]. If $\mbox{Ext}\,^{i}_{R}(M,R)=0$
for some $i>0$, then it is easy to see that
$\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$.
We frequently use the following Theorem of Auslander and Bridger.
###### Theorem 2.3.
[4, Theorem 2.8]_Let $M$ be an $R$–module and $n\geq 0$ an integer. Then there
are exact sequences of functors:_
$0\rightarrow\mbox{Ext}\,^{1}_{R}(\mathcal{T}_{n+1}M,-)\rightarrow\mbox{Tor}\,_{n}^{R}(M,-)\rightarrow\mbox{Hom}\,_{R}(\mbox{Ext}\,^{n}_{R}(M,R),-)\rightarrow\mbox{Ext}\,^{2}_{R}(\mathcal{T}_{n+1}M,-),$
$\mbox{Tor}\,_{2}^{R}(\mathcal{T}_{n+1}M,-)\rightarrow(\mbox{Ext}\,^{n}_{R}(M,R)\otimes_{R}-)\rightarrow\mbox{Ext}\,^{n}_{R}(M,-)\rightarrow\mbox{Tor}\,_{1}^{R}(\mathcal{T}_{n+1}M,-)\rightarrow
0.$
## 3\. the depth formula
Let $M$ and $N$ be $R$–modules. In the following, we investigate the
connection between the vanishing of homology modules,
$\mbox{Tor}\,_{i>0}^{R}(M,N)$, and the vanishing of cohomology modules,
$\mbox{Ext}\,^{i>0}_{R}(\mbox{Tr}\,M,N)$.
###### Lemma 3.1.
_Let $M$, $N$ be $R$–modules such that $M$ has reducible complexity. If $M$ is
of $G$-dimension zero, and $\mbox{Tor}\,_{i}^{R}(M,N)=0$ for all $i>0$ then
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$._
###### Proof.
Set $c=\mbox{cx}\,_{R}(M)$, we argue by induction on $c$. If $c=0$ then
$\mbox{Pd}\,_{R}(M)<\infty$ and so
$\mbox{Pd}\,_{R}(M)=\mbox{G-dim}\,_{R}(M)=0$. Therefore $\mbox{Tr}\,M=0$ and
we have nothing to prove. As $\mbox{Tor}\,_{i}^{R}(M,N)=0$ for all $i>0$,
$\mbox{Ext}\,^{1}_{R}(\mathcal{T}_{i+1}M,N)=0$ for all $i>0$ by Theorem 2.3.
Since $\mbox{Ext}\,^{i}_{R}(M,R)=0$ for all $i>0$ then
$\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$ for
all $i>0$ and so
(3.1) $\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{t}M,N)=0\text{ for all }t>1\text{ and
}1\leq i<t.$
Suppose that $c>0$ and that $\eta\in\mbox{Ext}\,^{*}_{R}(M,M)$ reduces the
complexity of $M$. Consider The exact sequence
(3.2) $0\rightarrow M\rightarrow
K_{\eta}\rightarrow\Omega^{q}_{R}(M)\rightarrow 0,$
where $|\eta|=q+1$ and $\mbox{cx}\,_{R}(K_{\eta})<c$. Note that
$\mbox{G-dim}\,_{R}(K_{\eta})=\mbox{G-dim}\,_{R}(M)=0$. The exact sequence
(3.2), induces the long exact sequence
$\cdots\rightarrow\mbox{Tor}\,_{i}^{R}(M,N)\rightarrow\mbox{Tor}\,_{i}^{R}(K_{\eta},N)\rightarrow\mbox{Tor}\,_{i+q}^{R}(M,N)\rightarrow\cdots,$
of homology modules. Therefore, $\mbox{Tor}\,_{i}^{R}(K_{\eta},N)=0$ for all
$i>0$ and so by induction hypothesis
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,K_{\eta},N)=0$ for all $i>0$. By [4, Lemma
3.9], from the exact sequence (3.2), we obtain the following exact sequence
$0\rightarrow{(\Omega^{q}_{R}(M))}^{*}\rightarrow{K_{\eta}}^{*}\rightarrow
M^{*}\rightarrow\mbox{T}\,(\Omega^{q}_{R}(M))\rightarrow\mbox{T}\,(K_{\eta})\rightarrow\mbox{T}\,(M)\rightarrow
0$
where $\underline{\mbox{T}\,(M)}\cong\underline{\mbox{Tr}\,M}$,
$\underline{\mbox{T}\,(K_{\eta})}\cong\underline{\mbox{Tr}\,K_{\eta}}$ and
$\underline{\mbox{T}\,(\Omega^{q}_{R}(M))}\cong\underline{\mathcal{T}_{q+1}M}$.
Since $\mbox{Ext}\,^{1}_{R}(\Omega^{q}_{R}(M),R)=0$, we get the exact sequence
(3.3)
$0\rightarrow\mbox{T}\,(\Omega^{q}_{R}(M))\rightarrow\mbox{T}\,(K_{\eta})\rightarrow\mbox{T}\,(M)\rightarrow
0.$
The exact sequence (3.3), induces a long exact sequence
(3.4)
$\cdots\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(M),N)\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(K_{\eta}),N)\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(\Omega^{q}_{R}(M)),N)\rightarrow\cdots$
of cohomology modules. As $\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,K_{\eta},N)=0$ for
all $i>0$, we obtain from the (3.4)
$\mbox{Ext}\,^{i+1}_{R}(\mbox{Tr}\,M,N)\cong\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{q+1}M,N)$
for all $i>0$ and since
$\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$ for
all $i>0$,
(3.5)
$\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{q+1}M,N)\cong\mbox{Ext}\,^{i+1}_{R}(\mbox{Tr}\,M,N)\cong\mbox{Ext}\,^{i+q+1}_{R}(\mathcal{T}_{q+1}M,N)\text{
for all }i>0.$
Therefore if $q>0$ then by (3.5) and (3.1)
(3.6) $\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{q+1}M,N)=0\text{ for }i\neq
j(q+1)\text{ and }j>0,$ (3.7)
$\mbox{Ext}\,^{q+1}_{R}(\mathcal{T}_{q+1}M,N)\cong\mbox{Ext}\,^{j(q+1)}_{R}(\mathcal{T}_{q+1}M,N)\text{
for all }j>0.$
By [8, Lemma 2.3], there exists an exact sequence
(3.8) $0\rightarrow\Omega^{q+1}_{R}(K_{\eta})\rightarrow K_{\eta^{2}}\oplus
F\rightarrow K_{\eta}\rightarrow 0,$
where $F$ is free. As $\mbox{G-dim}\,_{R}(K_{\eta})=0$, by [4, Lemma 3.9] we
obtain the following exact sequence
(3.9)
$0\rightarrow\mbox{T}\,(K_{\eta})\rightarrow\mbox{T}\,(K_{\eta^{2}}\oplus
F)\rightarrow\mbox{T}\,(\Omega^{q+1}_{R}(K_{\eta}))\rightarrow 0,$
where $\underline{\mbox{T}\,(K_{\eta})}\cong\underline{\mbox{Tr}\,K_{\eta}}$,
$\underline{\mbox{T}\,(K_{\eta^{2}}\oplus
F)}\cong\underline{\mbox{Tr}\,K_{\eta^{2}}}$ and
$\underline{\mbox{T}\,(\Omega^{q+1}_{R}(K_{\eta}))}\cong\underline{\mathcal{T}_{q+2}K_{\eta}}$.
The exact sequence (3.9), induces a long exact sequence
(3.10)
$\cdots\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(\Omega^{q+1}_{R}(K_{\eta})),N)\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(K_{\eta^{2}}\oplus
F),N)\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(K_{\eta}),N)\rightarrow\cdots$
of cohomology modules. Note that by the proof of [8, Proposition 2.2(ii)],
$\Omega^{q+1}_{R}(K_{\eta})$ has also reducible complexity. Therefore, by
induction hypothesis, $\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{q+2}K_{\eta},N)=0$
for all $i>0$ and so by (3.10),
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,K_{\eta^{2}},N)=0$ for all $i>0$. By [4,
Lemma 3.9], from the exact sequence $0\rightarrow M\rightarrow
K_{\eta^{2}}\rightarrow\Omega^{2q+1}_{R}(M)\rightarrow 0$, we obtain the
following exact sequence
(3.11)
$0\rightarrow{(\Omega^{2q+1}_{R}(M))}^{*}\rightarrow{(K_{\eta^{2}})}^{*}\rightarrow
M^{*}\rightarrow\mbox{T}\,(\Omega^{2q+1}_{R}(M))\rightarrow\mbox{T}\,(K_{\eta^{2}})\rightarrow\mbox{T}\,(M)\rightarrow
0,$
where $\underline{\mbox{T}\,(M)}\cong\underline{\mbox{Tr}\,M}$,
$\underline{\mbox{T}\,(\Omega^{2q+1}_{R}(M))}\cong\underline{\mathcal{T}_{2q+2}M}$
and
$\underline{\mbox{T}\,(K_{\eta^{2}})}\cong\underline{\mbox{Tr}\,K_{\eta^{2}}}$.
As $\mbox{Ext}\,^{2q+2}_{R}(M,R)=0$, we get the exact sequence
(3.12)
$0\rightarrow\mbox{T}\,(\Omega^{2q+1}_{R}(M))\rightarrow\mbox{T}\,(K_{\eta^{2}})\rightarrow\mbox{T}\,(M)\rightarrow
0.$
The exact sequence (3.12), induces a long exact sequence
(3.13)
$\cdots\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(M),N)\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(K_{\eta^{2}}),N)\rightarrow\mbox{Ext}\,^{i}_{R}(\mbox{T}\,(\Omega^{2q+1}_{R}(M)),N)\rightarrow\cdots$
of cohomology modules. As $\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,K_{\eta^{2}},N)=0$
and $\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$
for all $i>0$, by (3.13) we get the following isomorphisms.
(3.14)
$\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{2q+2}M,N)\cong\mbox{Ext}\,^{i+1}_{R}(\mbox{Tr}\,M,N)\cong\mbox{Ext}\,^{2q+i+2}_{R}(\mathcal{T}_{2q+2}M,N)\text{
for all }i>0.$
If $q=0$ then by (3.14), (3.5) and (3.1), it is obvious that
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$. Now if $q>0$ then by
(3.14) and (3.1),
$\mbox{Ext}\,^{2q+2}_{R}(\mathcal{T}_{q+1}M,N)\cong\mbox{Ext}\,^{3q+3}_{R}(\mathcal{T}_{2q+2}M,N)\cong\mbox{Ext}\,^{q+1}_{R}(\mathcal{T}_{2q+2}M,N)=0$
Therefore by (3.6) and (3.7), $\mbox{Ext}\,^{i}_{R}(\mathcal{T}_{q+1}M,N)=0$
for all $i>0$ and so $\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$. ∎
The following Theorem is a generalization of [6, Theorem 2.5], [9, Corollary
3.4] and also [9, Theorem 3.1].
###### Theorem 3.2.
_Let $M$ and $N$ be $R$–modules and let $\mbox{Tor}\,_{i}^{R}(M,N)=0$ for
$i\gg 0$. If $M$ has reducible complexity and
$q=\sup\\{i\mid\mbox{Tor}\,_{i}^{R}(M,N)\neq 0\\}$ then the following
statements hold true._
* (i)
If $\mbox{G-dim}\,_{R}(M)<\infty$ and $q=0$ then
$0pt_{R}(M)+0pt_{R}(N)=0pt_{R}(M\otimes_{R}N)+0ptR.$
* (ii)
If $q>0$, $0pt_{R}(\mbox{Tor}\,_{q}^{R}(M,N))\leq 1$ then
$0pt_{R}(M)+0pt_{R}(N)=0ptR+0pt_{R}(\mbox{Tor}\,_{q}^{R}(M,N))-q$
###### Proof.
(i) We argue by induction on $c=\mbox{cx}\,_{R}(M)$. If $c=0$ then
$\mbox{Pd}\,_{R}(M)<\infty$ and the formula holds by Auslander’s original
result, so suppose that $c>0$ and that $\eta\in\mbox{Ext}\,^{*}_{R}(M,M)$
reduces the complexity of $M$. The exact sequence $0\rightarrow M\rightarrow
K_{\eta}\rightarrow\Omega^{n}_{R}(M)\rightarrow 0$, induces a long exact
sequence
(3.15)
$\cdots\rightarrow\mbox{Tor}\,_{i}^{R}(M,N)\rightarrow\mbox{Tor}\,_{i}^{R}(K_{\eta},N)\rightarrow\mbox{Tor}\,_{i+n}^{R}(M,N)\rightarrow\cdots$
of homology modules. Therefore $\mbox{Tor}\,_{i}^{R}(K_{\eta},N)=0$ for all
$i>0$. As $\mbox{cx}\,_{R}(K_{\eta})<c$ and
$\mbox{G-dim}\,_{R}(K_{\eta})<\infty$,
(3.16) $0pt_{R}(K_{\eta})+0pt_{R}(N)=0pt_{R}(K_{\eta}\otimes_{R}N)+0ptR$
by induction hypothesis. Now by induction on $\mbox{G-dim}\,_{R}(M)$, we show
that the formula holds. If $\mbox{G-dim}\,_{R}(M)=0$ then by the Lemma 3.1,
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$. Hence by Theorem 2.3,
$M\otimes_{R}N\cong\mbox{Hom}\,_{R}(M^{*},N)$ and also by the exact sequence
(2.1), $\mbox{Ext}\,^{i}_{R}(M^{*},N)=0$ for all $i>0$. Therefore by [6, Lemma
4.1], $0pt_{R}(M\otimes_{R}N)=0pt_{R}(\mbox{Hom}\,_{R}(M^{*},N))=0pt_{R}(N)$
and so by the Auslander-Bridger formula,
$0pt_{R}(M)+0pt_{R}(N)=0pt_{R}(M\otimes N)+0ptR$.
Now let $\mbox{G-dim}\,_{R}(M)>0$, if $n=0$ then we obtain the following exact
sequence
(3.17) $0\rightarrow M\otimes_{R}N\rightarrow K_{\eta}\otimes_{R}N\rightarrow
M\otimes_{R}N\rightarrow 0.$
As $\mbox{G-dim}\,_{R}(M)>0$,
$\mbox{G-dim}\,_{R}(\Omega_{R}(M))=\mbox{G-dim}\,_{R}(M)-1$. Note that by the
proof of [8, Proposition 2.2(ii)], $\Omega_{R}(M)$ has also reducible
complexity. Therefore,
(3.18) $0pt_{R}(\Omega_{R}(M))+0pt_{R}(N)=0pt_{R}(\Omega_{R}(M)\otimes
N)+0ptR$
by induction hypothesis. From the exact sequence
$0\rightarrow\Omega_{R}(M)\rightarrow F\rightarrow M\rightarrow 0$, where $F$
is a free module, we obtain the exact sequence
$0\rightarrow\Omega_{R}(M)\otimes_{R}N\rightarrow F\otimes_{R}N\rightarrow
M\otimes_{R}N\rightarrow 0$. Therefore,
$0pt_{R}(M\otimes_{R}N)\geq\min\\{0pt_{R}(N),0pt_{R}(\Omega_{R}(M)\otimes_{R}N)-1\\}$,
by the depth Lemma. Now by (3.18),
$0pt_{R}(\Omega_{R}(M)\otimes_{R}N)-1=0pt_{R}(N)-\mbox{G-dim}\,_{R}(M)<0pt_{R}(N)$
and so $0pt_{R}(M\otimes_{R}N)\geq 0pt_{R}(N)-\mbox{G-dim}\,_{R}(M)$. On the
other hand, if $0pt_{R}(M\otimes_{R}N)>0pt_{R}(N)-\mbox{G-dim}\,_{R}(M)$ then
by the exact sequence (3.17), it is obvious that
$0pt_{R}(K_{\eta}\otimes_{R}N)>0pt_{R}(N)-\mbox{G-dim}\,_{R}(M)=0pt_{R}(N)-\mbox{G-dim}\,_{R}(K_{\eta})$,
which is a contradiction by (3.16). Hence
$0pt_{R}(M)+0pt_{R}(N)=0pt_{R}(M\otimes N)+0ptR$.
Now let $n>0$, then
$\mbox{G-dim}\,_{R}(\Omega^{n}_{R}(M))=\max\\{0,\mbox{G-dim}\,_{R}(M)-n\\}<\mbox{G-dim}\,_{R}(M)$.
Note that by the proof of the [8, Proposition 2.2(ii)], $\Omega^{n}_{R}(M)$
has also reducible complexity and so by induction hypothesis,
$0pt_{R}(\Omega^{n}_{R}(M)\otimes_{R}N)=0pt_{R}(N)-\mbox{G-dim}\,_{R}(\Omega^{n}_{R}(M))$.
Therefore, as $\mbox{G-dim}\,_{R}(M)=\mbox{G-dim}\,_{R}(K_{\eta})$,
$0pt_{R}(K_{\eta}\otimes_{R}N)<0pt_{R}(\Omega^{n}_{R}(M)\otimes_{R}N)$ by
(3.16) and so from the exact sequence $0\rightarrow M\otimes_{R}N\rightarrow
K_{\eta}\otimes_{R}N\rightarrow\Omega^{n}_{R}(M)\otimes_{R}N\rightarrow 0$, it
is obvious that $0pt_{R}(M\otimes_{R}N)=0pt_{R}(K_{\eta}\otimes_{R}N)$.
Therefore by (3.16), $0pt_{R}(M)+0pt_{R}(N)=0pt_{R}(M\otimes N)+0ptR$.
(ii) We argue by induction on $\mbox{cx}\,_{R}(M)=c$. If $c=0$ then
$\mbox{Pd}\,_{R}(M)<\infty$ and the formula holds by Auslander’s original
result, so suppose that $c>0$ and that $\eta\in\mbox{Ext}\,^{*}_{R}(M,M)$
reduces the complexity of $M$. The exact sequence $0\rightarrow M\rightarrow
K_{\eta}\rightarrow\Omega^{n}_{R}(M)\rightarrow 0$, induces a long exact
sequence
(3.19)
$\cdots\rightarrow\mbox{Tor}\,_{q}^{R}(M,N)\overset{f}{\rightarrow}\mbox{Tor}\,_{q}^{R}(K_{\eta},N)\rightarrow\mbox{Tor}\,_{q+n}^{R}(M,N)\rightarrow\cdots$
of homology modules. From (3.19), it is obvious that
$q=\sup\\{i\mid\mbox{Tor}\,_{i}^{R}(K_{\eta},N)\neq 0\\}$. If $n=0$ then from
(3.19), we obtain the following exact sequences
(3.20)
$0\rightarrow\mbox{Tor}\,_{q}^{R}(M,N)\rightarrow\mbox{Tor}\,_{q}^{R}(K_{\eta},N)\rightarrow\mbox{Coker}\,(f)\rightarrow
0,$ (3.21)
$0\rightarrow\mbox{Coker}\,(f)\rightarrow\mbox{Tor}\,_{q}^{R}(M,N).$
If $0pt_{R}(\mbox{Tor}\,_{q}^{R}(M,N))=0$, then by the exact sequence (3.20),
$0pt_{R}(\mbox{Tor}\,_{q}^{R}(K_{\eta},N))=0$. As
$\mbox{cx}\,_{R}(K_{\eta})<c$ , by induction hypothesis,
$0pt_{R}(N)+0pt_{R}(K_{\eta})=0ptR-q$ and since
$0pt_{R}(K_{\eta})=0pt_{R}(M)$, we are done. If
$0pt_{R}(\mbox{Tor}\,_{q}^{R}(M,N))=1$ then by the exact sequence (3.21),
$0pt_{R}(\mbox{Coker}\,(f))>0$ and so by the exact sequence (3.20),
$0pt_{R}(\mbox{Tor}\,_{q}^{R}(K_{\eta},N))=1$. Therefore by induction
hypothesis, $0ptR+1-q=0pt_{R}(K_{\eta})+0pt_{R}(N)=0pt_{R}(M)+0pt_{R}(N)$.
Now suppose that $n>0$, then from the exact sequence (3.19), it is obvious
that $\mbox{Tor}\,_{q}^{R}(M,N)\cong\mbox{Tor}\,_{q}^{R}(K_{\eta},N)$ and so
by induction hypothesis, we are done. ∎
The following lemma is useful for the rest of the paper.
###### Lemma 3.3.
_For an $R$–module $M$, $\mbox{CI-dim}\,_{R}(M)=0$ if and only if $\mbox{CI-
dim}\,_{R}(\mbox{Tr}\,M)=0$._
###### Proof.
If $\mbox{CI-dim}\,_{R}(M)=0$ then $\mbox{CI-dim}\,_{R}(M^{*})=0$ by [10,
Lemma 3.5] and so from the exact sequence (2.1), $\mbox{CI-
dim}\,_{R}(\mbox{Tr}\,M)<\infty$. As $\mbox{G-dim}\,_{R}(M)=0$,
$\mbox{G-dim}\,_{R}(\mbox{Tr}\,M)=0$ by [4, Lemma 4.1] and so $\mbox{CI-
dim}\,_{R}(\mbox{Tr}\,M)=0$ by [5, Theorem 1.4]. As
$\underline{M}\cong\underline{\mbox{Tr}\,\mbox{Tr}\,M}$, the other side is
obvious.
∎
Let $M$ and $N$ be $R$–modules. In the following, we investigate the
connection between complete intersection dimension of $M$ and the vanishing of
cohomology modules, $\mbox{Ext}\,^{i>0}_{R}(\mbox{Tr}\,M,N)$, and the
vanishing of homology modules, $\mbox{Tor}\,_{i>0}^{R}(M,N)$.
###### Proposition 3.4.
_Let $M$ and $N$ be $R$–modules such that $\mbox{CI-dim}\,_{R}(M)<\infty$. If
two of the following conditions hold true then the third one is also true._
* (i)
_$\mbox{Tor}\,_{i}^{R}(M,N)=0$ for all $i>0$_,
* (ii)
_$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$_,
* (iii)
_$\mbox{CI-dim}\,_{R}(M)=0$_.
###### Proof.
(i),(ii)$\Rightarrow$(iii) By [6, Theorem 2.5] and [5, Theorem 1.4],
$0pt_{R}(M\otimes_{R}N)=0pt_{R}(N)-\mbox{CI-dim}\,_{R}(M)$. As
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$,
$M\otimes_{R}N\cong\mbox{Hom}\,_{R}(M^{*},N)$ by Theorem 2.3 and also
$\mbox{Ext}\,^{i}_{R}(M^{*},N)=0$ for all $i>0$, by the exact sequence (2.1).
Therefore, $0pt_{R}(\mbox{Hom}\,_{R}(M^{*},N))=0pt_{R}(N)$ by [6, Lemma 4.1].
Hence $\mbox{CI-dim}\,_{R}(M)=0$.
(ii),(iii)$\Rightarrow$(i) Set $K=\mbox{Tr}\,M$ and $c=\mbox{cx}\,_{R}(M)$. By
Lemma 3.3, $\mbox{CI-dim}\,_{R}(K)=0$. By Theorem 2.3,
$\mbox{Tor}\,_{1}^{R}(\mathcal{T}_{i+1}K,N)=0$ for all $i>0$. As
$\mbox{Ext}\,^{i}_{R}(K,R)=0$ for all $i>0$,
$\underline{\mathcal{T}_{i}K}\cong\underline{\Omega\mathcal{T}_{i+1}K}$ for
all $i>0$ and so $\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{j}K,N)=0$ for all $j>1$
and $1\leq i<j$. As $\underline{M}\cong\underline{\mbox{Tr}\,K}$,
$\mbox{cx}\,_{R}(\mbox{Tr}\,K)=c$ and so $\mbox{cx}\,_{R}(\mathcal{T}_{i}K)=c$
for all $i>0$. Since $\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{c+2}K,N)=0$ for $1\leq
i\leq c+1$ and $\mbox{CI-dim}\,_{R}(\mathcal{T}_{c+2}K)=0$, then
$\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{c+2}K,N)=0$ for all $i>0$, by[14, Corollary
2,6]. Therefore, $\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{1}K,N)=0$ for all $i>0$.
As $\underline{M}\cong\underline{\mbox{Tr}\,K}$, $\mbox{Tor}\,^{i}_{R}(M,N)=0$
for all $i>0$.
(i), (iii)$\Rightarrow$(ii) By [5, Theorem
1.4]$,\mbox{G-dim}\,_{R}(M)=\mbox{CI-dim}\,_{R}(M)=0$ and by [8, Proposition
2.2(i)], $M$ has reducible complexity. Therefore,
$\mbox{Ext}\,^{i}_{R}(\mbox{Tr}\,M,N)=0$ for all $i>0$ by Lemma 3.1. ∎
## 4\. Vanishing results
Let $M$ and $N$ be $R$–modules. In [14], Jorgensen proved that the vanishing
of Ext for a certain sequence of numbers forces the vanishing of all the
higher Ext groups. More precisely, he proved that if $\mbox{CI-
dim}\,_{R}(M)<\infty$ and
$\mbox{Ext}\,^{i}_{R}(M,N)=\mbox{Ext}\,^{i+1}_{R}(M,N)=\cdots=\mbox{Ext}\,^{i+c}_{R}(M,N)=0$,
where $c=\mbox{cx}\,_{R}(M)$, then $\mbox{Ext}\,^{j}_{R}(M,N)=0$ for all
$j>\mbox{CI-dim}\,_{R}(M)$ [14, Corollary 2.6]. In [7], Bergh assumed the
vanishing of nonconsecutive Ext groups and generalized this result. For an
$R$–module $M$ of finite complete intersection dimension, he proved that if
there exist an odd number $q\geq 1$, and a number $n>\mbox{CI-dim}\,_{R}(M)$
such that $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for $i\in\\{n,n+q,\cdots,n+cq\\}$,
then $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>\mbox{CI-dim}\,_{R}(M)$, [7,
Theorem 3.1]. In this section, we are going to prove similar results, when
$\mbox{w.g.d}\,_{R}(M)<\infty$ and $N$ has finite complete intersection
dimension.
###### Proposition 4.1.
_Let $M$ and $N$ be $R$–modules and let $\mbox{CI-dim}\,_{R}(N)<\infty$ and
$\mbox{w.g.d}\,_{R}(M)<\infty$. Set $\mbox{cx}\,_{R}(N)=c$. If
$\mbox{Ext}\,^{i}_{R}(M,N)=\mbox{Ext}\,^{i+1}_{R}(M,N)=\ldots=\mbox{Ext}\,^{i+c}_{R}(M,N)=0$,
for some $i>\mbox{w.g.d}\,_{R}(M)$, then $\mbox{Ext}\,^{j}_{R}(M,N)=0$ for all
$j>\mbox{w.g.d}\,_{R}(M)$._
###### Proof.
We argue by induction on $c$. If $c=0$ then $\mbox{Pd}\,_{R}(N)<\infty$. As
$\mbox{Ext}\,^{i}_{R}(M,R)=0$ for all $i>\mbox{w.g.d}\,_{R}(M)$,
$\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$ for
all $i>\mbox{w.g.d}\,_{R}(M)$ and also by Theorem 2.3,
$\mbox{Ext}\,^{j}_{R}(M,N)\cong\mbox{Tor}\,_{1}^{R}(\mathcal{T}_{j+1}M,N)$ for
all $j>\mbox{w.g.d}\,_{R}(M)$. Set $t=\mbox{Pd}\,_{R}(N)$. Therefore
$\mbox{Ext}\,^{j}_{R}(M,N)\cong\mbox{Tor}\,_{t+1}^{R}(\mathcal{T}_{t+j+1}M,N)=0$
for all $j>\mbox{w.g.d}\,_{R}(M)$.
Now suppose $c$ is positive and set $n=\mbox{w.g.d}\,_{R}(M)$. As $\mbox{CI-
dim}\,_{R}(N)<\infty$, by [7, Lemma 2.1(i)], there exists a quasi deformation
$R\rightarrow A\twoheadleftarrow Q$ such that the $A$–module $A\otimes_{R}N$
has reducible complexity by an element
$\eta\in\mbox{Ext}\,^{2}_{A}(A\otimes_{R}N,A\otimes_{R}N)$. Set
$\hat{N}=A\otimes_{R}N$ and $\hat{M}=A\otimes_{R}M$. Note that
$\mbox{w.g.d}\,_{A}(\hat{M})=n$ and by the proof of [7, Lemma 2.1(i)],
$\mbox{CI-dim}\,_{A}(\hat{N})<\infty$. The exact sequence
$0\rightarrow\hat{N}\rightarrow
K_{\eta}\rightarrow\Omega_{A}\hat{N}\rightarrow 0$
induces a long exact sequence
(4.1)
$\cdots\rightarrow\mbox{Ext}\,^{j}_{A}(\hat{M},\hat{N})\rightarrow\mbox{Ext}\,^{j}_{A}(\hat{M},K_{\eta})\rightarrow\mbox{Ext}\,^{j}_{A}(\hat{M},\Omega_{A}(\hat{N})){\rightarrow}\cdots$
of cohomology modules. Now consider the exact sequence
$0\rightarrow\Omega_{A}(\hat{N})\rightarrow F\rightarrow\hat{N}\rightarrow 0$,
where $F$ is a free $A$–module. As $\mbox{Ext}\,^{k}_{A}(\hat{M},A)=0$ for all
$k>n$, we get the following isomorphism
(4.2)
$\mbox{Ext}\,^{j-1}_{A}(\hat{M},\hat{N})\cong\mbox{Ext}\,^{j}_{A}(\hat{M},\Omega_{A}(\hat{N}))$
for all $j>n+1$. Now from (4.1) and (4.2), it is obvious that
$\mbox{Ext}\,^{j}_{A}(\hat{M},K_{\eta})=0$ for $i+1\leq j\leq i+c$. As
$\mbox{cx}\,_{A}(K_{\eta})<\mbox{cx}\,_{A}(\hat{N})=c$, by induction
hypothesis we have $\mbox{Ext}\,^{j}_{A}(\hat{M},K_{\eta})=0$ for all $j>n$.
Therefore from (4.1) and (4.2), we get
$\mbox{Ext}\,^{j-1}_{A}(\hat{M},\hat{N})\cong\mbox{Ext}\,^{j}_{A}(\hat{M},\Omega_{A}(\hat{N}))\cong\mbox{Ext}\,^{j+1}_{A}(\hat{M},\hat{N})$
for all $j>n+1$. Now since $c>0$, it is obvious that
$\mbox{Ext}\,^{j}_{A}(\hat{M},\hat{N})=0$ for all $j>n$. Therefore
$\mbox{Ext}\,^{j}_{R}(M,N)=0$ for all $j>n$. ∎
Now we can generalize Proposition 4.1 as follows.
###### Theorem 4.2.
_Let $M$ and $N$ be $R$–modules and let $\mbox{CI-dim}\,_{R}(N)<\infty$ and
$\mbox{w.g.d}\,_{R}(M)<\infty$. Set $\mbox{cx}\,_{R}(N)=c$. If there exist an
odd number $q\geq 1$, and $i>\mbox{w.g.d}\,_{R}(M)$ such that
$\mbox{Ext}\,^{j}_{R}(M,N)=0$, for $j\in\\{i,i+q,\cdots,i+cq\\}$, then
$\mbox{Ext}\,^{j}_{R}(M,N)=0$ for all $j>\mbox{w.g.d}\,_{R}(M)$._
###### Proof.
We argue by induction on $c$. If $c=0$ then $\mbox{Pd}\,_{R}(N)<\infty$ and as
we have seen in the proof of Proposition 4.1, $\mbox{Ext}\,^{j}_{R}(M,N)=0$
for all $j>\mbox{w.g.d}\,_{R}(M)$.
Now let $c>0$, $q=2t-1$, $t\geq 1$. As $\mbox{CI-dim}\,_{R}(N)<\infty$, by [7,
Lemma 2.1(i)], there exists a quasi deformation $R\rightarrow
A\twoheadleftarrow Q$ such that the $A$–module $A\otimes_{R}N$ has reducible
complexity by an element
$\eta\in\mbox{Ext}\,^{2}_{A}(A\otimes_{R}N,A\otimes_{R}N)$. Set
$\hat{N}=A\otimes_{R}N$ and $\hat{M}=A\otimes_{R}M$. As $\mbox{CI-
dim}\,_{A}(\hat{N})<\infty$, by [7, Lemma 2.1(ii)], the element $\eta^{t}$
also reduces the complexity of $\hat{N}$. From the exact sequence
$0\rightarrow\hat{N}\rightarrow
K_{\eta^{t}}\rightarrow\Omega^{q}_{A}(\hat{N})\rightarrow 0$
we get the following long exact sequence of cohomology modules.
(4.3)
$\cdots\rightarrow\mbox{Ext}\,^{j}_{A}(\hat{M},\hat{N})\rightarrow\mbox{Ext}\,^{j}_{A}(\hat{M},K_{\eta^{t}})\rightarrow\mbox{Ext}\,^{j}_{A}(\hat{M},\Omega^{q}_{A}(\hat{N}))\rightarrow\cdots.$
As $\mbox{w.g.d}\,_{A}(\hat{M})=\mbox{w.g.d}\,_{R}(M)<\infty$, it is easy to
see that
(4.4)
$\mbox{Ext}\,^{j}_{A}(\hat{M},\Omega^{q}_{A}(\hat{N}))\cong\mbox{Ext}\,^{j-q}_{A}(\hat{M},\hat{N})$
for all $j>\mbox{w.g.d}\,_{A}(\hat{M})+q$. Now from (4.3) and (4.4), it is
obvious that $\mbox{Ext}\,^{j}_{A}(\hat{M},K_{\eta^{t}})=0$ for
$j\in\\{i+q,i+2q,\cdots,i+cq\\}$. As
$\mbox{cx}\,_{A}(K_{\eta^{t}})<\mbox{cx}\,_{A}(\hat{N})=c$, by induction
hypothesis we have $\mbox{Ext}\,^{j}_{A}(\hat{M},K_{\eta^{t}})=0$ for all
$j>\mbox{w.g.d}\,_{A}(\hat{M})$. Therefore from (4.3) and (4.4), we get
$\mbox{Ext}\,^{j}_{A}(\hat{M},\hat{N})\cong\mbox{Ext}\,^{j-1}_{A}(\hat{M},\Omega^{q}_{A}(\hat{N}))\cong\mbox{Ext}\,^{j-1-q}_{A}(\hat{M},\hat{N})$
for all $j>\mbox{w.g.d}\,_{A}(\hat{M})+q+1$. Now it is easy to see that,
$\mbox{Ext}\,^{j}_{A}(\hat{M},\hat{N})=0$ for $i+cq\leq j\leq i+cq+c$.
Therefore, by Proposition 4.1, $\mbox{Ext}\,^{j}_{A}(\hat{M},\hat{N})=0$ for
all $j>\mbox{w.g.d}\,_{A}(\hat{M})$, and so $\mbox{Ext}\,^{j}_{R}(M,N)=0$ for
all $j>\mbox{w.g.d}\,_{R}(M)$. ∎
###### Corollary 4.3.
_Let $M$ and $N$ be $R$–modules of finite complete intersection dimensions.
Set $c=\min\\{\mbox{cx}\,_{R}(M),\mbox{cx}\,_{R}(N)\\}$, if there exist an odd
number $q\geq 1$, and $i>\mbox{CI-dim}\,_{R}(M)$ such that
$\mbox{Ext}\,^{j}_{R}(M,N)=0$, for $j\in\\{i,i+q,\cdots,i+cq\\}$, then
$\mbox{Ext}\,^{j}_{R}(M,N)=0$ for all $j>\mbox{CI-dim}\,_{R}(M)$._
###### Proof.
By [5, Theorem 1.4], $\mbox{w.g.d}\,_{R}(M)=\mbox{G-dim}\,_{R}(M)=\mbox{CI-
dim}\,_{R}(M)$. Now the assertion is obvious by [7, Theorem 3.1] and Theorem
4.2. ∎
Let $M$ and $N$ be $R$–modules. It is well-known that if $M$ has finite
Gorenstein dimension and $\mbox{Pd}\,_{R}(N)<\infty$ then
$\mbox{G-dim}\,_{R}(M)=\sup\\{i\mid\mbox{Ext}\,^{i}_{R}(M,N)\neq 0\\}$. In the
following, we generalize this result for modules with reducible complexity.
###### Theorem 4.4.
_Let $M$, $N$ be nonzero $R$–modules. If $N$ has reducible complexity,
$\mbox{w.g.d}\,_{R}(M)<\infty$ and $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for $i\gg 0$
then_
* (i)
$\mbox{Ext}\,^{\tiny{\mbox{w.g.d}\,_{R}(M)}}_{R}(M,N)\cong\mbox{Ext}\,^{\tiny{\mbox{w.g.d}\,_{R}(M)}}_{R}(M,R)\otimes_{R}N$,
* (ii)
$\mbox{w.g.d}\,_{R}(M)=\sup\\{i\mid\mbox{Ext}\,^{i}_{R}(M,N)\neq 0\\}$.
###### Proof.
Set $n=\mbox{w.g.d}\,_{R}(M)$ and $c=\mbox{cx}\,_{R}(N)$. First we show that
$\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>n$. We argue by induction on $c$. If
$c=0$ then $\mbox{Pd}\,_{R}(N)<\infty$ and so as we have seen in the proof of
Proposition 4.1, $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>n$. Now let $c>0$
and $\eta\in\mbox{Ext}\,^{*}_{R}(N,N)$ reduces the complexity of $N$. Consider
the exact sequence
(4.5) $0\rightarrow N\rightarrow
K_{\eta}\rightarrow\Omega^{q}_{R}(N)\rightarrow 0,$
where $q=|\eta|-1$ and $\mbox{cx}\,_{R}(K_{\eta})<c$. The exact sequence
(4.5), induces a long exact sequence
(4.6)
$\cdots\rightarrow\mbox{Ext}\,^{i}_{R}(M,N)\rightarrow\mbox{Ext}\,^{i}_{R}(M,K_{\eta})\rightarrow\mbox{Ext}\,^{i}_{R}(M,\Omega^{q}_{R}(N))\rightarrow\cdots$
of cohomology modules. As $\mbox{Ext}\,^{i}_{R}(M,R)=0$ for $i>n$, it is easy
to see that
(4.7)
$\mbox{Ext}\,^{i}_{R}(M,\Omega^{q}_{R}(N))\cong\mbox{Ext}\,^{i-q}_{R}(M,N)$
for $i>n+q$. As $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for $i\gg 0$, from (4.6) and
(4.7) we see that $\mbox{Ext}\,^{i}_{R}(M,K_{\eta})=0$ for $i\gg 0$ and so by
induction hypothesis $\mbox{Ext}\,^{i}_{R}(M,K_{\eta})=0$ for all $i>n$.
Therefore by (4.6) and (4.7), we have
$\mbox{Ext}\,^{i-q}_{R}(M,N)\cong\mbox{Ext}\,^{i}_{R}(M,\Omega^{q}_{R}(N))\cong\mbox{Ext}\,^{i+1}_{R}(M,N)$
for $i\geq n+q+1$ and so $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>n$.
Now we show that if $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>n$ then
$\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{n+1}M,N)=0$ for all $i>0$. We argue by
induction on $c$. If $c=0$ then $\mbox{Pd}\,_{R}(N)<\infty$. As
$\mbox{Ext}\,^{i}_{R}(M,R)=0$ for all $i>n$,
$\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$ for
all $i>n$ and so it is obvious that
$\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{n+1}M,N)=0$ for all $i>0$. Now let $c>0$
and $\eta\in\mbox{Ext}\,^{*}_{R}(N,N)$ reduces the complexity of $N$. Consider
the exact sequence
(4.8) $0\rightarrow N\rightarrow
K_{\eta}\rightarrow\Omega^{q}_{R}(N)\rightarrow 0,$
where $\mbox{cx}\,_{R}(K_{\eta})<c$. As we have seen in the proof of the first
part $\mbox{Ext}\,^{i}_{R}(M,K_{\eta})=0$ for all $i>n$ and so by induction
hypothesis $\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{n+1}M,K_{\eta})=0$ for all
$i>0$. As $\mbox{Ext}\,^{i}_{R}(M,K_{\eta})=0$ for all $i>n$, by Theorem 2.3,
$\mbox{Tor}\,_{1}^{R}(\mathcal{T}_{i+1}M,K_{\eta})=0$ for all $i>n$ and since
$\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$ for
all $i>n$, it is easy to see that
$\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{j}M,K_{\eta})=0$ for all $i>0$ and $j\geq
n+1$. Set $t=q+n+2$. The exact sequence (4.8) induces the long exact sequence
(4.9)
$\cdots\rightarrow\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{t}M,N)\rightarrow\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{t}M,K_{\eta})\rightarrow\mbox{Tor}\,_{i+q}^{R}(\mathcal{T}_{t}M,N)\rightarrow\cdots$
of homology modules. Therefore
(4.10)
$\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{t}M,N)\cong\mbox{Tor}\,_{i+q+1}^{R}(\mathcal{T}_{t}M,N)\text{
for all }i>0.$
As $\mbox{Ext}\,^{i}_{R}(M,N)=0$ for all $i>n$,
$\mbox{Tor}\,_{1}^{R}(\mathcal{T}_{i}M,N)=0$ for all $i>n+1$ by Theorem 2.3.
As $\underline{\mathcal{T}_{i}M}\cong\underline{\Omega\mathcal{T}_{i+1}M}$ for
all $i>n$,
$\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{t}M,N)\cong\mbox{Tor}\,_{1}^{R}(\mathcal{T}_{t-i+1}M,N)=0$
for $1\leq i\leq q+1$ and so $\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{t}M,N)=0$ for
all $i>0$, by (4.10). Therefore $\mbox{Tor}\,_{i}^{R}(\mathcal{T}_{n+1}M,N)=0$
for $i>0$ and so
$\mbox{Ext}\,^{n}_{R}(M,R)\otimes_{R}N\cong\mbox{Ext}\,^{n}_{R}(M,N)$, by
Theorem 2.3. As $R$ is local and $N$, $\mbox{Ext}\,^{n}_{R}(M,R)$ are non-
zero, $\mbox{Ext}\,^{n}_{R}(M,N)\neq 0$ and so
$n=\sup\\{i\mid\mbox{Ext}\,^{i}_{R}(M,N)\neq 0\\}$. ∎
Acknowledgements. I am very grateful to Petter Andreas Bergh and my thesis
adviser Mohammad Taghi Dibaei for their assistance in the preparation of this
article. This work was done while the author was visiting Trondheim, Norway,
Autumn 2011. I thank the Algebra Group at the Institutt for Matematiske Fag,
NTNU, for their hospitality.
## References
* [1] F. W. Anderson and K. R. Fuller, _Rings and Categories of Modules_ , Second edition, Springer-Verlag, 1992.
* [2] M. Auslander, _Anneaux de Gorenstein, et torsion en algèbre commutative_ , in: Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, vol. 1966/67, Secrétariat mathématique, Paris, 1967.
* [3] M. Auslander,_Modules over unramified regular local rings_ , Illinois J. Math., 5:631 647, 1961.
* [4] M. Auslander and M. Bridger, _Stable module theory_ , Mem. of the AMS 94, Amer. Math. Soc., Providence 1969.
* [5] L. L. Avramov, V. N. Gasharov and I. V. Peeva, _Complete intersection dimension_ , Publ. Math. I.H.E.S. 86 (1997), 67-114.
* [6] T. Araya and Y. Yoshino, _Remarks on a depth formula, a grade inequality and a conjecture of Auslander_ , Comm. Algebra 26 (1998), 3793-3806.
* [7] P. Bergh, _On the vanishing of (co)homology over local rings_ , J. Pure Appl. Algebra 212 (2008), 262-270.
* [8] P. Bergh, _Modules with reducible complexity_ , J. Algebra 310 (2007), 132-147.
* [9] P. Bergh and D. Jorgensen, _The depth formula for modules with reducible complexity_. Illinois J. Math. to appear
* [10] P. Bergh and D. Jorgensen, _On the vanishing of homology for modules of finite complete intersection dimension_ , J. Pure Appl. Algebra 215 (2011), 242-252.
* [11] Hõ Dình Duâ’n, _A note on Gorenstein dimension and the Auslander-Buchsbaum formula_. Kodai Math. J. 17.(1994), no. 3, 390-394.
* [12] C. Huneke and R. Wiegand, _Tensor products of modules and the rigidity of Tor_ , Math. Ann. 299 (1994), 449-476.
* [13] S. Iyengar, _Depth for complexes, and intersection theorems_ , Math. Z. 230 (1999), no. 3, 545-567.
* [14] D. Jorgensen, _Vanishing of (co)homology over commutative rings_ , Comm. Alg. 29 (2001), 1883-1898.
* [15] D. Jorgensen and L. Şega, _Independence of the total reflexivity conditions for modules_. Algebr. Represent. Theory 9, 217 226 (2006).
* [16] V. Maşiek, _Gorenstein dimension and torsion of modules over commutative Noetherian rings_ , Comm. Algebra (2000), 5783-5812.
|
arxiv-papers
| 2012-04-18T13:44:02 |
2024-09-04T02:49:29.805875
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arash Sadeghi",
"submitter": "Arash Sadeghi",
"url": "https://arxiv.org/abs/1204.4083"
}
|
1204.4232
|
# Schauder Bases and Operator Theory III: Schauder Spectrums
Yang Cao Yang Cao, Department of Mathematics , Jilin university, 130012,
Changchun, P.R.China Caoyang@jlu.edu.cn , Geng Tian Geng Tian, Department
of Mathematics , Jilin university, 130012, Changchun, P.R.China
tiangeng09@mails.jlu.edu.cn and Bingzhe Hou Bingzhe Hou, Department of
Mathematics , Jilin university, 130012, Changchun, P.R.China houbz@jlu.edu.cn
(Date: Oct. 14, 2010)
###### Abstract.
In this paper, we study spectrums of Schauder operators. We show that we
always can choose a Schauder operator in a given orbit such that the Schauder
spectrum of it is empty.
###### Key words and phrases:
.
###### 2000 Mathematics Subject Classification:
Primary 47A10, 47A99; Secondary 40C05, 46A35
## 1\. Introduction
To study operators on $\mathcal{H}$ from a basis theory viewpoint, it is
naturel to consider the behavior of operators related by equivalent bases. For
examples, we show that there always be some strongly irreducible operators in
the orbit of equivalent Schauder matrices([12]). However, in the usual way a
spectral method consideration of operators in the equivalent orbit is also
important to the joint research both on operator theory and Schauder bases.
For this reason, we introduces the conception Schauder spectrum to do this
work. The main purpose of this paper is to show that we always can choose a
Schauder operator in a given orbit such that the Schauder spectrum of it is
empty. The operator theory description of bases on a separable Hilbert space
$\mathcal{H}$ developed in our paper [3] helps us to do this job.
Recall that a sequence of vectors $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ in
$\mathcal{H}$ is said to be a Schauder basis [22] for $\mathcal{H}$ if every
element $f\in\mathcal{H}$ has a unique series expansion $f=\sum c_{n}f_{n}$
which converges in the norm of $\mathcal{H}$. If
$\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is Schauder basic for $\mathcal{H}$, the
sequence space associated with $\psi$ is defined to be the linear space of all
sequences $\\{c_{n}\\}$ for which $f=\sum c_{n}f_{n}$ is convergent. Two
Schauder bases $\\{f_{n}\\}_{n=1}^{\infty}$ and $\\{g_{n}\\}_{n=1}^{\infty}$
are equivalent to each other if they have the same sequence space. Denote by
$\omega$ the countable infinite cardinal. In paper [3], we considered the
$\omega\times\omega$ matrix whose column vectors comprise a Schauder basis and
call them the Schauder matrix. An operator has a Schauder matrix
representation under some ONB is called a Schauder operator. Given an
orthonormal basis(ONB in short) $\varphi=\\{e_{n}\\}_{n=1}^{\infty}$, the
vector $f_{n}$ in a Schauder basis sequence $\psi=\\{f_{n}\\}_{n=1}^{\infty}$
corresponds an $l^{2}$ sequence $\\{f_{mn}\\}_{m=1}^{\infty}$ defined uniquely
by the series $f_{n}=\sum_{m=1}^{\infty}f_{mn}e_{m}$. The matrix
$F_{\psi}=(f_{mn})_{\omega\times\omega}$ is called the Schauder matrix of
basis $\psi$ under the ONB $\varphi$.
Assume that $\psi_{1},\psi_{2}$ are Schauder bases and
$T_{\psi_{1}},T_{\psi_{2}}$ are the operators defined by Schauder matrices
$F_{\psi_{1}}$ and $F_{\psi_{2}}$ respectively under the same ONB. These
operators $T_{\psi_{1}},T_{\psi_{2}}$ will be called equivalent Schauder
operators if and only if $\psi_{1},\psi_{2}$ are equivalent Schauder bases.
From the Arsove’s theorem([1], or theorem 2.12 in [3]), there is some
invertible operator $X\in L(\mathcal{H})$ such that
$XT_{\psi_{1}}=T_{\psi_{2}}$ holds. Hence it is an equivalence relation on
$L(\mathcal{H})$. For a Schauder basis $\psi=\\{f_{n}\\}_{n=1}^{\infty}$, the
set defined as
$\mathcal{O}_{gl}(\psi)=\\{X\psi;X\in gl(\mathcal{H})\\}$
in which $X\psi=\\{Xf_{n}\\}_{n=1}^{\infty}$ and $gl(\mathcal{H})$ consists of
all invertible operators in $L(\mathcal{H})$ contains exactly all equivalent
bases to $\psi$. Moreover, the set
$\mathcal{O}_{gl}(F_{\psi})=\\{M_{X}F_{\psi};M_{X}\hbox{ is the matrix of some
operator }X\in gl(\mathcal{H})\\}$
consists of all Schauder matrix equivalent to $F_{\psi}$. In the operator
level, we define
$\mathcal{O}_{gl}(T_{\psi})=\\{XT_{\psi};X\in gl(\mathcal{H})\\}.$
Then the set $\mathcal{O}_{gl}(T_{\psi})$ consists of operators related to
bases equivalent to $\psi$. Similarly, we consider following sets:
$\begin{array}[]{c}\mathcal{O}_{u}(\psi)=\\{U\psi;U\in U(\mathcal{H})\\},\\\
\mathcal{O}_{u}(F_{\psi})=\\{M_{U}F_{\psi};M_{U}\hbox{ is the matrix of some
unitary operator }U\\},\\\ \mathcal{O}_{u}(T_{\psi})=\\{UT_{\psi};U\in
U(\mathcal{H})\\}.\end{array}$
Roughly speaking, by these set we bind operators related to equivalent bases
of the basis $\psi$ with the same basis const. Since a Schauder operator
$T_{\psi}$ is injective and having a dense range in $\mathcal{H}$, if let
$T_{\psi}=UA_{\psi}$ denote the polar decomposition of $T_{\psi}$, then the
partial isometry $U$ must be a unitary operator. Hence, if $T_{\psi}$ is a
Schauder operator and $T_{\psi}=UA_{\psi}$ denote the polar decomposition of
$T$, then $\mathcal{O}_{U}(T_{\psi})=\mathcal{O}_{U}(A_{\psi})$, where
$A_{\psi}$ is an self-adjoint operator.
Now we state our main result in this paper.
###### Theorem 1.1.
For each Schauder operator $T$, there is an operator $T^{{}^{\prime}}\in
O_{u}(T)$ such that $\sigma_{S}(T^{{}^{\prime}})=\emptyset$.
Above theorem there may be notable differences between equivalent Schauder
operators $T_{\psi_{1}}$ and $T_{\psi_{2}}$ from the view of operator theory.
For example, a self-adjoint $A$ may satisfy $\sigma_{S}(A)=\sigma(A)$ while
there is some unitary operator $U$ such that $\sigma_{S}(UA)=\emptyset$ holds.
Moreover, we can choose a unitary operator $U$ as a unitary spread, which has
a nice basis understanding.
We organize this paper as follows. In section 2, we give some examples and a
description of Schauder spectrums of compact operators. In the case that the
Schauder operator $T$ is a compact shift, theorem 3.4 is easy to check(see
example 2.10). The proof of the general situation is the content of section 3.
## 2\. Schauder Spectrum
In this subsection, we consider the spectrum of operators from the viewpoint
of basis. Compare to the classical results, there are many similar conclusions
in the case of compact operators.
We begin with the following observation.
###### Theorem 2.1.
The operator $T\in L(\mathcal{H})$ is a Schauder operator if and only if $T$
is injective and its range is dense in $\mathcal{H}$.
###### Definition 2.2.
For a complex number $\lambda$, $\lambda$ will be called in the Schauder
spectrum denoted by $\sigma_{S}(T)$ if and only if there is no ONB such that
$\lambda I-T$ has a matrix representation as a Schauder matrix. The set
$\rho_{S}(T)=\mathbb{C}-\sigma_{S}(T)$ will be called the Schauder resolvent
set of $T$.
A direct result of theorem 2.1 is
###### Theorem 2.3.
$\lambda\notin\sigma_{S}(T)$ if and only if $T$ is both injective and having a
dense range in $\mathcal{H}$.
With above theorem, it is easy to check
###### Proposition 2.4.
For a self-adjoint operator $A$, we have
$\sigma_{S}(A)=\sigma(A)/\sigma_{p}(A)$.
###### Example 2.5.
Assume that $[a,b]$ is an interval and $A$ be a self-adjoint operator
satisfying $\sigma_{p}(A)=\emptyset,\sigma(A)=[a,b]$. Then we have
$\sigma_{S}(A)=\emptyset$.
###### Example 2.6.
Consider the diagonal operator $D=diag(m_{1},m_{2},\cdots,m_{k},\cdots)$ in
which $m_{k}\neq 0$ and $m_{k}\rightarrow 0$ for $k=0,1,\cdots$. Then we have
$\sigma_{S}(D)=\\{m_{k};k=1,2,\cdots\\}$. As an example, diagonal operator
$D=(1,\frac{1}{2},\frac{1}{3},\cdots)$ has Schauder spectrum
$\sigma_{S}(D)=\\{\frac{1}{k};k=1,2,\cdots\\}$.
More general, we have
###### Proposition 2.7.
For any operator $T\in L(\mathcal{H}),X\in Gl(\mathcal{H})$, we have
$0\in\sigma_{S}(T)$ if and only if $0\in\sigma_{S}(XT)$.
###### Corollary 2.8.
For self-adjoint operator $T$ we have $\sigma_{S}(T)\in\mathcal{R}$; For
compact operator $K$ we have $\sigma(K)/\\{0\\}\subseteq\sigma_{S}(K)$ and
$0\in\sigma_{S}(K)$ if and only if $0$ is in the set $\sigma_{p}(K)$ or
$\overline{ranK}\neq\mathcal{H}$.
###### Proof.
The assertion of first result of corollary is just the direct corollary of
theorem 2.3. If $K$ is a compact operator, then its spectrum consist of
$\\{0\\}$ and point spectrum $\sigma(K)$. In the case $0\in\sigma_{p}(K)$ or
$\overline{ranK}\neq\mathcal{H}$, $0$ is simply in the $\sigma_{S}(K)$; if it
is not true, then we have $K=UA$ in which $U$ is a unitary operator and $A$ is
a compact self-adjoint operator whose eigenvectors spans the Hilbert space
$\mathcal{H}$. ∎
###### Theorem 2.9.
For a compact self-adjoint operator $K$, $0\in\sigma_{S}(K)$ if and only if
$0\in\sigma_{p}(K)$.
###### Example 2.10.
In the case that $K$ is a compact operator but its spectrum is equivalent to
$\\{0\\}$, there is example in which $\sigma_{S}(K)=\emptyset$. Consider an
injective bilateral shift which also be a compact operator. For example, let
$\\{\tilde{e}_{j}\\}_{j\in\mathbb{Z}}$ be an ONB of $\mathcal{H}$ and
$w_{j}=\frac{1}{1+|j|}$ for $j\in\mathbb{Z}$. Then
$K\tilde{e}_{j}=w_{j}\tilde{e}_{j-1}$ is such a compact injective bilateral
weighted shift operator(CIBWS, in short). As well-known that the spectrum of a
weighted shift $T$ always be symmetric (see [5], corollary 1 and 2, p52), that
is, if $\lambda\in\sigma(T)$ then we have $e^{i\theta}\lambda\in\sigma(T)$. So
we must have $\sigma(K)=\\{0\\}$. Now to decide wether $\sigma_{S}(K)=\\{0\\}$
or not, we need more information given by polar decomposition of $K$. To avoid
complex computation on $K^{*}K$, we need to rearrange the ONB
$\\{e_{j}\\}_{j\in\mathbb{Z}}$ in some proper order as follows. We take all
$\tilde{e}_{j}$ with negative index $j<0$ as even integer and the one with
index $j>0$ as positive integer. More clarity, let
$e_{2k}=\tilde{e}_{-k},e_{2k+1}=\tilde{e}_{k}\hbox{ for k
}\in\mathbb{N}\cup\\{0\\}.$
Under the ONB $\\{e_{k}\\}_{k=1}^{\infty}$, the classical backward bilateral
shift is just the operator
$Se_{2k+1}=e_{2k-1},Se_{2k-2}=e_{2k}\hbox{ for }k=1,2,\cdots.$
Now the CIBWS $K$ defined above can be rewritten as
$Ke_{2k+1}=w_{k}e_{2k-1},Ke_{2k-2}=w_{-k}e_{2k}.$
Now consider the diagonal operator $D=diag(d_{j})$ with element $d_{j}$ on
diagonal line in which $d_{2j-1}=w_{j},d_{2j}=w_{-j}$, then we have $K=SD$
which implies that $0$ is not in the Schauder spectrum $\sigma_{S}(K)$.
Therefore we have $\sigma_{S}(K)=\emptyset$.
Compare to the classical Riesz’s theorem on compact operator(see [5],
theorem7.1, p219), we can characterize the Schauder spectrum of compact
operator as follows:
###### Theorem 2.11.
If $\mathcal{H}$ is a separable Hilbert space and $\dim\mathcal{H}=\infty$.
Then for a compact operator $K\in L(\mathcal{H})$, one and only one of the
following situations occurs:
1\. $\sigma_{S}(K)=\emptyset$;
2\. $\sigma_{S}(K)=\\{0\\}$;
3\. $\sigma_{S}(K)=\\{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\\}$ in which
$\lambda_{k}\neq 0$ and $\dim\ker(\lambda_{k}-K)<\infty$;
4\. $\sigma_{S}(K)=\\{0,\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\\}$ in
which $\lambda_{k}\neq 0$ and $\dim\ker(\lambda_{k}-K)<\infty$;
5\. $\sigma_{S}(K)=\\{\lambda_{1},\lambda_{2},\cdots\\}$. $0$ is the unique
limit point of $\lambda_{k}$ and $\lambda_{k}\neq
0,\dim\ker(\lambda_{k}-K)<\infty$.
6\. $\sigma_{S}(K)=\\{0,\lambda_{1},\lambda_{2},\cdots\\}$. $0$ is the unique
limit point of $\lambda_{k}$ and $\lambda_{k}\neq
0,\dim\ker(\lambda_{k}-K)<\infty$.
## 3\. The Orbits of Schauder Operators and Schauder matrices
### 3.1.
Now we fix an ONB $\\{e_{n}\\}_{n=1}^{\infty}$. For a Schauder operator $T$,
suppose $W$ be a unitary matrix(Hence a well defined operator under the ONB
fixed) such that $AW$ be a Schauder matrix. Now the set
$O_{gl}(T)=\\{XT;X\in Gl(\mathcal{H})\\}$
gives exactly the Schauder matrices(operators) $XTW$ whose corresponding basis
is equivalent to the basis consisting of the column vectors of $TW$. As well
known that the topology group $Gl(\mathcal{H})$ is connected under the norm
topology. Hence roughly speaking two equivalent basis can always “deform” to
each other. Moreover, if we ask that $X$ be a unitary operator, then this
“deformation” may have more nice properties. Following proposition is an
example.
###### Proposition 3.1.
Suppose that $F$ is a Schauder matrix and $U$ be a unitary operator. Then the
basis given by $F$ and $UF$ are equivalent basis with the same basis const.
Moreover, if $F$ is a unconditional basis, then they have the same
unconditional basis const.
###### Proof.
For any projection $P$, we have $||UPU^{*}||=||P||$. Then apply proposition
2.6 and 2.7 in [3]. ∎
Compare to the set $O_{gl}(T)$, we consider the set
$O_{u}(T)=\\{UT;U\in U(\mathcal{H})\\}.$
It just gives a part of equivalent basis of $A$ with the same basis const,
although not all in general. However, this situation is more interesting since
it have a natural operator theory understanding, so called, the polar
decomposition of operator. In fact, since $T$ is injective and having a dense
range in $\mathcal{H}$ (proposition 2.14, [3]), we know that the partial
isometry $U$ appearing in its polar decomposition $T=UA$ must be a unitary
operator. Hence we have $O_{u}(T)=O_{u}(A)$ if $T$ is a Schauder operator.
This fact suggest us that to study the Schauder operator we can begin with the
self-adjoint operators having a dense range and then consider their orbit
$O_{u}(A)$ (cf, papers [21], [20]).
A natural question is
###### Question 3.2.
Assume that $F_{1},F_{2}$ are equivalent Schauder bases and $T_{1},T_{2}$ be
the corresponding operators. Does there be some notable difference between the
operators $T_{1}$ and $T_{2}$?
From the operator theory viewpoint, the answer is affirmative. We shall show
that even in the case $T_{1}=UT_{2}$, their spectrum may be very different. In
fact, we have
###### Theorem 3.3.
Assume that $A$ is a self-adjoint operator such that $0\notin\sigma_{p}(A)$.
Then there is some unitary operator $U$ such that
$\sigma_{p}(UA)=\sigma_{p}(AU^{*})=\emptyset$. Moreover, we can choose the
unitary operator $U$ as a combination of unitary spreads.
By virtue of 3.3, we can always choose a good representative element from the
set $O_{u}(T)$. That is the following theorem.
###### Theorem 3.4.
For each Schauder operator $T$, there is an operator $T^{{}^{\prime}}\in
O_{u}(T)$ such that $\sigma_{S}(T^{{}^{\prime}})=\emptyset$.
###### Proof.
By virtue of theorem 2.13 in [3], we need only to verify the following claim:
###### Claim 3.5.
For a self-adjoint operator $A$, there be some unitary operator $U$ such that
the operator $\lambda I-UA$ always be injective and has a dense range in
$\mathcal{H}$ for each $\lambda\in\mathbb{C}$.
Now by virtue of theorem 3.3, there is some unitary operator $U$ such that
$\sigma_{p}(UA)=\sigma_{p}(AU^{*})=\emptyset$. From
$\sigma_{p}(UA)=\emptyset$, we know that the operator $\lambda I-A$ always be
injective; On the other side, basic operator theory result tell us
$\overline{Ran(\lambda I-UA)}=(\ker(\lambda I-UA)^{*})^{\perp}.$
So from $\sigma_{p}(AU^{*})=\emptyset$ we have
$\overline{Ran(\lambda
I-UA)}=(\ker(\bar{\lambda}I-AU^{*}))^{\perp}=\mathcal{H}.$
∎
We shall prove theorem 3.3 in later subsections. Prior to this, we give some
remarks at first.
###### Remark 3.6.
Relation to the “invariant subspace” problem.
1\. It is trivial to check that we have $\sigma_{S}(T)=\emptyset$ if $T$ has
no nontrivial subspace.
2\. Assume that there do have some operator $T$ having no nontrivial invariant
subspace. Then $T$ appear in some orbit $O_{u}(A)$ of some self-adjoint
operator $A$ since $T$ must be a Schauder operator(injective and having a
dense range). What can we say about the self-adjoint operator $A$? Clearly
there do exist some orbit $O_{u}(A)$ such that each operator in it must have a
nontrivial invariant subspace. A trivial example is the identity operator
$I$(cf, [2], or IX.9 [5]).
3\. Theorem 3.3 tell us that we can remove the eigen-subspaces, that is, the
most “trivial” nontrivial subspaces.
###### Remark 3.7.
Continuous “deformation” of Schauder bases. If we restrict to consider the
basis whose corresponding Schauder matrix represents a bounded operator, then
we can define the continuous deformation of bases as follows. A (continuous)
curve of bases is just a map
$\gamma:I\rightarrow L(\mathcal{H})$
satisfying the following properties:
1\. for each $t\in I$, $\gamma(t)$ is a Schauder matrix;
2\. $\gamma(t)$ represents a bounded operator;
3\. The map is continuous in the variable $t\in I$ under the norm topology
$L(\mathcal{H})$.
Here $I$ is an interval(either open or closed). Denote by $\mathcal{F}$ the
set of all Schauder matrices, we have the following question:
###### Question 3.8.
Does $\mathcal{F}$ must be a connected set?
Given a Schauder matrix $F$, denote by $O_{gl}(F)$ the set consisting of all
Schauder matrices equivalent to $F$. As well-known, invertible operators are
connected(Problem 141, [10], p76), so we have
###### Theorem 3.9.
The set $O_{gl}(F)$ is always path-connected for each Schauder matrix $F$.
Denote by $O_{gl}^{c}(F)$ the set of Schauder matrices $F$ which is a Schauder
matrix and there is a sequence $F_{k}\in O_{gl}(F)$ such that
$||F_{k}-F||\rightarrow 0$.
###### Question 3.10.
If $F$ is a conditional(unconditional) matrix, whether each matrix
$F^{{}^{\prime}}\in O^{c}_{gl}(F)$ must be also a conditional(unconditional)
matrix or not?
### 3.2.
Before going ahead, recall the definition of the “spread from $A$ to $B$”
given by W. T. Gowers and B. Maurey.
###### Definition 3.11.
([29], p549) Given an ONB $\\{e_{n}\\}_{n=1}^{\infty}$ and two infinite
subsets $A,B$ of $\mathbb{N}$. Let $c_{00}$ be the vector space of all
sequences of finite support. Let the elements of $A$ and $B$ be written in
increasing order respectively as $\\{a_{1},a_{2},\cdots\\}$ and
$\\{b_{1},b_{2},\cdots\\}$. Then $e_{n}$ maps to $0$ if $n\notin A$, and
$e_{a_{k}}$ maps to $e_{b_{k}}$ for every $k\in\mathbb{N}$. Denote this map by
$S_{A,B}$ and call it the spread from $A$ to $B$.
###### Example 3.12.
([29], p549) Let $A=\\{2,3,4,\cdots\\}$ and $B=\\{1,2,3\\}$, then $S_{A,B}$ is
just the backward unilateral shift operator(cf, [23]) which is defined as
$S(e_{n})=e_{n-1}$ for $n\geq 2$ and $Se_{1}=0$.
###### Example 3.13.
Using spread forms, we can write some unitary operator into their linear
combination. For example, let $\sigma$ be a bijection on $\mathbb{N}$(a
permutation of $\mathbb{N}$, so called in [22]) defined as $\sigma(2n)=2(n-1)$
for $n\geq 2$ and $\sigma(2)=1$ for even numbers and $\sigma(2n-1)=2n+1$ for
odd numbers. Then the operator $U_{\sigma}(e_{n})=e_{\sigma(n)}$ is a
bilateral shift and a unitary operator. Let
$A_{2}=\\{2,4,6,\cdots\\},B_{2}=\\{1,2,4,\cdots\\}$ and
$A_{1}=\\{1,3,5,\cdots\\}$ and $A_{2}=\\{3,5,7,\cdots\\}$. We have
$U_{\sigma}=S_{A_{1},B_{1}}+S_{A_{2},B_{2}}$.
###### Definition 3.14.
A unitary operator $U$ on $\mathcal{H}$ is said to be a unitary spread if
there is a sequence $\\{S_{A_{n},B_{n}}\\}_{n=1}^{\infty}$ of spreads such
that the series $\sum_{n=1}^{\infty}S_{A_{n},B_{n}}$ converges to $U$ in SOT.
Moreover, $U$ will be called a finite unitary spread if $U$ can be written as
a finite linear combination.
In the paper [30], we proved the following result.
###### Lemma 3.15.
For each bijection $\sigma$ on the set $\mathbb{N}$, the unitary operator
$U_{\sigma}$ is a unitary spread.
### 3.3.
Now we begin to prove theorem 3.3. Firstly we give an outline of the proof. By
the spectral theorem of normal operators, we write a self-adjoint operator
into the orthogonal diagonal direct sum $A=A_{0}\oplus A_{1}$ in which these
operators satisfy the following properties.
Property 1. The eigenvectors of $A_{0}$ defined on the Hilbert space
$\mathcal{H}_{1}$ span the whole Hilbert space $\mathcal{H}_{1}$;
Property 2. The operator $A_{1}$ defined on the Hilbert space
$\mathcal{H}_{1}$ has only a “small” point spectrum. The meaning of “small”
shall be clear in later proof.
Roughly speaking, $A_{0}$ represents the discrete case and $A_{1}$ the
continuous one. Moreover, in each situation, the spectrum containing the point
$0$ or not will be considered by different ways. We shall deal with the
discrete case in this subsection and then turn to the continuous one later.
###### Lemma 3.16.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
1\. $\sigma(A)=\sigma_{p}(A)\cup\\{0\\}$ and $0$ is the unique accumulation
point of $\sigma(A)$;
2\. For each $t\in\sigma_{p}(A)$, $\dim\ker{A-tI}=1$;
3\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary spread $U$ such that we have both
$\sigma_{p}(UA)=\emptyset$ and $\sigma_{p}(AU^{*})=\emptyset$.
###### Proof.
The self-adjoint operator satisfying the conditions appearing in the lemma has
a spectrum in the following form:
$\sigma(A)=\\{t_{1},t_{2},\cdots,t_{k},\cdots\\}\cup\\{0\\}$
where $t_{k}>t_{k+1}$ and $\\{0\\}$ is the only one accumulation point of the
sequence $\\{t_{k}\\}$. It is clear that $A$ is a compact operator. Moreover,
each $t_{k}$ is a point spectrum of $A$ since $A$ is a self-adjoint operator
and $t_{k}$ is a isolated point in $\sigma(A)$. Then $A$ has a diagonal form
as follows in an ONB:
$\begin{array}[]{cc}\left(\begin{array}[]{ccccc}t_{1}&0&\cdots&0&\cdots\\\
0&t_{2}&\cdots&0&\cdots\\\ \vdots&\vdots&\ddots&\vdots&\\\
0&0&\cdots&t_{k}&\\\
\vdots&\vdots&&&\ddots\end{array}\right)&\begin{array}[]{c}\mathcal{H}_{0}\\\
\mathcal{H}_{1}\\\ \vdots\\\ \mathcal{H}_{k}\\\
\vdots\end{array}\par\end{array}$
where by $\mathcal{H}_{k}$ we denote the 1-dimensional subspace
$\ker(t_{k}I-A)$.
Now we begin to construct the unitary spread $U$. For convenience, we denote
by $e_{k}$ the unit eigenvector of $\ker(t_{k}I-A)$. Let $U$ be the shift
constructed as follows:
$\begin{array}[]{rl}Ue_{n}=&\left\\{\begin{array}[]{ll}e_{1},&\hbox{ for
}n=2;\\\ e_{2k-2},&\hbox{ for }n=2k,k\geq 2\\\ e_{2k+1},&\hbox{ for
}n=2k-1,k\geq 1\end{array}\right.\end{array}.$
Clearly, the operator $U$ is just the unitary spread $U_{\sigma}$ defined in
example 3.13. We have
$\begin{array}[]{rl}UAe_{n}=&\left\\{\begin{array}[]{ll}t_{2}e_{1},&\hbox{ for
}n=2;\\\ t_{2k}e_{2k-2},&\hbox{ for }n=2k,k\geq 2\\\ t_{2k-1}e_{2k+1},&\hbox{
for }n=2k-1,k\geq 1\end{array}\right.\end{array}.$
Assume that $x=\sum_{k=0}^{\infty}x_{k}e_{k}$, then we have
$UAx=y=\sum_{k=0}^{\infty}y_{k}e_{k}$ where
$\begin{array}[]{rl}y_{n}=&\left\\{\begin{array}[]{ll}t_{2}x_{2},&\hbox{ for
}n=1;\\\ t_{2k+2}x_{2k+2},&\hbox{ for }n=2k,k\geq 1\\\
t_{2k-1}x_{2k-1},&\hbox{ for }n=2k+1,k\geq 1\end{array}\right.\end{array}.$
Now if $\lambda$ is an eigenvalue of $UA-\lambda I$, then we have
$\begin{array}[]{rcl}t_{2}x_{2}&=&\lambda x_{1}\\\ t_{2k+2}x_{2k+2}&=&\lambda
x_{2k}\\\ t_{2k-1}x_{2k-1}&=&\lambda x_{2k+1}\end{array}.$
Therefore we have
$\begin{array}[]{rl}x_{2k+1}&=x_{1}\cdot\lambda^{-k}\cdot\prod_{j=1}^{k}t_{2j-1},\\\
x_{2k}&=x_{1}\lambda^{k}\cdot\prod_{j=1}^{k}\frac{1}{t_{2j}}.\end{array}$
Now $\lambda^{k}\cdot\prod_{j=1}^{k}\frac{1}{t_{2j}}\rightarrow\infty$ for
$\lambda\neq 0$ as $k\rightarrow\infty$ since $t_{j}$ tends to 0, we must have
$x_{n}=0$ for $n=1,2,\cdots$. Therefore we must have $\sigma_{p}(\lambda
I-UA)=\emptyset$ for $\lambda\neq 0$. Moreover, by
$\overline{Ran(A)}=\mathcal{H}$ we have
$\ker(A)=\overline{Ran(A)}^{\perp}=\\{0\\}$. Hence we have
$\ker(UA)=\ker(A)=\\{0\\}$ and then $\sigma_{p}(UA)=\emptyset$ in turn. $UA$
is just a weighted bilateral shift operator(cf, [23]). Moreover, $UA$ is a
compact operator since $A$ is compact itself. By Riesz’s theorem on compact
operator, we have $\sigma(UA)=\\{0\\}$. So we just need to show
$\overline{Ran(UA)}=\mathcal{H}$ to finish the proof. But it is trivial by the
fact $Ran(UA)=Ran(A)$. ∎
###### Corollary 3.17.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
1\. $\sigma(A)=\sigma_{p}(A)\cup\\{0\\}$ and $0$ is the unique accumulation
point of $\sigma(A)$;
2\. For each $t\in\sigma_{p}(A)$, $\dim\ker{A-tI}=1$;
3\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary operator $U$ such that $UA$ has no point spectrum.
Moreover, we can ask that the unitary operator $U$ satisfies the following
property:
for any point $\lambda\in\mathbb{C}$, $\lambda I-UA$ have a dense range in
$\mathcal{H}$.
Now we get rid of the second condition of above lemma.
###### Lemma 3.18.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
1\. $\sigma(A)=\sigma_{p}(A)\cup\\{0\\}$ and $0$ is the unique accumulation
point of $\sigma(A)$;
2\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary spread $U$ such that both $UA$ and $AU^{*}$ have empty
point spectrum.
###### Proof.
Assume $\sigma_{p}(A)=\\{t_{k}\\}_{k=1}^{\infty}$ and $t_{k}>t_{k+1}$ for
$k\in\mathbb{N}$. Firstly we cut the integers set $\mathbb{N}$ into two
disjoint subsets:
$\begin{array}[]{l}E_{0}=\\{k\in\mathbb{N};\dim\ker{A-t_{k}I}<\infty\\},\\\
E_{1}=\\{k\in\mathbb{N};\dim\ker{A-t_{k}I}=\infty\\}.\end{array}$
Now define
$\begin{array}[]{l}\mathcal{H}_{0}=span\\{\ker(t_{k}I-A);k\in E_{0}\\};\\\
\mathcal{H}_{1}=span\\{\ker(t_{k}I-A);k\in E_{1}\\};\end{array}$
Moreover, denote by $\widetilde{\mathcal{H}}_{k}=\ker(t_{k}I-A)$ and $I_{k}$
be the identity operator on $\widetilde{\mathcal{H}}_{k}$. Let
$A_{1}=\oplus_{k\in E_{0}}\tilde{A}_{k}$ and $A_{1}=\oplus_{k\in
E_{1}}\tilde{A}_{k}$ in which we define $\tilde{A}_{k}=t_{k}I_{k}$. We can
write $A$ into the orthogonal direct sum $A=A_{0}\oplus A_{1}$.
From $E_{0}$ we construct a new set $E^{{}^{\prime}}_{0}$ as follows. If
$\dim\ker{A-t_{k}I}=k_{m}$, then we add $k_{m}-1$ copies of $t_{k}$ into
$E^{{}^{\prime}}_{0}$. Then for each $t^{{}^{\prime}}_{k}$, we can assign
exactly one unit vector $e_{k}^{(0)}\in\ker(t^{{}^{\prime}}_{k}I-A)$ such that
those vectors $\\{e_{k}^{(0)}\\}$ consists of an orthonormal subset. Arrange
the elements in $E^{{}^{\prime}}_{0}$ decreasingly as $t^{{}^{\prime}}_{0}\geq
t^{{}^{\prime}}_{1}\geq t^{{}^{\prime}}_{2}\geq\cdots\geq
t^{{}^{\prime}}_{k}\geq t^{{}^{\prime}}_{k+1}\geq\cdots$. If $E_{0}$ is a
finite subset, then we must have $\lim_{k\in
E_{1},k\rightarrow\infty}t_{k}=0$. Let $N=\max E_{0}$. For each $k>N$, fix a
unit vector $\tilde{e}_{k}^{(0)}\in\widetilde{\mathcal{H}}_{k}$ and add
$t^{{}^{\prime}}_{k}=t_{k}$ into the set $E^{{}^{\prime}}_{0}$. By replacing
$\mathcal{H}_{0}$ by the subspace
$\mathcal{H}_{0}^{{}^{\prime}}=span_{k>N}\\{\tilde{e}_{k}^{0}\\}\oplus\mathcal{H}_{0}$,
and $\mathcal{H}_{1}$ by
$\mathcal{H}_{1}^{{}^{\prime}}=(\mathcal{H}_{0}^{{}^{\prime}})^{\perp}$, we
always can assume that $E_{0}$ be a infinite subset and then we have
$\lim_{k\in E_{0},k\rightarrow\infty}t_{k}=0$. Moreover, let
$\widetilde{\mathcal{H}}^{{}^{\prime}}_{k}=(\tilde{e}^{(0)}_{k})^{\perp}\cap\widetilde{\mathcal{H}}_{k}$
for $k>N$ and
$\widetilde{\mathcal{H}}^{{}^{\prime}}_{k}=\widetilde{\mathcal{H}}_{k}$ for
$k\leq N$, then clearly we have
$\widetilde{\mathcal{H}}_{l}\perp\widetilde{\mathcal{H}}_{m}$ for $l\neq m$
and $\mathcal{H}_{1}^{{}^{\prime}}=\oplus_{k\in
E_{1}}\widetilde{\mathcal{H}}^{{}^{\prime}}_{k}$. Now the operator $A$ has the
following form
$\begin{array}[]{rll}A=&\left(\begin{array}[]{cc}A^{{}^{\prime}}_{0}&0\\\
0&A^{{}^{\prime}}_{1}\end{array}\right)&\begin{array}[]{c}\mathcal{H}^{{}^{\prime}}_{0}\\\
\mathcal{H}^{{}^{\prime}}_{1}\end{array}\end{array}.$
The operator $A_{0}^{{}^{\prime}}$ satisfies all conditions in lemma 3.16, by
modifying the unitary operator constructed in the proof of 3.16 by these new
indices we can get a unitary spread $U_{0}$ on the subspace
$\mathcal{H}^{{}^{\prime}}_{0}$ such that both
$\sigma_{p}(U_{0}A^{{}^{\prime}}_{0})=\emptyset$ and
$\sigma_{p}(A^{{}^{\prime}}_{0}U_{0}^{*})=\emptyset$ hold. Moreover, it is
easy to check that there is some bijection $\sigma_{0}$ on
$E_{0}^{{}^{\prime}}$ such that $U_{0}=U_{\sigma_{0}}$.
Now we consider the operator $A^{{}^{\prime}}_{1}$. It can be written as the
orthogonal direct sum $A=\oplus_{k\in E_{1}}\tilde{A}_{k}$ in which
$\tilde{A}_{k}$ is just the restriction of $A$ on the infinite dimensional
subspace $\widetilde{\mathcal{H}}^{{}^{\prime}}_{k}$. For each operator $k\in
E_{1}$, choose an ONB $\\{e^{(k)}_{l}\\}_{l=1}^{\infty}$ of the subspace
$\widetilde{\mathcal{H}}^{{}^{\prime}}_{k}$. Denote by
$A^{(k)}_{1}=A_{1},B^{(k)}_{1}=B_{1},A^{(k)}_{2}=A_{2}$ and
$B^{(k)}_{2}=B_{2}$ corresponding to the subsets of $\mathbb{N}$ defined in
example 3.13 and
$\widetilde{U}_{k}=S_{A^{(k)}_{1},B_{1}^{(k)}}+S_{A^{(k)}_{2},B^{(k)}_{2}}$.
Then we have $\widetilde{U}_{k}A_{k}=t_{k}\widetilde{U}_{k}$ which satisfies
$\sigma_{p}(t_{k}\widetilde{U}_{k})=\sigma_{p}(t_{k}\widetilde{U}_{k}^{*})=\emptyset$.
Clearly the operator defined as $U_{1}=\oplus_{k=1}^{\infty}\widetilde{U}_{k}$
is a unitary operator on $\mathcal{H}_{1}$ and also satisfying
$\sigma_{p}(U_{1}A_{1})=\sigma_{p}(A_{1}U_{1}^{*})=\emptyset$. Example 3.13
also tell us that there is some bijection $\sigma_{k}$ on $\mathbb{N}$ such
that $\widetilde{U}_{k}=U_{\sigma_{k}}$ for each $k\in E_{1}$.
Now we turn to verify that the unitary operator
$\begin{array}[]{rll}U=&\left(\begin{array}[]{cc}U_{0}&0\\\
0&U_{1}\end{array}\right)&\begin{array}[]{c}\mathcal{H}^{{}^{\prime}}_{0}\\\
\mathcal{H}^{{}^{\prime}}_{1}\end{array}\end{array}$
is the unitary spread we seek for. Clearly we only need to show that $U$ is a
unitary spread. Let
$\mathbb{N}^{{}^{\prime}}=E^{{}^{\prime}}_{0}\times(\times_{k\in
E_{1}}\mathbb{N})$, then clearly the set $\mathbb{N}^{{}^{\prime}}$ is just
$\mathbb{N}$ in a new order. The map defined as
$\sigma:\mathbb{N}\rightarrow\mathbb{N}$ by
$\sigma=\sigma_{0}\times(\times_{k\in E_{1}}\sigma_{k})$ is trivially a
bijection. Therefore we can apply lemma 3.15 to finish the proof. ∎
Now we turn to the more general situation.
###### Lemma 3.19.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
1\. $\sigma(A)=\sigma_{p}(A)$ is a finite set;
2\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary spread $U$ such that both $UA$ and $AU^{*}$ have empty
point spectrum.
###### Proof.
Now we have $\sigma(A)=\sigma_{p}(A)=\\{t_{1},t_{2},\cdots,t_{n}\\}$ and
$t_{k}\neq 0$. Let
$\begin{array}[]{l}E_{0}=\\{1\leq k\leq n;\dim\ker{A-t_{k}I}<\infty\\},\\\
E_{1}=\\{1\leq k\leq n;\dim\ker{A-t_{k}I}=\infty\\}.\end{array}$
Clearly we have $E_{1}\neq\emptyset$ since $\dim\mathcal{H}=\infty$. Without
loss of generality, assume $t_{1}\in E_{1}$. The subspace
$\widetilde{\mathcal{H}}_{0}=span_{k\in E_{0}}\ker(t_{k}I-A)$ is a finite
dimensional subspace, so we can pick an ONB
$\\{\tilde{e}^{(0)}_{l}\\}_{l=1}^{N}$ of it in which $N=\sum_{k\in
E_{0}}\dim\ker(t_{k}I-A)$. Choose an ONB $\\{e_{m}\\}_{m=1}^{\infty}$ of the
subspace $\ker(t_{1}I-A)$. Let $e^{(0)}_{l}=\tilde{e}^{(0)}_{l}$ for $1\leq
l\leq N$ and $e^{(0)}_{l}=e_{l-N}$ for $l\geq n$, then the sequence
$\\{e^{(0)}_{l}\\}_{l=1}^{\infty}$ is an ONB of the subspace
$\mathcal{H}_{0}=\widetilde{\mathcal{H}}_{0}\oplus\ker(t_{1}I-A)$. We also
have $\mathcal{H}_{1}=\mathcal{H}_{0}^{\perp}=\oplus_{k\in E_{1},k\neq
1}\ker(t_{k}I-A)$. Now we can rewrite the operator $A$ into the following form
$\begin{array}[]{rll}A=&\left(\begin{array}[]{cc}A_{0}&0\\\
0&A_{1}\end{array}\right)&\begin{array}[]{c}\mathcal{H}_{0}\\\
\mathcal{H}_{1}\end{array}\end{array}.$
Repeat the corresponding discussion in the proof of lemma 3.18, we need only
to prove the following
###### Claim 3.20.
There is a unitary spread $U_{0}$ on $\mathcal{H}_{0}$ such that we have both
$\sigma_{p}(U_{0}A_{0})=\emptyset$ and $\sigma_{p}(A_{0}U^{*}_{0})=\emptyset$.
To do this, let $U_{0}$ be the unitary spread $U_{\sigma}$ defined in example
3.13. Now it is trivial to check that operators $U_{0}A_{0}$ and $t_{1}U_{0}$
are similarity to each other (that is, there is some invertible operator $X\in
L(\mathcal{H}_{0})$ such that we have $XU_{0}A_{0}X^{-1}=t_{1}U_{0}$) by
theorem 2 of the paper [23](p54). And then claim holds by the fact
$\sigma_{p}(U_{0})=\sigma_{p}(U^{*}_{0})=\emptyset$. ∎
###### Corollary 3.21.
If $T$ is a compact Schauder operator, then there is a unitary operator $U$
such that $\sigma_{S}(UT)=\\{0\\}$.
###### Lemma 3.22.
Assume that $A$ is a self-adjoint operator, satisfying the following
properties:
1\. $span\\{\ker(t_{k}I-A);t_{k}\in\sigma_{p}(A)\\}=\mathcal{H}$;
2\. There is some point $t_{0}$ such that it is an accumulation point of
$\sigma(A)$.
3\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary operator $U$ such that both $UA$ and $AU^{*}$ have an
empty point spectrum.
###### Proof.
Firstly we cut the integers set $\mathbb{N}$ into two disjoint subsets:
$\begin{array}[]{l}E_{0}=\\{k\in\mathbb{N};\dim\ker{A-t_{k}I}<\infty\\},\\\
E_{1}=\\{k\in\mathbb{N};\dim\ker{A-t_{k}I}=\infty\\}.\end{array}$
And define
$\begin{array}[]{l}\mathcal{H}_{0}=span\\{\ker(t_{k}I-A);k\in E_{0}\\};\\\
\mathcal{H}_{1}=span\\{\ker(t_{k}I-A);k\in E_{1}\\}.\end{array}$
Then we can write $A$ into the form
$\begin{array}[]{rll}A=&\left(\begin{array}[]{cc}A_{0}&0\\\
0&A_{1}\end{array}\right)&\begin{array}[]{c}\mathcal{H}_{0}\\\
\mathcal{H}_{1}\end{array}\end{array}.$
Now with the same discussion on the part $A_{1}^{{}^{\prime}}$ in lemma 3.18,
we can remove $A_{1}$ since $t_{k}\neq 0$ by property 3. That is, we can
assume $E_{0}=\mathbb{N}$ and $E_{1}=\emptyset$.
Let $\\{a_{k}\\}_{k=1}^{\infty}$ be a sequence of positive numbers satisfying
$a_{k}\rightarrow 0$ decreasingly and $\sum_{k=1}^{\infty}a_{k}<\infty$.
Assume $\sigma(A)\subseteq[t_{0}-M,t_{0}+M]$ and $M>a_{1}$. Denote by
$\begin{array}[]{l}I_{1}=[t_{0}-M,t_{0}-a_{1})\cup(t_{0}+a_{1},t_{0}+M]\hbox{
and }\\\
I_{k}=[t_{0}-a_{k},t_{0}-a_{k+1})\cup(t_{0}+a_{k+1},t_{0}+a_{k}]\hbox{ for
}k\geq 2.\end{array}$
Then $\sigma_{p}(A)\cap I_{k}$ contains at most countable elements. We divide
$\mathbb{N}$ into two parts
$\begin{array}[]{l}G_{0}=\\{k\in\mathbb{N};Card\\{\sigma_{p}(A)\cap
I_{k}\\}<\infty\\}\hbox{ and }\\\
G_{1}=\\{k\in\mathbb{N};Card\\{\sigma_{p}(A)\cap
I_{k}\\}=\infty\\}.\end{array}$
According to the cardinal of the set $G_{1}$, we shall prove the lemma in
following two cases.
Case 1. If $G_{1}$ is an infinite subset, we can absorb the elements in
$G_{0}$ by the first next one in the set $G_{1}$ and then we can assume
$G_{1}=\mathbb{N}$ and $G_{0}=\emptyset$. Moreover, we can also ask that
$\dim\ker(t_{k}I=1)$ by adding at most countable copies to $t_{k}$ and
rearranging this new countable set. For each $k\in\mathbb{N}$, we arrange the
elements in the set $\sigma_{p}(A)\cap I_{k}$ as a sequence
$\\{t_{l}^{(k)}\\}_{l=1}^{\infty}$. Clearly we have
$\lim_{k\rightarrow\infty}t_{l}^{(k)}=t_{0}$. For each $t_{l}^{(k)}$, we
assign a unit vector $e^{(k)}_{l}$. Then by the spectral theorem we have
$e_{l}^{(k)}\perp e_{m}^{(j)}$ for $(l,k)\neq(m,j)$. Denote by
$\widetilde{\mathcal{H}}_{l}=span_{k\in\mathbb{N}}\\{e_{l}^{(k)}\\}$. Now we
can write $A$ into the orthogonal direct sum $A=\oplus_{l}\tilde{A}_{l}$ in
which the operator $\tilde{A}$ is the restriction of $A$ on the subspace
$\widetilde{\mathcal{H}}_{l}$. when $t_{0}=0$ then we have $\ker(A)=\\{0\\}$
by the property 3 of lemma, and apply lemma 3.16 to finish our proof of this
case. If $t_{0}\neq 0$, we need the following estimation:
###### Claim 3.23.
There are const $0<c<C<\infty$ such that for any $k,l>0$ we have
$c<\prod_{j=k}^{k+l}\frac{t_{0}-a_{j}}{t_{0}}<C.$
In fact, as well known the infinite product
$\prod_{j=1}^{\infty}\frac{t_{0}-a_{j}}{t_{0}}$ converges if the series
$\sum_{j=1}^{\infty}a_{j}$ converges(see, [26], p141). Now for the subspace
$\widetilde{\mathcal{H}}_{l}$ and its ONB $\\{e_{l}^{(k)}\\}_{k=1}^{\infty}$,
let $U_{l}$ be the unitary spread constructed in example 3.13, and we denote
the corresponding subsets by $A_{1}^{(l)},B_{1}^{(l)},A_{2}^{(l)}$ and
$B_{2}^{(l)}$. By the theorem 2 in the paper [23] and above claim, we know the
operators $U_{l}\tilde{A}_{l}$ and $t_{0}U_{l}$ are similarity to each others.
Hence the unitary spread $U_{l}$ satisfies
$\sigma_{p}(U_{l}\tilde{A}_{l})=\sigma_{p}(\tilde{A}_{l}U_{l}^{*})=\emptyset$.
Let $U$ be the corresponding orthogonal direct sum
$U=\oplus_{l=1}^{\infty}U_{l}$, then we have
$\sigma_{p}(UA)=\sigma_{p}(AU)=\emptyset$ and it is trivial to check that $U$
is also a unitary spread.
Case 2. Now we assume that $G_{1}$ is a finite subset of $\mathbb{N}$. In
virtue of lemma 3.19 and again by the spectral theorem, we can assume
$G_{0}=\mathbb{N}$ and $G_{1}=\emptyset$. Just by the same reason, we can also
assume $G_{0}$ is an infinite subset. For convenience, denote by
$\alpha_{k}=\dim\widetilde{\mathcal{H}}_{k}$. If
$\limsup_{k}\alpha_{k}=\infty$, then we can repeat our above discussion in
case 1 to finish the proof. So we just need to consider the situation
$m=\limsup_{k}\alpha_{k}<\infty$. Now we cut the subset $G_{0}$ into the
pieces
$L_{n}=\\{k\in\mathbb{N};\alpha_{k}=n\\},1\leq n\leq m.$
Clearly there is at least one subset $L_{n}$ such that it is an infinite
subset. We add all finite subset $L_{n}$ into a fixed infinite subset, said,
the set $L_{1}$. For the remaining infinite subsets except $L_{1}$, we can
repeat the discussion in case 1 to get an appropriate unitary spread. For the
infinite subset $L_{1}$, the same discussion also goes well if we apply the
theorem 2 in the paper [23] again and note that adjusting finite nonzero
weights into another nonzero ones does not change the similarity class of a
weighted bilateral shift operator. Hence we can also get a unitary spread
which is a finite or infinite orthogonal direct sum of unitary spreads
dependent on the condition $\limsup_{k}\alpha_{k}<\infty$ or not, such that
$\sigma_{p}(UA)=\sigma_{p}(AU)=\emptyset$. ∎
With a little more operator theory discussion, the proof of above lemma
implies the following result. Since it deviate our main aim in this paper, we
omit the proof and just state it here.
###### Theorem 3.24.
Assume that $A$ is a self-adjoint operator, satisfying the following
properties:
1\. $span\\{\ker(t_{k}I-A);t_{k}\in\sigma_{p}(A)\\}=\mathcal{H}$;
2\. There is some point $t_{0}\neq 0$ such that it is an accumulation point of
$\sigma(A)$.
3\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there are unitary spreads $U,U_{\sigma}$ and an invertible operator $X\in
L(\mathcal{H})$ such that $XUAX^{-1}=\oplus_{k=1}^{\infty}t_{0}U_{\sigma}$.
###### Theorem 3.25.
Assume that $A$ is a self-adjoint operator, satisfying the following
properties:
1\. $span\\{\ker(t_{k}I-A);t_{k}\in\sigma_{p}(A)\\}=\mathcal{H}$;
2\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary spread $U$ such that both $UA$ and $AU^{*}$ have an
empty point spectrum.
###### Proof.
If $\sigma_{p}(A)$ has no accumulation point then it is a finite set, and then
we apply lemma 3.19. If not, above lemma 3.22 holds. ∎
### 3.4.
Now we begin to consider the continuous case.
###### Lemma 3.26.
Assume that $A$ is a self-adjoint operator satisfying the following
properties:
1\. $\sigma(A)\subseteq(m,M)$ in which $m<M$ are positive finite real numbers;
2\. $\sigma_{p}(A)=\\{t_{0}\\}$ for some $t_{0}\in(m,M)$ and
$\dim\ker(t_{0}I-A)=1$.
Then there is a unitary spread such that
$\sigma_{p}(UA)=\sigma_{p}(AU^{*})=\emptyset$.
###### Proof.
By the classical spectral theory of normal operator(cf, [5], pp297-299), we
have following orthogonal decomposition of $A$:
$\begin{array}[]{rl}\left(\begin{array}[]{cc}t_{0}&0\\\
0&A_{1}\end{array}\right)&\begin{array}[]{l}\ker(t_{0}I-A)\\\
\ker(t_{0}I-A)^{\perp}\end{array}\end{array}.$
Then $A_{1}$ is a self-adjoint operator whose point spectrum must be void
since $\sigma_{p}(A)=\sigma_{p}(A_{1})\cup\\{t_{0}\\}$. Then $\sigma(A_{1})$
must be a closed set without isolated point because each isolated point must
be an eigenvalue of $A_{1}$ by the spectral theorem. Now we fixed a point
$\alpha\neq 0\in\sigma(A_{1})$. Then at least one of following assertions
holds:
1\. There is a sequence $\alpha_{n}\rightarrow\alpha$ such that we have
$\alpha_{n+1}>\alpha_{n}$ for each $n\geq 1$. Moreover, the range of spectral
projection $E_{[\alpha_{n},\alpha_{n+1}]}$ is an infinite subspace;
2\. There is a sequence $\alpha_{n}\rightarrow\alpha$ such that we have
$\alpha_{n+1}<\alpha_{n}$ for each $n\geq 1$. Moreover, the range of spectral
projection $E_{[\alpha_{n},\alpha_{n+1}]}$ is an infinite subspace;
We assume that the first assertion is true. The case that the second assertion
holds will be proved in the just same way. Now let $\alpha_{1}=||A||$ for
convenience. By picking a subsequence if need, we also can assume that the
sequence $\\{\alpha_{n}\\}_{n=1}^{\infty}$ satisfies the following properties
$\alpha(1-\frac{1}{2^{n}})<\alpha_{n}.$ It is easy to check:
###### Claim 3.27.
For each $\varepsilon>0$, there is a positive integer $N$ such that for any
subset $\Delta$ containing $k$ elements of $\mathbb{N}$ and satisfying
$\Delta\cap\\{1,2,\cdots,N-1\\}=\emptyset$ we have
$(1-\varepsilon)\alpha^{k}\leq\prod_{n_{k}\in\Delta}\alpha_{n_{k}}\leq(1+\varepsilon)\alpha^{k}.$
Now we rearrange these interval as follows.
$\begin{array}[]{l}I_{n}=[\alpha_{2n-1},\alpha_{2n})\cup(2\alpha-\alpha_{2n},2\alpha-\alpha_{2n-1}]\hbox{
for }n\geq 1,\\\
I_{n}=[\alpha_{-2(n+1)},\alpha_{-2n-1})\cup(2\alpha-\alpha_{-2n-1},2\alpha-\alpha_{-2(n+1)}]\hbox{
for }n\leq 0.\end{array}$
Denote $E_{n}=E_{I_{n}}$($E_{r}=E_{I_{r}}$) the spectral projection of $A_{1}$
on the interval $I_{n}$($I_{r}$) and by $\mathcal{H}_{n}=Ran(E_{n})$ for
$n\in\mathbb{Z}$,
$\mathcal{H}_{0}^{{}^{\prime}}=\mathcal{H}_{0}\cap\ker(t_{0}I-A)^{\perp}$ and
$E_{0}^{{}^{\prime}}$ be the orthogonal projection onto the subspace
$\mathcal{H}_{0}^{{}^{\prime}}$. Now we choose an ONB
$\\{e^{(n)}_{k}\\}_{k=1}^{\infty}$ of $\mathcal{H}_{n}$ for each
$n\in\mathbb{Z},n\neq 0$. For $\mathcal{H}_{0}$, we pick an ONB
$\\{\tilde{e}^{(0)}_{k}\\}_{k=1}^{\infty}$ of the subspace
$\mathcal{H}_{0}^{{}^{\prime}}$ and rearrange them and $e_{0}$ into an ONB of
$\mathcal{H}_{0}$ as follows:
$e^{(0)}_{1}=e_{0},e^{(0)}_{k}=\tilde{e}^{(0)}_{k-1}\hbox{ for }k\geq 2.$
It is trivial to check that the set
$\varphi=\\{e^{(n)}_{k};n\in\mathbb{Z},k\in\mathbb{N}\\}$ is an ONB of the
whole Hilbert space $\mathcal{H}$. Now let $U$ be the unitary operator defined
as
$Ue^{(n)}_{k}=e^{(n+1)}_{k},\hbox{ for }n\in\mathbb{Z}\hbox{ and
}k\in\mathbb{N}.$
By lemma 3.15, we know that $U$ is a unitary spread.
To finish the proof of lemma, now we prove that both $UA$ and $AU^{*}$ have no
eigenvalues. The proof of these facts are similar, so we only prove the first
part and omit the other one to save space. Since each $\mathcal{H}_{n}$ is a
reducing subspace of $A$, we can write $A$ into the direct sum:
$A=\oplus_{n=-\infty}^{\infty}A_{n}$
in which $A_{n}=AE_{n}=E_{n}AE_{n}$ for $n\neq 0$ and
$A_{0}=AE_{0}^{{}^{\prime}}\oplus
t_{0}I=E_{0}^{{}^{\prime}}AE_{0}^{{}^{\prime}}\oplus t_{0}I$. We have the
following estimation:
$\begin{array}[]{rl}\alpha_{2n-1}||x||\leq||A_{n}x||\leq(2\alpha-\alpha_{2n-1})||x||,&\hbox{
for }n\geq 1\\\
\alpha_{-2(n+1)}||x||\leq||A_{n}x||\leq(2\alpha-\alpha_{-2(n+1)})||x||,&\hbox{
for }n<0.\end{array}$
Moreover, by $\alpha<2\alpha-\alpha_{-1(n+1)}<(1+\frac{1}{2^{n}})\alpha$, we
have
$(1+\frac{1}{2^{n}})^{-1}\alpha^{-1}||x||<(2\alpha-\alpha_{-2(n+1)})^{-1}||x||\leq||A_{n}^{-1}x||\leq\alpha_{-2(n+1)}^{-1}||x||$
for $n<0$. For $A_{0}$, we have $m||x||\leq||A_{0}x||\leq M||x||$.
For a vector $x\in\mathcal{H}$, now under the ONB $\varphi$ it have a $l^{2}-$
sequence coordinate in the form
$x^{(n)}=E_{n}x=\sum_{k=1}^{\infty}x^{(n)}_{k}e^{(n)}_{k}\in\mathcal{H}_{n},x=\sum_{n\in\mathbb{Z}}x^{(n)},$
in which the series converges in the norm on $\mathcal{H}$ and
$\\{x^{(n)}_{k}\\}_{k=1}^{\infty}$ is also a $l^{2}-$sequence. Here we
emphasize that vectors $x^{(k)}$ and $x^{(j)}$ are orthogonal to each other
for $k\neq j$. Let $y=Ax$, then by the fact that $\mathcal{H}_{n}$ is a
reducing subspace of the operator $A$ we can also write $y$ into the same
form:
$y^{(n)}=A_{n}x^{(n)}=\sum_{k=1}^{\infty}y^{(n)}_{k}e^{(n)}_{k}\in\mathcal{H}_{n},y=\sum_{n\in\mathbb{Z}}y^{(n)}.$
Now simply we have
$UAx^{(n)}=Uy^{(n)}=\sum_{k=1}^{\infty}y^{(n)}_{k}e^{(n+1)}_{k}.$
We can identify $\mathcal{H}_{n}$ with a fixed separable infinite dimensional
Hilbert space $\mathcal{H}^{*}$ as follows. Fix an ONB
$\\{e_{k}\\}_{k=1}^{\infty}$, let $\widetilde{U}_{n}$ be the unitary operator
defined as $\widetilde{U}_{n}e^{(n)}_{k}=e_{k}$. Now
$\widetilde{U}_{n}x^{(n)}$ just the vector with the same $l^{2}-$coordinate in
$\mathcal{H}^{*}$. Moreover, each operator $A_{n}$ can be seen as the operator
$\widetilde{U}_{n}A_{n}\widetilde{U}_{n}^{*}$ in $L(\mathcal{H}^{*})$. Hence
$A$ is unitary equivalent to the operator
$\oplus_{n\in\mathbb{Z}}\widetilde{U}_{n}A_{n}\widetilde{U}_{n}^{*}$ on the
Hilbert space $\oplus_{-\infty}^{\infty}\widetilde{H}^{*}$ which is an
orthogonal direct sum of countable copies of $\mathcal{H}^{*}$. Denote by
$\tilde{A}_{n}=\widetilde{U}_{n}A_{n}\widetilde{U}_{n}^{*}$ for convenience.
Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have
$\lambda\widetilde{U}_{n}x^{(n)}=\widetilde{U}_{n-1}A_{n-1}x^{(n-1)}=\widetilde{U}_{n-1}A_{n-1}\widetilde{U}_{n-1}^{*}\widetilde{U}_{n-1}x^{(n-1)}=\tilde{A}_{n-1}\widetilde{U}_{n-1}x^{(n-1)}.$
Therefore, following equations hold:
$\begin{array}[]{l}\widetilde{U}_{n}x^{(n)}=\lambda^{-n}\tilde{A}_{n-1}\cdot\tilde{A}_{n-2}\cdots\tilde{A}_{0}\cdot\widetilde{U}_{0}x^{(0)},\hbox{
for }n\geq 1;\\\
\widetilde{U}_{n}x^{(n)}=\lambda^{|n|}\tilde{A}_{n}^{-1}\cdot\tilde{A}_{n+1}^{-1}\cdots\tilde{A}_{-1}^{-1}\cdot\widetilde{U}_{0}x^{(0)},\hbox{
for }n\leq-1.\end{array}$
Immediately we have $x^{(0)}\neq 0$. Moreover, by the fact
$||A_{k}^{-1}||=||\tilde{A}_{k}^{-1}||$ we have the following inequations:
$\begin{array}[]{l}||x^{(n)}||\geq
m||x^{(0)}||\lambda^{-n}\prod_{k=1}^{n-1}\alpha_{2k-1},\hbox{ for }n\geq 1;\\\
||x^{(n)}||\geq||x^{(0)}||\lambda^{|n|}\prod_{k=1}^{|n|}(1+\frac{1}{2^{n}})^{-1}\alpha^{-1},\hbox{
for }n\leq-1.\end{array}$
Now for a given $\varepsilon>0$, let $N$ be the integer defined in the claim
3.27, for $n\geq 1$ we have
$\begin{array}[]{rl}||x^{(n)}||&\geq
m||x^{(0)}||\lambda^{-n}(1-\varepsilon)\alpha^{n-N}\prod_{k=1}^{N-1}\alpha_{2k-1}\\\
&=(1-\varepsilon)m||x^{(0)}||\lambda^{-N}\cdot(\frac{\alpha}{\lambda})^{n-N}\prod_{k=1}^{N-1}\alpha_{2k-1}.\end{array}$
And for $n\leq 0$,
$\begin{array}[]{rl}||x^{(n)}||&\geq||x^{(0)}||\lambda^{|n|}\alpha^{n+N}\cdot\prod_{k=1}^{|n|}(1+\frac{1}{2^{n}})^{-1}\cdot\alpha^{-N}\\\
&=||x^{(0)}||(\frac{\lambda}{\alpha})^{N}\cdot(\frac{\alpha}{\lambda})^{n+N}\prod_{k=1}^{|n|}(1+\frac{1}{2^{n}})^{-1}\end{array}$
Now if $|\lambda|<|\alpha|$, then
$(\frac{\alpha}{\lambda})^{n+N}\rightarrow\infty$ as $n\rightarrow\infty$. If
$|\lambda|>|\alpha|$, then $(\frac{\alpha}{\lambda})^{n+N}\rightarrow\infty$
as $n\rightarrow-\infty$ since the infinite product
$\prod_{k=1}^{\infty}(1+\frac{1}{2^{n}})^{-1}$ converges to a nonzero number.
So we must have $|\lambda|=|\alpha|$. But this implies
$||x^{(n)}||\geq
m||x^{(0)}||(1-\varepsilon)\lambda^{-N}\prod_{k=0}^{N-1}\alpha_{2k-1}\hbox{
for all }n>N,$
which is impossible since we have $||x||=\infty$ in such case. ∎
###### Theorem 3.28.
Assume that $A$ is a self-adjoint operator satisfying:
1\. $\sigma_{p}=\\{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\\}$ and
$\dim\ker(\lambda_{k}I-A)<\infty$; and
2\. $R(A)$ is dense in the Hilbert space $\mathcal{H}$.
Then there is a unitary spread $U$ such that
$\sigma_{p}(UA)=\sigma_{p}(AU^{*})=\emptyset$.
###### Proof.
Clearly we have $\lambda_{k}\neq 0$ for $k=1,2,\cdots,n$. Let
$m_{k}=\dim\ker(\lambda_{k}I-A)$. We can assume that $m_{k}=1$ by adding
$m_{k}-1$ copies of $\lambda_{k}$ into $\sigma_{p}(A)$. Denote by
$\mathcal{H}_{0}=span_{1\leq k\leq n}\\{\ker(\lambda_{k}I-A)\\}$ and
$\mathcal{H}_{r}=\mathcal{H}_{0}^{\perp}$. Moreover denote by $A_{0}$ the
diagonal operator $A_{0}=diag(\lambda_{1},\cdots,\lambda_{n})$ and $A_{r}$ the
restriction of $A$ on the reducing subspace $\mathcal{H}_{r}$. By the spectrum
theorem, we can write $A$ into the orthogonal direct sum $A=A_{0}\oplus
A_{r}$.
Case 1. Assume $0\in\sigma(A)$. Then $\\{0\\}$ can not be an isolated point of
$\sigma(A)$. And for ant $\delta>0$, the projection $E_{\delta}$ on the
interval $(-\delta,\delta)$ has an infinite dimensional range by the spectral
theorem. Let $\\{\alpha_{k}\\}_{k=1}^{\infty}$ be a sequence satisfies the
following conditions:
1\. $\alpha_{k}>\alpha_{k+1}$ and $\alpha_{k}\rightarrow 0$;
2\. Let $I_{k}=[-\alpha_{k},-\alpha_{k+1})\cup(\alpha_{k+1},\alpha_{k}]$, the
spectral projection $E_{k}$ of $A_{r}$ on the subset $I_{k}$ has an infinite
dimensional range;
3\. $\cup_{k\geq 1}I_{k}\supset\sigma(A)-\\{0\\}$.
Now for $k\leq n$, let $\tilde{A}_{r}^{(k)}$ be the orthogonal direct sum
$\begin{array}[]{rll}\tilde{A}_{r}^{(k)}=&\left(\begin{array}[]{cc}\lambda_{k}&0\\\
0&E_{k}AE_{k}\end{array}\right)&\begin{array}[]{c}\ker(\lambda_{k}I-A)\\\
Ran(E_{k})\end{array}\end{array}.$
Then $\tilde{A}_{r}^{(k)}$ is an operator on the subspace
$\widetilde{H}_{k}=\ker(\lambda_{k}I-A)\oplus Ran(E_{k})$. Moreover, for $k>n$
we define $\tilde{A}_{r}^{(k)}=E_{k}AE_{k}$. Now we see that each
$\tilde{A}_{r}^{(k)}$ satisfies the requirements of lemma 3.26. So for each
$k$, there is some unitary spread $U_{k}$ on $\widetilde{H}_{k}$ such that we
have
$\sigma_{p}(U_{k}\tilde{A}_{r}^{(k)})=\sigma_{p}(\tilde{A}_{r}^{(k)}U_{k}^{*})=\emptyset$.
Moreover, again by the spectrum theorem, we can write $A$ into the orthogonal
direct sum $A=\oplus_{k=1}^{\infty}\tilde{A}_{r}^{(k)}$. Then the unitary
operator $U=\oplus_{k=1}^{\infty}U_{k}$ satisfies
$\sigma_{p}(UA)=\sigma_{p}(AU^{*})=\emptyset$. It is easy to check that $U$ is
a unitary spread by lemma 3.15.
Case 2. Assume $0\notin\sigma(A)$. This situation is more easy to deal with.
We just need to cut $\sigma(A_{r})$ into exact $n$ suitable pieces and then
repeat the above discussion. ∎
Now finally we can prove theorem 3.3.
###### Proof.
Let $\mathcal{H}_{0}=span\\{\ker(\lambda I-A);\lambda\in\sigma_{p}(A)\\}$ and
$\mathcal{H}_{1}=\mathcal{H}_{0}^{\perp}$. By the spectrum theory of normal
operator, we always can write $A$ into the form:
$\begin{array}[]{rl}\left(\begin{array}[]{cc}A_{0}&0\\\
0&A_{1}\end{array}\right)&\begin{array}[]{c}\mathcal{H}_{0}\\\
\mathcal{H}_{1}\end{array}\end{array}$
in which $\mathcal{H}_{0}=span\\{\ker(\lambda
I-A_{0});\lambda\in\sigma_{p}(A)\\}$ and $\sigma_{p}(A_{1})=\emptyset$. If
$\dim\mathcal{H}_{0}=\infty$ we can apply theorem 3.25; In the case
$\dim\mathcal{H}_{0}<\infty$ we apply theorem 3.28. ∎
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* [28] Gowers, W. T.; Maurey, B. The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), no. 4, 851-874.
* [29] Gowers, W. T.; Maurey, B. Banach spaces with small spaces of operators. Math. Ann. 307 (1997), no. 4, 543-568.
* [30] Cao Yang and el, On unitary operators spread $\mathbb{N}$, preprint.
* [31] Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. An introduction to the theory of numbers. Fifth edition. John Wiley and Sons, Inc., New York, 1991.
|
arxiv-papers
| 2012-04-19T01:27:01 |
2024-09-04T02:49:29.820085
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Cao, Geng Tian, Bingzhe Hou",
"submitter": "Cao Yang",
"url": "https://arxiv.org/abs/1204.4232"
}
|
1204.4272
|
Conformal transformations and doubling of the particle states
A. I. Machavariania,b
a Joint Institute for Nuclear Research, Moscow Region 141980 Dubna, Russia
b High Energy Physics Institute of Tbilisi State University, University Str. 9
###### Abstract
The 6D and 5D representations of the four-dimensional (4D) interacting fields
in the Heisenberg picture and the corresponding equations of motion are
studied using equivalence of the conformal transformations of the four-
momentum $q_{\mu}$ ($q^{\prime}_{\mu}=q_{\mu}+h_{\mu}$,
$q^{\prime}_{\mu}=\Lambda^{\nu}_{\mu}q_{\nu}$, $q^{\prime}_{\mu}=\lambda
q_{\mu}$ and $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$) and the corresponding
rotations on the 6D cone $\kappa_{A}\kappa^{A}=0$ $(A=\mu;5,6\equiv
0,1,2,3;5,6)$, where $q_{\mu}=M\ \kappa_{\mu}/(\kappa_{5}+\kappa_{6})$ and $M$
is the scale parameter. The 4D reduction of the 6D fields on the cone
$\kappa_{A}\kappa^{A}=0$ is unambiguously fulfilled by the intermediate 5D
projection into two 5D hyperboloids $q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}$ and
$q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}$ in order to cover the whole domains
$-\infty<q_{\mu}q^{\mu}<\infty$ and $q_{5}^{2}\geq 0$. The resulting 5D and 4D
fields in the coordinate space consist of two parts $\varphi_{1}(x,x_{5})$,
$\varphi_{2}(x,x_{5})$ and $\Phi_{1}(x)=\varphi_{1}(x,x_{5}=0)$,
$\Phi_{2}(x)=\varphi_{2}(x,x_{5}=0)$, where the Fourier conjugate of
$\varphi_{1}(x,x_{5})$ and $\varphi_{2}(x,x_{5})$ are defined on the
hyperboloids $q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}$ and
$q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}$ respectively. Consequently, the 4D reduction
of the 6D fields generate two kinds of the 5D and 4D fields
$\varphi_{\pm}=\varphi_{1}\pm\varphi_{2}$ and
$\varphi_{\pm}(x,x_{5}=0)=\Phi_{\pm}(x)=\Phi_{1}(x)\pm\Phi_{2}(x)$ with the
same quantum numbers but with the different masses and the sources. This
doubling of the 4D fields $\Phi_{\pm}=\Phi_{1}\pm\Phi_{2}$ can be applied for
unified description of the interacting electron and muon fields, $\pi$ and
$\pi(1300)$-mesons, $N$ and $N(1440)$-nucleons and other particles with the
same quantum numbers but different masses and interactions.
Introduction
The 5D extension of the 4D relativistic theories is the fruitful method that
has a long history. The Kaluza-Klein theory and their generalizations for the
gauge transformations [1, 2, 3, 4] allow to unify the electromagnetic and
gravitation theories. In the traditional Kaluza-Klein theory all partial
derivatives with respect to fifth coordinates have been equated to zero and
the extra spatial dimension was compacted to a small size circle. The rigorous
mathematical approach for the $N+1$ and $N$ dimensional manifolds (see ch. 2.2
in [3]) allow to embed the 4D equation of motion with the sources into the 5D
equation without sources. In the recent 5D field theoretical formulations [5,
6] the extra fifth dimension is required to solve the problems of the
renormalizable $SO(10)$ grand unification theories with the breakdown of the
gauge coupling. In this approach the fifth dimension enable to reproduce the
fermion generations, quark mass hierarchy, flavor mixing and Cabbibbo-
Kabayashi-Maskawa matrices and it is argued, that in virtue of the no go
theorems it is not possible to achieve these results in the 4D space. Other
kind of the 5D relativistic field theories were performed within the invariant
time method [7, 8, 9], where the fifth coordinate is the proper time
$x_{5}^{2}=x_{o}^{2}-{\bf x}^{2}$. In these theories $x_{5}$ is an auxiliary
variable and the sought 4D wave functions and fields are reproduced through
the 5D wave functions and fields via the boundary conditions for $x_{5}=0$ or
$x_{5}=\sqrt{t^{2}-{\bf x}^{2}}$ and the evolution over the fifth coordinates
were often described through the equation for the first derivatives of the
scalar and fermion fields $i\partial\phi/\partial x_{5}$ and
$i\partial\psi/\partial x_{5}$.
The general scheme for the 5D extensions of the 4D relativistic theories and
the 4D reductions of the 5D relativistic formulations presents the conformal
group of the 4D transformations that can be unambiguously represent through
the rotations on the 6D cone. In particular, the conformal transformations of
the four coordinate $x_{\mu}$ consists of the following independent motions
$x^{\prime}_{\mu}=x_{\mu}+a_{\mu}$,
$x^{\prime}_{\mu}=\Lambda^{\nu}_{\mu}x_{\nu}$, $x^{\prime}_{\mu}=\lambda
x_{\mu}$ and $x^{\prime}_{\mu}=-\ell^{2}x_{\mu}/x^{2}$ which can be performed
through the rotations on the 6D cone
$\xi_{A}\xi^{A}\equiv\xi_{\mu}\xi^{\mu}+\xi_{5}\xi^{5}-\xi_{6}\xi^{6}=0$ [10],
where $A=0,1,2,3;5,6\equiv\mu;5,6$, $x_{\mu}=\xi_{\mu}/\xi_{+}$,
$\xi_{\pm}=(\xi_{5}\pm\xi_{6})/\ell$ and $\ell$ is the dimension parameter.
The one-to-one relationship between the 4D conformal transformations and 6D
rotations allow to construct the one-to-one relationship between an
interacting 4D Heisenberg field $\Phi(x)$ and the corresponding 6D field
$\varsigma(\xi)\equiv\varsigma(\xi_{0},\xi_{1},\xi_{2},\xi_{3};\xi_{5},\xi_{6})$
as $\Phi(x)=\Bigl{[}\varsigma(x,\xi_{+},\xi_{-})\Bigr{]}_{\xi_{A}\xi^{A}=0}$
with the fixed scale parameter $\xi_{+}$ and
${\xi_{-}}/\xi_{+}=x^{2}/\ell^{2}$ [10]-[13]. The other 4D reduction of
$\varsigma(\xi)$ was used in the manifestly conformal invariant formulation
[14]-[24], where the homogeneity of the conformal invariant 6D fields
$\Bigl{[}\varsigma(\xi)\Bigr{]}_{\xi_{A}\xi^{A}=0}=\Bigl{[}\varsigma(x,\xi_{+},\xi_{A}\xi^{A}=0)$
over the scale variable $\xi_{+}$ is required, i.e.
$\varsigma(x,\xi_{+},\xi_{A}\xi^{A}=0)={\xi_{+}}^{d}{\phi}(x,\xi_{A}\xi^{A}=0)$
and the 4D conformal invariant field is
$\Phi(x)\equiv{\phi}(x,\xi_{A}\xi^{A}=0)$.
Location of $\varsigma(\xi)$ on the 6D cone $\xi_{A}\xi^{A}=0$ impose
additional condition by conformal transformations. For instance, the Fourier
transformation of an arbitrary field $\varsigma(\xi)$ on the 6D cone produces
the following condition
$\biggl{(}{{\partial^{2}}\over{\partial{\kappa}^{\mu}\partial{\kappa}_{\mu}}}+{{\partial^{2}}\over{\partial{\kappa}^{5}\partial{\kappa}_{5}}}-{{\partial^{2}}\over{\partial{\kappa}^{6}\partial{\kappa}_{6}}}\biggr{)}\int
d^{6}\xi
e^{i\kappa_{A}\xi^{A}}\delta\Bigl{(}\xi_{o}^{2}-\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}+\xi_{5}^{2}-\xi_{6}^{2}\Bigr{)}\varsigma(\xi)=0,$
$None$
where $\kappa_{A}$ are Fourier conjugate to $\xi_{A}$ and for derivation of
(I.1a) the condition $(\xi_{A}\xi^{A})\delta(\xi_{A}\xi^{A})=0$ was used. The
intermediate 5D projection of the 6D fields and the condition (I.1a) determine
the corresponding 5D fields and the 5D condition.
In order to obtain the 5D and 6D extensions of the 4D equations of motion it
is convenient to consider the conformal transformations of the four momentum
$q_{\mu}$ ($q^{\prime}_{\mu}=q_{\mu}+h_{\mu}$,
$q^{\prime}_{\mu}=\Lambda^{\nu}_{\mu}q_{\nu}$, $q^{\prime}_{\mu}=\lambda
q_{\mu}$ and $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$). The 6D representation of
the conformal transformations for the independent four components of the
momentum $q_{\mu}$ are similar with the conformal transformations in the
coordinate space. The principal difference between the conformal
transformations in the coordinate and momentum space is in the translation. In
the next section it is shown that the translation of the four momentum
$q^{\prime}_{\mu}=q_{\mu}+h_{\mu}$ for the Fourier conjugate of $\Phi(x)$
produces the gauge transformation
$\Phi^{\prime}(x)=e^{ih_{\mu}x^{\mu}}\Phi(x)$. According to the Dirac
geometrical model [10]-[18], each of the conformal transformations in the
momentum space is unambiguously determined via the appropriate 6D rotation
with the invariant 6D form
$\kappa_{A}\kappa^{A}\equiv\kappa_{\mu}\kappa^{\mu}+\kappa_{5}^{2}-\kappa_{6}^{2}=0,$
$None$
where the four momentum $q_{\mu}$ ($\mu=0,1,2,3$) is defined as
$q_{\mu}=\kappa_{\mu}/\kappa_{+}$ and $M$ is a scale parameter. The 6D cone
(I.2a) and the corresponding surface
$q_{\mu}q^{\mu}+M^{2}{{\kappa_{-}}\over{\kappa_{+}}}=0,\ \ \ with\ \ \
\kappa_{\pm}={{\kappa_{5}\pm\kappa_{6}}\over M}$ $None$
are invariant under any combination of the conformal transformations of
$q_{\mu}$.
In analogy with (I.1a) the conformal transformations of a 4D field $\Phi(x)$
in the momentum space can be performed via the 6D rotations of the
corresponding 6D field operator $\varsigma(\kappa)$ which is embedded into the
cone (I.2a). Therefore, location of $\varsigma(\kappa)$ on the same 6D cone
(I.2a) before and after the conformal transformations in the momentum space
impose the condition
$\biggl{(}{{\partial^{2}}\over{\partial{\xi}^{\mu}\partial{\xi}_{\mu}}}+{{\partial^{2}}\over{\partial{\xi}^{5}\partial{\xi}_{5}}}-{{\partial^{2}}\over{\partial{\xi}^{6}\partial{\xi}_{6}}}\biggr{)}\int{{d^{6}\kappa}\over{(2\pi)^{4}}}e^{i\kappa_{A}\xi^{A}}\delta\Bigl{(}\kappa_{\mu}\kappa^{\mu}+\kappa_{5}^{2}-\kappa_{6}^{2}\Bigr{)}\varsigma(\kappa)=0.$
$None$
This paper deals with consistency of the usual 4D equations of motion for 4D
interacting Heisenberg field $\Phi(x)$ and boundary conditions and constrains
for the 5D and 6D representations of $\Phi(x)$ which follows from the
conformal group of the transformations in the momentum space. Two particular
features generate the special interest to the conformal transformations in the
momentum space[20]. First, the observables of the particle interactions, like
the cross sections and polarizations are determined in the momentum space.
Secondly, the accuracy of the measurement of the particle coordinates is in
principle restricted by the Compton length of this particle. Moreover,
determination of the coordinate of the conformal invariant massless particles
produces additional essential troubles (see [19] ch. 20 and [26]). The
conformal transformations of the fields and the corresponding equations of
motion in the momentum space were considered in ref. [11, 13, 20, 25], where
the conformal transformations were performed in the configuration space and
followed relations in the momentum space were obtained using the Fourier
transformation.
The 4D reduction of the 6D operators
$\varsigma(\kappa)\equiv\varsigma(q,\kappa_{+},\kappa_{-})$ generates the 5D
operators as the intermediate 5D projections. There exists only two 5D De
Sitter spaces with the constant curvature which have the invariant forms
$q_{\mu}q^{\mu}\pm q_{5}^{2}\mp M^{2}=0$ ($q_{5}^{2}\geq 0$) of the $O(2,3)$
and $O(1,4)$ rotational groups [18, 24, 28]. The single 5D hyperboloid is not
enough for reproduction of the whole values of $-\infty<q^{2}<\infty$.
Therefore, we use the domains from the both 5D hyperboloids which are
connected by inversion $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$. Thus for the
intermediate 5D projections of the 6D cone (I.2a) we shall use the invariant
forms
$q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}\ \ \ with\ \ \ {q_{5}^{2}\over
M^{2}}={{\kappa_{-}}\over{\kappa_{+}}}+1,\ \ \ and\ \ q_{5}^{2}\geq 0$ $None$
$q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}\ \ with\ \ \ {q_{5}^{2}\over
M^{2}}=-{{\kappa_{-}}\over{\kappa_{+}}}+1\ \ \ and\ \ q_{5}^{2}\geq 0.$ $None$
In (I.3a,b) the fifth variable $q_{5}^{2}$ is positive $0\leq
q_{5}^{2}`\infty$. Consequently, for the positive $q^{2}\equiv
q_{\mu}q^{\mu}\geq 0$ the corresponding four-momenta $q_{\mu}$ are distributed
between the domains $0\leq q^{2}\leq M^{2}$ and $q^{2}>M^{2}$ on the
hyperboloids (I.3a) and (I.3b) respectively. These domains are connected by
inversion $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$. The values of the $q^{2}$ on
the hyperboloids (I.3a) and (I.3b) are also connected through the reflection
$q^{2}\longleftrightarrow-q^{2}$. Therefore, for the negative $q^{2}<0$ one
has $-M^{2}<q^{2}<0$ on the hyperboloid (I.3b) and $-\infty<q^{2}<-M^{2}$ is
on the hyperboloid (I.3a). More detailed the distributions of $q^{2}$ on the
hyperboloids (I.3a) and (I.3b) are listed in Table 1 of Section 2.
In order to determine the 5D and 4D projections of the 6D fields
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ (I.2a) we shall introduce the
following 5D fields $\varphi_{1}(x,x_{5})$ and $\varphi_{2}(x,x_{5})$
$\varphi_{1}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}+q_{5}^{2}-M^{2})\Bigl{[}\theta(q^{2})\theta(M^{2}-q^{2})+\theta(-q^{2})\theta(-M^{2}-q^{2})]\Bigr{]}\phi(q,q_{5}^{2}),$
$None$
$\varphi_{2}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}-q_{5}^{2}+M^{2})\Bigl{[}\theta(q^{2})\theta(-M^{2}+q^{2})+\theta(-q^{2})\theta(M^{2}+q^{2})\Bigr{]}\phi(q,q_{5}^{2}).$
$None$
where
$\phi(q,q_{5}^{2})={{M^{2}}\over
2}\int\kappa_{+}^{3}d\kappa_{+}\theta(\kappa_{+})\varsigma(q,q_{5}^{2},\kappa_{+}).$
$None$
For the sake of simplicity the scale variable $\kappa_{+}$ (I.2b) is taken in
positive i.e. $\phi(\kappa)=\theta(\kappa_{+})\phi(\kappa)$, where
$\theta(\kappa_{+})=1$ for $\kappa_{+}>0$ and $\theta(\kappa_{+})=0$ for
$\kappa_{+}<0$.
The Fourier conjugate of $\varphi_{1}(x,x_{5})$ and $\varphi_{2}(x,x_{5})$ are
located into hyperboloids (I.3a) and (I.3b) respectively. Therefore they
satisfy the conditions
$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\biggr{)}\varphi_{1}(x,x_{5})=0,\ \ \ \ \ \ \ \ \
\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}-{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}-M^{2}\biggr{)}\varphi_{2}(x,x_{5})=0.$ $None$
The fields $\varphi_{1}$ (I.4a) and $\varphi_{2}$ (I.4b) produce two
independent 5D fields
$\varphi_{+}(x,x_{5})=\varphi_{1}(x,x_{5})+\varphi_{2}(x,x_{5});\ \ \
\varphi_{-}(x,x_{5})=\varphi_{1}(x,x_{5})-\varphi_{2}(x,x_{5}),$ $None$
which Fourier conjugate are defined in the whole domains
$(-\infty<q_{\mu}<+\infty)$ and $(-\infty<q^{2}<+\infty)$.
The usual boundary condition for the 5D fields $\varphi_{\pm}(x,x_{5})$ at
$x_{5}=0$ allows to get the 4D fields $\Phi_{\pm}(x)$
$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0).$ $None$
According to (I.4a,b), (I.7) and (I.8) the Fourier conjugate of
$\Phi_{\pm}(x)$
$\Phi_{\pm}(q)=\int d^{4}xe^{iqx}\Phi_{\pm}(x)$ $None$
have the following structure
$\Phi_{\pm}(q)=\sum_{N=I,III}\Phi_{N}(q)\pm\sum_{N=II,IV}\Phi_{N}(q);$ $None$
where
$\Phi_{I}(q)=\theta(q^{2})\theta(M^{2}-q^{2})\phi(q,q^{2}_{5}=M^{2}-q^{2});\ \
\
\Phi_{II}(q)=\theta(q^{2})\theta(-M^{2}+q^{2})\phi(q,q^{2}_{5}=M^{2}+q^{2});$
$\Phi_{III}(q)=\theta(-q^{2})\theta(-M^{2}-q^{2})\phi(q,q^{2}_{5}=M^{2}-q^{2});\
\ \
\Phi_{IV}(q)=\theta(-q^{2})\theta(M^{2}+q^{2})\phi(q,q^{2}_{5}=M^{2}+q^{2}),$
$None$
where the lower index $I,\ II,\ III$ and $IV$ of $\Phi(q)$ corresponds to the
domains of $q^{2}$ which are listed in Table 1 of Section 2. The details of
the relationship between the 4D and 5D scalar field are given in in Section 3.
The equations (I.4a,b)-(I.10) presents the relationship between the 4D, 5D and
6D fields $\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$. The 4D interacting
Heisenberg fields $\Phi_{\pm}(x)$ and their 6D representations
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ are not invariant under the
conformal transformations. Nevertheless, location of
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ on the 6D cone (I.2a) and the
corresponding location of the Fourier conjugate of $\varphi_{\pm}(x,x_{5})$ on
the hyperboloids (I.3a,b) produce the conditions (I.1b) and (I.6) for any 6D
field and its 5D projections. The consistency of these conditions with the
equation of motion for $\varphi_{\pm}(x,x_{5})$ and $\Phi_{\pm}(x)$ and the
boundary conditions (I.8) are considered in Sect. 4 and 5.
The intermediate projection of the 6D field
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ on the invariant forms
(I.3a,b) of the $O(2,3)$ and $O(1,4)$ subgroups of the conformal group
$O(2,4)$ need to introduce the two independent 5D fields $\varphi_{+}$ and
$\varphi_{-}$ (I.7) which are constructed from the same parts $\varphi_{1}$
and $\varphi_{2}$ (I.4a,b). The invariant forms (I.3a,b) of the $O(2,3)$ and
$O(1,4)$ form the definition area of $\varphi_{1}$ and $\varphi_{2}$ and
correspondingly of the field $\Phi_{N}(q)$ in (I.10) and (I.11). The 4D fields
$\Phi_{+}$ and $\Phi_{-}$ (I.10) have the same quantum numbers. But $\Phi_{+}$
and $\Phi_{-}$ can have the different masses and the different sources. Other
details of the inversion and related constructions of the 4D fields are given
in the next Section.
The reduction formulas (I.5) of the 6D field $\varsigma(\kappa)$ on the cone
(I.2a)
$\varsigma(\kappa)\equiv\varsigma(\kappa_{\mu},\kappa_{+},\kappa_{A}\kappa^{A}\neq
0)$ $\equiv\varsigma(q,q_{5}^{2},\kappa_{+})$ differ from the reduction
formula in the manifestly covariant formulation [14]-[24], where the
homogeneity of
$\varsigma(\kappa)=\phi_{(}q_{\mu},\kappa_{+},\kappa_{A}\kappa^{A}=0)$ over
the scale variable $\kappa_{+}$ is required, i.e.
$\phi(q_{\mu},\kappa_{+},\kappa_{A}\kappa^{A}=0)={\kappa_{+}}^{d}{\Phi}(q_{\mu}),$
where $d$ is the scale dimension of $\phi(\kappa)$. In order to reproduce this
property in the present approach one can use an additional condition in (I.5)
$\varsigma(q,q_{5}^{2},\kappa_{+})=\delta(\kappa_{+}-{\cal
M})\phi(q,q_{5}^{2},{\cal M});\ \ \ \ \ \ or\ \ \ \ \ \ \
\varsigma(q,q_{5}^{2},\kappa_{+})=\theta(\kappa_{+}-{\cal
M})\phi(q,q_{5}^{2},{\cal M})$ $None$
where ${\cal M}$ is a fixed scale parameter.
The 5D quantum field theory with the invariant forms
$q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}$ or $q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}$
separately was firstly studied in refs. [27, 28, 29], where $M$ was
interpreted as the fundamental (maximal) mass and its inverse $1/M$ as the
fundamental (minimal) length [30, 31]. The conformal transformations in the
momentum space for the complete fields $\varphi_{+}=\varphi_{1}+\varphi_{2}$
was suggested in [32], where $M$ is determined via $m_{\pi}$ and $m_{Higgs}$
according to the chiral symmetry breaking mechanism within the 5D chiral
models. In the present paper is studied coupling between the 5D and 4D
equations of motion for the fields $\varphi_{+}$, $\Phi_{+}$ and
$\varphi_{-}$, $\Phi_{-}$. Besides the present paper contains more general and
self-consistent formulation of translations and inversions in the 4D momentum
space for the 4D charged and neutral fields.
In Section 1 the conformal transformations in the 4D momentum space and the
corresponding transformations of the interacting fields $\Phi(x)$ are
considered. The domains of the variables $q^{2}=q_{\mu}q^{\mu}$ and
$q^{2}_{5}$ (I.3a,b) are determined in Sect. 2. The 4D reduction of the 5D
fields and the related projections and convolution formulas are given in Sect.
3. In Sections 4 and 5 the 4D and 5D equations of motion for the scalar fields
and the corresponding constrains for $x_{5}$ are considered. Sections 6 is
devoted to the 5D and 4D Lagrangians. The 4D and 5D equations of motion for
the fermion fields with the electromagnetic interaction and the constrains for
the fifth coordinates $x_{5}$ are considered in Sect. 7 and 8. In Section 9
the 5D extension of the standard $SU(2)\times U(1)$ model for the electron and
muon fields is given as an example of the suggested doubling for the fermion
states. Besides in this Section is shortly discussed consistency of the
present scheme and the purely 5D models [5, 6] of the grand unification
theory. The generalized translations in the momentum space as the gauge
transformations are considered in Sect 10. The summary is given in Sect. 11.
1\. Conformal transformations in the 4D momentum space.
Conformal transformations of the four-momentum $q_{\mu}$ $(\mu=0,1,2,3)$
consists of
$translations\hfill{q_{\mu}\longrightarrow q_{\mu}^{\prime}=q_{\mu}+h_{\mu},\
\ \ \ \ (1.1a)}$
$rotations\hfill{q_{\mu}\longrightarrow
q_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}q_{\nu},\ \ \ \ \ (1.1b)}$
$dilatation\hfill{q_{\mu}\longrightarrow q_{\mu}^{\prime}=e^{\lambda}\
q_{\mu},\ \ \ \ \ (1.1c)}$
$and\ inversion\hfill{q_{\mu}\longrightarrow
q_{\mu}^{\prime}=-M^{2}q_{\mu}/q^{2},\ \ \ \ \ (1.1d)}$
where $M$ is a mass parameter that insures the correct dimension of $q_{\mu}$.
Translations and inversions form
$special\ conformal\ transformation\hfill{q_{\mu}\longrightarrow
q_{\mu}^{\prime}=\frac{\Large{q_{\mu}-{\hbar}_{\mu}q^{2}/M^{2}}}{\Large{1-2q_{\nu}{\hbar}^{\nu}/M^{2}+{\hbar}^{2}q^{2}/M^{4}}}.\
\ \ \ \ (1.1e)}$
$q_{\mu}$ in (1.1a)-(1.1e) is off mass shell, i.e. $q_{o}$ is an independent
variable and $q_{o}\neq\sqrt{{\bf q}^{2}+m^{2}}$.
According to the Dirac geometrical model [10], transformations (1.1a)-(1.1e)
are equivalent to the rotations on the 6D cone
$\kappa^{2}\equiv\kappa_{A}\kappa^{A}=0$ (I.2a) with the metric tensor
$g_{AB}=diag(+1,-1,-1,-1,+1,-1)$. In particular, translation (1.1a) and the
special conformal translation are generated by the combinations of the
rotations in the planes $(\mu,5)$ and $(\mu,6)$, dilatation is obtained via
the rotation in the plane (5,6) and inversion follows from transposition of
the of the variables $\kappa^{\prime}_{5}=\kappa_{5}$ and
$\kappa_{6}^{\prime}=-\kappa_{6}$
$translation:\ \ \ \ \ \ \ \ \ \ \ \ \ \
\kappa_{\mu}^{\prime}=\kappa_{\mu}+h_{\mu}\kappa_{+};\ \ \ \ \
\kappa_{+}^{\prime}=\kappa_{+};\ \ \ \ \
\kappa_{-}^{\prime}=-{{2h_{\mu}\kappa^{\mu}}\over{\kappa_{+}}}-{{\kappa_{\mu}\kappa^{\mu}}\over{\kappa_{+}}}$
$None$ $rotation:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \kappa_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}\kappa_{\nu};\ \ \ \ \
\kappa_{+}^{\prime}=\kappa_{+};\ \ \ \ \ \kappa_{-}^{\prime}=\kappa_{-},$
$None$ $dilatation:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \kappa_{\mu}^{\prime}=\kappa_{\mu};\ \ \ \ \
\kappa_{+}^{\prime}=e^{-\lambda}\kappa_{+};\ \ \ \ \
\kappa_{-}^{\prime}=e^{\lambda}\kappa_{-},$ $None$ $inversion:\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\kappa_{\mu}^{\prime}=\kappa_{\mu};\ \ \ \ \ \kappa_{+}^{\prime}=\kappa_{-};\
\ \ \ \ \kappa_{-}^{\prime}=\kappa_{+},$ $None$
where
$q_{\mu}={{\kappa_{\mu}}\over{\kappa_{+}}};\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\kappa_{\pm}={{\kappa_{5}\pm\kappa_{6}}\over M};\ \ \ \ \ \ \ \ \
\mu=0,1,2,3.$ $None$
The equivalence of the 4D and 6D conformal transformations implies that
translation, rotation, dilatation and inversion of the 4D field $\Phi(q)$
(I.9) are unambiguously (isomorphic) determined through the corresponding 6D
rotations of the 6D field
$\varsigma(\kappa)\equiv\varsigma(\kappa_{\mu};\kappa_{+},\kappa_{-})$
$\Phi(q_{\mu}^{\prime}=q_{\mu}+h_{\mu})\Longleftrightarrow\varsigma(\kappa_{\mu}=\kappa_{\mu}+h_{\mu}\kappa_{+},\kappa_{+},\kappa_{-}^{\prime}=\kappa_{-}-\frac{2h_{\mu}\kappa^{\mu}}{\kappa_{+}}-\frac{\kappa_{\mu}\kappa^{\mu}}{\kappa_{+}})$
$None$
,
$\Phi(q_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}q_{\nu})\Longleftrightarrow\varsigma(\kappa_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}\kappa_{\nu},\kappa_{+},\kappa_{-})$
$None$
$\Phi(q_{\mu}^{\prime}={\lambda}q_{\nu})\Longleftrightarrow\varsigma(\kappa_{\mu},e^{-\lambda}\kappa_{+},e^{\lambda}\kappa_{-})$
$None$
$\Phi(q_{\mu}^{\prime}=-q_{\mu}/q^{2})\Longleftrightarrow\varsigma(\kappa_{\mu},\kappa_{-},\kappa_{+})$
$None$
These relationships between the 4D and 6D operators $\Phi(q)$ and
$\varsigma(\kappa)$ is achieved in (I.1b) through
$\delta(\kappa_{A}\kappa^{A})$.
An interacting scalar field $\Phi(x)$ is usually decomposed in the positive
and in the negative frequency parts in the 3D Fock space
$\Phi(x)=\int{{d^{3}p}\over{(2\pi)^{3}2\omega_{{\bf p}}}}\Bigl{[}a_{{\bf
p}}(x_{0})e^{-ipx}+{b^{+}}_{{\bf p}}(x_{0})e^{ipx}\Bigr{]};\ \ \
p_{o}\equiv\omega_{{\bf p}}=\sqrt{{\bf p}^{2}+m^{2}},$ $None$
where in the asymptotic regions $a_{{\bf p}}(x_{0})$ and ${b^{+}}_{{\bf
p}}(x_{0})$ transforms into particle (antiparticle) annihilation (creation)
operators. On the other hand, $\Phi(x)$ can be decomposed in the 4D momentum
space as
$\Phi(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}\Bigl{[}{\Phi^{(+)}}(q)e^{-iqx}+{\Phi^{(-)}}^{+}(q)e^{iqx}\Bigr{]};\
\ \ or\ \ \ \Phi(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}\Phi(q)e^{-iqx}.$ $None$
where
$\Phi(q)=\Phi^{(+)}(q)+{\Phi^{(-)}}^{+}(-q)$ $None$
Comparison of (1.5) and (1.6a) gives
${{e^{-i\omega_{\bf p}x_{o}}}\over{2\omega_{\bf p}}}a_{\bf
p}(x_{o})=\int{{dq_{o}}\over{2\pi}}{\Phi^{(+)}}(q_{o},{\bf
p})e^{-iq_{o}x_{o}}$ $None$
and
${{e^{i\omega_{\bf p}x_{o}}}\over{2\omega_{\bf p}}}{b^{+}}_{\bf
p}(x_{0})=\int{{dq_{o}}\over{2\pi}}{\Phi^{(-)}}^{+}(q_{o},{\bf
p})e^{iq_{o}x_{o}}.$ $None$
The field operators $a_{{\bf p}}(x_{0})$ and ${b^{+}}_{{\bf p}}(x_{o})$ are
simply determined via the corresponding source operator $\partial a_{{\bf
p}}(x_{0})/\partial{x_{0}}=i\int d^{3}xe^{ipx}j(x)$, where
$\Bigl{(}{{\partial^{2}}/{\partial{x_{\mu}}\partial{x^{\mu}}}}+m^{2}\Bigr{)}\Phi(x)=j(x)$.
Moreover, these operators determine the transition ${\cal S}$-matrix
${\cal S}_{mn}\equiv<out;{\bf p^{\prime}}_{1},...,{\bf p^{\prime}}_{m}|{\bf
p}_{1},...,{\bf p}_{n};in>=\prod_{i=1}^{m}\Bigl{[}\int
d{{x^{0}}^{\prime}}_{i}{{d}\over{d{{x^{\prime}}^{0}}_{i}}}\Bigr{]}$
$\prod_{j=1}^{n}\Bigl{[}\int
d{x^{0}}_{j}{{d}\over{d{x^{0}}_{j}}}\Bigr{]}<0|T\Bigl{(}a_{{\bf
p^{\prime}}_{m}}({x^{0}}^{\prime}_{m}),...,a_{{\bf
p^{\prime}}_{1}}({x^{0}}^{\prime}_{1})a_{{\bf
p}_{n}}^{+}({x^{0}}_{n}),...,a_{{\bf p}_{1}}^{+}({x^{0}}_{1})\Bigr{)}|0>.$
$None$
The translation (1.1a) for the 4D field in the momentum space
$\Phi^{\prime}(q)=\Phi(q+h)$ generates the corresponding gauge transformation
for $\Phi(x)$. In particular, for the complex charged field the four-momentum
translation (1.1a) produces the simplest gauge transformation
${\Phi}^{\prime}(x)=e^{ih_{\mu}x^{\mu}}\Phi(x);\ \ \ \ \
i{{\partial}\over{\partial{x_{\mu}^{\prime}}}}=i{{\partial}\over{\partial{x_{\mu}}}}+h_{\mu},$
$None$
where generally $h_{\mu}$ is a complex constant.
For the changeless real fields $\Phi(x)$ the gauge transformation (1.9) is
consistently defined for the pure imaginary $h_{\mu}=ir_{\mu}$ with the real
$r_{\mu}$. In particular, for the real scalar fields we get
${\Phi}^{\prime}(x)=e^{-r_{\mu}x^{\mu}}\Phi(x);\ \ \ \ \
{{\partial}\over{\partial{x_{\mu}^{\prime}}}}={{\partial}\over{\partial{x_{\mu}}}}+r_{\mu},$
$None$
The generalization of the gauge transformations (1.10) for the real scalar
fields within the nonlinear $\sigma$ model was performed in [36, 16]. These
formulation we shall consider at end of Sect. 10.
The transformation of $\Phi(x)$ under the rotation and dilatation of $q_{\mu}$
(1.1b,c) can be reproduced through the Fourier transformations in (1.6a) using
the inverse rotation and dilatation of $x_{\mu}$ in $exp(-iqx)$.
The doubling of the 5D fields $\varphi_{\pm}(x,x_{5})$ in (I.4a,b) and (I.7)
is generated by the intermediate projections onto domains of the 5D
hyperboloids (I.3a,b). These domains cover unambiguously the whole area of
$q_{\mu}$ and $q^{2}$ and cover the whole domain of the variables of the
fields $\Phi_{\pm}(q)$ (I.10). Inversion
$q_{\mu}^{\prime}=-M^{2}q_{\mu}/q^{2}$ transform the domains of the variables
of $\varphi_{1}(q,q^{2}_{5})$ into the domain of the variables of
$\varphi_{1}(q,q^{2}_{5})$ and vice versa. But inversion transforms also these
fields, i.e. inversion replaces $\varphi_{1}(x,x_{5})$ (1.4a) and
$\varphi_{2}(x,x_{5})$ (I.4b) with the $\varphi_{2}^{(I)}$ and
$\varphi_{1}^{(I)}$
$\varphi_{1}{\stackrel{{\scriptstyle
inversion}}{{\Longleftrightarrow}}}\varphi^{(I)}_{2}\ \ \ i.e.\ \ \
\varphi_{+}{\stackrel{{\scriptstyle
inversion}}{{\Longleftrightarrow}}}\varphi^{(I)}_{+};\ \ \
\varphi_{-}{\stackrel{{\scriptstyle
inversion}}{{\Longleftrightarrow}}}-\varphi^{(I)}_{-},$ $None$
. where the upper index (I) denotes the inversion of the corresponding
operator. One needs to introduce this index because the 4D equation of motion
for the massive particles are not invariant under the inversion. For instance,
the equation of motion $(q^{2}-m^{2})\Phi(q)=j(q)$ after inversion transforms
into new type equation $(M^{2}-q^{2}\
m^{2}/M^{2})\phi^{(I)}(q)=q^{2}j^{(I)}(q)/M^{2}$.
2\. Domains of $q^{2}$ and $q^{2}_{5}$
The invariant form $\kappa_{A}\kappa^{A}=0$ (I.2a) of the $O(2,4)$ group can
be represented for $q^{2}$ as
$q^{2}+M^{2}{{\kappa_{-}}\over{\kappa_{+}}}=0,$ $None$
where
$q_{\mu}={{\kappa_{\mu}}\over{\kappa_{+}}};\ \ \ \
\kappa_{\pm}={{\kappa_{5}\pm\kappa_{6}}\over{M}}.$ $None$
It is convenient to use the auxiliary fifth momentum $q_{5}^{2}$ instead of
$\kappa_{5}/\kappa_{6}$ in (2.1a). This procedure implies a projection of the
6D rotational invariant form $\kappa_{A}\kappa^{A}=0$ into corresponding 5D
forms. In order to cover unambiguously the whole domain $-\infty\leq
q^{2}\equiv q_{\mu}q^{\mu}\leq\infty$ we have distributed $q^{2}$ and
corresponding $q_{5}^{2}$ between the domains of the two 5D hyperboloids
$q^{2}+q_{5}^{2}=M^{2}\ \ \ \ \ with\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
q_{5}^{2}=M^{2}{{2\kappa_{5}}\over{\kappa_{5}+\kappa_{6}}},$ $None$
and
$q^{2}-q_{5}^{2}=-M^{2}\ \ \ \ \ with\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
q_{5}^{2}=M^{2}{{2\kappa_{6}}\over{\kappa_{5}+\kappa_{6}}}.$ $None$
The hyperboloids (2.2a,b) presents the simple intermediate 5D projection of
the 6D cone (I.2a) into the 4D momentum space with only one auxiliary variable
$q_{5}^{2}$.
$\kappa_{+}$ and $q^{2}_{5}$ in hyperboloids (2.2a,b) are defined in positive.
Consequently, $\kappa_{5}$ on the hyperboloid (2.2a) and $\kappa_{6}$ in
(2.2b) are also positive. The domains of the variables $q^{2}$, $q^{2}_{5}$,
$\kappa_{5}$ and $\kappa_{6}$ defined on the 6D cone (2.1a) and on the
corresponding 5D hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ (2.2a,b) are
listed in Table 1. The border points $q^{2}=0$, $q^{2}=M^{2}$ and
$q^{2}=-M^{2}$ are included in the domain ${\bf I}$ and ${\bf IV}$.
It must be emphasized, that the domain $M^{2}<q_{5}^{2}<2M^{2}$ of the
variables of $\phi(q,q_{5}^{2})$ is excluded by construction of
$\varphi_{1,2}(x,x_{5})$ (I.4a,b). Correspondingly, the variables $\kappa_{5}$
and $\kappa_{6}$ cover also only the part of the 6D cone
$\kappa_{A}\kappa^{A}=0$ (I.2a). The principal restriction for the auxiliary
variables $\kappa_{5}$, $\kappa_{6}$ and $q_{5}^{2}$ is that the corresponding
$q^{2}$ must cover the whole domain $(-\infty,+\infty)$. This property allows
to construct unambiguously the 4D field
$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0)$.
Table 1 Domains of $q^{2}$, $q_{5}^{2}$, $\kappa_{5}$ and $\kappa_{6}$ placed
on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ (2.2a,b) and on the surface
(2.1a).
| I | II | III | IV
---|---|---|---|---
| $q^{2}+q_{5}^{2}=M^{2}$ | $q^{2}-q_{5}^{2}=-M^{2}$ | $q^{2}+q_{5}^{2}=M^{2}$ | $q^{2}-q_{5}^{2}=-M^{2}$
$q^{2}$ | $0\leq q^{2}═\leq═M^{2}$ | $M^{2}<q^{2}<\infty$ | $-\infty<q^{2}<-M^{2}$ | $-M^{2}\leq q^{2}<0$
$q_{5}^{2}$ | $0\leq q_{5}^{2}═\leq M^{2}$ | $2M^{2}<q_{5}^{2}<\infty$ | $2M^{2}<q_{5}^{2}<\infty$ | $0\leq q_{5}^{2}<M^{2}$
$\kappa_{5}\&\kappa_{6}$ | $0\leq\kappa_{5}\leq\kappa_{6}$ | $\kappa_{6}>0$; $\kappa_{5}<0$; $\kappa_{5}+\kappa_{6}>0$ | $\kappa_{5}>0$; $\kappa_{6}<0$; $\kappa_{5}+\kappa_{6}>0$ | $0\leq\kappa_{6}<\kappa_{5}$
The hyperboloids $q^{2}+q^{2}_{5}=M^{2}$ and $q^{2}-q^{2}_{5}=-M^{2}$ and the
corresponding domains ${\bf I,III}$ and ${\bf II,IV}$ transforms also into
each other by reflection $q^{2}\longleftrightarrow-q^{2}$ which is generated
by transposition of the variables $\kappa_{5}$ and $\kappa_{6}$
$q^{2}\longleftrightarrow-q^{2}\ \ \ \ \ \ \ \ \kappa_{5}=\kappa_{6},\ \ \
\kappa_{6}=\kappa_{5};\ \ \ \ \ \kappa_{+}=\kappa_{+},\ \ \
\kappa_{-}=-\kappa_{-}.$ $None$
The similar transpositions of the hyperboloids $q^{2}+q^{2}_{5}=M^{2}$ and
$q^{2}-q^{2}_{5}=-M^{2}$ produces inversion $q_{\mu}^{I}=-M^{2}q_{\mu}/q^{2}$
(1.1d) which is generated by transposition of the variables $\kappa_{+}$ and
$\kappa_{-}$ according to (1.2d).
The choice of the 5D hyperboloids is not unique. For instance, instead of the
two 5D hyperboloids (2.2a,b) one can take other hyperboloids $q^{2}\pm
q_{5}^{2}=M^{2}$ with $q_{5}^{2}={{\pm
2M^{2}\kappa_{5}}/{\kappa_{5}+\kappa_{6}}}$, where $\kappa_{+}$ is fixed
$\kappa_{+}={\cal M}/M$ and ${\cal M}$ is a mass parameter. But these domains
of $q^{2}$ are not symmetric under the reflection and inversion of $q^{2}$
unlike the domains in Table 1. Therefore we do not consider them.
An other choice of the intermediate 5D projections presents the stereographic
projection, where $\kappa_{6}=-1$, and the auxiliary momenta
$Q_{\mu}=q_{\mu}/(1-q^{2})$ and $Q_{4}=(1+q^{2})/(1-q^{2})$, i.e.
$q_{\mu}=Q_{\mu}/(1+Q_{4})$, $q^{2}=(Q_{4}-1)/(1+Q_{4})$ are introduced (see,
for example, eq. (13.43) in [7]). This choice of the variables require only
one 5D hyperboloid $Q^{2}-Q_{4}^{2}=-1$ with the 5 auxiliary variables
$Q_{\mu}$ and $Q_{4}$ for the intermediate 5D projections. Certainly, one can
represent this projection via the considered projections on the hyperboloids
$q^{2}\pm q_{5}^{2}=\pm M^{2}$ (2.2a,b) where only one auxiliary variable
$q^{2}_{5}$ is used.
3\. 4D reductions of the 5D fields.
The 5D fields $\varphi_{\pm}=\varphi_{1}\pm\varphi_{2}$ (I.7) consist of the
two parts $\varphi_{1}$ (I.4a) and $\varphi_{1}$ (I.4b). which satisfy the
conditions (I.6), because their Fourier conjugates are defined on the 5D
hyperboloids $q^{2}\pm q^{2}_{5}=\pm M^{2}$ (I.3a,b). For $\varphi_{\pm}$
these conditions can be represented as
${{\partial^{2}\varphi_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}\varphi_{\mp}(x,x_{5})=0.$ $None$
Integration over $q_{5}^{2}$ in (I.4a,b) yields
$\varphi_{\pm}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\Bigl{[}\phi(q,Q^{2}_{1})\Lambda_{1}(q^{2})e^{-iQ_{1}x_{5}}\pm\phi(q,Q^{2}_{2})\Lambda_{2}(q^{2})e^{-iQ_{2}x_{5}}\Bigr{]}.$
$None$
The expression (3.2) can be represented via the 5D convolution formula
$\varphi_{\pm}(x,x_{5})=\int d^{5}y\phi(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5}),$ $None$
where $\phi(x,x_{5})$ is the Fourier conjugate of the full 5D function
$\phi(q,q_{5}^{2})$ in (I.4a,b) and
$\Lambda_{a}(q^{2})=\left\\{\begin{array}[]{ll}\theta(q^{2})\theta(M^{2}-q^{2})+\theta(-q^{2})\theta(-M^{2}-q^{2})&\mbox{if
$a=1$};\\\
\theta(q^{2})\theta(-M^{2}+q^{2})+\theta(-q^{2})\theta(M^{2}+q^{2})&\mbox{if
$a=2$},\end{array}\right.$ $None$
$Q_{a}^{2}=\left\\{\begin{array}[]{ll}M^{2}-q^{2}&\mbox{if $a=1$};\\\
M^{2}+q^{2}&\mbox{if $a=2$},\end{array}\right.\ \ \ and\ \ \
Q_{a}=\sqrt{Q_{a}^{2}}.$ $None$
The operators
${\cal P}_{\pm}(x,x_{5})={\cal P}_{1}(x,x_{5})\pm{\cal P}_{2}(x,x_{5})$ $None$
consist of the two parts that are placed onto hyperboloids
$q^{2}+q_{5}^{2}=M^{2}$ and $q^{2}-q_{5}^{2}=-M^{2}$
${\cal P}_{a}(x,x_{5})=\left\\{\begin{array}[]{ll}\int
dq_{5}^{2}e^{-iq_{5}x_{5}}{{d^{4}q}/{(2\pi)^{4}}}e^{-iqx}\Lambda_{1}(q^{2})\delta(q^{2}+q_{5}^{2}-M^{2})&\mbox{if
$a=1$};\\\ \int
dq_{5}^{2}e^{-iq_{5}x_{5}}{{d^{4}q}/{(2\pi)^{4}}}e^{-iqx}\Lambda_{2}(q^{2})\delta(q^{2}-q_{5}^{2}+M^{2})&\mbox{if
$a=2$},\end{array}\right.$ $None$
that satisfy the orthogonality and completeness conditions at $x_{5}=0$
$\int d^{4}y{\cal P}_{a}(x-y,0){\cal P}_{b}(y,0)=\delta_{ab}{\cal
P}_{a}(x,0);$ $None$
${\cal P}_{+}(x,0)=\delta^{(4)}(x)$ $None$
The relation (3.3) presents the projections of the complete 5D field
$\phi(x,x_{5})$ onto the two independent 5D fields $\varphi_{\pm}(x,x_{5})$
which for $x_{5}=0$ produce the 4D fields $\Phi_{\pm}(x)$ (I.8). The Fourier
conjugate $\Phi_{\pm}(q)$ (I.9) and the Fourier conjugate of the 5D fields
$\varphi_{\pm}$ (3.3) consist of the same four parts that are given in (I.10)
and (I.11). Therefore from (3.3) we get
$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\Bigl{[}\phi(q,q_{5}^{2}=M^{2}-q^{2})\Lambda_{1}(q^{2})\pm\phi(q,q_{5}^{2}=M^{2}+q^{2})\Lambda_{2}(q^{2})\Bigr{]},$
$None$
The equations (I.5) and (3.9) presents the 4D reduction of the 6D field
$\varsigma(\kappa,\kappa_{-},\kappa_{+})$ and 5D field $\phi(q,q^{2}_{5})$
onto the 4D field $\Phi_{\pm}(q)$ (I.10).
It must be noted that $\phi(q,M^{2}<q^{2}_{5}<2M^{2})$ does not contribute in
the 4D fields $\Phi_{\pm}(x)$ because the region $M^{2}<q^{2}_{5}<2M^{2}$ is
excluded from the domains in Table 1.
(3.3) and (3.8) allow to represent (3.9) m as the 4D convolution formula
$\varphi_{\pm}(x,x_{5})=\int d^{4}y\Phi_{\pm}(x-y){\cal P}_{+}(y,x_{5}).$
$None$
that determines the two 5D fields $\varphi_{\pm}$ via the corresponding 4D
fields $\Phi_{\pm}$.
4\. 4D and 5D equations of motion.
According to the boundary condition (3.9) and the convolution formula (3.10)
the 5D and 4D scalar fields $\varphi_{\pm}$ and $\Phi_{\pm}$ satisfy the
similar equation of motion
$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+m_{\pm}^{2}\biggr{)}\varphi_{\pm}(x,x_{5})=j_{\pm}(x,x_{5}),$ $None$
$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+m_{\pm}^{2}\biggr{)}\Phi_{\pm}(x)=J_{\pm}(x)$ $None$
where
$J_{\pm}(x)=j_{\pm}(x,x_{5}=0).$ $None$
The 5D source operators $j_{\pm}(x,x_{5})$ in (4.1a) consist of the products
of the 5D fields $\varphi_{\pm}$ and their derivatives. But the Fourier
conjugate of the multiplication of $\varphi_{\pm}(x,x_{5})$ are not placed on
the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$. The source operators
$j_{\pm}(x,x_{5})$ in (4.1a) must be placed on the hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$ as well as $\varphi_{\pm}(x,x_{5})$ in (3.1).
Consequently, $j_{\pm}(x,x_{5})$ must satisfy the additional 5D constrains in
analogy with (3.1) for $\varphi_{\pm}(x,x_{5})$. In order to obtain these
conditions for $j_{\pm}$ we combine the equations (4.1a,b) and the conditions
(3.1) as
${{\partial^{2}j_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}j_{\mp}(x,x_{5})=$ $\Bigl{(}{{\partial^{2}}\over{\partial
x_{\mu}\partial x^{\mu}}}{{\partial^{2}}\over{\partial x_{5}\partial
x^{5}}}-{{\partial^{2}}\over{\partial x_{5}\partial
x^{5}}}{{\partial^{2}}\over{\partial x_{\mu}\partial
x^{\mu}}}\Bigr{)}\varphi_{\mp}(x,x_{5})+(m_{\pm}^{2}-m_{\mp}^{2}){{\partial^{2}\varphi_{\pm}(x,x_{5})}\over{\partial
x_{\mu}\partial x^{\mu}}}.$ $None$
For the independent variables $x_{\mu}$ and $x_{5}$ the operator
${{\partial^{2}}/{\partial x_{\mu}\partial x^{\mu}}}$ and
${{\partial^{2}}/{\partial x_{5}\partial x^{5}}}$ commute. Therefore we get
${{\partial^{2}{{\widetilde{j}}_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}{\widetilde{j}}_{\mp}(x,x_{5})=0,$ $None$
where
$\widetilde{j}_{\pm}(x,x_{5})=j_{\pm}(x,x_{5})-m^{2}_{\pm}\varphi_{\pm}(x,x_{5})$
$None$
Thus $\widetilde{j}_{\pm}$ satisfy also the same sourceless equation (3.1).
Using (4.1a) one can rewrite (4.4a) as
${{\partial^{2}\over{\partial x^{\nu}\partial
x_{\nu}}}}{{\partial^{2}{{\varphi}_{\pm}(x,x_{5})}\over{\partial
x^{\mu}\partial x_{\mu}}}}+{{\partial^{2}\over{\partial x^{\nu}\partial
x_{\nu}}}}\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}{\varphi}_{\mp}(x,x_{5})=0,$ $None$
which indicates that the Fourier conjugate of
${{\partial^{2}{{\varphi}_{\pm}(x,x_{5})}/{\partial x^{\mu}\partial
x_{\mu}}}}$ and ${\widetilde{j}}_{\pm}$ in (4.1a) are placed on the
hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$. Therefore, in analogy with
(I.4a,b) we get
$j_{1}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}+q_{5}^{2}-M^{2})\Bigl{[}\theta(q^{2})\theta(M^{2}-q^{2})+\theta(-q^{2})\theta(-M^{2}-q^{2})]\Bigr{]}{\cal
J}(q,q_{5}^{2}),$ $None$
$j_{2}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}-q_{5}^{2}+M^{2})\Bigl{[}\theta(q^{2})\theta(-M^{2}+q^{2})+\theta(-q^{2})\theta(M^{2}+q^{2})\Bigr{]}{\cal
J}(q,q_{5}^{2}).$ $None$
where ${\cal J}$ denote the full 5D source operator
${\cal J}(x,x_{5})=\int{{d^{5}q}\over{(2\pi)}}e^{-iqx-iq_{5}x_{5}}{\cal
J}(q.q_{5}^{2})$ $None$
Afterwards the 5D sources ${j}_{\pm}$ can be represented as
${{j}}_{\pm}(x,x_{5})=\int{{d^{4}qe^{-iqx}}\over{(2\pi)^{4}}}\Bigl{[}{\cal
J}(q,Q^{2}_{1})\Lambda_{1}(q^{2})e^{-iQ_{1}x_{5}}\pm{\cal
J}(q,Q^{2}_{2})\Lambda_{2}(q^{2})e^{-iQ_{2}x_{5}}\Bigr{]},$ $None$
that in analogy with (3.3) and (3.10) allow to construct the 5D sources
${{j}}_{\pm}(x,x_{5})$ through the 4D sources $J_{\pm}(x)$ or the full 5D
sources ${\cal J}$ using the following convolution formulas
${{j}}_{\pm}(x,x_{5})=\int d^{4}yJ_{\pm}(x-y){\cal P}_{+}(y,x_{5}),$ $None$
$j_{\pm}(x,x_{5})=\int d^{5}y{\cal J}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5}).$ $None$
According to (4.6a) the sources $J_{\pm}(x)$ and their Fourier conjugate
$J_{\pm}(q)$ consist of the four parts as well as the 4D field $\Phi_{\pm}(q)$
in (I.10)
$J_{\pm}(q)=\sum_{N=I,III}J_{N}(q)\pm\sum_{N=II,IV}J_{N}(q),$ $None$
where
$J_{I}(q)=\theta(q^{2})\theta(M^{2}-q^{2}){\cal J}(q,q^{2}_{5}=M^{2}-q^{2});\
\ \ \ \ J_{III}(q)=\theta(-q^{2})\theta(-M^{2}-q^{2}){\cal
J}(q,q^{2}_{5}=M^{2}-q^{2});$ $None$
$J_{II}(q)=\theta(q^{2})\theta(-M^{2}+q^{2}){\cal
J}(q,q^{2}_{5}=M^{2}+q^{2});\ \ \ \ \
J_{IV}(q)=\theta(-q^{2})\theta(M^{2}+q^{2}){\cal J}(q,q^{2}_{5}=M^{2}+q^{2}),$
$None$
where the lower index $N=I,II,III,IV$ in ${{J}_{N}}$ indicates the domains of
$q^{2}$ in Table 1.
In (4.6a) ${{j}}_{\pm}(x,x_{5})$ is constructed through the 4D sources
$J_{\pm}(q)$ (4.8a,b) that are defined via the 4D fields $\Phi_{\pm}$, i.e.
(4.6a) allow to determine the 5D source $j_{\pm}$ through the 4D fields
$\Phi_{\pm}$. In (4.6b) ${{j}}_{\pm}(x,x_{5})$ is constructed is constructed
via the 5D sources ${\cal J}(x,x_{5})$ (4.5c) which consists of the products
of the complete 5D fields $\phi(x,x_{5})$. The 5D source ${\cal J}$ (4.5c) can
be determined via the fields $\phi$ according to the 5D extension of the
Klein-Gordon equation
$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+{\widetilde{m}}^{2}\biggr{)}\phi(x,x_{5})={\cal J}(x,x_{5}).$ $None$
In the present formulation the terms with $\partial\varphi_{\pm}/\partial
x_{5}$ are determined through the constrains which ensure the consistency of
the condition (3.1) and the equation of motion (4.1a). These constrains are
considered in the next section. In (4.9) the projections of
$\partial\phi_{\pm}/\partial x_{5}$ can be included in ${\cal J}$. But these
terms must be consistent with the constrains for $\partial\varphi/\partial
x_{5}$.
The projections of the 5D field $\phi$ with mass ${\widetilde{m}}$ and source
${\cal J}$ into two 5D fields $\varphi_{\pm}$ with the two different masses
$m_{\pm}$ and the two sources $j_{\pm}$ are performed in (3.3) and (4.6c)
using the projection operators ${\cal P}_{\pm}$ (3.6a). In these projections
only the parts of the complete 5D field and its source are used for
construction of the 5D fields $\varphi_{\pm}$ and $j_{\pm}$ because the parts
of these fields in the domains $N=I,II,III,IV$ in Table 1 cover only the part
of the full 5D space.
The input 5D fields $\phi$, ${\cal J}$ and their projections $\varphi_{\pm}$,
$j_{\pm}$ can be constructed within the various relativistic invariant time
models [8, 9].
It must be noted, that $\partial\varphi_{\pm}(x,x_{5})/\partial x_{5}$,
$j_{\pm}(x,x_{5})$ and $m^{2}_{\pm}$ satisfy an additional conditions that can
be obtained combining (4.1a) and (3.1)
$M^{2}\biggl{[}1+\Bigl{(}{1\over M}{{\partial}\over{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}\varphi_{\pm}=m_{\mp}^{2}\varphi_{\mp}-j_{\mp}.$
$None$
The conditions (4.10) presents the relationship between $j_{\pm}$, $m_{\pm}$
and the operators $\biggl{[}1+\Bigl{(}{1\over M}{{\partial}/{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}\varphi_{\pm}$. The factorization of
$\biggl{[}1+\Bigl{(}{1\over M}{{\partial}/{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}$ allows to determine
$\partial\varphi_{\pm}/\partial x_{5}$ through $j_{\pm}(x,x_{5})$,
$m^{2}_{\pm}$ and vice versa. This problem will be considered in the next
section.
5\. Constrains for ${{\partial\varphi_{\pm}}/{\partial x^{5}}}$.
In order to obtain the constrains for ${{\partial\varphi_{\pm}}/{\partial
x^{5}}}$ we shall factorize (4.10). The general form of the sought linear
conditions for ${{\partial\varphi_{\pm}}/{\partial x_{5}}}$ of the scalar
charged fields are
${i\over M}{{\partial\varphi_{\pm}}\over{\partial
x_{5}}}=\alpha_{\pm}\varphi_{+}+\beta_{\pm}\varphi_{-}+C_{\pm},$ $None$
where $C_{\pm}$ consist of the products of $\varphi_{\pm}$. The Fourier
conjugate of these products are not located on the hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$. Therefore, $C_{\pm}$ are defined as
$C_{\pm}(x,x_{5})=\int{\widetilde{\cal C}}_{\pm}(x-y,x_{5}-y_{5})d^{5}y{\cal
P}_{+}(y,y_{5}).$ $None$
Substituting (5.1) in (4.10) one obtains
$m_{+}^{2}=-2M^{2}\alpha_{+}{{1-\alpha_{+}^{2}}\over{\beta_{+}}}\ \ \ \ \
j_{+}=M^{2}\biggl{[}{{1-\alpha_{+}^{2}}\over{\beta_{+}}}C_{+}+\alpha_{+}C_{-}+{i\over
M}{{\partial C_{-}}\over{\partial x_{5}}}\biggr{]};$ $None$
$m^{2}_{-}=-2M^{2}\alpha_{+}\beta_{+};\ \ \ \ \
j_{-}=M^{2}\biggl{[}\alpha_{+}C_{+}+\beta_{+}C_{-}+{i\over M}{{\partial
C_{+}}\over{\partial x_{5}}}\biggr{]}.$ $None$
where are using the conditions
$\alpha_{+}^{2}+\alpha_{-}\beta_{+}=1;\ \ \ \ \ \ \ \ \ \ \
\beta_{-}^{2}+\alpha_{-}\beta_{+}=-1$ $None$
that insures reproduction of the mass terms in (4.10).
From (5.4) follows that $\alpha_{+}=\beta_{-}$ and the condition
$m^{2}_{\pm}>0$ requires that $-1<\alpha_{+}<1$. If $\alpha_{+}=-\beta_{-}$,
then $m_{+}=m_{-}=0$.
The relations (5.3a,b) determine $m^{2}_{\pm}$ through the three parameters
$M^{2}$, $\alpha_{+}$ and $\beta_{+}$. The same parameters, $C_{\pm}$ and
${i/M}\ {{\partial C_{\pm}}/{\partial x_{5}}}$ determine the sources
$j_{\pm}$.
For the neutral particles the analogue of (5.1) is
${1\over M}{{\partial\varphi_{\pm}}\over{\partial
x_{5}}}=\alpha_{\pm}\varphi_{+}+\beta_{\pm}\varphi_{-}+C_{\pm},$ $None$
Substituting this condition in (4.10) we get
$m_{+}^{2}=-2M^{2}\alpha_{+}{{1+\alpha_{+}^{2}}\over{\beta_{+}}}\ \ \ \ \
j_{+}=M^{2}\biggl{[}{{1+\alpha_{+}^{2}}\over{\beta_{+}}}C_{+}+\alpha_{+}C_{-}+{1\over
M}{{\partial C_{-}}\over{\partial x_{5}}}\biggr{]};$ $None$
$m^{2}_{-}=2M^{2}\alpha_{+}\beta_{+};\ \ \ \ \
j_{-}=M^{2}\biggl{[}\alpha_{+}C_{+}+\beta_{+}C_{-}+{1\over M}{{\partial
C_{+}}\over{\partial x_{5}}}\biggr{]},$ $None$
where the following conditions for the parameters are used
$\alpha_{+}^{2}+\alpha_{-}\beta_{+}=-1;\ \ \ \ \ \ \ \ \ \ \
\beta_{-}^{2}+\alpha_{-}\beta_{+}=-1.$ $None$
The ratio $m_{+}^{2}/m_{-}^{2}=-(\alpha_{+}^{2}+1)/\beta_{+}^{2}$ in this case
is negative. Therefore, the constrains (5.5) can not generate the positive
$m_{+}^{2}$ and $m_{-}^{2}$ simultaneously for the neutral fields.
The positive masses $m_{\pm}^{2}\geq 0$ for the neuthral particles in (4.10)
can be reproduced using the other kind of the constrains
$\varphi_{-}=\alpha\varphi_{+}+{\em G}(\varphi_{+},\varphi_{-}),$ $None$
where ${\em G}$ does not contain the linear over $\varphi_{\pm}$ terms. For
the sake of simplicity the dependence over ${\partial\varphi_{\pm}}/{\partial
x_{\mu}}$ and ${\partial\varphi_{\pm}}/{\partial x_{5}}$ in ${\em G}$ (5.7a)
are omitted.
Acting with $\biggl{[}1+\Bigl{(}{1\over M}{{\partial}/{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}$ in (5.7a) and using the conditions
$m_{+}^{2}=\alpha^{2}m_{-}^{2};\ \ \ \ \ {\em
F}=M^{-2}\biggl{[}1+\alpha{{m_{-}^{2}}\over{M^{2}}}+\Bigl{(}{1\over
M}{{\partial}\over{\partial x_{5}}}\Bigr{)}^{2}\biggr{]}^{-1}{\em G}$ $None$
one determine $j_{-}$ via $j_{+}$ as
$\alpha j_{-}-j_{+}={\em F}.$ $None$
In particular, for the $\Phi_{+}^{4}$ model with the 4D source
$J_{+}=a\Phi_{+}^{2}+b\Phi_{+}^{3}$ and mass $m_{+}$ the relations (5.7a,b,c)
allow to reproduce the 5D equations (4.1a) with the 5D source operator
$j_{\pm}$ and the masses $m_{\pm}^{2}$. These 5D equation generates the 4D
equation of motion with $J_{-}=(J_{+}+{\em F})/\alpha$ and the real mass
$m_{-}=m_{+}/\alpha$.
For the charged fields it is convenient to use the following constrains
$\biggl{[}1-{i\over M}{{\partial}\over{\partial
x_{5}}}\biggr{]}\varphi_{\pm}={\em C}_{\pm}.$ $None$
where ${\em C}_{\pm}$ depends generally on the products of $\varphi_{\pm}^{+}$
and $\varphi_{\pm}$.
$M^{2}\biggl{[}1+{i\over M}{{\partial}\over{\partial x_{5}}}\biggr{]}{\em
C}_{\mp}=m_{\pm}^{2}\varphi_{\pm}-j_{\pm}.$ $None$
Present formulation has in common with the other 5D field-theoretical
approaches within the relativistic invariant time method [7, 8, 9] with
$x_{5}^{2}=x_{0}^{2}-{\bf x}^{2}\equiv x_{\mu}x^{\mu}$. Unlike these 5D
formulations our approach based on the invariance of the 6D and 5D forms
(I.2a,b) and (I.3a,b) under the conformal transformations in the momentum
space. The main difference between the our conditions for
$\partial\varphi_{\pm}/\partial x_{5}$ (4.10), (5.1) and (5.7c) and the
evaluation-type equations over the $x_{5}$ other authors is that in the
present formulation $\partial\varphi_{+}/\partial x_{5}$ is determined through
the source and mass of the coupling field $j_{-}$ and $m_{-}$. Nevertheless,
one can use the 5D models in [7, 8, 9] for input 5D field $\phi(q,q^{2}_{5})$
in (I.4a,b) and input 5D and 4D source operators
$J_{\pm}(x)=j_{\pm}(x,x_{5}=0)$ in (4.1,b).
6\. The 5D and 4D Lagrangians.
In this section we shall consider the 5D Lagrangians ${{\cal
L}}_{\pm}(x,x_{5})$ for the two interacted scalar 5D fields $\varphi_{\pm}$
with the same quantum numbers, but with the different masses $m_{\pm}$ and
sources $j_{\pm}(x,x_{5})$. These Lagrangians must reproduce the 4D
Lagrangians $L_{\pm}(x)$ at $x_{5}=0$ and the 5D equations of motion (4.1a).
The sought Lagrangian can be represented as
${{\cal L}}_{\pm}(x,x_{5})=({{\cal L}}_{\pm})_{o}(x,x_{5})+({{\cal
L}}_{\pm})_{INT}(x,x_{5})+({{\cal L}}_{\pm})^{C}(x,x_{5})$ $None$
where $({{\cal L}}_{\pm})_{o}$ and $({{\cal L}}_{\pm})_{INT}$ denote the non-
interacted and interacted parts of these Lagrangians. The third term $({{\cal
L}}_{\pm})^{C}$ reproduce the constrains for the
$\partial\varphi_{\pm}/\partial x_{5}$ in (5.1), (5.5), (5.7a) and (5.8a,b).
Any one of these constrains together with the 5D equation of motion (4.1a)
generate the conditions (3.1). In particular, $({{\cal L}}_{\pm})^{C}$ for the
constrains (5.8a,b) are
${\cal L}_{\pm}^{C}=M^{2}|{i\over M}{{\partial\varphi_{\pm}}\over{\partial
x_{5}}}-\varphi_{\pm}+{\em C}_{\pm}|^{2}+M^{2}|{i\over M}{{\partial{\em
C}_{\pm}}\over{\partial x_{5}}}+{\em
C}_{\pm}-{{m_{\mp}^{2}\varphi_{\mp}-j_{\mp}}\over{M^{2}}}|^{2},$ $None$
where $\delta{\cal L}_{\pm}^{C}/\delta{\em C}_{\pm}$ and $\delta{\cal
L}_{\pm}^{C}/\delta[{{\partial{\em C}_{\pm}}/{\partial x_{5}}}]$ reproduce
(5.8a) and (5.8b) correspondingly.
The 5D Lagrangians (6.1) can be simply constructed starting from the two 4D
Lagrangians $L_{\pm}(x)$ for the two interacted scalar fields $\Phi_{\pm}(x)$
$L_{\pm}(x)=(L_{\pm})_{o}(x)+({L}_{\pm})_{INT}(x),$ $None$
where the non-interacting part $(L_{\pm})_{o}$
$({{L}}_{\pm})_{o}(x)={{\partial{\Phi_{\pm}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\Phi_{\pm}}}\over{\partial
x^{\mu}}}-m_{\pm}^{2}{\Phi_{\pm}}^{+}{\Phi_{\pm}}$ $None$
determines the non-interacted part of the 5D Lagrangians (6.1)
$({{\cal L}}_{\pm})_{o}(x,x_{5})={{\partial{\varphi_{\pm}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\varphi_{\pm}}}\over{\partial
x^{\mu}}}-m_{\pm}^{2}{\varphi_{\pm}}^{+}{\varphi_{\pm}}.$ $None$
The 5D source $j_{\pm}$ in (4.1a) can be constructed from the 4D source
$J_{\pm}$ (4.1b) using the convolution formula (4.6b). The sought 5D
Lagrangian $({{\cal L}}_{\pm})_{INT}$ consists of the fields $\varphi_{\pm}$.
The replacement of the 4D fields $\Phi_{\pm}$ in $(L_{\pm})_{INT}$ with the 5D
fields $\varphi_{\pm}$ give
$({{\cal L}}_{\pm})_{INT}(x,x_{5})=\int d^{4}y({\sf
L}_{\pm})_{INT}(x-y,x_{5}){\cal P}_{+}(y,x_{5}),\ \ \ where\ \ \ ({\sf
L}_{\pm})_{INT}(x,x_{5})=({L}_{\pm})_{INT}\Bigl{(}\Phi_{\pm}\Longleftrightarrow\varphi_{\pm}\Bigr{)}.$
$None$
The 5D sources ${j}_{\pm}$ satisfy the Euler-Lagrange equations
${j}_{\pm}(x,x_{5})={{\partial{({\cal
L}_{\pm})_{INT}}}\over{\partial{\varphi_{\pm}}^{+}}}-{{d}\over{dx_{\mu}}}\biggl{(}{{\partial{({\cal
L}_{\pm})_{INT}}}\over{\partial{{[\partial{\varphi_{\pm}}^{+}}/{\partial
x^{\mu}}]}}}\biggr{)},$ $None$
The other kind of the 5D sources $j_{\pm}$ can be obtained from the general 5D
Lagrangian ${\ell}(\phi,\phi^{+})$ for the 5D scalar field $\phi(q,q_{5}^{2})$
that parts are used in (I.4a,b) and (3.2) for reproduction of $\varphi_{\pm}$
${\ell}(\phi,\phi^{+})={{\partial{\phi}^{+}}\over{\partial
x_{\mu}}}{{\partial{\phi}}\over{\partial
x^{\mu}}}+{\widetilde{m}}^{2}\phi^{+}\phi+{\ell}_{INT}(\phi,\phi^{+}),$ $None$
where the terms with $\partial\phi/\partial x_{5}$ are included in
${\ell}_{INT}$. The Lagrangian (6.7) generate the 5D equation of motion (4.9).
In order to reproduce $j_{\pm}$ from ${\ell}(\phi,\phi^{+})$ (6.7) one must
determine $\partial\phi/\partial x_{5}$ according to constrains for
$\partial\varphi_{\pm}/\partial x_{5}$ that are given in $({{\cal
L}}_{\pm})^{C}$ (6.1). For this aim we shall consider the projection of the
equation of motion (4.9) according to (3.3) and (4.6c)
$\int
d^{5}y\Bigr{[}{{\partial^{2}\phi(x-y,x_{5}-y_{5})}\over{\partial(x-y)^{\mu}\partial(x-y)_{\mu}}}-{{\cal
J}}(x-y,x_{5}-y_{5})+{\widetilde{m}}^{2}\phi(x-y,x_{5}-y_{5})\Bigl{]}{\cal
P}_{\pm}(y,y_{5})=0$ $None$
This equation has the form of the equation (4.1a) if
$\int d^{5}y{{\cal J}}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5})=j_{\pm}(x,x_{5})-\Bigl{(}m^{2}_{\pm}-{\widetilde{m}}^{2}\Bigr{)}\varphi_{\pm}(x,x_{5})$
$None$
The sources $j_{\pm}$ in (6.8a,b) are defined in (4.6c) via the projections of
${\cal J}$. The complete 5D source ${\cal J}$ consists of the products of the
fields $\phi$ and their derivatives. Therefore, ${\cal J}$ can not be reduced
to the combination of the fields $\varphi_{\pm}$ that are the parts of
$\phi(x,x_{5})$. Thus in opposite to $j_{\pm}$ constructed from the Lagrangian
(6.5), the projections of ${\cal J}$ in (6.8a,b) does not satisfy the equation
(6.6) and the Lagrangian . ${\ell}(\phi,\phi^{+})$ (6.7) can not be reduced to
the Lagrangians (6.1) $({\cal L}_{\pm}$. The consistency conditions of the
projections of ${\cal J}$ in (6.8a,b) in the equation of motion (4.1a) and the
condition (3.1) can be obtained using the constrains for
$\partial\varphi/\partial x_{5}$ considered in the previous section.
The third form of the 5D Lagrangians (6.1) reproduce exactly the constrains
(3.1) and (5.8a,b) with the a priory given source $j_{\pm}$
${\cal L}(x,x_{5})={{\partial{\varphi_{+}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\varphi_{+}}}\over{\partial
x^{\mu}}}+M^{2}{\varphi_{-}}^{+}{\varphi_{+}}+({{\cal
L}}_{+})^{C}+{{\partial{\varphi_{-}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\varphi_{-}}}\over{\partial
x^{\mu}}}+M^{2}{\varphi_{+}}^{+}{\varphi_{-}}+({{\cal L}}_{-})^{C}.$ $None$
where the Lagrangians $({{\cal L}}_{\pm})^{C}$ generate the conditions
(5.8a,b) and variation over ${\varphi_{\pm}}^{+}$ and
${{\partial{\varphi_{\pm}}^{+}}/{\partial x_{\mu}}}$ produces (3.1). The
combination of (3.1) and (5.8a.b) gives the equation of motion (4.1a).
7\. The 4D and 5D equation of motion for the fermion fields with the
electromagnetic interactions.
The 4D and 5D equation of motion for the two fermion fields
$\Psi_{+}(x)=\psi_{+}(x,x_{5}=0)$ and $\Psi_{-}(x)=\psi_{-}(x,x_{5}=0)$ with
the same quantum numbers but with the different masses $m_{\pm}$ and source
operators can be represented in analogy with the scalar fields. In order to
place the Fourier conjugate of the 5D Dirac fields
$\psi_{1,2}(x,x_{5})=1/2\Bigl{(}\psi_{\pm}(x,x_{5})\pm\psi_{\pm}(x,x_{5})\Bigr{)}$
on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ the fields
$\psi_{+}(x,x_{5})$ and $\psi_{-}(x,x_{5})$ must satisfy the condition (3.1)
${{\partial^{2}\psi_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}\psi_{\mp}(x,x_{5})=0.$ $None$
Similarly with $\varphi_{\pm}$ in (I.10), we divide the Fourier conjugate of
$\psi_{\pm}(x,x_{5})$ into four parts defined in the domains I,II,III and IV
in Table 1
$\psi_{I}(q,q_{5}^{2})=\theta(q^{2})\theta(M^{2}-q^{2})\theta(M^{2}-q_{5}^{2})\Upsilon(q,q_{5}^{2});\
\ \
\psi_{II}(q,q_{5}^{2})=\theta(q^{2})\theta(-M^{2}+q^{2})\theta(q_{5}^{2}-2M^{2})\Upsilon(q,q^{2}_{5});$
$\psi_{III}(q,q_{5}^{2})=\theta(-q^{2})\theta(-M^{2}-q^{2})\theta(q_{5}^{2}-2M^{2})\Upsilon(q,q^{2}_{5});\
\ \
\psi_{IV}(q,q_{5}^{2})=\theta(-q^{2})\theta(M^{2}+q^{2})\theta(M^{2}-q_{5}^{2})\Upsilon(q,q_{5}^{2}),$
$None$
Then as in (3.2) for $\varphi_{\pm}(x,x_{5})$, for $\psi_{\pm}(x,x_{5})$ we
have
$\psi_{\pm}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\psi_{N}(q,Q_{1}^{2})e^{-iQ_{1}x_{5}}\pm\sum_{N=II,IV}\psi_{N}(q,Q_{2}^{2})e^{-iQ_{2}x_{5}}\biggr{]},$
$None$
and $\psi_{\pm}(x,x_{5})$ satisfy automatically the condition (7.1).
For $x_{5}=0$ $\psi_{\pm}$ (7.3) generate the physical 4D fields
$\Psi_{\pm}(x)$
$\Psi_{\pm}(x)=\psi_{\pm}(x,x_{5}=0)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\Psi_{N}(q)\pm\sum_{N=II,IV}\Psi_{N}(q)\biggr{]},$
$None$
where
$\Psi_{I}(q)=\psi_{I}(q,q^{2}_{5}=M^{2}-q^{2});\ \ \
\Psi_{II}(q)=\psi_{II}(q,q^{2}_{5}=M^{2}+q^{2});$
$\Psi_{III}(q)=\psi_{III}(q,q^{2}_{5}=M^{2}-q^{2});\ \ \
\Psi_{IV}(q)=\psi_{IV}(q,q^{2}_{5}=M^{2}+q^{2})$ $None$
and
$\Psi_{\pm}(q)=\sum_{N=I,III}\Psi_{N}(q)\pm\sum_{N=II,IV}\Psi_{N}(q).$ $None$
According to (7.4) and (7.5a,b) the Fourier conjugate of the 5D fields
$\psi_{\pm}(x,x_{5})$ (7.3) contains the same four parts as the Fourier
conjugate of the 4D fields $\Psi_{\pm}(x)$. Therefore, one can represent the
$\psi_{\pm}(x,x_{5})$ through $\Psi(x)$ as
$\psi_{\pm}(x,x_{5})=\int d^{4}y\Psi_{\pm}(x-y){\cal P}_{+}(y,x_{5}).$ $None$
The convolution formula (7.6) is similar to (3.10) for the scalar fields
$\varphi_{\pm}(x,x_{5})$.
The equation of motion for the 5D fields $\psi_{\pm}$ can be derived using the
gauge transformation of the 5D Dirac equations for the two non-interacting
fields $\psi_{\pm}^{(o)}$ with the different masses $m_{\pm}$
$\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{\pm}\biggr{)}\psi_{\pm}^{(o)}(x,x_{5})=0,$ $None$
where
$\psi_{\pm}^{(o)}(x,x_{5})=\int
d^{4}y\exp{\Bigl{(}ie\Lambda(x-y,x_{5})\Bigr{)}}\psi_{\pm}(x-y,x_{5}){\cal
P}_{+}(y,x_{5});$ $None$
$ie{A}^{\mu}(x,x_{5})=\exp{\Bigl{(}-ie\Lambda(x,x_{5})\Bigr{)}}{{\partial}\over{\partial
x_{\mu}}}\exp{\Bigl{(}ie\Lambda(x,x_{5})\Bigr{)}}.$ $None$
Substituting (7.8a,b) into (7.7) we obtain
$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-e\gamma_{\mu}{A}^{\mu}(x-y,x_{5})-m_{\pm}\biggr{)}\psi_{\pm}(x-y,x_{5}){\cal
P}_{+}(y,x_{5})=0.$ $None$
According to (3.8) the projection operator ${\cal P}_{+}(y,x_{5})$ for
$x_{5}=0$ transforms into $\delta^{4}(y)$. Therefore, for $x_{5}=0$ the 5D
non-local equations (7.9) generate the usual 4D Dirac equation for the fermion
field with the electromagnetic interaction
$\biggl{(}i\gamma_{\mu}{{\partial}\over{\partial x_{\mu}}}-e\gamma_{\mu}{\sf
A}^{\mu}(x)-m_{\pm}\biggr{)}\Psi_{\pm}(x)=0.$ $None$
where ${\sf A}^{\mu}(x)={A}^{\mu}(x,x_{5}=0)$,
$\Psi_{\pm}^{(o)}(x)=\psi_{\pm}^{(o)}(x,x_{5}=0)$ are the asymptotic
${}^{\prime\prime}in^{\prime\prime}$ or ${}^{\prime\prime}out^{\prime\prime}$
fields which satisfy the equations
$\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{\pm}\biggr{)}\Psi_{\pm}^{(o)}(x)=0;$ $None$
$\Psi_{\pm}^{(o)}(x)=\exp{\Bigl{(}ie\Lambda(x,0)\Bigr{)}}\Psi_{\pm}(x);\ \ \ \
\ \ ie{\sf
A}^{\mu}(x)=\exp{\Bigl{(}-ie\Lambda(x,0)\Bigr{)}}{{\partial}\over{\partial
x_{\mu}}}\exp{\Bigl{(}ie\Lambda(x,0)\Bigr{)}}.$ $None$
The Fourier conjugate of the 5D gauge fields ${A}^{\mu}(x,x_{5})$ are embedded
on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ and they satisfy the
condition (3.1) As well as the scalar fields $\varphi_{\pm}$. Due to gauge
invariance one can use any combination of ${A}^{\mu}_{+}$ and ${A}^{\mu}_{-}$
in the gauge transformations (7.8b). Therefore, in (7.8a,b)-(7.11a,b) and in
the following formulas the lower indexes ${\pm}$ of $\Lambda$, ${\sf A}^{\mu}$
and ${A}^{\mu}$ are omitted.
The 5D representations of the interacted and non-interacted 5D fields
$\psi_{\pm}$ and $\psi_{\pm}^{(o)}$ are single-valued determined through the
their 4D reductions $\Psi_{\pm}$ and $\Psi_{\pm}^{(o)}$ according to (7.6).
Therefore $\psi_{\pm}^{(o)}$ satisfies also the condition (7.1)
${{\partial^{2}\psi_{\pm}^{(o)}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}\psi_{\mp}^{(o)}(x,x_{5})=0.$ $None$
The gauge transformation (7.8a) in (7.12) together with (7.1) yields
$\eta_{\pm}(x,x_{5})=-\eta_{\mp}^{(5)}(x,x_{5}),$ $None$
where
$\eta_{\pm}(x,x_{5})=\biggl{(}ie{{\partial}\over{\partial
x^{\mu}}}{A}^{\mu}(x,x_{5})+ie{A}^{\mu}(x,x_{5}){{\partial}\over{\partial
x^{\mu}}}-e^{2}{A}^{\mu}(x,x_{5}){A}_{\mu}(x,x_{5})\biggr{)}\psi_{\pm}(x,x_{5}),$
$None$
and the auxiliary source operator $\eta_{\pm}^{(5)}$ is defined via ${A}^{5}$
as
$\eta_{\pm}^{(5)}(x,x_{5})=\biggl{(}ie{{\partial}\over{\partial
x^{5}}}{A}^{5}(x,x_{5})+ie{A}^{5}(x,x_{5}){{\partial}\over{\partial
x^{5}}}-e^{2}{A}^{5}(x,x_{5}){A}_{5}(x,x_{5})\biggr{)}\psi_{\pm}(x,x_{5}),$
$None$
$ie{A}^{5}(x,x_{5})=\exp{\Bigl{(}-ie\Lambda(x,x_{5})\Bigr{)}}{{\partial}\over{\partial
x_{5}}}\exp{\Bigl{(}ie\Lambda(x,x_{5})\Bigr{)}}.$ $None$
Condition (7.13) allows to define ${A}^{5}$ through $\eta_{\pm}$ and
$\psi_{\pm}$.
The complete 5D fields $\Upsilon(x,x_{5})$ and their Fourier conjugate
$\Upsilon(q,q_{5}^{2})$ in (7.2) can be defined in the relativistic invariant
time models [8, 9]. The projections of $\Upsilon(x,x_{5})$ and
$\Upsilon^{(o)}(x,x_{5})$ onto hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ can
be determined also via the 5D convolution formulas
$\psi_{\pm}(x,x_{5})=\int d^{5}y{\Upsilon}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5});\ \ \ \psi_{\pm}^{(o)}(x,x_{5})=\int
d^{5}y{\Upsilon}^{(o)}(x-y,x_{5}-y_{5}){\cal P}_{\pm}(y,y_{5}),$ $None$
8\. Constrains for $\partial\psi_{\pm}/\partial x_{5}$.
The consistency condition between the 5D Dirac equations (7.9) and the
constrains (7.1) can be represent through the linear equations for
$\partial\psi_{\pm}/\partial x_{5}$. For this aim it is convenient to rewrite
the (7.9) in the form of the Klein-Gordon equations. Action of the operator
$i\gamma_{\mu}{{\partial}/{\partial x_{\mu}}}+m_{\pm}$ on (7.7) produces the
following Klein-Gordon equations
$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+m_{\pm}^{2}\biggr{)}\psi_{\pm}^{(o)}(x,x_{5})=0.$ $None$
Then the gauge transformation (7.8a) yields
$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}\Bigl{(}{{\partial^{2}}\over{\partial
x_{\mu}\partial
x^{\mu}}}-m_{\pm}^{2}\Bigr{)}\psi_{\pm}(x-y,x_{5})-\eta_{\pm}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0,$ $None$
where $\eta_{\pm}(x,x_{5})$ is defined in (7.14a).
One can replace (7.1) with
$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}{{\partial^{2}\psi_{\pm}(x-y,x_{5})}\over{\partial
x_{\mu}\partial x^{\mu}}}+\Bigl{(}M^{2}+{{\partial^{2}}\over{\partial
x_{5}\partial x^{5}}}\Bigr{)}\psi_{\mp}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0.$ $None$
Afterwards we obtain the following consistency condition of (8.3) and (8.2)
for the 5D Dirac fields
$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\Biggl{\\{}\biggl{[}M^{2}+\Bigl{(}{{\partial}\over{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}\psi_{\pm}(x-y,x_{5})-m_{\mp}^{2}\psi_{\mp}(x-y,x_{5})+\eta_{\mp}(x-y,x_{5})\Biggr{\\}}{\cal
P}_{+}(y,x_{5})=0.$ $None$
According to (8.4) $\Bigr{[}M^{2}+\bigl{(}\partial/{\partial
x_{5}}\bigr{)}^{2}\Bigr{]}\psi_{+}$ is determined by the source and mass of
the $\psi_{-}$ fields in (8.2) and vice versa. Unlike (4.10) the condition
(8.4) is given in the form of the 4D convolution formula.
One can factorise (8.4) in the same way as (4.10) using the constrain for
$\partial\ \psi_{\pm}/\partial x_{5}$. In particular,
${i\over M}{{\partial\over{\partial
x_{5}}}\psi_{\pm}}=\alpha_{\pm}\psi_{+}+\beta_{\pm}\psi_{-}+C_{\pm},$ $None$
where $C_{\pm}$ are defined onto hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$
according to (5.2) and $C_{\pm}$ does not contain the linear terms over
$\psi_{\pm}$.
The constrains (8.5) allow to construct the relationship between the masses
$m_{\pm}^{2}$, sources $j_{\pm}$ and the parameters $M^{2}$, $\alpha_{+}$,
$\beta_{+}$ and $C_{\pm}$ in the same way as in (5.3a,b). In particular, the
choice of the parameters according to (5.4) gives
${{m_{+}^{2}}\over{m_{-}^{2}}}={{1-\alpha_{+}^{2}}\over{\beta_{+}^{2}}};\ \ \
\ \ {{m_{+}^{2}m_{-}^{2}}\over{4M^{2}}}=\alpha_{+}^{2}(1-\alpha_{+}^{2});\ \ \
\ \ \alpha_{+}^{2}={1\over 2}\pm{1\over
2}\sqrt{1-{{m_{+}^{2}m_{-}^{2}}\over{M^{4}}}}$ $None$
The ratio $m_{+}^{2}/m_{-}^{2}$ and the product $m_{+}^{2}m_{+}^{2}/4M^{4}$ in
(8.6) are positive if
$0<\alpha_{+}^{2}<1;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{{m_{+}^{2}m_{+}^{2}}\over{M^{4}}}\leq 1.$ $None$
Therefore, the constrain (8.5) generate the positive masses $m_{+}^{2}$ and
$m_{-}^{2}$ simultaneously if $M^{2}\geq m_{+}m_{-}$.
9\. 5D extension of the standard $SU(2)\times U(1)$ model for the electron and
muon.
As example of unification of the 4D fields with the same quantum numbers and
different masses and sources, we shall consider the relationship between the
4D electron and muon fields $\Psi_{+}\equiv\Psi_{el}$ and
$\Psi_{-}\equiv\Psi_{muon}$, $\Psi_{+}\equiv\Psi_{el}$ and
$\Psi_{-}\equiv\Psi_{muon}$ according to (7.3)-(7.6). In the standard
Weinberg-Salam $SU(2)\times U(1)$ model [35] we have the following 4D
equations of motion
$\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{el}\Bigr{)}\Psi_{el}={\cal J}_{el};\ \ \ \ \ \ \ \ \ \ \ \ \
\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{muon}\Bigr{)}\Psi_{muon}={\cal J}_{muon};$ $None$ ${\cal
J}_{el}=\Bigl{(}-e\gamma_{\mu}{\sf
A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{\sf
Z}^{\mu}{{1+\gamma_{5}}\over 2}+g^{\prime}\gamma_{\mu}{\sf
Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\Psi_{el}+{g\over{\sqrt{2}}}\gamma_{\mu}{\sf
W}^{\mu}{{1+\gamma_{5}}\over 2}\nu_{el},$ $None$ ${\cal
J}_{muon}=\Bigl{(}-e\gamma_{\mu}{\sf
A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{\sf
Z}^{\mu}{{1+\gamma_{5}}\over 2}+g^{\prime}\gamma_{\mu}{\sf
Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\Psi_{muon}+{g\over{\sqrt{2}}}\gamma_{\mu}{\sf
W}^{\mu}{{1+\gamma_{5}}\over 2}\nu_{muon},$ $None$
where $\nu_{el}$ and $\nu_{muon}$ denote the corresponding neutrino fields,
${\sf W}^{\mu}$ and ${\sf Z}^{\mu}$ stands for the charged and neutral vector
fields, ${\sf A}^{\mu}$ is the photon field
$g=-{e\over{sin\theta}};\ \ \ \ \ g^{\prime}=-{e\over{cos\theta}};\ \ \ \ \
sin^{2}\theta=0.222\pm 0.011$ $None$
For derivation of (9.1a,b,c) was used the 4D gauge transformation of the
neutrino-electron and neutrino-muon doublets according to (21.3.12)-(21.3.20)
in [35].
$\sum_{\alpha}{\cal A}^{\mu}_{\alpha}(x){\em
t}_{\alpha}+y{B}^{\mu}=\exp{\Bigl{(}\Lambda(x)\Bigr{)}}{{\partial}\over{\partial
x_{\mu}}}\exp{\Bigl{(}\Lambda(x)\Bigr{)}}.$ $None$
${\sf W}^{\mu}={1\over{\sqrt{2}}}\Bigl{(}{\cal A}_{1}^{\mu}+i{\cal
A}_{2}^{\mu}\Bigr{)};\ \ \ {\sf Z}^{\mu}=cos\theta\ {\cal
A}_{3}^{\mu}+sin\theta\ {B}^{\mu};\ \ \ {\sf A}^{\mu}=-sin\theta\ {\cal
A}_{3}^{\mu}+cos\theta\ {B}^{\mu};$ $None$
The direct 5D extension of the 4D Dirac equations (9.1a) can be obtained in
the same way as (7.9) using the 4D gauge transformations (7.8a,b) with the
common parameter $x_{5}$
$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{el}\Bigr{)}\psi_{el}(x-y,x_{5})-{\widetilde{\eta}}_{el}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0.$ $None$
$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{muon}\Bigr{)}\psi_{muon}(x-y,x_{5})-{\widetilde{\eta}}_{muon}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0,$ $None$
where $\Lambda(x,x_{5}=0)=\Lambda(x)$ For $x_{5}=0$ the 5D fields and sources
in (9.3a,b) are reduced into the 4D fields and sources
$\Psi_{el,}(x)=\psi_{el}(x,x_{5}=0);\ \ \ \ \ \ \ \ \ \
\Psi_{muon}(x)=\psi_{muon}(x,x_{5}=0);$ $None$
${\widetilde{\eta}}_{el}(x,x_{5}=0)={\cal J}_{el}(x);\ \ \ \ \ \ \ \ \ \
{\widetilde{\eta}}_{muon}(x,x_{5}=0)={\cal J}_{muon}(x),$ $None$
where
${\widetilde{\eta}}_{el}(x,x_{5})=\Bigl{(}-e\gamma_{\mu}{A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{Z}^{\mu}{{1+\gamma_{5}}\over
2}+g^{\prime}\gamma_{\mu}{Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\psi_{el}+{g\over{\sqrt{2}}}\gamma_{\mu}{W}^{\mu}{{1+\gamma_{5}}\over
2}\nu_{el},$ $None$
${\widetilde{\eta}}_{muon}(x,x_{5})=\Bigl{(}-e\gamma_{\mu}{A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{Z}^{\mu}{{1+\gamma_{5}}\over
2}+g^{\prime}\gamma_{\mu}{Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\psi_{muon}+{g\over{\sqrt{2}}}\gamma_{\mu}{W}^{\mu}{{1+\gamma_{5}}\over
2}\nu_{muon},$ $None$
It is easy to check, that for $x_{5}=0$ equation (9.3a,b) transforms into
(9.1a) due to ${\cal P}_{+}(x,x_{5}=0)=\delta(x)$.
In order to place the Fourier conjugate of the 5D sources (9.4a,b) onto
hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ we shall use the following
convolution formula
$\eta_{el}(x,x_{5})=\int d^{4}y{\widetilde{\eta}}_{el}(x-y,x_{5}){\cal
P}_{+}(y,x_{5});\ \ \ \eta_{muon}(x,x_{5})=\int
d^{4}y{\widetilde{\eta}}_{muon}(x-y,x_{5}){\cal P}_{+}(y,x_{5}).$ $None$
The exact form of ${\cal P}_{+}$ (3.6a,b) allows to rewrite (9.5a) as
$\eta_{el}(x,x_{5})=\int d^{4}y{\cal J}_{el}(x-y){\cal P}_{+}(y,x_{5});\ \ \
\eta_{muon}(x,x_{5})=\int d^{4}y{\cal J}_{muon}(x-y){\cal P}_{+}(y,x_{5}).$
$None$
$\psi_{el}$ and $\psi_{muon}$ satisfy the condition (7.1). Therefore, there
are two different ways for construction of the 5D fields $\psi_{el}$ and
$\psi_{muon}$
1. If the electron and muon fields are completely independent, then the constrain (7.1) can be reproduced through doubling of the electron and muon fields separately, i.e. through $(\psi_{el})_{\pm}$ and $(\psi_{muon})_{\pm}$, where $(\psi_{el})_{-}$ and $(\psi_{muon})_{-}$ corresponds to the “electron” and “muon“ with the negative or imaginary mass. This doubling of the electron and muon states can be realized via the appropriate constrains (8.5) and the equations of motion (7.9).
2. If the 5D fields $\psi_{el}$ and $\psi_{muon}$ consists of the same parts, then in (7.1) and (7.9) $\psi_{el}\equiv\psi_{+}$ and $\psi_{muon}\equiv\psi_{-}$ and for their 4D reductions we have
$\Psi_{el}(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\Psi_{N}(q)+\sum_{N=II,IV}\Psi_{N}(q)\biggr{]},$
$None$
$\Psi_{muon}(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\Psi_{N}(q)-\sum_{N=II,IV}\Psi_{N}(q)\biggr{]}.$
$None$
The equations (9.6a,b) unify the 4D electron and muon Heisenberg fields which
satisfies the 4D equation of motion (9.1a). Despite of the mixing of the
$\Psi_{el}(x)$ and $\Psi_{muon}(x)$ in (9.6a,b) the 4D equations of motion
(9.1a) are the same as in the standard model [35], where the electron-muon
coupling is strongly suppressed. Therefore, the perturbation series
constructed in the framework of the Weinberg-Salam $SU(2)\times U(1)$ theory
and the perturbation series based on the equation of motion (9.1a,b,c) with
the mixed fields (9.6a,b) coincides.
The common structure of the interacted Heisenberg fields $\Psi_{el}(x)$ and
$\Psi_{muon}(x)$ in (9.6a,b) is important for the theories beyond Weinberg-
Salam model. In particular, within the non-perturbative formulation the
functional integrals with the 4D electron and muon fields $\int{\cal
D}(\Psi_{el}){\cal D}({\overline{\Psi}}_{el})\Bigl{[}...\Bigr{]}$ and
$\int{\cal D}(\Psi_{muon}){\cal
D}({\overline{\Psi}}_{muon})\Bigl{[}...\Bigr{]}$ are strongly correlated due
to (9.6a,b).
Doubling of the electron and muon fields (9.6a,b) can be examined within the
5D invariant time theories [8, 9] which allow to construct the independent 5D
fields $\Upsilon_{el}(x,x_{5})$ and $\Upsilon_{muon}(x,x_{5})$. If the parts
of the $\Upsilon_{el}(q,q^{2}_{5})$ and $\Upsilon_{muon}(q,q_{5}^{5})$ are not
the same as in (9.6a,b), then they are dubled and according to the present
approach appear the fields $(\Upsilon_{el})_{-}(x,x_{5})$ and
$(\Upsilon_{muon})_{-}(x,x_{5})$ with the negative or imaginary mass.
Nowadays unification of the different lepton families is performed within the
5D grand unification models [5]. In these models the fifth dimension is the
indivisible part of the particle interaction, the space-time is not
asymptotically flat, in the extra dimensions enter other than the gravitation
fields and is argued the breakdown of the gauge coupling unification. The
present formulation can be used for the 4D projections’ of the corresponding
5D electron and muon fields $\Upsilon_{el}$ and $\Upsilon_{muon}$. For this
aim we put the Fourier conjugate of the complete 5D fields $\Upsilon_{el}$ and
$\Upsilon_{muon}$ on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ using the
replacement of the integrals $\int d^{4}qdq^{2}_{5}\\{...\\}$ with $\int
d^{4}qdq^{2}_{5}\Bigl{[}\delta(q^{2}+q_{5}^{2}-M^{2})\\{...\\}\pm\delta(q^{2}-q_{5}^{2}+M^{2})\\{...\\}\Bigr{]}$.
Then in analogy with (I.4a,b) and (I.7) we get the 5D fields
$(\psi_{el})_{\pm}(x,x_{5})$ and $(\psi_{muon})_{\pm}(x,x_{5})$. In the low
energy region, where the asymptotically flat space is assumed,
$(\psi_{el})_{\pm}(x,x_{5}=0)$ and $(\psi_{muon})_{\pm}(x,x_{5}=0)$ determine
the 4D fields $(\Psi_{el})_{\pm}(x)$ and $(\Psi_{muon})_{\pm}(x)$. The gauge
invariant 4D fields $(\Psi_{el})_{+}(x)$ and $(\Psi_{el})_{-}(x)$ consist from
the same four parts as well as $(\Psi_{muon})_{+}(x)$ and
$(\Psi_{muon})_{-}(x)$. Thus in addition to $\Psi_{el}(x)$ $\Psi_{muon}(x)$ we
get two other fields $(\Psi_{el})_{-}(x)$ and $(\Psi_{muon})_{-}(x)$ with the
negative or imaginary mass.
10\. Gauge transformation as generalized translation
The generalized translation of the four-momentum
${q^{\mu}}^{\prime}=q^{\mu}-eA^{\mu}(q)$ in the equation of motion can be
performed through the gauge transformations. In particular, the 4D and 5D
gauge transformations (7.11a,b) and (7.8a,b) generates the corresponding
translations of the four momentum $q^{\mu}$ and $i\partial/\partial x^{\mu}$
${q^{\mu}}^{\prime}=q^{\mu}-e{A}^{\mu}_{\pm}(q,q_{5}^{2})\Longleftrightarrow
i{{\partial}\over{\partial{x^{\prime}}^{\mu}}}=i{{\partial}\over{\partial{x}^{\mu}}}-e{A}^{\mu}_{\pm}(x,x_{5})$
$None$
${q^{5}}^{\prime}=q^{5}-e{A}^{5}_{\pm}(q,q_{5}^{2})\Longleftrightarrow
i{{\partial}\over{\partial{x^{\prime}}^{5}}}=i{{\partial}\over{\partial{x}^{5}}}-e{A}^{5}_{\pm}(x,x_{5})$
$None$
with the 5D fields ${A}^{\mu}_{\pm}$, ${A}^{5}_{\pm}$, $\psi_{\pm}$ and
$\psi_{\pm}^{\prime}$.
One can construct ${A}^{\mu}_{\pm}$ and ${A}^{5}_{\pm}$ starting from the 6D
gauge translations
${\kappa_{C}}^{\prime}={\kappa_{C}}-ea_{C}(\kappa)\Longleftrightarrow
i{{\partial}\over{\partial{\xi^{\prime}}^{C}}}=i{{\partial}\over{\partial{\xi}^{C}}}-ea_{C}(\xi)$
$None$
$\psi^{\prime}(\xi)=\exp{\Bigl{(}ie\lambda(\xi)\Bigr{)}}\psi(\xi);\ \ \ \ \ \
\ \ \ \ \ \ \ \ \
a_{C}(\xi)=\exp{\Bigl{(}-ie\lambda(\xi)\Bigr{)}}{{\partial}\over{\partial{\xi}^{C}}}\exp{\Bigl{(}ie\lambda(\xi)\Bigr{)}},$
$None$
where $C\equiv\mu;5,6=0,1,2,3;5,6$ and the Fourier conjugate of
$\psi^{\prime}(\xi)$, $\psi(\xi)$, $\lambda(\xi)$ and $a_{C}(\xi)$ are not
placed on the cone $\kappa_{C}\kappa^{C}=0$.
According to invariance of the 6D cone $\kappa_{C}{\kappa}^{C}=0$ we have
$\kappa_{C}{\kappa}^{C}=\kappa^{\prime}_{C}{\kappa^{\prime}}^{C}=\Bigl{(}\kappa_{C}-ea_{C}(\kappa)\Bigr{)}\Bigl{(}\kappa^{C}-ea^{C}(\kappa)\Bigr{)}=0,$
$None$
where $\kappa_{+}$ is invariant under the translations as it is indicated in
(1.2a). The invariance of $\kappa_{+}$ under the 4D translations (10.2a)
requires that $a_{5}(k)=-a_{6}(k)$. Substituting this condition in (10.3) we
get
$a_{5}(k)={1\over{2M\kappa_{+}}}\Bigl{(}-a_{\nu}(\kappa)\kappa^{\nu}-\kappa^{\nu}a_{\nu}(\kappa)+ea_{\nu}(\kappa)a^{\nu}(\kappa)\Bigr{)};\
\ \ \ \ \ \ \ \
{{\kappa_{-}^{\prime}}\over{\kappa_{+}}}={{\kappa_{-}}\over{\kappa_{+}}}-e{{a_{5}(k)}\over{\kappa_{+}}}.$
$None$
In the considered formulation the 4D reduction of the 6D fields
$a_{\mu}(\kappa)$ and $\psi(\kappa)$ based on the intermediate projections of
the 6D cone $\kappa_{C}{\kappa}^{C}=0$ into 5D hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$ (I.3a,b) for the domains I, III and II, IV of the Table
1. The corresponding 5D reductions of the 6D fields $a_{\mu}(\kappa)$ and
$\psi(\kappa)$ are
$\Upsilon(q,q_{5}^{2})={{M^{2}}\over 2}\int
k_{+}^{3}d\kappa_{+}\theta(\kappa_{+})\psi(q,q_{5}^{2},\kappa_{+}),$ $None$
where $\Upsilon(q,q_{5}^{2})$ is Fourier conjugate of $\Upsilon(x,x_{5})$ in
(7.2) and
${\widetilde{A}}^{\mu}(q,q_{5}^{2})={{M^{2}}\over 2}\int
k_{+}^{3}d\kappa_{+}\theta(\kappa_{+})a^{\mu}(q,q_{5}^{2},\kappa_{+});\ \ \ \
\ \ {\widetilde{A}}^{5}(q,q_{5}^{2})={{M^{2}}\over 2}\int
k_{+}^{3}d\kappa_{+}\theta(\kappa_{+})a^{5}(q,q_{5}^{2},\kappa_{+}).$ $None$
The 5D gauge fields ${A}^{\mu}_{\pm}(x,x_{5})$ which Fourier conjugate are
placed on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ are constructed
through ${\widetilde{A}}^{\mu}$
${A}^{\mu}_{\pm}(x,x_{5})=\int
d^{5}y{\widetilde{A}}^{\mu}(x-y,x_{5}-y_{5}){\cal P}_{\pm}(y,y_{5}),$ $None$
${A}^{5}_{\pm}(x,x_{5})=\int d^{5}y{\widetilde{A}}^{5}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5}),$ $None$
The relationship between the fifth gauge field ${A}^{5}_{\pm}$ (10.7b) and
${A}^{\mu}_{\pm}$ (10.7a) is given in (7.13) and (7.14a,b).
The 4D gauge field ${\sf A}^{\mu}_{\pm}(x)={A}^{\mu}_{\pm}(x,0)$ are
determined through ${A}^{\mu}_{N}(q,q^{2}_{5})$ in the domains $N=I.II,III,IV$
as
${\sf
A}^{\mu}_{\pm}(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}\exp{(-iqx)}\Bigl{[}{\sf
A}^{\mu}_{I}(q)\pm{\sf A}^{\mu}_{II}(q)+{\sf A}^{\mu}_{III}(q)\pm{\sf
A}^{\mu}_{IV}(q)\Bigr{]}.$ $None$
The convolution formula (10.7) can be represented in the 4D form
${A}^{\mu}_{\pm}(x,x_{5})=\int d^{4}y{\sf A}^{\mu}_{\pm}(x-y){\cal
P}_{+}(y,x_{5}).$ $None$
This representation of ${A}^{\mu}_{\pm}(x,x_{5})$ indicates that the 5D and 4D
fields ${A}^{\mu}_{\pm}(x,x_{5})$ and ${\sf A}^{\mu}_{\pm}(x)$ satisfy the
same 4D equations of motion. On the other hand ${A}^{\mu}_{\pm}(x,x_{5})$
consists of the parts placed on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm
M^{2}$ and satisfy the similar to (3.1) 5D conditions
${{\partial^{2}{A}^{\mu}_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}{A}^{\mu}_{\mp}(x,x_{5})=0,$ $None$
The consistency condition of (10.10) and the corresponding equation of motion
have the same form as (4.10) for the scalar field. Thus the present 5D
formulation allows to construct simultaneously the 4D fields ${\sf
A}^{\mu}_{+}(x)$ and ${\sf A}^{\mu}_{-}(x)$ which consists from the same parts
${\sf A}^{\mu}_{I}$, ${\sf A}^{\mu}_{II}$, ${\sf A}^{\mu}_{III}$ and ${\sf
A}^{\mu}_{IV}$. ${\sf A}^{\mu}_{+}(x)$ and ${\sf A}^{\mu}_{-}(x)$ have the
same quantum numbers, but the different sources. For the photon field ${\sf
A}^{\mu}_{+}(x)$ the role of ${\sf A}^{\mu}_{-}(x)$ can play the Z-boson.
It must be noted that the gauge transformations can be performed also for the
neutral (uncharged) particles. Within the nonlinear $\sigma$ model [16, 36]
for the triplet of the neutral auxiliary pion fields $\pi^{\alpha}$
($\alpha=1,2,3$, $\pi^{\pm}=1/2(\pi^{1}\pm i\pi^{2})$; $\pi^{0}\equiv\pi^{3}$)
is replaced with the interpolating pion field
$\pi^{\alpha}(x)={\cal U}(x)\chi^{\alpha}(x),$ $None$
where in [16, 36] ${\cal
U}(x)=\Bigl{(}1+\chi^{2}(x)/4f_{\pi}^{2}\Bigr{)}^{-1}$, $f_{\pi}=93\ MeV$ is
the pion decay constant and
$\chi^{2}=\sum_{\alpha=1}^{3}\chi^{\alpha}\chi^{\alpha}$. The replacement
(10.11) generates the following transformations
${{\partial}\over{\partial x_{\mu}}}\pi^{\alpha}={\cal
U}\Bigl{[}{{\partial}\over{\partial x_{\mu}}}+{\cal
D}^{\mu}\Bigr{]}\chi^{\alpha};\ \ \ {\cal D}^{\mu}={\cal
U}^{-1}{{\partial}\over{\partial x_{\mu}}}{\cal U}$ $None$
which presents the extension of (7.11a,b) for the neutral fields. The
transformations (10.11) and (10.12) generalise also the gauge transformations
(1.11) for the neutral fields.
11\. Summary
The one-to-one relationship between the 4D and 5D fields and their equations
of motion, established by present article, based on the equivalence of the
conformal transformations of the four momentum $q_{\mu}$ (1.1a)-(1.1e) and the
6D rotations on the cone $\kappa_{A}\kappa^{A}=0$ (1.2a)-(1.2d) and its 5D
projections on the two invariant forms $q^{2}\pm q_{5}^{2}=\pm M^{2}$ of the
$O(2,3)$ and $O(1,4)$ subgroups of the conformal group $O(2,4)$. The 6D cone
$\kappa_{A}\kappa^{A}=0$ and its 5D projections $q^{2}\pm q_{5}^{2}=\pm M^{2}$
are invariant under the 6D rotations and the corresponding 4D conformal
transformations. Consequently, the 4D projection of the 6D field
$\varsigma(\kappa)$ with the intermediate projection on the two hyperboloids
$q^{2}\pm q_{5}^{2}=\pm M^{2}$ determine the two 5D fields
$\varphi_{1}(x,x_{5})$ (I.4a) and $\varphi_{2}(x,x_{5})$ (I.4b). The Fourier
conjugate of these 5D fields are placed on the hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$ before and after conformal transformations. Consequently
the 5D fields $\varphi_{+}=\varphi_{1}+\varphi_{2}$ and
$\varphi_{-}=\varphi_{1}-\varphi_{2}$ (I.7) are defined on the whole domain of
$-\infty<q^{2}<\infty$ and reproduce the 4D fields
$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0)$. In addition the fields
$\varphi_{\pm}(x,x_{5})$ satisfy the 5D constrains (3.1) that have the form of
the coupled sourceless 5D equations. The 5D equation of motion for
$\varphi_{\pm}$ (4.1a) and their 4D reductions (4.1b) are embedded into these
sourceless coupled 5D conditions (3.1). The consistency conditions of the 5D
equation of motion (4.1a) and the 5D constrains (3.1) generate the constrains
for $\partial\varphi_{\pm}/\partial x_{5}$. The similar sourceless 5D coupling
condition (7.1) are valid for the interacting fermion fields $\psi_{+}$ and
$\psi_{-}$.
This scheme can be used for the 5D extension of the 4D models and for the 4D
reductions of the 5D formulations. The parts of the 5D field
$\phi(q,q_{5}^{2}=M^{2}-q^{2})$ and $\phi(q,q_{5}^{2}=M^{2}+q^{2})$
unambiguously determine the Fourier conjugate of the 4D fields $\Phi_{+}(x)$
and $\Phi_{-}(x)$ (I.10)-(I.11). The same expressions determine also the
Fourier conjugate of the 5D fields $\varphi_{1}$ (I.4a), $\varphi_{2}$ (I.4b)
and $\varphi_{\pm}$ (I.7). These parts of the single 5D field
$\phi(q,q_{5}^{2})$ determine unambigously the two 4D fields $\Phi_{\pm}(x)$
with the same quantum numbers, but with the different masses and sources. And
vice versa, starting from the 4D field $\Phi_{+}(x)$ one can construct
$\Phi_{-}(x)$ and the parts of the 5D field $\phi(q,q_{5}^{2}=M^{2}-q^{2})$
and $\phi(q,q_{5}^{2}=M^{2}+q^{2})$. This doubling of the 4D fields is result
the intermediate projection of the 6D field placed on the 6D cone
$\kappa_{A}\kappa^{A}=0$ into the two 5D fields $\varphi_{\pm}$ embedded into
two invariant forms $q^{2}\pm q_{5}^{2}=\pm M^{2}$. The considered
intermediate 5D projections take into account the symmetry under the inversion
$q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$ and reflection
${q^{\prime}}^{2}=-q^{2}$ between the domains of these forms. As it is
mentioned in the last two paragraphs of Sect. 2, the stereographic and other
5D projections of the 6D cone $\kappa_{A}\kappa^{A}=0$ can be reproduced
through the considered projections on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm
M^{2}$.
The boundary conditions of the 5D fields $\varphi_{\pm}$ (I.8) and
$\psi_{\pm}$ (7.4) at $x_{5}=0$ can be extended for an arbitrary value
$x_{5}=t_{5}$ if one replaces $x_{5}$ by $x_{5}-t_{5}$ in the definitions of
the 5D fields (I.4a,b) and (i.7). In particular, $t_{5}=\sqrt{x_{0}^{2}-{\bf
x}^{2}}$ in the formulation within the relativistic invariant time theories[8,
9].
The common parts of the 4D interacted fields $\Phi_{+}(q)$ and $\Phi_{-}(q)$
in (I.10)-(I.11) allow to separate the $4!=24$ different fields with the same
quantum numbers and with the different masses and sources. This unification
scheme of the interacting Heisenberg fields can be applied for description of
the nucleons and resonances in the $P11$ states $N(1440)$, $N(1710)$, … [37]
based on an additional dynamical mechanism for the generation of the
resonances. Similarly, one can combine the interacting Heisenberg fields of
the pions and the resonances with the same quantum numbers $\pi(1300)$.
The particle states with the same quantum numbers and the different masses and
sources can be constructed within the various 5D relativistic invariant time
theories [8, 9]. The different 5D fields $\phi_{+}(q,q_{5}^{2})$ and
$\phi_{-}(q,q_{5}^{2})$ in these theories do not have the common parts
$\phi_{+}(q,q_{5}^{2}=M^{2}-q^{2})$ and $\phi_{-}(q,q_{5}^{2}=M^{2}+q^{2})$.
Nevertheless the present formulation requires doubling of the 5D fields, i.e.
instead of the $\phi_{+}(x,x_{5})$ and $\phi_{-}(x,x_{5})$ we get
$(\phi_{+})_{\pm}(x,x_{5})$ and $(\phi_{-})_{\pm}(x,x_{5})$, where
$(\phi_{+})_{-}(x,x_{5})$ and $(\phi_{-})_{-}(x,x_{5})$ must have the negative
or imaginary masses. The relationships between the different masses and
sources for the 5D fields $(\phi_{+})_{\pm}(x,x_{5})$ and
$(\phi_{-})_{\pm}(x,x_{5})$ are determined by constrains for the
$\partial(\phi_{+})_{\pm}/partialx_{5}$ and
$\partial(\phi_{-})_{\pm}/partialx_{5}$. Thus the present approach allows to
get the 4D projections of the 5D fields from [8, 9] for $x_{5}=0$.
It must be noted, that if the 4D fields $\Phi_{\pm}(x)$ are determined through
the parts of the 5D fields $\phi_{+}(q,q^{2}_{5}=M^{2}\mp q^{2})$ and
$\phi_{-}(q,q^{2}_{5}=M^{2}\mp q^{2})$, then the Fourier conjugate of the
observed 4D fields $\Phi_{\pm}(q)$ can have the jump singularities at
$q^{2}=0,\pm M^{2}$.
Presently the different lepton families including the electron and muon fields
$\Upsilon_{el}$ and $\Upsilon_{muon}$ are constructed within the 5D grand
unification theories [5, 6]. In this formulation the 5D electron and muon
fields $\Upsilon_{el}(q,q_{5}^{2}=M^{2}\mp q^{2})$ and
$\Upsilon_{muon}(q,q_{5}^{2}=M^{2}\mp q^{2})$ are independent. The present
scheme of the 4D reduction of the 5D fields allows to get the 4D projections
of these 5D fields with the doubling of the 4D electron and muon fields
$(\Psi_{el})_{\pm}(x)$ and $(\Psi_{muon})_{\pm}(x)$, where $(\Psi_{el})_{-}$
and $(\Psi_{muon})_{-}$, have the negative or imanary masses. These 4D
projections of the 5D fields restore the 4D gauge invariance, because the
gauge transformations are considered as the generalized 4D translation in the
momentum space (see Sect. 10) and translations of the four momentum preserve
invariance of the 6D cone $\kappa_{A}\kappa^{A}=0$ and their projections on
the 5D invariant forms.
Other kind of the unification of the 4D electron and muon fields in this
approach is considered in the Section 9 for the Standard Model [35], where
$\Phi_{el}(q)$ and $\Phi_{muon}(q)$ consist of the same parts (9.6a,b) of the
complete 5D field $\Upsilon(q,q^{2}_{5}=M^{2}\mp q^{2})$. The coupling between
the electron and muon fields are strongly suppressed in the Standard Model.
Therefore, the common structure of the electron and muon fields can not be
observed in the perturbation series of the corresponding 4D equations.
I thank V.G.Kadyshevsky for numerous helpful discussions. I am thankful to A.
Machavariani (junior) and G. Münster for the current interest in this work.
## References
* [1] T. Appelquist, A. Chodos and P.G.O. Freund. Modern Kaluza-Klein Theories, Addison-Wesley, Monte-Park, 1987.
* [2] D. Bailin and A. Love.Rep. Prog. Phys. 50 (1987) 1087.
* [3] P. S. Wesson. Space- Time - Matter; Modern Kaluza-Klein Theory, World Scientific, 2000.
* [4] F. Cianfrani, A. Marrocco and G. Montani. Int. Jour. Modern Phys. D14 (2005) 1195.
* [5] T. Fukuyama, Int. Jour. Mod. Phys. A28 (2013) 1330008.
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* [7] C. Itzykson and J. B. Zuber. Quantum Field Theory; New York, McGrew-Hill, 1980.
* [8] J. L. Fanchi, Found. Phys.22(1993)487.
* [9] M. C. Land, Found. Phys.27(1997) 19.
* [10] P.A.M. Dirac, Ann. Math. 37 (1936) 429.
* [11] H. A. Kastrup, Phys. Rev.150(1966) 1183;
* [12] L. Castell, Nuovo Cim.A46(1966) 1; L. Castell, Nuovo Cim.A49(1967) 285.
* [13] L. Castell, Nucl. Phys. B4(1967) 343.
* [14] G. Mack and A. Salam, Ann. Phys.53(1969) 174.
* [15] S. Ferrara, R. Gatto and A.F. Grilo, Ann. Phys. 76 (1973) 161; S. Ferrara, R. Gatto and A.F. Grilo,in Scale and Conformal Symmetry in Hadron Physics (ed. R. Gatto), Wiley, New York, 1974;
* [16] V. De Alfaro, S. Fubini, G. Furlan and C. Rosseti, Currents in Hadron Physics; (North-Holland, Amsterdam) 1973.
* [17] B.G. Konopelchenko,Sov. J. Elem. Part. and At. Nucl. (in Russian) 11(1977) 135.
* [18] J. Beckers, J. Harnad, M. Perroud and P. Winternitz, J. Math. Phys.19(1978) 2126.
* [19] A.O.Barut and R. Raczka, Theory of Group Representations and Applications;PWN,Warszawa, 1977.
* [20] P. Budinich and R. Raczka, Found. Phys.23(1993) 599.
* [21] R. Mirman, Quantum Field Theory Conformal Group Theory Conformal Field Theory: Mathematical And Conceptual Foundations Physical And Geometrical Applications;Paperback, Backinprint.com, 2005.
* [22] V.M. Braun, G. P. Korchemsky and D. Müller, Prog. Part. Nucl. Phys. 51 (2003) 311.
* [23] E. S. Fradkin and M.Y. Palchik, Conformal Quantum Field Theory in D-Dimensions, in: Mathematics and its Applications V.376 Kluwer,Dordrecht, Netherlands, New York, 1996; E. S. Fradkin and M.Y. Palchik, Phys.Rep.300 (1998) 1.
* [24] I. T. Todorov, Conformal Description of Spinning Particles; Springer, New York, 1986; Todorov I. T., Minchev M.C. and Petkova V.B. // Conformal covariance in quantum field theory; (Scuola Normale Superiore, Pisa, 1978.
* [25] P. Budinich, Found. Phys. 32(2002) 1347.
* [26] H. Bacry, Ann. Ins. H. Pouncare, 49(1988) 245; H. Bacry, Localizabelity and Space in Quantum Physics; Lect. Notes in Phys. 308 (Springer, Berlin Heidelberg, 1988).
* [27] V. G. Kadyshevsky, J.Exp. Theor. Phys. (in Russian)41(1961) 1885.
* [28] V. G. Kadyshevsky, Sov.J. Elem. Part. and At. Nucl. (in Russian) 11(1980) 5; Preprint JINR,(in Russian) ░2-84-753.
* [29] V. G. Kadyshevsky and M. D. Mateev, Nuovo Cim.87A(1985)324; M. V. Chizhov, A. D. Donkov, R. M. Ibadov, V. G. Kadyshevsky and M. D. Mateev, Nuovo Cim.87A(1985)351 and 373.
* [30] W. Heisenberg, Ann. Phys. (Leipzig)5(1938) 20; H. P. Dürr and W. Heisenberg, Z. Natur. 16a(1961) 726.
* [31] M. A. Markov, Suppl. Prog. Theor. Phys. Commamemory Issue for 30-th Anniveversary of Meson Theory by Dr. A. Yukawa (1965) 865; J.Exp. Theor. Phys. (in Russian) 51(1966) 878.
* [32] A.I.Machavariani, Preprint arXiv/math-ph/0611083v1 (2006) which is the renewed version of the lectures presented in arXiv/hep-th/0504030 (2005).
* [33] G. Mack and I. T. Todorov, Phys. Rev.D8(1973) 1764.
* [34] J. R. Bjorken and S. D. Drell. Relativistic Quantum Fields; New York, McGrew-Hill, 1963.
* [35] S. Weinberg, The Quantum Theory of Fields; Cambridge, University Press, 1995 and 1996.
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|
arxiv-papers
| 2012-04-19T07:54:03 |
2024-09-04T02:49:29.833597
|
{
"license": "Public Domain",
"authors": "A. I. Machavariani",
"submitter": "Alexander Machavariani",
"url": "https://arxiv.org/abs/1204.4272"
}
|
1204.4299
|
Prof. Stefano Vitale Dr. Mauro Hueller Department of Physics University of
Trento PhilosophiæDoctor (PhD) 26th March 2012
# Spacetime Metrology with LISA Pathfinder
Giuseppe Congedo
© 2012
Giuseppe Congedo
All Rights Reserved
LISA is the proposed ESA-NASA space-based gravitational wave detector in the
$0.1\,\mathrm{mHz}\text{--}0.1\,\mathrm{Hz}$ band. LISA Pathfinder is the
down-scaled version of a single LISA arm. In this thesis it is shown that the
arm – named Doppler link – can be treated as a differential accelerometer,
measuring the relative acceleration between test masses. LISA Pathfinder – the
in-flight test of the LISA instrumentation – is currently in the final
implementation and planned to be launched in 2014. It will set stringent
constraints, with unprecedented pureness, on the ability to put test masses in
geodesic motion to within the required differential acceleration of
$3\\!\times\\!10^{-14}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}$ and
track their relative motion to within the required differential displacement
measurement noise of
$9\\!\times\\!10^{-12}\,\mathrm{m\,Hz^{-\nicefrac{{1}}{{2}}}}$, at frequencies
relevant for the detection of gravitational waves. Given the scientific
objectives, it will carry out – for the first time with such high accuracy
required for gravitational wave detection – the science of spacetime
metrology, in which the Doppler link between two free-falling test masses
measures the spacetime curvature. This thesis contains a novel approach to the
calculation of the Doppler response to gravitational waves. It shows that the
parallel transport of 4-vectors records the history of gravitational wave
signals passing through photons exchanged between an emitter and a receiver.
In practice, the Doppler link is implemented with 4 bodies (two test masses
and two spacecrafts) in LISA and 3 bodies (two test masses within a
spacecraft) in LISA Pathfinder. Different non-idealities may originate in the
measurement process and noise sources couple the motion of the test masses
with that of the spacecraft. To compensate for such disturbances and stabilize
the system a control logic is implemented during the measurement. The complex
closed-loop dynamics of LISA Pathfinder can be condensed into operators acting
on the physical coordinates describing the relative motion. The formalism can
handle the couplings between the test masses and the spacecraft, the sensing
noise, as well as the cross-talk, and allows for the system calibration. It
suppresses the transients in the estimated residual acceleration noise between
the test masses. The scope of system identification is indeed the calibration
of the instrument and the compensation of different effects. After introducing
a model for LISA Pathfinder along the optical axis and an example of cross-
talk from other degrees of freedom to the optical axis, this thesis describes
some data analysis procedures applied to synthetic experiments and tested on a
realistic simulator provided by ESA. The same procedures will also be adopted
during the mission. Those identification experiments can also be optimized to
get an improvement in precision of the noise parameters that the performances
of the mission depend on. This thesis demonstrates the fundamental relevance
of system identification for the success of LISA Pathfinder in demonstrating
the principles of spacetime metrology needed for all future space-based
missions.
To my wife Laura,
source of inspiration and happiness.
###### Acknowledgements.
Firstly, I would like to acknowledge my research advisor Prof. Stefano Vitale
which offered me to work on the LISA Pathfinder project, gave me the chance to
know the many people around the Data Analysis team and, most of all, taught me
how to proficiently work as a researcher in physics. The co-advisor Mauro
Hueller for his guidance through the open problems in data analysis and the
organization of the research program. Luigi Ferraioli (now in APC, Universitè
Paris Diderot) for the scientific rigor he taught me, the many times he helped
me during these years with both research and traveling issues, without whom I
would never have realized a big part of this work. Fabrizio De Marchi (now in
University of Rome, Tor Vergata) for the many interesting discussions in
astronomy, science (and lots more), the scientific collaboration and the great
company during these years. Bill Weber who bore the reading of this thesis and
helped me with English. Prof. Giovanni Prodi for the initial involvement in
research just after my arrival in Trento. Renato Mezzena for the interesting
discussions during lunch time. Karine Frisinghelli for her strong support and
friendship. Abroad, Martin Hewitson (AEI, Max-Planck-Institut für
Gravitationsphysik und Universität Hannover) head of the LISA Pathfinder Data
Analysis team; Michele Armano (ESAC, ESA, Madrid) and Miquel Nofrarias (IEEC,
Universitat de Barcelona) for useful suggestions and discussions; Michele
Vallisneri (JPL NASA and California Institute of Technology, Pasadena) for
clarifications on the response of the LISA detector. Again, Prof. Stefano
Vitale for the teaching opportunity that increased my self-control and my
ability as a speaker, much more than I expected. For the teaching adventure,
Rita Dolesi, Barbara Rossi, David Tombolato and Fabrizio De Marchi with all
whom I collaborated successfully. Finally, I would like to thank my family in
Lecce for the support during the academic program in physics started years ago
– this thesis is its natural conclusion – and my wife for her role in
significantly contributing to both my physical and mental wellness during the
PhD.
###### Contents
1. 1 Introduction
1. 1.1 LISA, a space-borne gravitational wave detector
2. 1.2 LISA Pathfinder: spacetime metrology and verification of the detection principle
3. 1.3 LISA Technology Package
1. 1.3.1 Gravitational reference sensor
2. 1.3.2 Optical metrology system
3. 1.3.3 Star-trackers
4. 1.3.4 Drag-free and attitude control system
5. 1.3.5 Thrusters
4. 1.4 Outline of the work
2. 2 Spacetime metrology
1. 2.1 Metrology without noise
1. 2.1.1 Weak field limit
2. 2.2 Doppler link as differential accelerometer
3. 2.3 Metrology with noise
1. 2.3.1 Laser frequency noise
2. 2.3.2 Residual acceleration noise
3. 2.3.3 Readout noise
4. 2.3.4 Summary
4. 2.4 Dynamics of fiducial points
3. 3 Controlled dynamics
1. 3.1 Closed-loop formalism
1. 3.1.1 Coordinate definitions
2. 3.1.2 Controller
3. 3.1.3 Equation of motion
2. 3.2 Suppressing system transients
3. 3.3 Dynamical model along $x$
4. 3.4 Cross-talk from degrees of freedom other than the optical axis
5. 3.5 Dynamical model for $xy$ cross-talk
4. 4 System identification
1. 4.1 Dynamical model
1. 4.1.1 Anelasticity and damping
2. 4.2 Noise characterization
3. 4.3 Identification experiments
4. 4.4 Parameter estimation
1. 4.4.1 Review of the problem
2. 4.4.2 Estimation method
3. 4.4.3 Whitening
4. 4.4.4 Search algorithm
5. 4.4.5 ESA simulator
6. 4.4.6 Monte Carlo validation
7. 4.4.7 Non-standard scenario: under-performing actuators and under-estimated couplings
8. 4.4.8 Non-standard scenario: non-Gaussianities
5. 4.5 Estimation of total equivalent acceleration noise
6. 4.6 Suppressing transients in the total equivalent acceleration noise
5. 5 Design of optimal experiments
1. 5.1 Review of the problem
2. 5.2 Optimizing the identification experiments
3. 5.3 Multi-experiment, single-input
4. 5.4 Single-experiment, multi-input
6. 6 Conclusions and future perspectives
7. A Appendix
1. A.1 A single galactic binary in LISA noise
2. A.2 Non-pure free fall and Fermi-Walker transport
3. A.3 Calculation in metrology without noise
4. A.4 Linearized Einstein equations for Doppler link as differential accelerometer
5. A.5 Demonstration of noise non-stationarity
6. A.6 Time-frequency analysis of non-stationary noise
7. A.7 More on Monte Carlo validation
## 1 Introduction
40 years ago the binary pulsar 1913+16 [1] opened up a long series of
observations aimed at determining various relativistic effects – like the
periastron shift – that were confirmed to be in very good agreement with
General Relativity (GR). The discovery of the pulsar gave the first strong
indication of the existence of Gravitational Waves (GWs). Yet to date no
direct detection has been made, in spite of many efforts of disparate
experiments still in progress. The detection of GW signals requires the
development of sophisticated devices capable in accurately measuring very
small accelerations between nominally free-falling test particles subjected to
different noise sources. The same measurement principle, with slight
modifications, is shared among the 1st, the 2nd and the 3rd generation of
ground-based detectors, as well as the planned spaced-based detectors.
### 1.1 LISA, a space-borne gravitational wave detector
A passing GW would cause a change in the relative velocities between test
particles in nominal free fall. As a Michelson interferometer, a GW detector
measures such a physical quantity. Ground-based GW detectors have currently
reached almost their design sensitivities, and the 2nd generation, Adv. LIGO
[2], Adv. Virgo [3] and GEO-HF [4], promises an improvement in detection rates
and a wider horizon to be explored in the
$10\,\mathrm{Hz}\text{--}10\,\mathrm{kHz}$ band. The 3rd generation with the
Einstein Telescope [5, 6] will provide further enhancements in both
sensitivity and frequency band, especially toward the low-frequency end that,
at $1\,\mathrm{Hz}$, is limited by the Earth gravity noise. It’s not just by
chance that the proposed design for the Einstein Telescope is an underground
$100\,\mathrm{km}$-wide equilateral triangular scheme of Michelson
interferometers as the triangle can be considered the optimal configuration in
resolving both source polarization and position with extremely high
confidence. Years ago, the Laser Interferometer Space Antenna (LISA) [7, 8] –
a joint ESA [9] \- NASA [10] mission – was discovered to offer the possibility
of exploring a much lower frequency band,
$0.1\,\mathrm{mHz}\text{--}0.1\,\mathrm{Hz}$, expected to be saturated by the
huge population of GW binaries.
The key concept of LISA is the constellation flight of three SpaceCrafts (SCs)
– each hosting and protecting two Test Masses (TMs) in nominal free fall – in
a $5\\!\times\\!10^{6}\,\mathrm{km}$ sided equilateral triangle around the Sun
at $1\,\mathrm{AU}$ as shown in Figure 1.1. The arm length is approximately
constant within a fractional tolerance of few percent. The angles are allowed
to vary over the year within $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}1^{\text{o}}$ at most. No frequent orbit corrections are actually needed
and the formation follows the Earth with a trailing angle of $\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}20^{\text{o}}$, a compromise
solution between gravitational perturbations and communication/fuel
constraints 111Recently, due to funding cuts, the US side has withdrawn its
participation in a GW mission in the 2020-2025 timeframe. Meanwhile, the
European has started a feasibility study of a descoped version named eLISA/NGO
[11, 12] fitting the cost of an ESA L-class mission and at the same time
maintaining most of the scientific objectives. Some of the modifications
include a shorter lifetime, shorter arm lengths
($1\\!\times\\!10^{6}\,\mathrm{km}$), a smaller trailing angle and the
possible suppression of two Doppler links. The adopted “mother-daughter”
configuration would be the first Michelson interferometer in space allowing
for the detection of many continuous sources with revolutionary scientific
returns [13]. This mission is being evaluated by ESA at the time of writing
down this thesis. However, this thesis refers to LISA without any loss of
generality, while keeping in mind that all discussions and results are still
valid for any variant of LISA based on the same detection principle..
|
---|---
(a) | (b)
Figure 1.1: Scheme of the LISA orbit (not in scale) around the Sun and details
of a single SC. (a) the triangular formation follows the Earth and maintains
its arm length approximately constant within few percents. (b) each SC
contains two TMs and the relative displacements to the faraway counterparts
are detected by a laser-interferometric technique. 6 TMs constitute 6 Doppler
links, two per LISA arm, tracking the local curvature variations around the
Sun and are sensitive to GW signals in the
$0.1\,\mathrm{mHz}\text{--}0.1\,\mathrm{Hz}$ band.
In LISA the relative velocities between the TMs change as a GW passes through
the constellation. LISA is a combination of 3 quasi-independent Michelson
interferometers and, as such, detects oscillating signals. Given the very low
frequency band compared to the ground-based, LISA will be sensitive to
continuous signals arising from inspiral, merger and ringdown of binaries.
Among many astrophysical targets, the detection and characterization of the
following objects will be of fundamental importance during the nominal 5-year
mission:
1. 1.
Super-Massive Black Holes (SMBHs) with very high Signal-to-Noise Ratio (SNR),
out to redshift $z\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}15$,
from the merging of galactic nuclei;
2. 2.
a dozen of galactic verification binaries for each of which an electromagnetic
counterpart is available;
3. 3.
hundreds (or even thousands) of galactic binaries, continuous or chirping,
that can be distinctively resolved;
4. 4.
unresolved galactic binaries appearing as noise foreground at low frequency;
5. 5.
Extreme Mass Ratio Inspirals (EMRIs) to study GR in highly curved spacetimes;
6. 6.
stochastic cosmic background.
These scientific objectives make LISA a GW telescope with a potentially huge
impact in whole physics. Contrary to the ground-based detectors, LISA can be
considered a signal-dominated detector where the interferometric outputs are
three correlated time-series containing the superposition of many signals in
whole sky: its conceptual and practical complexities make the extraction of
such signals sophisticated. A typical feature of LISA is its ability in
resolving sources with very high position accuracy. This is due to a double
Doppler modulations induced by the revolution around the Sun and the
intrinsical rotation of the normal to the constellation plane (Appendix A.1
shows an example of the LISA response to a single galactic binary).
The LISA objectives in astrophysics requires that the TMs must be kept in free
fall with a residual acceleration noise as low as
$3\\!\times\\!10^{-15}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$ – a goal achievable thanks to the sophisticated design and
technology employed onboard.
### 1.2 LISA Pathfinder: spacetime metrology and verification of the
detection principle
In the last decade LISA Pathfinder (LPF) [14] was proposed to fly as a
targeted ESA mission [15] to verify the detection principle of LISA. LPF is a
down-scaled version of a single LISA arm to the size of about
$40\,\mathrm{cm}$. The main scope of LPF is to give an in-flight test of the
LISA instrumentation and demonstrate that parasitic forces are constrained
such that the measured differential acceleration between two TMs is below the
level of
$3\\!\times\\!10^{-14}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$.
Currently in the final implementation and planned to be launched in 2014 [16],
LPF will fly in a Lissajous orbit around the L1 Lagrange point
($1.5\\!\times\\!10^{6}\,\mathrm{km}$ away from the Earth toward the Sun). See
Figure 1.2 for reference. Even though such orbits are periodic, they are
unstable and station-keeping forces must be applied orthogonally to the orbit
plane (and parallel to the axis joining the two celestial bodies). The solar
array, also working as a shield to the SC underneath, will point the Sun to
within a few degrees. A residual spin around the same axis is kept lower than
3${}^{\text{o}}$ per day for scientific requirements. An alternate possibility
has been also considered as backup option in case the propulsion module may
fail in transferring the payload from the low Earth orbit to the target. The
SC may be injected in a highly eccentric orbit around the Earth with a period
of 27 days. Even though this solution does not allow for continuous
measurements at the optimal sensitivity close to the perigee for 2–3 days, it
is an interesting test-bench for utilizing the Moon as a calibrator of the
instrument [17].
|
---|---
(a) | (b)
Figure 1.2: Scheme of the LPF orbit (not in scale) around L1 and details of
the SC. (a) the SC is in a halo orbit and station-keeping forces must be
applied orthogonally for its stabilization. (b) the SC contains two TMs whose
relative displacements are detected by a laser-interferometric technique.
LPF is expected to provide an accurate noise model for LISA and put stringent
constraints, with unprecedented results, on [18]:
1. 1.
the ability to keep TMs in free fall – the so-called differential acceleration
noise requirement – to within the level of
$S_{\text{n},\delta a}^{\nicefrac{{1}}{{2}}}\lesssim
3\\!\times\\!10^{-14}\left[1+\left(\frac{f}{f_{0}}\right)^{2}\right]\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}\leavevmode\nobreak\
;$ (1.1)
2. 2.
the ability to track relative displacements between the TMs with a laser
interferometer – the so-called differential displacement noise requirement –
to within the level of
$S_{\text{n},\delta x}^{\nicefrac{{1}}{{2}}}\lesssim
9\\!\times\\!10^{-12}\left[1+\left(\frac{f_{0}}{f}\right)^{2}\right]\,\mathrm{m\,Hz^{-\nicefrac{{1}}{{2}}}}\leavevmode\nobreak\
;$ (1.2)
where $f_{0}=3\,\mathrm{mHz}$ and over the $1\text{--}30\,\mathrm{mHz}$ band.
The LPF requirements are relaxed by almost an order of magnitude to LISA. The
high frequency regime is dominated by the displacement requirement of
$9\,\mathrm{pm\,Hz^{-\nicefrac{{1}}{{2}}}}$, whereas the acceleration
requirement of $30\,\mathrm{fN\,Hz^{-\nicefrac{{1}}{{2}}}}$ has much more
importance to the low frequency assessment of the LISA noise. Figure 1.3
compares the requirements in Power Spectral Density (PSD) of the residual
acceleration noise for LISA and LPF. Even though LPF shares the same hardware
design with LISA, a relaxation in both acceleration noise level and frequency
band is allowed for the first.
Figure 1.3: Comparison between the residual acceleration noise requirement of
LPF and LISA. LPF is relaxed with respect to LISA by a factor $\raise
0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}7$ in amplitude. The required
LISA band ($0.1\,\mathrm{mHz}\text{--}0.1\,\mathrm{Hz}$) is extended toward
the low frequency compared to the LPF band ($1\text{--}30\,\mathrm{mHz}$).
Obviously, during the mission a lower acceleration and a wider frequency band
will be easily reached.
### 1.3 LISA Technology Package
LPF and its main scientific payload, the LISA Technology Package (LTP) [19],
will give an in-flight test of the LISA hardware and effectively measure the
differential acceleration noise that pollutes the sensitivity of LISA below
$3\\!\times\\!10^{-14}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$ – the minimum performance level for LISA to carry on its
science program in astrophysics. As said, the observational horizon of LISA
will include thousands of GW sources. Among all, those with the highest SNR
will be surely the SMBHs. However, there are sources that are expected to lay
at the limit of the LISA sensitivity for which an accurate assessment of the
instrumental noise is mandatory. The population of EMRIs [20] is the most
important example: they are a valuable instrument to test GR and curvature in
the strong gravity regime. Different EMRI search methods have been developed.
After having subtracted the highest signals (SMBHs and calibration binaries),
in order to extract the EMRI signatures, all methods strictly have to deal
with the instrumental noise level, for which the LPF mission has a crucial
role. In fact, a systematic error in the reconstructed noise shape would
dramatically affect the identification of such sources. This thesis shows the
importance of LPF and system identification for the correct assessment of the
noise parameters and the noise shape. A numerical example will be provided by
Chapter 4.
During the 3 months of operations, the LTP experiment on board LPF will be
used in an extensive characterization campaign to measure all force
disturbances and systematics, like the TMs couplings, the various cross-talks,
the TM charging due to cosmic particles and its interaction with the
electrostatic environment, the thermal and magnetic effects, etc. The impact
of the effects on the differential acceleration noise can be inferred by
simulations and through on-ground measurements. In fact, two facilities
(single-mass and 4-mass torsion pendulum) have been employed during the last
years to investigate the one-degree-of-freedom behavior of a replica of the
Au-Pt TM of $1.96\,\mathrm{kg}$ and its electrostatic housing, including all
sensing and actuation capacitive electrodes, entirely named Gravitational
Reference Sensor (GRS) [21]. A comprehensive review of the current status of
the on-going measurement activities and their extrapolations to LISA are given
in [19] and references therein.
The LTP experiment comprises the following key subsystems shown in Figure 1.4:
two GRSs, the Optical Metrology System (OMS) (InterFerOmeters (IFOs) and the
optical bench), Star-Trackers (STs), an on-board computer, the Drag-Free and
Attitude Control System (DFACS) and the Field Emission Electric Propulsion
(FEEP) thrusters. The experiment is also equipped with magnetometers,
thermometers and a cosmic charge counter. The sensors with the relative sensed
motions are reported in Table 1.1. The noise requirements are reported in
Table 1.2.
Figure 1.4: Scheme of the key subsystems of the LPF mission. The SC contains two GRSs and an optical bench with four interferometers. The relative displacements and attitudes between the TMs and the optical bench are read out by the interferometers and the capacitive sensors. The interferometric, capacitive and star-tracker readouts (solid lines) are fed into the DFACS that computes the forces that shall be actuated by the FEEP thrusters and the capacitive actuators (dashed lines). In the main science mode the reference TM is not actuated along the optical axis. Table 1.1: LTP sensors and the relative sensed motions. Sensor | Motion
---|---
GRS | linear and angular motion of the TMs relative to their housings
OMS | linear and angular motion of the reference TM relative to the optical bench
| linear and angular motion of the second TM relative to the reference TM
ST | absolute attitude of the SC
Table 1.2: LTP key subsystems and the main noise requirements around $1\,\mathrm{mHz}$. Subsystem | Requirement | Note
---|---|---
GRS | $1.8\,\mathrm{nm\,Hz^{-\nicefrac{{1}}{{2}}}}$ | displacement sensing
$20\,\mathrm{fN\,Hz^{-\nicefrac{{1}}{{2}}}}$ | actuation
OMS | $9\,\mathrm{pm\,Hz^{-\nicefrac{{1}}{{2}}}}$ | displacement sensing
$20\,\mathrm{nrad\,Hz^{-\nicefrac{{1}}{{2}}}}$ | attitude sensing
ST | $32\,\mathrm{{}^{\prime\prime}\,Hz^{-\nicefrac{{1}}{{2}}}}$ | -
DFACS | $5\text{--}6\,\mathrm{nm\,Hz^{-\nicefrac{{1}}{{2}}}}$ | displacement control
(main science mode)
$0.4\text{--}0.5\,\mathrm{\mu rad\,Hz^{-\nicefrac{{1}}{{2}}}}$ | attitude control
(main science mode)
FEEP | $0.1\,\mathrm{\mu N\,Hz^{-\nicefrac{{1}}{{2}}}}$ | -
#### 1.3.1 Gravitational reference sensor
Each GRS comprises an Au-Pt cubic TM of size $46\,\mathrm{mm}$ and a
surrounding electrostatic housing containing capacitive sensors and actuators
in all 6 degrees of freedom. Each GRS senses the relative displacement and
attitude of the TM to its housing and provides actuation along the same
degrees of freedom. Gaps between the TM and its housing are
$3\text{--}4\,\mathrm{mm}$, a compromise between noise minimization and
efficient sensing/actuation. The GRS vacuum chamber allows for a residual gas
pressure at the level of $10\,\mathrm{\mu Pa}$. UV light illumination is
utilized to control the accumulated charge with a discharging threshold of
$\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}10^{7}\,\mathrm{e}$ –
the accumulated charge in one day for an expected charging rate of $\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}10^{2}\,\mathrm{e\,s^{-1}}$. The
sensing requirements of each GRS are
$1.8\,\mathrm{nm\,Hz^{-\nicefrac{{1}}{{2}}}}$ in displacement and
$200\,\mathrm{nrad\,Hz^{-\nicefrac{{1}}{{2}}}}$ in attitude. The actuation
requirement is $20\,\mathrm{fN\,Hz^{-\nicefrac{{1}}{{2}}}}$ with a maximum
range of $2.5\,\mathrm{nN}$.
#### 1.3.2 Optical metrology system
The OMS [22] comprises: a Zerodur® monolithic optical bench, 4 Mach-Zehnder
heterodyne $1.024\,\mathrm{\mu m}$ interferometers and redundant quadrant
photodiodes. The first IFO, $X_{1}$, senses the relative displacement and
attitude of one reference TM to the optical bench itself. The differential
IFO, $X_{12}$, senses the relative displacement and attitude between the two
TMs. Relative displacements are measured by averaging among the four
quadrants, whereas relative angles are measured by taking the difference
between opposite quadrants (differential wavefront sensing). The “reference”
IFO is subtracted from the previous ones for compensating spurious fiber
optical path length variations before the first beam splitter. The “frequency”
IFO is utilized for laser frequency stabilization. The sensing requirements
are $9\,\mathrm{pm\,Hz^{-\nicefrac{{1}}{{2}}}}$ in displacement, as in (1.2),
and $20\,\mathrm{nrad\,Hz^{-\nicefrac{{1}}{{2}}}}$ in attitude with a maximum
range of $100\,\mathrm{\mu m}$. A rotation around the optical axis is not
sensed, but can be provided by the GRS.
#### 1.3.3 Star-trackers
The STs are small telescopes reading out the inertial attitude of the SC with
respect to the star field. The sensing requirement is
$32\,\mathrm{{}^{\prime\prime}\,Hz^{-\nicefrac{{1}}{{2}}}}$ ($160\,\mathrm{\mu
rad\,Hz^{-\nicefrac{{1}}{{2}}}}$).
#### 1.3.4 Drag-free and attitude control system
The outputs of all sensors, GRSs, OMS and STs, are elaborated by the on-board
computer and fed into the DFACS [23]. The DFACS has the responsibility of
computing the control forces that shall be passed to capacitive and thruster
actuators in order to stabilize the system and meet the acceleration
requirement in (1.1).
There are different operational control modes for the LPF mission. To avoid
large transients in the data, the transition between two modes is implemented
with overlapping sub-modes. In the accelerometer mode LPF acts as a standard
accelerometer in which the TMs are both electrostatically actuated along the
optical axis and controlled to follow the SC motion. The resulting noise is
much higher than the requirement. In the main science mode, the DFACS is
responsible in maintaining a reference TM in free fall along the optical axis
and forcing both the second TM and the SC to follow it by capacitive and
thruster actuation.
The need for the DFACS is explained not only by the scientific requirements,
but also by the fact that noise sources can destabilize the system on a time
scale of few minutes and the gaps between the TM and its housing are just
$3\text{--}4\,\mathrm{mm}$. One of the proposed activities, the free flight
experiment [24], is aimed at obtaining an improvement in differential
acceleration noise at low frequency by turning off the capacitive actuation
also on the second TM which is left in “parabolic” free fall and impulsively
kicked every $200\,\mathrm{s}$.
In the main science mode the DFACS is conceptually divided into three control
loops [25] with the following priority:
1. 1.
drag-free control loop, controlling the relative displacement and attitude of
the SC with respect to the reference TM through thruster actuation;
2. 2.
electrostatic suspension control loop, controlling the relative displacement
and attitude between the TMs through capacitive actuation on the second TM;
3. 3.
attitude control loop, controlling the inertial (absolute) attitude of the TMs
through capacitive actuation.
The drag-free requirement are
$5\text{--}6\,\mathrm{nm\,Hz^{-\nicefrac{{1}}{{2}}}}$ in displacement and
$0.4\text{--}0.5\,\mathrm{\mu rad\,Hz^{-\nicefrac{{1}}{{2}}}}$ in attitude.
#### 1.3.5 Thrusters
The FEEP is attained by an ensemble of 3 clusters, of 4 thrusters each,
attached to the SC. An electron flux keeps the SC neutral. The force
requirement is $0.1\,\mathrm{\mu N\,Hz^{-\nicefrac{{1}}{{2}}}}$ with a maximum
range of $100\,\mathrm{\mu N}$. The FEEP thruster authority is the only means
by which the reference TM can be maintained in free-fall along the optical
axis, hence mitigating the SC jitter at low frequency. The SC is also equipped
with colloid thrusters provided by NASA for complementary experiments.
Recently, ESA has considered the possibility to employ cold gas thrusters in
place of the FEEP. The new design is expected to perform to within the
requirements as well. However, the considerations and the results of this
thesis are still valid and are not appreciably affected by the possible change
in design.
### 1.4 Outline of the work
In LISA a total of 6 TMs, whose relative displacements 222Throughout this
thesis an extensive use (and abuse) of terms like “relative displacement”,
“frequency shift”, “phase difference”, etc. will be made without any relevant
distinction. The explanation is that a relative displacement is proportional
to a phase difference, $\delta r\simeq\lambda\,\delta\phi$ (with $\lambda$ the
light wavelength), and a relative velocity is proportional to a frequency
shift, $\delta v\simeq\lambda\,\delta\omega$. The two are obviously related by
a time derivative. The fractional frequency shift is also useful, as in the
next chapter, and its relation to phase difference is
$\delta\omega/\omega=\dot{\delta\phi}/\omega$. The following table shows the
equivalence between the mentioned quantities: Relative displacement Phase
shift Relative velocity Frequency shift Relative acceleration Frequency shift
rate are tracked by a laser-interferometric technique, constitute 6 Doppler
links, two per LISA arm, tracking the local curvature variations around the
Sun and sensing the small fluctuations induced by GW signals in the
$0.1\,\mathrm{mHz}\text{--}0.1\,\mathrm{Hz}$ band. LISA can be viewed as a
combination of three quasi-independent nominally equal-arm Michelson
interferometers with vertices at each SC. In the ground-based detectors the
laser frequency noise is common-mode between the two arms and can be
subtracted with very high accuracy. In LISA a relatively small difference
between two arms of order of a few percent makes such a subtraction impossible
and a laser frequency fluctuation noise as large as $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}10^{-13}\,\mathrm{Hz^{-\nicefrac{{1}}{{2}}}}$
around $1\,\mathrm{mHz}$ corrupts the GW detection. The Time-Delay
Interferometry (TDI) [26] provides for a solution of the problem: the Doppler
measurements are properly time-shifted, to take into account on the photon
flight times, and linearly combined, to get the suppression of the laser
frequency fluctuation noise by 7 orders of magnitudes. Scope of the entire LPF
mission is the accurate modeling of the unsuppressed part of the noise (except
for the relative motion between the SCs), the residual acceleration noise
affecting the geodesic motion of the TMs after the TDI compensation.
In LISA 6 TMs, whose frequency shifts are optically sensed along each arm of
the triangle, build up 6 Doppler links, two per single arm in both directions,
forth and back. The fundamental Doppler link can be described as a four-body
TM-SC-SC-TM sequence of measurements. Referring to Figure 1.5, the relative
velocity of one TM to the optical bench of its hosting SC is measured by a
local interferometer; at the same time, the laser signal is sent toward the
second SC; finally, a new local measurement is performed between the second TM
and the optical bench. Therefore, three measurements, TM to SC, SC to SC and
TM to SC, are combined to form the TM-to-TM Doppler link that carries the GW
signal. It is easy to recognize that the two local signals carry no GW
information, but they are affected by noise, mostly due to parasitic forces
that couple the TMs to the SC motion and interferometric sensing, which enter
into the noise budget of the Doppler link. The single LISA arm is efficiently
reformulated in Chapter 2 as a time-delayed differential accelerometer whose
input signals and noise sources are effectively described as equivalent
differential accelerations between the TMs. The most important disturbances
affecting the GW detection are due to:
1. 1.
real forces, relevant at low frequency, say below few $\mathrm{mHz}$, with red
spectrum;
2. 2.
readout sensing coming from all noise sources in the interferometric readout,
except for the frequency fluctuation subtracted by TDI;
3. 3.
mixing of motion from degrees of freedom other than the axis joining the TMs,
named cross-talk from other degrees of freedom into the main optical axis.
As the main aim is the measuring of the total equivalent differential
accelerations, for the rest, all disturbances above will be treated as
equivalent accelerations, inputs to a time-delayed differential accelerometer.
Figure 1.5: LISA measurement scheme. The solid arrows show the local links
measuring the relative motion of the TMs to their hosting SCs. The dashed
arrows show the links measuring the relative motion between the SCs.
LPF aims at estimating the residual noise affecting the LISA link through
measurements performed in closed loop. One (any) arm of LISA is virtually
shrunk [27] to $38\,\mathrm{cm}$ and implemented in the LPF mission with some
differences. LPF is essentially a SC carrying two TMs in nominal free fall and
employs a three-body TM-SC-TM sequence of measurements. It measures the
relative motion of a TM with respect to the SC and the relative motion between
the TMs. All TMs in LISA are controlled along the degrees of freedom
orthogonal to the measurement axes and the control is said off-axis. Instead,
as the measurement axis for LPF is within the SC, a TM must be controlled
along the same degree of freedom and the control is said on-axis. In this way
it is not yet possible to maintain both TMs in free fall along the optical
axis: while a reference TM is nominally in free fall, the second must be
actuated in order for the differential force disturbances can be compensated.
As the control has a fundamental importance in the system stabilization,
applied forces must be taken into account as inputs to the differential
accelerometer and subtracted from the data.
The LISA arm viewed as a time-delayed differential accelerometer is
practically implemented in LPF in a closed-loop differential measurement based
on three main concepts: dynamics, sensing and control. Chapter 3 will give an
extensive description of the equations governing the link, showing how known
couplings, cross-talks and control forces can be taken into account. In the
approximation of small TM motion and weak force couplings, the system is
linear and the dynamical equations can be rewritten as linear operators acting
on the relevant coordinates. As will be demonstrated, the construction of a
differential operator then allows:
1. 1.
the conversion of the sensed motion into total equivalent acceleration;
2. 2.
the subtraction of the couplings, the control forces and the cross-talk from
the data;
3. 3.
the suppression of the system transients, at least to within the accuracy to
which the system parameters have been measured.
The assessment of the final level of the total equivalent differential
acceleration noise – the key scientific target of LPF – is literally an
iterative process, since the quality of free fall achieved at a given stage of
the mission depends on the results of the previous experiments and the
accuracy and precision to which the noise parameters have been estimated.
Examples of the adopted data analysis procedures will be given in Chapter 4,
showing the relevance of system identification to achieve the free-fall level
needed for LISA. A whole data analysis pipeline will be described and applied
to data generated with the model described in Chapter 3 and a realistic
simulator provided by industry, hence putting constraints on the accuracy to
which the noise parameters can be estimated. The precision of those extracted
parameters can also be inferred and optimized as shown in Chapter 5. All
analysis has been performed under the framework of the LTP Data Analysis
(LTPDA) Toolbox [28], an objected-oriented extension of MATLAB® [29] that will
be extensively employed during the mission.
Chapter 2.
The chapter discusses on the Doppler link between two TMs in free fall and the
GW perturbation of the link through the parallel transport of the emitter
4-velocity. The chapter shows that the parallel transport induces a time delay
in the physical quantities. It presents a novel derivation of the response of
the Doppler link to the GW, an analogous result already found in literature.
The Doppler link can be reformulated as a time-delayed differential
accelerometer where all inputs (signals and noise) are equivalent differential
accelerations. In the end, it introduces the concept of cross-talk from other
degrees of freedom to the optical axis.
Chapter 3.
The mathematical description achieved so far is translated into equations
governing dynamics, sensing, and control for LPF, i.e. the implementation of a
single down-scaled LISA arm. The chapter introduces an operator formalism
capable of managing the complex and coupled equations in a compact form. The
main advantage of such an abstract formalism is that transfer matrices can be
easily extracted, in particular the one representing the conversion from the
sensed coordinates to the total equivalent acceleration. The extent to which
the suppression of system transients can be achieved is also a novel result of
this thesis. The cross-talk from other degrees of freedom can be viewed as a
first-order perturbation of the nominal dynamics and all relevant transfer
matrices are derived for this case. A model of LPF along the optical axis and
an example of cross-talk are given in the end of the chapter.
Chapter 4.
System identification is the key method for the calibration of the system
modeled by transfer matrices, allowing for confident noise projections and,
most of all, the unbiased estimation of the total equivalent acceleration
noise. The chapter discusses examples of the data analysis pipelines adopted
for the LPF mission. The relevance of system identification for non-standard
scenarios, its impact to the estimation of the total equivalent acceleration
noise and the suppression of system transients are given in the end of the
chapter.
Chapter 5.
Parameter accuracy is the main target of system identification, whereas
precision is the main target of the design of optimal experiments. The chapter
focuses on the search of optimal experiments for the LPF mission allowing for
a more precise identification of the system parameters that are crucial for
the estimation of the total equivalent acceleration noise.
## 2 Spacetime metrology
This chapter is devoted to discussing on the significance of the Doppler link
as a detector to track the spacetime curvature and show the road toward the
real detection of GWs. The Doppler link comprises two free-falling particles
exchanging photons. As a GW passes through that region, the relative velocity
between the particles changes as well and produces a frequency shift in the
detected photon. The calculation of the natural physical observable discussed
here – the fractional frequency shift – is formally equivalent to the well-
known integration of null geodesics found in literature. This thesis presents
a novel derivation by employing the fact that the underlying mathematical
operation producing the shift is the parallel transport of 4-vectors.
Subsequently, the chapter stresses that many problems may worsen the real
extraction of GW signals from Doppler measurements. In fact, (i) the particles
are nearly in free fall, which means that noise forces push the masses away
from the reference optimal geodesics; (ii) there are sensing inaccuracies;
(iii) the TMs are extended bodies; (iv) the SCs are extended body coupling
with the motion of the TMs. In realistic conditions like these, a useful
concept is to describe the Doppler link as a differential accelerometer whose
inputs are equivalent accelerations. Therefore, GW signals, real forces,
sensing noise, pointing inaccuracies and extended body dynamics can be all
treated as equivalent input accelerations. One more benefit is that
performances of different gravitational experiments whose measurement
principle is based on free-falling TMs can be compared at the level of
equivalent differential acceleration noise.
### 2.1 Metrology without noise
The fundamental measurement scheme of LISA and LPF is the Doppler link between
two free-falling TMs embedded into a gravitational field. This section
introduces the physics of the Doppler link, viewed as the rod to track the
spacetime curvature in a purely idealistic viewpoint where noise does not
affect the measurement and the TMs are in perfect free fall 111Otherwise, the
TMs would have non-zero acceleration and even in this idealistic situation
theory needs some care. See Appendix A.2 for a discussion.. An emitter sends a
photon to a faraway counterpart; the receiver measures the photon frequency
and compares it to a reference frequency of a locally emitted photon. The
comparison requires the emitter and receiver to have their clocks previously
synchronized to a common reference. As such possible error is a subject of
TDI, the following assumes a perfect synchronization.
Denoting with $k^{\mu}$ the photon wave 4-vector, the frequency of the photon
measured by any observer with 4-velocity $v^{\mu}$ is the scalar product
$\omega=k_{\mu}v^{\mu}$ [30]. The measured frequency shift of a photon
produced by an emitter with velocity $v^{\mu}_{\text{e}}$ at the event
$x^{\mu}_{\text{e}}$ and detected by a receiver with velocity
$v^{\mu}_{\text{r}}$ at the event $x^{\mu}_{\text{r}}$, both in free fall, is
given by [27, 31]
$\delta\omega_{\text{e}\rightarrow\text{r}}=k_{\mu}\Delta
v^{\mu}_{\text{e}\rightarrow\text{r}}\leavevmode\nobreak\ ,$ (2.1)
where all quantities are measured by the receiver and the operation $\Delta
v^{\mu}_{\text{e}\rightarrow\text{r}}$ implements the difference between
$v^{\mu}_{\text{r}}$ and $v^{\mu}_{\text{e}}$, parallel-transported from
$x^{\mu}_{\text{e}}$ to $x^{\mu}_{\text{r}}$
$\Delta
v^{\mu}_{\text{e}\rightarrow\text{r}}=v^{\mu}_{\text{r}}(x^{\alpha}_{\text{r}})-v^{\mu}_{\text{e}}(x^{\alpha}_{\text{e}}\xrightarrow{\text{\tiny{parallel}}}x^{\alpha}_{\text{r}})\leavevmode\nobreak\
,$ (2.2)
where by definition $v^{\mu}_{\text{e}}$ is parallel-transported along the
photon path if $v^{\mu}_{\text{e}\leavevmode\nobreak\ ;\alpha}k^{\alpha}=0$
and the photon path is defined by the null geodesic equation
$k^{\mu}_{\leavevmode\nobreak\ ;\alpha}k^{\alpha}=0$. As usual in GR, a
semicolon is a covariant derivative, whereas a comma is an ordinary
derivative. In (2.2) an $\alpha$-index is used for clearness, but it does not
have relevance for all tensor operations. A representative pictorial view of
the operation being performed is shown in Figure 2.1.
Figure 2.1: Pictorial view of the Doppler link. A free-falling emitter with
4-velocity $v^{\mu}_{\text{e}}$ sends a photon at the event
$x^{\mu}_{\text{e}}$. The photon has wave vector $k^{\mu}$ and is detected by
a free-falling receiver with 4-velocity $v^{\mu}_{\text{r}}$ at the event
$x^{\mu}_{\text{r}}$. In order for the Doppler frequency shift to be recorded,
$v^{\mu}_{\text{e}}$ must be parallel-transported from $x^{\mu}_{\text{e}}$ to
$x^{\mu}_{\text{r}}$, in this way tracking the spacetime curvature along the
null geodesic $\gamma$.
The formula (2.1) can be split into two terms that make the understanding
easier. In order to do that, it is necessary to integrate the equation
governing the parallel transport of $v^{\mu}_{\text{e}}$ in (2.2). Firstly, it
is worth observing that $k^{\mu}=\text{d}x^{\mu}/\text{d}\lambda$, where
$\lambda$ is an affine parameter and $x^{\mu}$ spans the photon geodesic.
Therefore, using the definition of the covariant derivative it holds
$\begin{split}0&=v^{\mu}_{\text{e}\,;\alpha}k^{\alpha}=\left(v^{\mu}_{\text{e}\,,\alpha}+\Gamma^{\mu}_{\alpha\beta}v^{\beta}_{\text{e}}\right)\frac{\text{d}{x^{\alpha}}}{\text{d}{\lambda}}\\\
&\implies\frac{\partial{v^{\mu}_{\text{e}}}}{\partial{x^{\alpha}}}=-\Gamma^{\mu}_{\alpha\beta}v^{\beta}_{\text{e}}\\\
&\implies\text{d}v^{\mu}_{\text{e}}=-\Gamma^{\mu}_{\alpha\beta}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}\leavevmode\nobreak\
,\end{split}$ (2.3)
where $\Gamma^{\mu}_{\alpha\beta}$ are the Christoffel symbols for the
underlying curved spacetime. Substituting the preceding in (2.2) the following
expression turns out
$\begin{split}\Delta v^{\mu}_{\text{e}\rightarrow\text{r}}=\delta
v^{\mu}_{\text{e}\rightarrow\text{r}}+\int_{\gamma}\Gamma^{\mu}_{\alpha\beta}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}\leavevmode\nobreak\
.\end{split}$ (2.4)
where $\gamma:\,x^{\mu}_{\text{e}}\rightarrow x^{\mu}_{\text{r}}$,
parameterized by $\lambda$, is the photon geodesic from the emitter to the
receiver and $\delta
v^{\mu}_{\text{e}\rightarrow\text{r}}=v^{\mu}_{\text{r}}(x^{\alpha}_{\text{r}})-v^{\mu}_{\text{e}}(x^{\alpha}_{\text{e}})$
is the difference in velocity without the parallel-transport of
$v^{\mu}_{\text{e}}$. Finally, the total frequency shift measured by the
receiver reads
$\delta\omega_{\text{e}\rightarrow\text{r}}=\delta\omega_{v}+\delta\omega_{\Gamma}\leavevmode\nobreak\
,$ (2.5)
where
$\displaystyle\delta\omega_{v}$ $\displaystyle=k_{\mu}\delta
v^{\mu}_{\text{e}\rightarrow\text{r}}\leavevmode\nobreak\ ,$ (2.6a)
$\displaystyle\delta\omega_{\Gamma}$
$\displaystyle=k_{\mu}\int_{\gamma}\Gamma^{\mu}_{\alpha\beta}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}\leavevmode\nobreak\
,$ (2.6b)
which correspond to the following two contributions:
1. 1.
the relativistic Doppler shift just due to the relative velocity between the
emitter and the receiver, as if it was in absence of gravity;
2. 2.
the parallel transport term written as a global path integral on the light
beam and dominated by the spacetime curvature between the emitter and the
observer.
Inspecting (2.6b), since $\Gamma^{\mu}_{\alpha\beta}$ goes like a space
derivative of the metric, it can be found that
$\Gamma^{\mu}_{\alpha\beta}v^{\alpha}_{\text{e}}$ goes like a time derivative
of the metric itself. The consequence is that the Doppler shift due to
curvature can be seen as the space integral of the first time derivative of
the metric over the light beam. It is worth noting that such operation of
comparing far apart vectors is not local. Indeed, in GR locality implies
flatness and, if the operation was local, gravity would have no influence on
it: the global behavior of the parallel transport gives gravity a central role
in the Doppler link.
#### 2.1.1 Weak field limit
To better understand the meaning of (2.5) and how curvature affects the
Doppler link through (2.6b), it is a good practice to take the weak field
limit of it. This is also of crucial importance since it shows how GWs can be
effectively detected.
The metric $g_{\mu\nu}$ can be expanded to first order like
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\leavevmode\nobreak\ ,$ (2.7)
with $h_{\mu\nu}$ a perturbation to the flat Minkowski metric $\eta_{\mu\nu}$.
The proper expansion of the Christoffel symbols to first order is
$\Gamma^{\mu}_{\alpha\beta}=\frac{1}{2}\left(h^{\mu}_{\leavevmode\nobreak\
\alpha\,,\beta}+h^{\mu}_{\leavevmode\nobreak\
\beta\,,\alpha}-h_{\alpha\beta}^{\quad,\mu}\right)\leavevmode\nobreak\ ,$
(2.8)
and all indices are raised up by means of $\eta_{\mu\nu}$.
The aim is to estimate the contribution of the perturbation $h_{\mu\nu}$ to
the Doppler shift $\delta\omega_{\Gamma}$, now renamed $\delta\omega_{h}$.
When the underlying spacetime metric is flat the photon geodesic connecting
emitter and receiver can be considered a straight line: hence, the only effect
that parallel transport can cause is a time delay on the emitter 4-vectors. In
this case, $k^{\mu}$ is constant all along the light path with good
approximation and (2.6b) becomes
$\begin{split}\text{d}\omega_{h}&=k_{\mu}\Gamma^{\mu}_{\alpha\beta}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}\\\
&=k_{\mu}\Gamma^{\mu}_{\alpha\beta}v^{\alpha}_{\text{e}}k^{\beta}C_{\lambda}\,\text{d}\tau\leavevmode\nobreak\
,\end{split}$ (2.9)
where $C_{\lambda}=\text{d}\lambda/\text{d}\tau$ is a constant for the linear
transformation [30] that connects the photon affine parameter to a reference
proper time assumed here to be the one measured by the receiver. Considering
that
$\begin{split}\Gamma^{\mu}_{\alpha\beta}k_{\mu}k^{\beta}&=\frac{1}{2}\left(h^{\mu}_{\leavevmode\nobreak\
\alpha\,,\beta}+h^{\mu}_{\leavevmode\nobreak\
\beta\,,\alpha}-h_{\alpha\beta}^{\quad,\mu}\right)k_{\mu}k^{\beta}\\\
&=\frac{1}{2}h^{\mu}_{\leavevmode\nobreak\
\beta\,,\alpha}k_{\mu}k^{\beta}\leavevmode\nobreak\ ,\end{split}$ (2.10)
since the first and third terms cancel out 222Indeed, the third term is
$h_{\alpha\beta}^{\quad,\mu}k_{\mu}k^{\beta}=h_{\alpha\beta\,,\mu}k^{\mu}k^{\beta}=h_{\alpha\leavevmode\nobreak\
,\mu}^{\leavevmode\nobreak\ \beta}k^{\mu}k_{\beta}$ which is exactly the first
term by considering that $\mu$ and $\beta$ are contracted indices and
$h_{\alpha\beta}$ is symmetric., then (2.9) can be recast as
$\text{d}\omega_{h}=\frac{1}{2}h^{\mu}_{\leavevmode\nobreak\
\beta\,,\alpha}k_{\mu}k^{\beta}v^{\alpha}_{\text{e}}C_{\lambda}\,\text{d}\tau\leavevmode\nobreak\
.$ (2.11)
The GW theory usually assumes the well-known traceless-transverse (TT) gauge
$h_{\mu\nu}=\begin{pmatrix}0&\lx@intercol\hfil\cdots\hfil\lx@intercol&0\\\
\hbox{\multirowsetup\vdots}&h_{+}&h_{\times}&\hbox{\multirowsetup\vdots}\\\
&h_{\times}&-h_{+}\\\
0&\lx@intercol\hfil\cdots\hfil\lx@intercol&0\end{pmatrix}\leavevmode\nobreak\
,$ (2.12)
which further simplifies the computation of (2.11). Moreover, the so-called
wave coordinate system can be readily exploited. The $z$ axis is the direction
of the incoming GW and $x$ and $y$ define the polarization plane. See Figure
2.2 for a graphical definition.
Figure 2.2: Definition of the instantaneous wave coordinate system. The GW
propagates along the direction $z$. $x$ and $y$ define the polarization plane.
The 3-vector $\bm{k}$ is firstly projected onto the polarization plane and
then to each of two polarization axes. The concept is better clarified in
(2.20).
Therefore, in the TT gauge and in the wave instantaneous coordinate system, it
holds (see Appendix A.3 for details)
$h^{\mu}_{\leavevmode\nobreak\
\beta\,,\alpha}k_{\mu}k^{\beta}=H_{,\alpha}\leavevmode\nobreak\ ,$ (2.13)
where $H$ is the response to the GW
$H=K_{+}h_{+}+K_{\times}h_{\times}\leavevmode\nobreak\ ,$ (2.14)
and the coefficients $K_{+}$ and $K_{\times}$ are defined by
$\displaystyle K_{+}$ $\displaystyle=k_{x}^{2}-k_{y}^{2}\leavevmode\nobreak\
,$ (2.15a) $\displaystyle K_{\times}$
$\displaystyle=2k_{x}k_{y}\leavevmode\nobreak\ .$ (2.15b)
The meaning of (2.14) is readily clarified: the photon wave vector is
decomposed along the two polarization states of the GW. To look for the
response of the Doppler link to the GW signal, (2.13) is substituted in (2.11)
and the following equation turns out
$\displaystyle\text{d}\omega_{h}$
$\displaystyle=\frac{1}{2}H_{,\alpha}v^{\alpha}_{\text{e}}C_{\lambda}\,\text{d}\tau$
(2.16a)
$\displaystyle=\frac{1}{2}\frac{\partial{H}}{\partial{x^{\alpha}}}\frac{\text{d}{x^{\alpha}_{\text{e}}}}{\text{d}{\tau}}C_{\lambda}\,\text{d}\tau$
(2.16b) $\displaystyle=\frac{1}{2}C_{\lambda}\,\text{d}H\leavevmode\nobreak\
.$ (2.16c)
The preceding can be easily integrated between the instants at which the
photon is emitted and received, $\tau_{\text{e}}$ and $\tau_{\text{r}}$. For
instance, the right-end side is
$\delta H=H(\tau_{\text{r}})-H(\tau_{\text{e}})\leavevmode\nobreak\ ,$ (2.17)
and the equation finally reads
$\delta\omega_{h}=\frac{1}{2}C_{\lambda}\delta H\leavevmode\nobreak\ .$ (2.18)
The result obtained above shows that an incoming GW induces a Doppler
frequency shift on a photon exchanged between two geodesics. The effect is
proportional to the difference between the GW signal at the time of the
receiver and the one, time-delayed, at the time of the emitter, as a strict
consequence of the parallel transport.
The formula in (2.18) can even be put in a more explicit and physically
interpretable form. So far, the following facts have been considered: (i)
weakness of the gravitational field, such that the underlying metric is flat;
(ii) calculation in the TT gauge and in the wave coordinate system. The last
reasonable assumption is about the non-relativistic regime of the test
particles. As a matter of fact, the emitter and the receiver can be assumed to
fall in the gravitational field at low velocities compared to $c$. Hence, all
4-vector equations can be rewritten in terms of 3-vectors. In this
approximation, the definition of the photon wave vector implies
$C_{\lambda}=c/k$ 333Indeed, from the definition of $k^{\mu}$ along the null
geodesics it follows $k^{\mu}\text{d}\lambda=\text{d}x^{\mu}$ and
differentiating with respect to the proper time of the receiver implies
$k^{\mu}\,\text{d}\lambda/\text{d}\tau=\text{d}x^{\mu}/\text{d}\tau$. In the
non-relativistic regime $\text{d}x^{\mu}/\text{d}\tau\rightarrow c$ and using
the definition of $C_{\lambda}$ the relation is finally demonstrated., where
$k$ is the module of $\bm{k}$, the space part of $k^{\mu}$. The GW
polarization responses are symmetric if the wave coordinate system is written
in spherical coordinates 444From the definitions,
$K_{+}=k_{x}^{2}-k_{y}^{2}=k^{2}\sin^{2}\theta\left(\cos^{2}\phi-\sin^{2}\phi\right)=k^{2}\sin^{2}\theta\cos
2\phi$ and
$K_{\times}=2k_{x}k_{y}=2k^{2}\sin^{2}\theta\sin\phi\cos\phi=k^{2}\sin^{2}\theta\sin
2\phi$.
$\displaystyle K_{+}(k,\theta,\phi)$
$\displaystyle=k^{2}\xi_{+}(\theta,\phi)\leavevmode\nobreak\ ,$ (2.19a)
$\displaystyle K_{\times}(k,\theta,\phi)$
$\displaystyle=k^{2}\xi_{\times}(\theta,\phi)\leavevmode\nobreak\ ,$ (2.19b)
and
$\displaystyle\xi_{+}(\theta,\phi)$
$\displaystyle=\sin^{2}\theta\cos{2\phi}\leavevmode\nobreak\ ,$ (2.20a)
$\displaystyle\xi_{\times}(\theta,\phi)$
$\displaystyle=\sin^{2}\theta\sin{2\phi}\leavevmode\nobreak\ ,$ (2.20b)
are the directional sensitivities to each of the two GW polarizations.
$\theta$ is the projection angle, named declination, of $\bm{k}$ onto the GW
polarization plane orthogonal to the $z$ axis defining the GW propagation
direction. Notice in (2.20) that the Doppler response is null, both in
$\xi_{+}$ and $\xi_{\times}$, for $\theta=0$, i.e., when the photon wave
vector is parallel to $z$, whereas is maximum for
$\theta=\nicefrac{{\pi}}{{2}}$, i.e., when is orthogonal to $z$. $\phi$ is the
projection angle, named polarization, onto the two polarization states. In
fact, when $\phi=0,\nicefrac{{\pi}}{{2}}$, then $\xi_{+}$ is maximum and
$\xi_{\times}=0$; when $\phi=\nicefrac{{\pi}}{{4}},\nicefrac{{3\pi}}{{4}}$,
then $\xi_{\times}$ is maximum and $\xi_{+}=0$. See Figure 2.3 for a graphical
interpretation. Since the degeneracy around $\bm{k}$, the right ascension is
not measured with a single photon, but it can be inferred from the modulation
induced by the rotation of the beam.
Figure 2.3: Graphical interpretation of the $\phi$ polarization angle
(measured counterclockwise around $z$). When $\phi=0,\nicefrac{{\pi}}{{2}}$,
then $\xi_{+}$ is maximum and $\xi_{\times}=0$ (dashed lines); when
$\phi=\nicefrac{{\pi}}{{4}},\nicefrac{{3\pi}}{{4}}$, then $\xi_{\times}$ is
maximum and $\xi_{+}=0$, as predicted by (2.20). Hence, any GW signal can be
decomposed into the $+$ and $\times$ polarization states in the $xy$ plane.
The polarization states can be viewed as two independent bases of the
fundamental decomposition
$h(t,\theta,\phi)=\xi_{+}(\theta,\phi)h_{+}(t)+\xi_{\times}(\theta,\phi)h_{\times}(t)\leavevmode\nobreak\
,$ (2.21)
where $h_{+}$ and $h_{\times}$ are the two GW polarization states, $\xi_{+}$
and $\xi_{\times}$ the two directional sensitivities of the Doppler link and
$h$ the Doppler response. (2.18) can be elaborated as
$\begin{split}\delta\omega_{h}&=\frac{1}{2}C_{\lambda}\delta H\\\
&=\frac{1}{2}\frac{c}{k}k^{2}\delta h\\\ &=\frac{1}{2}\omega_{\text{e}}\delta
h\leavevmode\nobreak\ .\end{split}$ (2.22)
where $\omega_{\text{e}}$ is the frequency of the emitted photon and $\delta
h$ denotes the difference between the signal evaluated at detection and
emission. The final result is the fractional frequency shift measured by the
Doppler link
$\frac{\delta\omega_{h}}{\omega_{\text{e}}}=\frac{1}{2}\delta
h\leavevmode\nobreak\ .$ (2.23)
Therefore, if $\delta x$ is the separation between two geodesics, the
fractional frequency shift – the natural physical observable – is proportional
to the difference between the GW response evaluated at the instant of
detection and the one time-delayed to the instant of emission,
$\delta h(t)=h(t)-h(t-\delta x/c)\leavevmode\nobreak\ .$ (2.24)
A particularly interesting discussion is about the long-wavelength limit, for
which the GW wavelength $\lambda\gg\delta x$. By taking the limit for
infinitely small $\delta x/c$, i.e., assuming that the two geodesics are
infinitely close each other or the photon flight time is infinitely small,
there is no parallel transport and $\delta h$ becomes a time derivative
$\delta h\simeq\frac{\delta x}{c}\dot{h}\leavevmode\nobreak\ .$ (2.25)
Analogously, in Fourier domain for $\omega\ll c/\delta x$ it holds
$\begin{split}\delta h&=\left(1-e^{-i\,\frac{\delta x}{c}\,\omega}\right)h\\\
&\simeq i\,\frac{\delta x}{c}\,\omega\,h\leavevmode\nobreak\ .\end{split}$
(2.26)
Therefore, the time delay due to the parallel transport can be effectively
ignored at low frequency. For example, in LISA the long-wavelength limit
applies below $60\,\mathrm{mHz}$ for a photon one-way trip. The fractional
frequency shift becomes proportional to the time derivative of the GW signal
or, equivalently, the phase shift becomes directly proportional to the GW
signal, in fact
$\begin{split}\frac{\dot{\delta\phi}_{h}}{\omega_{\text{e}}}&\simeq\frac{1}{2}\frac{\delta
x}{c}\dot{h}\implies\\\
\delta\phi_{h}&\simeq\frac{1}{2}\omega_{\text{e}}\frac{\delta
x}{c}h\leavevmode\nobreak\ .\end{split}$ (2.27)
This section has shown how the GW signal convolves with the Doppler link and
produces a frequency shift measured by the receiver. The results are well-
known in literature [32, 33], but the difference here is in the derivation.
Instead of integrating the null geodesic, the calculations have been performed
employing the parallel transport of 4-vectors, a very fundamental concept in
GR.
### 2.2 Doppler link as differential accelerometer
This section reformulates the Doppler link as a differential time-delayed
accelerometer. The result is that the Doppler link measures the spacetime
curvature between emitter and receiver, corrupted by differential parasitic
accelerations and non-inertial forces due to the particular choice of the
detector reference frame in which the measurement is performed.
Consider the frequency shift in (2.5), induced by the classical Doppler
contribution in (2.6a) and the contribution due to the parallel transport in
(2.6b). For LISA, in the weak-field limit the metric can be decomposed as
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu\,\odot}+h_{\mu\nu\,\oplus}+h_{\mu\nu}\leavevmode\nobreak\
,$ (2.28)
where $\eta_{\mu\nu}$ is the flat Minkowski metric, $|h_{\mu\nu\,\odot}|\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}2\\!\times\\!10^{-12}$ is the
perturbation due to the Sun gravity and $|h_{\mu\nu\,\odot}|\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}2\\!\times\\!10^{-17}$ is the
perturbation due to the Earth gravity. $h_{\mu\nu}$ is the perturbation due to
GWs; since $|h_{\mu\nu}|\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}|h_{\mu\nu\,\odot}|^{2}$, it is clearly smaller than the average local
gravity of the Solar System. Expanding the Christoffel symbols to second order
for the local gravity and to first order for the GW perturbation, it becomes
$\Gamma^{\mu}_{\alpha\beta}\rightarrow\Gamma^{\mu}_{\alpha\beta\,\odot}+\Gamma^{\mu}_{\alpha\beta\,\oplus}+\Gamma^{\mu}_{\alpha\beta}\leavevmode\nobreak\
,$ (2.29)
since the mixed products between $h_{\mu\nu}$ and $h_{\mu\nu\,\odot}$ are
negligible with respect to $h_{\mu\nu}$. Hence, the effect of the local
gravity within the Solar System can be separated from the effect of GWs.
Moreover, these effects intervenes at typical frequencies 555Around
$3\\!\times\\!10^{-8}\,\mathrm{Hz}$ for the revolution about the Sun and
around $4\\!\times\\!10^{-7}\,\mathrm{Hz}$ for the revolution of the Moon
about the Earth. below the LISA measurement band.
In the same way, for low velocities, i.e. small compared to $c$, all mixed
products between $h_{\mu\nu}$ and velocity are second order. Analogously, the
parallel transport of the acceleration contributes to second order. To first
order, $k^{\mu}$ is constant along the light path and differentiating (2.5),
with respect to the proper time of the receiver $\tau$, it holds
$\dot{\delta\omega}_{\text{e}\rightarrow\text{r}}=k_{\mu}\delta
a^{\mu}_{\text{e}\rightarrow\text{r}}+k_{\mu}\int_{\gamma}\frac{\text{d}{\Gamma^{\mu}_{\alpha\beta}}}{\text{d}{\tau}}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}\leavevmode\nobreak\
,$ (2.30)
where the derivative commutes with the integral as the variation of the
extremes of integration contributes to second order. Hence, the differential
time-delayed accelerometer measures the effect of the parallel transport and
differential parasitic accelerations between emitter and receiver. For the
rest, $\Gamma^{\mu}_{\alpha\beta}$ describes only the GW perturbation, bearing
in mind that there the gravity of the Solar System falls below the measurement
band.
To first approximation, the frequency shift is now evaluated in a reference
frame in where emitter and receiver appear at rest. Since a net relative
velocity is a Doppler effect, this is eventually included in the first term
and, in fact, in LISA it must be considered in the calculation, even though it
intervenes at frequencies again lower than the measurement band. In such a
reference frame, $v^{\alpha}_{\text{e}}=(c,0,0,0)$, the $x$-axis is aligned to
$k^{\mu}$ to have $k^{\mu}=k\,(1,1,0,0)$ and
$\text{d}x^{\beta}=(c\,\text{d}t,\text{d}x,0,0)$. In addition,
$\text{d}/\text{d}\tau=\text{d}t/\text{d}\tau\,\text{d}/\text{d}t$, where
$\text{d}t/\text{d}\tau=1$ is the Lorentz factor and
$\text{d}/\text{d}t=\partial/\partial t=c\,\partial_{0}$ for low relative
velocities. From the definition, $\text{d}x^{\beta}=k^{\beta}\text{d}\lambda$,
the second term in (2.30) becomes
$k_{\mu}\int_{\gamma}\frac{\text{d}{\Gamma^{\mu}_{\alpha\beta}}}{\text{d}{\tau}}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}=c^{2}k^{2}\int_{\gamma}\partial_{0}\left(\Gamma^{0}_{00}+\Gamma^{0}_{01}-\Gamma^{1}_{00}-\Gamma^{1}_{01}\right)\text{d}\lambda\leavevmode\nobreak\
.$ (2.31)
where a $c$ comes from the velocity and another one from $\partial_{0}$. Using
the expansion of the Christoffel symbols in (2.8) it follows that
$\Gamma^{0}_{00}+\Gamma^{0}_{01}-\Gamma^{1}_{00}-\Gamma^{1}_{01}=\frac{1}{2}\left(h_{00,0}+2h_{01,0}+h_{11,0}\right)\leavevmode\nobreak\
.$ (2.32)
Applying the derivative, the integrand becomes
$\partial_{0}\left(\Gamma^{0}_{00}+\Gamma^{0}_{01}-\Gamma^{1}_{00}-\Gamma^{1}_{01}\right)=\frac{1}{2}\left(h_{00,00}+2h_{01,00}+h_{11,00}\right)\leavevmode\nobreak\
.$ (2.33)
In these approximations, the only independent component of the Riemann tensor
that can be observed along the beam is $R_{0110}$ 666The number of independent
components of the Riemann tensor are $\nicefrac{{1}}{{12}}\,n^{2}(n^{2}-1)$,
where $n$ is the number of dimensions. The particular choice of the reference
frame is equivalent to working within a 2-dimensional space. that, to first
order, is given by
$R_{0110}=\frac{1}{2}\left(h_{00,11}-2h_{01,01}+h_{11,00}\right)\leavevmode\nobreak\
.$ (2.34)
The integral can be recast as
$\begin{split}k_{\mu}&\int_{\gamma}\frac{\text{d}{\Gamma^{\mu}_{\alpha\beta}}}{\text{d}{\tau}}v^{\alpha}_{\text{e}}\,\text{d}x^{\beta}=\\\
&c^{2}k\int_{\gamma}\left[R_{0110}+\frac{1}{2}\left(h_{00,00}+2h_{01,00}+2h_{01,01}-h_{00,11}\right)\right]\text{d}x\leavevmode\nobreak\
.\end{split}$ (2.35)
Dividing by $\omega_{\text{e}}$ the result is the derivative of the fractional
frequency shift; multiplying this by $c$ the result is the equivalent input
acceleration in terms of curvature
$\delta a_{R}=c^{2}\int_{\gamma}R_{0110}\,\text{d}x\leavevmode\nobreak\ ,$
(2.36)
The equivalent input acceleration in terms of the additional contribution is
$\delta
a_{\text{gauge}}=\frac{1}{2}c^{2}\int_{\gamma}\left(h_{00,00}+2h_{01,00}+2h_{01,01}-h_{00,11}\right)\,\text{d}x\leavevmode\nobreak\
.$ (2.37)
Solving the linearized Einstein equations in the $(ct,x)$ coordinates, it
follows $h_{00,11}=h_{11,11}$ (see Appendix A.4); since this does not simplify
the above formula, the additional contribution is interpreted merely as a
gauge effect depending on the particular choice of the reference frame. The
fixing of a proper gauge should be able, in principle, to suppress those
terms. A local gauge transformation in $h_{\mu\nu}$ is defined by
$h^{\prime}_{\mu\nu}=h_{\mu\nu}-\xi_{\mu,\nu}-\xi_{\nu,\mu}\leavevmode\nobreak\
,$ (2.38)
where $h^{\prime}_{\mu\nu}$ is the transformed perturbation and $\xi_{\mu}$
are infinesimal shifts in the coordinates
$x^{\prime\mu}=x^{\mu}+\xi^{\mu}\leavevmode\nobreak\ .$ (2.39)
As the above is a tranformation between two reference frames, the gauge terms
are interpreted as non-inertial forces.
The conclusion of the section is that the LISA arm can be viewed as a
differential time-delayed accelerometer measuring equivalent input
acceleration. It measures the spacetime curvature between emitter and receiver
along the light beam. The measurement is corrupted by: (i) parasitic
differential forces affecting the geodesic motion of the TMs; (ii) the
curvature due to the Solar System at frequencies below the measurement band;
(iii) non-inertial forces mainly due to the rotation of the arm.
### 2.3 Metrology with noise
This section presents a series of issues in the actual measurement of
frequency shifts by means of the Doppler link. The results of Section 2.1 can
be summarized in (2.5), (2.18) and subsequently in (2.23), but have been
obtained in very idealistic conditions.
There are many points where noise, non-idealities, etc., may enter into the
measurement. However, taking a look on (2.30), noise sources and disturbances
corrupt the GW detection at the level of differential accelerations. The
emitter and the receiver are faraway of being in free fall because of the
presence of many external non-gravitational forces. The environment can be
chosen to be as quiet as possible, but in reality many disturbances can take
the emitter and the receiver away from the purely gravitational geodesic
$\frac{\text{d}^{2}x^{\mu}}{\text{d}\tau^{2}}+\Gamma^{\mu}_{\alpha\beta}\frac{\text{d}{x^{\alpha}}}{\text{d}{\tau}}\frac{\text{d}{x^{\beta}}}{\text{d}{\tau}}=\frac{f^{\mu}}{m}\leavevmode\nobreak\
,$ (2.40)
where $m$ is the particle mass and $f^{\mu}$ are the external non-
gravitational noise forces affecting the exact knowledge of
$x^{\mu}_{\text{e}}$ and $x^{\mu}_{\text{r}}$. In this way the photon geodesic
is determined only to within a given uncertainty given by the noise in the
coordinates. Actually, emitter and receiver are not pointlike, but are
extended bodies introducing more degrees of freedom in the dynamics and an
extra source of indetermination as it is discussed in the next section. In
addition, the future position of the receiver can not be determined a priori
with absolute precision and there are surely pointing misalignments affecting
the measurement.
To defend the TM from the “polluted environment” in which it is embedded, an
isolating box, the SC, contains and protects it. This prevents the TM from
being disturbed by external non-gravitational forces, but introduces a series
of parasitic couplings to the SC, mostly electromagnetic and self-gravity,
which must be measured and compensated.
The classical Doppler shift is a deterministic signal that does contribute,
but at much lower frequencies and can be effectively subtracted from the data.
In LISA the Doppler effect is minimized in advance in the experimental design
by optimizing the SC orbits, so that the maximum allowed relative fluctuation
of the arm lengths is few percents.
Table 2.1 summarizes some types of disturbances, starting from the most
relevant ones, playing the role of imperfections for the detection of GWs
through the Doppler link in LISA.
Table 2.1: Sources of indetermination for the GW detection through the Doppler link in LISA. Disturbances | Note
---|---
classical Doppler shift | minimized in orbit design, but out of band
laser frequency fluctuation | abated in post-processing by 7 orders of magnitude with TDI
differential forces | mostly coupling forces between the TM and the SC, estimated and characterized by LPF
displacement sensing between the TM and the SC | readout noise, pointing inaccuracies, estimated and characterized by LPF
displacement sensing between two SCs | readout noise, pointing inaccuracies, peculiarity of LISA
extended body dynamics | dynamical, sensing and actuation cross-talk, estimated and characterized by LPF
clock stability | required by TDI
The next subsections introduce in turn the three most relevant noise
contributions in LISA: (i) the frequency noise due to laser instability and
largely compensated on-ground through TDI; (ii) the acceleration noise due to
force couplings between the TM and the SC; (iii) the readout noise due to the
interferometric sensing.
#### 2.3.1 Laser frequency noise
The practical implementation of the Doppler link between two faraway TMs in
nominal free fall, like in LISA and all spaced-based missions, has a
fundamental problem. The laser interferometry on ground is based on equal arms
and power recycling: therefore it can not be of any help for the space-based
detectors. In fact, there are mostly two reasons for this. On one hand, it is
impossible to put two satellites in space with fixed and constant separation
without taking into account of the Keplerian evolution. On the other hand,
there is a huge light power dispersion among million of kilometers preventing
the same signal of being bounced back in order to do the usual interferometry.
In a LISA arm, the light signal is sent toward the other SC where it is
compared to a local reference signal. Therefore, a LISA arm, as shown in
Figure 2.4, is obtained by a combination of lower-level measurements between
four bodies: two TMs and two SCs. In the language of the preceding section the
emitter coincides nominally with $\text{TM}_{2}$ and the receiver coincides
nominally with $\text{TM}_{1}$. Hence, the TM-to-TM link can be effectively
depicted with three interferometric measurements: $\text{TM}_{2}$ to its
hosting $\text{SC}_{2}$, $\text{SC}_{2}$ to $\text{SC}_{1}$, and
$\text{TM}_{1}$ to its hosting $\text{SC}_{1}$. The frequency shift between
two faraway TMs for a light ray from $\text{TM}_{2}$ to $\text{TM}_{1}$, can
be constructed as follows
$\begin{split}\frac{\delta\omega_{2\rightarrow
1}}{\omega_{\text{e}}\leavevmode\nobreak\
}&=\frac{1}{c}\hat{\bm{k}}\cdot\left(\bm{v}_{\text{TM}_{2}}-\bm{v}_{\text{TM}_{1}}\right)\\\
&=\frac{1}{c}\hat{\bm{k}}\cdot\left(\bm{v}_{\text{TM}_{2}}^{(\text{SC})}+\bm{v}_{\text{SC}_{2}}-\bm{v}_{\text{TM}_{1}}^{(\text{SC})}-\bm{v}_{\text{SC}_{1}}\right)\\\
&=\frac{1}{c}\hat{\bm{k}}\cdot\left(\delta\bm{v}_{\text{SC}}+\bm{v}_{\text{TM}_{2}}^{(\text{SC})}-\bm{v}_{\text{TM}_{1}}^{(\text{SC})}\right)\leavevmode\nobreak\
,\end{split}$ (2.41)
where $\delta\bm{v}_{\text{SC}}$ is the measurement between the two SCs
containing the time delay due to the photon flight time (about
$17\,\mathrm{s}$ for LISA); $\bm{v}_{\text{TM}_{1}}^{(\text{SC})}$ is the
local measurement between $\text{TM}_{1}$ and its hosting SC;
$\bm{v}_{\text{TM}_{2}}^{(\text{SC})}$ is the local measurement between
$\text{TM}_{2}$ and its hosting SC, but time-delayed by the photon flight
time. Obviously, the three measurements contain noise sources at different
levels, but the GW signal is masked within the first one.
|
---|---
(a) | (b)
Figure 2.4: Scheme of the LISA constellation. (a) the single arm is made of
two TMs, each contained in two faraway SCs; the Doppler link is obtained by
three independent measurements: two local measurements (TMs to their optical
benches) and a faraway measurement (SC to SC). (b) the constellation comprises
6 Doppler links, forth and back for every arm.
When extended to whole LISA configuration with 6 TMs, 6 faraway links and 6
local links, as shown in Figure 2.4, the adopted scheme contains an
unavoidable large laser frequency fluctuation noise of $\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}10^{-13}\,\mathrm{Hz^{-\nicefrac{{1}}{{2}}}}$ due to arm length imbalances
of a few percent. Such disturbance can be mitigated by TDI [26, 34] in data
post-processing allowing for the compensation of arm length imbalances and
optical bench vibrations (1st generation TDI), as well as arm flexing (2nd
generation TDI).
A more abstract notation can be introduced for describing the problem.
Referring to Figure 2.4 and assuming the standard naming convention of TDI,
the SCs are numbered clockwise with index $k$, each arm is labeled with the
number of the opposing SC, each TM is numbered as the hosting SC, but it is
primed if it is on the right side of the SC. The photodiode outputs
corresponding to the local measurements are named, $s_{k}$ for the $k$-th TM
and $s_{k}^{\prime}$ for the $k^{\prime}$-th TM. The photodiode outputs
corresponding to the faraway incoming link between the SCs are named, $S_{k}$
for the side of the $k$-th TM and $S_{k}^{\prime}$ for the side of the
$k^{\prime}$-th TM. The result in (2.41) can be generalized to any incoming
Doppler links on the left and right sides of the $k$-th SC
$\displaystyle\sigma_{k}(t)$
$\displaystyle=s_{k}(t)+S_{k}(t)-s_{p[k]}^{\prime}(t-T_{p^{2}[k]})\leavevmode\nobreak\
,$ (2.42a) $\displaystyle\sigma_{k}^{\prime}(t)$
$\displaystyle=s_{k}^{\prime}(t)+S_{k}^{\prime}(t)-s_{p^{2}[k]}(t-T_{p[k]})\leavevmode\nobreak\
,$ (2.42b)
where $p[k]$ is the cyclic permutation of $(123)$ and $p^{2}[k]=p[p[k]]$.
$T_{k}$ is the time delay in the $k$-th arm assumed constant within the 1st
generation TDI. Notice the symmetry of the preceding equations: an unprimed
index goes to a primed one (and vice-versa) and $p[k]$ goes to $p^{2}[k]$ (and
vice-versa).
The 1st generation TDI solution corresponding to an unequal-arm Michelson
interferometer with the $k$-th SC at its vertex is a linear combination of
time-shifted photodiode outputs given by
$\begin{split}X_{k}(t)&=\sigma_{k}^{\prime}(t)+\sigma_{p^{2}[k]}(t-T_{p[k]})+\sigma_{k}(t-2T_{p[k]})+\sigma_{p[k]}^{\prime}(t-T_{p^{2}[k]}-2T_{p[k]})\\\
&\quad-\left[\sigma_{k}(t)+\sigma_{p[k]}^{\prime}(t-T_{p^{2}[k]})+\sigma_{k}^{\prime}(t-2T_{p^{2}[k]})+\sigma_{p^{2}[k]}(t-T_{p[k]}-2T_{p^{2}[k]})\right]\leavevmode\nobreak\
,\end{split}$ (2.43)
which contains the round-trip delay of $\sigma_{k}$ in the $p[k]$-th arm and
$\sigma_{k}^{\prime}$ in the $p^{2}[k]$-th arm. Such combinations are able to
cancel out the frequency fluctuation noise of arm length imbalances and
optical bench vibrations 777The 2nd generation TDI solution can be derived
considering that the photon flight times are not constant and the time delays
do not commute anymore. Such combinations can compensate the arm flexing, but
introduces much more complexity in the system. to $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}10^{-20}\,\mathrm{Hz^{-\nicefrac{{1}}{{2}}}}$
corresponding to the differential acceleration requirement of
$3\\!\times\\!10^{-15}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$ 888In fact,
$3\\!\times\\!10^{-15}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}/[(2\pi\times
1\,\mathrm{mHz})^{2}\times
5\\!\times\\!10^{6}\,\mathrm{km}]=1.5\\!\times\\!10^{-20}\,\mathrm{Hz^{-\nicefrac{{1}}{{2}}}}$..
#### 2.3.2 Residual acceleration noise
The first remaining contribution after the TDI compensation of the frequency
fluctuation noise is the residual acceleration noise, also frequently named
force (per unit mass) noise, whose characterization is one of the main
scientific targets of the LPF mission.
Considering the low velocity regime of (2.5), whose GW signal is given by
(2.23), the Doppler link expressed as the time derivative of the fractional
frequency shift is
$\frac{\dot{\delta\omega}_{\text{e}\rightarrow\text{r}}}{\omega_{\text{e}}\leavevmode\nobreak\
}=\frac{1}{c}\delta a_{\parallel}+\frac{1}{c}\delta
a_{\bot}+\frac{1}{2}\dot{\delta h}\leavevmode\nobreak\ ,$ (2.44)
where the first two terms are accelerations, parallel and orthogonal to the
line of sight $\hat{\bm{k}}$ defined by the light beam
$\displaystyle\delta a_{\parallel}$
$\displaystyle=\hat{\bm{k}}\cdot\delta\bm{a}_{\text{e}\rightarrow\text{r}}\leavevmode\nobreak\
,$ (2.45a) $\displaystyle\delta a_{\bot}$
$\displaystyle=\dot{\hat{\bm{k}}}\cdot\delta\bm{v}_{\text{e}\rightarrow\text{r}}\leavevmode\nobreak\
.$ (2.45b)
This shows that the Doppler link reads the GW signal, but also accelerations
longitudinal and transversal to the line of sight.
The deep meaning of (2.44) is that signals and all unwanted noise sources
effectively enter into the Doppler link as equivalent time-delayed
accelerations that can be modeled as
$\frac{\dot{\delta\omega}_{\text{e}\rightarrow\text{r}}}{\omega_{\text{e}}\leavevmode\nobreak\
}=\frac{1}{c}\left(\delta a_{\text{n}}+\delta a_{h}\right)\leavevmode\nobreak\
,$ (2.46)
analogous to the reformulation of the Doppler link as a differential
accelerometer in (2.30). In fact, the GW signal equivalent acceleration is
$\delta a_{h}=\frac{c}{2}\dot{\delta h}\leavevmode\nobreak\ ,$ (2.47)
which becomes proportional to the second time-derivative of the GW signal in
the long-wavelength limit (2.25)
$\delta a_{h}\simeq\frac{\delta x}{2}\ddot{h}\leavevmode\nobreak\ .$ (2.48)
Spurious sources are overall contained in $\delta a_{\text{n}}$ and expressed
as equivalent accelerations. They can all be categorized in two types of
contributions following the idea of (2.44): those along the light path – the
most important contribution due to real differential forces (per unit mass)
$\nicefrac{{\delta f}}{{m}}$ acting between the TMs, with a typical spectral
shape $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}\omega^{-n}$,
$n\simeq 1,2,4$ – and those orthogonal – the cross-talk from other degrees of
freedom to the optical axis introduced with some examples in Section 2.4.
#### 2.3.3 Readout noise
The second noise contribution after the TDI compensation is the
interferometric sensing noise due to various unsuppressed frequency
fluctuations. The interferometric sensing is usually expressed in terms of
displacement $\delta x$ having the typical spectral shape of (1.2), i.e., flat
at high frequency and $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}\omega^{-2}$ at low frequency.
As already discussed for the GW signal, even the readout noise can be
expressed in terms of equivalent acceleration as input to the differential
accelerometer, by multiplying the displacement spectrum by $\omega^{2}$. In
fact, if the noise PSD in displacement is
$S_{\text{n},\delta x}^{\nicefrac{{1}}{{2}}}(\omega)=\delta
x_{0}\left[1+\left(\frac{\omega_{0}}{\omega}\right)^{2}\right]\leavevmode\nobreak\
,$ (2.49)
where $\delta x_{0}$ and $\omega_{0}$ are two scaling constants, the
corresponding equivalent acceleration is found to be
$S_{\text{n},\delta a_{x}}^{\nicefrac{{1}}{{2}}}(\omega)=\delta
a_{0}\left[1+\left(\frac{\omega}{\omega_{0}}\right)^{2}\right]\leavevmode\nobreak\
,$ (2.50)
where $\delta a_{0}=\omega_{0}^{2}\,\delta x_{0}$.
The equivalent acceleration to the readout noise can be summed up to the
acceleration noise and assuming the two contributions are uncorrelated, the
noise PSD of the total equivalent acceleration noise is
$S_{\text{n},\delta a}(\omega)=S_{\text{n},\nicefrac{{\delta
f}}{{m}}}(\omega)+\omega^{4}S_{\text{n},\delta x}(\omega)\leavevmode\nobreak\
.$ (2.51)
As a matter of fact, the total equivalent acceleration – the main focus of
this thesis, whose results can be easily extrapolated to LISA and all space-
based missions – is dominated by sensing at high frequency due to the
$\omega^{4}$ factor and differential forces at low frequency. The preceding
shows again that the sensing can be described as input equivalent acceleration
to the LISA arm viewed as differential accelerometer.
#### 2.3.4 Summary
The Doppler link is a de facto differential time-delayed accelerometer: it
measures relative time-delayed accelerations between nominal freely-falling
particles, where the accelerations come from direct forces at low frequency,
sensing at high frequency and the cross-talk from other degrees of freedom
that couples with the dynamics along the optical axis. This approach has two
very practical and useful consequences:
1. 1.
it puts signals, force noise, readout noise and whatever noise sources at the
same level, treating them as equivalent differential accelerations, and
provides for a benchmark to compare them all; even though the aim of this
thesis is not to give a comprehensive review of all noise sources and
systematics, nor a full noise projection, a general idea is given throughout
this work.
2. 2.
it is a means by which very disparate gravitational experiments, ground-based
and space-based missions, with different scientific targets and frequency
bands, can be really qualified within a unified viewpoint; for example, see
[18] (Figure 5 and references therein) for a comparison of the experimental
performances of few missions on gravitational physics, based at some extent on
the ability of putting test particles in geodesics motion.
### 2.4 Dynamics of fiducial points
This section describes in more details two important effects that enter into
the Doppler measurement previously introduced: the body finite extension and
the pointing inaccuracies due to misalignments in the optical device. Both
cases can be traced back to the fact that the fiducial points in which light
is reflected do not coincide with the centers of mass.
A toy model is now introduced in order to give a first understanding of the
problem. Let $x$ be the Doppler measurement axis and $y$ and $z$ the
respective orthogonal ones. Consider a single cubic TM of latus $l$, subjected
to:
1. 1.
a small rotation $\delta\phi$ due to an unsuppressed torque along $z$;
2. 2.
a small translation $\delta y$ due to an unsuppressed force along $y$.
Conversely, both cases may correspond to small misalignments of the optical
bench performing the measurement along $x$. Figure 2.5 gives the proper
geometrical representation where the effects are purposely enlarged for the
sake of clarity.
Figure 2.5: Geometrical representation of misalignments in the measurement
axis. The actual measurement $o_{\text{actual}}$ contains small imperfections
due to unsuppressed TM translations and rotations to the optimal
$o_{\text{optimal}}$ direction pointing the TM center of mass. The fiducial
points where light is reflected are highlighted as big dots.
$o_{\text{actual}}$ differs from $o_{\text{optimal}}$ by $\delta x=\delta
y\tan{\delta\phi}-\left(\frac{l}{2}\frac{1}{\cos{\delta\phi}}-\frac{l}{2}\right)$.
The optimal measurement to the TM center of mass is named
$o_{\text{optimal}}$; the actual misaligned measurement $o_{\text{actual}}$
differs from this by a small amount $\delta x$,
$o_{\text{actual}}=o_{\text{optimal}}+\delta x\leavevmode\nobreak\ ,$ (2.52)
and $\delta x$ contains the cross-talk from both type of imperfections
$\delta x=\delta x_{y}+\delta x_{\phi}\leavevmode\nobreak\ .$ (2.53)
With simple considerations (see Figure 2.5), to second order, it turns out
$\displaystyle\delta x_{y}$ $\displaystyle\simeq\delta
y\delta\phi\leavevmode\nobreak\ ,$ (2.54a) $\displaystyle\delta x_{\phi}$
$\displaystyle\simeq-\frac{l}{4}\delta\phi^{2}\leavevmode\nobreak\ .$ (2.54b)
In fact, when the measurement is performed along the optimal axis, but the TM
is rotated, then only $\delta x_{\phi}$ survives and the contribution is
negative since it subtracts displacement to $o_{\text{optimal}}$. Instead,
when the TM is not rotated $\delta x_{y}$ vanishes. In the general situation
when the TM is both translated and rotated $\delta x_{y}$ is intrinsically
coupled with the $\phi$ motion. The above will be referred as dynamical cross-
talk.
This simple calculation suggests that any detector measuring the relative
motion between two extended bodies reads out a fake signal due to an
unavoidable cross-talk from other degrees of freedom to the optical axis.
Figure 2.6 shows a scheme of a misalignment between a TM and its hosting SC,
affecting the local link within the LISA arm. As said, the local link is
corrupted by force noise coupling the SC motion with the TM. However, small
misalignments of the optical bench and the (linear and angular) motion of the
TM enter into the link. Among the things, the local link will be characterized
by the LPF mission.
Figure 2.6: Scheme of a misalignment between a TM and its hosting SC in the
LISA arm. The (linear and angular) motion of the TM relative to its hosting SC
couples with the Doppler link.
Analogously, Figure 2.7 shows a scheme of a misalignment between two SCs,
affecting the faraway link within the LISA arm. Again, the (linear and
angular) motion between the SCs corrupts the link. Despite to LISA, in LPF
there is only one SC, so LPF will not characterize the link between the SCs.
Figure 2.7: Scheme of a misalignment between two SCs in the LISA arm. The
(linear and angular) motion of a SC relative to the other couples with the
Doppler link.
It is worth to stress that the Doppler link implemented between two faraway
extended bodies measures an unavoidable acceleration coming from the
differential (time-delayed) dynamics of fiducial points.
The LISA arm (4-body system) is a sensor measuring the relative motion of the
TMs with respect to the hosting SCs and the relative motion between the two
SCs. Instead, its down-scaled version to LPF (3-body system) is a sensor
measuring only the relative motion of the TMs with respect to the common
hosting SC. Such difference between LISA and LPF implies that the total number
of degrees of freedom is 24 for LISA and 18 for LPF. However, the linear
motion of the center of mass must be subtracted from this figure as it is
common-mode. Since the relative motion between the TMs is the scientific
degree of freedom, the spurious degrees of freedom are 20 for LISA and 14 for
LPF. Table 2.2 shows each contribution affecting the differential measurement.
As expected, LPF reproduces the LISA arm up to the two local measurements
between the TMs and the SC, but the differential motion between the SCs is a
peculiarity of LISA.
Table 2.2: Spurious sources coming from the dynamics of other degrees of freedom and affecting the main sensitive axis of LISA and LPF. The interferometric arm respectively reads 20 and 14 spurious degrees of freedom in LISA and LPF. The main difference is that in LPF the TMs fit a common SC. Extra-contribution | Degrees of freedom
---|---
LISA | LPF
Linear motion between the SCs | $2$ | -
Linear motion of the TMs | $6$ | $5$
Angular motion of the SCs | $6$ | $3$
Angular motion of the TMs | $6$ | $6$
Total | $20$ | $14$
## 3 Controlled dynamics
As said in the Introduction, LPF is aimed at demonstrating the geodesic motion
of TMs within a single SC reproducing a down-scaled version of the LISA arm.
The previous chapter discussed the fundamental physics of the Doppler link and
the way external forces can be measured by frequency shifts of photons
exchanged between two TMs. The GW signal, non-gravitational disturbances and
all noise sources can be effectively viewed as input equivalent accelerations
to a differential time-delayed accelerometer. In a step by step discussion it
was shown that many effects may corrupt the measurement and, among all, there
is the fact that the link is actually implemented in a dynamical system of 3
extended bodies, whose relative motions are optically tracked with inevitable
pointing inaccuracies and misalignments.
This chapter introduces a further concept: the control. In LISA the drag-free
controller acts as a shield for the external disturbances. In the adopted
scheme, the SCs are actuated to follow the free-falling TMs along the
measurement axes, whereas the TMs are actuated along the degrees of freedom
orthogonal to those axes. This concept is implemented and verified in LPF with
a difference. In LISA each SC contains two TMs belonging to different
measurement axes, the links to the faraway SCs. In LPF, as shown in Figure
3.1, there is only one measurement axis, therefore the SC can not follow both
TMs independently. While the SC follows the reference TM, the other TM must be
capacitively actuated to follow the reference TM. This is the target
configuration named science mode.
Figure 3.1: Simplified scheme of Figure 1.4. In spite of LISA, in LPF there is
only one measurement axis. As the reference TM is in free fall and the SC is
forced to follows it to compensate for external disturbances, the second TM is
forced to follow the reference TM.
Scope of this chapter is to step into the details of the measurement scheme of
LPF. A unified formalism is introduced to describe dynamics, sensing and
control as a whole in view of defining a fundamental operator that:
1. 1.
converts the sensed TM relative motion into total equivalent input relative
acceleration;
2. 2.
subtracts known force couplings, control forces and the cross-talk (sensing,
dynamical and actuation);
3. 3.
suppresses system transients.
Section 3.1 introduces the formalism describing the closed-loop implementation
of the LISA arm in LPF. Section 3.2 discusses on the suppression of system
transients in the total reconstructed equivalent acceleration noise as a
natural consequence of the formalism. Section 3.3 discusses the first
application: a dynamical model of LPF along the optical axis. Section 3.4
presents the mathematical description of the cross-talk from nominally
orthogonal degrees of freedom to the optical axis. Section 3.5 discusses the
second application: an example of cross-talk.
### 3.1 Closed-loop formalism
The formalism developed in this section is effective in mapping a complex
dynamics into a simple equation, treating different aspects of the system at
the same time as a whole, and allowing for the reconstruction of the total
input differential acceleration from the interferometrically-sensed motion.
Like every physical dynamical system, LPF can be described by three main
conceptual parts:
1. 1.
free dynamics;
2. 2.
sensing;
3. 3.
control and actuation.
The first one is the natural free evolution of the system. This gives the
dynamical evolution of the TMs as they were left alone in their flight.
However, small unwanted disturbances can take each TM away from the ideal
geodesic, the reference trajectory. On-ground measurements and models predict
that to first order the TMs are electrostatically coupled with the SC through
negative force gradients described by unstable oscillators. If the TMs were
left to follow their free evolution, the system would exponentially
destabilize in a very small timescale. Referring to Figure 3.1, in the main
science mode the sensed motion between the TM and the interferometer and the
sensed relative motion between the TMs is fed into the DFACS controller to
command actuation on the SC and the second TM to both follow the reference TM.
In this way, one would say that the controller utilizes the sensed relative
motion to suppress the disturbances by “pushing” a body toward the reference
trajectory, i.e., by actuating it along specifical degrees of freedom.
In turn, Section 3.1.1 lists the relevant coordinates in LPF, the sensors, the
control laws and the actuators for each degree of freedom; Section 3.1.2
provides for a general description of the control philosophy; Section 3.1.3
describes the generalized equation of motion for LPF.
#### 3.1.1 Coordinate definitions
As pointed out at the end of the previous chapter, LPF is a 3-body dynamical
system composed by a SC containing two TMs, whose relative motion is sensed by
an interferometer and the capacitive sensors, as described in the
Introduction. As LPF characterizes the relative motion between those bodies,
the inertial acceleration of the SC is not sensed. Therefore, the degrees of
freedom of the system are:
1. 1.
the relative translations of the TM with respect to the SC, $3+3$;
2. 2.
the relative attitudes of the TM with respect to the SC, $3+3$;
3. 3.
additionally, the absolute (inertial) attitude of the SC with respect to the
celestial frame, 3.
The naming convention for the sensed coordinates in LPF in science mode can be
found in Figure 3.2. There are 15 control laws implemented by the DFACS, 12
for the TM relative motions and 3 for the SC absolute attitude. A coordinate
guiding the drag-free loop, i.e., a thruster actuation on the SC, is named
drag-free coordinate. Analogously, a coordinate guiding the electrostatic
suspension loop, i.e., a capacitive actuation on the TMs, is named
electrostatic suspension coordinate. Finally, a coordinate guiding the
attitude loop, i.e., a capacitive actuation on the TMs to maintain the
inertial orientation, is named attitude coordinate. The names of the control
loops, the sensor readouts used as inputs to the control laws and the
actuators are reported in Table 3.1 for all controlled degrees of freedom in
the main science mode.
Figure 3.2: Coordinate naming convention for the 3-body LPF system. The $x$-axis is the laser sensitive translational degree of freedom, as well as the $\eta$ and $\phi$ angles are optically detected. The $\theta$ angle is not interferometrically detectable. Other coordinates can be read out by capacitive sensors, especially along $y$ and $z$. Table 3.1: List of all controlled degrees of freedom for the LPF mission in the main science mode. The drag-free, electrostatic suspension and attitude control loops, together with the interferometer, capacitive and star-tracker sensors and the thruster and capacitive actuators are reported for each coordinate. Interferometric sensing is used in place of the capacitive whenever possible. Notice that the interferometer measures the relative linear and angular motion between the TMs, i.e., $x_{12}=x_{2}-x_{1}$, $\eta_{12}=\eta_{2}-\eta_{1}$ and $\phi_{12}=\phi_{2}-\phi_{1}$. The SC absolute position is not sensed. Coordinate | Control | Sensor | Actuator
---|---|---|---
$x_{1}$ | Drag-free | $o_{1}\,=\,\text{IFO}[x_{1}]$ | FEEP
$y_{1}$ | Drag-free | $o_{y_{1}}\,=\,\text{GRS}[y_{1}]$ | FEEP
$z_{1}$ | Drag-free | $o_{z_{1}}\,=\,\text{GRS}[z_{1}]$ | FEEP
$\theta_{1}$ | Drag-free | $o_{\theta_{1}}\,=\,\text{GRS}[\theta_{1}]$ | FEEP
$\eta_{1}$ | Elect. suspension | $o_{\eta_{1}}\,=\,\text{IFO}[\eta_{1}]$ | GRS
$\phi_{1}$ | Elect. suspension | $o_{\phi_{1}}\,=\,\text{IFO}[\phi_{1}]$ | GRS
$x_{2}$ | Elect. suspension | $o_{12}\,=\,\text{IFO}[x_{12}]$ | GRS
$y_{2}$ | Drag-free | $o_{y_{2}}\,=\,\text{GRS}[y_{2}]$ | FEEP
$z_{2}$ | Drag-free | $o_{z_{2}}\,=\,\text{GRS}[z_{2}]$ | FEEP
$\theta_{2}$ | Elect. suspension | $o_{\theta_{2}}\,=\,\text{GRS}[\theta_{2}]$ | GRS
$\eta_{2}$ | Elect. suspension | $o_{\eta_{12}}\,=\,\text{IFO}[\eta_{12}]$ | GRS
$\phi_{2}$ | Elect. suspension | $o_{\phi_{12}}\,=\,\text{IFO}[\phi_{12}]$ | GRS
$\theta_{\text{SC}}$ | Attitude | $o_{\theta_{\text{SC}}}\,=\,\text{ST}[\theta_{\text{SC}}]$ | GRS
$\eta_{\text{SC}}$ | Attitude | $o_{\eta_{\text{SC}}}\,=\,\text{ST}[\eta_{\text{SC}}]$ | GRS
$\phi_{\text{SC}}$ | Attitude | $o_{\phi_{\text{SC}}}\,=\,\text{ST}[\phi_{\text{SC}}]$ | GRS
Basically, in the main science mode all optical readings are used whenever
possible and:
1. 1.
along $x$: guided by the optical $x_{1}$, the SC is forced to follow the
reference TM through thruster actuation; guided by the optical $x_{12}$ the
second TM is forced to follow the reference TM through capacitive actuation;
2. 2.
along orthogonal degrees of freedom: guided by the average linear motion of
the TMs read out by the capacitive sensors, the SC is forced to follow both
TMs through thruster actuation; guided by the star-tracker inertial attitude
the TMs are oriented through capacitive actuation;
3. 3.
along rotational degrees of freedom: guided by the differential linear motion
of the TMs read out by the capacitive sensors, the SC is forced to follow both
TMs through thruster actuation; guided by the optical TM attitudes both TMs
are oriented through capacitive actuation.
#### 3.1.2 Controller
The controller is a dynamical system (see Figure 3.3), in general
multidimensional, taking the difference between the measured and the reference
trajectories as inputs and producing forces to be applied to the bodies as
outputs. If $\bm{o}$ is the sensed motion, the error signals for all
controlled degrees of freedom are
$\bm{e}=\bm{o}-\bm{o}_{\text{i}}\leavevmode\nobreak\ ,$ (3.1)
where $\bm{o}_{\text{i}}$ are named reference set-point signals or simply
guidance signals.
Figure 3.3: Block diagram of the controller. It takes the differences between
the measured coordinates $\bm{o}$ and the reference coordinates
$\bm{o}_{\text{i}}$ and calculates control forces $\bm{f}_{\text{c}}$ to be
applied to the SC and the TMs.
The DFACS is responsible of the minimization of the error signals. In this
way, it compensates for negative force gradients and makes the system stable.
It utilizes the sensed relative motion along different degrees of freedom,
contained in the error signal, to compute actuation forces
$\bm{f}_{\text{c}}$. The discrete implementation of the $n$-th value of the
commanded force $f_{\text{c},n}$, for a generic control law in LPF [35]
controlling a single degree of freedom, is a linear combination of the past
values of the force $f_{\text{c},n-1},f_{\text{c},n-2},\ldots$ and the present
and past values of the error signal (the innovations) $e_{n},e_{n-1},\ldots$
$f_{\text{c},n}=\sum_{i}q_{i}f_{\text{c},n-i}+\sum_{j}p_{j}e_{n-j}\leavevmode\nobreak\
,$ (3.2)
where $i=1,\ldots,N_{q}$ and $N_{q}$ is the order of the autoregressive
filter; $j=0,\ldots,N_{p}$ and $N_{p}$ is the order of the moving average
filter. The $z$-transform of the above gives the well-known autoregressive
moving average model of the discrete control law
$C(z)=\frac{\sum_{j}p_{j}z^{-j}}{1-\sum_{i}q_{i}z^{-i}}\leavevmode\nobreak\ .$
(3.3)
The control design assures: (i) the compensation of negative force gradients;
(ii) the asymptotic stability; (iii) the mitigation of system resonances;
(iii) the minimal-cost performance, i.e., the control computes the minimum
actuation forces that allow the TMs to reach the reference signals to within
the given accuracy of $5\,\mathrm{nm\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$ (for the relative displacement control as reported in Table
1.2), whose unsuppressed part contributes to the residual noise budget.
ASTRIUM [36] – the main industry contractor of LPF – has provided only the
continuous representation of the controller as a rational function in the
$s$-domain (of maximum order 6), used for system modeling, simulation and
analysis shown in Chapter 4.
#### 3.1.3 Equation of motion
This section describes the formalism on the basis of the modeling of the
closed-loop LPF system. The most important assumption concerns on the
linearity of the equations, i.e., that all physical quantities characterizing
the motion enter linearly into the equations. Here is a list of the involved
limitations:
1. 1.
the force couplings between the TMs and the SC are mainly caused by
electrostatics and SC self-gravity: those forces decay as the inverse of the
distance at most; they are treated to first order as spring-like forces;
2. 2.
the interferometric sensing involves reflections and transmissions through
optical elements: even in geometric optics the equations must involve
trigonometric expressions of the angles; it is assumed that trigonometric
functions confuses with angles, whenever applicable;
3. 3.
the angular motion of a rigid body is described by the Euler equations: they
are non-linear with respect to the angular velocities; if the angular motion
is small, non-linearities are second-order effect.
Since the controller forces the motion around the reference trajectories, it
also assures that the motion is small enough that all forces and non-linear
terms can be expanded to first order with good approximation. In this way, the
coupling forces are modeled as negative spring-like constants; the non-
linearities due to optics and the angular motion can be effectively ignored.
In general, the linearized equations of motion must contain terms to within
the order of an imperfection multiplied by a noise contribution. In fact,
other combinations like a noise contribution multiplied by another noise
contribution are second-order effects and must be neglected. The accuracy to
which linearity is achieved depends on: (i) the assumption that the controller
does not itself introduce non-linearities in the system; (ii) the unsuppressed
noisy motion in the error signals is to within the requirement figure of the
controller.
With these assumptions, LPF is viewed as a closed-loop Multi-Input-Multi-
Output (MIMO) linear time-invariant dynamical system described by vector
equations with operators modeling dynamics, sensing and control. The
linearized equations for LPF are [37], [38, 39] and more recently [40]
$\displaystyle\bm{D}\,\bm{q}$ $\displaystyle=\bm{g}\leavevmode\nobreak\ ,$
(3.4a) $\displaystyle\bm{g}$
$\displaystyle=\bm{f}_{\text{n}}+\bm{A}\left[\bm{f}_{\text{i}}-\bm{C}\,(\bm{o}-\bm{o}_{\text{i}})\right]\leavevmode\nobreak\
,$ (3.4b) $\displaystyle\bm{o}$
$\displaystyle=\bm{S}\,\bm{q}+\bm{o}_{\text{n}}\leavevmode\nobreak\ .$ (3.4c)
The total forces (per unit mass) $\bm{g}$ produce the motion through the
acting of the dynamics operator $\bm{D}$ onto the physical coordinates
$\bm{q}$. The natural physical coordinates for LPF are given by the TM
relative linear and angular motion. $\bm{D}$ is a differential operator
containing time derivatives and the modeled coupling coefficients (the
negative spring constants due to the linearization), and the dynamical cross-
talk from other degrees of freedom to the sensitive axis as well. Section 3.4
generalizes this concept by decoupling the dynamics along the measurement axis
(the nominal dynamics) from the dynamics along other degrees of freedom (the
first-order perturbation). The external forces can be split into pure noise
sources $\bm{f}_{\text{n}}$ – mostly from the SC jitter and within the TM
housings; applied biases $\bm{f}_{\text{i}}$ – directly on each TM and the SC;
applied biases through $\bm{o}_{\text{i}}$ – the controller guidance signals
already discussed in the preceding subsection. $\bm{C}$ is the operator
containing the control laws. By changing the controller guidance signals, net
forces on each body are commanded to the actuators
$\bm{f}_{\text{c}}=-\bm{C}\,(\bm{o}-\bm{o}_{\text{i}})\leavevmode\nobreak\ ,$
(3.5)
where $\bm{o}$ is the closed-loop measurement. Therefore, the application of
biases in the controller guidance signals is equivalent to the application of
explicit forces on the bodies. In this description, the application of the
forces is modeled by an actuation operator $\bm{A}$. All force biases and
control forces are fed into such an operator, responsible of the force
dispatching on all bodies. In the main science mode, along the measurement
axis, this implies a thruster actuation on the SC to follow the reference TM
in free fall and a capacitive actuation on the second TM to follow the
reference TM. Finally, the physical coordinates $\bm{q}$ are converted into
the system readouts $\bm{o}$ (from interferometric and capacitive sensors)
through the sensing operator $\bm{S}$, mostly diagonal, and corrupted by the
readout noise $\bm{o}_{\text{n}}$. $\bm{S}$ is nominally an identity operator,
but in reality there is a sensing cross-talk between different readout
channels and miscalibrations as well. Figure 3.4 shows the block diagram of
the closed-loop dynamics for LPF where all operators act on their own inputs
and produce their outputs for the dynamical equations in (3.4); deterministic
and stochastic inputs are also distinguished for clearness.
Figure 3.4: Block diagram for the three main conceptual steps of LPF:
dynamics, sensing and control. There are two different noise sources,
$\bm{f}_{\text{n}}$ and $\bm{o}_{\text{n}}$, and biases to inject,
$\bm{f}_{\text{i}}$ and $\bm{o}_{\text{i}}$. The open loops are defined by the
transfers from forces to readouts. The forces produce the motion in the
$\bm{q}$ coordinates through the inverse of $\bm{D}$. The coordinates are
converted into sensed coordinates $\bm{o}$ through $\bm{S}$. The controller
closes the loop in order to minimize the error signals, through $\bm{C}$
applied to the sensed coordinates. The calculated forces are then converted
into actuation forces through $\bm{A}$.
The full equation of motion in vector form and expressed in terms of the
sensed relative coordinates, $\bm{o}$, can be obtained by manipulating the
three equations in (3.4). The idea is to substitute (3.4c) and (3.4b) in
(3.4a) and rearrange the equation so that the deterministic and stochastic
inputs are on the right-hand side. The result is the equation of motion in the
sensed coordinates
$\bm{\Delta}\,\bm{o}=\bm{f}_{\text{n}}+\bm{D}\,\bm{S}^{-1}\bm{o}_{\text{n}}+\bm{A}\left(\bm{f}_{\text{i}}+\bm{C}\,\bm{o}_{\text{i}}\right)\leavevmode\nobreak\
,$ (3.6)
where four terms are clearly recognized: force noise, readout noise, force
bias and controller guidance bias, that all constitute the noise budget of LPF
in terms of total equivalent acceleration. The second-order differential
operator on the right-hand side is defined as
$\bm{\Delta}=\bm{D}\,\bm{S}^{-1}+\bm{A}\,\bm{C}\leavevmode\nobreak\ .$ (3.7)
The deep meaning of the operator is that it allows for the reconstruction of
the total equivalent input acceleration from the sensed relative motion and at
the same time isolating and subtracting dynamics, sensing, control and
actuation. Indeed, by looking at Figure 3.4, $\bm{D}\,\bm{S}^{-1}$ is the open
loop from the sensed relative motion to input forces (inverting the direction
of an arrow the corresponding operator must be inverted); whereas
$\bm{A}\,\bm{C}$ is the control loop consisting of all control laws commanding
the force actuation.
In (3.6) two transfer operators can be naturally identified
$\displaystyle\bm{T}_{o\rightarrow f}$
$\displaystyle=\bm{\Delta}\leavevmode\nobreak\ ,$ (3.8a)
$\displaystyle\bm{T}_{o_{\text{i}}\rightarrow o}$
$\displaystyle=\bm{\Delta}^{-1}\bm{A}\,\bm{C}\leavevmode\nobreak\ .$ (3.8b)
The second one solves the equation of motion for deterministic guidance
signals and substituted into (3.5) gives the following transfer operator
$\bm{T}_{o_{\text{i}}\rightarrow
f_{\text{c}}}=-\bm{C}\,(\bm{T}_{o_{\text{i}}\rightarrow
o}-\bm{1})\leavevmode\nobreak\ ,$ (3.9)
converting the bias injections $\bm{o}_{\text{i}}$ into the calculated control
forces that the actuators must apply in order to stabilize the motion toward
the reference signal.
The first transfer operator $\bm{T}_{o\rightarrow f}$ has fundamental
relevance as it shows that the differential operator allows for the estimation
of the total out-of-loop equivalent acceleration noise [41] on noisy
interferometric data, i.e., when all explicit stimuli are set to zero, whose
modeling in terms of force noise and readout noise is provided by the equation
of motion (3.6). However, the evaluation requires the calibration of the
dynamics $\bm{D}$, the sensing $\bm{S}$ and the actuation $\bm{A}$ operators
overall depending on many system parameters. This critical procedure, named
system identification, which the performances of the LPF mission depend on,
will be outlined in Chapter 4. It mainly consists on calibrating the second
transfer operator $\bm{T}_{o_{\text{i}}\rightarrow o}$ and estimating all
system parameters in dedicated experiments.
### 3.2 Suppressing system transients
As the main target of LPF is the estimation of the total equivalent input
acceleration between the TMs, the transfer operator (3.8a), once calibrated,
allows for the compensation of the force gradients, but also for any system
transients. Indeeed, the formalism of the previous section can be applied to
understand in what sense system transients can be suppressed and the extent to
which the suppression is effective.
In the approximation of small relative motion, the dynamical evolution of the
TMs in LPF can be described by a linear differential equation with constant
coefficients
$\bm{\Delta}\,\bm{o}=\bm{f}\leavevmode\nobreak\ ,$ (3.10)
where the external forces $\bm{f}$ produce the motion in the sensed
coordinates $\bm{o}$ through the acting of the second-order differential
operator $\bm{\Delta}$. In LPF the operator also contains negative force
gradients modeled as spring-like constants, the control laws, the actuation
and the sensing conversion between physical coordinates and sensed
coordinates. As it is shown in the next section this complex structure can be
further described with the introduction of targeted parameters modeling the
system. Those parameters may vary in time so that the equation has no longer
constant coefficients: in principle, such behavior could be observable at very
low frequency. Even in this case, there are theorems [42] ensuring the
existence and uniqueness of solutions, at least locally, for reasonable
conditions often met in practice.
The particular solution of (3.10) is provided to satisfy
$\bm{\Delta}\,\bm{o}_{\text{s}}=\bm{f}\leavevmode\nobreak\ ,$ (3.11)
and gives the steady state of the system where the evolution is completely
driven by the external forces, noise in any form viewed as equivalent
acceleration and any possible applied bias. The above equation is usually
well-understood and easily solved, for example, in frequency domain.
On the contrary, the homogeneous solution of (3.10) is provided to satisfy
$\bm{\Delta}\,\bm{o}_{\text{t}}=0\leavevmode\nobreak\ ,$ (3.12)
and gives the transient state of the system. The transient state is not
influenced by the steady state and vice-versa.
The operator kernel is defined by the set of all solutions satisfying (3.12).
It may be natural to question about the dimensionality of the kernel. There
are two possible alternatives [42]:
1. 1.
the kernel is trivial and the only possible homogeneous solution is the null
solution;
2. 2.
the kernel is non-trivial;
Excluding the trivial case, whenever the operator is linear it is proved that
the kernel itself is a vector space – that in case can be provided with a norm
or an inner dot – where any combination of basis functions $\bm{\phi}_{k}$ in
that space
$\bm{o}_{\text{t}}=\sum_{k}c_{k}\,\bm{\phi}_{k}\leavevmode\nobreak\ ,$ (3.13)
is still a solution of the homogeneous equation for $k$ running through the
dimension of the space. $c_{k}$ are some constants depending on the initial
(or boundary) conditions of the system.
So far, an extensive use of inversion operations – in particular of the
$\bm{\Delta}$ operator – has been made in all calculations concerning
dynamics. In general, this may not be completely allowed when the kernel is
non-trivial. In this case the operator is singular by definition and can not
be inverted. Equivalently, transients exist independently from the steady-
state solution driven by the external forces.
The solution of the differential equation (3.10) exists and is unique for
given input forces $\bm{f}$, initial conditions $c_{k}$ and suitable
assumptions. The general solution is a sum of: (i) the steady state
$\bm{o}_{\text{s}}$ proportional to the input forces; (ii) the transient state
$\bm{o}_{\text{t}}$ set by the initial conditions. Hence, by applying the
differential operator to both the steady state and the transient state, it
holds
$\begin{split}\bm{\Delta}\,\bm{o}&=\bm{\Delta}\left(\bm{o}_{\text{s}}+\bm{o}_{\text{t}}\right)\\\
&=\bm{\Delta}\,\bm{o}_{\text{s}}+\sum_{k}c_{k}\,\bm{\Delta}\,\bm{\phi}_{k}\\\
&=\bm{f}\leavevmode\nobreak\ ,\end{split}$ (3.14)
where $k$ spans the kernel. Since $\bm{\Delta}\,\bm{\phi}_{k}=0$ for any
$\bm{\phi}_{k}$ lying in the kernel, the operator automatically suppresses any
system transients, if present.
For example, suppose that a system is modeled by a mono-dimensional harmonic
oscillator, with $\Delta=\text{d}^{2}/\text{d}t^{2}-\omega_{0}^{2}$, where
$\omega_{0}^{2}$ is the spring constant. An external force (per unit mass)
$f(t)$ produces the sensed motion $o(t)$, from which the external force must
be estimated. It is well-known that the transient solution is a combination of
exponentials $o_{\text{t}}(t)=c_{1}\,\exp(-t/\tau)+c_{2}\,\exp(t/\tau)$, where
$c_{1}$ and $c_{2}$ are constants depending on the initial conditions and
$\tau=1/\omega_{0}^{2}$. By definition $\Delta o_{\text{t}}(t)=0$ and $\Delta
o(t)$ provides an estimate of $f(t)$.
The accuracy to which the suppression of system transients is effective
depends on the accuracy to which the system parameters have been previously
calibrated. If $\bm{p}_{\text{est}}$ is the vector of parameter estimates
approximating the “true” parameter values $\bm{p}_{\text{true}}$ modeling the
system up to the inaccuracies $\delta\bm{p}$, i.e.,
$\bm{p}_{\text{est}}\simeq\bm{p}_{\text{true}}+\delta\bm{p}$, then to first
order
$\bm{\Delta}_{\text{est}}\simeq\bm{\Delta}_{\text{true}}+\delta\bm{\Delta}$,
where $\bm{\Delta}_{\text{true}}=\bm{\Delta}(\bm{p}_{\text{true}})$ and
$\bm{\Delta}_{\text{est}}=\bm{\Delta}(\bm{p}_{\text{est}})$. Therefore, it
follows
$\begin{split}\bm{\Delta}_{\text{est}}\,\bm{o}&\simeq\left(\bm{\Delta}_{\text{true}}+\delta\bm{\Delta}\right)\left(\bm{o}_{\text{s}}+\bm{o}_{\text{t}}\right)\\\
&=\bm{\Delta}_{\text{true}}\,\bm{o}_{\text{s}}+\bm{\Delta}_{\text{true}}\,\bm{o}_{\text{t}}+\delta\bm{\Delta}\left(\bm{o}_{\text{s}}+\bm{o}_{\text{t}}\right)\\\
&=\bm{f}_{\text{true}}+\delta\bm{f}\leavevmode\nobreak\ ,\end{split}$ (3.15)
where the first term gives the true forces, the second is identically zero by
definition and the last one gives the systematic errors in the reconstructed
forces
$\delta\bm{f}=\delta\bm{\Delta}\left(\bm{o}_{\text{s}}+\bm{o}_{\text{t}}\right)\leavevmode\nobreak\
.$ (3.16)
From the preceding equation, two cases can be distinguished:
1. 1.
$\bm{o}_{\text{t}}\ll\bm{o}_{\text{s}}$, transients are negligible, but
inaccuracies in the operator still produce systematic errors in the estimated
total equivalent acceleration noise; a similar argument will be used in
Section 4.5 to demonstrate that biases in the estimated parameters may produce
biases in the estimated total equivalent acceleration noise;
2. 2.
$\bm{o}_{\text{t}}\gg\bm{o}_{\text{s}}$, transients are important and
inaccuracies in the operator makes impossible their complete suppression; much
more important, biases in the estimated parameters may even amplify the
transients.
Once more, the suppression of transients is assured to the level by which the
operator itself is calibrated. It is worth stressing the importance of the
result. In LPF the main scientific target is the estimation of the total out-
of-loop equivalent acceleration from the sensed relative motion. The damping
is fundamentally governed by the controller design, i.e., the efficiency to
which the controller responds and stabilizes the system toward the zero-
reference signal. As the controller is designed to mostly compensate the
expected force gradients that are roughly $|\omega^{2}|\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}1\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$, this figure gives a typical
timescale of $\tau\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}6\\!\times\\!10^{3}\,\mathrm{s}$ (almost 2 hours) for the damping of
initial transients. Since the mission operations are very time-constrained, it
is not possible to wait for the steady state and the total equivalent
acceleration noise must be estimated when the system is fully dominated by
transients. The considerations enlightened in this section, together with the
procedures of system identification described in Chapter 4, assure that in the
estimation of the total out-of-loop equivalent acceleration noise, even though
transients could almost certainly last for hours, they can be mitigated with
good and reasonable confidence if the $\bm{\Delta}$ operator is calibrated on
fiducial values of the system parameters.
### 3.3 Dynamical model along $x$
In what follows a model for the LPF mission is elaborated in terms of the two
main degrees of freedom along the optical axis: the relative motion of the
reference TM to the optical bench and the differential motion between the TMs.
In this formulation the relative motion is sensed with the interferometer –
the reference measurement for scientific operations – while keeping in mind
that the capacitive sensors could even be used in place of the interferometer
as a backup option, even though such a measurement would be two orders of
magnitude worse, especially at high frequency. However, along the other
orthogonal axes the capacitive sensors are the only means by which the TM
relative motion can be measured.
By tracing back the equations to Section 3.1, the formalism developed so far
allows for a straightforward description of LPF as a doubly closed-loop
dynamical system in which the effect of the modeled couplings and control laws
must be isolated and subtracted from the data when estimating the total
equivalent acceleration noise. Figure 3.5 shows a sketch of a LPF model, in
the main science mode, along the optical axis that is discussed here in
details.
Figure 3.5: Scheme of the LPF model along the optical axis in the main science
mode. The first TM is in free fall along $x$ and its displacement to the
optical bench ($o_{1}$) is sensed by the interferometer (IFO) and fed into the
controller ($C_{\text{df}}$) to force the SC to follow the TM through thruster
actuation (drag-free loop). Analogously, the sensed differential displacement
between the two TMs ($o_{12}$) is fed into the controller ($C_{\text{sus}}$)
to force the TM to follow the reference one through capacitive actuation
(suspension loop). The critical system parameter are the TM spring-like
couplings to the SC ($\omega_{1}^{2}$ and $\omega_{2}^{2}$), the sensing
cross-talk ($S_{21}$) and the actuation gains ($A_{\text{df}}$ and
$A_{\text{sus}}$). The system can be excited by injecting signals as direct
forces on the masses ($f_{\text{i},1}$, $f_{\text{i},2}$ and
$f_{\text{i},\text{SC}}$) or controller guidance signals ($o_{\text{i},1}$ and
$o_{\text{i},12}$).
Referring to Figure 3.2 and Figure 3.5, $x$ is the interferometric axis.
$x_{\text{SC}}$ is the absolute SC position and $x_{1}$, $x_{2}$ are the
relative TM positions with respect to the SC;
$m_{\text{SC}}=422.7\,\mathrm{kg}$ and $m_{1}=m_{2}=1.96\,\mathrm{kg}$ are the
respective masses; $\tilde{m}_{1}=\tilde{m}_{2}=5\\!\times\\!10^{-3}$ are the
masses normalized to $m_{\text{SC}}$; $f_{1}$, $f_{2}$ and $f_{\text{SC}}$ are
the total forces (per unit mass) containing noise in any form and applied
biases.
In the linear approximation (small motion, small forces, as already discussed)
the 3-body dynamics is described by a linear system of differential equations.
In frequency domain and assuming null initial conditions the equations of
motion are
$\displaystyle
s^{2}\,x_{1}+s^{2}\,x_{\text{SC}}+\omega_{1}^{2}\,x_{1}+\Gamma_{x}\left(x_{2}-x_{1}\right)$
$\displaystyle=f_{1}\leavevmode\nobreak\ ,$ (3.17a) $\displaystyle
s^{2}\,x_{2}+s^{2}\,x_{\text{SC}}+\omega_{2}^{2}\,x_{2}-\Gamma_{x}\left(x_{2}-x_{1}\right)$
$\displaystyle=f_{2}-C_{\text{sus}}(s)\,o_{12}\leavevmode\nobreak\ ,$ (3.17b)
$\displaystyle\begin{split}s^{2}\,x_{\text{SC}}-\tilde{m}_{1}\,\omega_{1}^{2}\,x_{1}-\tilde{m}_{2}\,\omega_{2}^{2}\,x_{2}&=f_{\text{SC}}+C_{\text{df}}(s)\,o_{1}\\\
&\quad-\tilde{m}_{1}\,f_{1}-\tilde{m}_{2}\,f_{2}\\\
&\quad+\tilde{m}_{2}\,C_{\text{sus}}(s)\,o_{12}\leavevmode\nobreak\
,\end{split}$ (3.17c)
where $\omega_{1}^{2}\simeq\omega_{2}^{2}\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}-1\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$ are spring
constants modeling oscillator-like force couplings between the TMs and the SC,
named parasitic stiffness. As the dominating part of such force gradients is
due to electrostatics, the oscillators are unstable: that is the reason why a
controller is employed. $\Gamma_{x}\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}4\\!\times\\!10^{-9}\,\mathrm{s^{-2}}$ is the
gravity gradient (per unit mass) between the TMs corresponding to a nominal
separation of $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}38\,\mathrm{cm}$. All terms containing normalized masses are back-
reactions that can be neglected to zeroth order.
In writing the dynamics the control in the science mode is implicitly assumed,
where the SC is forced to follow a reference TM in free fall along the optical
axis and the other TM is forced to follow the reference TM along the same
axis. As declared by Table 3.1, the interferometric readout $o_{1}$ ($x_{1}$
coordinate) is a drag-free coordinate and is the input to the drag-free
control law $C_{\text{df}}(s)$ assuring thruster actuation. The
interferometric readout $o_{12}$ ($x_{12}=x_{2}-x_{1}$ coordinate) is an
electrostatic suspension coordinate and is the input to the electrostatic
suspension control law $C_{\text{sus}}(s)$ assuring capacitive actuation on
the second TM.
The first step is to rearrange the equations so that to eliminate the
unmeasurable absolute position $x_{\text{SC}}$ and rewrite them in terms of
the two main degrees of freedom $x_{1}$ and $x_{12}$. In fact, by taking the
difference between (3.17b) and (3.17a) the SC acceleration vanishes. Then, the
SC acceleration in (3.17c) is substituted in (3.17a). The structure of the
equations suggests to define the differential forces $f_{12}=f_{2}-f_{1}$ and
the differential parasitic stiffness
$\omega_{12}^{2}=\omega_{2}^{2}-\omega_{1}^{2}$. The equations can be finally
condensed into the formalism of (3.4a), where the dynamics operator has the
following matrix representation
$\bm{D}=\begin{pmatrix}s^{2}+\left(1+\tilde{m}_{1}+\tilde{m}_{2}\right)\omega_{1}^{2}+\tilde{m}_{2}\,\omega_{12}^{2}&\Gamma_{x}+\tilde{m}_{2}\left(\omega_{1}^{2}+\omega_{12}^{2}\right)\\\
\omega_{12}^{2}&s^{2}+\omega_{1}^{2}+\omega_{12}^{2}-2\,\Gamma_{x}\end{pmatrix}\leavevmode\nobreak\
,$ (3.18)
that acting on the system coordinates
$\bm{q}=\begin{pmatrix}x_{1}\\\ x_{12}\end{pmatrix}\leavevmode\nobreak\ ,$
(3.19)
produces the external forces
$\bm{g}=\begin{pmatrix}\left(1+\tilde{m}_{1}+\tilde{m}_{2}\right)f_{1}+\tilde{m}_{2}\,f_{12}-f_{\text{SC}}-C_{\text{df}}(s)\,o_{1}-\tilde{m}_{2}\,C_{\text{sus}}(s)\,o_{12}\\\
f_{12}-C_{\text{sus}}(s)\,o_{12}\end{pmatrix}\leavevmode\nobreak\ .$ (3.20)
The preceding contains force noise sources and injected biases. Neglecting all
back-reactions it shows that the first degree of freedom $x_{1}$ is dominated
by the thruster noise and the drag-free actuation; the second degree of
freedom $x_{12}$ is dominated by the differential force noise and the
capacitive actuation on the second TM. The identified control operator of
(3.4b) is given by
$\bm{C}=\begin{pmatrix}C_{\text{df}}(s)&\tilde{m}_{2}\,C_{\text{sus}}(s)\\\
0&C_{\text{sus}}(s)\end{pmatrix}\leavevmode\nobreak\ ,$ (3.21)
where the off-diagonal quantity is the back-reaction from the suspension to
the drag-free loop.
The dynamical equations shown above assume a perfect actuation. This implies
that $\bm{A}$ is an identity. Otherwise, actuation gains $A_{\text{df}}$ and
$A_{\text{sus}}$ may be conveniently introduced to model the efficiency to
which commanded forces are converted to actual applied forces by the
corresponding loops.
The expression in (3.4c) gives the sensing conversion between the physical
coordinates $\bm{q}$ and the interferometric readouts
$\bm{o}=\begin{pmatrix}o_{1}\\\ o_{12}\end{pmatrix}\leavevmode\nobreak\ ,$
(3.22)
being fed up into the controller. The perfect conversion is represented by an
identity matrix. The imperfect conversion is due to both miscalibrations (the
diagonal terms) or cross-talk contributions (the off-diagonal terms). The on-
ground characterization and the theoretical modeling of the interferometer
[43, 44] suggests that the most relevant is the cross-talk from $o_{1}$ to
$o_{12}$ that mixes the two nominally independent degrees of freedom in the
following way
$\bm{S}=\begin{pmatrix}1&0\\\ S_{21}&1\end{pmatrix}\leavevmode\nobreak\ .$
(3.23)
The cross-talk is explained by a tiny difference in the incidence angles with
which light reflects on the TM surface for the two readings. The photon phase
$\phi$ is built up by taking the difference between the averaged measurement
of the 4 quadrants of each photodiode and the reference phase used for common-
mode rejection of any residual optical path length variation. A generic
displacement readout is proportional to the measured phase
$o=\frac{\lambda}{4\pi\cos{\delta}}\phi\leavevmode\nobreak\ ,$ (3.24)
where $\lambda=1.064\,\mathrm{\mu m}$ is the laser wavelength and
$\delta\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}4.5\,\mathrm{{}^{\circ}}$ is the nominal incidence angle. The sensing
cross-talk due to a small mismatch in incidence angles is
$S_{21}=\frac{\delta_{2}-\delta_{1}}{\delta_{1}}\leavevmode\nobreak\ .$ (3.25)
Therefore, a measured difference of $(\delta_{2}-\delta_{1})\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}5\,\mathrm{{}^{\prime\prime}}$
produces a sensing cross-talk as large as $S_{21}\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}{3\\!\times\\!10^{-4}}$.
The model described in this section, with some slight improvements, has been
extensively used for simulations and analysis – and examples with references
will be discussed in details in the next chapter – to test the algorithms
aimed at estimating the TM couplings, the sensing cross-talk and other
relevant parameters needed for system calibration. Such calibration is also
critical for the unbiased estimation of the total equivalent acceleration
noise. The same model is planned to be employed during the identification
experiments of the LPF mission.
### 3.4 Cross-talk from degrees of freedom other than the optical axis
Section 3.1 has introduced a general formalism – the three master equations of
(3.4) – capable in describing the evolution of LPF as a physical system
continuously subjected to a digital control. Subsequently, in Section 3.3 the
formalism has been applied in the derivation of a model of LPF along the two
main optical degrees of freedom. However, there are sources of non-idealities
in this description. The knowledge of the operators $\bm{D}$, $\bm{A}$ and
$\bm{S}$ might lack because of a poor calibration, misalignments, pointing
inaccuracies as in Section 2.4, force gradients and force noise along
different axes. Such errors couple with the main interferometric axis and
manifest themselves as a cross-talk from other degrees of freedom. Despite
describing the system in its full complexity with all degrees of freedom at
once, in the hypothesis of small motion, the absence of strong non-linearities
assures that the contribution from other degrees of freedom is a small
perturbation to the nominal dynamics along the optical axis.
For the sake of clarity, the approach used here in describing such effects is
based on a first-order perturbation theory, where the zeroth-order dynamics,
named the nominal dynamics, is fully known. The readouts $\bm{o}$ can then be
split into a nominal response of the system to the $x$-dynamics, say
$\bm{o}_{0}$, and a small perturbation coming from other degrees of freedom
($y$, $z$ or some torsional angles), say $\delta\bm{o}$, therefore
$\bm{o}\simeq\bm{o}_{0}+\delta\bm{o}\leavevmode\nobreak\ .$ (3.26)
The leading correcting terms can be embedded into the dynamics as
imperfections to the operators introduced in the previous section. For
example, if $\bm{D}_{0}$ is the nominal dynamics operator, $\delta\bm{D}$ is
the relative imperfection.
To first order the dynamics is a generalization of (3.4) [45]
$\displaystyle(\bm{D}_{0}+\delta\bm{D})\,\bm{q}$
$\displaystyle=\bm{g}\leavevmode\nobreak\ ,$ (3.27a) $\displaystyle\bm{g}$
$\displaystyle=\bm{f}_{\text{n}}+(\bm{A}_{0}+\delta\bm{A})\left[\bm{f}_{\text{i}}-\bm{C}_{0}\,(\bm{o}-\bm{o}_{\text{i}})\right]\leavevmode\nobreak\
,$ (3.27b) $\displaystyle\bm{o}$
$\displaystyle=(\bm{S}_{0}+\delta\bm{S})\,\bm{q}+\bm{o}_{\text{n}}\leavevmode\nobreak\
.$ (3.27c)
where the control laws in $\bm{C}_{0}$ are exactly known from the original
design. Figure 3.6 shows a block diagram of the above set of equations, in
where a dashed arrow stands for a cross-talk contribution. Every time a
physical quantity must be passed throughout an operator, then the relative
imperfection mixes it up among many degrees of freedom.
Figure 3.6: An update of the block diagram shown in Figure 3.4 where the
cross-talks (dynamics, sensing and actuation) are introduced as imperfections
(dashed arrows) to the nominal operators. For example,
$\bm{D}\simeq\bm{D}_{0}+\delta\bm{D}$.
The nominal solution $\bm{o}_{0}$ is provided by (3.6) in which all operators
have the proper subscript indicating a zeroth-order term. The differential
operator is obtained from the definition (3.7) by means of simple algebra and
using the inversion lemma
111$\left(\bm{S}_{0}+\delta\bm{S}\right)^{-1}=\bm{S}_{0}^{-1}-\bm{S}_{0}^{-1}\,\delta\bm{S}\left(\bm{1}+\bm{S}_{0}^{-1}\,\delta\bm{S}\right)^{-1}\,\bm{S}_{0}^{-1}\simeq\bm{S}_{0}^{-1}-\bm{S}_{0}^{-1}\,\delta\bm{S}\,\bm{S}_{0}^{-1}$.
$\begin{split}\bm{\Delta}&\simeq\left(\bm{D}_{0}+\delta\bm{D}\right)\left(\bm{S}_{0}+\delta\bm{S}\right)^{-1}+\left(\bm{A}_{0}+\delta\bm{A}\right)\bm{C}_{0}\\\
&\simeq\left(\bm{D}_{0}+\delta\bm{D}\right)\bm{S}_{0}^{-1}\left(\bm{1}-\delta\bm{S}\,\bm{S}_{0}^{-1}\right)+\left(\bm{A}_{0}+\delta\bm{A}\right)\bm{C}_{0}\\\
&\simeq\left(\bm{D}_{0}\,\bm{S}_{0}^{-1}+\bm{A}_{0}\,\bm{C}_{0}\right)+\left(\delta\bm{D}\,\bm{S}_{0}^{-1}-\bm{D}_{0}\,\bm{S}_{0}^{-1}\,\delta\bm{S}\,\bm{S}_{0}^{-1}+\delta\bm{A}\,\bm{C}_{0}\right)\end{split}$
(3.28)
where the nominal operator and its imperfection are defined as
$\displaystyle\bm{\Delta}_{0}$
$\displaystyle=\bm{D}_{0}\,\bm{S}_{0}^{-1}+\bm{A}_{0}\,\bm{C}_{0}\leavevmode\nobreak\
,$ (3.29a) $\displaystyle\delta\bm{\Delta}$
$\displaystyle=\delta\bm{D}\,\bm{S}_{0}^{-1}-\bm{D}_{0}\,\bm{S}_{0}^{-1}\,\delta\bm{S}\,\bm{S}_{0}^{-1}+\delta\bm{A}\,\bm{C}_{0}\leavevmode\nobreak\
.$ (3.29b)
There are two ways to obtain the evolution of the first-order correction
$\delta\bm{o}$. The first method involves a direct computation on (3.27)
similar to the reasoning of Section 3.1. The idea is to combine the three
equations, substitute back the zeroth-order and keep only first-order terms.
The second method is more straightforward and is based on the elaboration of
the equation of motion (3.6). Following this idea, the expansion of (3.6) is
$\begin{split}\left(\bm{\Delta}_{0}+\delta\bm{\Delta}\right)\left(\bm{o}_{0}+\delta\bm{o}\right)&\simeq\bm{f}_{\text{n}}\\\
&\quad+\left(\bm{D}_{0}+\delta\bm{D}\right)\left(\bm{S}_{0}+\delta\bm{S}\right)^{-1}\,\bm{o}_{\text{n}}\\\
&\quad+\left(\bm{A}_{0}+\delta\bm{A}\right)\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)\leavevmode\nobreak\
,\end{split}$ (3.30)
Now, to first order the equation becomes
$\begin{split}\bm{\Delta}_{0}\,\bm{o}_{0}+\bm{\Delta}_{0}\,\delta\bm{o}+\delta\bm{\Delta}\,\bm{o}_{0}&\simeq\bm{f}_{\text{n}}\\\
&\quad+\left(\bm{D}_{0}\,\bm{S}_{0}^{-1}+\delta\bm{D}_{0}\,\bm{S}_{0}^{-1}-\bm{D}_{0}\,\bm{S}_{0}^{-1}\,\delta\bm{S}\,\bm{S}_{0}^{-1}\right)\bm{o}_{\text{n}}\\\
&\quad+\left(\bm{A}_{0}+\delta\bm{A}\right)\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)\leavevmode\nobreak\
,\end{split}$ (3.31)
which is further simplified by assuming the knowledge of the nominal dynamics.
Indeed, $\bm{\Delta}_{0}\,\bm{o}_{0}$ on the left-hand side is canceled out by
$\bm{f}_{\text{n}}+\bm{D}_{0}\,\bm{S}_{0}^{-1}\,\bm{o}_{\text{n}}+\bm{A}_{0}\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)$
on the right-hand side. Therefore, the first-order dynamics is given by
$\begin{split}\bm{\Delta}_{0}\,\delta\bm{o}&\simeq-\delta\bm{\Delta}\,\bm{o}_{0}+\left(\delta\bm{D}\,\bm{S}_{0}^{-1}-\bm{D}_{0}\,\bm{S}_{0}^{-1}\,\delta\bm{S}\,\bm{S}_{0}^{-1}\right)\bm{o}_{\text{n}}\\\
&\quad+\delta\bm{A}\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)\leavevmode\nobreak\
,\end{split}$ (3.32)
where the operator in front of $\bm{o}_{\text{n}}$ is exactly
$\delta\bm{\Delta}-\delta\bm{A}\,\bm{C}_{0}$ (it can be checked by comparing
to (3.29b)). The equation also shows that the first-order dynamics is given in
terms of the zeroth-order $\bm{o}_{0}$. To write down an explicit form of the
above formula, it is necessary to substitute the expression of $\bm{o}_{0}$ in
(3.6) and get
$\begin{split}\bm{\Delta}_{0}\,\delta\bm{o}&\simeq-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\left[\bm{f}_{\text{n}}+\bm{D}_{0}\,\bm{S}_{0}^{-1}\,\bm{o}_{\text{n}}+\bm{A}_{0}\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)\right]\\\
&\quad+\left(\delta\bm{\Delta}-\delta\bm{A}\,\bm{C}\right)\bm{o}_{\text{n}}+\delta\bm{A}\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)\\\
&=-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\,\bm{f}_{\text{n}}+\left(\delta\bm{\Delta}-\delta\bm{A}\,\bm{C}_{0}-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\,\bm{D}_{0}\,\bm{S}_{0}^{-1}\right)\bm{o}_{\text{n}}\\\
&\quad+\left(\delta\bm{A}-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\,\bm{A}_{0}\right)\left(\bm{f}_{\text{i}}+\bm{C}_{0}\,\bm{o}_{\text{i}}\right)\leavevmode\nobreak\
.\end{split}$ (3.33)
The operator in front of $\bm{f}_{\text{n}}$ is the force noise cross-talk,
whose transfer to total equivalent acceleration is denoted with
$\delta\bm{T}_{f_{\text{n}}\rightarrow
f}=-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\leavevmode\nobreak\ ;$ (3.34)
the operator in front of $\bm{o}_{\text{n}}$ is the readout noise cross-talk
and can be rewritten as
$\begin{split}\delta\bm{T}_{o_{\text{n}}\rightarrow
f}&=\delta\bm{\Delta}-\delta\bm{A}\,\bm{C}_{0}-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\,\bm{D}_{0}\,\bm{S}_{0}^{-1}\\\
&=\delta\bm{\Delta}-\delta\bm{A}\,\bm{C}_{0}-\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\left(\bm{\Delta}_{0}-\bm{A}_{0}\,\bm{C}_{0}\right)\\\
&=\delta\bm{\Delta}\,\bm{\Delta}_{0}^{-1}\,\bm{A}_{0}\,\bm{C}_{0}-\delta\bm{A}\,\bm{C}_{0}\\\
&=-\left(\delta\bm{A}+\delta\bm{T}_{f_{\text{n}}\rightarrow
f}\,\bm{A}_{0}\right)\bm{C}_{0}\leavevmode\nobreak\ ;\end{split}$ (3.35)
the operator in front of $\bm{f}_{\text{i}}$ is the force actuation cross-talk
$\delta\bm{T}_{f_{\text{i}}\rightarrow
f}=\delta\bm{A}+\delta\bm{T}_{f_{\text{n}}\rightarrow
f}\,\bm{A}_{0}\leavevmode\nobreak\ ;$ (3.36)
and the operator in front of $\bm{o}_{\text{i}}$ is the bias actuation cross-
talk
$\delta\bm{T}_{o_{\text{i}}\rightarrow
f}=\left(\delta\bm{A}+\delta\bm{T}_{f_{\text{n}}\rightarrow
f}\,\bm{A}_{0}\right)\bm{C}_{0}\leavevmode\nobreak\ ,$ (3.37)
that is exactly the readout noise cross-talk modulo a sign. As reasonable to
expect, the symmetry between the various cross-talk terms comes from the
linearization of the problem.
The equation of motion for the first-order cross-talk is finally given by
$\bm{\Delta}_{0}\,\delta\bm{o}\simeq\delta\bm{T}_{f_{\text{n}}\rightarrow
f}\,\bm{f}_{\text{n}}+\delta\bm{T}_{o_{\text{n}}\rightarrow
f}\,\bm{o}_{\text{n}}+\delta\bm{T}_{f_{\text{i}}\rightarrow
f}\,\bm{f}_{\text{i}}+\delta\bm{T}_{o_{\text{i}}\rightarrow
f}\,\bm{o}_{\text{i}}\leavevmode\nobreak\ ,$ (3.38)
that enlightens the various contributions to the overall cross-talk from other
degrees of freedom. On one side, in a pure noise measurement during the LPF
mission (no application of forces or biases) a noise cross-talk sums up to the
nominal dynamics along the main degrees of freedom. On the other side,
whatever a bias is applied to the system, a perturbation is produced along the
sensitive axis. As it is clear, a non-negligible cross-talk at any level of
the system (dynamics, sensing and actuation) actually breaks the nominal
orthogonality between different degrees of freedom.
### 3.5 Dynamical model for $xy$ cross-talk
The application of the idea of the preceding section is a further development
of the model along the optical axis described in Section 3.3, i.e., the cross-
talk from other nominally orthogonal degrees of freedom to the optical axis.
The example discussed in this section is the cross-talk from the $y$ degree of
freedom to $x$.
Referring to Figure 3.2 for the usual coordinate naming convention, the $xy$
cross-talk can be viewed as a first-order perturbation of the dynamics in the
$xy$ plane to the zeroth-order dynamics along $x$. Obviously, the dynamics in
$xy$ contains the rotation $\phi$ about the $z$ axis.
The control design of the main science mode for this simplified model requires
a minimal number of drag-free, electrostatic suspension and attitude
coordinates as inputs to the same control loops. Table 3.2 presents a list of
those coordinates relevant for the $xy$ cross-talk as taken from Table 3.1.
Table 3.2: List of the 7 controlled degrees of freedom for the $xy$ cross-talk of the LPF mission in the main science mode. Refer to Table 3.1 for a comprehensive description of all coordinates. Notice that $o_{\phi_{2}}=o_{\phi_{1}}+o_{\phi_{12}}$ is used in the equations for clearness and simplicity. Coordinate | Control | Sensor | Actuator
---|---|---|---
$x_{1}$ | Drag-free | $o_{1}\,=\,\text{IFO}[x_{1}]$ | FEEP
$y_{1}$ | Drag-free | $o_{y_{1}}\,=\,\text{GRS}[y_{1}]$ | FEEP
$\phi_{1}$ | Elect. suspension | $o_{\phi_{1}}\,=\,\text{IFO}[\phi_{1}]$ | GRS
$x_{2}$ | Elect. suspension | $o_{12}\,=\,\text{IFO}[x_{12}]$ | GRS
$y_{2}$ | Drag-free | $o_{y_{2}}\,=\,\text{GRS}[y_{2}]$ | FEEP
$\phi_{2}$ | Elect. suspension | $o_{\phi_{12}}\,=\,\text{IFO}[\phi_{12}]$ | GRS
$\phi_{\text{SC}}$ | Attitude | $o_{\phi_{\text{SC}}}\,=\,\text{ST}[\phi_{\text{SC}}]$ | GRS
The control is such that:
1. 1.
along $x$: guided by the optical $x_{1}$, the SC is forced to follow the
reference TM through thruster actuation; guided by the optical $x_{12}$, the
second TM is forced to follow the reference TM through capacitive actuation;
2. 2.
along $y$: guided by the capacitive $(y_{1}+y_{2})/2$, the SC is forced to
follow both TMs through thruster actuation; guided by the star-tracker
$\phi_{\text{SC}}$, the TMs are oriented along $\phi$;
3. 3.
along $\phi$: guided by the capacitive $(y_{1}-y_{2})/2$, the SC is forced to
follow both TMs through thruster actuation; guided by the optical $\phi_{1}$
and $\phi_{2}$, both TMs are oriented along $\phi$ through capacitive
actuation.
As already pointed out, the cross-talk can be described by a first-order
perturbation of the nominal dynamics along $x$. Three different types of
cross-talks can be identified:
1. 1.
dynamical cross-talk;
2. 2.
actuation cross-talk;
3. 3.
sensing cross-talk.
All equations must be written to within linear terms of an imperfection or
noise contribution.
Concerning the dynamics along the $x$ axis, in the approximation of small
motion, the stiffness constant has been introduced to model residual
oscillator-like couplings between the TMs and the SC. The generalization in
three dimensions is straightforward. Since there are electrodes all around the
TMs and the most important coupling is indeed due to electrostatics, in place
of a single oscillator, 6 coupled harmonic oscillators along the translational
and rotational degrees of freedom must be considered. Therefore, the stiffness
constant becomes a $6\times 6$ quasi-diagonal matrix, the stiffness matrix.
For the $xy$ cross-talk, since it makes sense to inspect the cross-stiffness
from $y$ to $x$ or $\phi$ to $x$, the structure of the matrix for the first TM
is
$\bm{\kappa}_{1}=\begin{pmatrix}m_{1}\,\omega_{1,x}^{2}&\delta_{1,xy}\,m_{1}\,\omega_{1,y}^{2}&\delta_{1,x\phi}\,I_{1,z}\,\omega_{1,\phi}^{2}\\\
0&m_{1}\,\omega_{1,y}^{2}&0\\\
0&0&I_{1,z}\,\omega_{1,\phi}^{2}\end{pmatrix}\leavevmode\nobreak\ ,$ (3.39)
where $m_{1}=1.96\,\mathrm{kg}$ is the TM mass,
$I_{1,z}=\nicefrac{{1}}{{6}}\,m_{1}\,l^{2}\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}7\\!\times\\!10^{-4}\,\mathrm{kg\,m^{2}}$ is the
inertia matrix about $z$ and $l=46\,\mathrm{mm}$ is the TM size. A
$\delta$-coefficient, with the number of the TM and the names of two
coordinates as subscripts, denotes a dynamical cross-talk imperfection
typically $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}1\\!\times\\!10^{-3}$, an assumption based on on-ground measurements and
theoretical models of the GRS.
The second type is the actuation cross-talk due to misalignments in the setup
of the thrusters and electrodes. The idea is that every time a force is
actuated along a nominally orthogonal degree of freedom it couples with $x$. A
$\delta$-coefficient, with the names of the actuation and the coordinate,
denotes an actuation cross-talk imperfection.
Finally, the third type is the sensing cross-talk due to miscalibrations and
misalignments in the sensors. For the capacitive sensing, misalignments in the
electrodes produces a mixing in the sensed coordinates. For the optical
sensing, misalignments in the optical elements produce an analogous result. As
in Section 3.3, the 7 physical coordinates
$\bm{q}=\scalebox{0.8}{$\begin{pmatrix}x_{1}\\\ x_{12}\\\ y_{1}\\\ y_{2}\\\
\phi_{1}\\\ \phi_{2}\\\ \phi_{\text{SC}}\end{pmatrix}$}\leavevmode\nobreak\ ,$
(3.40)
are converted into the sensed coordinates
$\bm{o}=\scalebox{0.8}{$\begin{pmatrix}o_{1}\\\ o_{12}\\\ o_{y_{1}}\\\
o_{y_{2}}\\\ o_{\phi_{1}}\\\ o_{\phi_{2}}\\\
o_{\phi_{\text{SC}}}\end{pmatrix}$}\leavevmode\nobreak\ ,$ (3.41)
with an inevitable mixing. Assuming a sensing operator which is nominally
identity, the relative imperfection operator has the following matrix
representation
$\delta\bm{S}=\left(\begin{smallmatrix}0&0&\delta_{S,1y_{1}}&\delta_{S,1y_{2}}&\frac{l}{2}\,\delta_{S,1\phi_{1}}&\frac{l}{2}\,\delta_{S,1\phi_{2}}&\frac{L}{2}\,\delta_{S,1\phi_{\text{SC}}}\\\
S_{21}&0&\delta_{S,2y_{1}}&\delta_{S,2y_{2}}&\frac{l}{2}\,\delta_{S,2\phi_{1}}&\frac{l}{2}\,\delta_{S,2\phi_{2}}&\frac{L}{2}\,\delta_{S,2\phi_{\text{SC}}}\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{smallmatrix}\right)\leavevmode\nobreak\ ,$ (3.42)
where $l=46\,\mathrm{mm}$ is the TM size and $L=38\,\mathrm{cm}$ is the
nominal separation between the TM centers of mass. $S_{21}$ is the sensing
cross-talk between $o_{1}$ and $o_{12}$, already discussed, and a generic
$\delta$-coefficient denotes a sensing cross-talk imperfection from an
orthogonal degree of freedom to $x$. Since the target of the investigation is
the cross-talk to the two optical degrees of freedom, no cross-talk to other
coordinates is considered as this is second order.
In the derivation of the equations of motion the gravity/torque gradients
between the TMs can be neglected without loss of generality, as well as the
back-reactions, since they are all second order effects. The first set of
equations is along the $x$ degree of freedom. With the same considerations of
Section 3.3, i.e., assuming small motion, small forces and null initial
conditions, the linear equations of motion, per unit mass, in frequency domain
are
$\displaystyle\begin{split}s^{2}\,x_{1}+s^{2}\,x_{\text{SC}}+\omega_{1,x}^{2}\,x_{1}\quad&\\\
\quad+\,\delta_{1,xy}\,\omega_{1,y}^{2}\,y_{1}+\delta_{1,x\phi}\,\frac{l}{2}\,\omega_{1,\phi}^{2}\,\phi_{1}&=f_{1,x}\\\
&\quad+\delta_{\text{sus},\,y_{1}}\,\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad+\delta_{\text{sus},\,\phi_{1}}\,\frac{l}{2}\,C_{\text{sus},\,\phi_{1}}(s)\,o_{\phi_{1}}\leavevmode\nobreak\
,\end{split}$ (3.43a)
$\displaystyle\begin{split}s^{2}\,x_{2}+s^{2}\,x_{\text{SC}}+\omega_{2,x}^{2}\,x_{2}\quad&\\\
\quad+\,\delta_{2,xy}\,\omega_{2,y}^{2}\,y_{2}+\delta_{2,x\phi}\,\frac{l}{2}\,\omega_{2,\phi}^{2}\,\phi_{2}&=f_{2,x}-C_{\text{sus},\,x}(s)\,o_{12}\\\
&\quad-\delta_{\text{sus},\,y_{2}}\,\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad+\delta_{\text{sus},\,\phi_{2}}\,\frac{l}{2}\,C_{\text{sus},\,\phi_{2}}(s)\,o_{\phi_{2}}\leavevmode\nobreak\
,\end{split}$ (3.43b)
$\displaystyle\begin{split}s^{2}\,x_{\text{SC}}&=f_{\text{SC},x}+C_{\text{df},\,x}(s)\,o_{1}\\\
&\quad+\delta_{\text{df},\,y_{\text{SC}}}\,\frac{1}{2}\left[C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}+C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}\right]\\\
&\quad-\delta_{\text{df},\,\phi_{\text{SC}}}\,\frac{1}{2}\left[C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}-C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}\right]\leavevmode\nobreak\
,\end{split}$ (3.43c)
The second row of the left-hand side of the equations for the TMs contain the
dynamical cross-talk due to the stiffness matrix (imperfections
$\delta_{1,xy}$, $\delta_{1,x\phi}$, $\delta_{2,xy}$ and $\delta_{2,x\phi}$).
Instead, the last two rows of the right-hand side of the equations are
actuation cross-talk. Notice that for the TMs there is a cross-talk from the
SC inertial attitude control ($\delta_{\text{sus},\,y}$) and one from the TM
attitude control ($\delta_{\text{sus},\,\phi}$). For the SC there is a cross-
talk from the $y$ and $\phi$ drag-free actuation
($\delta_{\text{df},\,y_{\text{SC}}}$ and
$\delta_{\text{df},\,\phi_{\text{SC}}}$). Other relevant quantities have been
already defined in the previous section and are not further discussed here.
The second set of equations describes the dynamics along the nominally
orthogonal $y$ axis
$\displaystyle
s^{2}\,y_{1}+s^{2}\,y_{\text{SC}}+\omega_{1,y}^{2}\,y_{1}-\frac{L}{2}\,s^{2}\,\phi_{\text{SC}}$
$\displaystyle=f_{1,y}+\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\leavevmode\nobreak\
,$ (3.44a) $\displaystyle
s^{2}\,y_{2}+s^{2}\,y_{\text{SC}}+\omega_{2,y}^{2}\,y_{2}+\frac{L}{2}\,s^{2}\,\phi_{\text{SC}}$
$\displaystyle=f_{2,y}-\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\leavevmode\nobreak\
,$ (3.44b) $\displaystyle s^{2}\,y_{\text{SC}}$
$\displaystyle=f_{\text{SC},y}+\frac{1}{2}\left[C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}+C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}\right]\leavevmode\nobreak\
.$ (3.44c)
The inertial attitude control is implemented as electrostatic suspension
actuation on the TMs along $y$ (notice the opposite signs) through the
$C_{\text{sus},\,\phi_{\text{SC}}}(s)$ control law. $C_{\text{df},\,y_{1}}(s)$
and $C_{\text{df},\,y_{2}}(s)$ are the two drag-free control laws along $y$.
On the left-end side of the equations the SC absolute angular acceleration
$s^{2}\,\phi_{\text{SC}}$ also appears as a strict consequence of the coupling
between the $\phi$ motion with $y$.
The third set of equations describes the dynamics along the nominally
orthogonal $\phi$ angle
$\displaystyle
s^{2}\,\phi_{1}+s^{2}\,\phi_{\text{SC}}+\omega_{1,\phi}^{2}\,\phi_{1}$
$\displaystyle=\tau_{1,z}-C_{\text{sus},\,\phi_{1}}(s)\,o_{\phi_{1}}\leavevmode\nobreak\
,$ (3.45a) $\displaystyle
s^{2}\,\phi_{2}+s^{2}\,\phi_{\text{SC}}+\omega_{2,\phi}^{2}\,\phi_{2}$
$\displaystyle=\tau_{2,z}-C_{\text{sus},\,\phi_{2}}(s)\,o_{\phi_{2}}\leavevmode\nobreak\
,$ (3.45b) $\displaystyle s^{2}\,\phi_{\text{SC}}$
$\displaystyle=\tau_{\text{SC},z}+\frac{1}{L}\left[C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}-C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}\right]\leavevmode\nobreak\
,$ (3.45c)
where $\tau$ denotes a generic component of torque per unit of inertia
$I_{\text{TM},z}\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}7\\!\times\\!10^{-4}\,\mathrm{kg\,m^{2}}$ and $I_{\text{SC},z}\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}1\\!\times\\!10^{3}\,\mathrm{kg\,m^{2}}$ [46]. On the left-end side, the
angular stiffness constants are evident. On the right-end side, there are the
$C_{\text{sus},\,\phi}(s)$ electrostatic suspension control law and the
$C_{\text{df},\,y}(s)$ drag-free control law. The SC attitude is controlled by
actuating along the differential TM positions along $y$.
As said, along $y$ the SC follows the average $y$ motion of the TMs sensed
with the capacitive $o_{y_{1}}$ and $o_{y_{2}}$; the TMs are oriented
following the star-tracker $o_{\phi_{\text{SC}}}$. Analogously, along $\phi$
the SC is oriented following the differential $y$ motion of the TMs sensed
with the capacitive $o_{y_{1}}$ and $o_{y_{2}}$; the TMs are oriented
following the optical $o_{\phi_{1}}$ and $o_{\phi_{2}}$.
The SC absolute linear acceleration is not measurable, whereas the SC absolute
attitude is measured by the ST with respect to the celestial inertial frame.
Therefore, the system of 9 equations turns into 7 equations because the SC
acceleration along $x$ and $y$ must be canceled out. By doing so in (3.43) and
defining the differential TM displacement, the dynamics along $x$ becomes
$\displaystyle\begin{split}s^{2}\,x_{1}+\omega_{1,x}^{2}\,x_{1}\quad&\\\
\quad+\,\delta_{1,xy}\,\omega_{1,y}^{2}\,y_{1}+\delta_{1,x\phi}\,\frac{l}{2}\,\omega_{1,\phi}^{2}\,\phi_{1}&=f_{1,x}-f_{\text{SC},x}-C_{\text{df},\,x}(s)\,o_{1}\\\
&\quad+\delta_{\text{sus},\,y_{1}}\,\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad+\delta_{\text{sus},\,\phi_{1}}\,\frac{l}{2}\,C_{\text{sus},\,\phi_{1}}(s)\,o_{\phi_{1}}\\\
&\quad-\delta_{\text{df},\,y_{\text{SC}}}\,\frac{1}{2}\left[C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}+C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}\right]\\\
&\quad+\delta_{\text{df},\,\phi_{\text{SC}}}\,\frac{1}{2}\left[C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}-C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}\right]\leavevmode\nobreak\
,\end{split}$ (3.46a)
$\displaystyle\begin{split}s^{2}\,x_{12}+\omega_{2,x}^{2}\,x_{2}-\omega_{1,x}^{2}\,x_{1}\quad&\\\
\quad+\,\delta_{2,xy}\,\omega_{2,y}^{2}\,y_{2}+\delta_{2,x\phi}\,\frac{l}{2}\,\omega_{2,\phi}^{2}\,\phi_{2}\\\
\quad-\,\delta_{1,xy}\,\omega_{1,y}^{2}\,y_{1}-\delta_{1,x\phi}\,\frac{l}{2}\,\omega_{1,\phi}^{2}\,\phi_{1}&=f_{2,x}-C_{\text{sus},\,x}(s)\,o_{12}\\\
&\quad-\delta_{\text{sus},\,y_{2}}\,\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad+\delta_{\text{sus},\,\phi_{2}}\,\frac{l}{2}\,C_{\text{sus},\,\phi_{2}}(s)\,o_{\phi_{2}}\\\
&\quad-\delta_{\text{sus},\,y_{1}}\,\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad-\delta_{\text{sus},\,\phi_{1}}\,\frac{l}{2}\,C_{\text{sus},\,\phi_{1}}(s)\,o_{\phi_{1}}\leavevmode\nobreak\
.\end{split}$ (3.46b)
Analogously, substituting the SC acceleration along $y$ and $\phi$ into the
dynamics along $y$
$\displaystyle\begin{split}s^{2}\,y_{1}+\omega_{1,y}^{2}\,y_{1}&=f_{1,y}-f_{\text{SC},y}+\frac{L}{2}\tau_{\text{SC},z}+\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad-C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}\leavevmode\nobreak\ ,\end{split}$
(3.47a)
$\displaystyle\begin{split}s^{2}\,y_{2}+\omega_{2,y}^{2}\,y_{2}&=f_{2,y}-f_{\text{SC},y}-\frac{L}{2}\tau_{\text{SC},z}-\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\,o_{\phi_{\text{SC}}}\\\
&\quad-C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}\leavevmode\nobreak\ .\end{split}$
(3.47b)
$y_{1}$ and $y_{2}$ are named drag-free coordinates as they guide a drag-free
actuation; $\phi_{\text{SC}}$ is an electrostatic suspension coordinate as it
guides a capacitive actuation. At the same time the dynamics along $\phi$ is
$\displaystyle\begin{split}s^{2}\,\phi_{1}+\omega_{1,\phi}^{2}\,\phi_{1}&=\tau_{1,z}-\tau_{\text{SC},z}-C_{\text{sus},\,\phi_{1}}(s)\,o_{\phi_{1}}\\\
&\quad-\frac{1}{L}\left[C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}-C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}\right]\leavevmode\nobreak\
,\end{split}$ (3.48a)
$\displaystyle\begin{split}s^{2}\,\phi_{2}+\omega_{2,\phi}^{2}\,\phi_{2}&=\tau_{2,z}-\tau_{\text{SC},z}-C_{\text{sus},\,\phi_{2}}(s)\,o_{\phi_{2}}\\\
&\quad-\frac{1}{L}\left[C_{\text{df},\,y_{2}}(s)\,o_{y_{2}}-C_{\text{df},\,y_{1}}(s)\,o_{y_{1}}\right]\leavevmode\nobreak\
.\end{split}$ (3.48b)
and the one for $\phi_{\text{SC}}$ which remains unchanged. $\phi_{1}$ and
$\phi_{2}$ are electrostatic suspension coordinates as they guide a capacitive
actuation.
The equations of motion presented above can now be mapped to the formalism of
the previous section. Therefore, the $7\times 7$ nominal dynamics operator has
the following matrix representation
$\bm{D}_{0}=\scalebox{0.8}{$\begin{pmatrix}s^{2}+\omega_{1,x}^{2}&0&0&0&0&0&0\\\
\omega_{12,x}^{2}&s^{2}+\omega_{1,x}^{2}+\omega_{12,x}^{2}&0&0&0&0&0\\\
0&0&s^{2}+\omega_{1,y}^{2}&0&0&0&0\\\ 0&0&0&s^{2}+\omega_{2,y}^{2}&0&0&0\\\
0&0&0&0&s^{2}+\omega_{1,\phi}^{2}&0&0\\\
0&0&0&0&0&s^{2}+\omega_{2,\phi}^{2}&0\\\
0&0&0&0&0&0&s^{2}\end{pmatrix}$}\leavevmode\nobreak\ ,$ (3.49)
which is a natural generalization of the same operator (3.18) written for the
model along $x$. The first-order perturbation due to the dynamical cross-talk
is given by
$\delta\bm{D}=\scalebox{0.8}{$\begin{pmatrix}0&0&\delta_{1,xy}\,\omega_{1,y}^{2}&0&\delta_{1,x\phi}\,\frac{l}{2}\,\omega_{1,\phi}^{2}&0&0\\\
0&0&-\delta_{1,xy}\,\omega_{1,y}^{2}&\delta_{2,xy}\,\omega_{2,y}^{2}&-\delta_{1,x\phi}\,\frac{l}{2}\,\omega_{1,\phi}^{2}&\delta_{2,x\phi}\,\frac{l}{2}\,\omega_{2,\phi}^{2}&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{pmatrix}$}\leavevmode\nobreak\ .$ (3.50)
Analogously, the control operator can be identified as
$\bm{C}_{0}=\scalebox{0.8}{$\begin{pmatrix}C_{\text{df},\,x}(s)&0&0&0&0&0&0\\\
0&C_{\text{sus},\,x}(s)&0&0&0&0&0\\\
0&0&C_{\text{df},\,y_{1}}(s)&0&0&0&-\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\\\
0&0&0&C_{\text{df},\,y_{2}}(s)&0&0&\frac{L}{2}\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\\\
0&0&-\frac{1}{L}\,C_{\text{df},\,y_{1}}(s)&\frac{1}{L}\,C_{\text{df},\,y_{2}}(s)&C_{\text{sus},\,\phi_{1}}(s)&0&0\\\
0&0&-\frac{1}{L}\,C_{\text{df},\,y_{1}}(s)&\frac{1}{L}\,C_{\text{df},\,y_{2}}(s)&0&C_{\text{sus},\,\phi_{2}}(s)&0\\\
0&0&-\frac{1}{L}\,C_{\text{df},\,y_{1}}(s)&\frac{1}{L}\,C_{\text{df},\,y_{2}}(s)&0&0&0\end{pmatrix}$}\leavevmode\nobreak\
,$ (3.51)
where possible actuation gains can be intended as multiplicative factor of
each single control law. The first-order perturbation due to the control
actuation cross-talk is given by
$\delta\bm{A}\,\bm{C}_{0}=\frac{1}{2}\scalebox{0.7}{$\begin{pmatrix}0&0&\delta_{\text{df}}^{-}\,C_{\text{df},\,y_{1}}(s)&\delta_{\text{df}}^{+}\,C_{\text{df},\,y_{2}}(s)&-\delta_{\text{sus},\,\phi_{1}}\,l\,C_{\text{sus},\,\phi_{1}}(s)&0&-\delta_{\text{sus},\,y_{1}}\,L\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\\\
0&0&0&0&\delta_{\text{sus},\,\phi_{1}}\,l\,C_{\text{sus},\,\phi_{1}}(s)&-\delta_{\text{sus},\,\phi_{2}}\,l\,C_{\text{sus},\,\phi_{2}}(s)&\delta_{\text{sus}}^{+}\,L\,C_{\text{sus},\,\phi_{\text{SC}}}(s)\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{pmatrix}$}\leavevmode\nobreak\ ,$ (3.52)
where
$\delta_{\text{df}}^{-}=\left(\delta_{\text{df},\,y_{\text{SC}}}-\delta_{\text{df},\,\phi_{\text{SC}}}\right)$,
$\delta_{\text{df}}^{+}=\left(\delta_{\text{df},\,y_{\text{SC}}}+\delta_{\text{df},\,\phi_{\text{SC}}}\right)$
and
$\delta_{\text{sus}}^{+}=\delta_{\text{sus},\,y_{1}}+\delta_{\text{sus},\,y_{2}}$
are three new definitions of effective cross-talk coefficients.
The imperfection matrices $\delta\bm{D}$, $\delta\bm{A}\,\bm{C}_{0}$ and
$\delta\bm{S}$, together with the nominal matrices $\bm{D}_{0}$, $\bm{C}_{0}$
and $\bm{S}_{0}=\bm{1}$, allows for a simplification of the nominal
differential operator and its imperfection in (3.29)
$\displaystyle\bm{\Delta}_{0}$
$\displaystyle=\bm{D}_{0}+\bm{C}_{0}\leavevmode\nobreak\ ,$ (3.53a)
$\displaystyle\delta\bm{\Delta}$
$\displaystyle=\delta\bm{D}-\bm{D}_{0}\,\delta\bm{S}+\delta\bm{A}\,\bm{C}_{0}\leavevmode\nobreak\
.$ (3.53b)
The second equation explains the fact that the imperfection to the
differential operator converting sensed coordinates into total equivalent
acceleration is given by three terms: the dynamical cross-talk, the sensing
cross-talk and the control actuation cross-talk. The application of the first
matrix to the first-order correction to the sensed coordinates finally gives
the various cross-talk contributions in the equation of motion (3.38).
Inverting $\bm{\Delta}_{0}$, and applying it to the various cross-talk
contributions of (3.38), it allows for a modeling of the response of the
system along the optical axis to noise sources affecting the nominally
orthogonal degrees of freedom.
## 4 System identification
This chapter focuses on a topic that can be considered the core of the whole
LPF mission in view of characterizing the total equivalent acceleration noise
affecting each single LISA arm. In system identification LPF is modeled as a
matrix of parametric transfer functions. Targeted experiments where the system
is stimulated on each degree of freedom can be used to infer the values of the
critical parameters contained in those functions.
The preceding chapter described the closed-loop dynamics underlying LPF, the
methods to handle and subtract the applied control forces, the sensing and the
dynamical couplings between the TMs and the SC, the extent to which system
transients can be suppressed and the estimation of the equivalent out-of-loop
acceleration noise can be made possible.
This chapter shows an application of the ideas in a mission-like fashion with
numerical applications. It assumes a model for LPF along the optical axis,
which gives the description of the dynamics to first approximation. The aim is
to simulate and analyze the data as they will be released during the mission.
To simplify the discussion, only two experiments are considered, allowing for
a complete identification of the system along the optical axis. As the methods
developed in this chapter are general, they can also be applied to the study
of more sophisticated experiments. Examples are the cross-talk experiments
from orthogonal degrees of freedom to the optical axis: the modeled transfer
functions are different, the dimensionality of the system is different, but
the approach is the same. In the end, all experiments analyzed with the
methods described in this chapter will hopefully provide a coherent
understanding of the system, contributing to the final success of the LPF
mission.
In turn, this chapter discusses: the dynamical model assumed for simulations
and analysis; the noise characterization of the system; the simulated
identification experiments; the parameter estimation method, the validation
and the robustness to non-standard scenarios. Finally, it demonstrates the
impact of system identification on the estimation of the residual equivalent
acceleration noise and the suppression of transients in data produced by a
simulator provided by ESA.
### 4.1 Dynamical model
Section 3.3 provided a model of LPF along $x$, the optical axis. In the main
science mode, the reference TM is in free fall along $x$. The other TM and the
SC are, respectively, forced by capacitive and thruster actuation to follow
the reference TM along the same axis. The interferometer keeps track of the
relative motion between the reference TM and the SC ($o_{1}$) and between the
two TMs ($o_{12}$). The two readouts expressed in displacement are fed into
the DFACS to command force actuation, hence minimizing the relative motion.
Figure 4.1 and Figure 4.2 show the frequency dependence of the two control
laws converting the sensed displacements to commanded forces to the thrusters
(drag-free loop) and the electrostatic suspensions (electrostatic suspension
loop), respectively. At low frequency, the drag-free gain is very high due to
the need for suppressing the SC jitter that dominates $o_{1}$. Instead, the
electrostatic suspension gain is designed to suppress the force couplings
between the TMs and the SC that dominate $o_{12}$. The control laws used in
this thesis are provided by ASTRIUM [35] – the main industry contractor of
LPF.
Figure 4.1: Frequency dependence of the drag-free loop controller per unit SC
mass. The very high gain at low frequency is explained by the need for
removing the thruster noise. Following (3.9), $1\,\mathrm{\mu m}$ sensed
displacement of the first TM relative to the SC produces a thruster actuation
of $\raise 0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}0.02\,\mathrm{\mu N}$
at $1\,\mathrm{mHz}$. Figure 4.2: Frequency dependence of the electrostatic
suspension loop controller per unit TM mass. Here there is no such a huge
variation in the order of magnitude as for the drag-free controller in Figure
4.1. The control law is designed in particular to suppress the force couplings
between the TMs and the SC at low frequency. Following (3.9), $1\,\mathrm{\mu
m}$ sensed displacement of the second TM relative to the first one produces a
capacitive actuation of $\raise 0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}0.1\,\mathrm{nN}$ at $1\,\mathrm{mHz}$.
As described in Section 3.3, the system can be modeled by the operators
$\bm{D}$ (dynamics), $\bm{S}$ (sensing) and $\bm{A}$ (actuation) representing
different non-idealities in the practical implementation of the closed-loop
LISA arm. The operators contain all system parameters describing the dynamics
along the optical axis. One last source of indetermination introduced here is
a delay in the application of the guidance signals
$\bm{T}=\begin{pmatrix}e^{-s\,\Delta t_{1}}&0\\\ 0&e^{-s\,\Delta
t_{2}}\end{pmatrix}\leavevmode\nobreak\ ,$ (4.1)
whose possible causes may be either due to the digitalization of the
continuous control laws or to bus delays, a possibility not considered in a
previous model [47]. With the introduction of the delays, the model (3.8b)
becomes now
$\bm{T}_{o_{\text{i}}\rightarrow
o}=\bm{\Delta}^{-1}\bm{A}\,\bm{C}\,\bm{T}\leavevmode\nobreak\ ,$ (4.2)
where the differential operator $\bm{\Delta}$, defined in (3.8a), converts the
sensed motion into total equivalent acceleration.
Figure 4.3 shows the transfer gains of the model
$\bm{T}_{o_{\text{i}}\rightarrow o}$, whereas the dynamical cross-talk from
the differential channel to the first one is definitely negligible with peak
gain of about $4\\!\times\\!10^{-6}$ at $30\,\mathrm{mHz}$. The diagonal
elements have respectively peak gains of almost 3 at $0.1\,\mathrm{Hz}$ and
about 2 at $0.8\,\mathrm{mHz}$. The dynamical cross-talk from the first
channel to the differential one has peak gain of about $5\\!\times\\!10^{-2}$
at $0.5\,\mathrm{mHz}$. The above transfer matrix is used to both model the
outputs of the system subjected to bias injections and perform system
identification.
Figure 4.3: Frequency dependence of the transfer matrix
$\bm{T}_{o_{\text{i}}\rightarrow o}$ used for system identification. The
transfer function $T^{11}_{o_{\text{i}}\rightarrow
o}=T_{o_{\text{i,1}}\rightarrow o_{1}}$ has peak gain of almost 3 at
$0.1\,\mathrm{Hz}$. The transfer function $T^{22}_{o_{\text{i}}\rightarrow
o}=T_{o_{\text{i,12}}\rightarrow o_{12}}$ has peak gain of about 2 at
$0.8\,\mathrm{mHz}$, then it quickly decays. The dynamical cross-talk
$T^{12}_{o_{\text{i}}\rightarrow o}=T_{o_{\text{i,1}}\rightarrow o_{12}}$ has
peak gain of about $5\\!\times\\!10^{-2}$ at $0.5\,\mathrm{mHz}$. The other
dynamical cross-talk is negligible since has peak gain of about
$4\\!\times\\!10^{-6}$ at $30\,\mathrm{mHz}$.
Throughout this chapter bias injections at the level of controller guidance
signals $\bm{o}_{\text{i}}$ 111Following (3.9), a bias in the guidance signals
is equivalent to a commanded force bias directly applied onto the SC through
thruster actuation and the second TM through capacitive actuation. are
considered and the transfer matrix in (4.2) models the response of the system
to those signals. As the modeled system parameters appear in the operators,
$\bm{T}_{o_{\text{i}}\rightarrow o}$ is parameter-dependent. The modeled
system response is then parameter-dependent. The parameters can be arranged in
a vector that will be abstractly referred to $\bm{p}$
$\bm{p}=\scalebox{0.8}{$\begin{pmatrix}\omega_{1}^{2}\\\ \omega_{12}^{2}\\\
S_{21}\\\ A_{\text{df}}\\\ A_{\text{sus}}\\\ \Delta t_{1}\\\ \Delta t_{2}\\\
\end{pmatrix}$}\leavevmode\nobreak\ ,$ (4.3)
where Table 4.1 provides a description of the above system parameter with
initial plausible estimates coming from on-ground measurements and theoretical
modeling.
Table 4.1: List of the modeled system parameters, introduced in Section 3.3,
except for $\Delta t_{1}$ and $\Delta t_{2}$, with descriptions and initial
estimates. The parameters that are fitted to data are $\omega_{1}^{2}$,
$\omega_{12}^{2}$, $S_{21}$, $A_{\text{df}}$, $A_{\text{sus}}$, $\Delta
t_{1}$, $\Delta t_{2}$.
Parameter | Description | Note | Estimate
---|---|---|---
$\omega_{1}^{2}$, $\omega_{12}^{2}$ | parasitic stiffness constants modeling residual oscillator-like couplings between the SC and the reference TM and between the two TMs | must be estimated from experiments | $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}-1\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$
$S_{21}$ | sensing cross-talk between $o_{1}$ and $o_{12}$ interferometric readouts | must be estimated from experiments | $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}1\\!\times\\!10^{-4}$
$A_{\text{df}}$, $A_{\text{sus}}$ | actuation gains for the application of forces by the thrusters and the electrostatic suspensions | must be estimated from experiments | $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}1$
$\Delta t_{1}$, $\Delta t_{2}$ | delays in the application of biases to the controller computing the actuation | must be estimated from experiments | $\lesssim 1\,\mathrm{s}$
$\Gamma_{x}$ | gravity gradient between the two TMs | could be estimated from experiments with different actuation stiffness but difficult, considered fixed | $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}4\\!\times\\!10^{-9}\,\mathrm{s^{-2}}$
$m_{1}$, $m_{2}$, $m_{\text{SC}}$ | masses of TMs and SC | considered fixed | $1.96\,\mathrm{kg}$, $422.7\,\mathrm{kg}$
The aim of system identification, as thoroughly described in this chapter, is
the estimation of these system parameters with targeted experiments.
#### 4.1.1 Anelasticity and damping
The parameters defined above are implicitly assumed to be independent from
frequency. For example, the parasitic stiffness constant may show a frequency
dependence due to anelasticity (an “internal” dissipation of the string
constant) or a damping effect.
For the sake of clarity, $\omega_{0}^{2}$ is the (negative) stiffness constant
not to be confused with the Fourier angular frequency $\omega$. An
anelasticity can be modeled as a frequency dependence in the imaginary part
[48] of a complex stiffness constant
$\tilde{\omega}^{2}(\omega)=\omega_{0}^{2}\left[1+i\phi(\omega)\right]\leavevmode\nobreak\
,$ (4.4)
where $\phi$ is named the loss angle modeling the dissipation. Sources of
dissipation are the dielectric losses in the surface of the electrodes facing
the TM that can be modeled by a constant
$\phi_{\epsilon}(\omega)=-\delta_{\epsilon}\leavevmode\nobreak\ ,$ (4.5)
such that it produces a force proportional to displacement and in phase with
velocity 222Thanks to the imaginary unit. The minus sign is due to the fact
that the stiffness constant is usually negative.. The other source of
dissipation is the residual gas damping that can be modeled by a function
proportional to frequency
$\phi_{\text{g}}(\omega)=\frac{\omega}{\omega_{0}^{2}\tau}\leavevmode\nobreak\
,$ (4.6)
where $\tau$ is the damping characteristic time. The above produces a force
proportional to velocity 333In fact, the damped harmonic oscillator in
frequency domain is $(-\omega^{2}+i\gamma\omega+\omega_{0}^{2})x=f$, where
$\gamma=1/\tau$ is the damping coefficient. Then, the complex stiffness
constant is given by
$\tilde{\omega}^{2}=\omega_{0}^{2}+i\gamma\omega=\omega_{0}^{2}+i\omega/\tau$..
The loss angle function may show other interesting features beyond the ones
reported here. To first approximation, the following analysis assumes that all
parameters are independent from frequency, at least within the frequency band
of interest.
### 4.2 Noise characterization
One of the objectives of the LPF mission is to provide a full noise projection
of the total equivalent differential acceleration noise between the TMs. As
this is well beyond the scope of this thesis, the following presents a hint of
the problem. Moreover, a theoretical projection of the observed displacement
noise is needed in advance in order to identify the dominant effects in the
noise and produce the generating filters used for all simulations. The noise
projections shown in this section are given by plausible noise shapes
implemented in the simulator provided by ESA (that will be specifically
introduced in the first paragraph of Section 4.4.5).
Figure 4.4 shows a theoretical noise projection of the equivalent acceleration
noise affecting the $x_{1}$ degree of freedom. Evidently, the thruster
actuation noise dominates the total noise budget in the frequency band of
interest. Other important noise sources are the infrared thermal emission of
the SC external surface and the $o_{1}$ sensing noise.
Figure 4.4: Theoretical noise projection of the residual equivalent
acceleration noise of the relative motion between the SC and the first TM for
the nominal dynamics along $x$. The thruster actuation noise dominates the
total noise budget (dashed line) in the frequency band of interest. Other
important noise sources are the infrared thermal emission of the SC external
surface and the $o_{1}$ sensing noise.
Figure 4.5 shows the second and most important projection of the equivalent
differential acceleration noise affecting the $x_{12}$ degree of freedom. A
turning point around $6\,\mathrm{mHz}$ between two regimes is clearly evident.
At high frequency, the $o_{1}$ sensing noise dominates the total noise budget.
At low frequency, $\nicefrac{{2}}{{3}}$ of the total noise budget (in units of
$\sqrt{\text{PSD}}$) is due to force couplings between the SC and the TMs.
Other important noise sources, intervening at low frequency, are the
capacitive actuation noise on the second TM, forces on the TMs coming from
outside the SC and the $o_{12}$ and $o_{1}$ sensing noises.
Figure 4.5: Theoretical noise projection of the residual equivalent
acceleration noise of the relative motion between the TMs for the nominal
dynamics along $x$. At high frequency, the $o_{12}$ sensing noise dominates
the total noise budget (dashed line). At low frequency, $\nicefrac{{2}}{{3}}$
of the total noise budget is due to force couplings between the SC and the
TMs. Other important noise sources are the capacitive actuation noise on the
second TM, forces on the TMs coming from outside the SC and the $o_{12}$ and
$o_{1}$ sensing noises.
The above acceleration noise projections are the equivalent acceleration
inputs to LPF coming from reasonable noise shapes, producing a characteristic
output in the interferometric readouts. Figure 4.6 and Figure 4.7 contain the
relative projections for the two interferometric readouts, $o_{1}$ and
$o_{12}$ along $x$, produced with a plausible transfer model. Analogously to
the equivalent acceleration noise, for $o_{1}$ the thruster actuation noise
dominates the total noise budget in the frequency band of interest. As
previously pointed out, for $o_{12}$ there is a turning point around
$6\,\mathrm{mHz}$. At high frequency, the $o_{1}$ sensing noise dominates the
total noise budget. At low frequency, $\nicefrac{{2}}{{3}}$ of the total noise
budget is due to force couplings between the SC and the TMs. Secondary
sources, intervening at low frequency, are the capacitive actuation noise on
the second TM, forces on the TMs coming from outside the SC, the $o_{12}$ and
$o_{1}$ sensing noises and the thruster actuation noise.
Figure 4.6: Theoretical noise projection of the $o_{1}$ data channel sensing
the relative motion between the SC and the first TM for the nominal dynamics
along $x$. The thruster actuation noise dominates the total noise budget
(dashed line) in the frequency band of interest. Secondary sources are the
infrared thermal emission of the SC external surface and the $o_{1}$ sensing
noise. Figure 4.7: Theoretical noise projection of the $o_{12}$ data channel
sensing the relative motion between the TMs for the nominal dynamics along
$x$. At high frequency, the $o_{12}$ sensing noise dominates the total noise
budget (dashed line). At low frequency, $\nicefrac{{2}}{{3}}$ of the total
noise budget is due to force couplings between the SC and the TMs. Secondary
sources are the capacitive actuation noise on the second TM, forces on the TMs
coming from outside the SC, the $o_{12}$, $o_{1}$ sensing noises and the
thruster actuation noise.
The noise shapes of the interferometric readouts (with their cross-
correlation) are also used for simulation purposes. From those models, noise
shaping filters are derived and integrated into a multi-channel cross-
correlated noise generator [49]. Figure 4.8 reports an example of a noise run
lasting 12 hours and obtained by coloring an input zero-mean
$\delta$-correlated (white) Gaussian noise with those filters. $o_{12}$ shows
a huge red component caused by the increase of the PSD at low frequency, due
to forces on the TMs, as predicted by Figure 4.7. While $o_{1}$ is dominated
by the thruster jitter, $o_{12}$ becomes much less noisy at high frequency,
being dominated by readout noise only. The red noise shape of $o_{12}$ is an
expected feature during the experiments of the LPF mission.
Figure 4.8: A simulated noise run of about 12 hours. $o_{1}$ and $o_{12}$ are
the two interferometer readings. Notice the behavior of $o_{12}$ at low
frequency – an expected feature during the LPF mission – showing a huge red
component caused by force couplings between the TMs and the SC. At high
frequency, $o_{12}$ becomes much less noisy than $o_{1}$, the former being
dominated by only interferometer readout noise and the latter by thruster
noise.
### 4.3 Identification experiments
Among the series of experiments characterizing the LPF mission, a few of
capital importance will tackle system identification. This thesis considers
two identification experiments allowing for a complete identification of the 7
most important system parameters introduced in Section 4.1. As said,
considering bias injections at the level of controller guidance signals is
completely equivalent to applying direct force stimuli through the equivalence
given by (3.5). In the nominal $x$-dynamics two experiments are defined:
1. 1.
an injection into the controller guidance of the $o_{1}$ channel, namely
$o_{\text{i},1}$, producing forces on the SC through thruster actuation;
2. 2.
an injection into the controller guidance of the $o_{12}$ channel, namely
$o_{\text{i},12}$, producing forces on the second TM through capacitive
actuation.
To naively understand how the parameters can be determined from the above
experiments and the model described in Section 4.1, it is useful to make a
projection of the differential operator, whose inverse enters into the
transfer matrix through (4.2).
Figure 4.9 contains the projection of the differential operator from the first
channel to equivalent acceleration in terms of: (i) dynamics and sensing; (ii)
control. Clearly, the control dominates the transfer for almost the entire
frequency band, in order to attenuate the SC jitter. For this reason,
injecting a signal into the first controller guidance (i.e., applying a
thruster actuation on the SC) allows for the identification, in turn, of: the
actuation gain, $A_{\text{df}}$, the first TM coupling to the SC,
$\omega_{1}^{2}$, as well as a possible delay in the application of the same
bias, $\Delta t_{1}$.
Figure 4.9: Frequency dependence of the differential operator for the transfer
from $o_{1}$ to equivalent acceleration. The control dominates the transfer
for almost the entire frequency band, in order to attenuate the SC jitter.
Analogously, Figure 4.10 contains the projection of the differential operator
from the differential channel to equivalent acceleration. Below
$1\,\mathrm{mHz}$, the control dominates the transfer in order to compensate
the differential force disturbances. Above $1\,\mathrm{mHz}$, dynamics and
sensing dominate the transfer. For this reason, injecting a signal into the
second controller guidance (i.e., applying a capacitive actuation on the
second TM) allows for the identification, in turn, of: the actuation gain,
$A_{\text{sus}}$, the differential coupling between the TMs,
$\omega_{12}^{2}$, as well as a possible delay in the application of the same
bias, $\Delta t_{2}$. Given the cross-talk elucidated in Figure 4.3 at low
frequency, the sensing cross-talk, $S_{21}$, can also be determined.
Figure 4.10: Frequency dependence of the differential operator for the
transfer from $o_{12}$ to equivalent acceleration. The control dominates the
transfer at low frequency, in order to compensate the differential force
disturbances.
To conclude the discussion on the projection of the differential operator, it
is worth noting that the off-diagonal terms contribute with a figure of
$1\\!\times\\!10^{-7}\,\mathrm{s^{-2}}$. In particular, even if not shown in
any figures, the transfer from the first channel to the equivalent
differential acceleration is dominated by dynamics and sensing; the other by
control at low frequency and dynamics and sensing at high frequency. As the SC
motion is common-mode and the first and differential channel are correlated,
the estimation of the differential acceleration noise can not be performed
independently of the first channel, which is the only means by which the SC
jitter can be measured and subtracted. The details of such an estimation will
be given in Section 4.5.
The next section is devoted to the estimation of the 7 system parameters by
means of a MIMO approach that maximizes the overall information. The
identification experiments defined at the beginning of this section are
simulated for a total duration of almost 3 hours each – a suitable timescale
for the mission – by injecting stimulating biases. The following facts are
assumed:
1. 1.
the noise $\bm{o}_{\text{n}}$ is generated as in Section 4.2, independently
from the noise-only run which is used for noise characterization, and is
Gaussian and stationary;
2. 2.
the signals $\bm{o}_{\text{s}}$ are simulated in time domain with a MIMO
approach by means of (3.8b), i.e., by anti-Fourier transforming 444The
numerical implementation of the direct and inverse Fourier transform are the
Fast Fourier Transform (FFT) and Inverse FFT (IFFT). Being circular
operations, the input time-series needs to be zero-padded to avoid systematic
errors caused by wrapped-around data [50]. A conservative default value of one
data length is assumed. with $\mathcal{F}^{-1}$ the deterministic input
signals
$\bm{o}_{\text{s}}(t,\bm{p}_{\text{true}})=\mathcal{F}^{-1}\left[\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p}_{\text{true}})\,\bm{o}_{\text{i}}(\omega)\right](t)\leavevmode\nobreak\
,$ (4.7)
where $\bm{p}_{\text{true}}$ is the set of assumed true system parameter
values to be estimated from the analysis and which the estimation of residual
equivalent acceleration noise depends on;
3. 3.
the superposition principle of signals and noise holds true in the hypothesis
of small motion and in absence of non-linearities in the system, so that the
“experimental” data are simulated by
$\bm{o}_{\text{exp}}=\bm{o}_{\text{s}}+\bm{o}_{\text{n}}\leavevmode\nobreak\
.$ (4.8)
The underlying idea in parameter estimation is to excite the system with
proper high SNR signals so that the modeled parameters can be measured. A
typical injected bias is a series of sine waves of logarithmically increasing
frequency, with integer number of cycles, divided by gaps of $150\,\mathrm{s}$
to allow for system relaxation. The sine stretches last $1200\,\mathrm{s}$
each. The amplitudes are conservatively selected not to exceed $1\%$ of the
operating range of the interferometer, corresponding to a maximum sensed
displacement of $1\,\mathrm{\mu m}$, and $10\%$ of the maximum allowed force
authority, corresponding to $10\,\mathrm{\mu N}$ of thruster actuation and
$0.25\,\mathrm{nN}$ of capacitive actuation. The biases are parameterized in
Table 4.2 and referred to the standard input signals used for the rest of the
analysis. Instead, Chapter 5 will focus on the optimization of the same input
signals.
Table 4.2: Controller guidance signals injected as biases for system identification. The sine stretches last $1200\,\mathrm{s}$ each and are separated by gaps of $150\,\mathrm{s}$. The sine waves perform an integer number of cycles, from 1 to 64. The amplitudes are selected to not exceed $1\%$ of the operating range of the interferometer and $10\%$ of the maximum force authority. $o_{\text{i},1}$ for Exp. 1 | $o_{\text{i},12}$ for Exp. 2
---|---
$f\,\mathrm{[mHz]}$ | $a\,\mathrm{[\mu m]}$ | $f\,\mathrm{[mHz]}$ | $a\,\mathrm{[\mu m]}$
$0.83$ | $1.0$ | $0.83$ | $0.80$
$1.7$ | $1.0$ | $1.7$ | $0.48$
$3.3$ | $1.0$ | $3.3$ | $0.19$
$6.6$ | $1.0$ | $6.6$ | $0.088$
$13$ | $0.59$ | $13$ | $0.096$
$27$ | $0.28$ | $27$ | $0.18$
$53$ | $0.14$ | $53$ | $0.46$
Data are simulated at $10\,\mathrm{Hz}$ and decimated to $1\,\mathrm{Hz}$ to
ease data processing. During the mission, data will be collected at a sample
rate between $1$ and $10\,\mathrm{Hz}$, depending on the experiment and
available down-link bandwidth. The simulation of the first experiment, with
injection of the $o_{i,1}$ signal of Table 4.2, is shown in Figure 4.11. The
response of the system in $o_{1}$ is approximately equal to $o_{i,1}$, except
at high frequency where there is a modest gain due to the particular shape of
the first diagonal element of the transfer function at that frequency. A
residual signal in $o_{12}$ of absolute peak $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}4\\!\times\\!10^{-8}\,\mathrm{m}$ is also visible
and due to dynamical cross-talk. As said before, the gaps allow for system
relaxation, particularly at high frequency.
Figure 4.11: Exp. 1 synthetic data. An injection of sine-wave signals lasting
for almost 3 hours into the first controller guidance $o_{\text{i},1}$
produces a different response in the two interferometer readings. The response
in $o_{1}$ is approximately equal to $o_{\text{i},1}$ (dashed line), except at
high frequency where there is a modest gain. A residual signal in $o_{12}$ of
absolute peak $\raise 0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}4\\!\times\\!10^{-8}\,\mathrm{m}$ is due to dynamical cross-talk (see
inset at the left bottom side). Gaps between two cycles of injection allow for
system relaxation (see inset at the right top side).
The simulation of the second experiment, with injection of the $o_{i,12}$
signal of Table 4.2, is shown in Figure 4.12. The response of the system in
$o_{12}$ is evidently phase delayed to $o_{\text{i},12}$. At high frequency,
the very low gain of the transfer function almost suppresses the signal. Since
the transfer from $o_{i,12}$ to $o_{1}$ is negligible, in this experiment
$o_{1}$ has signal contribution completely hidden by noise. For this reason,
during the mission the $o_{1}$ readout will serve as a useful sanity check for
a first understanding of the model.
Figure 4.12: Exp. 2 synthetic data. An injection of sine-wave signals lasting
for almost 3 hours into the second controller guidance $o_{\text{i},12}$
produces a different response in the two interferometer readings. The response
in $o_{12}$ is evidently phase delayed to $o_{\text{i},12}$. At high
frequency, the very low gain of the transfer function almost suppresses the
signal (see inset). The $o_{1}$ data channel has negligible contribution
hidden by the noise.
### 4.4 Parameter estimation
During the mission, noise runs will be used to characterize the noise itself
and estimate the total equivalent input acceleration. The estimation of the
total equivalent acceleration is possible if LPF is properly modeled. For this
in the various experiments, signals will be injected along different degrees
of freedom to study the response of the system. Along $x$, LPF will be
characterized giving, as a first approximation, the nominal dynamics. Instead,
along others degrees of freedom, LPF will be characterized in terms of the
many cross-talk contributions arising from the dynamical couplings, the
imperfections in the sensing conversion and the imperfections in the
actuation.
This section handles the general problem of estimating the LPF parameters
modeled as a MIMO dynamical system, where different inputs enters into the
system and produce a response in different outputs. For the sake of
simplicity, for the rest only the two experiments introduced above – the
characterization of the nominal dynamics along the optical axis – are
considered, bearing in mind that the method is general enough to handle more
complex experiments. An example would be the identification of the $xy$ cross-
talk, in where guidance or force bias signals are injected, in turn, along
$y_{1}$, $y_{2}$, $\phi_{1}$, $\phi_{2}$ and $\phi_{\text{SC}}$ to study the
response along the optical axis.
Finally, this section develops and validates the estimation procedures on the
two most important experiments described in the previous section. It also
shows the application to a couple of non-standard scenarios that may happen
during the real LPF mission.
#### 4.4.1 Review of the problem
The experimental data (either simulated or from the mission) can be modeled
superimposing deterministic signals with noise
$\bm{o}_{\text{exp}}=\bm{o}_{\text{s}}+\bm{o}_{\text{n}}\leavevmode\nobreak\
,$ (4.9)
where $\bm{o}_{\text{n}}$ is the output noise with cross PSD matrix
$\bm{S}_{\text{n}}$ and
$\bm{o}_{\text{s}}(t,\bm{p})=\mathcal{F}^{-1}\left[\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p})\,\bm{o}_{\text{i}}(\omega)\right](t)\leavevmode\nobreak\ ,$
(4.10)
are the so-called template signals obtained by injecting bias guidance signals
$\bm{o}_{\text{i}}$ into the system modeled by the transfer matrix
$\bm{T}_{o_{\text{i}}\rightarrow o}$.
It is useful to think that the experimental data depends on the true parameter
values
$\bm{o}_{\text{exp}}=\bm{o}_{\text{exp}}(t,\bm{p}_{\text{true}})\leavevmode\nobreak\
,$ (4.11)
that need to be estimated from fitting procedures. In the case of simulated
experiments, the true values are exactly those used in data generation. In the
case of real mission experiments, the true values are actually those giving
the best possible description of the data, the one that perfectly subtracts
the deterministic signals, hence recovering the instrumental noise shapes.
In the same way, the observed noise (either simulated or from the mission)
depends on the parameter values
$\bm{o}_{\text{n}}=\bm{o}_{\text{n}}(t,\bm{p}_{\text{true}})\leavevmode\nobreak\
;$ (4.12)
but can be considered constant with respect to the parameter values for the
timescale of an identification experiment where only high SNR signals will be
injected.
The scope of parameter estimation is to recover the best possible description
of the experimental data. If the residuals between the experimental data and
the modeled template signals are defined by
$\bm{o}_{\text{r}}=\bm{o}_{\text{exp}}-\bm{o}_{\text{s}}\leavevmode\nobreak\
,$ (4.13)
the best possible description of the experimental data is given by
$\bm{o}_{\text{r}}(t,\bm{p}_{\text{est}})\simeq\bm{o}_{\text{n}}(t,\bm{p}_{\text{true}})\leavevmode\nobreak\
,$ (4.14)
implying that the residuals evaluated at the estimated parameter values
$\bm{p}_{\text{est}}$ recover the true instrumental noise.
#### 4.4.2 Estimation method
LPF is a MIMO dynamical system for which each experiment has a unique set of
meaningful parameters. Hence, for two generic experiments two sets of
parameters can be independently determined. Sometimes a subset may be shared
between the two; sometimes there could be parameters that can be estimated by
only a particular experiment. Moreover, each experiment has multiple readouts
sensitive to different parameters. Section 4.3 has given an intuitive hint of
such an idea.
The first approach is to build an information-weighted average [40, 47] of
different parameter estimates coming from all readouts and experiments. If
$\bm{p}_{ij}$ are the parameter estimates of the $i$-th experiment and $j$-th
readout, the corresponding Fisher information matrix [51]
$\bm{\mathcal{I}}_{ij}=\int{\nabla_{\bm{p}}\bm{o}_{\text{r}}^{(ij)}(\omega,\bm{p}_{\text{est}})}^{*}\,\bm{S}_{\text{n}}^{(ij)}(\omega)^{-1}\,\nabla_{\bm{p}}\bm{o}_{\text{r}}^{(ij)}(\omega,\bm{p}_{\text{est}})\,\text{d}\omega\leavevmode\nobreak\
,$ (4.15)
where $\bm{S}_{\text{n}}^{(ij)}$ is the noise PSD of $i$-th experiment and
$j$-th readout, $\bm{o}_{\text{r}}^{(ij)}$ is the corresponding vector of
residuals, $\nabla_{\bm{p}}$ is the gradient with respect to the parameters
and ∗ is the conjugate transpose. The final combined parameter estimates are
given by
$\bm{p}=\bm{\mathcal{I}}^{-1}\sum_{i=1}^{N_{\text{exp}}}\sum_{j=1}^{N_{o}}\bm{\mathcal{I}}_{ij}\,\bm{p}_{ij}\leavevmode\nobreak\
,$ (4.16)
where $N_{\text{exp}}$ is the number of experiments and $N_{o}$ the number of
readouts per experiment assumed the same across the experiments. The combined
Fisher information matrix is
$\bm{\mathcal{I}}=\sum_{i=1}^{N_{\text{exp}}}\sum_{j=1}^{N_{o}}\bm{\mathcal{I}}_{ij}\leavevmode\nobreak\
.$ (4.17)
Notice that the estimates $\bm{p}_{ij}$ may have different dimension depending
of the $i$-th experiment and $j$-th interferometric readout; the same happens
for the corresponding information matrices. The issue can be easily solved by
inserting zeros where there is no information.
An example can readily show that the definition of (4.16) is not robust. In
fact, suppose that the estimation of the system parameters is performed
independently on each readout and one of those parameters has a biased value
for an inaccuracy of the transfer matrix model. Therefore, the information
matrix for that estimate is biased and the combined one in (4.17) as well. The
numerical inversion in (4.16) inexorably amplifies that bias to the combined
parameter estimates. To overcome the problem, one could try removing the
failing estimates (which is possible only if one has good indication of what
the real values are, for example, from ground measurements or previous
independent experiments), but in doing so information and precision would
definitely be lost.
The only solution is to attack the problem by a complete MIMO approach where
the poor information coming from the biased model of a readout is continuously
compensated by the others as the optimization goes on. One other advantage is
that a joint information can likely remove or, at least, reduce the effect of
parameter degeneracies.
The MIMO-Multi-Experiment joint log-likelihood of the system is a
generalization of the standard definition [51] and is given by
$\chi^{2}(\bm{p})=\int{\bm{o}_{\text{r}}(\omega,\bm{p})}^{*}\,\bm{S}_{\text{n}}(\omega)^{-1}\,\bm{o}_{\text{r}}(\omega,\bm{p})\,\text{d}\omega\leavevmode\nobreak\
,$ (4.18)
where
$\bm{o}_{\text{r}}(\omega,\bm{p})=\bm{o}_{\text{exp}}(\omega)-\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p})\,\bm{o}_{\text{i}}(\omega)\leavevmode\nobreak\ ,$ (4.19)
are the residuals between the experimental data $\bm{o}_{\text{exp}}$ and the
modeled system response. $\bm{o}_{\text{i}}$ are the controller biases,
$\bm{T}_{o_{\text{i}}\rightarrow o}$ the transfer matrix depending on all
system parameters $\bm{p}$ (stiffness constants, sensing cross-talk, etc.),
$\bm{S}_{\text{n}}$ the cross output noise PSD matrix assumed constant to the
system parameters. For two experiments and two interferometric readouts each,
$\bm{o}_{\text{i}}$ is a 4-vector, null in the second and third element, since
the injection is in $o_{\text{i},1}$ (first experiment) and $o_{\text{i},12}$
(second experiment); $\bm{T}_{o_{\text{i}}\rightarrow o}$ is a block diagonal
$4\times 4$-matrix replicating the same $2\times 2$-matrix;
$\bm{S}_{\text{n}}$ is a $4\times 4$-matrix of cross PSDs between different
readouts and experiments; $\bm{o}_{\text{exp}}$ is a 4-vector of all
experimental readouts.
Assuming that all readouts are sampled at the same rate and last for the same
duration, the overall number $\nu$ of degrees of freedom for the problem is
defined as
$\nu=N_{\text{exp}}\times N_{o}\times
N_{\text{data}}-N_{p}\leavevmode\nobreak\ ,$ (4.20)
where $N_{\text{exp}}$ is the number of experiments; $N_{o}$ is the number of
readouts per experiment (assumed the same across the experiments);
$N_{\text{data}}$ is the number of data points per readout; $N_{p}$ is the
dimension of the parameter space. For example, $\nu\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}4\\!\times\\!10^{4}$ for two experiments, two
readouts each, lasting for about 3 hours and sampled at $1\,\mathrm{Hz}$. For
the rest, if not otherwise stated, the reduced log-likelihood $\chi^{2}/\nu$
will be used in place of the standard definition, as its expectation value is
1.
Notice that system identification may be also implemented in the domain of
equivalent acceleration. If the $\bm{\Delta}$ operator is invertible, the two
approaches – identification in acceleration and displacement – are completely
equivalent. In fact,
$\begin{split}\chi^{2}&=\int{\left(\bm{f}_{\text{exp}}-\bm{f}_{\text{mdl}}\right)}^{*}\,\bm{S}_{\text{n},f}^{-1}\left(\bm{f}_{\text{exp}}-\bm{f}_{\text{mdl}}\right)\,\text{d}\omega\\\
&=\int{\left(\bm{o}_{\text{exp}}-\bm{o}_{\text{mdl}}\right)}^{*}\,{\bm{\Delta}}^{*}\left(\left.{\bm{\Delta}}^{*}\right.^{-1}\bm{S}_{\text{n},o}^{-1}\,\bm{\Delta}^{-1}\right)\bm{\Delta}\left(\bm{o}_{\text{exp}}-\bm{o}_{\text{mdl}}\right)\,\text{d}\omega\\\
&=\int{\left(\bm{o}_{\text{exp}}-\bm{o}_{\text{mdl}}\right)}^{*}\,\bm{S}_{\text{n},o}^{-1}\,\left(\bm{o}_{\text{exp}}-\bm{o}_{\text{mdl}}\right)\;\text{d}\omega\leavevmode\nobreak\
.\end{split}$ (4.21)
where $\bm{f}_{\text{mdl}}$ and $\bm{f}_{\text{exp}}$ are the modeled and
experimental equivalent accelerations; $\bm{o}_{\text{mdl}}$ and
$\bm{o}_{\text{exp}}$ are the modeled and experimental displacement readouts.
In the preceding equation, $\bm{\Delta}$ and $\bm{\Delta}^{-1}$ are used to
transform the sensed relative motion into equivalent acceleration (and vice-
versa) and contain the dependence on the modeled parameters. The main benefit
of working with accelerations is the automatic subtraction of system
transients as described in Section 3.2 and that is numerically demonstrated at
the end of this chapter. Despite the identification in displacement where the
parameters are explicit in the modeled template signal, in the identification
in acceleration the parameters are implicit in the estimated acceleration.
Even though there is no real experimental acceleration because this must be
estimated from the displacement readouts, system identification in
acceleration domain can be still carried out numerically with a non-standard
approach based upon a closed-loop optimization over the estimated acceleration
data, whereas the modeled forces are the injected bias signals. For the rest,
the following discussion employs the estimation in the domain of displacement
readouts, as the other approach is currently under investigation.
The MIMO-Multi-Experiment Fisher information matrix for the parameter
estimates $\bm{p}_{\text{est}}$ is the local curvature of the log-likelihood
surface around the minimum and is given by
$\bm{\mathcal{I}}=\int{\bm{o}_{\text{i}}(\omega)}^{*}\,{\nabla_{\bm{p}}\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p}_{\text{est}})}^{*}\,\bm{S}_{\text{n}}(\omega)^{-1}\,\nabla_{\bm{p}}\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p}_{\text{est}})\,\bm{o}_{\text{i}}(\omega)\,\text{d}\omega\leavevmode\nobreak\
,$ (4.22)
where $\nabla_{\bm{p}}$ is the gradient with respect to all 7 system
parameters. As above, if $\bm{T}_{o_{\text{i}}\rightarrow o}$ is a $4\times
4$-matrix, then $\nabla_{\bm{p}}\bm{T}_{o_{\text{i}}\rightarrow o}$ is a
$7\times 4\times 4$-tensor and the information is a $7\times 7$-matrix as
required. The very high SNR regime of the signals in Figure 4.11 and Figure
4.12 assures that the linear approximation of (4.22) holds true and no
corrective terms arise as pointed out by [52] and more recently by [53]. As
the inverse of the information matrix provides the estimated covariance
matrix, the validity of the linear approximation is checked a posteriori in
Section 4.4.6 by inspecting the statistics of a Monte Carlo simulation.
#### 4.4.3 Whitening
The colored noise behavior of a typical LPF run makes mandatory to decorrelate
the data used for system identification in order for a generic statistical
estimator be unbiased. Consider for example a stationary noisy time-series
$o(t)$ with noise PSD $S_{\text{n}}(\omega)$. The SNR of the signal [51] can
be recast as
$\begin{split}\rho^{2}&=\int\frac{{o}^{*}(\omega)\,o(\omega)}{S_{\text{n}}(\omega)}\,\text{d}\omega\\\
&=\int
o^{*}_{\text{w}}(\omega)\,o_{\text{w}}(\omega)\,\text{d}\omega\leavevmode\nobreak\
,\end{split}$ (4.23)
which can be viewed as the acting of the whitening filter
$W(\omega)=1/\sqrt{S_{\text{n}}(\omega)}$ on $o(\omega)$ to produce the
whitened series
$o_{\text{w}}(\omega)=W(\omega)\,o(\omega)\leavevmode\nobreak\ .$ (4.24)
Here “whitened” is equivalent to saying that the noise PSD of the filtered
series is approximately frequency-independent. The discrete time-domain
version of the preceding involves the noise covariance matrix
$\bm{C}_{\text{n}}$
$\begin{split}\rho^{2}&={\bm{o}}^{\mathsf{T}}\,\bm{C}_{\text{n}}^{-1}\bm{o}\\\
&={\bm{o}}^{\mathsf{T}}_{\text{w}}\,\bm{\Lambda}_{\text{n}}^{-1}\bm{o}_{\text{w}}\leavevmode\nobreak\
,\end{split}$ (4.25)
which again can be viewed as the acting of the whitening filter $\bm{W}$, an
orthogonal matrix satisfying
$\bm{C}_{\text{n}}^{-1}={\bm{W}}^{\mathsf{T}}\bm{\Lambda}_{\text{n}}^{-1}\bm{W}$
555In fact, if
$\bm{C}_{\text{n}}={\bm{U}}^{\mathsf{T}}\bm{\Lambda}_{\text{n}}\bm{U}$ where
$\bm{U}$ is an orthogonal matrix and $\bm{\Lambda}_{\text{n}}$ is the eigen-
decomposition of $\bm{C}_{\text{n}}$, then it turns out that
$\bm{U}^{-1}={\bm{W}}^{\mathsf{T}}$., on $\bm{o}$ to produce the whitened
unit-variance series
$\bm{o}_{\text{w}}=\bm{W}\bm{o}\leavevmode\nobreak\ .$ (4.26)
As above, “whitened” means that the process diagonalizes the covariance
matrix, so that $\bm{\Lambda}_{\text{n}}$ effectively becomes an identity
matrix.
For simulation and analysis purposes, whitening a time-series is formally the
inverse process of noise generation. Whitening filters are obtained by
performing a fit in the $z$-domain to the inverse of the estimated PSD
666Throughout this thesis, if not otherwise stated, it is assumed that a PSD
is estimated by means of the Welch (modified periodogram) method [54] using a
4-sample 92-dB Blackman-Harris window [55], 16-segments averaged, $66\%$
overlap and mean detrended.. Figure 4.13 reports an example of whitening
777Data filtering can produce fake transients at the beginning of the filtered
time-series. To avoid this possibility, an initial segment of data is usually
cut away. a typical 28-hour run of interferometric noise. The effect of the
whitening filters, as required, is to flatten the noise shapes, i.e., to
decorrelate the time-series.
Figure 4.13: Whitening of a simulated noise run. $o_{1}$ and $o_{12}$ are the
two interferometer readings with PSD reported on the basis of the scale on the
left end side. $o_{1,\text{w}}$ and $o_{12,\text{w}}$ are the whitened
counterparts with PSD reported on the basis of the scale at the right end
side. They show how the whitening filters can flatten the noise shapes. The
convolution with a low-pass filter of the data resampling from 10 to
$1\,\mathrm{Hz}$ is the cause of the drop around $0.5\,\mathrm{Hz}$.
Despite the PSD shapes which seem reasonably good at first sight, a residual
red component still persists. Table 4.3 reports two higher-order moments
(skewness and excess kurtosis) of the empirical distribution together with
their uncertainties [50]. By inspecting the values, it turns out that the
sample mean of the differential channel $o_{12}$ is not compatible with zero,
as one would expect. Usually, a first or second order polynomial fit is
necessary to subtract that residual component. The result is not surprising:
the intrinsical difficulty is that the whitening process is performed on a
restricted frequency band (the one of the estimated PSD) and low-frequency
components may survive after the filtering.
Table 4.3: Sample mean $\mu$, standard deviation $\sigma$ and higher moments, the sample skewness $\gamma_{1}$ and the excess kurtosis $\gamma_{2}$, for the whitened data channels $o_{1}$ and $o_{12}$. Assuming Gaussian-distributed data, the approximate standard deviations are $\sigma_{\mu}\simeq\sigma/\sqrt{N}$, $\sigma_{\sigma}\simeq\sigma/\sqrt{2N}$, $\sigma_{\gamma_{1}}\simeq\sqrt{\nicefrac{{6}}{{N}}}$, $\sigma_{\gamma_{2}}\simeq\sqrt{\nicefrac{{24}}{{N}}}$, with $N$ the number of data samples. Data | $\mu$ | $\sigma$ | $\gamma_{1}$ | $\gamma_{2}$
---|---|---|---|---
$o_{1,\text{w}}$ | $0.008\,\pm\,0.003$ | $0.970\,\pm\,0.002$ | $(-5\,\pm\,8)\\!\times\\!10^{-3}$ | $(0\,\pm\,2)\\!\times\\!10^{-2}$
$o_{12,\text{w}}$ | $-0.254\,\pm\,0.003$ | $1.002\,\pm\,0.002$ | $(0\,\pm\,8)\\!\times\\!10^{-3}$ | $(3\,\pm\,2)\\!\times\\!10^{-2}$
The extent to which the idea of this section holds true depends on the
assumption of stationarity and Gaussianity. Even though for LPF the
interferometric noise is not explicitly dependent on the system parameters, it
may depend implicitly through the coupling between the external force noise
and the system response. Yet, as it will be discussed later in this chapter,
the estimated equivalent acceleration noise depends explicitly on the system
parameters through the transfer matrix given by the differential operator.
As a matter of fact, a non-stationarity in any of the system parameters
implies a non-stationarity in the noise. In fact, if $o=o\left(t,p(t)\right)$
is a generic interferometer readout depending, for simplicity, on just one
parameter fluctuating of $\delta p$ around the nominal value $p_{0}$, then to
first order $o\simeq o_{0}+o^{\prime}\,\delta p$, where $o_{0}=o(t,p_{0})$ and
$o^{\prime}=\left.\partial o(t,p)/\partial o\right|_{p_{0}}$. For a zero-mean
process the total variance is
$\text{Var}[o]\simeq\text{Var}[o_{0}]+\text{Var}^{\prime}[o_{0}]\,\delta
p+\text{Var}[o^{\prime}]\,\delta p^{2}\leavevmode\nobreak\ ,$ (4.27)
where the linear and quadratic terms come from the covariance between $o_{0}$
and $o^{\prime}$ and the variance of $o^{\prime}$ itself (see Appendix A.5 for
details). Therefore, if any of the system parameters fluctuates, noise is
likely to become non-stationary. In LPF all PSDs must be estimated piecewise
along data segments approximately stationary on a timescale given by the one
of the fluctuating parameter. The converse, i.e., a non-stationarity in the
noise implies a non-stationarity in any of the parameters is not assured,
since other effects, independent from those parameters, may still be relevant.
For example, Section 4.4.8 describes the possible existence of glitches, a
non-stationary behavior in the noise, and its impact to system identification.
Instead, Appendix A.6 introduces the time-frequency approach in the study of
non-stationarity noise.
#### 4.4.4 Search algorithm
The joint log-likelihood (4.18) for two experiments, two readouts each, is
implemented in time domain by means of FFT/IFFT the whitened time-series. The
relevant iteration steps of the process taking to the final estimates of the
system parameters, in loop of increasing accuracy, are:
1. 1.
the whitening filters are estimated on a long noise run, as in Section 4.4.3;
2. 2.
the interferometric readouts of each experiments are whitened;
3. 3.
the templates are generated according to (4.10) for the current parameter
values;
4. 4.
the templates are whitened;
5. 5.
the log-likelihood is evaluated, i.e., “models fit the data”, for the current
parameter values;
6. 6.
the parameter values are updated according to the adopted optimization scheme.
From the optimization viewpoint, the log-likelihood is named the merit
function, i.e., the one being minimized as the parameter values are updated.
Figure 4.14 shows a sketch of the whole process of system identification. The
data production provides for the noise run and the experiments, with both
interferometric readouts and injected biases. Instead, the modeling provides
for the proper transfer matrix being used for simulating the template signals.
Finally, the data analysis concerns the estimation of the whitening filters
and the algorithm for the log-likelihood minimization.
Figure 4.14: Sketch of the system identification process for the two simulated
experiments along the optical axis. Noise run and experiments pertain to data
production. The modeling provides for the transfer matrix being used for
simulation and analysis. For system identification, data analysis comprises
the estimation of whitening filters and the log-likelihood optimization. The
estimated parameters are output together with their covariance matrix.
The algorithm performs a log-likelihood minimization by taking advantage of
the most recent developments in numerical non-linear optimization [50]. During
the work for this thesis, an investigation of different optimization
algorithms was carried out. Non-standard schemes like the simulated annealing,
genetic algorithms and the pattern search, with or without a multi start (an
initial Monte-Carlo-like exploration of the parameter space in which the
initial most likely points are taken into account for further processing),
were considered for the purposes of system identification. They were also
compared to a mixed strategy employing more standard and widely-used
optimization algorithms applied in sequence:
1. 1.
the preconditioned conjugate gradient search (alternatively, the quasi-Newton
method) explores the parameter space to large scales;
2. 2.
the derivative-free simplex allows to reach the required numerical accuracy.
The key advantage of mixing different approaches is that the global structure
of the parameter space can be explored while keeping the numerical accuracy.
Such an investigation proved that for the LPF system identification non-
standard schemes have comparable performances with respect to the one proposed
above which is assumed for the rest. The optimization is numerically
controlled and stopped until either the function tolerance or the average
parameter tolerance meets the requirement of $1\\!\times\\!10^{-4}$. The final
parameter estimates are output from the fitting tool, together with the
estimated covariance matrix, obtained by inverting the Fisher information
matrix (4.22) around the minimum.
Before showing the application to data simulated specifically for LPF, the
tool was checked against more simpler cases like the linear fit (which is
analytically solvable to any order), the chirped sine and the harmonic
oscillator [56, 57]. The result is that there are no systematic errors and
parameter uncertainties are in accordance to a Monte Carlo simulation. A
similar check for LPF is discussed in Section 4.4.6.
#### 4.4.5 ESA simulator
A very important test-bench on both system modeling and validation of the
estimation techniques is the analysis of realistic data, closer to the actual
LPF mission than the ones simulated and shown in this work. A real LPF
simulator, named Off-line Simulation Environment (OSE), provided by ESA and
written by ASTRIUM has given the chance to promptly analyze the data as they
were realistically produced during the mission. The OSE is a state-space
representation of a 3-dimensional LPF model written under the MATLAB® and
Simulink® [58] environments. It contains the most relevant disturbances and
noise sources, the same actuation algorithms for drag-free, electrostatic
suspension and attitude controls (DFACS) embedded in LPF, all couplings within
the dynamics along the optical axis and between different degrees of freedom.
The OSE was written to mainly check all procedures, the mission timeline, the
experiments and validate the noise budget.
Several extended data analysis operational exercises were called in the past 2
years, where parameter estimation had a pivotal role, and therefore very
similar to a mock data challenge, where data production is strictly separated
from data analysis. The operational exercises culminated with the sixth one
targeted to parameter estimation using a linear fit with singular value
decomposition, a Markov-chain Monte Carlo method and the one described in this
thesis. The first application of system identification on that operational
exercise is contained in [59]. The final conclusion of the activity on the
same exercise is recently described in [60]. The three methods are apparently
in good agreement to each other, particularly the first and third approaches,
but an investigation of the fit residuals, like the one in Figure 4.4.7, shows
a mismatch in the first experiment between data and model at high frequency.
The fact is confirmed by a statistical comparison between the residual PSDs to
a noise-only measurement with a very general and model-independent method
based on the Kolmogorov-Smirnov test [61]. The explanation of such a mismatch
will be given in the near future with further operational exercises and much
more detailed knowledge of the simulator.
#### 4.4.6 Monte Carlo validation
The aim of this section is to statistically validate the estimation method
presented so far. A Monte Carlo simulation of $1000$ different noise
realizations is used to check for consistency of the method. The estimation is
identically repeated at each step, enabling fine tuning and the study of the
statistics for every system parameter.
Table 4.4 reports on the comparison between the mean best-fit values and the
true values: the accordance is at the level of at most 2 standard deviations
and demonstrates that the estimation method is statistically unbiased.
Secondarily, it shows the best-fit standard deviations, i.e., the parameter
fluctuations due to noise, compared to the mean expected standard deviations
(the mean fit errors).
Table 4.4: Monte Carlo validation of 1000 independent noise realizations on which parameter estimation is repeated identically at each step. The mean best-fit values are compatible with the true values within 2 standard deviations. The terms in brackets are the error relative to the rightmost digit. The mean expected standard deviations (estimated from the fit) and the best-fit standard deviations are approximately the same order of magnitude. The mean log-likelihood is $\chi^{2}=0.96$ with $\nu=79993$. Parameter | True | Mean | Best-fit | Mean
---|---|---|---|---
| best-fit | st. dev. | exp. st. dev.
$\omega_{1}^{2}\,[10^{-6}\,\text{s}^{-2}]$ | $-1.303$ | $-1.303006(7)$ | $2\times 10^{-4}$ | $1\times 10^{-3}$
$\omega_{12}^{2}\,[10^{-6}\,\text{s}^{-2}]$ | $-0.698$ | $-0.697998(6)$ | $2\times 10^{-4}$ | $5\times 10^{-4}$
$S_{21}\,[10^{-4}]$ | $0.9$ | $0.90004(9)$ | $3\times 10^{-3}$ | $4\times 10^{-3}$
$A_{\text{df}}$ | $1.003$ | $1.00297(1)$ | $4\times 10^{-4}$ | $4\times 10^{-4}$
$A_{\text{sus}}$ | $0.9999$ | $0.9999001(1)$ | $4\times 10^{-6}$ | $2\times 10^{-5}$
$\Delta t_{1}\,[\text{s}]$ | $0.06$ | $0.059995(3)$ | $9\times 10^{-5}$ | $3\times 10^{-4}$
$\Delta t_{12}\,[\text{s}]$ | $0.05$ | $0.05000(3)$ | $8\times 10^{-4}$ | $1\times 10^{-3}$
Figure 4.15 shows a more in-depth analysis of all parameter statistics. The
accordance between the sample statistics of the Monte Carlo simulation and the
scaled theoretical Gaussian Probability Density Function (PDF) (evaluated at
the sample mean and standard deviation) is self-evident and demonstrates that:
(i) the estimation is statistically unbiased; (ii) the parameters are Gaussian
distributed.
| |
---|---|---
(a) | (b) | (c)
| | |
---|---|---|---
(d) | (e) | (f) | (g)
Figure 4.15: Monte Carlo validation of 1000 independent noise realizations on
which parameter estimation is repeated identically at each step. The plots
show the statistics for all parameter estimates (a)-(g). The scaled Gaussian
PDF is evaluated at the sample mean (dashed vertical lines) and sample
standard deviation (half horizontal bars), which are compared to the true
values (solid vertical lines).
Analogously, Figure 4.16 shows the statistics for the estimated variances.
Theory prescribes that the variance must be $\chi^{2}$ distributed, but for
$\nu=79993$ the $\chi^{2}$ distribution tends to a Gaussian distribution with
very good approximation, as is clear from the plots.
| |
---|---|---
(a) | (b) | (c)
| | |
---|---|---|---
(d) | (e) | (f) | (g)
Figure 4.16: Monte Carlo validation of 1000 independent noise realizations on
which parameter estimation is repeated identically at each step. The plots
show the statistics for all parameter variances (a)-(g). The scaled Gaussian
PDF is evaluated at the sample mean and standard deviation.
Appendix A.7 also discusses some other interesting features of the Monte Carlo
statistics, like the parameter correlation, related to the rotation of the
log-likelihood paraboloid principal axis around the minimum, and the scatter
of the estimation chains due to the noise fluctuation.
The final, and most remarkable check, is the comparison between the fit
$\chi^{2}$ log-likelihood and the one calculated on pure noise data contained
in Figure 4.17. It is worth stressing that both the fit and the noise
$\chi^{2}$ showed agreement between each other, but they were both positively
skewed in a preliminary Monte Carlo simulation. The following facts explain
why. Section 4.4.3 has discussed the practical method to implement the
diagonalization of the noise covariance matrix with its main limitation. This
consists in the impossibility of filtering out the lowest frequencies, due to
the finiteness of the data stretches from which whitening filters are derived
and which causes the skewness. Transparently, the application of a high-pass
filter to the data has solved the issue. The comparison in the plot provides
for an important twofold test: on one side, the parameter variances are
statistically distributed as the fit $\chi^{2}$ log-likelihood, as required;
on the other, the fit $\chi^{2}$ log-likelihood is in agreement with the noise
$\chi^{2}$ log-likelihood, showing that the estimation method has
statistically suppressed the deterministic signals and recovered the noise
statistic with no extra bias.
|
---|---
(a) | (b)
Figure 4.17: Monte Carlo validation of 1000 independent noise realizations on
which parameter estimation is repeated identically at each step. The plots
show the statistic for (a) the fit $\chi^{2}$ log-likelihood and (b) the noise
$\chi^{2}$ log-likelihood. The agreement between the two demonstrates that the
deterministic signals are statistically suppressed out of the data.
#### 4.4.7 Non-standard scenario: under-performing actuators and under-
estimated couplings
System identification has a key role in compensating the SC jitter and the TM
couplings. Even in the unlikely (but possible) situation of under-performing
actuators or under-estimated force couplings, it is still possible to retrieve
the actual parameter values and allow for a precise estimation of the total
equivalent acceleration noise without loosing sensibility and getting into
systematic errors. The impact on the estimation of the total equivalent
acceleration noise will be illustrated in Section 4.5.
To introduce the problem, suppose that the predicted TM couplings are
$\omega_{1}^{2}=-1.3\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$ and
$\omega_{12}^{2}=-0.7\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$ and during the LPF
mission:
1. 1.
the actual TM couplings are about two times the predicted ones, due to
unexpected/unmodeled stronger forces, like
$\omega_{1}^{2}=-3\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$ and
$\omega_{12}^{2}=-2\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$;
2. 2.
the thruster and capacitive actuators unfortunately misfunction, due to both a
breakdown of one or more thruster clusters and a loss of efficiency in the
capacitive actuators on the second TM; this situation can be described by
gains sensitively lower than one, like $A_{\text{df}}=0.62$ and
$A_{\text{sus}}=0.6$;
3. 3.
the interferometer introduces an extra cross-talk,
$S_{21}=1.5\\!\times\\!10^{-3}$, ten times the expected one $S_{21}\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}1\\!\times\\!10^{-4}$.
In this very unfortunate situation, system identification, see Table 4.5,
allows for the estimation of the true values within 1 standard deviation from
the true values, so maintaining precision, even though the optimizations
starts from initial guesses which are typically $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}10^{3}$ standard deviations away, so guaranteeing
accuracy too.
Table 4.5: Robustness to a non-standard scenario: under-performing actuators / under-estimated couplings. Initial estimates (guess) at $\chi^{2}=1.3\\!\times\\!10^{5}$, $\nu=79193$; best-fit values at $\chi^{2}=0.99$. The term in brackets is the error relative to the rightmost digit. In curly brackets the bias (absolute deviation from the real value in units of standard deviation) for each estimate. Parameter | True | Best-fit | Guess
---|---|---|---
$\omega_{1}^{2}\,[10^{-6}\,\text{s}^{-2}]$ | $-3$ | $-2.9998(2)$ | $\\{1.1\\}$ | $-1.3$ | $\\{7.8\times 10^{3}\\}$
$\omega_{12}^{2}\,[10^{-6}\,\text{s}^{-2}]$ | $-2$ | $-2.0000(1)$ | $\\{0.32\\}$ | $-0.7$ | $\\{1.0\times 10^{4}\\}$
$S_{21}\,[10^{-3}]$ | $-1.5$ | $-1.4998(1)$ | $\\{0.55\\}$ | $0$ | $\\{4.7\times 10^{3}\\}$
$A_{\text{df}}$ | $0.62$ | $0.61994(8)$ | $\\{0.77\\}$ | $1$ | $\\{4.9\times 10^{3}\\}$
$A_{\text{sus}}$ | $0.6$ | $0.599990(8)$ | $\\{1.3\\}$ | $1$ | $\\{5.1\times 10^{4}\\}$
$\Delta t_{1}\,[\text{s}]$ | $0.6$ | $0.6013(7)$ | $\\{1.8\\}$ | $0$ | $\\{8.4\times 10^{2}\\}$
$\Delta t_{12}\,[\text{s}]$ | $0.4$ | $0.398(2)$ | $\\{0.95\\}$ | $0$ | $\\{2.3\times 10^{2}\\}$
Figure 4.18 elucidates much more the results, showing the overall performances
of the estimation. The $\chi^{2}$ is reduced from $1\\!\times\\!10^{5}$ to
$\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}1$ – the required
optimum – within the given tolerances (set to $1\\!\times\\!10^{-4}$ in both
log-likelihood and parameter values), while keeping both accuracy and
precision. The figure reports two examples of estimation chains (for
$\omega_{1}^{2}$ and $\omega_{12}^{2}$), showing the correlation with the big
jumps in the $\chi^{2}$ chain and how the parameters saturate to the optimum
values. The estimation, as already said, is divided into two phases: a
gradient-based search, spanning the global structure of the parameter space,
and a simplex search, improving the final accuracy.
---
(a)
|
(b) | (c)
Figure 4.18: Robustness to a non-standard scenario: under-performing actuators
/ under-estimated couplings. The estimation performances relative to the log-
likelihood minimization (a) from $\raise 0.58554pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}1\\!\times\\!10^{5}$ to the optimum $\raise
0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}1$ and two examples of
estimation chains for (b) $\omega_{1}^{2}$ and (c) $\omega_{12}^{2}$ showing
the correlation with the big jumps in the $\chi^{2}$ chain. A preliminary
global gradient search is followed by a local simplex. The process lasts for
1636 iterations and stops when the required tolerance is met.
The final and most important discussion is the analysis of residuals
summarized in Figure 4.19 for both identification experiments and
interferometric readouts. The estimated PSDs of both initial and best-fit
residuals are compared to the PSDs of an independent noise run. It is clear
that the deterministic signals are completely subtracted from the data, hence
recovering the noise shapes for all experiments and readouts. The improvement
is mostly evident at low frequency: for $o_{12}$ the residuals are suppressed
by $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}3$ orders of
magnitude around $1\,\mathrm{mHz}$. The same happens for $o_{1}$ in the first
experiment where the improvement is $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}2$ orders of magnitude. Only $o_{1}$ in the second
experiment shows no improvement for the reason already discussed in Section
4.3 (the signal is negligible).
|
---|---
(a) | (b)
|
(c) | (d)
Figure 4.19: Robustness to a non-standard scenario: under-performing actuators
/ under-estimated couplings. Analysis of residuals for all simulated
identification experiments and interferometric readouts. Initial and best-fit
residuals are compared to the expected noise shapes estimated from an
independent run. For $o_{12}$ the improvement in both experiments (b) and (d)
is of $\raise 0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}3$ orders of
magnitude around 1 mHz; (a) for $o_{1}$ in the first experiment is $\raise
0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}2$ orders of magnitude; (c)
contains no signal.
#### 4.4.8 Non-standard scenario: non-Gaussianities
This section is devoted to showing the impact of non-Gaussianities in the
noise to parameter estimation. The main realistic behavior of experimental
noise is the possible presence of outliers: consequently, the sampling
distribution of the data may show some prominent tails. An example of such
outliers is the manifestation of glitches, very short noise transients due to
anomalous response in the readout/circuitry.
Given the non-Gaussian components in the noise, the log-likelihood defined so
far is no longer well-behaved. Because of the intrinsical assumption of
Gaussianity, it usually overweighs the outliers, and a systematic error may
arise. A standard approach, named local L-estimate [50] 888“L” stands for
“likelihood”., requires the generalization of the definition of log-
likelihood. The idea is to properly take care of the outliers by regularizing
the usual square of whitened residuals with other similar definitions by means
of a weighting function $\rho$
$\chi^{2}=\sum_{i}\rho(r_{\text{w},i})\leavevmode\nobreak\ ,$ (4.28)
where, as an example, three possible choices, the squared, absolute and
logarithmic deviations, are considered
$\rho(r_{\text{w},i})=\begin{cases}r_{\text{w},i}^{2}&\text{mean squared
dev.}\\\ \left|r_{\text{w},i}\right|&\text{mean absolute dev.}\\\
\log(1+r_{\text{w},i}^{2})&\text{mean logarithmic
dev.}\end{cases}\leavevmode\nobreak\ ,$ (4.29)
corresponding to the cases of data distributed according to Gaussian, log-
normal and Lorentzian distribution, respectively. The subscript $i$ is a
generalized index counting the data available from all experiments and
interferometric readouts and $r_{\text{w},i}$ is the whitened time-series of
residuals. Figure 4.20 compares the three weighing functions for residuals out
to 5 standard deviations. As is clear, the squared deviation overweighs the
outliers. The absolute deviation gives a slightly better weight at high
deviations, but performs poorly at low deviations. The logarithmic deviation
has much more flexibility as it behaves like the squared deviation at low
deviations and performs better than the absolute deviation.
Figure 4.20: Comparison of the three weighing functions of (4.29) for the the
proper weighing of outliers in the data. The logarithmic deviation is the most
accurate as it behaves like the squared deviation at low deviations and
performs better than the absolute deviation.
The method can be successfully applied to data with glitches. Noise glitches
are unpredictable high-frequency noise transients mostly due to failures in
the circuitry. Such outliers usually fall well beyond 3 standard deviations
and produce an excess at the tails of the statistic. Since the output of the
interferometer might be subject to similar phenomena, this section presents
the results of the investigation of a realistic experiment containing
glitches. Such transients are modeled as sine-Gaussian functions
$o_{\text{gl}}(t)=a\sin\left[2\pi
f_{0}(t-t_{0})\right]\exp\left[-\frac{(t-t_{0})^{2}}{\tau^{2}}\right]\leavevmode\nobreak\
,$ (4.30)
where the glitch parameters span a wide (uniformly distributed) range of
values. In particular, the glitch frequency, $f_{0}$, covers the whole
bandwidth $(10^{-4}\text{--}0.45)\,\mathrm{Hz}$; the injection time, $t_{0}$,
is distributed all along the time-series; the characteristic time, $\tau$,
giving the typical duration of the pulse is $(1\text{--}2)\,\mathrm{s}$; the
amplitude, $a$, falls outside the Gaussian statistic by $(3\text{--}20)$ noise
standard deviations. Moreover, the number of glitch injections is fixed as a
fractional part of the whole data series, conventionally choosing
$f_{\text{gl}}=N_{\text{gl}}/N_{\text{data}}=1\%$, since higher values are
very unlikely. Notice that this value represents only the number of
injections: the actual fraction of corrupted data is the order of
$3\,\text{E}[\tau]\,f_{\text{gl}}\simeq 5\%$.
Glitchy noise is readily produced by coloring a white, zero-mean, unitary
standard deviation input time-series, as in Section 4.2, corrupted by random
injections of glitches. Figure 4.21 shows how glitches appear in the
interferometric differential readout and in the estimated PSDs, compared to
the original noise stretches. The effect of glitches is that the PSD of the
simulated noise scales linearly with the frequency, up to
$4\\!\times\\!10^{-9}\,\mathrm{m\,Hz^{-\nicefrac{{1}}{{2}}}}$ and
$6\\!\times\\!10^{-11}\,\mathrm{m\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$0.2\,\mathrm{Hz}$ for the first and differential readout, respectively. This
excess noise sums up to the original one and is shown as high-frequency
components. Obviously, the noise statistic contains an excess at the tails.
For example, $o_{1}$ has an excess kurtosis of $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}19$, compared to the original one of
$-9\\!\times\\!10^{-3}$. No significant difference in skewness is detected
since the statistic does not loose symmetry with the glitch injections.
|
---|---
(a) | (b)
Figure 4.21: Robustness to a non-standard scenario: non-Gaussianities. (a)
simulated original and glitchy noise for $o_{12}$; (b) PSDs of the simulated
original and glitchy noise for $o_{1}$ and $o_{12}$. The level of data
corruption is evident and glitches appear as high-frequency bumps around
$0.2\,\mathrm{Hz}$.
Whitening filters are derived from the glitchy noise stretches with the same
procedure described in Section 4.4.3. However, since the whitening process
works assuming stationarity, glitches are not filtered out from the data.
Table 4.6 shows the results of three different parameter estimations with the
definitions of the weighting functions in (4.29).
Table 4.6: Robustness to a non-standard scenario: non-Gaussianities. The
comparison between three parameter estimations with the three definitions in
(4.29). $\nu=79193$. The term in brackets is the error relative to the
rightmost digit. In curly brackets the bias (absolute deviation from the real
value in units of standard deviation) for each estimate.
Parameter | Real | Best-fit | Best-fit | Best-fit | Guess
---|---|---|---|---|---
| (mean sq. dev.) | (mean abs. dev.) | (mean log. dev.)
| $\chi^{2}=10$ | $\chi^{2}=2.1$ | $\chi^{2}=0.95$
$\omega_{1}^{2}\,[10^{-6}\,\text{s}^{-2}]$ | $-1.32$ | $-1.320(1)$ | {0.061} | $-1.3188(6)$ | {2.0} | $-1.3192(4)$ | {2.0} | $-1.3$
$\omega_{12}^{2}\,[10^{-6}\,\text{s}^{-2}]$ | $-0.68$ | $-0.6798(7)$ | {0.29} | $-0.68000(3)$ | {0.011} | $-0.6804(2)$ | {1.8} | $-0.7$
$S_{21}\,[10^{-4}]$ | $1.1$ | $1.10(2)$ | {0.074} | $1.113(7)$ | {1.8} | $1.116(5)$ | {3.4} | $0$
$A_{\mathrm{df}}$ | $1.01$ | $1.011(3)$ | {0.29} | $1.010(1)$ | {0.23} | $1.0109(8)$ | {1.2} | $1$
$A_{\mathrm{sus}}$ | $0.99$ | $0.99000(5)$ | {0.035} | $0.98959(2)$ | {20} | $0.99001(1)$ | {0.99} | $1$
$\Delta t_{1}\,[\text{s}]$ | $0.1$ | $0.100(3)$ | {0.045} | $0.090(1)$ | {8.3} | $0.1007(8)$ | {0.90} | $0$
$\Delta t_{12}\,[\text{s}]$ | $0.1$ | $0.098(5)$ | {0.36} | $-0.0290(2)$ | {58} | $0.098(2)$ | {1.2} | $0$
The most conservative least square estimator provides overestimated errors
since they scale as $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}\sqrt{\chi^{2}}$. The absolute and logarithmic deviations provide better
statistics and lower errors, but the first gives biased estimates of
$A_{\text{sus}}$, $\Delta t_{1}$ and $\Delta t_{12}$ and the last one a
slightly biased estimate of $S_{21}$. The analysis of residuals demonstrates
that the three methods recover the noise shapes and are in agreement with each
other, so the systematic errors are only in the estimated parameters. These
estimators are also $30\%$ and $9\%$ faster than the Gaussian (mean squared
deviation), as the outliers have less influence on the estimation chains.
By inspecting the results, it turns out that there is no absolute rule that
can be applied when dealing with glitches. However, from the differences
between the estimates it is possible to infer the sensitivity of each single
parameter to glitches. For example, adopting the ratio between the biases as
the a-posteriori criterion for comparing two methods, it tends to one if that
parameter is not sensitive to glitches; otherwise, it tends to a very small or
very large number. In view of this consideration, the comparison between the
mean squared deviation and the mean logarithmic deviation gives that $S_{21}$
is the most sensitive parameter, whereas $\Delta t_{12}$ the least.
Starting from the fact that the three methods give the same results for purely
Gaussian noise, a proposed recipe is the following:
1. 1.
apply the conservative approach (the ordinary mean squared deviation) directly
to corrupted time series and try with different estimators (mean absolute
deviation, mean logarithmic deviation, etc.);
2. 2.
start removing some outliers giving them negligible weight;
3. 3.
redo the analysis with all estimators;
4. 4.
check for convergence and agreement between the estimators.
The overall process can be actually viewed as a reweighing analysis providing
for robust uncertainties and, at the same time, the removal of outliers in a
step-by-step smooth readjustment. Even though it would be possible in
principle to clean up the data just before the estimation, in that case the
results would likely be dependent on the statistical criterion used for such
cleaning. Even though it is beyond the scope of this thesis to implement the
idea, it is worth observing that the two main advantages of the preceding
recipe are its robustness in definition and the fact that data polishing is
smooth and model independent.
### 4.5 Estimation of total equivalent acceleration noise
This section justifies the efforts in developing the techniques introduced so
far with all tests and validation runs, showing the impact of system
identification on the estimation of the total equivalent acceleration noise.
As said throughout this thesis, the main objective of the LPF mission in view
of a real GW astronomy with spaced-based detectors is the characterization of
the Doppler link as the fundamental spacetime meter in terms of equivalent
differential acceleration. Even if LPF is different in design with respect to
LISA – no faraway optical measurement between two SCs is actually implemented
– yet the principle and, most of all, the performances in sensitivity can be
extrapolated and gather more confidence in the scientific scopes of any
spaced-based GW detector.
Assessing the performance in sensitivity as equivalent input acceleration
noise is a very effective way to put dynamics, sensing and control on the same
footing as described in Section 2.3. This can be achieved by means of the
$\bm{\Delta}$ operator of Section 3.1, connecting interferometric displacement
readouts to total equivalent acceleration and at the same time compensating
for TM couplings, SC jitter and sensing cross-talk.
Suppose that $\bm{S}_{\text{n},o}(\omega,\bm{p}_{\text{true}})$ is the
measured interferometric noise PSD. Then, the estimated total equivalent
acceleration noise PSD is given by
$\bm{S}_{\text{n},f}(\omega,\bm{p}_{\text{est}})=\bm{\Delta}(\omega,\bm{p}_{\text{est}})\,\bm{S}_{\text{n},o}(\omega,\bm{p}_{\text{true}})\,{\bm{\Delta}(\omega,\bm{p}_{\text{est}})}^{*}\leavevmode\nobreak\
,$ (4.31)
where $\bm{\Delta}(\omega,\bm{p}_{\text{est}})$ models the transfer from
interferometric displacement readouts to total equivalent acceleration and
$\bm{p}_{\text{est}}$ are the parameter estimates as obtained by system
identification. It is worth noting that if $\bm{S}_{\text{n},o}$ was assumed
constant to the parameter values in first approximation, the transfer to total
equivalent acceleration would anyhow couple the output noise with the dynamics
so that the estimated total equivalent acceleration noise becomes explicitly
dependent on the parameter values. This shows that parameter estimation serves
not only for system identification, but also for the actual identification of
the total equivalent acceleration noise.
Furthermore, suppose that
$\bm{p}_{\text{est}}\simeq\bm{p}_{\text{true}}+\delta\bm{p}$, with
$\delta\bm{p}$ the parameter biases being much larger than the statistical
uncertainties on $\bm{p}_{\text{est}}$. It is easy to show that the parameter
biases propagate to the differential operator
$\bm{\Delta}_{\text{est}}\simeq\bm{\Delta}_{\text{true}}+\delta\bm{\Delta}$,
where $\bm{\Delta}_{\text{true}}=\bm{\Delta}(\omega,\bm{p}_{\text{true}})$ and
$\bm{\Delta}_{\text{est}}=\bm{\Delta}(\omega,\bm{p}_{\text{est}})$. Systematic
errors found in the parameter values produce systematic errors in the
recovered total equivalent acceleration noise
$\delta\bm{S}_{\text{n},f}\simeq\delta\bm{\Delta}\,\bm{S}_{\text{n},o}\,{\bm{\Delta}}^{*}+\bm{\Delta}\,\bm{S}_{\text{n},o}\,{\delta\bm{\Delta}}^{*}\leavevmode\nobreak\
,$ (4.32)
where the subscript “true” is dropped out for clearness. As pointed out in
[62], the statistical uncertainty on the parameter values are masked by the
statistical uncertainty on the estimated spectrum. Despite this, systematic
errors in the estimated parameters can fall well outside the confidence levels
of the optimal spectrum and show themselves as not mere excess noise, but
producing really different noise shapes. Hence, it is expected that the
estimation of the total equivalent acceleration noise is biased if the
parameter values are not correctly assessed from system identification.
To demonstrate the impact of system identification on the estimation of the
total equivalent acceleration noise, a very long noise run, $\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}6$ days, is simulated with the
same procedures of Section 4.2, i.e., by coloring a sequence of white Gaussian
input time-series with cross-correlating noise shaping filters. The
interferometric displacement noise model is derived in a non-standard
configuration of LPF, as in Section 4.4.7, namely in the case of stronger-
than-expected TM couplings, malfunctioning actuators and a higher sensing
cross-talk. In this case, the estimation of the total equivalent acceleration
noise with naively guessed parameter values will surely contain systematic
errors.
The estimation of the total equivalent acceleration noise is readily performed
on the multi-channel interferometric run with a scheme described in details in
[62, 41], by applying a time-domain version of the $\bm{\Delta}$ operator of
Section 3.1. The issues connected to numerical derivatives in LPF are
extensively discussed and solved in [63]. As said, system identification
effectively helps in the calibration of the operator. In support of the
statement, the numerical estimation of the total equivalent acceleration noise
is performed assuming three different parameter sets that can be found in
Table 4.5:
1. 1.
the initial guess values, as it was without a preliminary system
identification: $\omega_{1}^{2}=-1.3\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$,
$\omega_{12}^{2}=-0.7\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$, $S_{21}=0$,
$A_{\text{df}}=1$, $A_{\text{sus}}=1$ (typically $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}10^{4}$ standard deviations away from the real
values);
2. 2.
the best-fit values, as it was with a preliminary system identification, i.e.,
after having calibrated the differential operator:
$\omega_{1}^{2}=-2.9998(2)\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$,
$\omega_{12}^{2}=-2.0000(1)\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$,
$S_{21}=-1.4998(1)\\!\times\\!10^{-3}$, $A_{\text{df}}=0.61994(8)$,
$A_{\text{sus}}=0.599990(8)$;
3. 3.
the true values, used for consistency checks:
$\omega_{1}^{2}=-3\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$,
$\omega_{12}^{2}=-2\\!\times\\!10^{-6}\,\mathrm{s^{-2}}$,
$S_{21}=-1.5\\!\times\\!10^{-3}$, $A_{\text{df}}=0.62$, $A_{\text{sus}}=0.6$.
The result of the analysis is contained in Figure 4.22, showing the total
equivalent differential acceleration noise, both numerically estimated and
modeled, for the three different cases.
Figure 4.22: Total equivalent differential acceleration noise numerically
estimated on synthetic data and compared to theoretical noise models obtained
by a full projection of fundamental noise sources. The estimation of the total
out-of-loop equivalent acceleration can be performed either with a preliminary
system identification or without it. The PSD estimated with a preliminary
system identification completely overlaps the one of a hypothetical estimation
assuming the knowledge of the true parameter values. The observed difference
shows that a preliminary system identification is mandatory to avoid
systematic errors in the reconstructed total equivalent acceleration noise.
The solid thinner lines indicate the reasons of such a discrepancy. Around
$50\,\mathrm{mHz}$ the bump is due to unsuppressed thruster noise exceeding
the interferometric $o_{12}$ readout noise. At low frequency and around
$0.4\,\mathrm{mHz}$, the two major contributions are the unsuppressed force
couplings between the TMs and the SC and the capacitive actuation noise.
Thanks to system identification, an improvement in performance of a factor 4
at low frequency is evident.
First, the agreement between modeled and estimated total equivalent
acceleration noise PSDs states that: (i) the generation of the interferometric
noise is accurate to the assumed models at least to within the statistical
uncertainty of the spectra; (ii) the numerical estimation of the total
equivalent acceleration in time-domain is accurately explained by the
frequency-domain transfer matrix from interferometric readouts to the total
equivalent acceleration.
Second but more important, the total equivalent acceleration noise estimated
with a preliminary system identification completely overlaps the one of a
hypothetical estimation assuming the complete knowledge of the true values.
Therefore it demonstrates that it is still possible to meet the sensitivity
requirements during under-performing mission operations.
The observed systematic errors in the total equivalent acceleration noise
estimated without identification show that system identification is strictly
mandatory to avoid such problems and guarantee the scientific objectives. The
systematic errors can be explained by the fact that the naive initial guess
values are sensitively different from the true values. Since the operator is
not calibrated on fiducial parameter values, it is not effective in
compensating, in turn, the SC jitter due to the thruster actuation noise, the
TM couplings and the capacitive actuation noise. In particular, around
$50\,\mathrm{mHz}$ the bump is the unsuppressed thruster noise exceeding the
interferometric $o_{12}$ readout noise: the effect is due to the uncalibrated
drag-free gain $A_{\text{df}}$. At low frequency and around
$0.4\,\mathrm{mHz}$, the major contributions are the coupling forces between
the TMs and the SC (two contributions, accounting for
$1.8\\!\times\\!10^{-13}\,\mathrm{m\,s^{-2}}$, almost the whole noise budget)
and the capacitive actuation noise
($7\\!\times\\!10^{-14}\,\mathrm{m\,s^{-2}}$): the effect is due to the
uncalibrated stiffness constants $\omega_{1}^{2}$ and $\omega_{12}^{2}$ and
the suspension gain $A_{\text{sus}}$.
The final improvement in the estimated total equivalent acceleration noise
with system identification is a factor $4$ around $0.4\,\mathrm{mHz}$ and a
factor $2$ around $50\,\mathrm{mHz}$ in units of $\sqrt{\text{PSD}}$. The
conclusion is that without a preliminary system identification – robust to
non-standard parameter values – the performance of the mission and the
characterization of the total equivalent acceleration noise would seriously be
compromised.
### 4.6 Suppressing transients in the total equivalent acceleration noise
This final section discusses on the suppression of system transients for
realistic data produced by the OSE and provided by ESA. Section 3.2 and in
particular (3.16) demonstrate that system transients can be suppressed to
within the accuracy to which the differential operator $\bm{\Delta}$ has been
calibrated on parameter values representative of the system. A supporting
example is provided in what follows.
Figure 4.23 shows the first 3 hours of a typical noise run of the OSE. In
complete realism, just after the TM release the system is firstly turned into
accelerometer mode, then into science mode (around
$1\\!\times\\!10^{4}\,\mathrm{s}$) 999It is worth recalling that in
accelerometer mode the TMs are both electrostatically suspended, whereas in
the main science mode one of the two is in drag-free. The resulting noise is
at least one order of magnitude lower in the second case, especially in the
differential readout.: transients appears as a direct consequence of the non-
zero initial conditions. In fact, the initial positions are $0.24\,\mathrm{\mu
m}$ ($o_{1}$) and $0.36\,\mathrm{\mu m}$ ($o_{12}$), whereas the estimated
velocities 101010Since only an order of magnitude is needed, a two point
forward difference is applied together with a low-pass filter with frequency
cut at $100\,\mathrm{mHz}$. are about $-500\,\mathrm{pm\,s^{-1}}$ ($o_{1}$)
and $-4\,\mathrm{pm\,s^{-1}}$ ($o_{12}$). The transient in $o_{1}$ lasts for
half a hour and in $o_{12}$ for about 2 hours – the timescale of typical
transients as predicted by Section 3.2.
Figure 4.23: The first 3 hours of a typical noise run of the OSE. After the TM
release, the system is firstly turned into accelerometer mode, then into
science mode around $1\\!\times\\!10^{4}\,\mathrm{s}$. The transient due to
non-zero initial conditions lasts for half a hour in $o_{1}$ and about 2 hours
in $o_{12}$.
The estimation of the total equivalent differential acceleration noise is
performed twice on the same data, including the initial transitory, assuming
each time a different set of parameter values modeling the system. On one
side, a fair approximation of those parameters – the so-called initial guess –
reproduces the situation in which the estimation of the total equivalent
acceleration is performed without a preliminary system identification, as in
the previous section. On the other side, a fiducial approximation of those
parameters – the so-called best-fit – reproduces the situation in which the
estimation of the total equivalent acceleration is performed with a
preliminary system identification. Moreover, the estimation is performed
parallelly on two data segments lasting $3\\!\times\\!10^{4}\,\mathrm{s}$
each: the first one just after the system is turned into science mode and
containing the transient state; the second one follows it and is driven by the
steady state. Figure 4.24 shows the estimated total equivalent differential
acceleration noise for the two segments and for the two sets of parameter
values. The comparison shows that the transient is suppressed, and there is no
relevant difference between the two segments.
|
---|---
(a) | (b)
Figure 4.24: Total equivalent differential acceleration noise numerically
estimated on synthetic data produced by the OSE and shown in Figure 4.23. The
data are split into two segments: the first one just after the system is
turned into science mode and containing the transient state; the second one
follows it and is driven by the steady state. The estimation of the total
equivalent out-of-loop acceleration can be performed either with a preliminary
system identification (fiducial parameter values) or without it (approximate
parameter values) (lines with different colors) and apparently there is no
difference. (a) the transient is suppressed, compared to (b) where the system
is dominated by the steady state.
Figure 4.25 reports the PSDs of the estimated total equivalent differential
acceleration noise for the above time-series, i.e., assuming the two sets of
parameter values for both segments. At low frequency, the noise level of the
segment containing the transient is higher then the subsequent segment, but
system identification helps in suppressing part of the noise around
$1\,\mathrm{mHz}$. Below $0.7\,\mathrm{mHz}$ there is an evidence that there
is an unsuppressed residual transitory in the data.
Figure 4.25: Total equivalent differential acceleration noise numerically
estimated on synthetic data produced by the OSE and shown in Figure 4.24. The
estimation of the total equivalent out-of-loop acceleration can be performed
either with a preliminary system identification (fiducial parameter values) or
without it (approximate parameter values) on both segments: the first one
dominated by the transient state and the second one dominated by the steady
state. System identification helps in suppressing the transient around
$1\,\mathrm{mHz}$.
The results of this section demonstrate how transients due to initial
conditions can be suppressed with reasonably good approximation in the total
equivalent acceleration time-series. The accuracy to which the suppression is
effective depends on the accuracy to which the parameter values of the system
are known. As shown, system identification helps in mitigating the effect due
to transients in the data.
## 5 Design of optimal experiments
The previous chapter introduced system identification and its relevance for
the unbiased estimation of the total equivalent acceleration noise of the LPF
mission. A standard series of sine injections spanning the frequency band was
utilized therein. Such bias injections make possible the estimation of the
modeled system parameters the total equivalent acceleration noise depends on.
Clearly, the goal of system identification is the parameter accuracy. A
possible approach is also to search for optimized stimuli to assess the system
parameters with better precision, with the final aim of a better estimate of
the equivalent acceleration noise. The relevance is worth that this chapter
addresses the question and provides for a solution.
### 5.1 Review of the problem
Aiming at discriminating among different designs of the same system
identification experiment, it is a rather natural consequence to enter into
the field of the optimal design of experiments [64, 65]. This matter tries to
answer to those physical problems characterized by a design matrix that shall
be maximized in order to perform a targeted measurement with an optimized
precision. This links to some very recent examples of practical applications
of the optimal design theory, multidisciplinary and covering very different
research fields: from dynamical systems [66], to geophysics [67], quantum
state tomography [68] and even magnetic resonance in medical engineering [69].
A general review can be found in [70, 71]. In what follows the same philosophy
is applied to the LPF mission with its peculiarities: the level of complexity
is worth as the very example of a MIMO multi-degree-of-freedom dynamical
system with coupled closed loops and subjected to various constraints.
As described in the previous chapter, system identification is targeted to
measuring the system parameters $\bm{p}$ appearing within the transfer matrix
$\bm{T}_{o_{\text{i}}\rightarrow o}(\omega,\bm{p})$ connecting applied
controller biases to interferometric readouts, for the case of the
investigation along the optical axis. If the inputs $\bm{o}_{\text{i}}$ are
parameterized by a set of parameters $\bm{\theta}$, then the Fisher
information matrix in (4.22) becomes
$\bm{\mathcal{I}}(\bm{\theta})=\int{\bm{o}_{\text{i}}(\omega,\bm{\theta})}^{*}\,{\nabla_{\bm{p}}\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p}_{\text{est}})}^{*}\,\bm{S}_{\text{n}}(\omega)^{-1}\,\nabla_{\bm{p}}\bm{T}_{o_{\text{i}}\rightarrow
o}(\omega,\bm{p}_{\text{est}})\,\bm{o}_{\text{i}}(\omega,\bm{\theta})\,\text{d}\omega\leavevmode\nobreak\
.$ (5.1)
Requiring that the estimates $\bm{p}_{\text{est}}$ should be given with the
optimal precision implies that the preceding matrix must be optimized with
respect to the design given by $\bm{\theta}$.
Theory provides for a solution of the problem. In fact, the optimal design is
attained by building up a scalar estimator on the information matrix,
$\phi[\bm{\mathcal{I}}]$, which is mathematically a functional over that
matrix. With the parametrization introduced above, the functional simply
becomes a scalar function of $\bm{\theta}$
$\phi[\bm{\mathcal{I}}]=\phi(\bm{\theta})\leavevmode\nobreak\ ,$ (5.2)
for given noise PSDs, interferometric readouts and estimated system
parameters. Hereafter, three different choices of the functional $\phi$ are
considered
$\phi(\bm{\theta})=\begin{cases}\det(\bm{\mathcal{I}}(\bm{\theta}))&\text{D
optimality}\\\ \min(\text{eig}(\bm{\mathcal{I}}(\bm{\theta})))&\text{E
optimality}\\\ \text{tr}(\bm{\mathcal{I}}(\bm{\theta}))&\text{T optimality}\\\
\end{cases}\leavevmode\nobreak\ ,$ (5.3)
and the corresponding for the covariance matrix, obtained directly inverting
(5.1), since maximum information is equivalent to minimum variance. The
interpretation of each single criterion is readily discussed. The D optimality
is the determinant of the information matrix and averages the information
along all terms, diagonal and off-diagonal. The E optimality takes the minimum
eigenvalue and tries to balance it with the others, hence regularizing the
conditional number of the matrix 111The conditional number is defined as the
ratio between the minimum and maximum eigenvalues. It expresses the
sensitivity of the matrix to numerical inversions. Round-off errors affect the
operation when the conditional number is either very small or very large. A
number of order 1 is considered stable to inversions.. The T optimality gives
to the diagonal the highest weight and corresponds to the averaged information
along all parameters. Even though quite different in definitions, the criteria
share the same philosophy: maximizing/minimizing the information/covariance
volume in the system parameter space around the minimum.
For LPF there is one more point adding much more complexity. The typical
constraints that must be met during all operations and especially for the
experiment design are:
1. 1.
the general shape of the biases being injected;
2. 2.
the dynamical range of the interferometer, $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}100\,\mathrm{\mu m}$;
3. 3.
the force authority for thruster actuation, $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}100\,\mathrm{\mu N}$;
4. 4.
the force authority for capacitive actuation, $\raise 0.73193pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}2.5\,\mathrm{nN}$.
Concerning the first one, the typical duration of an identification experiment
shall not exceed $T\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}3$
hours, mostly to ensure noise stationarity. The system can be stimulated with
a series of sine-waves of constant duration each $\delta t\simeq
1200\,\mathrm{s}$, as already described in Section 4.3. To simplify the
problem, the duration is kept fixed during the optimization. Furthermore, the
requirement of avoiding possible system transients at the beginning and the
end of each cycle, suggests to set null Dirichlet boundary conditions (i.e.,
null initial and final values of the signals) and leave gaps of $\delta
t_{\text{gap}}\simeq 150\,\mathrm{s}$. The general expression of a guidance
signal is a windowed series of sines
$o_{\text{i}}(t)=\sum_{n=1}^{N_{\text{inj}}}a_{n}\,\sin(2\pi
f_{n}\,t)\,\theta(t-t^{\prime}_{n})\,\theta(t^{\prime\prime}_{n}-t)\leavevmode\nobreak\
,$ (5.4)
where $\theta$ is the Heaviside unit-step, $f_{n}=n/\delta t$ is the injected
frequency of the $n$-th cycle and $a_{n}$ the corresponding amplitude, through
the maximum number of injections $N_{\text{inj}}=7$, and
$t^{\prime}_{n}=t_{0}+(n-1)(\delta t+\delta t_{\text{gap}})$ and
$t^{\prime\prime}_{n}=t_{0}+n\,(\delta t+\delta t_{\text{gap}})-\delta
t_{\text{gap}}$ the initial and final instants of the $n$-th injection cycle,
with $t_{0}$ the starting instant of the experiment. Clearly, the frequencies
$f_{n}$ are set by general requirements on the experiment duration, whereas
the amplitudes $a_{n}$ by the other three requirements (dynamical range and
force authority). It is rather obvious that the information matrix scales as
the SNR of the signal, hence as $a_{n}$, so the amplitudes are chosen to be
the maximum allowed, not exceeding namely $1\%$ of the operating range of the
interferometer and $10\%$ of the maximum force authority.
The optimal design problem for LPF can now be stated as the following. The
functional $\phi$ must be optimized for given noise PSDs and transfer matrix
around the estimated system parameters, with respect to the design parameters
$\bm{\theta}$ containing the frequencies of the injected biases. As the
frequency changes, the amplitudes are updated accordingly while preserving the
constraints elucidated above.
The dependence of the information matrix to the parameters of the injected
bias signals is somewhat implicit and masked by the integral and the Fourier
transform in (5.1). It should be also noticed that the criteria in (5.3) are
de-facto producing a matrix that is as much diagonal as possible with respect
to the choice of $\bm{\theta}$. The implicit parametric diagonalization of the
information matrix is equivalent to the simultaneous diagonalization of noise
and transfer matrices. In light of this, optimal design appears somehow
related to an eigen-decomposition of the system with respect to the
differential operator and noise at the same time.
### 5.2 Optimizing the identification experiments
Referring to Section 3.3 and Section 4.1 for the description of a LPF model
along the optical axis, this section shows the improvement in the measured
precision of the stiffness constants, $\omega_{1}^{2}$ and $\omega_{12}^{2}$,
the sensing cross-talk, $S_{21}$, and the actuation gains, $A_{\text{df}}$ and
$A_{\text{sus}}$. Possible delays in the application of guidance signals are
left out from the analysis without lose of generality.
The two standard identification experiments described in Section 4.3 can be
optimized independently once an estimate of the parameter values is given by a
preliminary system identification. The scheme proposed here – to be followed
during the mission – is:
1. 1.
estimate the parameter values with standard experiments as in the preceding
chapter;
2. 2.
optimize the experiments around the parameter estimates;
3. 3.
estimate the parameter values with optimized experiments, as in this chapter,
to get more confidence in the recovered total equivalent acceleration noise.
As said, the design parameters on which the information matrix is optimized
are the frequencies of the injected bias signals. Instead, the amplitudes are
updated accordingly by meeting the requirements on the interferometer sensing
range and force authorities. By means of the transfer matrices in Section 3.1,
the maximum amplitudes are conservatively computed by taking the minimum
between the requirements in interferometer range and force authority
$\displaystyle a_{o_{\text{i},1}}$
$\displaystyle=\min\\{T_{o_{\text{i,1}}\rightarrow
o_{1}}\,o_{1,\text{max}}\,,\,T_{o_{\text{i,1}}\rightarrow
f_{\text{c},1}}\,f_{1,\text{max}}\\}\leavevmode\nobreak\ ,$ (5.5a)
$\displaystyle a_{o_{\text{i},12}}$
$\displaystyle=\min\\{T_{o_{\text{i,12}}\rightarrow
o_{12}}\,o_{12,\text{max}}\,,\,T_{o_{\text{i,12}}\rightarrow
f_{\text{c},12}}\,f_{12,\text{max}}\\}\leavevmode\nobreak\ ,$ (5.5b)
where $o_{1,\text{max}}=o_{12,\text{max}}=1\,\mathrm{\mu m}$ ($1\%$ of the
interferometer range), $f_{1,\text{max}}=10\,\mathrm{\mu N}$ ($10\%$ of
thruster authority) and $f_{12,\text{max}}=0.25\,\mathrm{nN}$ ($10\%$ of
electrostatic suspension authority). For example,
$T_{o_{\text{i,1}}\rightarrow o_{1}}$ represents the transfer from the
guidance signal $o_{\text{i,1}}$ to the interferometric readout $o_{1}$;
analogously, $T_{o_{\text{i,1}}\rightarrow f_{\text{c},1}}$ represents the
transfer from the guidance signal $o_{\text{i,1}}$ to the commanded thruster
force $f_{\text{c},1}$.
Figure 5.1 shows how the amplitudes so far determined depend on the injection
frequencies, for the first identification experiment. Analogously, Figure 5.1
shows the same relationship for the second identification experiment. The
interferometer range is the most stringent requirement, whereas force
authority may limit at high frequency, especially for $o_{i,1}$. For this
reason, only the first requirement is actually considered in the analysis,
however limiting the maximum frequency to $50\,\mathrm{mHz}$
($\nicefrac{{1}}{{10}}$ of the Nyquist frequency for data sampled at
$1\,\mathrm{Hz}$).
Figure 5.1: Maximum allowed amplitude for bias injection $o_{i,1}$. The
amplitude is limited by the interferometer operating range for almost the
entire frequency band. Above $20\,\mathrm{mHz}$ it starts to be limited by
thruster authority. Maximum amplitudes do not exceed $1\,\mathrm{\mu m}$ in
interferometer range, $10\,\mathrm{\mu N}$ in thruster authority and
$0.25\,\mathrm{nN}$ in electrostatic suspension authority. The combination of
both requirements is shown as a dashed line. Figure 5.2: Maximum allowed
amplitude for bias injection $o_{i,12}$. The amplitude is limited by the
interferometer operating range. Maximum amplitudes do not exceed
$1\,\mathrm{\mu m}$ in interferometer range, $10\,\mathrm{\mu N}$ in thruster
authority and $0.25\,\mathrm{nN}$ in electrostatic suspension authority. The
combination of both requirements is shown as a dashed line.
The analysis of two experiments requires two independent optimizations on
7-dimensional discrete spaces, spanning the frequencies of each injection
cycle. Despite the previous chapter where the optimization variables (the
system parameters) were continuous, the injection frequency space must be
discrete. In fact, each wave is required to start and stop at zero, so that
transients can be avoided.
Discrete optimization is always more mathematically complicated than
continuous optimization. The first can invoke refined mathematical techniques
like graph theory. On the contrary, standard well-known numerical optimization
algorithms frequently assume continuity and smoothness in the independent
variable being optimized. Since investigating in sophisticated methods is out
of the scope of this thesis, a trick is found here to overcome the problem of
discrete numerical optimization. First of all, the choice naturally falls to
direct methods, like the simplex and pattern search [50]. Those methods (i) do
not make use of analytical derivatives, as such an evaluation for this problem
introduces a very high level of complexity and (ii) are more robust to
function discontinuities than other algorithms. The trick consists on
overlapping a discrete grid to the continuous space, whose nodes are the pole
of attraction for the independent continuous variables. The merit function
consists of three main calculations:
1. 1.
the information matrix $\bm{\mathcal{I}}$ for given noise, transfer matrix and
parametric input signals, following (5.1);
2. 2.
the functional $\phi$ in (5.3);
3. 3.
the rounding of the injection frequencies to the nearest grid node as the
optimization carries on.
Every time the merit function is called, the frequencies are forced to lay on
the grid, but the side effect is that the surface becomes highly irregular.
However, the optimization can be implemented with the standard direct search
algorithms.
### 5.3 Multi-experiment, single-input
In view of comparing the performances of the 6 optimization criteria contained
in (5.3) (both information and covariance matrices) for two identification
experiments in a mission-like manner, here is the adopted analysis procedure:
1. 1.
two standard identification experiments are simulated and the system
parameters estimated according to the methods of the previous chapter;
2. 2.
6 independent optimizations around the best-fit values allow to find optimized
experimental designs of the injection biases;
3. 3.
6 system identifications are performed along with those designs;
4. 4.
optimal best-fit values and standard deviations are extracted from each fit.
Table 5.1 shows the results of the investigation, by comparing the standard
experiment to the optimized ones. The standard deviations as estimated from
the fit quantifies the precision of that design, whereas the estimate biases
(deviation of the best-fit value from the real value in units of standard
deviations) quantifies the accuracy. By inspecting the results, the T
optimality criterion for the information matrix gives, in average, the best
precision and accuracy. The estimate biases are within 1–2 standard deviations
and the fit standard deviations are lower than the standard by a factor 2 for
$\omega_{1}^{2}$ and $\omega_{1}^{2}$, 4 for $S_{21}$, 5 for $A_{\text{sus}}$
and 7 for $A_{\text{df}}$. Other criteria may worsen the measurement,
especially for the covariance matrix: this is an indication that the numerical
matrix inversion introduces an extra source of indetermination.
Table 5.1: Comparison of performances for different optimal designs. The fit
standard deviations for all 5 parameters are reported for the 6 optimal design
approaches, based on information and covariance matrices. In curly brackets
the bias (absolute deviation from the real value in units of standard
deviation) for each estimate. The T optimality criterion for the information
matrix gives, in average, the best precision and accuracy.
Parameter | Standard | Information | Covariance
---|---|---|---
st. dev. | | D | E | T | D | E | T
$\sigma_{\omega_{1}^{2}}\,[\text{s}^{-2}]$ | $4\times 10^{-10}\\{1.4\\}$ | $3\times 10^{-10}\\{0.48\\}$ | $8\times 10^{-9}\\{0.24\\}$ | $2\times 10^{-10}\\{0.68\\}$ | $9\times 10^{-10}\\{1.1\\}$ | $3\times 10^{-10}\\{2.1\\}$ | $2\times 10^{-9}\\{0.13\\}$
$\sigma_{\omega_{12}^{2}}\,[\text{s}^{-2}]$ | $2\times 10^{-10}\\{0.41\\}$ | $2\times 10^{-10}\\{1.6\\}$ | $8\times 10^{-9}\\{0.23\\}$ | $1\times 10^{-10}\\{2.0\\}$ | $9\times 10^{-10}\\{0.97\\}$ | $2\times 10^{-10}\\{2.7\\}$ | $2\times 10^{-9}\\{0.16\\}$
$\sigma_{S_{21}}$ | $4\times 10^{-7}\\{0.086\\}$ | $1\times 10^{-7}\\{0.55\\}$ | $2\times 10^{-7}\\{0.77\\}$ | $1\times 10^{-7}\\{1.1\\}$ | $1\times 10^{-7}\\{0.047\\}$ | $1\times 10^{-7}\\{0.056\\}$ | $1\times 10^{-7}\\{0.58\\}$
$\sigma_{A_{\text{df}}}$ | $7\times 10^{-4}\\{1.6\\}$ | $2\times 10^{-4}\\{0.61\\}$ | $1\times 10^{-4}\\{1.0\\}$ | $1\times 10^{-4}\\{0.50\\}$ | $3\times 10^{-4}\\{0.36\\}$ | $2\times 10^{-4}\\{2.1\\}$ | $1\times 10^{-4}\\{1.8\\}$
$\sigma_{A_{\text{sus}}}$ | $1\times 10^{-5}\\{1.7\\}$ | $1\times 10^{-6}\\{0.24\\}$ | $1\times 10^{-6}\\{0.27\\}$ | $2\times 10^{-6}\\{0.28\\}$ | $2\times 10^{-6}\\{0.38\\}$ | $1\times 10^{-6}\\{0.95\\}$ | $1\times 10^{-6}\\{1.5\\}$
Choosing the T criterion for the information matrix as the reference for
further comments, Table 5.2 reports the optimal input frequencies and
amplitudes compared to the standard ones for both experiments. Transparently,
the system relaxes to only two relevant frequencies: the lowest,
$0.83\,\mathrm{mHz}$, and the highest allowed, $49\,\mathrm{mHz}$. The result
should not surprise since the previous chapter implicitly took to a similar
conclusion: the two frequencies are indeed the two maxima of the transfer
matrix in Figure 4.3. The transfer from $o_{i,1}$ and $o_{i,12}$ to $o_{12}$
are maximized at little less than $1\,\mathrm{mHz}$; the transfer from
$o_{i,1}$ to $o_{1}$ is maximized at around $0.1\,\mathrm{Hz}$.
Table 5.2: Comparison of input frequencies and amplitudes for the standard and optimal experiments. The injection cycles last $1200\,\mathrm{s}$ each and are separated by gaps of $150\,\mathrm{s}$. The system relaxes to only two relevant frequencies $0.83\,\mathrm{mHz}$ and $49\,\mathrm{mHz}$, namely the lowest and the highest allowed. Standard Exp. 1 | Optimal Exp. 1 | Standard Exp. 2 | Optimal Exp. 2
---|---|---|---
$f\,\mathrm{[mHz]}$ | $a\,\mathrm{[\mu m]}$ | $f\,\mathrm{[mHz]}$ | $a\,\mathrm{[\mu m]}$ | $f\,\mathrm{[mHz]}$ | $a\,\mathrm{[\mu m]}$ | $f\,\mathrm{[mHz]}$ | $a\,\mathrm{[\mu m]}$
$0.83$ | $1.0$ | $0.83$ | $1.0$ | $0.83$ | $0.80$ | $0.83$ | $0.55$
$1.7$ | $1.0$ | $0.83$ | $1.0$ | $1.7$ | $0.48$ | $49$ | $52$
$3.3$ | $1.0$ | $49$ | $0.55$ | $3.3$ | $0.19$ | $49$ | $52$
$6.6$ | $1.0$ | $49$ | $0.55$ | $6.6$ | $0.088$ | $49$ | $52$
$13$ | $0.59$ | $0.83$ | $1.0$ | $13$ | $0.096$ | $49$ | $52$
$27$ | $0.28$ | $0.83$ | $1.0$ | $27$ | $0.18$ | $49$ | $52$
$53$ | $0.14$ | $49$ | $0.55$ | $53$ | $0.46$ | $49$ | $52$
In Figure 5.3 the optimal bias signals are shown together with the system
responses in both interferometric readouts for the two identification
experiments. As usual, the bias signals are made of a series of sine-waves,
whose frequencies and amplitudes are the ones described in Table 5.2. By
inspecting the response in the second experiment (panel (d)) it naturally
turns out that the big jumps are produced by the first derivative
discontinuity of the Heaviside unit-step in (5.4). At that frequency, the
discontinuity gives rise to a transient overlapping to the injected signal.
However, the information on the system parameters is mostly carried by the
injection frequency and not by the discontinuities. In fact, the simulation of
another experiment with approximately the same duration and constituted by an
injection of the same signal without the gaps proved that the same parameter
precision can be attained.
|
---|---
(a) | (b)
|
(c) | (d)
Figure 5.3: Synthetic data for optimal design of Exp. 1 (compare to Figure
4.11) and Exp. 2 (compare to compare to Figure 4.12). The input signals, (a)
and (c), and the interferometric readouts, (b) and (d), show that a better
precision on the measurement of the system parameters can be attained by
injecting only two relevant frequencies: the lowest and highest allowed.
A very interesting feature of the optimal design is its ability in improving
the fit performances. It allows for the recovering of the best-fit values in
fewer iterations than the standard design. It can be explained by the fact
that the optimal design mitigates parameter correlations (the diagonalization
of the information matrix implies lower correlation). Table 5.3 recaps some
examples showing a clear improvement, 4 through 7 times better than the
standard design, apart for $\text{Corr}[\omega_{1}^{2},\omega_{12}^{2}]$ that
remains unchanged.
Table 5.3: Different examples of parameter correlations. In some cases, the optimal design is capable in lowering the parameter correlation. Correlation | Standard | Optimal
---|---|---
$\text{Corr}[S_{21},\omega_{12}^{2}]$ | $-0.2$ | $-0.03$
$\text{Corr}[S_{21},\omega_{1}^{2}]$ | $0.09$ | $0.02$
$\text{Corr}[A_{\text{sus}},\omega_{1}^{2}]$ | $-0.7$ | $-0.2$
$\text{Corr}[\omega_{1}^{2},\omega_{12}^{2}]$ | $-0.5$ | $-0.5$
### 5.4 Single-experiment, multi-input
The preceding section has shown the optimization of the LPF experiments
independently, by exploring the 7-dimensional input frequency space. The
results are the improvement in precision, lower parameter correlation and the
fact that only two input frequencies are actually needed. This section
investigates on the possibility of defining a unique experiment in which bias
signals are injected both at the same time.
Instead of two independent optimizations in 7-dimensional frequency spaces,
for the experiment defined so far an optimization in a 14-dimensional
frequency space is now needed. To actuate this program, the experiment, namely
Exp. 3, is defined for the simultaneous injection of $o_{\text{i},1}$ and
$o_{\text{i},12}$. Table 5.4 reports the identification with such an optimized
experiment, compared to the standard design, on one side, and the
independently optimized designs, on the other side.
Table 5.4: Performances of optimal Exp. 3 (simultaneous injection in both guidance signals), compared to the optimal Exp. 1 & Exp. 2 of Section 5.3 and the standard ones. The T optimality criterion is considered. The fit standard deviations for all 5 parameters are reported for the three cases. In curly brackets the bias (absolute deviation from the real value in units of standard deviation) for each estimate. Parameter | Standard | Optimal | Optimal
---|---|---|---
st. dev. | Exp. 1 & Exp. 2 | Exp. 1 & Exp. 2 | Exp. 3
$\sigma_{\omega_{1}^{2}}\,[\text{s}^{-2}]$ | $4\times 10^{-10}$ | {1.4} | $2\times 10^{-10}$ | {0.68} | $1\times 10^{-10}$ | {1.9}
$\sigma_{\omega_{12}^{2}}\,[\text{s}^{-2}]$ | $2\times 10^{-10}$ | {0.41} | $1\times 10^{-10}$ | {2.0} | $8\times 10^{-12}$ | {0.42}
$\sigma_{S_{21}}$ | $4\times 10^{-7}$ | {0.086} | $1\times 10^{-7}$ | {1.1} | $3\times 10^{-8}$ | {0.57}
$\sigma_{A_{\text{df}}}$ | $7\times 10^{-4}$ | {1.6} | $1\times 10^{-4}$ | {0.50} | $8\times 10^{-5}$ | {0.73}
$\sigma_{A_{\text{sus}}}$ | $1\times 10^{-5}$ | {1.7} | $2\times 10^{-6}$ | {0.28} | $1\times 10^{-6}$ | {0.16}
The remarkable point to stress is the improvement in precision of an order of
magnitude for almost all parameters with respect to the optimal experiments
considered so far. Notice that the comparison should be taken with care. Since
the information scale as the integration time $\mathcal{I}\propto T$ and the
standard deviation scales as $\sigma\propto T^{-\nicefrac{{1}}{{2}}}$, then to
compare the result of the third experiment to the other two experiments (that
is approximately half long the total integration time of two independent
experiments), its fit standard deviations must be divided by the factor
$\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}\sqrt{2}$. It should be
also pointed out that parameter correlation does not improve, in fact a
simultaneous injection may not be the best approach to disentangle
degeneracies between the system parameters; a philosophy of the type the
simpler the stimulus, the better the understanding of the system should be
adopted whenever possible.
Figure 5.4 shows the signals being injected and the system responses in the
interferometric readouts. As is clear, the level of numerical and conceptual
complexity involved in the optimization of 14 input frequencies in the same
experiments makes the interpretation very difficult. Contrary to the case of
two independent injections, the optimization does not appear relaxing to two
distinct frequencies. The reason could be conceptually matched to the
simultaneous injection or due to intrinsical difficulties in the optimization.
A mix of both causes is the most plausible explanation. Most important, the
high amplitudes suggests that the constraints in interferometer range and
force authority in (5.5) should be rewritten in a more suitable form for
promptly handling the problem.
|
---|---
(a) | (b)
Figure 5.4: Synthetic data for optimal design of Exp. 3 (simultaneous
injection in both guidance signals). (a) the input signals and (b) the
interferometric readouts. The system does not appear relaxing to only two
frequency as in the case of two independent injections.
The investigation on an experiment in which there is a simultaneous injection
of both guidance signals shows a higher level of complexity in the
optimization and the conceptual understanding of the system. As correlation
could not be resolved in this experiment, i.e., the parameters could still
remain correlated, such an experiment may not worth to be implemented during
the LPF mission. Moreover, as the cross-talk from one degree of freedom to the
optical axis is better identified with independent injections, this case seems
much more controllable and easier to interpret than the case of simultaneous
injections. The procedures developed in this chapter suggest that particular
designs can even be found, at least for the simpler case of independent
injections, providing in principle the measurement of the system parameters
with optimized uncertainty/correlation.
## 6 Conclusions and future perspectives
As a conclusion, it is worth to stress the key points presented in this
thesis. This work can be ideally divided into the following parts:
1. 1.
a theoretical contribution to the foundations of spacetime metrology that will
be demonstrated with LPF, in which TMs are required to free fall with
unprecedented pureness, to within
$3\\!\times\\!10^{-14}\,\mathrm{m\,s^{-2}\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$, and whose relative motion must be optically tracked with an
accuracy better than
$9\\!\times\\!10^{-12}\,\mathrm{m\,Hz^{-\nicefrac{{1}}{{2}}}}$ around
$1\,\mathrm{mHz}$;
2. 2.
a theoretical modeling of the dynamics of the LISA arm implemented in LPF;
3. 3.
a description of the procedures developed for system identification, crucial
for an accurate estimation of the total equivalent acceleration noise, and for
the success of LPF.
In particular, Chapter 2 showed a derivation of the Doppler link response to
GW signals, different from the well-known integration of null geodesics. The
parallel transport of the emitter 4-velocity along the photon geodesic induces
a time delay into the physical observable, the frequency shift. Hence, time
delays track the effect of GWs on the Doppler link. The Doppler link is the
measurement element of all space-based GW detectors, like LISA. The chapter
also showed how curvature directly affects the frequency shift along the beam
– a measurement that is concurrent to both parasitic differential
accelerations and non-inertial forces due to the particular choice of the
reference frame. Moreover, there are many sources of non-idealities to be
taken into account. The link is actually implemented with lower-measurements
between four bodies in LISA and three bodies in LPF, so the body extension and
misalignments in the optical elements couple with the main measurement axis,
still affected by parasitic acceleration and non-inertial forces. It is useful
for the discussion to treat all signals and noise sources as equivalent
differential accelerations, input to the Doppler link reformulated as a time-
delayed differential accelerometer.
LPF is the in-flight test of a down-scaled version of a single LISA arm. Most
of the control philosophy, actuation and dynamics is inherited from the LISA
design, with slight differences discussed in the text. As the control plays a
crucial role in LPF for the compensation of the differential forces of the two
TMs toward the SC, Chapter 3 described the sophisticated closed-loop dynamics
of two TMs within a hosting SC, whose relative motion is sensed by an
interferometer and capacitive sensors. The LPF dynamics can be modeled as
vector equations in which operators describe dynamics, sensing and control –
three essential constituents of the system. In view of deriving a generalized
equation of motion, a differential operator was identified. The operator has a
twofold relevance: on one side, it allows for the conversion of the sensed
relative motion into the total equivalent acceleration; on the other side,
such an operation requires the calibration of the system through another
operator exactly defined from the differential operator itself. The formalism
effectively helps in the subtraction of the couplings, the control, the SC
jitter and the system transitory. The chapter novelly showed that the accuracy
to which transients can be suppressed depends on the accuracy to which the
modeled system parameters have been estimated from targeted experiments. The
chapter presented a dynamical model for LPF along the optical axis that was
used in the analysis of this thesis and is planned to be employed during the
mission. As the characterization of the dynamics along the optical axis – the
main measurement axis – is the first target of the mission, the formalism was
employed to derive the equations governing the cross-talk, with a supporting
example, from other degrees of freedom to the nominal dynamics along the
optical axis.
The estimation of the total equivalent acceleration noise requires the
calibration of the differential operator converting the sensed motion into
equivalent acceleration. The operator contains critical parameters modeling
different non-ideality contributions like spring-like couplings between the
TMs and the SC, sensing cross-talk coefficients, actuation gains and delays in
the actual application of forces. The goal of Chapter 4 was to describe the
methods proposed, developed and tested to simulated experiments aimed at
system identification, i.e., the identification of those parameters crucial
for the estimation of the total equivalent acceleration noise, the
substraction of couplings, control forces, cross-talk and system transients in
the recovered acceleration time-series. The methods were applied to data
simulated with the same model for validation purpose (Monte Carlo simulation),
but also to data released by the OSE, the realistic simulator provided by ESA.
In a mission-like approach, different non-standard scenarios were considered:
under-performing actuators, under-estimated couplings and an example of non-
Gaussianities. Since the estimated equivalent acceleration noise depends on
the estimated system parameters, this chapter showed for the first time that
system identification is mandatory for the estimation of the equivalent
acceleration noise. Otherwise, systematic errors like the ones described in
this chapter might compromise the scientific objectives of the mission. As
said, system identification allows for mitigating transients in the data. In
the end, the chapter showed an example of application – completely in the
transitory regime – to data released by the OSE.
Since parameter estimation has fundamental importance for the achievement of
the mission requirements, Chapter 5 investigated on the design of optimal
identification experiments. This allows for the estimation of the system
parameters with better precision. Intuitively, better precision in the
estimated parameters is equivalent to better confidence in the estimated
equivalent acceleration noise. The standard theory of optimal design was
applied by taking into account the peculiarities, constraints and complexity
of LPF. The result was that the system can be stimulated with only two
frequencies, obtaining a gain in precision, to within an order of magnitude in
parameter standard deviation. It was also found that the two frequencies
stimulate the system into two regimes: the high-frequency regime dominated by
the sensing and the low-frequency regime dominated by the force couplings.
Evidently, this work covered only a restricted part of the experiments, the
investigations, the measurements and the scientific returns of the LPF
mission. First, more work could be done developing the theoretical description
of the Doppler link as a differential accelerometer in Chapter 2. Second, the
methods described in Chapter 4 might be also employed, as they are, in an
extensive investigation of the various possible cross-talk experiments and the
calibration of LISA-like data. Chapter 4 showed the robustness of the methods
to a couple of non-standard scenarios that might happen during the mission.
Additional investigation may be required for the possibility that the measured
noise would contain non-Gaussian, non-stationary and transient components,
even in the form of unmodeled transient signals. Finally, as a conclusion to
the investigation of Chapter 5, the optimized designs should be also checked
out with the OSE simulator.
This thesis showed the relevance of system identification for the correct
assessment – and the subtraction of various disturbances – of the total
equivalent differential acceleration. The total equivalent acceleration
characterizes the performance in sensitivity of the LISA arm, viewed as a
differential time-delayed accelerometer. Therefore, system identification is
crucial for the success of LPF in demonstrating the principles of spacetime
metrology needed for all future space-based missions.
## A Appendix
### A.1 A single galactic binary in LISA noise
Figure A.1 shows the response of the detector [72] to the injection of a
single galactic binary around $1\,\mathrm{mHz}$, weakly chirping at the rate
of $10\,\mathrm{\mu Hz}$ over 2 years, in the $X$ (1st generation TDI) channel
[26]. Noise is simulated according to the model described in [73]. The
estimated PSD 111Refer to footnote on pag. 6 for a brief description of the
employed method for spectral estimation. is also compared to the noise model
showing the self-consistency of the data generation process [49]. Notice the
convolution of the signal with the detector inducing the annual modulation.
The Doppler modulations are responsible of such complex LISA response, but
allows for a very precise identification of parameters like the source
position and polarization. LISA will be able in detecting thousands of such
sources superimposed to the variety of signals as briefly described in the
Introduction.
---
(a)
|
(b) | (c)
Figure A.1: Simulation of a single galactic binary around $1\,\mathrm{mHz}$,
weakly chirping at the rate of $10\,\mathrm{\mu Hz}$ over 2 years, in the
detector noise as seen by the $X$ LISA interferometer. (a) the signal appears
as a spike in the spectrum as large as the effect of the chirping is more
prominent. (b) the relative time-domain signature containing the noisy time-
series and the signal itself. (c) details of the source signal, where the
annual modulation due to the revolution of the constellation around the Sun is
evident. LISA will be able in detecting thousands of such sources, including
signals from the merging of SMBHs (with overwhelmingly large SNR), the
galactic binary foreground at low frequency and the EMRIs (with very low SNR).
### A.2 Non-pure free fall and Fermi-Walker transport
The are two main differences between a realistic and an idealistic description
of free-falling TMs making in practice a Doppler link:
1. 1.
the bodies are nearly in free fall, i.e., accelerations are very small, but
not zero;
2. 2.
the bodies have a finite extension coupling with the Doppler response and
producing extra-acceleration.
The Fermi-Walker transport (FW) is the same underlying principle. In fact, the
FW transport comes out every time the acceleration is different from zero and
can even vary with time. In such situations, the best implementation of a
local co-moving frame – also defining the body reference frame – is the one
having gyroscopes attached to the three space axes. This construction prevents
the space coordinates from rotating and forces them to be fixed as the time
flows. In this reference frame, any 4-vector $x^{\mu}$ is differentiated with
respect to the proper time of the geodesic following the rule [30]
$\frac{\text{d}{x^{\mu}}}{\text{d}{\tau}}=\Omega^{\mu\nu}x_{\nu}\leavevmode\nobreak\
.$ (A.1)
It is easy to recognize the ordinary cross product between the angular
velocity and the vector itself in the non-relativistic regime, so the FW
reference frame provides a generalization of the notion of angular velocity in
GR. The antisymmetric tensor $\Omega^{\mu\nu}$ contains all Lorentz
transformations (rotations and boosts), but no space rotations, and is given
by
$\Omega^{\mu\nu}=\frac{1}{c^{2}}\left(v^{\mu}a^{\nu}-v^{\nu}a^{\mu}\right)\leavevmode\nobreak\
,$ (A.2)
where $v^{\mu}$ and $a^{\mu}$ are the body velocity and acceleration. When
(A.2) holds true together with (A.1), then $x^{\mu}$ is said to be FW
transported along the same geodesic. Hence, four orthogonal vectors being FW
transported along the accelerated body geodesic define the local FW co-moving
frame.
In LPF the philosophy is different: there no gyroscopes attached to the TMs,
as the small linear and angular motion are indeed used to gather information
on the acceleration noise affecting the TM geodesic. As pointed out in the
thesis, the situation is more complicated since the TM-to-TM Doppler link is
carried out with three independent lower-level measurements evidently
introducing new couplings.
### A.3 Calculation in metrology without noise
This section demonstrates the formula (2.13) with a detailed calculation in
the TT gauge and in the instantaneous wave coordinate system. Indeed,
$\begin{split}h^{\mu}_{\leavevmode\nobreak\
\beta\,,\alpha}k_{\mu}k^{\beta}&=h_{\mu\beta\,,\alpha}k^{\mu}k^{\beta}\\\
&=h_{1\beta\,,\alpha}k^{1}k^{\beta}+h_{2\beta\,,\alpha}k^{2}k^{\beta}\\\
&=h_{11\,,\alpha}k^{1}k^{1}+h_{12\,,\alpha}k^{1}k^{2}+h_{21\,,\alpha}k^{2}k^{1}+h_{22\,,\alpha}k^{2}k^{2}\\\
&=h_{+\,,\alpha}k_{x}^{2}+h_{\times\,,\alpha}k_{x}k_{y}+h_{\times\,,\alpha}k_{y}k_{x}-h_{+\,,\alpha}k_{y}^{2}\\\
&=\left(k_{x}^{2}-k_{y}^{2}\right)h_{+\,,\alpha}+2k_{x}k_{y}\,h_{\times\,,\alpha}\leavevmode\nobreak\
,\end{split}$ (A.3)
where the coefficients of $h_{+\,,\alpha}$ and $h_{\times\,,\alpha}$ are
$K_{+}$ and $K_{\times}$.
### A.4 Linearized Einstein equations for Doppler link as differential
accelerometer
In this section the linearized Einstein equations are solved for the
calculation of Section 2.2. The linearized Einstein equations [30] are given
by
$\begin{split}h_{\mu\alpha,\nu}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}&+h_{\nu\alpha,\mu}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}-h_{\mu\nu,\alpha}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}-h^{\alpha}_{\leavevmode\nobreak\
\alpha,\mu\nu}\\\
&-\eta_{\mu\nu}\left(h_{\alpha\beta}^{\quad,\alpha\beta}-h^{\alpha\quad\beta}_{\leavevmode\nobreak\
\alpha,\beta}\right)=0\leavevmode\nobreak\ ,\end{split}$ (A.4)
where $\eta_{\mu\nu}$ is the flat Minkowski metric, $h_{\mu\nu}$ is the first-
order perturbation and the gauge is arbitrary.
In the $(ct,x)$ coordinates
$\displaystyle h^{\alpha}_{\leavevmode\nobreak\ \alpha}$
$\displaystyle=h_{00}-h_{11}\leavevmode\nobreak\ ,$ (A.5a) $\displaystyle
h_{\alpha\beta}^{\quad,\alpha\beta}$
$\displaystyle=h_{00,00}-2h_{01,01}+h_{00,11}\leavevmode\nobreak\ ,$ (A.5b)
$\displaystyle\begin{split}h^{\alpha\quad\beta}_{\leavevmode\nobreak\
\alpha,\beta}&=h^{\alpha}_{\leavevmode\nobreak\
\alpha,00}-h^{\alpha}_{\leavevmode\nobreak\ \alpha,11}\\\
&=h_{00,00}-h_{11,00}-h_{00,11}+h_{11,11}\leavevmode\nobreak\ .\end{split}$
(A.5c)
For $\mu=\nu=0$, the Einstein equations provide
$\begin{split}0&=2h_{0\alpha,0}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}-h_{00,\alpha}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}-h^{\alpha}_{\leavevmode\nobreak\ \alpha,00}\\\
&\quad-
h_{\alpha\beta}^{\quad,\alpha\beta}+h^{\alpha\quad\beta}_{\leavevmode\nobreak\
\alpha,\beta}\\\
&=2h_{00,00}-2h_{01,01}-h_{00,00}+h_{00,11}-h_{00,00}+h_{11,00}\\\ &\quad-
h_{00,00}+2h_{01,01}-h_{00,11}+h_{00,00}-h_{11,00}-h_{00,11}+h_{11,11}\\\
&=-h_{00,11}+h_{11,11}\leavevmode\nobreak\ .\end{split}$ (A.6)
For $\mu=0$ and $\nu=1$,
$\begin{split}0&=h_{0\alpha,1}^{\quad\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha}+h_{1\alpha,0}^{\quad\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha}-h_{01,\alpha}^{\quad\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha}-h^{\alpha}_{\leavevmode\nobreak\ \alpha,01}\\\
&=h_{00,01}-h_{01,11}+h_{01,00}-h_{11,01}-h_{01,00}+h_{01,11}-h_{00,01}+h_{11,01}\\\
&=0\leavevmode\nobreak\ .\end{split}$ (A.7)
For $\mu=\nu=1$,
$\begin{split}0&=2h_{1\alpha,1}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}-h_{11,\alpha}^{\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha}-h^{\alpha}_{\leavevmode\nobreak\ \alpha,11}\\\
&\quad+h_{\alpha\beta}^{\quad,\alpha\beta}-h^{\alpha\quad\beta}_{\leavevmode\nobreak\
\alpha,\beta}\\\
&=2h_{01,01}-2h_{11,11}-h_{11,00}+h_{11,11}-h_{00,11}+h_{11,11}\\\
&\quad+h_{00,00}-2h_{01,01}+h_{00,11}-h_{00,00}+h_{11,00}+h_{00,11}-h_{11,11}\\\
&=h_{00,11}-h_{11,11}\leavevmode\nobreak\ .\end{split}$ (A.8)
Therefore, in the approximations of Section 2.2, the Einstein equations reduce
to
$h_{00,11}-h_{11,11}=0\leavevmode\nobreak\ .$ (A.9)
### A.5 Demonstration of noise non-stationarity
This section demonstrates the validity of (4.27), i.e., that the fluctuation
of any of the system parameter produces non-stationary noise. Expanding the
noise around some nominal parameter value $p_{0}$, up to first order, and
computing the variance of the interferometric noise, it reads
$\begin{split}\text{Var}[o]&\simeq\text{Var}\left[o_{0}\right]+\text{Var}\left[o^{\prime}\delta
p\right]+2\text{Cov}\left[o_{0},o^{\prime}\delta p\right]\\\
&=\text{Var}\left[o_{0}\right]+\text{Var}\left[o^{\prime}\right]\delta
p^{2}+2\text{Cov}\left[o_{0},o^{\prime}\right]\delta p\leavevmode\nobreak\
,\end{split}$ (A.10)
where $\text{Var}\left[o^{\prime}\right]$ and
$\text{Cov}\left[o_{0},o^{\prime}\right]$ are the variance of the noise first
derivative and the covariance between the zeroth order and the first
derivative. So, for a zero-mean process with finite second moment, it holds
$\begin{split}\text{Cov}\left[o_{0},o^{\prime}\right]&=\text{E}\left[o_{0}o^{\prime}\right]-\text{E}\left[o_{0}\right]\text{E}\left[o^{\prime}\right]\\\
&=\text{E}\left[\frac{1}{2}\frac{\partial}{\partial p}n^{2}\right]\\\
&=\frac{1}{2}\frac{\partial}{\partial p}\text{Var}[n]\leavevmode\nobreak\
.\end{split}$ (A.11)
Substituting this result back into (A.10), (4.27) is finally demonstrated.
### A.6 Time-frequency analysis of non-stationary noise
Noise stationarity is the most important assumption taken by the standard
spectral estimation. There are cases where the estimated PSD or even more
advanced techniques like the Kolmogorov-Smirnov test [61] – aimed at comparing
the cumulative distribution function of the noise PSD compared to a reference
(either another noise measurement or a model expectation) – may fail in
detecting noise excesses or small transient signals concentrated in very
narrow time segments. To explain why, this section is devoted to showing an
example where such a problem can be found and how to promptly deal with it by
employing fast and efficient tools like the wavelet analysis [74].
Without loss of generality, in what follows it is considered that a short
transient force (per unit mass) gradient is modeled as a Gaussian-shaped
signal
$f_{12,\text{tr}}(t)=a\,\exp\left[-\frac{(t-t_{0})^{2}}{\tau^{2}}\right]\leavevmode\nobreak\
,$ (A.12)
of total duration $2\tau\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}1$ hour ($\tau=1\\!\times\\!10^{3}\,\mathrm{s}$), is turned on after about
8 hours ($t_{0}=3\\!\times\\!10^{4}\,\mathrm{s}$) during a LPF noise run of
about 12 hours and with amplitude
$a=1.6\\!\times\\!10^{-13}\,\mathrm{m\,s^{-2}}$. The gradient might be either
due to anomalous transient force couplings temporarily entering into the noise
budget, or effectively unexpected signals.
An example is the prediction of a gradient surplus as a deviation from
Newtonian gravity, described in [75, 76], where a flyby of the saddle point of
the Sun-Earth potential surface is proposed for the natural conclusion of the
LPF mission toward the escaping trajectory to test for alternative theories of
gravity. Even though there are different models claiming that a gradient with
high SNR could be detected if the SC would cross the “bubble” around the
saddle point with sufficient small impact parameter, in practice it is likely
that the SC orbit will never reach such an accuracy. Hence, it is a good idea
to start by looking for very small departures and putting thresholds to the
observability of noise transients. So much far beyond the objectives of this
section, the following gives a very first address of the problem.
Figure A.2 shows a simulation of a noise run of about 12 hours in the
differential interferometer readout, together with the system response to the
external signal (i.e., the gravity gradient excess as in the example above) of
absolute peak $\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$
}$$}5\,\mathrm{nm}$. It is evident that: (i) the signal could be easily
confused with the intrinsical noise fluctuation; (ii) PSD estimation can warn
of a change in the shape of the spectrum (a sign of non-stationarity),
sometimes by a huge amount, sometimes by a negligible amount as in this
example; but it does not say much about where is changing and on what time
scale, as the location is fundamentally important for the analysis of
transients.
Figure A.2: A simulated noise run of the differential interferometer readout
lasting for about 12 hours compared to the same with a gradient force injected
into the system. The signal, of absolute peak $\raise 0.58554pt\hbox{$$
\mbox{\scriptsize$\sim$ }$$}5\,\mathrm{nm}$, can be easily confused with the
noise. PSD estimation is not capable to quantitatively assess the significance
of the excess, both in term of frequency and position of the transient.
A solution is provided by the continuous wavelet transform that gives a full
time-frequency representation of the data series. Without going through the
mathematical details, the data stretch is decomposed into continuous waves,
the wavelets, that are the equivalent to the Fourier sines. The Fourier
transform is a function of frequency; the wavelet transform is function of
both time and scale. The time dependency gives the energy content with respect
to the wavelet location. The scale dependency gives the energy content with
respect to the wavelet compression. Therefore, the scale is inversely
proportionally to the frequency and, in fact, it is possible to associate an
approximate frequency to the scale of a given wavelet. A detailed discussion
can be found in [74].
Figure A.3 reports the time-frequency representation, the spectrogram, of
Figure A.2 for second-order Daubechies wavelets. The power is scaled to the
total energy in the time-frequency bands, so that the spectrogram is
normalized to one. The transient signal is visible as the narrow and darker
line around the instant of injection. Notice that its power is more than two
times the other peaks, so it can be easily identified in a quick-look search
of unmodeled transient signals.
Figure A.3: Wavelet-based spectrogram of a simulated noise run of the
differential interferometer readout lasting for about 12 hours in which a tiny
force signal of peak amplitude $1.6\\!\times\\!10^{-13}\,\mathrm{m\,s^{-2}}$
is turned on after about 8 hours and producing the interferometric response
showed in Figure A.2. The transient signal is visible as the narrow and darker
line at the instant of injection. The method allows for the identification of
short unmodeled transient signals and excess noise.
It is worth noting that an extensive investigation on this thematic – and in
particular on de-noising techniques with the discrete wavelet transform –
would surely improve the understanding of the non-stationary behavior of the
LPF noise, in view of a fast identification of unmodeled transient signals.
### A.7 More on Monte Carlo validation
This section investigates a little further on the Monte Carlo simulation of
Section 4.4.6, that demonstrated that all parameters are unbiased and Gaussian
distributed, as well as their variances.
Surprisingly, the correlations are also Gaussian distributed with good
approximation. See Figure A.4 for two examples.
|
---|---
(a) | (b)
Figure A.4: Monte Carlo validation of 1000 independent noise realizations on
which parameter estimation is repeated identically at each step. The
statistics is shown for two parameter correlations. The scaled Gaussian PDF is
evaluated at the sample mean and standard deviation.
The correlation between two parameters is somehow related to the rotation of
the $\chi^{2}$ paraboloid principal axes around the minimum. To support this
statement, Figure A.5 shows few examples of projections of the $7$-dimensional
surface onto two parameters at a time, around the best-fit values. Weakly
correlated parameters, like $S_{21}$ and $\omega_{1}^{2}$ ($\raise
0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}20\%$) (panel (b)), typically
have the principal axes of the contour curves aligned with the $x$ and $y$
axis. Highly correlated parameters, like $A_{\text{sus}}$ and $\omega_{1}^{2}$
($\raise 0.73193pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}-70\%$) (panel (d)),
have the principal axes that are significantly rotated.
|
---|---
(a) | (b)
|
(c) | (d)
Figure A.5: $\chi^{2}$ log-likelihood curvature around the best-fit values.
The 7-dimensional surface are projected onto two parameters at a time for some
examples. Correlation is the reason why the surface can be rotated.
Figure A.6 shows a record history of all Monte Carlo estimation chains. The
scatter of the chains is due to the noise fluctuation along the Monte Carlo
iterations. There are clearly some chains that are far away from the
accumulation zone: this behavior is quite unexpected as one would think the
noise to have little impact on the chain locations. Despite the big scatter,
the asymptotic distribution is Gaussian, as elucidated in Figure 4.17.
Figure A.6: Monte Carlo fit $\chi^{2}$ chains. The processes typically last
for $\raise 0.58554pt\hbox{$$ \mbox{\scriptsize$\sim$ }$$}1000$ iterations and
stop when either the function or the variable tolerance is below
$1\\!\times\\!10^{-4}$.
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|
arxiv-papers
| 2012-04-19T09:53:15 |
2024-09-04T02:49:29.854310
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Giuseppe Congedo",
"submitter": "Giuseppe Congedo",
"url": "https://arxiv.org/abs/1204.4299"
}
|
1204.4347
|
# Change-Of-Bases Abstractions for Non-Linear Systems.
Sriram Sankaranarayanan
Department of Computer Science
University of Colorado, Boulder, CO, USA.
srirams@colorado.edu
###### Abstract
We present abstraction techniques that transform a given non-linear dynamical
system into a linear system or an algebraic system described by polynomials of
bounded degree, such that, invariant properties of the resulting abstraction
can be used to infer invariants for the original system. The abstraction
techniques rely on a change-of-basis transformation that associates each state
variable of the abstract system with a function involving the state variables
of the original system. We present conditions under which a given change of
basis transformation for a non-linear system can define an abstraction.
Furthermore, the techniques developed here apply to continuous systems defined
by Ordinary Differential Equations (ODEs), discrete systems defined by
transition systems and hybrid systems that combine continuous as well as
discrete subsystems.
The techniques presented here allow us to discover, given a non-linear system,
if a change of bases transformation involving degree-bounded polynomials
yielding an algebraic abstraction exists. If so, our technique yields the
resulting abstract system, as well. This approach is further extended to
search for a change of bases transformation that abstracts a given non-linear
system into a system of linear differential inclusions. Our techniques enable
the use of analysis techniques for linear systems to infer invariants for non-
linear systems. We present preliminary evidence of the practical feasibility
of our ideas using a prototype implementation.
## 1 Introduction
In this paper, we explore a class of abstractions for non-linear autonomous
systems (continuous, discrete and hybrid systems) using _Change-of-Bases_
(CoB) transformations. CoB transformations are obtained for a given system by
expressing the dynamics of the system in terms of a new set of variables that
relate to the original system variables through the CoB transformation. Such a
transformation is akin to studying the system under a new set of “bases”. We
derive conditions on the transformations such that (a) the CoB transformations
also define an _autonomous system_ and (b) the resulting system abstracts the
original system: i.e., all invariants of the abstract system can be
transformed into invariants for the original system. Furthermore, we often
seek abstract systems through CoB transformations whose dynamics are of a
simpler form, more amenable to automatic verification techniques. For
instance, it is possible to use CoB transformations that relate an ODE with
non-linear right-hand sides to an affine ODE, or transformations that reduce
the degree of a system with polynomial right-hand sides. If such
transformations can be found, then safety analysis techniques over the simpler
abstract system can be used to infer safety properties of the original system.
In this paper, we make two main contributions: (a) we define CoB
transformations for continuous, discrete and hybrid systems and provide
conditions under which a given transformation is valid; (b) we provide search
techniques for finding CoB transformations that result in a polynomial system
whose right-hand sides are degree limited by some limit $d\geq 1$.
Specifically, the case $d=1$ yields an affine abstraction; and (c) we provide
experimental evidence of the application of our techniques to a variety of
ordinary differential equations (ODEs) and discrete programs.
The results in this paper extend our previously published results that
appeared in HSCC 2011 [34]. The contributions of this paper include (a) an
extension from linearizing CoB transformations to degree-bounded polynomial
CoB transformations, (b) extending the theory from purely continuous system to
discrete and hybrid systems, and (c) an improved implementation that can
handle hybrid systems with some evaluation results using this implementation.
On the other hand, our previous work also included an extension of the theory
to differential inequalities and iterative techniques over cones. These
extensions are omitted here in favor of an extended treatment of the theory of
differential equation abstractions for continuous, discrete and hybrid
systems.
### 1.1 Motivating Examples
In this section, we motivate the techniques developed in this paper by means
of a few illustrative examples involving purely continuous ODEs and purely
discrete programs .
Our first example concerns a continuous system defined by a system of Ordinary
Differential Equations (ODEs):
###### Example 1.1.
Consider a continuous system over $\\{x,y\\}$: $\dot{x}=xy+2x,\ \
\dot{y}=-\frac{1}{2}y^{2}+7y+1$, with initial conditions given by the set
$x\in[0,1],\ y\in[0,1]$. Using the transformation
$\alpha:(x,y)\mapsto(w_{1},w_{2},w_{3})$ wherein $\alpha_{1}(x,y)=x$,
$\alpha_{2}(x,y)=xy$ and $\alpha_{3}(x,y)=xy^{2}$, we find that the dynamics
over $\vec{w}$ can be written as
$\dot{w_{1}}=2w_{1}+w_{2},\
\dot{w_{2}}=w_{1}+9w_{2}+\frac{1}{2}w_{3},\dot{w_{3}}=2w_{2}+16w_{3}$
Its initial conditions are given by $w_{1}\in[0,1],\ w_{2}\in[0,1],\
w_{3}\in[0,1]$. We analyze the system using the TimePass tool as presented in
our previous work [37] to obtain polyhedral invariants:
$\begin{array}[]{l}-w_{1}+2w_{2}\geq-1\ \land\ w_{3}\geq 0\ \land\ w_{2}\geq
0\ \land\\\ -16w_{1}+32w_{2}-w_{3}\geq-17\ \land\ 32w_{2}-w_{3}\geq-1\
\land\\\ 2w_{1}-4w_{2}+17w_{3}\geq-4\ \land\ 286w_{1}-32w_{2}+w_{3}\geq-32\
\land\\\ \cdots\end{array}$
Substituting back, we can infer polynomial inequality invariants on the
original system including,
$\begin{array}[]{l}-x+2xy\geq-1\ \land\ xy^{2}\geq 0\ \land\ -16x+32xy-
xy^{2}\geq-17\\\ x\geq 0\ \land\ 2x-4xy+17xy^{2}\geq-4\ \land\
\cdots\end{array}$
Finally, we integrate the linear system to infer the following conserved
quantity for the underlying non-linear system:
$\begin{array}[]{l}\left(\frac{e^{-9t}}{51}+\frac{1}{102}\left(50+7\sqrt{51}\right)e^{\left(-9+\sqrt{51}\right)t}+\frac{1}{102}\left(50-7\sqrt{51}\right)e^{-\left(9+\sqrt{51}\right)t}\right)\
x+\\\
\left(-\frac{1}{102}e^{-9t-\left(9+\sqrt{51}\right)t}\left(\begin{array}[]{l}7e^{9t}-\sqrt{51}e^{9t}-14e^{\left(9+\sqrt{51}\right)t}+\\\
7e^{9t+\left(-9+\sqrt{51}\right)t+\left(9+\sqrt{51}\right)t}+\\\
\sqrt{51}e^{9t+\left(-9+\sqrt{51}\right)t+\left(9+\sqrt{51}\right)t}\end{array}\right)\right)\
xy+\\\
\left(\frac{1}{204}e^{-9t-\left(9+\sqrt{51}\right)t}\left(e^{9t}-2e^{\left(9+\sqrt{51}\right)t}+e^{9t+\left(-9+\sqrt{51}\right)t+\left(9+\sqrt{51}\right)t}\right)\right)xy^{2}\end{array}$
Finally, if $x(0)\not=0$, the map $\alpha$ is invertible and therefore, the
ODE above can be integrated.
Note that not every transformation yields a linear abstraction. In fact, most
transformations will not define an abstraction. The conditions for an
abstraction are discussed in Section 2. $\blacktriangle$
⬇ proc computeP(int k) int x,y; assert( K > 0); x := y := 0; while ( y < k ){ x := x + y * y; y := y + 1; } end-function | ⬇ proc computePAbs(int k) int x,y,y2; assert( K > 0); x := y := y2 := 0; while ( y < k ){ x := x + y2; y2 := y2 + 2 * y + 1; y := y + 1; } end-function
---|---
Figure 1: Program showing a benchmark example proposed by Petter [28] and its
abstraction obtained by a change of basis $(x\mapsto x,y\mapsto y,y2\mapsto
y^{2})$.
Next, we motivate our approach on purely discrete programs, showing how CoB
transformations can linearize a discrete program with non-linear assignments,
modeled by a _transition system_ [21]. In turn, we show how invariants of the
abstract linearized program can be transferred back.
###### Example 1.2.
Figure 1 shows an example proposed originally by Petter [28] that considers a
program that sums up all squares from $1$ to $K^{2}$ for some input $K\geq 0$.
Consider a very simple change of basis transformation wherein we add a new
variable “y2” that tracks the value of $y^{2}$ as the loop is executed. It is
straightforward to write assignments for “y2” in terms of itself, $x,y$. Doing
so for this example does not necessitate the tracking of higher degree terms
such as $y^{3},x^{2}y^{2}$ and so on. Finally, the resulting program has
affine guards and assignments, making it suitable for polyhedral abstract
interpretation [10, 16]. The polyhedral analysis yields linear invariants at
the loop head and the function exit in terms of the variables $x,y,y2$. We may
safely substitute $y^{2}$ in place of $y2$ and obtain invariants over the
original program. The non-linear invariants obtained at the function exit are
shown below:
$\begin{array}[]{c}4x+18y-7y^{2}\geq 11\ \land\ 4\leq 2x+7y-3y^{2}\ \land\
9\leq x+12y-3y^{2}\ \land\ 1\leq y\ \land\\\ 3y-y^{2}\leq 2\ \land\
5y-y^{2}\leq 6\ \land\ 6y-y^{2}\leq 9\ \land\ k=y\end{array}$
In this example, the change of basis to $y^{2}$ can, perhaps, be inferred from
the syntax of this program. However, we demonstrate other situations in this
paper, wherein the change of basis cannot be inferred from the expressions in
the program using syntactic means.
The invariant
$6x=2k^{3}+3k^{2}+k\,,$
discovered by Petter and many other subsequent works such as the complete
approach for P-solvable loops by Kovacs [18] can also be discovered by Karr’s
analysis when the term $y^{3}$ is introduced into the change-of-basis
transformations in addition to $y^{2}$. $\blacktriangle$
### 1.2 Related Work
Many different types of _discrete abstractions_ have been studied for hybrid
systems [1] including predicate abstraction [39] and abstractions based on
invariants [25]. The use of counter-example guided iterative abstraction-
refinement has also been investigated in the past (Cf. Alur et al. [2] and
Clarke et al. [6], for example). In this paper, we consider continuous
abstractions for continuous systems specified as ODEs, discrete systems and
hybrid systems using a change of bases transformation. As noted above, not all
transformations can be used for this purpose. Our abstractions for ODEs bear
similarities to the notion of topological semi-conjugacy between flows of
dynamical systems [23].
Previous work on invariant generation for hybrid system by the author
constructs invariants by assuming a desired template form (ansatz) with
unknown parameters and applying the “consecution” conditions such as _strong
consecution_ and _constant scale_ consecution [38]. Matringe et al. present
generalizations of these conditions using morphisms [22]. Therein, they
observe that strong and constant scale consecution conditions correspond to a
linear abstraction of the original non-linear system of a restrictive form.
Specifically, the original system is abstracted by a system of the form
$\frac{dx}{dt}=0$ for strong consecution, and a system of the form
$\frac{dx}{dt}=\lambda x$ for constant-scale consecution. This paper builds
upon this observation by Matringe et al. using fixed-point computation
techniques to search for a general linear abstraction that is related to the
original system by a change of basis transformation. Our work is also related
to the technique of differential invariants proposed by Platzer et al. [29].
At a high level Platzer et al. attempt to prove an invariant $p=0$ for a
continuous system (often a subsystem of a larger hybrid system) using
differential invariant rule wherein the state assertion $\frac{dp}{dt}=0$ is
established. Likewise, to prove $p\leq 0$, it seeks to establish
$\frac{dp}{dt}\leq 0$. In this paper, we may view the same process through a
CoB transformation $w\mapsto p(x)$ that allows us to write the abstract
dynamics as $\frac{dw}{dt}=0$. Going further, we seek to compute
$\vec{w}\mapsto\alpha(\vec{x})$ that maps the dynamics to an affine or a
polynomial system. On the other hand, differential invariants allow us to
reason about Boolean combinations of assertions and embed into a rich dynamic-
logic framework combining discrete and continuous actions on the state. The
work here and its extension to differential inequalities [34] can be utilized
in such a framework.
Fixed point techniques for deriving invariants of differential equations have
been proposed by the author in previous papers [37, 33] These techniques have
addressed the derivation of polyhedral invariants for affine systems [37] and
algebraic invariants for systems with polynomial right-hand sides [33]. In
this technique, we employ the machinery of fixed-points. Our primary goal is
not to derive invariants, per se, but to search for abstractions of non-linear
systems into linear systems.
##### Discrete Systems:
There has been a large body of work focused on the use of algebraic techniques
for deriving invariants of programs. Previous work by the author focuses on
deriving polynomial equality invariants for programs, automatically, by
setting up template polynomial invariants with unknown coefficients and
deriving constraints on values of these coefficients to ensure invariance [38,
35]. Carbonell et al. present loop invariant generation techniques by solving
recurrences and computing polynomial ideas to capture algebraic properties of
the reachable states [32] and subsequently using the descending abstract
interpretation over ideals with widening over ideals to ensure termination
[31]. The approach is extended to polyhedral cones generated by polynomial
inequalities to generate polynomial inequality invariants [3]. Another set of
related techniques concern the use of linear invariant generation techniques
for polynomial equality invariant generation. Müller-Olm and Seidl explore the
use of linear algebraic techniques, wherein a vector space of matrices are
used to summarize the transformation from the initial state of a program to a
given location. This space is then used to generate polynomial invariants of
the program [24]. Likewise, the work of Colón explores degree-bounded
restrictions to Nullstellensatz to enable linear algebraic techniques to
generate polynomial invariants [9]. More recently, the work of Kovacs uses
sophisticated techniques for solving recurrence equations over so-called
P-solvable loops to generate polynomial invariants for them [18].
Finally, our approach is closely related to _Carlemann embedding_ that can be
used to linearize a given differential equation with polynomial right-hand
sides [19]. The standard Carlemann embedding technique creates an infinite
dimensional linear system, wherein, each dimension corresponds to a monomial
or a basis polynomial. In practice, it is possible to create a linear
approximation with known error bounds by truncating the monomial terms beyond
a degree cutoff. Our approach for differential equation abstractions can be
_roughly_ seen as a search for a “finite submatrix” inside the infinite matrix
created by the Carleman linearization. The rows and columns of this submatrix
correspond to monomials such that the derivative of each monomial in the
submatrix is a linear combination of monomials that belong the submatrix.
Note, however, that while Carleman embedding is defined using some basis for
polynomials (usually power-products), our approach can derive transformations
that may involve polynomials as opposed to just power-products.
##### Organization:
The rest of this paper presents our approach for Ordinary Differential
Equations in Section 2. The ideas for discrete systems are presented in
Section 3 by first presenting the theory for simple loops and then extending
it to arbitrary discrete programs modeled by transition systems. The
extensions to hybrid systems are presented briefly by suitably merging the
techniques for discrete programs with those for ODEs. Finally, Section 4
presents an evaluation of the ideas presented using our implementation that
combines an automatic search for CoB transformations with polyhedral invariant
generation for continuous, discrete and hybrid systems [10, 16, 37].
## 2 Abstractions for ODEs
We first present some preliminary definitions for continuous systems defined
by Ordinary Differential Equations (ODEs).
### 2.1 Preliminaries: Continuous Systems
Let $\mathbb{R}$ denote the field of real numbers. Let $x_{1},\ldots,x_{n}$
denote a set of variables, collectively represented as $\vec{x}$. The set
$\mathbb{R}[\vec{x}]$ denotes the ring of multivariate polynomials over
$\mathbb{R}$.
A _power-product_ over $\vec{x}$ is of the form
$x_{1}^{r_{1}}x_{2}^{r_{2}}\cdots x_{n}^{r_{n}}$, succinctly written as
$\vec{x}^{\vec{r}}$, wherein each $r_{i}\in\mathbb{N}$. The _degree_ of a
monomial $\vec{x}^{\vec{r}}$ is given by
$\sum_{i=1}^{n}r_{i}=\vec{1}\cdot\vec{r}$. A _monomial_ is of the form $c\cdot
m$ where $c\in\mathbb{R}$ and $m$ is a power-product. A multivariate
polynomial $p$ is a sum of finitely many monomial terms:
$p=\sum_{\vec{r}\in\mathbb{R}^{n}}c_{r}\vec{x}^{\vec{r}}$. The degree of a
multivariate polynomial $p$ is the maximum over the degrees of all monomial
terms $m$ that _occur_ in $p$ with a non-zero coefficient.
We assume some basic familiarity with the basics of computational algebraic
geometry [11] and elementary linear algebra [17].
##### Vector Fields:
A _vector field_ $F$ over a manifold $M\subseteq\mathbb{R}^{n}$ is a map
$F:M\mapsto\mathbb{R}^{n}$ from each $\vec{x}\in M$ to a vector
$F(\vec{x})\in\mathbb{R}^{n}$, wherein $F(\vec{x})\in T_{M}(\vec{x})$, the
tangent space of $M$ at $\vec{x}$.
A vector field $F$ is continuous if the map $F$ is continuous. A _polynomial
vector field_ $F\in(\mathbb{R}[\vec{x}])^{n}$ is specified by a tuple
$F(\vec{x})=\left\langle
p_{1}(\vec{x}),p_{2}(\vec{x}),\ldots,p_{n}(\vec{x})\right\rangle$, wherein
$p_{1},\ldots,p_{n}\in\mathbb{R}[\vec{x}]$.
A system of (coupled) ordinary differential equations (ODE) specifies the
evolution of variables $\vec{x}:(x_{1},\ldots,x_{n})\in M$ over time $t$:
$\frac{dx_{1}}{dt}=p_{1}(x_{1},\ldots,x_{n}),\ \cdots,\
\frac{dx_{n}}{dt}=p_{n}(x_{1},\ldots,x_{n})\,,$
The system implicitly defines a vector field $F(\vec{x}):\left\langle
p_{1}(\vec{x}),\ldots,p_{n}(\vec{x})\right\rangle$. We assume that all vector
fields $F$ considered in this paper are (locally) Lipschitz continuous over
the domain $M$. In general, all polynomial vector fields are locally Lipschitz
continuous, but not necessarily _globally_ Lipschitz continuous over an
unbounded domain $X$. The Lipschitz continuity of the vector field $F$,
ensures that given $\vec{x}=\vec{x}_{0}$, there exists a time $T>0$ and a
unique time trajectory $\tau:[0,T)\mapsto\mathbb{R}^{n}$ such that
$\tau(t)=\vec{x}_{0}$ [23].
###### Definition 2.1.
For a vector field $F:\ \left\langle f_{1},\ldots,f_{m}\right\rangle$, the
_Lie derivative_ of a smooth function $f(\vec{x})$ is given by
$\mathcal{L}_{F}(f)=(\nabla f)\cdot
F(\vec{x})=\mathop{\sum}_{i=1}^{n}\left(\frac{\partial f}{\partial x_{i}}\cdot
f_{i}\right)$
Henceforth, wherever the vector field $F$ is clear from the context, we will
drop subscripts and use $\mathcal{L}(p)$ to denote the Lie derivative of $p$
w.r.t $F$.
###### Definition 2.2.
A continuous system over variables $x_{1},\ldots,x_{n}$ consists of a tuple
$\mathcal{S}:\left\langle X_{0},\mathcal{F},X_{I}\right\rangle$ wherein
$X_{0}\subseteq\mathbb{R}^{n}$ is the set of initial states, $\mathcal{F}$ is
a vector field over the domain represented by a manifold
$X_{I}\subseteq\mathbb{R}^{n}$.
Note that in the context of hybrid systems, the set $X_{I}$ is often referred
to as the _state invariant_ or the _domain_ manifold.
### 2.2 Change-of-Bases for Continuous Systems
In this section, we will present change-of-bases (CoB) transformations of
continuous systems and some of their properties.
Consider a map $\alpha:\mathbb{R}^{k}\mapsto\mathbb{R}^{l}$. Given a set
$S\subseteq\mathbb{R}^{k}$, let $\alpha(S)$ denote the set obtained by
applying $\alpha$ to all the elements of $S$. Likewise, the inverse map over
sets is $\alpha^{-1}(T):\ \\{s\ |\ \alpha(s)\in T\\}$. Let
$\mathcal{S}:\left\langle X_{0},\mathcal{F},X_{I}\right\rangle$ be a
continuous system over variables $\vec{x}:\ (x_{1},\ldots,x_{n})$ and
$\mathcal{T}:\left\langle Y_{0},\mathcal{G},Y_{I}\right\rangle$ be a
continuous system over variables $\vec{y}:(y_{1},\ldots,y_{m})$.
###### Definition 2.3.
We say that $\mathcal{T}$ _simulates_ $\mathcal{S}$ iff there exists a smooth
mapping $\alpha:\ \mathbb{R}^{n}\mapsto\mathbb{R}^{m}$ such that
1. 1.
$Y_{0}\supseteq\alpha(X_{0})$ and $Y_{I}\supseteq\alpha(X_{I})$.
2. 2.
For any trajectory $\tau:[0,T)\mapsto X_{I}$ of $\mathcal{S}$,
$\alpha\circ\tau$ is a trajectory of $\mathcal{T}$.
A simulation relation implies that any time trajectory of $\mathcal{S}$ can be
mapped to a trajectory of $\mathcal{T}$ through $\alpha$. However, since
$\alpha$ need not be invertible, the converse need not hold. I.e,
$\mathcal{T}$ may exhibit time trajectories that are not mapped onto by any
trajectory in $\mathcal{S}$.
Let $\mathcal{S}$ and $\mathcal{T}$ be defined by Lipschitz continuous vector
fields. The following theorem enables us to check given $\mathcal{S}$ and
$\mathcal{T}$, if $\mathcal{T}$ simulates $\mathcal{S}$.
###### Theorem 2.1.
$\mathcal{T}$ simulates $\mathcal{S}$ if the following conditions hold:
1. 1.
$Y_{0}\supseteq\alpha(X_{0})$.
2. 2.
$Y_{I}\supseteq\alpha(X_{I})$.
3. 3.
$\mathcal{G}(\alpha(\vec{x}))=J_{\alpha}.\mathcal{F}(\vec{x})$, wherein,
$J_{\alpha}$ is the Jacobian matrix
$J_{\alpha}(x_{1},\ldots,x_{n})=\left[\begin{array}[]{rcl}\frac{\partial\alpha_{1}}{\partial
x_{1}}&\cdots&\frac{\partial\alpha_{1}}{\partial x_{n}}\\\
\vdots&\ddots&\vdots\\\ \frac{\partial\alpha_{m}}{\partial
x_{1}}&\cdots&\frac{\partial\alpha_{m}}{\partial x_{n}}\\\
\end{array}\right]\,,$
and $\alpha(\vec{x})=(\alpha_{1}(\vec{x}),\cdots,\alpha_{m}(\vec{x})),\
\alpha_{i}:\mathbb{R}^{n}\mapsto\mathbb{R}$.
###### Proof.
Let $\tau_{x}$ be a trajectory over $\vec{x}$ for system $\mathcal{S}$. Note
that at any time instant $t\in[0,t)$,
$\frac{d\tau_{x}}{dt}=\mathcal{F}(\tau(t))$.
We wish to show that $\tau_{y}(t)=\alpha(\tau_{x}(t))$ is a time trajectory
for the system $\mathcal{T}$. Since, $\tau_{x}(0)\in X_{0}$, we conclude that
$\tau_{y}(0)=\alpha(\tau_{x}(0))\in Y_{0}$. Since $\tau_{x}(t)\in X_{I}$ for
all $t\in[0,T)$, we have that $\tau_{y}(t)=\alpha(\tau_{x}(t))\in Y_{I}$.
Differentiating $\tau_{y}$ we get,
$\begin{array}[]{rclclcl}\frac{d\tau_{y}}{dt}&=&\frac{d\alpha(\tau_{x}(t))}{dt}&=&J_{\alpha}\cdot\frac{d\tau_{x}}{dt}&=&J_{\alpha}\cdot\mathcal{F}(\tau_{x}(t))\\\
&=&\mathcal{G}(\alpha(\tau_{x}(t)))&=&\mathcal{G}(\tau_{y}(t))\,.\\\
\end{array}$
Therefore $\tau_{y}=\alpha\circ\tau_{x}$ conforms to the dynamics of
$\mathcal{T}$. By Lipschitz continuity of $\mathcal{G}$, we obtain that
$\tau_{y}$ is the unique trajectory starting from $\alpha\circ\tau(0)$. ∎
Theorem 2.1 shows that the condition
$\mathcal{G}(\alpha(\vec{x}))=J_{\alpha}.\mathcal{F}(\vec{x})$
relating vector fields $\mathcal{F}$ and $\mathcal{G}$ suffices to guarantee
that time trajectories (integral curves) of $\mathcal{F}$ are related to those
in $\mathcal{G}$ through the map $\alpha$. In differential geometric terms,
this condition can be stated as $\mathcal{F}$ is $\alpha$-related to
$\mathcal{G}$ [20].
Note that, in general, a trajectory $\tau_{y}(t)=\alpha(\tau_{x}(t))$ may
exist for a longer interval of time than the interval $[0,T)$ over which
$\tau_{x}$ is assumed to be defined.
###### Theorem 2.2.
Let $\mathcal{T}$ simulate $\mathcal{S}$ through a map $\alpha$. If
$Y\subseteq Y_{I}$ is a positive invariant set for $\mathcal{T}$ then
$\alpha^{-1}(Y)\cap X_{I}$ is a positive invariant set for $\mathcal{S}$.
###### Proof.
Assuming otherwise, let $\tau_{x}$ be a time trajectory that starts from
inside $\alpha^{-1}(Y)\cap X_{I}$ and has a time instant $t$ such that
$\tau_{x}(t)\not\in\alpha^{-1}(Y)\cap X_{I}$. Since we defined time
trajectories so that $\tau_{x}(t)\in X_{I}$, it follows that
$\tau_{x}(t)\not\in\alpha^{-1}(Y)$. As a result, $\alpha(\tau_{x}(t))\not\in
Y$. Therefore, corresponding to $\tau_{x}$, we define a new trajectory
$\tau_{y}=\alpha\circ\tau_{x}$ which violates the positive invariance of $Y$.
This leads to a contradiction. ∎
Let $\varphi[\vec{y}]$ be an assertion representing an invariant of the system
$\mathcal{T}$ that simulates $\mathcal{S}$ through CoB transformation
$\alpha$. The assertion $\varphi[\vec{y}\mapsto\alpha(\vec{x})]$ obtained by
substituting $\alpha(\vec{x})$ in place of occurrences of $\vec{y}$ is an
invariant for the original system. In other words, inverting the map $\alpha$
simply boils down to substituting $\alpha(\vec{x})$ in the invariants of the
abstract system. An application of the Theorem above is illustrated in Example
1.1.
###### Example 2.1.
Consider a mechanical system $\mathcal{S}$ expressed in generalized position
coordinates $(q_{1},q_{2})$ and momenta $(p_{1},p_{2})$ defined using the
following vector field:
$F(p_{1},p_{2},q_{1},q_{2}):\left\langle\begin{array}[]{l}-2q_{1}q_{2}^{2},\
-2q_{1}^{2}q_{2},\ 2p_{1},\ 2p_{2}\end{array}\right\rangle$
with the initial conditions: $(p_{1},p_{2})\in[-1,1]\times[-1,1]\ \land\
(q_{1},q_{2}):(2,2)$. Using the transformation
$\alpha(p_{1},p_{2},q_{1},q_{2}):p_{1}^{2}+p_{2}^{2}+q_{1}^{2}q_{2}^{2}$, we
see that $\mathcal{S}$ is simulated by a linear system $\mathcal{T}$ over $y$,
with dynamics given by $\frac{dy}{dt}=0,\ y(0)\in[16,18]$.
Incidentally, the form of the system $\mathcal{T}$ above indicates that
$\alpha$ is an expression for a conserved quantity (in this case, the
Hamiltonian) of the system. $\blacktriangle$
The main goal of this work is to study CoB transformations that “simplify” the
system’s dynamics either (a) casting a non-algebraic vector field into one
defined algebraically or (b) reducing the degree of a given algebraic vector
field by means of an abstraction. A special case consists of _linearizing CoB
transformations_ that map a non-linear system to one defined by affine
dynamics.
Recall that a system $\mathcal{T}$ is algebraic if it is described by a
polynomial vector field. Furthermore, $\mathcal{T}$ is _affine_ if it is
described by an affine vector field $\frac{d\vec{y}}{dt}=A\vec{y}+\vec{b}$ for
an $m\times m$ matrix $A$ and an $m\times 1$ vector $\vec{b}$.
###### Definition 2.4.
Let $\mathcal{S}$ be a (non-linear) system. We say that $\alpha$ is an
_algebraizing CoB transformation_ if it maps $\mathcal{S}$ to an algebraic
system $\mathcal{T}$.
We say that $\alpha$ is a _linearizing CoB transformation_ if it maps each
trajectory of $\mathcal{S}$ to that of an affine system $\mathcal{T}$.
###### Example 2.2.
Consider the vector field $\mathcal{F}$
$\frac{dx}{dt}=x^{3}-2x^{2}+y^{2}+xy,\ \frac{dy}{dt}=2x-3x^{2}+2y^{3}\,.$
Let $\alpha:(x,y)\rightarrow(w_{1},w_{2},w_{3},w_{4})$ be defined as
$\alpha(x,y):(x,y,x^{2},y^{2})$
We can verify that using $\alpha$, we note that $\mathcal{F}$ is simulated by
the vector field $\mathcal{G}$:
$\begin{array}[]{ll}\frac{dw_{1}}{dt}=w_{1}w_{3}-2w_{3}+w_{4}+w_{1}w_{2},&\frac{dw_{2}}{dt}=2w_{1}-3w_{3}+2w_{2}w_{4}\\\
\frac{dw_{3}}{dt}=-4w_{1}w_{3}+2w_{3}^{2}+2w_{2}w_{3}+2w_{1}w_{4},&\frac{dw_{4}}{dt}=4w_{1}w_{2}-6w_{2}w_{3}+4w_{4}^{2}\\\
\end{array}$
Note that while $\mathcal{F}$ is a cubic vector field over $\mathbb{R}^{2}$,
$\mathcal{G}$ is a quadratic vector field over $\mathbb{R}^{4}$.
$\blacktriangle$
Example 1.1 illustrates a linearizing CoB transformation.
The above definition of an algebraizing or linearizing CoB seems useful, in
practice, only if $\alpha$ and $\mathcal{T}$ are already known. We may then
use known techniques for reasoning over algebraic systems or affine systems
for safely bounding the reachable set of an affine system, given some initial
conditions, and transform the result back through substitution to obtain a
bound on the reachable set for $\mathcal{S}$.
We now present a technique that searches for a map $\alpha$ to obtain an
algebraic system $\mathcal{T}$ that simulates a given system $\mathcal{S}$
through $\alpha$ such that the vector field describing $\mathcal{T}$ is degree
bounded by a given degree limit $d>0$. In particular, if the degree limit $d$
is set to $1$, then the resulting transformation $\alpha$ is linearizing.
We ignore the initial condition and invariant, for the time being, and simply
focus on obtaining the dynamics of $\mathcal{T}$. In other words, we will
search for a map ${\alpha}:\ (\alpha_{1},\ldots,\alpha_{m})$ that maps
$\mathbb{R}^{n}$ into $\mathbb{R}^{m}$ so that
$J_{\alpha}(\vec{x})\cdot\mathcal{F}(\vec{x})=G(\alpha(\vec{x}))\,.$
Having found such a map, we may find appropriate over-approximate initial and
invariance conditions for the simulating system $\mathcal{T}$, so that
Definition 2.3 holds. Specifically, we are interested in finding
transformations $\alpha$ that ensure that (a) $G$ is a polynomial vector field
and (b) the degrees of polynomials describing $G$ are degree bounded by the
degree limit $d>0$.
### 2.3 Multilinear Abstractions through Dimension Copying
We first show that any polynomial system of ODEs can be abstracted by a
_multilinear_ system. However, doing so may require $\alpha$ to have many
repeated components wherein $\alpha_{i}(\vec{x})=\alpha_{j}(\vec{x})$ for
$i\not=j$.
###### Definition 2.5.
A polynomial $p$ is defined to be _multilinear_ if and only if each power-
product in $p$ is of the form $x_{1}^{r_{1}}x_{2}^{r_{2}}\cdots x_{n}^{r_{n}}$
wherein each $r_{i}=0\ \mbox{or}\ 1$.
###### Example 2.3.
As an example, the polynomial $p=2x_{1}x_{2}x_{3}+x_{1}x_{3}+4x_{1}-2x_{2}-1$
is multilinear. On the other hand, the polynomial $q=2x_{2}^{2}+x_{1}+x_{3}$
is not, owing to the $x_{2}^{2}$ power product.
We first observe that any polynomial ODE may be equivalently written by means
of a multilinear system using a suitably defined $\alpha$.
###### Theorem 2.3.
Let $\mathcal{F}$ be a polynomial vector field over
$\vec{x}\in\mathbb{R}^{n}$. There is a transformation
$\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, that maps $\mathcal{F}$ to a
multilinear system $\mathcal{G}$.
###### Proof.
Let us write $\mathcal{F}(\vec{x}):(p_{1},\ldots,p_{n})$ for multivariate
polynomials $p_{1},\ldots,p_{n}$. We will assume that the vector field
$\mathcal{F}$ is not already multi-linear. Therefore, some $p_{j}$ has a power
product that is divisible $x_{k}^{r}$ for some $r\geq 2$. The idea is to use
$r$ different functions $\alpha_{k,1}=\alpha_{k,2}=\cdots=\alpha_{k,r}=x_{k}$
so that in the transformed system the term $x_{k}^{r}$ appears as a
multilinear product $y_{k,1}y_{k,2}\cdots y_{k,r}$.
In the worst case, the transformation $\alpha$ involves $n\times K$
components, wherein
$K=\max(\mathsf{degree}(p_{1}),\ldots,\mathsf{degree}(p_{n}))\,.$
Each component $\alpha_{i,k}:x_{i}$ is simply a “copy” of the variable $x_{i}$
that ensures multilinearity of the transformed system. ∎
###### Example 2.4.
Consider the one dimensional system defined by
$\frac{dx}{dt}=2x^{5}+3x^{2}+x-5\,.$
We use the transformation $\alpha:\mathbb{R}\rightarrow\mathbb{R}^{5}$ wherein
$\alpha_{1}(x)=\alpha_{2}(x)=\cdots=\alpha_{5}(x)=x$. Using this
transformation, we derive an abstract system defined by the ODE
$\frac{dy_{j}}{dt}=2y_{1}y_{2}y_{3}y_{4}y_{5}+3y_{1}y_{2}+y_{1}-5\,,\
j=1,2,\ldots,5.$
$\blacktriangle$
Even though there are efficient algorithms for analyzing multi-linear systems
[4], the transformation in Theorem 2.3 faces two potential problems: (a) the
dimensionality of the transformed system $\mathcal{T}$ can be as large as the
dimensionality of the original system times the maximum degree of the
polynomials in the RHS of the vector field, and (b) ignoring the implicit
equality relationships between the various dimensions results in a very coarse
abstraction while taking them into account simply gives us the original system
back (albeit in a different form).
### 2.4 Independent Transformations
The rest of this paper, will focus on _independent transformations_
$\alpha:(\alpha_{1},\ldots,\alpha_{N})$ wherein each $\alpha_{i}$ cannot be
written as a linear combination of the remaining $\alpha_{j}$s for $j\not=i$.
Assuming independence automatically rules out the constructions used in
Theorem 2.3.
In general, computing independent transformations $\alpha$ for any given ODE
is a hard problem. In this paper, we will focus on solutions that involve
searching for an appropriate map $\alpha$, wherein $\alpha$ is specified to be
the linear combination of some fixed, finite set of basis functions
$g_{1},\ldots,g_{N}$. The initial basis is assumed to be given to our
algorithm by the user. Starting from this initial basis of functions, our
algorithm searches for transformations $\alpha$ whose components can be
written as linear combinations $\sum_{i=1}^{N}\lambda_{j}g_{j}$.
The basis functions could be specified implicitly as the set of all power
products over $\vec{x}$ of degree up to some limit $K>0$ or the set of all
power products involving the variables $x_{i}$ and various non-algebraic
functions $\sin(z),\cos(z)$ and $e^{z}$ applied to these power products.
Having chosen a basis $B=\\{g_{1},\ldots,g_{N}\\}$ for $\alpha$, we will cast
the search for the map $\alpha$ as a vector space iteration.
Let $\alpha(\vec{x}):(\alpha_{1}(\vec{x}),\ldots,\alpha_{m}(\vec{x}))$ be a
smooth mapping $\alpha:\mathbb{R}^{n}\mapsto\mathbb{R}^{m}$, wherein each
$\alpha_{i}:\mathbb{R}^{n}\mapsto\mathbb{R}$. Recall that
$\mathcal{L}_{F}(\alpha_{i}(\vec{x}))=(\nabla\alpha_{i})\cdot\mathcal{F}(\vec{x})$
denotes the Lie derivative of the function $\alpha_{i}(\vec{x})$ w.r.t vector
field $\mathcal{F}$.
###### Lemma 2.1.
$J_{\alpha}\cdot\mathcal{F}(\vec{x})=\left(\begin{array}[]{c}\mathcal{L}_{F}(\alpha_{1}(\vec{x}))\\\
\mathcal{L}_{F}(\alpha_{2}(\vec{x}))\\\ \vdots\\\
\mathcal{L}_{F}(\alpha_{m}(\vec{x}))\\\ \end{array}\right)$.
###### Proof.
Recall the definition of the Jacobian matrix $J_{\alpha}$:
$J_{\alpha}(x_{1},\ldots,x_{n})=\left[\begin{array}[]{rcl}\frac{\partial
y_{1}}{\partial x_{1}}&\cdots&\frac{\partial y_{1}}{\partial x_{n}}\\\
\vdots&\ddots&\vdots\\\ \frac{\partial y_{m}}{\partial
x_{1}}&\cdots&\frac{\partial y_{m}}{\partial x_{n}}\\\
\end{array}\right]=\left[\begin{array}[]{c}\nabla\alpha_{1}\\\ \vdots\\\
\nabla\alpha_{m}\\\ \end{array}\right]\,.$
Therefore,
$J_{\alpha}.\mathcal{F}=\left(\begin{array}[]{c}(\nabla\alpha_{1})\cdot(\mathcal{F})\\\
(\nabla\alpha_{2})\cdot(\mathcal{F})\\\ \vdots\\\
(\nabla\alpha_{m})\cdot(\mathcal{F})\\\
\end{array}\right)=\left(\begin{array}[]{c}\mathcal{L}_{F}(\alpha_{1}(\vec{x}))\\\
\mathcal{L}_{F}(\alpha_{2}(\vec{x}))\\\ \vdots\\\
\mathcal{L}_{F}(\alpha_{m}(\vec{x}))\\\ \end{array}\right)$. ∎
##### Note:
For the rest of this section, we will fix a vector field $\mathcal{F}$
belonging to a system $\mathcal{S}$ as the original system for which we seek
an abstraction. We will simply write $\mathcal{L}(g)$ to denote the Lie-
derivative of a given function $g$ in place of $\mathcal{L}_{F}(g)$.
### 2.5 Vector Space Closure
We first define the vector spaces that will be used in our search.
###### Definition 2.6.
Let $B=\\{g_{1},\ldots,g_{k}\\}$ be some finite set of functions wherein
$g_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ for some fixed $n,m>0$. The
_vector space_ spanned by $G$ denoted $\mathit{Span}(B)$ consists of all
functions that are linear combinations of $g_{i}$:
$\mathit{Span}(B)=\left\\{\sum_{i=1}^{k}\lambda_{i}g_{i}\ |\
\lambda_{i}\in\mathbb{R}\right\\}\,.$
We assume, without loss of generality, that the elements in $B$ are linearly
independent. I.e., no $g_{i}\in B$ can be written as a linear combination of
the remaining $g_{j}\in B$, for $j\not=i$.
Let $\mathbf{1}$ represent the constant function
$\mathbf{1}(\vec{x})=\vec{1}\in\mathbb{R}^{m}$. Given a vector space
$V=\mathit{Span}(B)$, we define the space of power products of $V$ up to a
degree limit $d\geq 1$ as
$V^{\left\langle d\right\rangle}=\mathit{Span}\left(\left\\{g_{i_{1}}\times
g_{i_{2}}\times\cdots\times g_{i_{d}}\ |\ g_{i_{1}},\ldots,g_{i_{d}}\in
B\cup\\{\mathbf{1}\\}\right\\}\right)\,.$
In particular, note that $V^{\left\langle
1\right\rangle}=\mathit{Span}(V\cup\\{\mathbf{1}\\})$.
###### Example 2.5.
Let $B=\\{x,\sin(y)\\}$ be our basis set. The vector space $V:\
\mathit{Span}(B)$ is given by $\\{a_{1}x+a_{2}\sin(y)\ |\
a_{1},a_{2}\in\mathbb{R}\\}$. The space $V^{\left\langle 2\right\rangle}$ is
the set
$\left\\{a_{0}+a_{1}x+a_{2}\sin(y)+a_{3}x\sin(y)+a_{4}x^{2}+a_{5}\sin^{2}(y)\
|\ a_{0},\ldots,a_{5}\in\mathbb{R}\right\\}\,.$
This space is generated by the functions
$\mathbf{1},x,\sin(y),x\sin(y),x^{2},\sin^{2}(y)$. It consists of all
polynomials of degree at most $2$ formed by the functions $x$, $\sin(y)$. The
purpose of adding the function $\mathbf{1}$ is to enable terms of degree $1$
and $0$ to be considered. $\blacktriangle$
Roughly, the main idea behind our approach is to find a vector space $U$ that
satisfies the following closure property:
$(\forall\ f\in U)\ \mathcal{L}(f)\in U^{\left\langle d\right\rangle}\,.$
In other words, we will search for a vector space $U$, such that taking the
Lie derivative of any element of $U$ yields an element in $U^{\left\langle
d\right\rangle}$. Such a vector space $U$ will be called $d-\mbox{closed}$.
Let $U=\mathit{Span}\left(\left\\{h_{1},\ldots,h_{m}\right\\}\right)$ be a
$d-\mbox{closed}$ vector space. We will prove that $\alpha:\
(h_{1},\ldots,h_{m})$ maps the original system $\mathcal{S}$ to an algebraic
system $\mathcal{T}$ with a vector field of degree at most $d$.
###### Definition 2.7.
A vector space $V$ is said to be $d-\mbox{closed}$ under the application of
Lie derivatives iff $(\forall\ f\in V)\ \mathcal{L}(f)\in V^{\left\langle
d\right\rangle}$.
In order to check whether a given space $V=\mathit{Span}(B)$ is
$d-\mbox{closed}$, it suffices to verify the property in Definition 2.7 for
the elements in $B$.
###### Lemma 2.2.
A vector space
$U=\mathit{Span}\left(\left\\{h_{1},\ldots,h_{m}\right\\}\right)$ be
$d-\mbox{closed}$ under Lie derivatives if and only if $\mathcal{L}(h_{i})\in
U^{\left\langle d\right\rangle}$ for $i\in\\{1,\ldots,m\\}$.
###### Proof.
If $U$ is $d-\mbox{closed}$ under Lie derivatives then by definition, the Lie
derivatives of its basis elements $h_{i}$ should lie in $U^{\left\langle
d\right\rangle}$. We will prove the reverse direction. Let $U$ be such that
for each basis element $h_{i}$, we have $\mathcal{L}(h_{i})\in U^{\left\langle
d\right\rangle}$. Any element of $U$ can be written as
$f=\sum_{j=1}^{k}a_{j}h_{j}$ for $a_{j}\in\mathbb{R}$. We have
$\mathcal{L}(f)=\sum_{j=1}^{k}a_{j}\mathcal{L}(h_{j})$. Since each
$\mathcal{L}(h_{j})\in U^{\left\langle d\right\rangle}$, we have that
$\mathcal{L}(f)\in U^{\left\langle d\right\rangle}$. This completes the proof.
∎
Next, we relate $d-\mbox{closed}$ vector spaces to algebraizing CoB
transformations. Let $B=\left\\{h_{1},\ldots,h_{m}\right\\}$ and
$U=\mathit{Span}\left(B\right)$ be a $d-\mbox{closed}$ vector space. Let
$\alpha$ be the map from $\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ defined as
$\alpha:(h_{1},\ldots,h_{m})$.
###### Theorem 2.4.
The map $\alpha$ formed by the basis elements of a $d-$closed vector field is
an algebraizing transformation from the original system $\mathcal{S}$ to a
system $\mathcal{T}$ defined by a polynomial vector field of degree at most
$d$.
###### Proof.
Since $U$ is $d-\mbox{closed}$, we note that for each $h_{i}$ in the basis of
$U$, we have $\mathcal{L}(h_{i})\in U^{\left\langle d\right\rangle}$. In other
words, we may write $\mathcal{L}(h_{i})$ as a linear combination of power
products as shown below:
$\mathcal{L}(h_{i}):\ \sum_{j=1}^{K}a_{ij}h_{i,j,1}\times
h_{i,j,2}\times\cdots\times h_{i,j,d}\,,\ \mbox{wherein}\ h_{i,j,k}\in
B\cup\\{\mathbf{1}\\}$ (1)
We define the system $\mathcal{T}$ over variables $y_{1},\ldots,y_{m}$. We
will use variable $y_{i}$ to correspond to $h_{i}(\vec{x})$. The dynamics are
obtained as
$\frac{dy_{i}}{dt}=\sum_{j=1}^{K}a_{ij}y_{i_{1}}\times
y_{i_{2}}\times\cdots\times y_{i_{k}}\,,$
by substituting the variable $y_{j}$ wherever the function $h_{j}$ occurs in
Equation (1). Let $G$ be the resulting vector field on $\vec{y}$. It is easy
to see that (a) $G$ is a polynomial vector field and (b) of degree at most
$d$.
From Lemma 2.1, we note that
$J_{\alpha}\mathcal{F}(\vec{x})=(\mathcal{L}(h_{1}),\ldots,\mathcal{L}(h_{m}))$.
We verify that
$(\mathcal{L}(h_{1}),\ldots,\mathcal{L}(h_{m}))=G(h_{1}(\vec{x}),\ldots,h_{m}(\vec{x}))$.
This is directly evident from the construction of $G$ from Equation (1). Thus,
the key condition (3) of Theorem 2.1 is seen to hold. By finding the right
sets $Y_{0},Y_{I}$ given $\alpha$, we take care of the remaining conditions as
well. ∎
##### Note:
The trivial space $V=\mathit{Span}(\\{0\\})$ consisting of the constant
function that maps all inputs to $\vec{0}$ is always $d-$closed. This space
yields $\alpha:(0)$ that maps all states $\vec{x}$ to the zero vector. As
such, the map $\alpha$ is not very useful in practice for inferring
invariants.
###### Example 2.6.
Consider the ODE from Example 1.1 recalled below:
$\begin{array}[]{rcl}\frac{dx}{dt}&=&xy+2x\\\
\frac{dy}{dt}&=&-\frac{1}{2}y^{2}+7y+1\\\ \end{array}$
We claim that the vector space $V$ generated by the set of functions
$\\{x,xy,xy^{2}\\}$ is $1-$closed. To verify, we compute the Lie derivative of
a function of the form $c_{1}x+c_{2}xy+c_{3}xy^{2}$ to obtain
$c_{1}(xy+2x)+c_{2}(\frac{1}{2}xy^{2}+9xy+x)+c_{3}(16xy^{2}+2xy)$
which is seen to belong to $V^{\left\langle 1\right\rangle}$. As a result, we
obtain the CoB abstraction $\alpha(x,y):(x,xy,xy^{2})$ that maps the vector
field to an affine vector field (polynomial of degree $1$).
The abstract system over $(w_{1},w_{2},w_{3})\in\mathbb{R}^{3}$ has dynamics
given by
$\begin{array}[]{rcl}\frac{dw_{1}}{dt}&=&2w_{1}+w_{2}\\\
\frac{dw_{2}}{dt}&=&\frac{1}{2}w_{3}+9w_{2}+w_{1}\\\
\frac{dw_{3}}{dt}&=&16w_{3}+2w_{2}\\\ \end{array}$
The mapping between original and abstract system is given by
$w_{1}\ \mapsto\ x,\ w_{2}\ \mapsto\ xy,\ w_{3}\ \mapsto\ xy^{2}\,.$
$\blacktriangle$
### 2.6 Finding Closed Vector Spaces
We will now describe a search technique for finding a map $\alpha$ and the
associated abstraction $\mathcal{T}$, such that the dynamics of $\mathcal{T}$
are described by polynomials with degree bound $d$. If $d=1$, the dynamics of
$\mathcal{T}$ are affine. The inputs to our search procedure are
1. 1.
The original system $\mathcal{S}$ described by a vector field $\mathcal{F}$,
2. 2.
The degree limit $d$ for the desired vector field $\mathcal{T}$, and
3. 3.
An initial basis $B_{0}=\\{h_{1},\ldots,h_{N}\\}$ of continuous and
differentiable functions. We may regard the linear combination
$c_{1}h_{1}(\vec{x})+c_{2}h_{2}(\vec{x})+\ldots+c_{N}h_{N}(\vec{x})\,,$
as an _ansatz_ or a template for each component $\alpha_{j}$ of the map
$\alpha:(\alpha_{1},\ldots,\alpha_{m})$, that we are searching for. However,
we do not fix the number of components $m$ of the transformation $\alpha$,
_apriori_ , or guarantee that a non-trivial $\alpha$ (with $m>0$) can be
found.
The initial basis $B_{0}$ is often specified as consisting of all power
products of the variables in $\vec{x}$ with a given degree limit $M$. This
limit $M$ is chosen independent of the limit $d$ for the desired abstraction
$\mathcal{T}$.
Our overall approach is to start with the initial vector space
$V_{0}:\mathit{Span}(B_{0})$ and iteratively refine $V_{0}$ to construct a
sequence of vector spaces
$V_{0}\supseteq V_{1}\supseteq V_{2}\cdots\supseteq V_{k}=V_{k+1}=V^{*}$
wherein, (1) $V_{j+1}\subseteq V_{j}$, for $j\in[1,k-1]$, and (2)
$V_{k}=V_{k+1}$. The iterative scheme is designed to guarantee that the
converged result $V^{*}$ is $d-$ closed. If $V^{*}$ has a non-zero basis, then
the basis elements of $V^{*}$ form the components of the map $\alpha$ and the
abstraction $\mathcal{T}$ whose dynamics have the desired form.
The main step of iteration is to derive $V_{i+1}$ from $V_{i}$. This is
performed as follows:
$~{}V_{i+1}=\\{g\in V_{i}\ |\ \mathcal{L}(g)\in V_{i}^{\left\langle
d\right\rangle}\\}\,.$ (2)
In other words, $V_{i+1}$ retains those functions $g\in V_{i}$ whose Lie
derivatives also lie inside $V_{i}^{\left\langle d\right\rangle}$.
###### Lemma 2.3.
(1) $V_{i+1}$ is a sub-space of $V_{i}$. (2) $V_{i}$ is $d-$closed iff
$V_{i}=V_{i+1}$.
###### Proof.
We prove the two parts (1) and (2) as follows.
(1) Since by Eq. (2), $V_{i+1}\subseteq V_{i}$, it suffices to show that
$V_{i+1}$ is a vector space. Let $g_{1},\ldots,g_{k}\in V_{i+1}$. We have that
$g_{1},\ldots,g_{k}\in V_{i}$. Furthermore, since $V_{i}$ is a vector space,
any linear combination $g:\ \sum_{j=1}^{k}\lambda_{j}g_{j}\in V_{i}$. The lie
derivative $\mathcal{L}(g)$ can be written as
$\sum_{j=1}^{k}\lambda_{j}\mathcal{L}(g_{j})$. Since $\mathcal{L}(g_{j})\in
V_{i}^{\left\langle d\right\rangle}$, we have
$\mathcal{L}(g)=\sum_{j=1}^{k}\lambda_{j}\mathcal{L}(g_{j})\in
V_{i}^{\left\langle d\right\rangle}$. Therefore, by definition $g\in V_{i+1}$
as well. The linear combination of any finite subset of elements from
$V_{i+1}$ also belongs to $V_{i+1}$, proving that it is a sub-space of
$V_{i}$.
(2) If $V_{i}=V_{i+1}$, it is easy to check that $V_{i}$ satisfies the
definition of being $d-$ closed. For the other direction, let us assume that
$V_{i}$ is $d-$closed. Then for each $g\in V_{i}$, we have $\mathcal{L}(g)\in
V_{i}^{\left\langle d\right\rangle}$. Thus $g\in V_{i+1}$. This proves that
$V_{i+1}\supseteq V_{i}$. Combining with the fact that $V_{i+1}\subseteq
V_{i}$, we obtain equality. ∎
We now focus on calculating $V_{i+1}$ from $V_{i}$. Let $V_{i}:\
\mathit{Span}(B_{i})$ for a finite set $B_{i}$. Any element of $V_{i}$ can be
represented as $\sum_{h_{j}\in B_{i}}c_{j}h_{j}$ for some multipliers $c_{j}$.
The Lie derivative is expressed as $\sum_{h_{j}\in
B_{i}}c_{j}\mathcal{L}(h_{j})$. The procedure for calculating $V_{i+1}$
reduces to finding the set of multipliers $(c_{1},\ldots,c_{M})$ where
$M=|B_{i}|$ such that $\sum_{h_{j}\in B_{i}}c_{j}\mathcal{L}(h_{j})\in
V_{i}^{\left\langle d\right\rangle}$.
The key challenge lies in comparing two elements of the form
$\sum_{j}c_{j}\mathcal{L}(h_{j})$ and $\sum_{k}d_{k}g_{k}$, for unknowns
$c_{j}$ and $d_{k}$, where $h_{j}\in B_{i}$ and $g_{k}\in V_{i}^{\left\langle
d\right\rangle}$. If both the functions are polynomials over $\vec{x}$, the
comparison is performed by equating the coefficients of corresponding
monomials. This is illustrated using the example below:
###### Example 2.7.
Consider once again the ODE from Example 1.1 and 2.6. We seek to find an
affine system $\mathcal{T}$ that abstracts this system. Let us consider the
space $V_{0}$ generated by the basis $B_{0}:\\{x,y,xy,x^{2},y^{2}\\}$ of all
degree $2$ monomials. Any element in $V_{0}$ can be written as
$p(c_{1},\ldots,c_{5}):\ c_{1}x+c_{2}y+c_{3}xy+c_{4}x^{2}+c_{5}y^{2}\,.$
Its Lie derivative is given by
$\begin{array}[]{l}c_{1}(xy+2x)+c_{2}(-\frac{1}{2}y^{2}+7y+1)+c_{3}x(-\frac{1}{2}y^{2}+7y+1)\\\
+c_{3}y(xy+2x)+c_{4}(2x)(xy+2x)+c_{5}(2y)(-\frac{1}{2}y^{2}+7y+1)\end{array}$
This can be simplified as
$p^{\prime}(c_{1},\ldots,c_{5}):\
\left[\begin{array}[]{l}c_{2}+(2c_{1}+c_{3})x+(7c_{2}+2c_{5})y+(c_{1}+9c_{3})xy+4c_{4}x^{2}+\\\
(14c_{5}-\frac{1}{2}c_{2})y^{2}+\frac{1}{2}c_{3}xy^{2}+2c_{4}x^{2}y-c_{5}y^{3}\end{array}\right]\,.$
We require the Lie derivative to belong to $V^{\left\langle
1\right\rangle}=\mathit{Span}(B_{0}\cup\\{1\\})$. This yields the constraints:
$(\exists d_{0},d_{1},\ldots,d_{5})\ (\forall\ x,y)\
d_{0}+d_{1}x+d_{2}y+d_{3}xy+d_{4}x^{2}+d_{5}y^{2}=p^{\prime}(c_{1},\ldots,c_{5})\,.$
We use the lemma that two polynomials are identical iff their coefficients on
corresponding power-products are. This yields the following system of linear
equations:
$\begin{array}[]{l}c_{2}=d_{0},\ 2c_{1}+c_{3}=d_{1},\ 7c_{2}+2c_{5}=d_{2},\
c_{1}+9c_{3}=d_{3},\\\ 4c_{4}=d_{4},14c_{5}-\frac{1}{2}c_{2}=d_{5},\ c_{3}=0,\
2c_{4}=0,\ c_{5}=0\\\ \end{array}$
Eliminating $d_{0},\ldots,d_{5}$, we obtain the constraints
$c_{3}=c_{4}=c_{5}=0$. The new basis $B_{1}$ is $\\{x,y\\}$. $\blacktriangle$
On the other hand, if the basis $B_{i}$ involves non-polynomials
(trigonometric or exponential functions), then encoding equality by matching
up coefficients of syntactically identical terms is _incomplete_ : I.e, not
all solutions can be found by equating coefficients of matching terms. In
general, deciding if two expressions involving trigonometric functions is
identically zero is undecidable 111 This follows from Richardson’s theorem
[27].. In practice, we may continue to handle trigonometric functions using
the same syntactic matching technique that is complete for polynomials. If a
$d-$closed basis is discovered this way, then it may be used to derive a valid
abstraction. On the other hand, the process may be unable to find a vector
space starting from the initial set of functions even if one such exists.
###### Example 2.8.
Consider a simple example with the ODE
$\frac{dx}{dt}=\sin(x+y),\ \ \ \frac{dy}{dt}=x+y\,.$
Consider the space $V$ spanned by the basis
$B=\\{x,y,\sin(x),\sin(y),\cos(x),\cos(y)\\}\,.$
Our goal is to check if $V$ is $3-$closed. Any element of $V$ can be written
as
$c_{1}x+c_{2}y+c_{3}\sin(x)+c_{4}\sin(y)+c_{5}\cos(x)+c_{6}\cos(y)\,.$
Its Lie derivative can be written as
$\begin{array}[]{c}c_{1}\sin(x+y)+c_{2}(x+y)+c_{3}\cos(x)\sin(x+y)+c_{4}\cos(y)(x+y)\\\
-c_{5}\sin(x)\sin(x+y)-c_{6}\sin(y)(x+y)\end{array}\,.$
Our goal is to check if the Lie derivative belongs to $V^{\left\langle
3\right\rangle}$. We note that a syntactic check for membership yields the
constraints $c_{1}=c_{3}=c_{5}=0$. On the other hand, substituting the
trigonometric identity
$\sin(x+y)\equiv\sin x\cos y+\sin y\cos x\,,$
we may indeed verify that the Lie derivative of any element of $V$ belongs to
$V^{\left\langle 3\right\rangle}$. This yields a degree $3$ algebraization
given by $\alpha(x,y):(x,y,\sin(x),\sin(y),\cos(x),\cos(y))$ with the abstract
system having the dynamics
$\begin{array}[]{rcl}\frac{dw_{1}}{dt}&=&w_{3}w_{6}+w_{4}w_{5}\\\
\frac{dw_{2}}{dt}&=&w_{1}+w_{2}\\\
\frac{dw_{3}}{dt}&=&w_{3}w_{5}w_{6}+w_{5}^{2}w_{4}\\\
\frac{dw_{4}}{dt}&=&w_{6}w_{1}+w_{6}w_{2}\\\
\frac{dw_{5}}{dt}&=&-w_{3}^{2}w_{6}-w_{3}w_{4}w_{5}\\\
\frac{dw_{6}}{dt}&=&-w_{4}w_{1}-w_{4}w_{2}\\\ \end{array}$
Here $w_{1},\ldots,w_{6}$ correspond to the components of the map $\alpha$
above. $\blacktriangle$
###### Theorem 2.5.
Given an initial vector space $V_{0}$ and vector field $\mathcal{F}$, the
iterative procedure using Eqn. (2) converges in finitely many steps to a
subspace $V^{*}\subseteq V_{0}$. Let $\alpha_{1},\ldots,\alpha_{m}$ be the
basis functions that generate $V^{*}$.
1. 1.
The transformation $\alpha:(\alpha_{1},\ldots,\alpha_{m})$ generated by the
basis functions of the final vector space leads to an abstract system whose
dynamics are described by polynomials of degree at most $d$.
2. 2.
For every CoB transformation $\beta:(\beta_{1},\ldots,\beta_{k})$, wherein
each $\beta_{i}\in V_{0}$ and $\beta$ yields a polynomial abstraction of
degree at most $d$, it follows that $\beta_{i}\in V^{*}$.
###### Proof.
Let us represent the iterative sequence as
$V_{0}\supseteq V_{1}\supseteq V_{2}\cdots$
The convergence of the iteration follows from the observation that if
$V_{i+1}\subset V_{i}$, the dimension of $V_{i+1}$ is at least one less than
that of $V_{i}$. Since $V_{0}$ is finite dimensional, the number of iterations
is upper bounded by the number of basis functions in $V_{0}$.
Statement 1 follows directly from Theorem 2.4.
Finally, us assume that a transformation $\beta$ exists such that
$\beta_{i}\in V_{0}$. We note that the space $U$ generated by
$\mathbf{1},\beta_{1},\ldots,\beta_{k}$ is a subset of $V_{0}$ and is
$d-$closed. We can now prove by induction that $U\subseteq V_{i}$ for each
$i$. The base case is true since $U\subseteq V_{0}$.
Next, we show that if $U\subseteq V_{i}$ then $U\subseteq V_{i+1}$. This
follows from Eq. 2 since for each $p\in U$, we have $p\in V_{i}$ and
$\mathcal{L}(p)\in U^{\left\langle d\right\rangle}$. This gives us
$\mathcal{L}(p)\in V_{i}^{\left\langle d\right\rangle}$. Therefore, $p\in
V_{i+1}$.
As a result, we prove by induction that $U\subseteq V_{i}$ for each $i$. This
also means that $U\subseteq V^{*}$. ∎
Note that it is possible for the converged result $V^{*}$ to be trivial. I.e,
it is generated by the constant function $\mathbf{1}$.
###### Example 2.9.
Consider the Vanderpol oscillator whose dynamics are given by
$\dot{x}=y,\ \dot{y}=\mu(y-\frac{1}{3}y^{3}-x)\,.$
Our search for polynomials ($\mu=1$) of degree up to 20 did not yield a non-
trivial linearizing transformation.
For a trivial system, the resulting affine system $\mathcal{T}$ is
$\frac{dy}{dt}=0$ under the map $\alpha(\vec{x})=0$. Naturally, this situation
is not quite interesting but will often result, depending on the system
$\mathcal{S}$ and the initial basis chosen $V_{0}$. We now discuss common
situations where the vector space $V^{*}$ obtained as the result is guaranteed
to be non-trivial.
### 2.7 Strong and Constant Scale Consecution
The notion of “strong” consecution, “constant scale” consecution and
“polynomial scale” consecution were defined for equality invariants of
differential equations in our previous work [38] and subsequently expanded
upon by Matringe et al. [22] using the notion of morphisms. We now show that
the techniques presented in this section can generalize strong and constant
scale consecutions, ensuring that all the systems handled by the techniques
presented in our previous work [38] can be handled by the techniques here (but
not vice-versa).
###### Definition 2.8.
A function $f$ satisfies the _strong scale_ consecution requirement for a
vector field $\mathcal{F}$ iff $\mathcal{L}_{F}(f)=0$. In other words, $f$ is
a conserved quantity. Similarly, $f$ satisfies the _constant scale_
consecution iff $\exists\lambda\in\mathbb{R},\ \mathcal{L}_{F}(f)=\lambda f$.
The following theorem is a corollary of Theorem 2.5 and shows that the ideas
presented in this section can capture the notion of strong and constant scale
consecution without requiring quantifier elimination, solving an eigenvalue
problem [38] or finding roots of a univariate polynomial [22].
###### Theorem 2.6.
The result of the iteration $V^{*}$ starting from an initial space $V_{0}$
contains all the strong and constant scale invariant functions in $V_{0}$.
###### Proof.
This is a direct consequence of Theorem 2.5 by noting that for a constant
scale consecuting function $f$, the subspace $U\subseteq V_{0}$ spanned by $f$
is closed under Lie derivatives. ∎
Furthermore, if such functions exist in $V_{0}$ the result after convergence
$V^{*}$ is guaranteed to be a non-trivial vector space (of positive
dimension). Finally, constant scale and strong scale functions can be
extracted by computing the affine equality invariants of the linear system
$\mathcal{T}$ that can be extracted from $V^{*}$.
#### 2.7.1 Stability
We briefly address the issue of deducing stability (or instability) of a
system $\mathcal{S}$ using an abstraction to a system $\mathcal{T}$. Since
$\alpha$ satisfies the identity
$\mathcal{G}(\alpha(\vec{x}))=J_{\alpha}.\mathcal{F}(\vec{x})\,.$
Every equilibrium of $\mathcal{S}$ ($\mathcal{F}(\vec{x})=0$) maps onto an
equilibrium of $\mathcal{T}$ ($\mathcal{G}(\vec{x})=0$), but not vice-versa.
Furthermore, the map $\alpha(\vec{x})=(\mathbf{0},\ldots,\mathbf{0})$ is an
abstraction from any non-linear system to one with an equilibrium at origin.
Therefore, unless restrictions are placed on $\alpha$, we are unable to draw
conclusions on liveness properties for $\mathcal{S}$ based on $\mathcal{T}$.
If $\alpha$ has a continuous inverse, then $\mathcal{T}$ is topologically
diffeomorphic to $\mathcal{S}$ [23]. This allows us to correlate equilibria of
$\mathcal{T}$ with those of $\mathcal{S}$. The preservation of stability under
mappings of state variables has been studied by Vassilyev and Ul’yanov [41].
We are currently investigating restrictions that will allow us to draw
conclusions about liveness properties of $\mathcal{S}$ from those of
$\mathcal{T}$.
The issue of stability preserving maps between continuous and hybrid systems
was recently addressed by the work of Prabhakar et al. [30].
### 2.8 Affine CoB Abstraction: Existence
We will now focus on the special case of CoB transformations that lead to
linear abstractions of the form $\frac{d\vec{w}}{dt}=A\vec{w}$ (and affine
abstractions of the form $\frac{d\vec{w}}{dt}=A\vec{w}+\vec{b}$).
Let $\mathcal{S}$ be a non-linear system over $\vec{x}$ that has a CoB
transformation $\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ with $m>0$
that maps to a linear system $\frac{d\vec{w}}{dt}=A\vec{w}$.
###### Lemma 2.4.
The system $\mathcal{S}$ has $m$ conserved quantities given by the components
of the vector valued function $e^{-tA}\alpha(\vec{x})$.
###### Proof.
Our goal is to prove that the Lie derivative of each component of
$e^{-tA}\alpha(\vec{x})$ equals zero. Since $\alpha$ is a linearizing CoB, we
have $\mathcal{L}(\alpha(\vec{x}))=A\alpha(\vec{x})$.
The Lie derivative of $e^{-tA}\alpha(\vec{x})$ is given by
$e^{-tA}\mathcal{L}(\alpha(\vec{x}))+\partial_{t}e^{-tA}\alpha(\vec{x})=e^{-tA}A\alpha(\vec{x})-e^{-tA}A\alpha(\vec{x})=0\,.$
Thus we see that the Lie derivative of $e^{-tA}\alpha(\vec{x})$ vanishes.
Therefore, each component of $e^{-tA}\alpha(\vec{x})$ is a conserved quantity.
∎
Conversely, whenever the original system $\mathcal{S}$ has conserved
quantities, it trivially admits the linearization $\frac{d\vec{w}}{dt}=0$
using a transformation $\alpha$ that is formed by its conserved quantity.
###### Theorem 2.7.
A system $\mathcal{S}$ has an independent, linearizing CoB transformation
$\alpha:\mathbb{R}^{n}\mapsto\mathbb{R}^{m}$ if and only if it has $m$
linearly independent conserved quantities.
The theorem extends to affine CoB transformations that yield abstract systems
of the form $\frac{d\vec{w}}{dt}=A\vec{w}+\vec{b}$. While conservative
mechanical and electromagnetic systems naturally have conserved quantities
(eg., conservation of momentum, energy, charge, mass), many systems
encountered are dissipative. Such cases are handled by extending the approach
presented here to differential inequality abstractions [34].
Furthermore, even in a setting where conservative quantities exist, the
advantages of searching for a CoB transformation as opposed to directly
searching for a conserved quantity from an ansatz are not clear at a first
glance. The advantage of the techniques presented here lies in the fact that
existing techniques that search for conserved quantities focus for the most
part on finding polynomial conserved quantities. Whereas, searching for a CoB
transformation allows us to implicitly obtain conserved quantities that may
involve exponentials, sines and cosines in addition to polynomial conserved
quantities by focusing purely on reasoning with vector spaces generated by
polynomials.
###### Example 2.10.
We observed the following conserved quantity for the system in Example 1.1
$\begin{array}[]{l}\left(\frac{e^{-9t}}{51}+\frac{1}{102}\left(50+7\sqrt{51}\right)e^{\left(-9+\sqrt{51}\right)t}+\frac{1}{102}\left(50-7\sqrt{51}\right)e^{-\left(9+\sqrt{51}\right)t}\right)\
x+\\\
\left(-\frac{1}{102}e^{-9t-\left(9+\sqrt{51}\right)t}\left(\begin{array}[]{l}7e^{9t}-\sqrt{51}e^{9t}-14e^{\left(9+\sqrt{51}\right)t}+\\\
7e^{9t+\left(-9+\sqrt{51}\right)t+\left(9+\sqrt{51}\right)t}+\\\
\sqrt{51}e^{9t+\left(-9+\sqrt{51}\right)t+\left(9+\sqrt{51}\right)t}\end{array}\right)\right)\
xy+\\\
\left(\frac{1}{204}e^{-9t-\left(9+\sqrt{51}\right)t}\left(e^{9t}-2e^{\left(9+\sqrt{51}\right)t}+e^{9t+\left(-9+\sqrt{51}\right)t+\left(9+\sqrt{51}\right)t}\right)\right)xy^{2}\end{array}$
This is one of the three conserved quantities obtained by computing
$e^{-tA}\alpha(\vec{x})$, where
$\alpha:(x,xy,xy^{2})\ \mbox{and}\ A=\left(\begin{array}[]{ccc}2&1&0\\\
1&9&\frac{1}{2}\\\ 0&2&16\\\ \end{array}\right)\,.$
We are unaware of techniques that can directly generate such conserved
quantities. $\blacktriangle$
Finally, we conclude by noting that conserved quantities such as the one
described above seem less useful for reasoning about the dynamics of the
underlying system when compared to the CoB transformation and the resulting
abstraction that gave rise to them.
## 3 Abstractions for Discrete and Hybrid Systems
In this section, we will discuss how the techniques of the previous sections
can be extended to find CoB transformations of purely discrete programs. In
particular, our focus will be on transforming loops in programs to infer
abstractions that are of a simpler form. Our presentation will first focus on
simple loops consisting of a single location. The combination of loops with
multiple locations and continuous dynamics will be handled in the subsequent
section.
### 3.1 Transition System Models
We will first define transition system models and the action of CoB
transformations on these models. Let $\vec{x}\in X$ represent real valued
system variables, where $X\subseteq\mathbb{R}^{n}$. Transition systems will
form our basic models for loops in programs [21].
###### Definition 3.1.
A _transition system_ $\Pi$ is defined by a tuple $\left\langle
X,L,\mathcal{T},X_{0},\ell_{0}\right\rangle$, wherein,
1. 1.
$X\subseteq\mathbb{R}^{n}$ represents the continuous state-space. We will
denote the system variables by $\vec{x}\in\mathbb{R}^{n}$.
2. 2.
$L$ denotes a finite set of _locations_.
3. 3.
$\mathcal{T}$ represents a finite set of _transitions_. Each transition
$t_{j}\in\mathcal{T}$ is a tuple
$\left\langle\ell_{j},m_{j},G_{j},F_{j}\right\rangle$, where
* •
$\ell_{j}\in L$ is the pre-location of the transition, and $m_{j}\in L$ is the
post-location.
* •
$G_{j}\subseteq\mathbb{R}^{n}$ is the guard condition on the system variables
$\vec{x}$.
* •
$F_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is the update function.
4. 4.
$X_{0}\subseteq X$ represents the possible set of initial values and
$\ell_{0}\in L$ represents the starting location.
$\ell_{0}$$t_{1}$$t_{2}$$\begin{array}[]{rclp{.5cm}rcl}\vec{x}&:&(x,y,k)\\\
L&:&\\{\ell_{0}\\}\\\ \mathcal{T}&:&\left\\{\begin{array}[]{l}t_{1}:\
(\ell_{0},\ell_{0},G_{1},F_{1}),\\\ t_{2}:\
(\ell_{0},\ell_{0},G_{2},F_{2})\end{array}\right\\}\\\ X_{0}&:&\\{(x,y,k)\ |\
x=y=0\ \land\ k>0\\}\,.\\\\[5.0pt] G_{1}&:&\\{(x,y,k)\ |\
y<k\\}&&G_{2}&:&\\{(x,y,k)\ |\ y\geq k\\}\\\ F_{1}&:&\lambda(x,y,k).\
(x+y^{2},y+1,k)&&F_{2}&:&\lambda(x,y,k).\ (x,y,k)\\\ \end{array}$ Figure 2:
Transition system model for the loop in Example 1.2.
###### Example 3.1.
Figure 2 shows an example of a transition system derived from a simple program
that computes the sum of the first $k$ squares. The transition system consists
of a single location $\ell_{0}$, transitions
$t_{1}:(\ell_{0},\ell_{0},G_{1},F_{1})$ and
$t_{2}:(\ell_{0},\ell_{0},G_{2},F_{2})$. $\blacktriangle$
A _state_ of the transition system is a tuple
$\sigma:\left\langle\ell,\vec{x}\right\rangle$ where $\ell$ is the _current_
location and $\vec{x}\in X$ are the values of the continuous variables.
A _run_ is a finite or infinite sequence of states
$\sigma_{0}\xrightarrow{t_{0}}\sigma_{1}\xrightarrow{t_{1}}\cdots\rightarrow\sigma_{j}\xrightarrow{t_{j}}\sigma_{j+1}\cdots\,,$
where each $\sigma_{j}:(\ell_{j},\vec{x}_{j})$ is a state and $t_{j}$ a
transition, satisfying the following conditions:
1. 1.
The starting state $\sigma_{0}:(\ell_{0},\vec{x}_{0})$ is initial. I.e.,
$\ell_{0}$ is the initial location of $\Pi$ and $\vec{x}_{0}\in X_{0}$.
2. 2.
The state $\sigma_{i+1}:(\ell_{i+1},\vec{x}_{i+1})$ is related to the state
$\sigma_{i}:(\ell_{i},\vec{x}_{i})$ in the following way:
1. (a)
The transition $t_{i}\in\mathcal{T}$ is of the form
$(\ell_{i},\ell_{i+1},G_{i},F_{i})$, leading from $\ell_{i}$ to $\ell_{i+1}$.
2. (b)
The valuation $\vec{x}_{i}$ of the continuous variables satisfy the guard
$G_{i}$ and the valuation $\vec{x}_{i+1}$ is obtained by executing the
assignments in $F_{i}$ on $\vec{x}_{i}$:
$\vec{x}_{i}\in G_{i}\;\mbox{and}\;\vec{x}_{i+1}=F_{i}(\vec{x}_{i})\,.$
A special class of “simple loop” transition systems that have a single
location are defined below.
###### Definition 3.2.
A transition system $\Pi$ is called a _simple loop_ if it has a single
location. I.e., $L=\\{\ell\\}$. All transitions of a simple loop are self-
loops around this location $\ell$.
The transition system in Example 3.1 is a simple loop. It consists of a single
location. In general, simple loops can have multiple transitions that “loop”
around this single location.
We will now discuss the pre-image operator fpre induced by a transition. Let
$g(\vec{x})$ be some function over the state variables and $t:\ (\ell,m,G,F)$
be a transition.
###### Definition 3.3.
The _functional pre-image_ $\mbox{{fpre}}(g,t)$ is defined as $g(F(\vec{x}))$.
##### Note:
The standard precondition operator works over assertions over the state
variables, involving computing the pre-image using $F$ and computing the
intersection of the result with the guard. The functional precondition defined
here is defined over functions $g(\vec{x})$ over the state variables.
###### Example 3.2.
Consider the transition
$t:(\ell,m,G,F),\ \mbox{wherein}\ G:\ \\{(x,y)\ |\ x\geq y\\},F:\
\lambda(x,y).\ (x^{2},y^{2}-x^{2}))\,.$
The functional pre-image of the function $g(x,y):x+y$, denoted
$\mbox{{fpre}}(x+y,t)$, is given by
$\mbox{{fpre}}(x+y,t):\ (x^{2})+(y^{2}-x^{2})=y^{2}\,.$
To contrast with the standard pre-condition operator, which applies to
assertions over states, let us consider the assertion $x+y\geq 0$. We have
$\mbox{{pre}}(x+y\geq 0,t):\ y^{2}\geq 0\ \land\ x\geq y\,.$
$\blacktriangle$
We now show that fpre is a linear operator over functions.
###### Lemma 3.1.
For any transition $t$ and functions $g_{1},g_{2},g$ over $\vec{x}$, we have
$\mbox{{fpre}}(g_{1}+g_{2},t)=\mbox{{fpre}}(g_{1},t)+\mbox{{fpre}}(g_{2},t)$
and further, $\mbox{{fpre}}(\lambda g)=\lambda\mbox{{fpre}}(g)$ for any
$\lambda\in\mathbb{R}$.
###### Proof.
Proof follows by directly applying Def. 3.3. ∎
Let us consider any run of the transition system
$r:\sigma_{0}\xrightarrow{t_{0}}\sigma_{1}\rightarrow\cdots\rightarrow\sigma_{i}\xrightarrow{t_{i}}\sigma_{i+1}\cdots\,.$
Let $t_{i}:(\ell_{i},\ell_{i+1},G_{i},F_{i})$ denote the transition between
$\sigma_{i}:(\ell_{i},\vec{x}_{i})$ and
$\sigma_{i+1}:(\ell_{i+1},\vec{x}_{i+1})$. Finally, let $g(\vec{x})$ be any
function over the state variables of the transition system.
###### Lemma 3.2.
The following identity holds for all successive pairs of states
$(\ell_{i},\vec{x}_{i})\xrightarrow{t_{i}}(\ell_{i+1},\vec{x}_{i+1})$
encountered in a run of the transition system and for all functions
$g(\vec{x})$:
$\mbox{{fpre}}(g,t_{i})(\vec{x}_{i})\equiv g(\vec{x}_{i+1})\,$
###### Proof.
We may write $\mbox{{fpre}}(g,t_{i})(\vec{x}_{i})=g(F(\vec{x}_{i}))$. We know
that $\vec{x}_{i+1}=F(\vec{x}_{i})$. Therefore,
$g(\vec{x}_{i+1})=g(F(\vec{x}_{i}))=\mbox{{fpre}}(g,t_{i})(\vec{x}_{i})$. ∎
We will now discuss change-of-basis abstractions for transition systems. The
discussion will focus on defining change-of-basis abstractions for simple
loops, which are represented by a transition system with a single location
$\ell$ (Cf. Definition 3.2). The subsequent sections will extend this concept
to arbitrary transition systems.
### 3.2 CoB Abstractions For Simple Loops
Consider a simple loop $\Pi$ over $\vec{x}\in\mathbb{R}^{n}$ with a single
location $\ell$, transitions $\\{t_{1},\ldots,t_{k}\\}$, and initial condition
$X_{0}$. We seek to abstract $\Pi$ with another simple loop $\Xi$ over
$\vec{y}\in\mathbb{R}^{l}$ with a single location $m$, transitions
$\\{t_{1}^{\prime},\ldots,t_{k}^{\prime}\\}$ and initial condition $Y_{0}$.
###### Definition 3.4.
Simple loop $\Xi$ is a CoB abstraction of $\Pi$ iff there is a continuous
function $\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{l}$ such that
1. 1.
The initial condition $Y_{0}\supseteq\alpha(X_{0})$,
2. 2.
For each transition $t_{i}:(\ell,\ell,G_{i},F_{i})$ in $\Pi$, there is a
corresponding transition $t_{i}^{\prime}:(m,m,G_{i}^{\prime},F_{i}^{\prime})$
in $\Xi$ such that
1. (a)
$G_{i}^{\prime}\supseteq\alpha(G_{i})$,
2. (b)
$\forall\ \vec{x}\ F_{i}^{\prime}(\alpha(\vec{x}))=\alpha(F_{i}(\vec{x}))$.
We will now present an example of CoB abstraction for simple loops.
###### Example 3.3.
Consider the simple loop from Example 3.1 (also Fig. 2). We note that the map
$\alpha:\mathbb{R}^{3}\rightarrow\ \mathbb{R}^{4},\ \mbox{where}\
\alpha=\lambda(x,y,k).(x,y,k,y^{2})\,,$
yields an abstract transition system $\Xi$ over variables
$\vec{w}:(w_{1},w_{2},w_{3},w_{4})$. Informally, the variables
$(w_{1},w_{2},w_{3},w_{4})$ are place holders for the expressions
$(x,y,k,y^{2})$, respectively. The resulting transition system $\Xi$ is
$\begin{array}[]{rcl}\vec{w}&:&(w_{1},\ldots,w_{4})\\\ L&:&\\{m\\}\\\
\mathcal{T}&:&\\{t_{1}^{\prime}:\
(m,m,G_{1}^{\prime},F_{1}^{\prime}),t_{2}^{\prime}:(m,m,G_{2}^{\prime},F_{2}^{\prime})\\}\\\
X_{0}&:&w_{1}=w_{2}=w_{4}=0\ \land\ w_{3}\geq 1\\\
G_{1}^{\prime}&:&\\{\vec{w}\ |\ w_{2}<w_{3}\\}\\\ G_{2}^{\prime}&:&\\{\vec{w}\
|\ w_{2}\geq w_{3}\\}\\\ F_{1}^{\prime}&:&\lambda\ \vec{w}.\
(w_{1}+w_{4},w_{2}+1,w_{3},w_{4}+2w_{2}+1)\\\ F_{2}^{\prime}&:&\lambda\
\vec{w}.\ \vec{w}\\\ \end{array}$
The various requirements laid out in Definition 3.4 can be easily verified. We
will verify the requirement for $F_{1}^{\prime}$:
$F_{1}^{\prime}(\alpha(x,y,k))=\alpha(F_{1}(x,y,k))$, as follows:
$\begin{array}[]{rclcl}F_{1}^{\prime}(\alpha(x,y,k))&=&F_{1}^{\prime}(x,y,k,y^{2})&=&(\underset{w_{1}+w_{4}}{\underbrace{x+y^{2}}},\underset{w_{2}+1}{\underbrace{y+1}},\underset{w_{3}}{\underbrace{k}},\underset{w_{4}+2w_{2}+1}{\underbrace{y^{2}+2y+1}})\\\\[5.0pt]
&=&\alpha(x+y^{2},y+1,k)&=&\alpha(F_{1}(x,y,k))\\\ \end{array}\,.$
$\blacktriangle$
The definition of CoB abstraction immediately admits the following key
theorem.
###### Theorem 3.1.
For any run
$\sigma_{0}:(\ell,\vec{x}_{0})\xrightarrow{t_{0}}(\ell,\vec{x}_{1})\xrightarrow{t_{1}}(\ell,\vec{x}_{2})\xrightarrow{t_{2}}\cdots$
the corresponding sequence of $\Xi$-states
$\gamma_{0}:(m,\alpha(\vec{x}_{0}))\xrightarrow{t_{0}^{\prime}}(m,\alpha(\vec{x}_{1}))\xrightarrow{t_{1}^{\prime}}(m,\alpha(\vec{x}_{2}))\xrightarrow{t_{2}^{\prime}}\cdots\,,$
is a run of $\Xi$.
###### Proof.
Proof uses the property that whenever the move
$(\ell,\vec{x}_{j})\xrightarrow{t_{j}}(\ell,\vec{x}_{j+1})$ is enabled in
$\Pi$ then the move
$(m,\alpha(\vec{x}_{j}))\xrightarrow{t_{j}^{\prime}}(m,\alpha(\vec{x}_{j+1}))$
is enabled in $\Xi$.
Let $t_{j}$ be described by the guard $G_{j}$ and the functional update
$F_{j}$. Likewise, let $t_{j}^{\prime}$ be described by $G_{j}^{\prime}$ and
$F_{j}^{\prime}$. We note that $\alpha(G_{j})\subseteq G_{j}^{\prime}$. Since
$\vec{x}_{j}$ satisfies the guard of $t_{j}$, $\alpha(\vec{x}_{j})$ satisfies
that of $t_{j}^{\prime}$. The state obtained after the transition is given by
$F^{\prime}(\alpha(\vec{x}_{j}))=\alpha(F(\vec{x}_{j}))=\alpha(\vec{x}_{j+1})\,.$
We have proved that whenever the move
$(\ell,\vec{x}_{j})\xrightarrow{t_{j}}(\ell,\vec{x}_{j+1})$ is possible in
$\Pi$ then the move
$(m,\alpha(\vec{x}_{j}))\xrightarrow{t_{j}^{\prime}}(m,\alpha(\vec{x}_{j+1}))$
is possible in $\Xi$. The rest of the proof extends this to trace containment
through induction over prefixes of the traces. ∎
As a direct consequence, we may state a theorem that corresponds to Theorem
2.2 for the case of vector fields.
###### Theorem 3.2.
Let $[[\varphi]]$ be an invariant set for the abstract system $\Xi$. Then,
$\alpha^{-1}([[\varphi]])$ is an invariant of the original system $\Pi$.
###### Proof.
First, we note from Theorem 3.1 that if $(\ell,\vec{x})$ is reachable in $\Pi$
then $(m,\alpha(\vec{x}))$ is reachable in $\Xi$. Since $\varphi$ is an
invariant for $\Xi$, we have $(m,\alpha(\vec{x}))\in[[\varphi]]$. Therefore
for any reachable state $(\ell,\vec{x})$ in $\Pi$, we have
$(\ell,\vec{x})\in\alpha^{-1}([[\varphi]])$. Thus $\alpha^{-1}([[\varphi]])$
is an invariant set for $\Pi$. ∎
Given an invariant $\varphi[\vec{y}]$ for $\Xi$ in the form of an assertion,
the invariants for the original system are obtained simply by substituting
$\alpha(\vec{x})$ in the place of $\vec{y}$ in $\varphi$.
###### Example 3.4.
Consider the transition system $\Pi$ from Example 3.1 and its abstraction
$\Xi$ in Example 3.3. We note that $\Xi$ has affine guards and updates.
Therefore, we may use a standard polyhedral analysis tool to compute
invariants over $\Xi$ [10, 16, 36]. Some of the invariants obtained include
$\begin{array}[]{l}13w_{4}\leq 9w_{1}+24w_{2}\ \land\ 7w_{4}\leq
6w_{1}+11w_{2}\ \land\ 4w_{1}+7w_{2}-7w_{4}+11w_{3}\geq 11\\\
2w_{1}+3w_{2}-3w_{4}+4w_{3}\geq 4\ \land\ w_{4}\leq 2w_{1}+w_{2}\ \land\
3w_{4}\leq w_{1}+12w_{2}\\\ 9-w_{1}-3w_{2}+3w_{4}-9w_{3}\leq 0\ \land\
w_{2}\geq 0\ \land\ 1\leq w_{3}\ \land\ w_{2}-w_{3}\leq 0\\\ \end{array}$
By substituting $w_{1}\mapsto x,w_{2}\mapsto y,w_{3}\mapsto k,w_{4}\mapsto
y^{2}$ on these invariants, we conclude invariants for the original system.
For instance, we conclude facts such as
$13y^{2}-24y-9x\geq 0\ \land\ 7y^{2}-11y-6x\geq 0\ \land\ 11k-7y^{2}+7y+4x\geq
11\,.$
$\blacktriangle$
The goal, once again, is to find an abstraction $\alpha$ and an abstract
system $\Xi$ starting from a description of the system $\Pi$. Furthermore, we
require that the update functions $F_{j}^{\prime}$ in $\Xi$ are all
polynomials whose degrees are smaller than some given limit $d>0$. In
particular, if we set $d=1$, we are effectively requiring all the updates in
$\Xi$ to be affine functions over $\vec{y}$.
Our strategy will be to find a map
$\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$. For convenience, we will
write $\alpha$ as $(\alpha_{1},\ldots,\alpha_{k})$, wherein each component
function $\alpha_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}$. Let $V$ be the
vector space spanned by the components of $\alpha$, i.e,
$V=\mathit{Span}(\\{\alpha_{1},\ldots,\alpha_{k}\\})$. Our goal will be to
ensure that for each transition $t$ in $\Pi$ and for each $\alpha_{i}$,
$\forall\ \vec{x},\ \mbox{{fpre}}(\alpha_{i}(\vec{x}),t)\ \in\ V^{\left\langle
d\right\rangle}\,.$ (3)
Let $V$ be a vector space that satisfies Eq. (3) for each transition $t$ in
$\Pi$. We will say that the space $V$ is _$d$ -closed_ w.r.t $\Pi$.
###### Theorem 3.3.
Let $V:\mathit{Span}(g_{1},\ldots,g_{k})$ be $d$-closed w.r.t $\Pi$ for
continuous functions $g_{1},\ldots,g_{k}$. The map
$\alpha:(g_{1},\ldots,g_{k})$ is a CoB transformation defining an abstract
system $\Xi$, wherein each transition of $\Xi$ has a polynomial update
function involving polynomials of degree at most $d$.
###### Proof.
We construct the abstract system $\Xi$ with variables $w_{1},\ldots,w_{k}$
representing the functions $g_{1},\ldots,g_{k}$ that are the components of
$\alpha$. $\Xi$ has a single location $m$ and for each transition
$t_{i}\in\Pi$, we construct a corresponding transition $t_{i}^{\prime}\in\Xi$
as follows.
Let $G_{i},F_{i}$ be the guard set and update function for $t_{i}$,
respectively. The guard set for $t_{i}^{\prime}$ is given by $\alpha(G_{i})$
or an over-approximation thereof. Likewise, the update $F_{i}^{\prime}$ for
$t_{i}^{\prime}$ is derived as follows. We note that
$\mbox{{fpre}}(g_{j},t_{i})=\sum_{r}c_{r_{1},r_{2},\ldots,r_{k}}g_{1}^{r_{1}}g_{2}^{r_{2}}\cdots
g_{k}^{r_{k}}\,,$
wherein $0\leq r_{1}+r_{2}+\ldots+r_{k}\leq d$. The corresponding update for
$w_{j}$ in the abstract system is given by
$F_{i}^{\prime}(w_{j})=\sum_{r}c_{r_{1},r_{2},\ldots,r_{k}}w_{1}^{r_{1}}w_{2}^{r_{2}}\cdots
w_{k}^{r_{k}}\,.$
Note that each function $F_{i}^{\prime}(w_{j})$ is a polynomial of degree at
most $d$ over $w_{1},\ldots,w_{k}$. ∎
Since the operator fpre used to define the closure in Eq. (3) is a linear
operator (Cf. Lemma 3.1), we may check the closure property for a given vector
space $V$ by checking if its basis functions satisfy the property.
###### Lemma 3.3.
The vector space $V:\mathit{Span}(\\{g_{1},\ldots,g_{k}\\})$ is $d$-closed
w.r.t $\Pi$ iff for each basis element $g_{i}$ of $V$, and for each transition
$t$ in $\Pi$, $\mbox{{fpre}}(g_{i},t)\in V^{\left\langle d\right\rangle}$.
###### Proof.
For the non-trivial direction, let $V$ be a space where for each basis element
$g_{i}$ of $V$, and for each transition $t$ in $\Pi$,
$\mbox{{fpre}}(g_{i},t)\in V^{\left\langle d\right\rangle}$. An arbitrary
element $g\in V$ can be written as a linear combination of its basis elements:
$g=\sum_{j}\lambda_{j}g_{j}$. We have
$\mbox{{fpre}}(g,t)=\sum_{j}\lambda_{j}\mbox{{fpre}}(g_{j},t)$ from Lemma 3.1.
Since $\mbox{{fpre}}(g_{j},t)\in V^{\left\langle d\right\rangle}$, which is a
vector space itself, we have that $\mbox{{fpre}}(g,t)$ is a linear combination
of elements in $V^{\left\langle d\right\rangle}$ and thus
$\mbox{{fpre}}(g,t)\in V^{\left\langle d\right\rangle}$. Thus $V$ is
$d$-closed. ∎
###### Example 3.5.
Once again, consider the system $\Pi$ in Example 3.1 and the map
$\alpha:(x,y,k,y^{2})$ from Example 3.3. The components of this map are the
functions $\alpha_{1}:x,\alpha_{2}:y,\alpha_{3}:k,\mbox{and}\
\alpha_{4}:y^{2}$. We may verify that the vector space
$V:\mathit{Span}(\\{x,y,k,y^{2}\\})$ satisfies the closure property in Eq. (3)
for $d=1$. The table below shows the results of applying fpre on each of the
basis elements.
$\begin{array}[]{|l|l|l|}\hline\cr\mbox{Basis function}\
g_{j}&\mbox{{fpre}}(g_{j},t_{1})&\mbox{{fpre}}(g_{j},t_{2})\\\ \hline\cr
x&x+y^{2}&x\\\ y&y+1&y\\\ k&k&k\\\ y^{2}&y^{2}+2y+1&y^{2}\\\
\hline\cr\end{array}$
Thus, $\mbox{{fpre}}(g_{j},t_{k})$ belongs to $V^{\left\langle
1\right\rangle}=\mathit{Span}(\\{1,x,y,k,y^{2}\\})$. $\blacktriangle$
##### Searching for Abstractions:
The procedure for finding abstractions is identical to that used for vector
fields with the caveat that closure under Lie-derivative is replaced by
closure under $\mbox{{fpre}}(\cdot,t_{j})$ for every transition $t_{j}$ in the
system. The procedure takes as input an initial basis of functions $B_{0}$ and
iteratively refines the vector space $V_{i}:\mathit{Span}(B_{i})$ by removing
all the functions that do not satisfy the closure property.
###### Example 3.6.
Consider the system $\Pi$ in Example 3.1 and the initial basis consisting of
all monomials of degree at most $2$ over variables $x,y,k$. We obtain the
basis $B_{0}:\\{x,y,k,x^{2},y^{2},k^{2},xy,yk,xk\\}$ and the space
$V_{0}:\mathit{Span}(B_{0})$. An element of $V_{0}$ can be written as
$p:\
\left[\begin{array}[]{c}c_{1}x+c_{2}y+c_{3}k+c_{4}x^{2}+c_{5}y^{2}+c_{6}k^{2}\\\
+c_{7}xy+c_{8}yk+c_{9}xk\,.\end{array}\right]$
We consider the transition $t_{1}$ with update
$F_{1}:\lambda(x,y,k).(x+y^{2},y+1,k)$. Transition $t_{2}$ is ignored as its
update is simply the identity relation. We have $\mbox{{fpre}}(p,t_{1})$ as
$\mbox{{fpre}}(p,t_{1}):\
\left[\begin{array}[]{c}(c_{2}+c_{5})+(c_{1}+c_{7})x+(c_{2}+2c_{5})y+(c_{3}+c_{8})k+c_{4}x^{2}+\\\
(c_{1}+c_{5}+c_{7})y^{2}+c_{6}k^{2}+c_{7}xy+c_{7}y^{3}+c_{4}y^{4}+2c_{4}xy^{2}+\\\
c_{8}yk+c_{9}xk+c_{9}y^{2}k\end{array}\right]$
The “overflow” terms $c_{7}y^{3}$, $c_{4}y^{4}$, $c_{9}y^{2}k$ immediately
yield the constraints $c_{4}=c_{7}=c_{9}=0$. The refined basis is
$B_{1}:\\{x,y,k,y^{2},k^{2},yk\\}$. The iterative process converges with
$V_{1}:\mathit{Span}(B_{1})$ yielding a linearization. $\blacktriangle$
### 3.3 Abstractions for General Transition Systems
Thus far, we have presented CoB abstractions for simple loops consisting of a
single location. The ideas seamlessly extend to systems with multiple
locations with a few generalizations that will be described in this section.
Let $\Pi$ be a system with a set of locations
$L=\\{\ell_{1},\ldots,\ell_{k}\\}$ and transitions $\mathcal{T}$. We will
assume that $|L|\geq 2$ so that the system is no longer a simple loop. The
main idea behind change of basis (CoB) transformations for systems with
multiple locations is to allow a different map for each location. In other
words, the abstraction is defined by a maps $\alpha_{\ell}(\vec{x})$ for each
location $\ell\in L$.
The maps for two different locations $\ell_{1}$ and $\ell_{2}$ are of the type
$\alpha_{\ell_{1}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m_{1}}$ and
$\alpha_{\ell_{2}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m_{2}}$. In general,
we may assume that $m_{1}\not=m_{2}$. This discrepancy can be remedied by
padding each $\alpha_{\ell_{i}}$ with extra components that map to the
constant function $0$. While, this transformation violates the linear
independence requirement between the various components in $\alpha$, it makes
the resulting abstract system easier to describe. Without loss of generality,
we assume that all the maps $\alpha_{\ell}$ for each $\ell\in L$ are of the
form $\alpha_{\ell}:\ \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ for a fixed
$m>0$.
###### Definition 3.5.
A system $\Xi$ is a CoB abstraction of $\Pi$ through a collection of maps
$\alpha_{\ell_{1}},\ldots,\alpha_{\ell_{k}}$ each of the type
$\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, corresponding to locations
$\ell_{1},\ldots,\ell_{k}$, iff
1. 1.
$\Xi$ has locations $m_{j}$ corresponding to $\ell_{j}\in L$ for $1\leq j\leq
k$ , and transitions $t_{i}^{\prime}$ corresponding to transition
$t_{i}\in\mathcal{T}$.
2. 2.
For each transition
$t_{i}:\left\langle\ell_{pre},\ell_{post},G_{i},F_{i}\right\rangle$ in $\Pi$,
the corresponding transition $t_{i}^{\prime}:\left\langle
m_{pre},m_{post},G_{i}^{\prime},F_{i}^{\prime}\right\rangle$ is such that
1. (a)
$m_{pre}$ and $m_{post}$ correspond to $\ell_{pre}$ and $\ell_{post}$,
respectively,
2. (b)
$G_{i}^{\prime}\supseteq\alpha_{\ell_{pre}}(G_{i})$,
3. (c)
$(\forall\ \vec{x})\
F_{i}^{\prime}(\alpha_{\ell_{pre}}(\vec{x}))=\alpha_{post}(F_{i}(\vec{x}))$.
We note that for a simple loop with a single location, the definition above is
identical to Def. 3.4.
⬇ int x,y,z; // .. initialize.. while (x + y - z <= 100){ (x,y):=( x + z * (x - y) , y + z * (y - x)); // x,y,z unmodified here (x,y,z) := (z+1 , x+y -1 , z+x+y -1 ); } | $\ell_{1}$$\ell_{2}$$\ell_{3}$$t_{1}$$t_{3}$$t_{2}$$\begin{array}[]{rcl}L&:&\\{\ell_{1},\ell_{2},\ell_{3}\\}\\\ \mathcal{T}&:&\\{t_{1},t_{2},t_{3}\\}\\\ t_{1}&:&\left\langle\ell_{1},\ell_{2},G_{1},F_{1}\right\rangle\\\ t_{2}&:&\left\langle\ell_{2},\ell_{1},G_{2},F_{2}\right\rangle\\\ t_{3}&:&\left\langle\ell_{1},\ell_{3},G_{3},F_{3}\right\rangle\\\\[5.0pt] G_{1}&:&\\{(x,y,z)\ |\ x+y-z\leq 100\\}\\\ F_{1}&:&\left(x+zx-zy,y+zy-zx,z\right)\\\\[5.0pt] G_{2}&:&\mathbb{Z}^{3}\\\ F_{1}&:&\left(z+1,x+y-1,z+x+y-1\right)\\\\[5.0pt] G_{3}&:&\\{(x,y,z)\ |\ x+y-z>100\\}\\\ F_{3}&:&(x,y,z)\end{array}$
---|---
Figure 3: An example program fragment with multiple locations and its
transition system.
###### Example 3.7.
Figure 3 shows an example of a transition system with multiple locations.
Consider the following CoB transformation:
$\begin{array}[]{rcl}\alpha_{\ell_{1}}&:&(z^{2},yz,xz,z,y^{2},xy,y,x^{2},x)\\\
\alpha_{\ell_{2}}&:&(z^{2},yz+xz,z,y,y^{2}+2xy+x^{2},x,0,0,0)\\\
\alpha_{\ell_{3}}&:&(z^{2},yz,xz,z,y^{2},xy,y,x^{2},x)\\\ \end{array}$
The transformation yields an abstraction $\Xi$ of the original system. The
abstract system has $9$ variables $w_{0},\ldots,w_{8}$. The structure of $\Xi$
mirrors that of $\Pi$ with three locations $m_{1},m_{2},m_{3}$ corresponding
to $\ell_{1},\ell_{2},\ell_{3}$, respectively and three transitions
$t_{1}^{\prime},t_{2}^{\prime}$ and $t_{3}^{\prime}$ corresponding to
$t_{1},t_{2}$ and $t_{3}$ in $\Pi$. The guards and updates of the transition
$t_{1}^{\prime}$ are
$\begin{array}[]{rcl}G_{1}^{\prime}&:&\\{(w_{0},\ldots,w_{8})\ |\
w_{8}+w_{6}-w_{3}\leq 100\\},\\\
F_{1}^{\prime}&:&(w_{0},w_{1}+w_{2},w_{3},w_{1}-w_{2}+w_{6},w_{4}+2w_{5}+w_{7},-w_{1}+w_{2}+w_{8},0,0,0)\\\
\end{array}$
We verify the key condition that ensures that $t_{1}^{\prime}$ is an
abstraction of $t_{1}$:
$\alpha_{\ell_{2}}(F_{1}(x,y,z))=F_{1}^{\prime}(\alpha_{\ell_{1}}(x,y,z))\,.$
The LHS $\alpha_{\ell_{2}}(F_{1}(x,y,z))=\alpha_{\ell_{2}}(x+zx-zy,y+zy-zx,z)$
is given by
$(z^{2},zx+zy,z,y+zy-zx,x^{2}+2xy+y^{2},x+zx-zy,0,0,0)\,.$
The RHS
$F_{1}^{\prime}(\alpha_{\ell_{1}}(x,y,z))=F_{1}^{\prime}(z^{2},yz,xz,z,y^{2},xy,y,x^{2},x)$
is given by
$(z^{2},xz+yz,z,y-zx+zy,y^{2}+2xy+x^{2},x+zx-zy,0,0,0)\,.$
The identity of LHS and RHS is thus verified. $\blacktriangle$
Our goal once again is to search of a collection of transformations
$\alpha_{\ell}$, for each $\ell\in L$ such that the resulting system is
described by polynomial updates of degree at most $d$. The case where $d=1$
corresponds to affine updates. Once again, we generalize the notion of a
$d-$closed vector space. Consider a collection of vector spaces
$V_{\ell}:\mathit{Span}(B_{\ell})$ for each location $\ell\in L$.
###### Definition 3.6.
We say that the collection $V_{\ell},\ell\in L$ is $d-$closed for transition
system $\Pi$ if and only if for each transition
$t_{j}:\left\langle\ell_{pre},\ell_{post},G_{j},F_{j}\right\rangle$ and for
each element $p\in V_{post}$, we have $\mbox{{fpre}}(p,t_{j})\in
V_{pre}^{\left\langle d\right\rangle}$.
The notion of $d-$closed vector spaces can be related to CoB transformations
and resulting abstractions whose updates are defined by means of polynomials
of degree at most $d$.
###### Theorem 3.4.
Let $V_{\ell},\ell\in L$ be a collection of vector spaces that are $d-$closed
for a system $\Pi$. The basis elements of $V_{\ell}$ yields a collection of
maps $\alpha_{\ell},\ \ell\in L$ that relate $\Pi$ to a CoB abstraction $\Xi$.
The update maps of $\Xi$ are all polynomials of degree at most $d$.
###### Example 3.8.
Consider the transition system described in Example 3.7 and Figure 3. We wish
to discover an affine abstraction for this system automatically. Starting from
the initial collection of vector spaces that maps each location to the space
of all polynomials of degree at most $2$ over $x,y,z$, we obtain the
transformations $\alpha_{\ell_{1}},\alpha_{\ell_{2}},\alpha_{\ell_{3}}$
described in the same example. This yields an abstract system over variables
$w_{0},\ldots,w_{8}$.
### 3.4 Combining Discrete and Continuous Systems
As a final step, we extend our approach to hybrid systems that combine
discrete and continuous dynamics. We define hybrid systems briefly and extend
the results from Sections 2 and 3 to address hybrid systems.
###### Definition 3.7.
A hybrid system consists of a discrete transition system $\Pi:\left\langle
X,L,\mathcal{T},X_{0},\ell_{0}\right\rangle$ and a mapping that associates
each location $\ell_{i}\in L$ with a continuous subsystem
$\mathcal{S}_{i}:\left\langle\mathcal{F}_{i},X_{i}\right\rangle$ over the
state-space $X$, consisting of a vector field $\mathcal{F}_{i}$ and location
invariant $X_{i}$.
A state $\sigma$ of the hybrid system consists of a tuple
$\left\langle\ell,\vec{x},T\right\rangle$ where $\ell\in L$ is the current
location, valuations to the continuous variables $\vec{x}\in X$ and the
current time $T\geq 0$.
Given a time $\delta\geq 0$, we write
$\left\langle\ell,\vec{x},T\right\rangle\ \underset{\delta}{\leadsto}\
\left\langle\ell,\vec{y},T+\delta\right\rangle$ to denote that starting from
state $\left\langle\ell,\vec{x},T\right\rangle$ the hybrid system _flows_
continuously according to the continuous subsystem $\mathcal{S}_{\ell}$
corresponding to the location $\ell$. Likewise, we write
$\left\langle\ell,\vec{x},T\right\rangle\xrightarrow{t_{j}}\left\langle\ell^{\prime},\vec{x}^{\prime},T\right\rangle$
to denote a _jump_ between two states upon taking a discrete transition
$t_{j}$ from $\ell$ to $\ell^{\prime}$. Note that no time elapses upon taking
a jump.
A run $R$ of the hybrid system is given by a countable sequence of alternating
flows (evolution according to the ODE inside a location) and jumps (discrete
transition to a different location) starting from an initial state:
$\sigma_{0}:\left\langle\ell_{0},\vec{x}_{0},0\right\rangle\underset{\delta_{0}}{\leadsto}\sigma_{0}^{\prime}:\left\langle\ell_{0},\vec{y}_{0},\delta_{0}\right\rangle\xrightarrow{t_{1}}\
\sigma_{1}:\left\langle\ell_{1},\vec{x}_{1},\delta_{0}\right\rangle\underset{\delta_{1}}{\leadsto}\
\sigma_{1}^{\prime}:\left\langle\ell_{1},\vec{y}_{0},\delta_{0}+\delta_{1}\right\rangle\xrightarrow{t_{2}}\cdots$
To avoid Zenoness, we require that the summation of the dwell times in the
individual modes $\sum_{j=0}^{\infty}\delta_{j}$ diverges.
We now define CoB abstractions for hybrid systems. Our definitions simply
combine aspects of the definition for transition systems 3.5 and continuous
systems 2.3.
A CoB abstraction of the hybrid system is obtained through a collection of
maps $\alpha_{\ell_{1}},\ldots,\alpha_{\ell_{k}}$ corresponding to the
locations $\ell_{1},\ldots,\ell_{k}$ of the hybrid system. It is assumed that
by padding with $0$s, we obtain each $\alpha_{\ell_{i}}$ as a function
$\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$.
###### Definition 3.8.
A system $\Xi$ is a CoB abstraction of $\Pi$ through a collection of maps
$\alpha_{\ell_{1}},\ldots,\alpha_{\ell_{k}}$ each of the type
$\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, corresponding to locations
$\ell_{1},\ldots,\ell_{k}$, iff
1. 1.
$\Xi$ has locations $m_{j}$ corresponding to $\ell_{j}\in L$ for $1\leq j\leq
k$ , and transitions $t_{i}^{\prime}$ corresponding to transition
$t_{i}\in\mathcal{T}$. Each location $m_{j}$ in $\Xi$ has an associated
continuous system $\mathcal{T}_{j}$.
2. 2.
For each corresponding location pair $\ell_{j},m_{j}$, the system
$\mathcal{T}_{j}$ is a CoB abstraction of $\mathcal{S}_{j}$ through the
transformation $\alpha_{\ell_{j}}$.
3. 3.
For each transition
$t_{i}:\left\langle\ell_{pre},\ell_{post},G_{i},F_{i}\right\rangle$ in $\Pi$,
the corresponding transition $t_{i}^{\prime}:\left\langle
m_{pre},m_{post},G_{i}^{\prime},F_{i}^{\prime}\right\rangle$ are such that
1. (a)
$m_{pre}$ and $m_{post}$ correspond to $\ell_{pre}$ and $\ell_{post}$,
respectively,
2. (b)
$G_{i}^{\prime}\supseteq\alpha_{\ell_{pre}}(G_{i})$,
3. (c)
$(\forall\ \vec{x})\
F_{i}^{\prime}(\alpha_{\ell_{pre}}(\vec{x}))=\alpha_{post}(F_{i}(\vec{x}))$.
Once again, we focus on searching for an abstraction $\Xi$ of a given hybrid
system wherein the continuous abstraction for each location and that of each
transition is expressed by means of polynomials degree bounded by some fixed
bound $d$. The case where the bound is $d=1$ specifies an affine hybrid
abstraction $\Xi$. We translate this into a $d-$closure condition for vector
spaces. Consider a collection of vector spaces
$V_{\ell}:\mathit{Span}(B_{\ell})$ for each location $\ell\in L$.
###### Definition 3.9.
We say that the collection $V_{\ell},\ell\in L$ is $d-$closed for hybrid
system $\Pi$ if and only if
1. 1.
For each location $\ell\in L$, the corresponding vector space $V_{\ell}$ is
$d$-closed w.r.t to the vector field $\mathcal{F}_{\ell}$ for the continuous
subsystem $\mathcal{S}_{\ell}$.
2. 2.
For each transition
$t_{j}:\left\langle\ell_{pre},\ell_{post},G_{j},F_{j}\right\rangle$ and for
each element $p\in V_{post}$, we have $\mbox{{fpre}}(p,t_{j})\in
V_{pre}^{\left\langle d\right\rangle}$.
Once again, the approach for finding a $d$-closed collection $V_{\ell},\
\ell\in L$ starts from an initial basis $V_{\ell}^{(0)}$ at each location
$\ell$ and refines the basis. Two types of refinements are applied (a)
refinement of $V_{\ell}$ to enforce closure w.r.t the Lie derivative of its
basis elements for the vector field $\mathcal{F}_{\ell}$ and (b) refinement of
$V_{m}$ w.r.t a transition $t:\left\langle\ell,m,G,F\right\rangle$ incoming at
location $m$.
## 4 Implementation and Evaluation
We have implemented the ideas described in this paper to derive affine
abstractions for (a) continuous systems described by ODEs with polynomial
right-hand sides, (b) discrete systems with assignments that have polynomial
RHS and (b) hybrid systems with polynomial ODEs and discrete transition
updates. Our approach takes as inputs the system description, a degree limit
$k>0$ that is used to construct the initial basis. Starting from this initial
basis, our approach iteratively applies refinement until convergence. Upon
convergence, we print the basis inferred along with the resulting abstraction.
Currently, our implementation does not abstract the guard sets of the
transitions and the invariant sets of the ODEs. However, once the basis is
inferred, the abstractions for the guards of the transition and mode
invariants are obtained using quantifier elimination techniques (which is
quite expensive in practice) [7, 8, 13] or optimization techniques such as
Linear programming or SOS programming [26]. Our implementation currently
relies on manual translation of invariant and guard assertions into the new
basis to form the abstract transition system.
If a non-trivial abstraction is discovered by our iterative scheme, we may use
a linear invariant generator on the resulting affine system to infer
invariants that relate to the original transition system.
Our implementation and the benchmarks used in the evaluation presented in this
section may be obtained upon request.
### 4.1 Continuous Systems
We first describe experimental results obtained for continuous systems
described by ODEs. Figure 4 summarizes the results on continuous system
benchmarks. We collected nearly $15$ benchmark systems and ran our
implementation to search for a linearizing CoB transformation. We report on
the degree of the monomials in the initial basis, time taken to converge and
the number of polynomials in the final basis that form the transformation to
the abstract system.
##### Trivial Transformations Found:
Some of the benchmarks attempted resulted in trivial final transformations.
Examples include the well-known Fitzhugh-Nagumo neuron model, the vanderpol
oscillators and similar small but complex systems that are known to be non-
integrable.
We now highlight some of the interesting results, while summarizing all
benchmarks in Table 4.
##### Toda Lattice with Boundary Particles:
The Toda lattice models an infinite array of point particles such that the
position and velocity of the $n^{th}$ particle are affected by its neighbors
the $(n-1)^{th}$ and $(n+1)^{th}$ particle for $n\in\mathbb{Z}$ 222See
description by Göktas and Hereman [15] and references therein.. We consider a
finite version of this lattice with $2$ fixed boundary particles that are
constrained to have a fixed position and zero velocity and $K$ particles in
the middle. The dynamics for $K=2$ non-fixed particles are given by position
variables $y_{1},y_{2}$, velocities $v_{1},v_{2}$ and extra state variables
$u_{1},u_{2}$ to model the interaction with neighbors.
$\begin{array}[]{rcl rcl rcl
}\frac{dx_{1}}{dt}&=&v_{1}&\frac{dv_{1}}{dt}&=&v_{1}(u_{1}-u_{2})&\frac{du_{1}}{dt}&=&-v_{1}\\\
\frac{dx_{2}}{dt}&=&v_{2}&\frac{dv_{2}}{dt}&=&v_{2}u_{2}&\frac{du_{2}}{dt}&=&v_{1}-v_{2}\\\
\end{array}$
In addition, we add time $t$ as a variable to the model with dynamics
$\frac{dt}{dt}=1$. Our approach initialized with polynomials of degree $2$
discovers a basis with $10$ polynomials:
$\begin{array}[]{l}w_{1}:\ -2v_{2}-2v_{1}-u_{2}^{2}+2x_{1}u_{1}+x_{2}^{2},\\\
w_{2}:\
-2v_{2}-2v_{1}-u_{2}^{2}+u_{1}u_{2}+x_{2}u_{1}+x_{1}u_{2}+x_{1}u_{1}+x_{1}x_{2}\\\
w_{3}:\ -2v_{2}-2v_{1}+2u_{1}u_{2}+2x_{2}u_{2}+2x_{2}u_{1}+x_{2}^{2},\ w_{4}:\
u_{1}+x_{1},\\\ w_{5}:\ 2v_{2}+2v_{1}+u_{1}^{2}+u_{2}^{2},\ w_{6}:\
u_{2}+x_{2}-x_{1},\ w_{7}:\ t,\\\ w_{8}:\ u_{1}t+x_{1}t,\ w_{9}:\
u_{2}t+u_{1}t+x_{2}t,\ w_{10}:\ t^{2}\end{array}$
The resulting abstract system has linear dynamics given by:
$\frac{dw_{j}}{dt}=0,\ 1\leq j\leq 6,\ \frac{dw_{7}}{dt}=1,\
\frac{dw_{8}}{dt}=w_{4},\ \frac{dw_{9}}{dt}=w_{4}+w_{6},\
\frac{dw_{10}}{dt}=2w_{7}\,.$
Results for larger instances are reported in Table 4.
##### Quadratic Fermi-Pasta-Ulam-Tsingou System:
Consider a system considered by Fermi et al. [14]. The system consists of a
chain of particles at positions $x_{1},\ldots,x_{N}$ with fixed boundary
particles $x_{0}=0$ and $x_{N+1}=N+1$. The dynamics are given by
$\frac{d^{2}x_{i}}{dt^{2}}=(x_{i+1}+x_{i-1}-2x_{i})+\alpha((x_{i+1}^{2}-x_{i}^{2})-(x_{i}-x_{i-1})^{2})\,,\
1\leq i\leq N$
We consider an instantiation with $N=3$, searching for CoB transformations
with an initial basis of monomials of degree up to $4$. We obtain a
transformation representing a conserved quantity
$\begin{array}[]{c}\frac{1}{2}(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-3x_{3}(1+3a-ax_{3})\\\
-x_{2}x_{3}(1+ax_{3}-ax_{2})-x_{1}x_{2}(1+ax_{2}-ax_{1})\end{array}\,.$
The abstract system is given by $\frac{dw_{1}}{dt}=0$.
##### Two Mass Spring System:
Consider the dynamics of two masses connected by a spring to each other and to
two fixed walls. The state variables are $(x_{1},x_{2},v_{1},v_{2})$
indicating the position and velocity of the masses while the spring constant
$k$ is a parameter. The dynamics are given by
$\begin{array}[]{rcl
rcl}\frac{dx_{1}}{dt}&=&v_{1}&\frac{dx_{2}}{dt}&=&v_{2}\\\
\frac{dv_{1}}{dt}&=&kx_{2}-2kx_{1}&\frac{dv_{2}}{dt}&=&k(x_{1}-x_{2})\\\
\end{array}$
Our procedure yields a change of basis transformation
$w_{1}:\ v_{2}^{2}+v_{1}^{2}+kx_{2}^{2}-2kx_{1}x_{2}+2kx_{1}^{2},\ w_{2}:\
v_{1}v_{2}-\frac{1}{2}v_{1}^{2}-\frac{1}{2}kx_{2}^{2}+2kx_{1}x_{2}-\frac{3}{2}kx_{1}^{2}$
Both $w_{1},w_{2}$ represent conserved quantities, yielding the abstraction
$\frac{dw_{1}}{dt}=\frac{dw_{2}}{dt}=0\,.$
##### Biochemical reaction network:
We consider a biochemical reaction network benchmark from Dang et al. [12].
The ODE along with the values are parameters in our model coincide with those
used by Dang et al. The ODE consists of $12$ variables and roughly $14$
parameters. Our search for degree bound $\leq 3$ discovers a transformation
generated by five basis functions (in roughly $3$ seconds).
##### Collision Avoidance
We consider the algebraic abstraction of the roundabout mode of a collision
avoidance system analyzed recently by Platzer et al. [29] and earlier by
Tomlin et al. [40]. The two airplane collision avoidance system consists of
the variables $(x_{1},x_{2})$ denoting the position of the first aircraft,
$(y_{1},y_{2})$ for the second aircraft, $(d_{1},d_{2})$ representing the
velocity vector for aircraft 1 and $(e_{1},e_{2})$ for aircraft $2$.
$\omega,\theta$ abstract the trigonometric terms. In addition, the parameters
$a,b,r_{1},r_{2}$ are also represented as system variables. The dynamics are
modeled by the following differential equations:
$\begin{array}[]{ccccccccc}x_{1}^{\prime}=d_{1}&x_{2}^{\prime}=d_{2}&d_{1}^{\prime}=-\omega
d_{2}&d_{2}^{\prime}=\omega d_{1}\\\
y_{1}^{\prime}=e_{1}&y_{2}^{\prime}=e_{2}&e_{1}^{\prime}=-\theta
e_{2}&e_{2}^{\prime}=\theta e_{1}\\\
a^{\prime}=0&b^{\prime}=0&r_{1}^{\prime}=0&r_{2}^{\prime}=0\\\ \end{array}$
A search for transformations of degree $2$ yields a closed vector space with
27 basis functions within $0.2$ seconds. The basis functions include
$a,b,r_{1},r_{2}$ and all degree two terms involving these. Removing these
from the basis, gives us $14$ basis functions that yield a transformation to a
$14$ dimensional affine ODE.
ID | #V | Deg. | #B0 | Time | #B* | #B0 | Time | #B*
---|---|---|---|---|---|---|---|---
Brusselator | 2 | 3 | 3 | 0.01 | 0 | 25 | 2.8 | 0
Fitz-Nagumo | 2 | 3 | 3 | 0.01 | 0 | 25 | 2.6 | 0
Vanderpol | 2 | 3 | 3 | 0.01 | 0 | 25 | 1.9 | 0
Proj-drag | 4 | 2 | 3 | 0.02 | 8 | 10 | 9.7 | 64
Circular | 4 | 2 | 3 | 6 | 0.01 | 10 | 10.6 | 83
Hamiltonian | 5 + 1 | 2 | 3 | 0.02 | 5$\dagger$ | 5 | 1.3 | 20$\dagger$
Two-spring | 4 + 1 | 2 | 3 | 0.03 | 2$\dagger$ | 5 | 0.5 | 6$\dagger$
Toda-2 | 7 | 2 | 3 | 0.1 | 22 | 5 | 4.6 | 82
Toda-3 | 10 | 2 | 3 | 0.5 | 38 | 5 | 95 | 169
Toda-5 | 16 | 2 | 3 | 6 | 90 | 5 | 6373 | 559
Toda-10 | 31 | 2 | 3 | 301.5 | 375 | 5 | dnf
FPUT-3 | 6 + 1 | 3 | 3 | 0.05 | 0$\dagger$ | 5 | 3.7 | 2 $\dagger$
FPUT-5 | 10 + 1 | 3 | 3 | 0.4 | 0 | 5 | 231 | 2
Bio-network | 13 | 2 | 3 | 0.07 | 5 | 5 | 4800 | 20
Roundabout | 10 + 4 | 2 | 3 | 1.5 | 68$\dagger$ | 5 | 890 | $\geq 600\ \dagger$
Figure 4: Experimental evaluation results on non linear polynomial ODE
benchmarks at a glance. Legend: #V denotes number of system variables +
parameters, Deg.: max. degree of the RHS, #B0: degree limit for monomials in
the initial basis, Time: timing in seconds, #B*: number of elements in the
final basis, $\dagger$: some elements of the basis involving just the
parameters were discarded from the count and dnf: did not finish in 2hrs or
out of memory crash.
### 4.2 Discrete Systems
We now describe experimental results on some discrete programs. We used a set
of benchmark programs that require non-linear invariants to prove correctness
compiled by Enric Carbonell 333The benchmark instances are available on-line
at http://www.lsi.upc.edu/~erodri/webpage/polynomial_invariants/list.html..
Our evaluation focuses on a subset of benchmarks that have non-linear
assignments or guards in them. The methods presented here converge in a single
step with the initial basis whenever the program being considered already has
affine updates.
⬇ int fermat(int N, int R) pre (N >= 0 && R >= 0); int u,v,r; u := 2*R -1; v := 1; r := R*R -N; 1: while ( r != 0 ){ 2: while (r > 0) (r,v) := (r-v, v+2); 3: while (r < 0) (r,v) := (r+u, u+2); } end | $\begin{array}[]{l}-4r-v^{2}-4Nv+2v+u^{2}-2u\ \leq\ 0\ \land\\\ -r-Nu\ \leq\ 0\ \land\ 1-v^{2}\ \leq\ 0\ \land\\\ 1-uv\ \leq\ 0\ \land\ -Rv+R\ \leq\ 0\ \land\\\ 1-v\ \leq\ 0\ \land\ 1-u^{2}\ \leq\ 0\ \land\\\ -Ru+2R^{2}+R\ \leq\ 0\ \land\ -Nu\ \leq\ 0\ \land\\\ 1-u\ \leq\ 0\ \land\ -R^{2}\ \leq\ 0\ \land\ -NR\ \leq\ 0\ \land\\\ -R\ \leq\ 0\ \land\ -N^{2}\ \leq\ 0\ \land\\\ v^{2}-2v-u^{2}+2u\ \leq\ 0\ \land\\\ 1+r-u-R^{2}\ \leq\ 0\ \land\ 1+4r-u^{2}\ \leq\ 0\ \land\\\ 4r+v^{2}-2v-u^{2}+2u\ \leq\ 0\ \land\\\ 2+6r-uv-u^{2}-2R^{2}\ \leq\ 0\ \land\\\ 4r+v^{2}-2v-u^{2}+2u+4N=0\end{array}$
---|---
Figure 5: Fermat’s algorithm for prime factorization taken from Bressoud [5]
and invariants computed at location 1 using polyhedral analysis of the
linearization.
##### Fermat Factorization:
Figure 5 shows a program for finding a factor of a number $N$ near its square
root taken from a book by Bressoud [5]. Our analysis initialized with
monomials of degree up to $2$ over the program variables yields a final basis
consisting of $17$ polynomials. The resulting affine system is analyzed by a
polyhedral analyzer using abstract interpretation to yield invariants. Some of
the invariants obtained at the loop head are shown in Figure 5. The equality
invariant
$4r+v^{2}-2v-u^{2}+2u+4N=0$
is obtained at locations $1,2$ and $3$ in the program. This forms a key part
of the program’s partial correctness proof.
⬇ int productBR(int x, int y) pre (x >= 0 && y >= 0); int a,b,p,q; (a,b,p,q) := (x,y,1,0); 1: while ( a >= 1 && b >= 1 ){ if ( a mod 2 == 0 && b mod 2 == 0) (a,b,p) := (a/2, b/2, 4 * p); elsif (a mod 2 == 1 && b mod 2 == 0) (a,q) := (a-1, q+ b*p); elsif (a mod 2 == 0 && b mod 2 == 1) (b,q) := (b-1, q + a*p); else (a,b,p) := (a-1, b-1, q + (a+b-1)*p); } end | $\begin{array}[]{l}1-p^{2}\leq 0\ \land\ -yp+y\leq 0\ \land\\\ -xp+x\leq 0\ \land\ 1-p\leq 0\ \land\\\ -1-b\leq 0\ \land\ -1-a\leq 0\ \land\\\ -y\leq 0\ \land\ -x\leq 0\ \land\\\ 2ap-xp-2a+x\leq 0\ \land\\\ 7ap-xp-7a-14x\leq 0\ \land\\\ 7ap-xp+8a-14x\leq 0\ \land\\\ 16ap-3xp-16a-12x\leq 0\ \land\\\ 2bp-yp-2b+y\leq 0\ \land\\\ 7bp-yp-7b-14y\leq 0\ \land\\\ 7bp-yp+8b-14y\leq 0\ \land\\\ 16bp-3yp-16b-12y\leq 0\end{array}$
---|---
Figure 6: A multiplication algorithm and loop invariant computed using
polyhedral analysis on the linearization.
##### Product of Numbers:
Consider the benchmark shown in Figure 6 that seeks to compute the product of
its arguments $x,y$. Our approach initialized using degree $2$ monomials
computes an abstract system with $20$ basis polynomials that in turn yields an
affine transition system with $20$ variables. Figure 6 shows the invariants
computed using polyhedral abstract interpretation. The invariant $q-abp=0$
cannot be established by our technique with degree $2$ monomials. On the other
hand, it can be established by considering degree $3$ monomials in the initial
basis. The resulting system however has $60$ variables, making polyhedral
analysis of the system as a whole hard.
⬇ int geoSum(int a, int r, int n ) int s := 0; int p := a; int k := 0; while (k < n) s := s + p; p := p * r; k := k + 1; end | $\begin{array}[]{l}-k^{2}+k\ \leq\ 0\ \land\ -2-k^{2}+3k\ \leq\ 0\ \land\\\ -6-k^{2}+5k\ \leq\ 0\ \land\ -9-k^{2}+6k\ \leq\ 0\ \land\\\ -k\ \leq\ 0\ \land\ -r^{2}\ \leq\ 0\ \land\ -n\ \leq\ 0\ \land\\\ -1+k-n\ \leq\ 0\ \land\ -p+s-rs+a=0\end{array}$
---|---
Figure 7: Geometric summation program and computed loop invariant.
##### Geometric Summation:
Consider the geometric summation program in Figure 7. Our approach computes a
linearization with $5$ variables in the abstract system. Polyhedral analysis
of the resulting program yields the invariant $(1-r)s=a-p$. This invariant
together with the invariant $p=ar^{k}$ (which cannot be obtained through
algebraic reasoning) suffices to prove the partial correctness of the program.
System | Linearization | Analysis
---|---|---
ID | #V | #Trs | Deg | B0 | #B* | Time | Time | #I
Petter2 | 2 | 1 | 2 | 2 | 3 | 0.02 | $\leq$ 0.01 | 10
Petter3 | 2 | 1 | 2 | 3 | 1 | 0.02 | $\leq$ 0.01 | 2
Petter3 | 2 | 1 | 3 | 3 | 4 | 0.02 | $\leq$ 0.01 | 25
Geo | 6 | 2 | 2 | 2 | 6 | 0.02 | $\leq$ 0.01 | 9
Fermat | 5 | 6 | 2 | 2 | 17 | 0.04 | 0.5 | 26
Prodbr | 7 | 5 | 2 | 2 | 20 | 0.06 | 2.0 | 19
Euclidex1 | 11 | 5 | 2 | 2 | 51 | 0.66 | DNF
Figure 8: Timings for computing abstractions of discrete systems and analyzing
the resulting abstractions. Legend: #V denotes number of system variables,
#Trs: number of transitions, Deg.: max. degree of the RHS, B0: degree limit
for monomials in the initial basis, Time: timing in seconds, #B*: number of
elements in the final basis, #I: invariants computed and DNF: did not finish
in 2hrs or out of memory crash.
## 5 Conclusion and Future Directions
Thus far, we have presented an approach that uses Change-Of-Bases
transformation for inferring abstractions of continuous, discrete and hybrid
systems. We have explored the theoretical underpinnings of our approach, its
connections to various invariant generation techniques presented earlier. Our
previous work presents an extension of the approach presented in this paper to
infer differential inequality abstractions [34]. Similar extensions for
discrete systems remain unexplored. Furthermore, the use of the abstractions
presented here to establish termination for transition systems is also a
promising line of future research. Future research will also focus on the use
of Lie symmetries to reduce the size of the ansatz or templates used in the
search for conserved quantities and CoB transformations [15].
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|
arxiv-papers
| 2012-04-19T13:40:45 |
2024-09-04T02:49:29.896656
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sriram Sankaranarayanan",
"submitter": "Sriram Sankaranarayanan",
"url": "https://arxiv.org/abs/1204.4347"
}
|
1204.4381
|
# Mapping the Linearly Polarized Spectral Line Emission around the Evolved
Star IRC+10216
J.M. Girart11affiliation: Institut de Ciències de l’Espai, (CSIC-IEEC), Campus
UAB, Facultat de Ciències, C5p 2, 08193 Bellaterra, Catalunya, Spain;
girart@ice.cat N. Patel22affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA W.
Vlemmings33affiliation: Department of Earth and Space Sciences, Chalmers
University of Technology, Onsala Space Observatory, SE-439 92 Onsala, Sweden
Ramprasad Rao44affiliation: Submillimeter Array, Academia Sinica Institute of
Astronomy and Astrophysics, 645 N. Aohoku Place, Hilo, HI 96720, USA
###### Abstract
We present spectro-polarimetric observations of several molecular lines
obtained with the Submillimeter Array (SMA)111The SMA is a joint project
between the Smithsonian Astrophysical Observatory and the Academia Sinica
Institute of Astronomy and Astrophysics, and is funded by the Smithsonian
Institution and the Academia Sinica. toward the carbon rich AGB star
IRC+10216. We have detected and mapped the linear polarization of the CO 3–2,
SiS 19–18 and CS 7–6 lines. The polarization arises at a distance of $\simeq
450$ AU from the star and is blueshifted with respect the Stokes I. The SiS
19–18 polarization pattern appears to be consistent with a locally radial
magnetic field configuration. However, the CO 3–2 and CS 7–6 line polarization
suggests an overall complex magnetic field morphology within the envelope.
This work demonstrates the feasibility of using spectro-polarimetric
observations to carry out tomographic imaging of the magnetic field in
circumstellar envelopes.
stars: AGB and post-AGB; stars: individual (IRC+10216, CW Leo); circumstellar
matter; polarization; submillimeter: stars
## 1 Introduction
High mass loss during the AGB phase is one of the main contributors to the
return of nucleo-synthesized material into the interstellar medium. A proper
understanding of AGB mass loss is thus crucial for the study of the chemical
evolution of the Galaxy. However, the exact mechanism or mechanisms
responsible for the AGB mass loss is still not clear. After the AGB phase, the
star evolves towards the Planetary Nebula (PN) phase. During this transition,
fast winds are launched that interact with the earlier circumstellar envelope
(CSE) that was created during the AGB. A large fraction of PNe are observed to
be aspherical, and the origin of the asphericity is attributed to the
influence of a binary companion, a disk, a magnetic field, or a combination of
these. The exact onset of asphericity is still unknown, and high angular
resolution molecular line observations indicate that several CSEs of AGB stars
already display various degrees of asymmetry. The mechanism responsible for
the creation of asymmetries is likely closely linked with that driving the
mass loss and can be directly probed by high angular resolution observations
of CSEs. In particular molecular line polarization observations are a unique
tool to study potential asymmetries in the CSE and/or determine the shape of
the circumstellar magnetic field.
The critical role of magnetic fields in star formation has been probed with
direct observations of polarized dust continuum emission toward both low and
high-mass star-forming regions (Girart et al., 2006, 2009; Rao et al., 2009).
Similar techniques are difficult to apply for AGB stars due to the need for
extremely high angular resolution ($<0.^{\prime\prime}5$) and high
sensitivity. A large number of studies have already been made of the magnetic
field induced polarization of maser lines. These studies have revealed
magnetic fields are present throughout the entire envelope. The SiO, H2O and
OH maser observations indicate that magnetic fields appear well ordered and
the Zeeman splitting indicates the field strengths range from several Gauss
close to the stellar surface to several mG at a few thousand AU (e.g., Etoka &
Diamond, 2004; Vlemmings et al., 2005, 2011; Herpin et al., 2006; Kemball et
al., 2009; Amiri et al., 2011). However, masers probe only a limited number of
lines of sight through the CSE and in most cases it is thus impossible to
fully reconstruct the magnetic field morphology throughout the envelope.
Furthermore, the most abundant masers are predominantly found around oxygen-
right (M-type) limiting the available source sample. Non-masing molecular
lines are however also predicted to be linearly polarized (e.g. Goldreich, &
Kylafis, 1981, 1982; Morris et al., 1985) to the level of a few percent, and
can provide more extensive probes of the entire envelope.
Polarized emission has however conclusively been detected from for example CO
in a number of star forming regions (e.g., Girart et al., 1999; Lai et al.,
2003; Cortes et al., 2005; Forbrich et al., 2008). The molecular line
polarization is due to the anisotropic radiation field from the central star
imparting angular momentum on the molecules or from anisotropic level
populations in the molecular magnetic substates when coupled with a magnetic
field. Without a (non-radial) magnetic field, and assuming a spherically
symmetric stellar wind, linear polarization should be radial or tangential and
should only be detectable at lines of sight away from the central star (Morris
et al., 1985). Thus, when these criteria are not met, molecular line
polarization provides a unique diagnostic of magnetic field morphology and
potential non-radial asymmetries in the stellar radiation field. Very
recently, Vlemmings et al. (2012) have reported the detection of CO polarized
emission toward IK Tau.
IRC+10216 (CW Leo) is a well studied AGB star with a high mass-loss rate,
$3\times 10^{-5}$ M⊙ yr-1, and a terminal velocity of 15 km s-1 (e.g. Young et
al., 1993). Due to the relatively close distance of 150 pc (Crosas & Menten,
1997), this star provides an ideal laboratory for studies of circumstellar
chemistry (Tenenbaum et al., 2010; Patel et al., 2011). Unlike typical Mira
variables, there are no masers associated with this source, except perhaps a
transition of SiS which may be a weak maser (Fonfría Expósito et al., 2006).
The dusty envelope of IRC+10216 shows arc-like structures in scattered light
(Mauron & Huggins, 1999) which extend over more than 1′ in angular radius.
Closer to the star, dust emission shows asymmetrical structures over scales of
2′′ (Men’shchikov et al., 2001; Menut et al., 2007). Optical and NIR (JHK)
interferometry reveals complex and time-varying structures on subarcsecond
scales close to the star (Tuthill et al., 2000, 2005; Weigelt et al., 2002).
Non-spherical structures were also revealed in linear polarization maps
produced by dust scattering at $0\farcs 25$ resolution in H band (Murakawa et
al., 2005).
A previous polarization detection in IRC+10216 was reported by (Glenn et al.,
1997) for the CS 2–1 line. In this letter, we present the detection of the
linear polarized emission for the CO 3–2 SiS 19–18 and CS 7–6 lines in
IRC+10216. This is the first time that maps of the linear polarized emission
in IRC+10216 are presented.
## 2 Observations
The SMA observations were taken on 2010 November 24 in the compact
configuration. The receiver was tuned to cover the 330.6-334.5 and 342.6-346.5
GHz frequencies in the lower (LSB) and upper side band, (USB) respectively.
The phase center of the telescope was RA(J2000.0)$=9^{\rm h}47^{\rm m}57\fs
38$ and DEC(J2000.0)$=13\arcdeg 16\arcmin 43\farcs 70$. The correlator
provided a spectral resolution of about 0.8 MHz (i.e., 0.7 km s-1 at 345 GHz).
The gain calibrators were QSOs J0854+201 and J1058+015 . The bandpass and
polarization calibrator was 3c454.3, which was observed in a parallactic angle
range of $\sim 120\arcdeg$. The absolute flux scale was determined from
observations of Titan. The flux uncertainty was estimated to be $\sim 20$%.
The data were reduced using the MIRIAD software package (Wright & Sault,
1993). The SMA conducts polarimetric observations by cross correlating
orthogonal circularly polarizations (CP). The CP is produced by inserting
quarter wave plates in front of the receivers which are inherently linearly
polarized. A detailed description of the instrumentation techniques as well as
calibration issues is discussed in Marrone & Rao (2008) and Marrone et al.
(2006). In order to obtain a more accurate polarization calibration, we solve
for the leakage solution independently for the strongest detected lines (CO
and 13CO 3–2, H13CN 3–2, CS 7–6 and SiS 19–18) by selecting a frequency range
of $\simeq 1.5$ GHz centered within 0.1 GHz with respect to the rest frequency
of each line. We found polarization leakages between 1 and 2% for the USB,
while the LSB leakages were between 2 and 4%. These leakages were measured to
an accuracy of 0.1%. Self-calibration was performed independently for the USB
and LSB on the continuum emission of IRC+10216. All maps were done with
Natural weighting in order to maximize the sensitivity, which yielded a
synthesized beam of $2\farcs 6\times 1\farcs 6$ with a position angle of
PA$\simeq 0\arcdeg$ (see caption of Figure 2 for more specific values).
Significant polarization was only detected in the CO 3–2, CS 7–6 and SiS 7–6
lines, so this paper presents and discuss the detection significance for these
three lines.
Figure 1: Spectra of the Stokes $I$ (top, black line), $U$ (center, blue line)
and $Q$ (bottom, red line) emission of the CO $J$=3–2 (left panel), SiS
$J$=19–18 (central panel) and CS $J$=7–6 lines (right panel). For each line
this spectrum was taken at the position where the maximum polarized emission
is detected, after convolving the maps with a Gaussian having a FWHM of
$4^{\prime\prime}\times 3^{\prime\prime}$.
## 3 Results and analysis
Figure 1 shows the Stokes $I$, $Q$ and $U$ obtained at the positions of
maximum polarized intensity for the CO 3–2, CS 7–6 and SiS 19–18 lines. Figure
2 shows the polarization maps for the emission of these lines averaged over
the velocity range that maximizes the polarized emission, which is different
for each line.
Figure 2: Color image of the linearly polarized intensity of the CO 3–2 (left
panel), SiS 19–18 (central panel) and CS 7–6 lines (right panel), overlapped
with the contour maps of the I emission for the respective lines. The orange
bars represent the polarization vectors. The CS and CO maps show the emission
at the $v_{\rm LSR}$ velocity of $-29$ km s-1 averaged over 16 km s-1. The SiS
map shows the emission at $v_{\rm LSR}$ $=$-31.5 km s-1 averaged over 20 km
s-1. The contour levels are 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95% of the
peak intensity. The wedge shows the polarized intensity scale in units of Jy
beam-1 . The synthesized beam is shown in the bottom left corner of each
panel.
The CO 3-2 lines is the strongest line detected in our observations (the line
is 50% brighter than the other two lines in the shortest baselines of the
visibility plane). This line emission is known to be spatially very extended,
much beyond the primary beam of the SMA antennas (Truong-Bach et al., 1991).
However the SMA filters out the CO emission that arises from structures larger
than about $10^{\prime\prime}$ (Girart et al., 2006), so the detected emission
appears relatively compact. Indeed, the CO averaged emission over the
blueshifted component shown in Fig. 2 has a relatively compact component (with
a radius of $\simeq 2^{\prime\prime}$), surrounded by a weak components that
extends up $5^{\prime\prime}$ from the center at an intensity level of about
5% of the maximum. The CS and SiS show also a compact component with similar
dimensions, but they lack the weaker and extended component. The Stokes I
emission of the 3 lines has a similar brightness, $\simeq 120$ K for the CS
and SiS and 150 K for CO.
The $rms$ noise of the Stokes Q and U emission appears to slightly increase in
the channels where the line emission is brightest. This effect is seen in the
three lines, though at different levels, from a maximum increase of 5%, 13%
and 20% for the CS, SiS and CO lines, respectively. This increase is probably
produced by the residual leakage (estimated to be of $\simeq 0.1$%, Marrone &
Rao, 2008) as the total intensity is very strong in the central channels. The
larger increase for the CO may be due because it is the most extended and the
brightest line (specially the shortest baselines). Taking into account this
increase of noise, there is still significant emission in the Stokes $Q$ and
$U$ maps of the SiS at the $\simeq 6$-$\sigma$ level, and in CO and CS lines
at the $\simeq 5$-$\sigma$ level. The linear polarization maps were computed
by using a 3-$\sigma$ cutoff, where $\sigma$ is the rms noise in the map where
the polarization is detected.
The CO linear polarization arises from both the U and Q components, being
relatively bright in the later. The CS and SiS lines show mainly polarized
emission in the U component. Interestingly, the polarized emission in the
three lines appears to be blueshifted with respect to the total intensity.
In order to derive the polarization pattern in the plane of the sky, we have
computed polarization maps with the emission averaged over the velocity range
where the polarization intensity is detected (see Figure 2). The polarization
degree at the position where it is strongest is of $\simeq 2$% for the CO and
SiS lines, and of $\simeq 4$% for the CS. The CO 3–2 polarization arises from
two spots, one at the center of IRC+10216 and the other located $\simeq
3^{\prime\prime}$ to the East. The polarization vectors are oriented roughly
North-South, changing slightly from a position angle of $PA=-11\arcdeg$ at the
eastern spot, to $-25\arcdeg$ at the central spot. The SiS 19–18 polarized
emission arises mainly from the north-eastern quadrant of the IRC+10216
envelope (the polarization peak is located $\simeq 2\farcs 6$ from the
center’s envelope). There are two other small spots, with a polarized emission
too marginal to be further considered here. The polarization of the main
component has a mean position angle of about $-41\arcdeg$, but the pattern of
the polarization vectors appear to form an arc, following roughly the contours
of the Stokes I emission. The CS 7–6 polarization arises from the envelope’s
south-western quadrant (the polarization peak is located $\simeq 2\farcs 9$
from the center’s envelope). The polarization $PA$ pattern is quite uniform
with an averaged value of $\simeq 48\arcdeg$.
The SiS polarization vectors’ pattern suggest a radial distribution.
Therefore, we have compared the polarization vector direction with the
expected radial direction (with respect to the envelope center) at the
position the vectors. Figure 3 shows the difference between the polarization
vectors and the radial directions. On one hand, the SiS polarization vectors
are all almost perpendicular to the radial direction, i.e., they form a nearly
perfect concentric arc-like pattern with respect to the envelope’s center. On
the other hand and despite the low polarization statistics, this is not the
case for the CO and CS polarization vectors.
## 4 Discussion and conclusions
The detailed analysis of population of the magnetic sub-levels in rotational
lines show that the highest polarized emission is expected for volume
densities similar to the critical density of the observed transition, and
depending on the transition and on the molecule, the polarization can still be
significant even at densities ten times higher (Deguchi & Watson, 1984). This
suggests that the polarization detected in the CO 3–2 line should arise at
volume densities of $\sim 10^{4}$ cm-3, so at the outer regions of the shell,
whereas the the SiS 19–18 and CS 7–6 polarization is expected to trace inner
regions, at densities of $\sim 10^{7}$ cm-3.
One of the interesting features is that in the three lines the linear
polarization is blueshifted with respect to the total emission (this effect is
more significant in the SiS line). Considering that the envelope is expanding,
this suggest that the polarized emission is being detected at the side of a
shell facing us and with the aforementioned volume densities. In addition,
most of the polarized emission arises about $3^{\prime\prime}$ offset ($\simeq
450$ AU in projection) of the envelope’s center. Thus the optical depth is
probably playing an important role. Indeed, subarcsecond resolution maps in
the IR (Menut et al., 2007; Leão et al., 2006) and HCN 3–2 emission (Shinnaga
et al., 2009) show that the molecular distribution is asymmetrical. This
suggests the anisotropy in the radiation field to be a cause for the
polarization pattern to be not distributed spherically. This is also in
agreement with the single-dish detection of the CS 2–1 line polarization
towards the center of IRC+10216, which suggests that there is a non-radial
polarization pattern (Glenn et al., 1997).
Figure 3: Distribution of the difference between the position angle of the
SiS, CS and CO polarization vectors and the radial direction with respect to
the center of the envelope. We have used a Nyquist sample of the polarization
vectors to compute this difference, excluding the vector closer than $\simeq
1^{\prime\prime}$ from the envelope center.
In circumstellar envelopes and for mm molecular lines the Zeeman splitting is
much larger than the collision and spontaneous rates, even for magnetic fields
strength of only few $\mu$G. Therefore, the polarization should be aligned
parallel or perpendicular to the magnetic field (Kylafis, 1983; Morris et al.,
1985). An overall radial magnetic field is expected if it is weak enough to be
energetically irrelevant, i.e., the magnetic energy is significantly smaller
than the mechanical energy of the stellar wind:
$B\ll(\dot{m}v_{t})^{1/2}(D\Omega)^{-1}$, where $\dot{m}$ is the mass loss
rate, $v_{t}$ the terminal velocity, $D$ the distance of IRC+10216 and
$\Omega$ the angular radius (Glenn et al., 1997). Using the measured values in
IRC+10216 (see § 1) and at the distance where the polarization is detected
($\simeq 3^{\prime\prime}$, 450 AU), this condition is satisfied if $B\ll 8$
mG. CN Zeeman splitting observations in IRC+10216 indicate a strength of
$\simeq 9$ mG at a larger distance (2500 AU, Herpin et al., 2009) . For a
solar-type and toroidal magnetic field configuration ($B\propto r^{-2}$ and
$r^{-1}$, respectively; Vlemmings, 2011) the expected strength where the
polarization is detected would be in the 50 to 300 mG range. Therefore, the
magnetic field is strong enough to not be radially shaped by the wind.
The measured polarized vectors can be either parallel or perpendicular to the
projected magnetic field direction in plane-of-the-sky (Kylafis, 1983). A
proper radiative transfer analysis and a model of the physical conditions of
the envelope (including the magnetic field configuration) is needed to solve
for this degeneracy. This applies for the measured CO and CS polarization
maps. Nevertheless, in the region where the SiS polarization is detected, the
polarization pattern is indicative of a radial magnetic field (see Fig. 3).
Theoretical studies show that in an expanding circumstellar envelope with a
radial magnetic field, the polarization will be parallel (or perpendicular) to
the radial direction if it arises at radii lower (or higher) than at certain
impact parameter, $R_{J}$ (Deguchi & Watson, 1984; Morris et al., 1985).
$R_{J}$ is the radius where the spontaneous emission rate for the
$J\rightarrow J-1$ transition is equal to the IR absorption rate. According to
Morris et al. (1985), in IRC+10216 the value of $R_{J}$ for the SiS 2–1 line
is $R_{2}\simeq 5\times 10^{16}$ cm. The spontaneous emission rate increases
with J as $J^{4}/(2J+1)$, and the IR absorption rate goes as $R^{-2}$, so
$R_{19}\simeq 2\times 10^{15}$ cm. The region where SiS 19–18 polarization is
detected arises at a radius of $\simeq 5\times 10^{15}$ cm from the star.
Therefore the SiS 19–18 polarization pattern is in agreement with the
theoretical predictions if the magnetic field is radial in the region where
SiS the polarization is detected.
In summary, we have detected and mapped the polarization pattern for the first
time in IRC+10216, through spectro-polarimetric observations of the CO 3–2,
SiS 19–18 and CS 7–6 lines. Although, the data obtained so far lack the
sensitivity to allow us to make specific predictions on the magnetic
configuration of the IRC+10216 envelope, the polarization pattern measured
discards that the magnetic field configuration has a global radial pattern
(this is only observed locally where the SiS 19–18 polarization is detected),
but it possibly has a rather complex magnetic field morphology. In addition,
the polarization detection in three different molecular lines implies that
with the higher sensitivity and angular resolution that the Atacama Large
Millimeter Array is going to provide, it would be possible to carry out
spectro-polarimetric observations for a tomographic imaging of the magnetic
field in circumstellar envelopes. However, the ambiguity between the
polarization direction with respect to the magnetic field direction, implies
that in order to properly relate the polarization pattern with the magnetic
field, a complete radiative transfer analysis should be made.
We thank all members of the SMA staff that made these observations possible.
JMG is supported by the Spanish MICINN AYA2008-06189-C03 and the Catalan AGAUR
2009SGR1172 grants. WV acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG) through the Emmy Noether Research grant VL
61/3-1.
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|
arxiv-papers
| 2012-04-19T15:35:26 |
2024-09-04T02:49:29.912847
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. M. Girart, N. Patel, W. Vlemmings and Ramprasad Rao",
"submitter": "Josep Miquel Girart",
"url": "https://arxiv.org/abs/1204.4381"
}
|
1204.4386
|
# Heat conduction: hyperbolic self-similar shock-waves in solids
I. F. Barnaa and R. Kersnerb a Energy Research Centre of the Hungarian Academy
of Sciences,
(KFKI-AEKI), H-1525 Budapest, P.O. Box 49, Hungary,
bUniversity of Pécs, PMMK, Department of Mathematics and Informatics,
Boszorkány u. 2, Pécs, Hungary
###### Abstract
Analytic solutions for cylindrical thermal waves in solid medium is given
based on the nonlinear hyperbolic system of heat flux relaxation and energy
conservation equations. The Fourier-Cattaneo phenomenological law is
generalized where the relaxation time and heat propagation coefficient have a
general power law temperature dependence. From such laws one cannot form a
second order parabolic or telegraph-type equation. We consider the original
non-linear hyperbolic system itself with the self-similar Ansatz for the
temperature distribution and for the heat flux. As results continuous and
shock-wave solutions are presented. For physical establishment numerous
materials with various temperature dependent heat conduction coefficients are
mentioned.
###### pacs:
44.90.+c, 02.30.Jr
In contemporary heat transport theory (ever since Maxwell’s paper maxw ) it is
widely accepted in the literature that only for stationary and weakly non-
stationary temperature fields the constitutive equation assumes that a
temperature gradient $\nabla T$ instantaneously produces heat flux ${\bf{q}}$
according to the Fourier law
${\bf{q}}({\bf{x}},t)=-\kappa{\bf{\nabla}}T({\bf{x}},t).$ (1)
Combining this equation with the energy conservation law the usual parabolic
heat conduction equation is given. Heat conduction mechanisms can be
classified via the temperature dependence of the coefficient $\kappa\sim
T^{\nu}$. There are three different cases of thermal conductivity, normal heat
conduction which obeys the Fourier law $(\nu=0)$, slow $(\nu>0)$ and fast heat
conduction $-2<\nu<0$.
In plasma physics if the temperature range is between $10^{5}$K and $10^{8}$ K
then the coefficient of the heat conductivity $\kappa$ depends on the
temperature and density of the material. It is usually assumed to have a power
dependence $\kappa=\kappa_{0}T^{\nu}v^{\mu}$ where $v=1/\rho$ is the specific
volume the coefficient $\kappa_{0}$ and the exponents $\nu,\mu$ depend on the
heat conduction mechanism zeld . With radiation heat conduction one has
$4\leq\nu\leq 6,\hskip 5.69054pt1\leq\mu\leq 2$; with electron heat conduction
and fully ionized plasma $\nu=5/2,\hskip 5.69054pt\mu=0$. For magnetically
confined non-neutral plasma the classical heat conduction coefficient is the
following dubin $\kappa\approx\frac{c_{1}}{\sqrt{T}}ln[c_{2}T^{3/2}]$.
Parabolic thermal wave theory is based on this approach zeld ; zk . In plasmas
heat conduction is strongly coupled to flow properties which we will not
consider in the following. The linear parabolic theory predicts infinite speed
of propagation which is known as the ”paradox of heat conduction” (PHC). The
following two theories resolve this contradiction.
However, if the time scale of local temperature variation is very small, Eq.
(1) is replaced by
${\bf{q}}({\bf{x}},t+\tau)=-\kappa\nabla T({\bf{x}},t)$ (2)
where $\tau$ is called the thermal relaxation time. This is a thermodynamic
property of the materials which was determined experimentally for large number
of materials. Although $\tau$ turns out to be very small in many instances
e.g. is of order of picoseconds for most metals, there are several materials
where this is not the case, most notably sand (21 s), H acid (25 s), NaHCO3
(29 s), and biological tissue (1-100 s) ind .
Unlike the Fourier’s heat conduction law, this constitutive equation is non-
local in time. The desired local character can be restored with the Taylor
expansion of ${\bf{q}}$ by time which is usually truncated at the first order
namely
${\bf{q}}({\bf{x}},t)+\tau\frac{\partial{\bf{q}}({\bf{x}},t)}{\partial
t}=-\kappa\nabla T({\bf{x}},t).$ (3)
This is the well-known Cattaneo heat conduction law cat the second term on
the left hand side is known as the ”thermal intertia”. (Unfortunately, this
form is not Galilean invariant, and gives a paradoxial results if the media is
in motion, this problem was eliminated in by christov .) Combining this
constitutive equation with the energy conservation yields the hyperbolic
telegraph heat conduction equation where $\tau$ and $\kappa$ are constants.
Hyperbolic equations usually ensure finite propagation velocity.
Unfortunately, telegraph equations has no self-similar solutions which would
be a desirable physical property. In the work of barn a non-autonomous
telegraph-type heat conduction equation is presented with self-similar non-
oscillating compactly supported solutions. A review with a large number of
physical models of heat waves can be found in ind ; prec . A recent work on
the speed of heat waves was published by makai .
Our starting point is the following
$\displaystyle q_{t}$ $\displaystyle=$
$\displaystyle-\frac{q}{\tau}-\frac{\kappa}{\tau}T_{r},$ (4) $\displaystyle
c_{0}T_{t}$ $\displaystyle=$ $\displaystyle-q_{r}-\frac{q}{r}.$ (5)
The first equation of the system is the generalized Fourier-Cattaneo heat
conduction law and the second one is the energy conservation condition for the
radial coordinate. The heat flux $q=q(r,t)$ and the temperature dependence
$T=T(r,t)$ have radial coordinate and time dependence. The subscripts r and t
notate the partial derivatives with respect to the radial coordinate and the
time, respectively. (From now on we investigate the radial coordinate of a
cylindrical symmetric problem as spatial dependence.) The parameter
$c_{0}=\rho c$ where $\rho$ is the mass density and $c$ is the specific heat.
Second order effects such as compressibility are neglected ($\rho$ and are c
constants during the process).
In the following we shall suppose that the heat conduction coefficient and the
thermal relaxation depend on temperature on the following way
$\kappa=\kappa_{0}T^{\omega},\hskip 28.45274pt\tau=\tau_{0}T^{-\epsilon}.$ (6)
The $\kappa_{0}$ and $\tau_{0}$ are real numbers with the proper physical
dimensions. Now our dimensionless system reads
$\displaystyle q_{t}$ $\displaystyle=$
$\displaystyle-T^{\epsilon}q-T^{\epsilon+\omega}T_{r},$ (7) $\displaystyle
T_{t}$ $\displaystyle=$ $\displaystyle-q_{r}-\frac{q}{r}.$ (8)
There are various phenomenological heat conduction laws available for all kind
of solids, without completeness we mention some well-known examples. For pure
metals according to jones (Page 275 Eq. 27.3) the Wiedemann-Franz low the
thermal conductivity is proportional with the electrical conductivity $\sigma$
times the temperature $\kappa=\sigma LT.$ The proportionality constant L is
the so called Lorentz number with the approximate numerical value of
$2.44\times 10^{-8}W\Omega K^{-2}$. For exact numerical data for various
metals see ashroft . The relaxation time $\tau$ is proportional to the heat
conduction coefficient divided by the temperature. For metals with impurities
the thermal resistivity (inverse of the thermal conductivity) is
$\kappa^{-1}=AT^{2}+BT^{-1}$ where A and B can be obtained from microscopic
calculation based on quantum mechanics jones (Page 297 Eq. 40.11).
A hard-sphere model for dense fluids from netl derives a relation where the
heat flux $q(x,t)=a\nabla T(x,t)+q^{2}(x,t)$ which certainly means a non-
linear heat propagation process. For the heat conduction in nanofluid
suspensions vadasz derives the $\kappa\approx c/(T_{2}-T_{1})$ law with
additional time dependence. Another exotic and very promising new materials
are the carbon nanotubes which have exotic heat conduction properties. Small
et al. small performed heat conductivity measurements and found that at low
temperatures there are two distinct regimes $\kappa(T)\sim T^{2.5}\>\>(T<50K)$
and $\kappa(t)\sim T^{2}\>\>(50<T<150K)$. Beyond this regime there is a
deviance from this quadratic temperature dependence and the maximum $\kappa$
value lies at 320 K. Above this value - at large temperatures - there is a
$\kappa(T)\sim 1/T$ dependence according to berber . Additional nanoscale
systems (like silicon films, or multiwall carbon nanotubes) have exotic
temperature dependent heat conduction coefficients as well, for more see
cahill . For encased graphene the heat conduction coefficient is $\kappa\sim
T^{\beta}$ where $1.5<\beta<2$ at low temperature $(T<150K)$ jang . A recent
review of thermal properties of graphene and nanostructured carbon materials
can be found in nature .
Our model is presented to describe the heat conduction of any kind of solid
state without additional restrictions, therefore room or even higher
temperature can be considered with large negative $\omega$ exponents.
Even from these examples we can see that it has a need to investigate the
general heat conduction problem, where the coefficients have general power law
dependence.
We look for the solutions of (7,8) in the most general self-similar form
$T=t^{-\alpha}f(\eta),\hskip 28.45274ptq=t^{-\delta}g(\eta).$ (9)
For a better transparency in the following we introduce a new variable
$\eta=\frac{r}{t^{\beta}}$, where $\alpha,\beta,\delta$ are all real numbers.
The similarity exponents $\alpha,\delta$ and $\beta$ are of primary physical
importance since $\alpha,\delta$ represents the rate of decay of the magnitude
T or q, while $\beta$ is the rate of spread (or contraction if $\beta<0$ ) of
the space distribution as time goes on. Self-similar solutions exclude the
existence of any single time scale in the investigated system.
We substitute (9) into (7) and (8). It can be checked that
$\alpha=\frac{1}{\omega+1},\hskip 8.53581pt\beta=\frac{1}{2(\omega+1)},\hskip
8.53581pt\delta=\frac{2\omega+3}{2(\omega+1)},\hskip
8.53581pt\epsilon=\omega+1.$ (10)
Then we can obtain the shape functions f and g the following ordinary
differential equation (ODE) system
$\displaystyle\delta g+\beta\eta g^{\prime}$ $\displaystyle=$ $\displaystyle
gf^{\omega+1}+f^{2\omega+1}f^{\prime},$ (11) $\displaystyle(\eta g)^{\prime}$
$\displaystyle=$ $\displaystyle\beta(\eta^{2}f)^{\prime}$ (12)
where prime means derivation with respect to $\eta$.
The first lucky moment is that (12) relates f and g in a simple way
$g=\beta\eta f$ (13)
if the $\alpha=2\beta$ universality relation is fulfilled.
Note, that we can immediately read how the self-similar solutions of the
temperature distribution T and the heat flux q depend on $\omega$
$T=t^{\frac{-1}{\omega+1}}f\left(\frac{r}{t^{\frac{1}{2(\omega+1)}}}\right),\hskip
22.76219ptq=t^{\frac{2\omega+3}{2(\omega+1)}}g\left(\frac{r}{t^{\frac{1}{2(\omega+1)}}}\right).$
(14)
The parameter dependence of the complete heat conduction coefficient and
relaxation time can be expressed via $\omega$ as well
$\kappa=\kappa_{0}t^{\frac{-\omega}{\omega+1}}f^{\omega}\left(\frac{r}{t^{\frac{1}{2(\omega+1)}}}\right),\hskip
22.76219pt\tau=\kappa_{0}t^{-1}f^{\omega+1}\left(\frac{r}{t^{\frac{1}{2(\omega+1)}}}\right).$
(15)
Recall that $\omega>-1$. These are already very informative and useful
relations to investigate the global properties of the solutions, note that
such kind of analysis are available for large number of complex mechanical and
flow problems sed .
Substituting these relations back to Eq. (11) after some algebra we arrive at
the following non-linear first-order ODE
$\frac{df}{d\eta^{2}}\left(\beta^{2}\eta^{2}-f^{2\omega+1}\right)=\frac{\beta
f}{2}[f^{\omega+1}-(2\beta+1)].$ (16)
Put $y=\eta^{2}$ and $x=f$. With this notation eq. (16) becomes linear for
$y(x)$ (this is the second lucky moment of investigation):
$\frac{dy}{dx}=\frac{y(x)-4(\omega+1)^{2}x^{2\omega+1}}{x[(\omega+1)x^{\omega+1}-\omega-2]}.$
(17)
Plainly, $f\equiv 0$ is a solution to eq. (16). If $y(x)$ the solution of eq.
(17) is strictly monotonic then so is the inverse function $f=x$ and no
discontinuity. However if $y(x)$ is not monotonic on some interval
$(x_{1},x_{2})$ and has a turning point at $x_{0}\epsilon(x_{1},x_{2})$ then
the inverse $(f=x)$ has sense on $[0,y(x)]$ only. One sets $f=0$ for
$y>y(x_{0})$ and the discontinuity a $y(x_{0})$ is apparent. The analytical
investigation of the linear equation (17) is, in general easier than of eq
(16). In some cases (for some $\omega$s) one can have more explicit or almost
explicit solutions.
There are two examples:
The first case is for $\omega=0,(\alpha=1,\beta=1/2,\delta=3/2,\epsilon=1)$.
This example was studied by choi in some details. The corresponding ODE (17)
reads $y^{\prime}=(y-4x)/x(x-2)$ which has a solution
$y=8+[(x-2)/x]^{1/2}[c_{1}-8ln(\sqrt{x}+\sqrt{x-2})]$ where $c_{1}$ is a
constant.
It is clear that must be $x\geq 2$ and $y(x)$ is monotonic for $x>2$ until
$x_{0}$ where $y=0$. This means that $x(y)$ exists and monotonic on some
interval $[0,y_{0}],x(y_{0})=2;$ for $y\geq y_{0}$ we have $x(y)=0$ so the
discontinuity. For a better understanding Figure 1a presents the graph of
solution of Eq. (17) through the point (3,0.5). The inverse of this function
for $x>2$ is shown on Fig. 1b (the nonzero part). The solid line is a solution
through the $f(0)=10.8$ point. Figure 2 presents the shock-wave propagation of
the temperature distribution $T(r,t)$ for $\omega=0$.
The second case is for $\omega=-1/2,(\alpha=2,\beta=1,\delta=2,\epsilon=1/2)$.
Now Eq. (17) takes the form of $\frac{dy}{dx}=2(y-1)/[x(\sqrt{x}-3)]$. It can
be checked that $y=c_{2}x^{-2/3}(x^{1/2}-3)^{4/3}$ is a solution for any
$c_{2}>0.$ Take $c_{2}=1$. The function $y(x)$ is monotonic on $(0,9),y(9)=0$.
Returning to original variables we have $f=9/[(\eta^{2})^{3/4}+1]^{2}$ (which
is plainly less than 9!) According to eq. (14) temperature and heat flux
distributions are
$T=\frac{9t}{(r^{3/2}+t^{3/2})^{2}},\hskip
56.9055ptq=\frac{9r}{(r^{3/2}+t^{3/2})^{2}}.$ (18)
These solutions are not discontinuous. Analytical and numerical calculus
suggest that $\omega=-1/2$ is a critical exponent: for $-1<\omega\leq-1/2$ the
solutions are continuous, for $\omega>-1/2$ the shocks always appear.
In summary
We presented a hyperbolic model for heat conduction in solids where the
relaxation time and heat conduction coefficient is a power law function of
time. There are basically two different regimes available for different power
laws. For $\\-1<\omega\leq-1/2$ the solutions are continuous for all positive
time and radial coordinate, for $\omega>-1/2$ the solutions are only
continuous on a finite and closed $[0:\eta_{0}]$ interval and have a finite
jump at the the endpoint $\eta_{0}$. As physical interpretation numerous
materials and solid state systems mentioned with temperature dependent heat
conduction coefficients.
The paper is dedicated to Annabella Barna who was born on 20th of December
2011.
a) b)
Figure 1: The direction field of a) Eq. (17) for $\omega=0$ and b) Eq. (16)
for $\omega=0$ The solid line presents numerical solutions for a) $y(3)=0.5$
and for b) $f(0)=10.8$.
Figure 2: The shock-wave propagation of the temperature distribution of
$T(r,t)$ for $\omega=0$
## References
* (1) J.C. Maxwell, Phil. Trans. R. Soc. Lond. 157, 49 (1867).
* (2) Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Academic New York 1966).
* (3) D.H.E. Dubin and T.M. O’Neil, Phys. Rev. Lett. 78, 3868 (1997).
* (4) Y.B. Zeldovich and A.S. Kompaneets, Collection Dedicated to the 70th Birthday of A.F. Joffe, Izdat. Akad. Nauk SSSR 1950, p.61.
* (5) D.S. Chandrasekharaiah, Appl. Mech. Rev. 39, 355 (1986); ibid 51, 705 (1989).
* (6) C. Cattaneo, Sulla conduzione del calore Atti. sem Mat. Fis. Univ. Modena 3, 83 (1948).
* (7) C.I. Christov and P.M. Jordan, Phy. Rev. Lett. 94, 154301 (2005).
* (8) I.F. Barna and R. Kersner, J. Phys. A Math. Theor. 43, 375210 (2010), Adv. Studies Theor. Phys. 5, 193 (2011).
* (9) D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, 41 1989; ibid 62, 375 (1990).
* (10) M. Makai, Eur. Phys. Lett. 96, 40010 (2011).
* (11) H.E. Wilhelm and S.H. Choi, J. Chem. Phys. 63, 2119 (1975).
* (12) H. Jones, Handb. Phys. 19, 227 (1956).
* (13) N.W. Ashcroft and D.N. Mermin, Solid State Physics Thomson Learning Inc.
* (14) R.E. Nettleton, J. Phys. A: Math. Gen 20, 4017 (1987).
* (15) P. Vadasz, J. Heat Trans. 128, 465 (2006).
* (16) J. P. Small, L. Shi and P. Kim, Solid. State. Commun. 127, 181 (2003).
* (17) S. Berber, Y.-K. Kwon and D. Tománek, Phys. Rev. Lett. 84, 4613 (2003).
* (18) D.G. Cahill et al., Journ. Appl. Phys. 93, 793 (2003).
* (19) W. Jang et al. Nano. Lett. 10, 3909 (2010).
* (20) A.A. Balandin, Nature Materials, 10, 569 (2011).
* (21) L. Sedov, Similarity and Dimensional Methods in Mechanics, CRC Press (1993).
|
arxiv-papers
| 2012-04-19T15:50:29 |
2024-09-04T02:49:29.920480
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Imre Ferenc Barna, Robert Kersner",
"submitter": "Imre Ferenc Barna Dr.",
"url": "https://arxiv.org/abs/1204.4386"
}
|
1204.4488
|
# Stabilité et simplicité positives
Mohammed Belkasmi
Université de Lyon
Université Lyon 1
CNRS UMR 5208 Institut Camille Jordan
43 blvd du 11 novembre 1918
69622 Villeurbanne cedex, France
###### Abstract
Dans la première partie de cet aricle on étudie la notion d’extension
universel en théorie des modèles positive, on donnons une preuve plus simple
au théorème de la conservation de la séparation par passage au restrictions
élémentaires. Elle nous permet aussi de donner une constructions du domaine
universel.
Dans le reste de l’article nous continuons l’étude de la stabilité positive
déja entamé par Ben Yaacov dans [3]. Nous ajoutons une autre caractérisation
de la stabilité par une propriété d’ordre postive, et nous étudions quelques
conséquences de la stabilité positive. A la fin de l’article nous proposons
une forme de stabilité faible compatible avec la notions de la simplicité
positive au sens de Pillay.
## 1 Notions de la théorie des modèles positive
cette section, nous rappellerons certaines définitions et notions de la
logique positive. Pour plus de détails, [4], [1] sont des sources suffisamment
complètes.
### 1.1 Outiles de la logique positive
###### Définition 1
:
* •
Une formule positive est une fomule existentielle de la forme
$\exists\bar{x}\bigvee_{i}\bigwedge_{j}\varphi_{ij}(\bar{x})$
où $\varphi_{ij}$ sont des formules atomiques.
* •
Un énoncé h-universel est un énoncé de la forme
$\neg\exists\bar{x}\varphi(\bar{x})$, où $\varphi$ est une formule positive
libre.
* •
Un énoncé h-inductif est une conjonction d’énoncés de la forme
$\forall\bar{x}\,\varphi(\bar{x})\rightarrow\psi(\bar{x})$, où $\varphi$ et
$\psi$ sont des formules positives.
* •
Une théorie h-inductive est un ensemble d’énoncés h-inductifs consistant.
* •
Soient $A,B$ deux L-structures et $f$ un homomorphisme de $A$ dans $B$, $f$
est une immersion si pour toutes formules positves $\varphi$ et uples
$\bar{a}\in A$ on a
$A\models\varphi(\bar{a})\Leftrightarrow B\models\varphi(f(\bar{a}))$
.
Une classe de modèles d’une théorie h-inductive qui elle seule représente la
théorie et où presque la totalité des études de la théorie des modèles
positive sont faites est la classe des existentiellement clos positives de la
théorie.
On retiendra les notations $\models$ et $\vdash$ pour noter la satisfaction et
le fait d’être conséquence respectivement. Parfois, afin de faciliter la
lecture, nous utiliserons la notation $A\models\neg\varphi(\bar{a})$ pour
remplacer $A\nvDash\varphi(\bar{a})$ dans le cas d’une formule positive
$\varphi(\bar{x})$ et d’un uple $\bar{a}$ extrait de la structure $A$.
###### Définition 2
Un modèle $A$ d’une théorie h-inductive $T$ est un existensiellement clos
positif (pec) si tout homomorphisme d’un modèle de $T$ dans $A$ est une
immersion.
Dans toute la suite on utilise le mot pec pour abréger existentiellement clos
positif.
Exemple:
Soient $L=\\{=\\}$, et $T$ la théorie h-inductive de l’égalité. Un pec de $T$
ne peut pas avoir plus d’un élément. En effet
{a, b,….} {a}
est un homomorphisme non injectif. $\square$
###### Définition 3
Une théorie h-inductive est dite modéle-compléte si tout modéle de $T$ est
pec.
Deux théories h-inductives sont dites compagnes si elles ont les mêmes pec.
Toute théorie h-inductive $T$ a:
* •
Une compagne maximale dite enveloppe de Kaiser notée $Tk(T)$, qui est la
théorie h-inductive des pec de $T$.
* •
Une compagne minimale notée $Tu(T)$ qui est la théorie h-universelle des pec
de T.
###### Définition 4
[14] Soient $A,B$ deux L-structures, on dit que $B$ est une extension
élémentaire positive de $A$, si $B$ est un pec de la théorie $Tk(A)$ dans le
langage $L(A)$.
### 1.2 Techniques d’amalgamation
nous abordons les techniques d’amalgamations proprement dites. L’efficacité de
cette notion sera illustrée par son application dans les différentes sections
de cet article.
###### Fait 1 ([4], Amalgamation asymétrique )
Soient $A$, $B$, $C$, des $L$-structures, $g$ une immersion de $A$ dans $B$ et
$h$ un homomorphisme de $A$ dans $C$ , alors il existe $D$, un modèle de
$Tk(C)$, un homomorphisme $g^{\prime}$ de $B$ dans $D$, et une immersion
$h^{\prime}$ de $C$ dans $D$ tels que $g^{\prime}\circ g=h^{\prime}\circ h$ .
###### Fait 2 (, Amalgamation kaisserienne)
Soient $A$ une L-structure, $B$ un modèle de $Tk(A)$ et $C$ une $L$-structure
dans laquelle $A$ s’immerge. Alors il existe $D$ un modèle de $Tk(C)$, et deux
immersions $\varphi$, $\psi$, telles que le diagramme suivant est commutatif
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\scriptstyle{im}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{{D}}$
###### Définition 5
Soit $T$ une théorie h-inductive. Un modèle $A$ de $T$ est dit une base
d’amalgamation, si pour tous $B,C$ modèles de $T$, où $A$ se continue par des
homomorphismes $f$ et $g$, il existe $D$ un modèle de $T$, et
$f^{\prime},g^{\prime}$ des homomorphismes tels que le diagramme suivant est
commutatif
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{{D}}$
On dit que la théorie $T$ a la propriété d’amalgamation si tous les modèles de
$T$ sont des bases d’amalgamation.
La section suivante sera un terrain d’application des techniques
d’amalgamation et où nous rappellerons une caractérisation des bases
d’amalgamation par la notion d’extension universelle.
### 1.3 Espaces de types positifs
###### Définition 6
Soit $T$ une théorie h-inductive. Un type en n variable est l’ensemble des
formules positives satisfaites par un n-uple dans un pec de $T$.
On note $S_{n}(T)$ l’espace des n-types, et $S(T)$ l’espace de tous les types
de $T$.
Topologie de $S_{n}(T)$
* •
La Topologie de $S_{n}(T)$ est définie par la famille des fermés élémentaires
$F_{\varphi}=\\{p\in S_{n}(T)|p\vdash\varphi\\}$
où $\varphi$ parcourt l’ensemble des formules positives.
* •
L’espace $S_{n}(T)$ est compacte, mais en général il n’est pas séparé.
###### Définition 7
[4] Une théorie h-inductive $T$ est dite séparée si pour tout $n\in\mbox{\Bbbb
N}$, $S_{n}(T)$ est séparée.
###### Fait 3 (, [4])
Une théorie h-inductive $T$ est séparée si et seulement si $Tk(T)$ a la
propriété d’amalgamation.
## 2 Extensions universelles
La notion d’extension universelle est réminiscente d’objets universels en
théorie des catégories. Dans notre contexte, la limite inductive d’extensions
universelles généralise la notion de saturation, comme cela se fait dans
l’étude des classes élémentaires abstraites.
Dans cette section on étudie certaines propriétés de cette notion
### 2.1 Notion d’extension universelle
###### Définition 8
Soient $A,B$ deux modèles d’une théorie h-inductive $T$, et $h$ un
homomorphisme de $A$ dans $B$. On dit que $(B,h)$ est une extension
universelle de $A$, si pour tout modèle $C$ de $T$ de cardinal $\leq|A|$ où
$A$ se continue par un homomorphisme $f$, il existe un homomorphisme $g$ de
$C$ dans $B$ tel que le diagramme suivant commute
$\textstyle{{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{f}$$\textstyle{{B}}$
Remarque:
Soient $(B,h)$ est une extension universelle de $A$, et $g$ un homomorphisme
de $B$ dans un modèle $C$ de $T$. Alors $(C,g\circ h)$ est aussi une extension
universelle de $A$. En particulier, $A$ admet une extension universelle
$(B_{e},h^{\prime})$, avec $B_{e}$ un pec de $T$.
###### Définition 9
Soient $T$ une théorie $h$-inductive et $\alpha$ un ordinal. Une chaîne
universelle de longueur $\alpha$ de $T$ est une famille inductive de modèles
$\\{A_{i}:i<\alpha\\}$ (resp. $\\{A_{i}:i\leq\alpha\\}$ si $\alpha$ est
successeur) de $T$ avec une famille d’homomorphismes $\\{f_{ij}:i\leq
j<\alpha\\}$ (resp. $\\{f_{ij}:i\leq j<\alpha\\}$ si $\alpha$ est successeur)
telle que pour tout ordinal $\beta<\alpha$, $(A_{\beta+1},f_{\beta,\beta+1})$
est une extension universelle de $A_{\beta}$ et que si $\beta\leq\alpha$ est
un ordinal limite alors $A_{\beta}$ est la limite inductive des $A_{i}$ avec
$i<\beta$, $f_{i\beta}$ étant défini comme l’application canonique de $A_{i}$
dans $A_{\beta}$.
###### Fait 4 ([1])
Soit $\\{A_{i};f_{ij}:\,i\leq j<\alpha\\}$ une chaîne universelle de la
théorie h-inductive $T$. on suppose que pour tout $i\leq\alpha$ ordinal
limite, $A_{i}$ est un pec de $T$. Dans ce cas, si $j\leq i$, l’application
$h_{ji}$, qui par construction des limites inductives est l’application
canonique de $A_{j}$ vers $A_{i}$, alors $(A_{i},h_{ji})$ est une extension
universelle de $A_{j}$.
### 2.2 Application de la notion d’extension universelle
Dans la suite de cette section nous proposons quelques applications de la
notion d’extension universelle en théorie des modèles positive.
La première application que nous proposons est la caractérisation des bases
d’amalgamation qui nous sera utile dans la deuxième étude portée sur la
topologie des espaces des types.
###### Fait 5 ([1])
Soit $A$ un modèle d’une théorie h-inductive $T$. Alors $A$ admet une
extension universelle si et seulement si $A$ est une base d’amalgamation.
La deuxième application est la donnée d’une autre preuve de la conservation de
la séparation topologique par passage au restriction élémentaires positives.
###### Lemme 1
Soit $A$ une L-structure et $B$ une extension élémentaire positive de $A$
séparée, alors $A$ est séparée
Preuve. Par le fait 3 il suffit de montrer que la théorie $Tk(A)$ a la
propriété d’amalgamation, ce qui est équivalent d’après le fait 5 à dire que
tout modèle de $Tk(A)$ admet une extension universelle.
Soient $A_{1}$ et $A_{2}$ deux modèles de $Tk(A)$ tels que
$|A_{2}|\leq|A_{1}|$ et $A_{1}$ se continue dans $A_{2}$ par un homomorphisme
$f$. Par amalgamation Kiesserienne ou asymétrique on déduit l’existence de
$B_{1}$ un modèle de $Tk(B)$ où $A_{1}$ s’immerge, ainsi on obtient le
diagramme commutatif suivant
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{i_{1}}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{B_{1}}}$
avec $i,i_{1},i_{2}$ des immersions et $e$ l’immersion élémentaire de $A$ dans
$B$. Maintenant par amalgamation asymétrique on obtient le diagramme
commutatif suivant
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{i_{1}}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{{B_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{{A^{\prime}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{B_{2}}$
avec $A^{\prime}_{2}$ un modèle de $Tk(A)$ et $im$ une immersion. Ensuite
comme $B$ est un pec de $Tk(A)$ et $A^{\prime}_{2}$ un modèle de $Tk(A)$,
alors $f^{\prime}\circ i_{2}$ est une immersion, ce qui implique que
$A^{\prime}_{2}$ est un modèle de $Tu(B)$. Ce qui nous permet de continuer
$A^{\prime}_{2}$ dans $B_{2}$ un modèle de $Tk(B)$ par un homomorphisme $g$,
en plus on peut prendre $B_{2}$ de cardinal inférieur au cardinal de $B_{1}$.
Maintenant comme $B_{1}$ est un modèle de $Tk(B)$ et $B$ est séparée, alors
$B_{1}$ admet une extension universelle $(B^{\star}_{2},h)$. Ainsi on obtient
le diagramme commutatif suivant
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{i_{1}}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{{B_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{h}$$\textstyle{B^{\star}_{2}}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{{A^{\prime}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$
Par cette construction on déduit que $(B^{\star}_{2},h\circ i)$ est une
extension universelle de $A_{1}$. Ainsi tout modèle $A_{1}$ de $Tk(A)$ est une
base d’amalgamation dans la classe des modèle de $Tk(A)$. $\square$
La dernière application de la notion des extensions universelles est la
contructions de domaine universel d’une théorie h-inductive complète non
bornée.
###### Définition 10
:
Une théorie h-inductive est dite complète, si ella a la propriété de la
continuation commune (pour tous $A,B$ modèles de $T$, il existe $C\models T$
où $A$ et $B$ se continuent.)
Une théorie h-inductive est bornée s’il existe une borne sur le cardinal de
ses pec.
Exemples:
* •
Soit $A$ une structure la théorie $Tk(A)$ est complète.
* •
La théorie h-inductive de la structure $(\mbox{\Bbbb Q},\leq)$ dans le langage
$L(\mbox{\Bbbb Q})=\\{\leq,q\,|q\in\mbox{\Bbbb Q}\\}$ est bornée.
Un domaine universel, de façon similaire aux modèles monstres en théorie des
modèles usuelle, est une structure suffisamment homogène et saturée. Dans
cette section, nous montrerons l’existence des $\lambda$-domaines universels
dans le contexte positif en passant par les chaînes universelles (définition
9).
###### Définition 11 ([12])
Soient $T$ une théorie h-inductive, et $\lambda$ un cardinal. Un modèle $M$ de
$T$ est dit un $\lambda$-domaine universel s’il vérifie les deux propriétés
suivantes:
1. 1.
$\lambda$-universel: tout type partiel $p(\bar{x})$ avec paramètres dans une
partie $A$ de $M$ de cardinal $<\lambda$, et finiment satisfaisable dans $M$,
est réalisé dans $M$.
2. 2.
$\lambda$-homogène: pour tous $A,B$ des modèles de $T$ qui s’immergent dans
$M$ et de cardinal $<\lambda$, et $f$ un isomorphisme entre $A$ et $B$ (ie une
bijection telle que $f$ et $f^{-1}$ sont des immersions). Alors il existe un
automorphisme de $M$ qui prolonge $f$.
Le fait suivant nous montre que dans un $\lambda$-domaine universel la
$\lambda$-homogénéité est équivalente à la propriété suivante: toute paire
d’uples de longueur strictement inférieure à $\lambda$ et de même type se
correspondent par automorphisme du domaine universel.
###### Fait 6 ([4])
Soient $M$ un $\lambda$-domaine universel et $\bar{a}$, $\bar{b}$ des uples de
longueur strictement inférieure à $\lambda$ et de même type. Alors il existe
un automorphisme $f$ de $M$ tel que $f(\bar{a})=\bar{b}$.
Remarque: Il s’ensuit de la définition 11 que les domaines universels sont
pec.
###### Exemple 7
Soit $T$ une théorie h-inductive bornée et complète, alors le pec maximal de
théorie $T$ est un domaine universel en son cardinal.
Le théorème suivant donne la construction d’un $\lambda$-domaine universel
comme celui-ci était défini ci-dessus. La raisonnement utilise les extensions
universelles (la définition 8) auxquelles on aboutit en construisant une
chaîne universelle de modèles pec de longueur $\omega$. Cette longueur nous
permet d’adopter une notation simplifée en réduisant la famille inductive
nécessaire pour la construction à une chaîne.
###### Théorème 1
Soit $T$ une théorie $h$-inductive, complète, non bornée. Soit
$\\{A_{i},h_{i}|i<\omega\\}$ une chaîne universelle telle que
$cf(|A_{0}|)>max(\aleph_{0},|L|)$ et que pour tout $i<\omega$, $A_{i}$ est un
pec de $T$. Soit $A$ la limite inductive de la chaîne universelle, et on pose
$\lambda=|A|$. Alors $A$ est un $cf(\lambda)$-domaine universel.
Preuve. D’abord nous allons montrer que $A$ est $cf(\lambda)$-universel. Soit
$B$ une partie de $A$ de cardinal $<cf(\lambda)$, et $\pi$ une famille de
formules positives finiment satisfaisable dans $A$. Alors il existe $C$ un
modèle de $T$ où $A$ se continue par un homomorphisme $f$ et qui réalise $\pi$
par un uple $\bar{c}$. Par définition de la cofinalité, il existe $m<\omega$
tel que $B\subset A_{m}$. Soient $g$ la restriction de $f$ à $A_{m}$ et
$C^{\star}$ le plus petit modèle de $T$ qui contient
$g(A_{m})\cup\\{\bar{c}\\}$ de cardinal $\leq|A_{m}|$ et qui s’immerge dans
$C$. Par conséquent, il existe $h$ un homomorphisme de $C^{\star}$ vers $A$
tel que le diagramme suivant est commutatif
$\textstyle{{C^{\star}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{A_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{g}$$\textstyle{{A}}$
Alors $h(\bar{c})$ est une réalisation de $\pi$ dans $A$.
Maintenant montrons que $A$ est $cf(\lambda)$-homogène. Soient $C_{1}$ et
$C_{2}$ deux modèles de $T$ inclus dans $A$ par immersion, de cardinal
$<cf(\lambda)$ et qui sont isomorphes. Par la définition de la cofinalité, il
existe $m<\omega$ tel que $C_{i}\subset A_{m}$. Alors par le lemme 2, il
existe $D$ un modèle de $T$ et $f,g$ des immersions tels que le diagramme
suivant est commutatif
$\textstyle{{C_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{is}$$\scriptstyle{im}$$\textstyle{{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{A_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{{D}}$
Remarquons qu’on peut remplacer $D$ dans le diagramme par $B_{0}=f(A_{m})$
sans perte de généralité. Ainsi on a le diagramme commutatif suivant:
$\textstyle{{C_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{is}$$\scriptstyle{im}$$\textstyle{{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{{A_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{B_{0}}$
Maintenant $A_{m}$ et $B_{0}$ sont isomorphes, on répète le même raisonnement
et on obtient le diagramme commutatif suivant:
$\textstyle{{C_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{is}$$\scriptstyle{im}$$\textstyle{{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{{A_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{h_{m}}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\textstyle{A_{m+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{B_{1}}$
Donc $A_{m+1}$ et $B_{1}$ sont isomorphes et $(B_{1},g_{0})$ est une extension
universelle de $B_{0}$. Ainsi on définit une deuxième chaîne universelle
$\\{B_{i},g_{i}|\beta\leq i<\omega\\}$, isomorphe à la chaîne
$\\{A_{i},h_{i}|m\leq i<\omega\\}$. Soit $B$ la limite de cette chaîne. et on
a le diagramme commutatif suivant
$\textstyle{{C_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{is}$$\scriptstyle{im}$$\textstyle{{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{{A_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{h_{m}}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\textstyle{A_{m+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
Alors l’application $f$ est un isomorphisme de $A$ dans $B$ qui prolonge
l’isomorphisme entre $C_{1}$ et $C_{2}$.
Montrons que $A=B$. D’abord rappelons que si
$\\{C^{\prime}_{k},f_{kl}|k<l<\omega\\}$ est une sous-suite inductive extraite
de la suite inductive $\\{C_{i},f_{ij}|i<j<\omega\\}$, alors les deux suites
ont la même limite inductive. En se basant sur cette propriété et en
construisant une suite inductive qui alterne deux suites extraites
respectivement des chaînes $\\{A_{i},h_{i}|m\leq i<\omega\\}$ et
$\\{B_{i},g_{i}|i<\omega\\}$, on montrera que $A=B$. Reprenons le diagramme
commutatif construit dans l’étape précédente
$\textstyle{{C_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{is}$$\scriptstyle{im}$$\textstyle{{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{{A_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{h_{m}}$$\scriptstyle{k_{1}}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\textstyle{A_{m+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{h_{m+1}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{k_{2}}$$\textstyle{A_{m+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{h_{m+2}}$$\scriptstyle{k_{3}}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{A_{m+3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{3}}$$\textstyle{B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
avec $k_{1}=g_{0}\circ f_{0}$, $k_{2}=f_{2}^{-1}\circ g_{1}=h_{m+1}\circ
f^{-1}_{1}$ et $k_{3}=g_{2}\circ f_{2}$. Soit la suite inductive
$\textstyle{A_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{1}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{2}}$$\textstyle{A_{m+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{3}}$$\textstyle{B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Alors on a $k_{2}\circ k_{1}=h_{m+1}\circ h_{m}$ et $k_{3}\circ
k_{2}=g_{2}\circ g_{1}$. Ainsi la suite inductive construite est une
alternation des deux sous-suites inductives extraites respectivement de
$\\{A_{i},hi|m\leq i<\omega\\}$ et $\\{B_{i},g_{i}|i<\omega\\}$. Par suite on
déduit que les limites de ces deux suites, qui sont respectivement $A$ et $B$,
sont égales. $\square$
###### Corollaire 1
Soit $T$ une théorie h-inductive complète et non bornée, alors pour tout
cardinal $\alpha>max(\aleph_{0},|L|)$ il existe un $\lambda$-domaine universel
de $T$ avec $\lambda$ un cardinal supérieur à $\alpha$.
Preuve. Soient $\alpha>max(\aleph_{0},|L|)$ et $\beta$ un cardinal tel que
$cf(\beta)>\alpha$. Comme la théorie $T$ est non bornée, il existe $A_{0}$ un
pec de cardinal supérieur à $\lambda$. D’après le théorème 5, il existe
$(A_{1},h_{1})$ une extension universelle de $A_{0}$, avec $A_{1}$ un pec de
cardinal $\leq 2^{|A_{0}|}$. En répétant le même raisonnement, on construit
une chaîne universelle $\\{A_{i},h_{i}|i<\alpha\\}$ où $\alpha$ est un ordinal
limite. D’après le théorème 1, la limite inductive $A$ de la chaîne
universelle construite est $cf(|A|)-$domaine universelle. $\square$
###### Théorème 2
(Erdös-Rado) Soit $\lambda$ un cardinal infini et $k\in\mathbb{N}$. Alors
$\beth_{k}(\lambda)^{+}\longrightarrow{(\lambda^{+})}_{\lambda}^{k+1}\ .$
###### Fait 8
Soient $T$ une théorie h-inductive, non bornée avec jocp ,
$\lambda>|S_{k}(T)|$ et $\mu=\beth_{\lambda^{+}}$. Soit $M$ un pec de $T$, tel
que $|M|\geq\mu$. Alors pour toute suite $(\bar{a}_{i}|i<\mu)$ de k-uples, il
existe $M^{\star}$ un modèle pec de $T$ de même cardinal que $M$ et où $M$
s’immerge, et une suite $(\bar{b}_{i}|i<\omega)$ indiscernable de $M^{\star}$
, telle que pour toute $n<\omega$ il existe $i_{0}<\cdots<i_{n-1}<\mu$ pour
laquelle
$tp(\bar{a}_{i_{0}},\cdots,\bar{a}_{i_{n-1}})=tp(\bar{b}_{0},\cdots,\bar{b}_{n-1})$
## 3 Stabilité positive
Dans cette section, nous adoptons les définitions et les résultats de Ben
Yaacov dans [3], et nous poursuivons le travail dans le cadre positif. L’une
des nouveautés est une notion d’ordre définissable. Sans surprise, la
différence fondamentale par rapport à la théorie des modèles usuelle est
l’absence de la négation, indispensable pour ordonner un ensemble
définissablement. Nous proposons une définition de la propriété de l’ordre en
utilisant des paires de formules contradictoires (définition 15). Cette
approximation positive à la négation caractérise la stabilité. Une adaptation
positive d’un théorème de Shelah (théorème 3) s’avère crucial dans le cas des
théories non bornées.
L’étude de la stabilité positive est réservée aux pec des théories
h-inductives complètes, ce qui met en défaut le théorème de compacité. En
effet, ce théorème n’est utile que quand la classe des pec est élémentaire.
Ceci nous contraint à utiliser des techniques de la combinatoire des
cardinaux, ce qui nous permet de ne pas sortir de la classe des pec.
Soulignons que l’absence de notre travail des caractérisations, pourtant
naturelles, de la stabilité par les (co)héritiers est liée à ce problème.
### 3.1 Rang de Shelah positif
Dans cette section on reprend la définition du rang de Shelah positif comme il
a été donné dans [3]. On en étudie les caractérisations dans le cadre positif
en reproduisant ceux qui existent dans [16].
###### Définition 12
Soient $T$ une théorie h-inductive, $A$ un sous-ensemble d’un modèle de $T$,
et $\Gamma(\bar{x},A)$ une famille de formules positives à paramètres dans
$A$. On dit que $\Gamma$ est un type partiel si il existe $B$ un modèle de
$T$, et $\bar{b}$ un uple de $B$ tel que $B\models\varphi(\bar{b},\bar{a})$
pour toute formule $\varphi(\bar{x},\bar{a})$ de $\Gamma$.
Remarque: De façon équivalente $\Gamma(\bar{x},A)$ est un type partiel si et
seulement si dans le langage $L\cup\\{A,\bar{x}\\}$ la famille
$T\cup\Gamma(\bar{x},A)$ est consistante.
###### Définition 13
Soit $p(\bar{x})$ un type partiel (éventuellement à paramètres dans un
ensemble $A$), $\varphi(\bar{x},\bar{y})$ une L-formule positive, et
$\psi(\bar{x},\bar{y})\in Res_{T}(\varphi(\bar{x},\bar{y}))$. Le rang de
Shelah de $p$, par rapport au couple $(\varphi,\psi)$, qu’on note
$R(p,\varphi,\psi)$, est défini par induction comme suit:
1. 1.
$R(p,\varphi,\psi)\geq 0$ si est seulement si $p$ est consistant,
2. 2.
Si $\beta$ est un ordinal limite, $R(p,\varphi,\psi)\geq\beta$ si et seulement
si pour tout $\alpha<\beta$, $R(p,\varphi,\psi)\geq\alpha$
3. 3.
$R(p,\varphi,\psi)\geq\alpha+1$ si et seulement si il existe $\bar{b}$ tel que
$\left\\{\begin{array}[]{rcl}R(p(\bar{x})\cup\\{\varphi(\bar{x},\bar{b})\\},\varphi,\psi)\geq\alpha\\\
R(p(\bar{x})\cup\\{\psi(\bar{x},\bar{b})\\},\varphi,\psi)\geq\alpha.\end{array}\right.$
###### Lemme 2
Soient $p(\bar{x})$ un type partiel, $\varphi_{0}(\bar{x},\bar{y})$ une
formule positive (où $\bar{x}$ est un m-uple et $\bar{y}$ un n-uple), et soit
$\varphi_{1}\in Res_{T}({\varphi_{0}}(\bar{x},\bar{y}))$. Alors
$R(p,\varphi_{0},\varphi_{1})\geq n$ si et seulement si il existe une famille
de n-uples $\\{a_{\eta}:\,\eta\in^{n\geq}2\\}$ tels que:
1. 1.
Pour tout $\eta\in^{n>}2$ on a
$a_{\eta\mbox{\textasciicircum}0}=a_{\eta\mbox{\textasciicircum}1}$, et
$\bar{a}_{0}=\bar{a}_{1}$
2. 2.
Pour tout $\eta\in^{n}2$, la famille de formules positives
$p(\bar{x})\cup\\{\varphi_{\eta(l)}(\bar{x},a_{\eta_{\upharpoonright
l}})|\quad l<n\\}$
est consistante.
Preuve. Par induction, supposons que $R(p,\varphi_{0},\varphi_{1})\geq n+1$.
Par définition du rang 13 il existe $\bar{b}$ telle que
$\left\\{\begin{array}[]{rcl}R(p(\bar{x})\cup\\{\varphi_{0}(x,\bar{b})\\},\varphi_{0},\varphi_{1})\geq
n\\\ R(p(\bar{x})\cup\\{\varphi_{1}(x,\bar{b})\\},\varphi_{0},\varphi_{1})\geq
n.\end{array}\right.$
Par hypothèse d’induction, il existe $(c_{\eta}|\,\eta\in\mbox{}^{n\geq}2)$ et
$(d_{\eta}|\,\eta\in\mbox{}^{n\geq}2)$ deux suites de n-uples qui vérifiant la
propriété (1) et telles que pour tout $\eta\mbox{}\in^{n}2$
$(\star)\left\\{\begin{array}[]{rcl}p(x)\cup\\{\varphi_{0}(\bar{x},\bar{b})\\}\cup\\{\varphi_{\eta(l)}(\bar{x},\bar{c}_{\eta_{\upharpoonright
l}})|\quad l<n\\}\,\text{est consistante}\\\
p(x)\cup\\{\varphi_{1}(\bar{x},\bar{b})\\}\cup\\{\varphi_{\eta(l)}(\bar{x},\bar{d}_{\eta_{\upharpoonright
l}})|\quad l<n\\}\,\text{est consistante}.\end{array}\right.$
Soit $\\{\bar{b}_{\rho}|\,\rho\in\mbox{}^{n+1\geq}2\\}$ la famille définie
comme suit: pour tout $\rho\in\mbox{}^{n+1\geq}2$, soit
$\eta=\rho\upharpoonright{\\{1,\cdots,n\\}}$
$\left\\{\begin{array}[]{rcl}\bar{b}_{\rho}=\bar{b}\mbox{\textasciicircum}\bar{c}_{\eta}\quad\text{si
}\quad\rho(0)=0\\\
\bar{b}_{\rho}=\bar{b}\mbox{\textasciicircum}\bar{d}_{\eta}\quad\text{si}\quad\rho(0)=1.\end{array}\right.$
D’après $(\star)$ pour tout $\rho\in\mbox{}^{n+1}2$, la famille
$p(\bar{x})\cup\\{\varphi_{\rho(l)}(\bar{x},\bar{b}_{\rho_{\upharpoonright
l}})|\quad l<n+1\\}$ est consistante. La propriété (1) est évidente par la
construction de la famille.
L’autre direction est évidente. $\square$
Remarque: Pour tout paire $p(\bar{x}),q(\bar{x})$ de types partiels tels que
$p\vdash q$ et pour tout couple de formule $(\varphi,\psi)$ qui vérifie les
conditions de la définition 13 on a $R(p,\varphi,\psi)\leq R(q,\varphi,\psi).$
###### Corollaire 2
Soient $p$ un type partiel, $\varphi_{0}(\bar{x},\bar{y})$ une L-formule
positive avec $l(\bar{x})=n,l(\bar{y})=m$ et $\varphi_{1}\in
Res_{T}(\varphi_{0})$. Supposons que $R(p,\varphi_{0},\varphi_{1})\geq\omega$
alors pour tout ordinal $\mu$ la famille
$\Gamma=p(\bar{x}_{\eta})\cup\\{\varphi_{\eta(\alpha)}(\bar{x}_{\eta},\bar{y}_{\eta\upharpoonright\alpha})\mid\quad\eta\mbox{}\in^{\mu}2,\,\alpha<\mu\\}\cup\\{\bar{y}_{\eta\mbox{\textasciicircum}0}=\bar{y}_{\eta\mbox{\textasciicircum}1}\mid\eta\in\mbox{}^{\mu>}2\\}\cup\\{\bar{a}_{0}=\bar{a}_{1}\\}$
est consistante, avec $x_{\eta}$ un n-uple et $y_{\eta}$ un m-uple.
Preuve. Un fragment fini $\Sigma$ de $\Gamma$ est la donnée de
$\alpha_{1}<\cdots<\alpha_{n}<\mu$, et $\eta_{1},\cdots,\eta_{m}$ de
$\mbox{}^{\mu}2$ tels que
$\Sigma=\\{\varphi_{\eta_{i}(\alpha_{j})}(\bar{x}_{\eta_{i}},\bar{y}_{\eta_{i}\upharpoonright\alpha_{j}}),\,\bar{y}_{\eta_{i}\mbox{\textasciicircum}0}=\bar{y}_{\eta_{i}\mbox{\textasciicircum}1}\mid\,1\leq
j\leq n,\quad 1\leq i\leq m\\}$
Comme $R(\bar{x}=\bar{x},\varphi_{0},\varphi_{1})\geq m.n$, par le lemme 2 il
existe une famille
$\\{\bar{a}_{\eta},\bar{b}_{\eta}|\,\eta\in\mbox{}^{nm\geq}2\\}$ tels que:
1. 1.
pour tout $\eta\in^{nm>}2$ on a
$\bar{b}_{\eta\mbox{\textasciicircum}0}=\bar{b}_{\eta\mbox{\textasciicircum}1}$
,
2. 2.
pour tout $\eta\in\mbox{}^{nm}2$, la famille de formules positives
$p(\bar{x})\cup\\{\varphi_{\eta(l)}(\bar{x},b_{\eta_{\upharpoonright
l}}),\quad l<nm\\}$
est réalisé par $\bar{a}_{\eta}$.
En interprétant les $y_{\eta_{i}}$ par les éléments de $\bar{b}_{\eta}$, et
les $x_{\eta_{j}}$ par la réalisation $\bar{a}_{\eta}$ avec $\eta$ dans
${}^{m.n\geq}2$, on obtient la consistance du fragment fini $\Sigma$, et par
suite la consistance de $\Gamma$. $\square$
###### Corollaire 3
Soient $p(\bar{x})$ un type partiel (eventuellement avec paramètres),
$\varphi_{0}(\bar{x},\bar{y})$ une formule positive et $\varphi_{1}\in
Res_{T}(\varphi_{0})$, supposons que $R(p,\varphi_{0},\varphi_{1})=n$ alors il
existe $\psi$ telle que $p\vdash\psi$ et $R(\psi,\varphi_{0},\varphi_{1})=n$
Preuve. Par l’absurde supposons que pour toute formule positive $\psi$ telle
que $p\vdash\psi$, on a $R(\psi,\varphi_{0},\varphi_{1})\geq n+1$. Alors
$\Gamma_{\psi}=\\{\psi(\bar{x}_{\eta})\wedge\varphi_{\eta(l)}(\bar{x}_{\eta},\bar{y}_{\eta\upharpoonright
l})|\quad\eta\in\mbox{}^{n+1}2,\,l\leq
n+1\\}\cup\\{\bar{y}_{\eta\mbox{\textasciicircum}0}=\bar{y}_{\eta\mbox{\textasciicircum}1}\mid\eta\in\mbox{}^{n>}2\\}$
est consistante, ce qui implique la consistance de
$\Gamma=p(\bar{x}_{\eta})\cup\\{\varphi_{\eta(l)}(\bar{x}_{\eta},\bar{y}_{\eta\upharpoonright
l})\mid\quad\eta\in\mbox{}^{n+1}2,\,l\leq
n+1\\}\cup\\{\bar{y}_{\eta\mbox{\textasciicircum}0}=\bar{y}_{\eta\mbox{\textasciicircum}1}\mid\eta\in\mbox{}^{n>}2\\}$.
Ainsi par le lemme 2 on déduit que $R(p,\varphi_{0},\varphi_{1})\geq n+1$,
contradiction. Par conséquent il existe $\psi$ dans $p$ telle que
$R(\psi,\varphi_{0},\varphi_{1})\leq n$. $\square$
###### Corollaire 4
Soient $\psi(\bar{x},\bar{z})$ et $\varphi(\bar{x},\bar{y})$ deux L-formules
positives, et $\varphi_{1}\in Res_{T}(\varphi_{0})$, et soient $\bar{a}$ et
$\bar{b}$ deux uples (dans un pec de $T$) qui ont le même type. Pour tout
entier naturel $n$, si $R(\psi(\bar{x},\bar{a}),\varphi_{0},\varphi_{1})\geq
n$ alors $R(\psi(\bar{x},\bar{b}),\varphi_{0},\varphi_{1})\geq n$.
Preuve. Supposons que $R(\psi(\bar{x},\bar{a}),\varphi_{0},\varphi_{1})\geq
n$. D’après le lemme 2
$\\{\psi(\bar{x}_{\eta},\bar{a})\\}\cup\\{\varphi_{\eta(l)}(\bar{x}_{\eta},\bar{y}_{\eta\upharpoonright
l})|l\leq
n,\eta\in\mbox{}^{n}2\\}\cup\\{\bar{y}_{\eta\mbox{\textasciicircum}0}=\bar{y}_{\eta\mbox{\textasciicircum}1}|\eta\in\mbox{}^{n>}2\\}$
est consistante, par conséquent la formule positive
$\exists_{l\leq n}\bar{x}_{\eta}\bar{y}_{\eta\upharpoonright
l}(\psi(\bar{x}_{\eta};\bar{z})\wedge\bigwedge_{l\leq
n}\varphi_{\eta(l)}(\bar{x}_{\eta},\bar{y}_{\eta\upharpoonright
l})\wedge\bigwedge_{\eta\in^{n>}2}\bar{y}_{\eta\mbox{\textasciicircum}0}=\bar{y}_{\eta\mbox{\textasciicircum}1})$
est dans le type de $\bar{a}$. Par suite elle appartient aussi au type de
$\bar{b}$, ce qui implique par le lemme 2 que
$R(\psi(\bar{x},\bar{b}),\varphi_{0},\varphi_{1})\geq n$. $\square$
### 3.2 Stabilité positive
Notre définition de la stabilité positive suit celle naturelle qui fait
intervenir le comptage des types. Soulignons quand-même que, la notion de type
positif n’ayant un sens que dans les pec d’une théorie $h$-inductive, nous
nous restreindrons à cette classe de modèles. En partant des travaux de Ben
Yaacov dans [3], on réussit à obtenir des caractérisations de cette notion
bien connues en théorie des modèles usuelle. Il faut souligner quand même que
la caractérisation par la présence d’un ensemble infini et ordonné
positivement nécessite des études séparées pour les théories bornées et non
bornées. Ceci est lié à l’utilisation des techniques combinatoires, en
particulier le théorème d’Erdös-Rado (théorème 2).
###### Définition 14
Soient $T$ une théorie h-inductive complète, et $\lambda$ un cardinal. On dit
que $T$ est $\lambda$-stable si pour tout pec $A$ de $T$ de cardinal
$\lambda\geq|L|$ on a:
$\mid S(A)\mid\leq\lambda$
La théorie $T$ est dite stable si elle est $\lambda$-stable pour un certain
$\lambda$.
Une structure $M$ est dite stable si la théorie $Tk(M)$ est stable.
Notre premier exemple d’une théorie stable provient d’une classe de structures
particulières à la théorie des modèles positive: les structures bornées. et on
commence par étudier la stabilité positive. Nous remercions Poizat pour cette
suggestion.
###### Lemme 3
Toute théorie h-inductive complète bornée est stable.
Preuve. Comme $T$ est une théorie bornée, le cardinal
$\lambda=max\\{|M|\mid\,M\,\text{un pec de }\,T\\}$
existe. Nous vérifions que $T$ est $\lambda-$stable. Sinon, il existe $M$ un
pec de $T$ de cardinal $\lambda$ tel que $|S(M)|>\lambda$, ce qui implique
l’existence d’une extension élémentaire positive $N$ de $M$ qui est aussi un
pec de $T$ (lemme LABEL:extensionegalpec) et qui réalise tous les types de
$S(M)$. Ainsi $M$ est un pec de $T$ de cardinal $>\lambda$, une contradiction.
$\square$
Un exemple de théorie h-inductive complète stable non bornée est celui de la
théorie des corps de caractéristique fixée dans le langage usuel des corps.
Cette théorie est complète positivement. Ses pec sont les corps algébriquement
clos de même caractéristique. Vu que la théorie de cette classe est complète
et stable au sens de la théorie des modèles usuelle, on déduit la stabilité
positive en appliquant directement la définition 14. Ainsi, la théorie des
corps de caractéristique fixée est stable positivement.
Précisons que cette théorie n’est même pas complète dans la logique usuelle.
Par conséquent, il n’est même pas possible de parler de sa stabilité dans la
logique usuelle. Nous reviendrons sur la comparaison des stabilités dans la
logique positive et la logigue usuelle dans la section 3.5.
###### Lemme 4
Soient $T$ une théorie h-inductive complète, et $(\varphi_{0},\varphi_{1})$ un
couple de formules positives telles que $\varphi_{1}\in Res_{T}(\varphi_{0})$
et $R(\bar{x}=\bar{x},\varphi_{0},\varphi_{1})\geq\omega$. Alors pour aucun
cardinal $\lambda$, $T$ n’est $\lambda$-stable.
Preuve. Soient $\lambda$ un cardinal infini, et
$\mu=min\\{\mu|2^{\mu}>\lambda\\}$, alors
$\sum_{\alpha<\mu}2^{\alpha}\leq\lambda$. Par le corollaire 2 la famille
$\Gamma=\\{\varphi_{\eta(\alpha)}(\bar{x}_{\eta},\bar{y}_{\eta\upharpoonright\alpha})\mid\quad\eta\in\mbox{}^{\mu}2,\,\alpha<\mu\\}\cup\\{\bar{y}_{0}=\bar{y}_{1}\\}$
est consistante avec $\bar{x}_{\eta}$ un n-uple, et $\bar{y}_{\eta}$ un
$\mu$-uple.
Soient $B=\\{b_{\eta}|\,\,\eta\in\mbox{}^{\mu>}2\\}$ et
$A=\\{\bar{a}_{\eta}|\,\eta\in\mbox{}^{\mu}2\\}$ un couple qui réalise
$\Gamma$. Par définition de $\mu$, on a $\mid A\mid\leq\lambda$. Maintenant
montrons que si $\eta\neq\nu\in\mbox{}^{\mu}2$, alors
$tp(\bar{a}_{\eta}\diagup A)\neq tp(\bar{a}_{\nu}\diagup A)$. En effet soit
$\rho=\eta\wedge\nu$, en d’autres termes, il existe $\beta<\alpha$ tels que
$\rho\in\mbox{}^{\beta}2$ et
$\eta_{\upharpoonright\beta}=\nu_{\upharpoonright\beta}=\rho$ et
$\eta(\beta+1)\neq\nu(\beta+1)$, tel que
$\rho\mbox{\textasciicircum}0=\eta_{\upharpoonright\beta+1}$,
$\rho\mbox{\textasciicircum}1=\nu_{\upharpoonright\beta+1}$ et
$\rho\in\mbox{}^{\gamma}2$. Donc par définition de $\Gamma,A$ et $B$, on a
$D\models\varphi_{0}(\bar{x}_{\eta},\bar{b}_{\eta\upharpoonright\gamma+1})$,
et $D\models\varphi_{1}(\bar{x}_{\nu},\bar{b}_{\nu\upharpoonright\gamma+1})$,
comme
$\bar{b}_{\eta\upharpoonright\gamma+1}=\bar{b}_{\nu\upharpoonright\gamma+1}$,
(lemme 2). Alors $tp(\bar{a}_{\eta}\diagup B)\neq tp(\bar{a}_{\nu}\diagup B)$.
Par suite on déduit que $|S(B)|\geq|^{\mu}2|>\lambda$, alors que
$|B|\leq\lambda$, d’où la non $\lambda$-stabilité de $T$. $\square$
### 3.3 Propriété de l’ordre positive et stabilité.
Dans cette sous-section on propose une notion d’ordre propre à la théorie des
modèles positive: on définit l’ordre avec deux formules positives
contradictoires. Cette notion d’ordre nous permettra comme dans la théorie des
modèles usuelle de caractériser l’instabilité positive. La caractérisation se
fera suivant deux chemins bien différents selon la nature de la théorie en
question. Si la théorie est bornée, les lemmes 3 et 7 permettent de conclure
rapidement grâce au caractère très particulier des théories bornées.
Dans le cas où la théorie est non bornée, le raisonnement est nettement plus
compliqué. D’abord, on démontre que la propriété de l’ordre positive implique
la non finitude du rang de Shelah (lemme 9). Ensuite on reprend le théorème
2.10 de [18] dans le cadre positif qui nous pemettra de conclure (corollaire
6).
###### Définition 15
Soit $T$ une théorie h-inductive dans un langage $L$. On dit qu’une
$L$-formule positive $\varphi(\bar{x},\bar{y})$ (avec
$l(\bar{x})=l(\bar{y})=n$) définit un ordre, si il existe $\psi\in
Res_{T}(\varphi(\bar{x},\bar{y}))$, et une suite $(\bar{a}_{i}|\quad
i<\omega)$ de n-uples distincts dans un modèle de $T$ tels que
$\left\\{\begin{array}[]{rcl}\varphi(\bar{a}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
\psi(\bar{a}_{i},\bar{a}_{j})\quad\text{si}\,\,j<i.\end{array}\right.$
On dit que le couple $(\varphi,\psi)$ ordonne la suite
$(\bar{a}_{i}|i<\omega)$.
La théorie $T$ a la propriété de l’ordre, si il existe une formule positive
dans le langage de $T$, qui définit un ordre sur un modèle de $T$.
Dans le reste, sauf mention contraire, nous utiliserons l’appellation
“propriété de l’ordre” au lieu de “propriété de l’ordre positive”.
###### Lemme 5
Soient $T$ une théorie h-inductive,
$(\varphi(\bar{x},\bar{y}),\psi(\bar{x},\bar{y}))$ un couple de formules
positives (éventuellement $l(\bar{x})\neq l(\bar{y})$) telles que $\psi\in
Res_{T}(\varphi)$ et qu’il existe $\\{(\bar{c}_{i},\bar{a}_{j})\mid\quad
i,j<\omega\\}$ une suite d’uples dans un modèle de $T$ telle que
$\left\\{\begin{array}[]{rcl}\varphi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
\psi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
Alors $T$ a la propriété de l’ordre.
Preuve. Supposons qu’il existe $\\{(\bar{c}_{i},\bar{a}_{j})\mid\quad
i,j<\omega\\}$ une suite d’uples dans un modèle de $T$ telle que
$\left\\{\begin{array}[]{rcl}\varphi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
\psi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
Soit $\theta$ la formule positive définie par
$\theta((\bar{x},\bar{x}^{\prime}),(\bar{y},\bar{y}^{\prime}))\equiv\varphi(\bar{x},\bar{y}^{\prime})$,
alors la formule $\theta^{\prime}$ définie par
$\theta^{\prime}((\bar{x},\bar{x}^{\prime}),(\bar{y},\bar{y}^{\prime}))\equiv\psi(\bar{x},\bar{y}^{\prime})$
est contradictoire avec $\theta$, en d’autres termes, $\theta^{\prime}\in
Res_{T}(\theta)$, et le couple de formules positives
$(\theta,\theta^{\prime})$ ordonne la suite
$\\{(\bar{c}_{i},\bar{a}_{i})\mid\quad i<\omega\\}$. $\square$
###### Lemme 6
Soit $T$ une théorie h-inductive sans la propriété de l’ordre. Alors pour tout
couple de formules positives
$(\varphi(\bar{x},\bar{y}),\psi(\bar{x},\bar{y}))$, avec $\psi\in
Res_{T}(\varphi(\bar{x},\bar{y}))$, il existe un entier naturel $N$ (qui
dépend du couple de formules) tel que pour tout modèle $M$ de $T$, il n’existe
pas $\\{(\bar{c}_{i},\bar{a}_{j})\mid i,j\leq N\\}$ dans $M$ tel que:
$(\star)\left\\{\begin{array}[]{rcl}\varphi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
\psi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
Preuve. Par l’absurde supposons que pour tout $N\in\mbox{\Bbbb N}$ il existe
$\\{(\bar{c}_{i},\bar{a}_{j})\mid i,j\leq N\\}$ dans un modèle de $T$ tel que:
$(\star)\left\\{\begin{array}[]{rcl}\varphi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
\psi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
Alors par compacité positive il existe $\\{(\bar{c}_{i},\bar{a}_{j})\mid
i,j<\omega\\}$ tel que:
$(\star)\left\\{\begin{array}[]{rcl}\varphi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
\psi(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
Par le lemme 5, $T$ a la propriété de l’ordre, contradiction. $\square$
###### Lemme 7
Si $T$ est une théorie h-inductive bornée, alors il n’a pas la propriété de
l’ordre.
Preuve. Supposons que $T$ a la propriété de l’ordre. Alors, il existe $A$ un
pec de $T$ et $(\bar{a}_{i}|\,i<\omega)$ une suite extraite de $A$ ordonnée
par un couple de formules $(\varphi,\psi)$ tel que $\psi\in Res_{T}(\varphi)$.
Ceci implique que la famille d’énoncés h-inductifs suivante est consistante:
$T\cup\\{\exists\bar{x}_{i}\bar{x}_{j}\varphi(\bar{x}_{i},\bar{x}_{j})\wedge\psi(\bar{x}_{j},\bar{x}_{i})|i<j<\omega\\}\
.$
Par le théorème de compacité positive, on déduit que pour tout cardinal
$\lambda$ la famille d’énoncés suivante est consistante
$T\cup\\{\exists\bar{x}_{i}\bar{x}_{j}\varphi(\bar{x}_{i},\bar{x}_{j})\wedge\psi(\bar{x}_{j},\bar{x}_{i})|i<j<\lambda\\}\
.$
Soient $B$ un modèle de cette famille d’énoncés et $(\bar{b}_{i}|\,i<\lambda)$
une suite extraite de $B$ ordonnée par le couple $(\varphi,\psi)$. Soit
$B_{e}$ un pec de $T$ où $B$ se continue par un homomorphisme $f$. Comme la
suite $(\bar{b}_{i}|\,i<\lambda)$ est ordonnée par $(\varphi,\psi)$ et que
$\psi\in Res_{T}(\varphi)$, on déduit que pour tout $i,j<\lambda$,
$f(\bar{b}_{i})=f(\bar{b}_{j})\Rightarrow\bar{b}_{i}=\bar{b}_{j}\ .$
Ainsi, $B_{e}$ est un pec de $T$ de cardinal au moins $\lambda$. Ceci
contredit que $T$ est bornée. $\square$
###### Définition 16
On appelle l’entier $N$ du corollaire 6 le nombre d’alternance du couple
$(\varphi(\bar{x},\bar{y}),\psi(\bar{x},\bar{y}))$, par rapport à $T$.
###### Lemme 8
Soit $T$ une théorie h-inductive non bornée. Une formule
$\varphi(\bar{x},\bar{y})$ définit un ordre si et seulement si, pour tout
ordinal $\alpha$, il existe une suite $(\bar{a}_{i}:\,i<\alpha)$ dans un pec
$A$ de $T$ qui vérifie:
$A\models\varphi(\bar{a}_{i},\bar{a}_{j})\quad\text{si et seulement si}\quad
i<j.$
Preuve. On suppose d’abord que la formule $\varphi(\bar{x},\bar{y})$ définit
un ordre. Alors il existe une formule positive $\psi\in Res_{T}(\varphi)$
telle que la famille suivante est consistante
$T\cup\\{\exists\bar{x}_{i}\bar{x}_{j}\varphi(\bar{x}_{i},\bar{x}_{j})\wedge\psi(\bar{x}_{j},\bar{x}_{i})|i<j<\omega\\}\
.$
Par compacité positive on déduit que pour tout ordinal $\alpha$ la famille
suivante est consistante
$T\cup\\{\exists\bar{x}_{i}\bar{x}_{j}\varphi(\bar{x}_{i},\bar{x}_{j})\wedge\psi(\bar{x}_{j},\bar{x}_{i})|i<j<\alpha\\}$
Soient $A$ un modèle de cette famille d’énoncés inductifs et
$(\bar{a}_{i}|i<\alpha)$ une suite extraite de $A$ ordonnée par le couple de
formules $(\varphi,\psi)$. Soit $A_{e}$ un pec de $T$ où $A$ se continue.
Comme $\psi\in Res_{T}(\varphi)$, l’image de $(\bar{a}_{i}|i<\alpha)$ dans
$A_{e}$ est de taille $\alpha$ et elle est ordonnée par le même couple de
formules.
Maitenant montrons l’autre sens. Soient $\lambda=|Res_{T}(\varphi)|$ et
$\alpha=\beth^{+}(\lambda)$, $A$ un pec de $T$ de cardinal $\geq\alpha$ et
$(\bar{a}_{i}|i<\alpha)$ une suite de $A$ tels que
$A\models\varphi(\bar{a}_{i},\bar{a}_{j})\quad\text{si et seulement si}\quad
i<j<\alpha.$
À tout couple $(i,j)$, on associe la formule positive $\psi_{ij}\in
Res_{T}(\varphi)$ telle que
$A\models\psi_{ij}(\bar{a}_{max(i,j)},\bar{a}_{min(i,j)})$. Soit $f$
l’application qui envoie $\\{i,j\\}$ vers $\psi_{ij}$. Par le théorème
d’Erdös-Rado (théorème 2), il existe $\psi\in Res_{T}(\varphi)$ et
$(\bar{a}_{i}|i<\lambda)$ une suite extraite de la suite
$(\bar{a}_{i}|i<\alpha)$ telle que
$\left\\{\begin{array}[]{rcl}A\models\varphi(\bar{a}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
A\models\psi(\bar{a}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
Ainsi le couple $(\varphi,\psi)$ ordonne la suite $(\bar{a}_{i};i<\lambda)$.
$\square$
Le lemme suivant est une adaptation du lemme 2.14 [10] au contexte positif. Il
affirme que avoir la propriété de l’ordre implique que le rang est infini.
###### Lemme 9
Soient $T$ une théorie h-inductive et complète. $(\varphi_{0},\varphi_{1})$ un
couple de formules contradictoires qui ordonne une suite dans un modèle de
$T$. Alors $R(\bar{x}=\bar{x},\varphi_{0},\varphi_{1})\geq\omega$.
Preuve. Soit $(\bar{a}_{i}|i<\omega)$ une suite d’un modèle $A$ de $T$. Soient
$n\in\mbox{\Bbbb N}$ et $\psi$ une formule positive telles que
$A\models\psi(\bar{a}_{i})$, pour tout $i<2^{n}$. Alors nous affirmons que
$R(\psi,\varphi_{0},\varphi_{1})\geq n$. En effet par induction, supposons le
résultat vrai pour $n$. Montrons qu’il est vrai pour $n+1$. Supposons que
$A\models\psi(\bar{a}_{i})$ pour tout $i<n+1$ et
$\left\\{\begin{array}[]{rcl}i<j\,\Rightarrow
A\models\varphi_{0}(\bar{a}_{i},\bar{a}_{j})\\\ i>j\,\Rightarrow
A\models\varphi_{1}(\bar{a}_{i},\bar{a}_{j}).\\\ \end{array}\right.$
Alors la formule $\psi(\bar{x})\wedge\varphi_{0}(\bar{a}_{2^{n}},\bar{x})$ est
réalisé par tous les $(\bar{a}_{i}|\,2^{n}<i\leq 2^{n+1})$, et de même la
formule $\psi(\bar{x})\wedge\varphi_{1}(\bar{a}_{2^{n}},\bar{x})$ est réalisée
par tous les $(\bar{a}_{i}:\,0\leq i<2^{n})$. Par hypothèse d’induction on
obtient que:
$\left\\{\begin{array}[]{rcl}R(\psi(\bar{x})\wedge\varphi_{0}(\bar{a}_{2^{n}},\bar{x}),\varphi_{0},\varphi_{1})\geq
n\\\
R(\psi(\bar{x})\wedge\varphi_{1}(\bar{a}_{2^{n}},\bar{x}),\varphi_{0},\varphi_{1})\geq
n.\\\ \end{array}\right.$
Ainsi par définition du rang on a $R(\psi,\varphi_{0},\varphi_{1})\geq n+1$.
Comme pour tout $i<\omega$ , la formule $\bar{x}=\bar{x}$ est réalisé par tous
les $\bar{a}_{i}$, donc $R(\bar{x}=\bar{x},\varphi_{0},\varphi_{1})\leq n$
pour tout $n$, et par suite
$R(\bar{x}=\bar{x},\varphi_{0},\varphi_{1})\geq\omega$. $\square$
Dans le reste de cette section, on montrera l’équivalence entre l’instabilité
et la propriété de l’ordre positive, pour cela on commence par rappeler et
introduire les outils dont on aura besoin.
###### Définition 17
Soit $T$ une théorie h-inductive. Soient $B$ un pec de $T$, $A$ une partie de
$B$, et $\varphi$ une L-formule positive. On note $S^{m}_{\varphi}(A)$
l’ensemble des m-types de $S(A)$ qui représentent la formule $\varphi$, en
d’autres termes $p\in S^{m}(A)$ tel que $p\vdash\varphi$.
Soit $\bar{a}\in A$, on note par $tp_{\varphi}(\bar{a}\diagup B)$ la famille
des formules positives $\varphi(\bar{x},\bar{b})$ telles que $\bar{b}\in B$ et
$A\models\varphi(\bar{a},\bar{b})$.
Une théorie h-inductive complète et non bornée est dite $\varphi$-instable
s’il existe $A$ un pec de $T$ tel que $\mid S_{\varphi}(A)\mid>\mid A\mid$.
###### Définition 18
Soient $T$ une théorie h-inductive, $A$ un pec de $T$, $\psi,\varphi$ deux
L-formules positives $p\in S_{n}(A)$, et $B\subset A$. On dit que le type $p$
est $(\psi,\varphi)$-sindé sur $B$ s’il existe $\bar{a},\bar{c}$ deux uples de
$A$ tels que
1. 1.
$tp_{\psi}(\bar{a}\diagup B)=tp_{\psi}(\bar{c}\diagup B)$;
2. 2.
$p\vdash\varphi(\bar{x},\bar{a})$ et $p\nvdash\varphi(\bar{x},\bar{c})$
###### Lemme 10
Soit $T$ une théorie h-inductive instable, alors il existe $\varphi$, une
L-formule positive $\varphi$ telle que la théorie $T$ est $\varphi$-instable.
Preuve. Comme $T$ n’est pas stable, par le lemme 3 on déduit que pour tout
cardinal $\lambda\geq max(|L|;\aleph_{0})$ il existe $A$ un pec de $T$ et
$n\in\mathbb{N}$ tel que $|A|=\lambda$ et $|S_{n}(A)|>|A|$. Soit $f$ une
application de $S_{n}(A)$ dans l’ensemble des formules positives qui à chaque
n-type $p$ associe une L-formule $\varphi(\bar{x},\bar{y})$ telle que
$p\vdash\varphi(\bar{x},\bar{a})$ avec $\bar{a}\in A$.
Comme $|S_{n}(A)|\geq|L|^{+}$ on déduit qu’il existe $\Sigma$ une partie de
$S_{n}(A)$ de taille $>|A|$, et $\varphi$ une formule positive telles que
$f(\Sigma)=\varphi$. Ainsi on obtient le résultat
$|S_{\varphi}(A)|>|A|.$
$\square$
###### Corollaire 5
Une théorie h-inductive $T$ complète est instable si et seulement si il existe
une formule positive $\varphi$ telle que $T$ soit $\varphi$-instable.
Le théorème suivant est la version positive du théorème 2.10 de [18]. C’est
l’outil principal pour montrer que l’instabilité implique la propriété de
l’ordre. Comme l’étude de la stabilité positive se fait dans la classe des pec
d’une théorie h-inductive complète, le théorème de compacité positive est
inefficace. En effet, ce théorème ne permet pas de contrôler le modèle dans
lequel vit les réalisations d’un ensemble d’énoncés h-inductifs consistant
avec la théorie. Par conséquent, dans le cas où la théorie n’est pas bornée,
nous sommes contraints à utiliser les techniques de la combinatoire des
cardinaux. La preuve du théorème suivant est une illustration de cet usage.
###### Théorème 3 ([18], théorème 2.10 )
Soient $T$ une théorie h-inductive complète, $\varphi$ une L-formule positive,
et $A$ un pec de $T$. Supposons que
$|S_{\varphi}(A)|>\sum_{0\leq\mu<\lambda}|A|^{\mu}+2^{2^{\mu}}$, où $\lambda$
est un cardinal tel que $\lambda\longrightarrow(\chi)_{2}^{2}$ et $\chi$ un
cardinal. Alors il existe $\theta$ une formule positive et
$(\bar{d}_{i}|i<\chi)$ une suite dans un pec $A$ de $T$, telle que
$A\models\theta(\bar{d}_{i},\bar{d}_{j})$ si et seulement si $i<j$.
Preuve. Soient $A$ un pec de $T$ et $\varphi(\bar{x},\bar{y})$ une L-formule
positive (avec $l(\bar{x})=m,l(\bar{y})=n$) tels que $\mid
S_{\varphi}(A)\mid>\mid A\mid=\kappa$ où $\kappa=\sum_{0\leq\mu<\lambda}\mid
A\mid^{\mu}+2^{2^{\mu}}$. Soit $(c^{i}|i<\kappa^{+})$ une suite de
réalisations des types de $S_{\varphi}(A)$ dans un pec assez large. Posons
$\psi(\bar{x},\bar{y})=\varphi(\bar{y},\bar{x})$. Définissons par induction
sur $\alpha\leq\lambda$ une suite croissante de pec $A_{\alpha}$ tels que
* •
$A_{0}=A$,
* •
Pour $\beta$ limite, $A_{\beta}=\bigcup_{\alpha<\beta}A_{\alpha}$
Pour definir $A_{\alpha+1}$ on procède comme suit: pour tout $p\in
S^{m}_{\varphi}(A_{\alpha})\cup S^{n}_{\psi}(A_{\alpha})$ et $B\subset
A_{\alpha}$ tel que $|B|<|\alpha|^{+}+\aleph_{0}$, $A_{\alpha+1}$ est le pec
engendré par les réalisations $p\upharpoonright B$.
Montrons que $|A_{\alpha}|\leq|A|^{|\alpha+1|}$, par induction sur
$\alpha<\lambda$. Supposons le résultat vrai pour $\alpha$, et montrons qu’il
est vrai pour $\beta=\alpha+1$. En effet comme $|S^{m}_{\varphi}(B)\cup
S^{n}_{\psi}(B)|\leq 2^{|B|}$ et le nombre de choix d’ensembles de cardinal
$\leq|\alpha|+\aleph_{0}$ de $A_{\alpha}$ est inférieur à
$|A_{\alpha}|^{|\alpha|+\aleph_{0}}$, on déduit que
$|A_{\beta}|\leq|A_{\alpha}|^{|\alpha|+\aleph_{0}}.2^{|\alpha|+\aleph_{0}}$.
Par hypothèse d’induction,
$|A_{\alpha+1}|\leq|A|^{|\alpha+1|.(|\alpha|+\aleph_{0})}.2^{|\alpha|+\aleph_{0}}$
Ainsi $|A_{\beta}|\leq|A|^{|\beta+1|}$. De même on remarque que
$|A_{\lambda}|\leq\kappa.$
Maintenant montrons la proposition suivante:
$(\textasteriskcentered)$ Il existe $i<\kappa^{+}$ tel que pour tous
$\alpha<\lambda$ et $B\subset A_{\alpha};|B|<|\alpha|^{+}+\aleph_{0}$,
$tp(c^{i}\diagup A_{\alpha+1})$ est ($\psi,\varphi$)-sindé sur $B$.
Preuve de $(\textasteriskcentered)$: Par l’absurde, supposons que pour tout
$i<\kappa^{+}$ il existe $\alpha_{i}<\lambda$, $B_{i}\subset A_{\alpha_{i}}$
de cardinal $<|\alpha_{i}|^{+}+\aleph_{0}$ tels que $tp(c^{i}\diagup
A_{\alpha_{i}+1})$ n’est pas ($\psi,\varphi$)-scindé sur $B_{i}$.
Comme à tout $i<\kappa^{+}$ on associe un ordinal
$\alpha_{i}<\lambda\leq\kappa$ et que $cf(\kappa^{+})=\kappa^{+}$, on déduit
qu’il existe un sous-ensemble $\Gamma$ de la famille $(c^{i}|i<\kappa^{+})$ de
cardinal $\kappa^{+}$ tel que pour tout $c^{i}\in\Gamma$,
$\alpha_{i}=\alpha<\lambda$.
De même comme le nombre des parties $B$ de $A_{\alpha}$ telles que
$|B|<|\alpha|^{+}+\aleph_{0}$ est au plus $\kappa$, alors il existe $B$ et
$\Gamma^{\prime}\subset\Gamma$ de cardinal $\kappa^{+}$ tel que pour tout
$c^{i}\in\Gamma^{\prime}$, $tp(c^{i}\diagup A_{\alpha+1})$ n’est pas
($\psi,\varphi$)-scindé sur $B$.
Dans la suite $B$ et $\alpha$ sont ceux trouvés dans les paragraphes
précédents.
Soit $\mu=|B|^{m}\leq|\alpha|+\aleph_{0}\leq\lambda$, par définition de
$|A_{\alpha+1}|$ il existe $C\subset A_{\alpha+1}$ qui réalise tous les types
de $S^{m}_{\psi}(B)$, donc $|C|\leq 2^{|\mu|}$, par suite
$|S^{n}_{\varphi}(C)|\leq 2^{|C|}\leq 2^{2^{\mu}}\leq\kappa$. Par le même
raisonnement que précedemment il existe $p\in S^{n}_{\varphi}(C)$ et
$\Gamma^{\prime\prime}\subset\Gamma^{\prime}$ de cardinal $\kappa^{+}$ tels
que pour tout $c^{i}\in\Gamma^{\prime\prime}$ $tp_{\varphi}(c^{i}\diagup
C)=p$.
Dans toute la suite les $c^{i}$ choisis sont dans $\Gamma^{\prime\prime}$.
Comme $tp_{\varphi}(c^{0}\diagup A)\neq tp_{\varphi}(c^{1}\diagup A)$ il
existe $\bar{a}\in A$ tel que
$(\star)\left\\{\begin{array}[]{rcl}tp_{\varphi}(c^{0}\diagup
A)\vdash\varphi(\bar{x},\bar{a})\\\ tp_{\varphi}(c^{1}\diagup
A)\nvdash\varphi(\bar{x},\bar{a}).\end{array}\right.$
Par définition de $C$ il existe $\bar{a}^{\prime}\in C$ tel que
$tp_{\psi}(\bar{a}\diagup B)=tp_{\psi}(\bar{a}^{\prime}\diagup B)$. Comme pour
tout $i<\kappa^{+}$, $tp(c^{i}\diagup A_{\alpha+1})$ ne ($\psi,\varphi$)-split
pas sur $B$, en particulier pour $i=0,1$, et que $tp_{\psi}(\bar{a}\diagup
B)=tp_{\psi}(\bar{a}^{\prime}\diagup B)$, pour $i=0,1$ on a
$(\textasteriskcentered\textasteriskcentered)\quad tp(c^{i}\diagup
A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a})\mbox{ si et seulement si
}tp(c^{i}\diagup A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a}^{\prime})$
du fait que $tp(c^{0}\diagup A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a})$ on
obtient que $tp(c^{0}\diagup
A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a}^{\prime})$. Comme pour tout $c^{i}$
dans $\Gamma^{\prime\prime}$ $tp(c^{i}\diagup C)=p$ alors $tp(c^{0}\diagup
C)=tp(c^{1}\diagup C)$, et comme $tp(c^{0}\diagup
A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a}^{\prime})$ alors $tp(c^{1}\diagup
C)\vdash\varphi(\bar{x},\bar{a}^{\prime})$, ce qui implique $tp(c^{1}\diagup
A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a}^{\prime})$. Ainsi par
$(\textasteriskcentered\textasteriskcentered)$ on obtient $tp(c^{1}\diagup
A_{\alpha+1})\vdash\varphi(\bar{x},\bar{a})$, ce qui contredit $(\star)$.
Ainsi on a démontré $(\textasteriskcentered)$.
Dans toute la suite posons $A^{\star}$ la limite inductive de la suite
$A_{\alpha}$ où $\alpha\leq\lambda$
Maintenant par induction sur $\alpha<\lambda$ on définit une suite de
$(\bar{a}_{\alpha},\bar{b}_{\alpha},\bar{c}_{\alpha})\in A_{2\alpha+2}$ comme
suit. Supposons que pour tout $\beta<\alpha$,
$(\bar{a}_{\beta},\bar{b}_{\beta},\bar{c}_{\beta})$ sont définies. Soit
$B_{\alpha}=\bigcup_{\beta<\alpha}\bar{a}_{\beta}\mbox{\textasciicircum}\bar{b}_{\beta}\mbox{\textasciicircum}\bar{c}_{\beta}$.
D’après $(\textasteriskcentered)$ le type $tp(c^{i}\diagup A_{2\alpha+1})$ est
($\psi,\varphi$)-split sur $B_{\alpha}$. Alors il existe
$\bar{a}_{\alpha},\bar{b}_{\alpha}\in A_{2\alpha+1}$ tels que
$tp_{\psi}(\bar{a}_{\alpha},B_{\alpha})=tp_{\psi}(\bar{b}_{\alpha},B_{\alpha})$
et
$\left\\{\begin{array}[]{rcl}tp(c^{i}\diagup
A_{2\alpha+1})\vdash\varphi(\bar{x},\bar{a}_{\alpha})\\\ tp(c^{i}\diagup
A_{2\alpha+1})\nvdash\varphi(\bar{x},\bar{b}_{\alpha}).\end{array}\right.$
On prend $c_{\alpha}\in A_{2\alpha+2}$ qui par définition de $A_{2\alpha+2}$
réalise $tp(c^{i}\diagup
B_{\alpha}\cup\\{\bar{a}_{\alpha},\bar{b}_{\alpha}\\})$. Ainsi on a
$A^{\star}\models\varphi(\bar{c}_{\alpha},\bar{a}_{\alpha})$ et
$A^{\star}\models\neg\varphi(\bar{c}_{\alpha},\bar{b}_{\alpha})$. On remarque
que pour tous $\alpha\leq\beta<\lambda$ on a
$(\ast)\left\\{\begin{array}[]{rcl}A^{\star}\models\varphi(\bar{c}_{\beta},\bar{a}_{\alpha})\\\
A^{\star}\models\neg\varphi(\bar{c}_{\beta},\bar{b}_{\alpha}).\end{array}\right.$
Soit $f$ l’application coloriage de $\lambda$ définie par: pour tous
$\alpha,\beta<\lambda$
$f(\alpha,\beta)=\left\\{\begin{array}[]{rcl}0\quad\text{si}\quad
A^{\star}\models\varphi(\bar{c}_{min(\alpha,\beta)},\bar{a}_{max(\alpha,\beta)})\\\
1\quad\quad\quad\quad\text{sinon}.\end{array}\right.$
Comme par hypothèse on a $\lambda\longrightarrow(\chi)_{2}^{2}$ alors il
existe $H\subset\lambda$ de taille $\chi$ telle que $f\upharpoonright H^{2}$
est constante. Supposons que $f(H^{2})=1$ si $\alpha,\beta\in H$, alors
d’après l’hypothèse $f(H^{2})=1$ et $(\ast)$, on a
$A^{\star}\models\neg\varphi(\bar{c}_{\alpha},\bar{a}_{\beta})$ si et
seulement si $\alpha<\beta$. Maintenant si $f(H^{2})=0$, alors pour tous
$\alpha<\beta<\lambda$ on a
$A^{\star}\models\varphi(\bar{c}_{\alpha},\bar{a}_{\beta})$, comme
$\bar{a}_{\alpha}$ et $\bar{b}_{\alpha}$ ont le même type sur $B_{\alpha}$ et
$\bar{c}_{\beta}\in B_{\alpha}$ alors on a pour tous $\alpha<\beta<\lambda$,
$A^{\star}\models\varphi(\bar{c}_{\alpha},\bar{b}_{\beta})$. Ainsi et d’après
$(\ast)$ on deduit que dans le cas où $f(H^{2})=0$
$A^{\star}\models\varphi(\bar{c}_{\alpha},\bar{b}_{\beta})\quad\text{si et
seulement si}\quad\alpha<\beta$
Par conséquent la formule positive $\theta$ définie par
$\theta((\bar{x},\bar{x}^{\prime}),(\bar{y},\bar{y}^{\prime}))\equiv\varphi(\bar{x},\bar{y}^{\prime})$
ordonne la suite $(\bar{c}_{\alpha},\bar{a}_{\alpha})$, où $\alpha,\beta\in
H$. $\square$
###### Corollaire 6
Si $T$ est une théorie h-inductive complète non bornée et instable, alors elle
a la propriété de l’ordre.
Preuve. Pour pouvoir appliquer le théorème 3 et comme $T$ est instable on
prend $\lambda\rightarrow(\chi)_{2}^{2}$, avec $\chi=\beth^{+}(T)$, et
$\lambda=(2^{\beth^{+}(T)})^{+}$.
Il existe $A$ un pec de $T$ de cardinal $\kappa=2^{2^{\lambda}}$, et $\varphi$
une formule positive tels que $|S_{\varphi}(A)|>|A|=\kappa$ et
$\kappa=\sum_{0\leq\mu<\lambda}\mid A\mid^{\mu}+2^{2^{\mu}}$.
Ainsi par le théorème 3, il existe $\theta(\bar{x},\bar{y})$ une formule
positive et $(\bar{a}_{\alpha},\alpha<\beth^{+}(T))$ une suite ordonnée par
$\theta$. Donc pour tout $\beta<\alpha<\beth^{+}(T)$,
$\models\neg\theta(\bar{a}_{\alpha},\bar{a}_{\beta})$ donc il existe
$\theta_{\alpha,\beta}$ une formule positive dans la résultante de $\theta$
telle que $\models\theta_{\alpha,\beta}(\bar{a}_{\alpha},\bar{a}_{\beta})$.
Soit l’application coloriage $f$ de $\beth^{+}(T))$ définie comme suit: Pour
tous $\alpha,\beta<\beth^{+}(T)$,
$f(\\{\alpha,\beta\\})=\theta_{max(\alpha,\beta),min(\alpha,\beta)}$ comme le
cardinal de la résultante est inférieur à $|T|$ ou ($|L|$), alors par le
théorème de Erdös-Rado il existe une suite de taille infinie et
$\theta^{\prime}$ telle que pour tout $\\{\alpha,\beta\\}$ de la nouvelle
suite on a $f(\\{\alpha,\beta\\})=\theta^{\prime}$. Ainsi le couple
$(\theta,\theta^{\prime})$ définit l’ordre cherché. $\square$
###### Corollaire 7
Soit $T$ une théorie h-inductive complète. Alors $T$ est stable si et
seulement si elle n a pas la propriété de l’ordre.
Preuve. La preuve se divise en deux cas, suivant si $T$ est bornée ou non.
Supposons d’abord que $T$ est non bornée. Si $T$ a la propriété de l’ordre,
par le lemme 9 il existe un couple de formules contradictoires
$(\varphi_{0},\varphi_{1})$ telles que
$R(x=x,\varphi_{0},\varphi_{1})\geq\omega$, ce qui implique par le lemme 4 que
$T$ est instable. L’autre direction découle directement du corollaire 6.
Dans le cas où la théorie est bornée la conclusion découle des lemmes 3 et 7.
$\square$
### 3.4 Autres caractérisations
Dans [3] Ben Yaacov a étudié et caractérisé la stabilité positive par la
finitude du rang de Shelah et la type-définissabilité des types ( définition
19). Dans le fait 9, on rappelle les caractérisations données par Ben Yaacov,
avec leurs démonstrations.
###### Définition 19 ( [3])
Soient $p(\bar{x})\in S(A)$ un type à paramètres dans un pec $A$ de $T$ et
$\varphi(\bar{x},\bar{y})$ une formule positive. On dit que $p(\bar{x})$ est
$\varphi$-définissable si il existe $q_{\varphi}(\bar{y})$ un type partiel
avec paramètres dans $A$ qui contient au plus $|T|$ L-formules, tel que
$p\vdash\varphi(\bar{x},\bar{a})\quad\text{si et seulemement si}\quad A\models
q_{\varphi}(\bar{b})$
On note aussi le type partiel $q_{\varphi}$ par $dp(\varphi)$ Un type $p$ est
dit définissable si il est $\varphi$-définissable pour toute formule positive
$\varphi$.
###### Fait 9 ([3])
Soit $T$ une théorie h-inductive, complète. Les propriétés suivantes sont
équivalentes:
1. 1.
Pour toute formule positive $\varphi$ et pour tout $\psi\in Res_{T}(\varphi)$,
$R(\bar{x}=\bar{x},\varphi,\psi)<\omega$.
2. 2.
Pour tout pec $A$ de $T$, tous les types de $S(A)$ sont définissables.
3. 3.
Pour tout pec $A$, $|S(A)|\leq(|A|+|T|)^{|T|}$.
4. 4.
Il existe un cardinal $\lambda$ tel que $T$ est $\lambda$-stable.
###### Lemme 11
Soit $T$ une théorie h-inductive complète et stable, soit $A$ un pec de $T$.
Alors la théorie $Tk(A)$ et ses compagnes sont stables.
Preuve. Soit $B$ un pec de la théorie complète $Tk(A)$. Alors $B$ est aussi un
pec de $T$. D’après le fait 9 on a $|S(B)|\leq(|B|+|T|)^{|T|}$, ce qui nous
permet de conclure que la théorie $Tk(A)$ est stable (fait 9). $\square$
### 3.5 Une comparaison entre les stabilités positive et usuelle.
Dans cette section nous étudions un exemple d’une structure qui est stable
positivement et qui ne l’est pas au sens de la théorie des modèles usuelle.
Soient $M_{1}=(\mbox{\Bbbb Q},\leq)$ dans le langage $L_{1}=\\{c_{q}\
(q\in\mbox{\Bbbb Q}),\leq\\}$. Comme la structure $M_{1}$ est bornée, elle est
stable (lemme 3). Remarquons aussi qu’elle est instable au sens de la théorie
des modèles usuelle puisqu’on peut ordonner un ensemble infini. Soient $M_{2}$
un pec d’une théorie h-inductive complète non bornée et positivement stable
(Par exemple la théorie des corps de caractéristique fixée), et $L_{2}$ le
langage de le théorie de $M_{2}$ augmenté en ajoutant un symbole de constante
pour chaque élément de $M_{2}$.
Soit $L^{\star}$ la réunion de $L_{1}$ et $L_{2}$ à laquelle nous ajoutons
deux symboles relationnels $m_{1},m_{2}$. Soit $T$ la théorie h-inductive
formée par $Tk(M_{1})\cup Tk(M_{2})$ et l’ensemble des énonces h-inductifs
$\\{\neg\exists xm_{1}(x)\wedge m_{2}(x),\forall xm_{1}(x)\vee m_{2}(x)\\}$ $\
\ \ \ \ \ \ \ \ \ \ \ \ \cup\ \\{m_{1}(a):\,a\in M_{1}\\}\ \cup\
\\{m_{2}(b):\,b\in M_{2}\\}.$
Une $L^{\star}$-formule positive est une combinaison booléenne de formules de
la forme $(\varphi(\bar{x})\wedge m_{1}(\bar{x}))\wedge(\psi(\bar{y})\wedge
m_{2}(\bar{y}))$ et $(\varphi^{\prime}(\bar{x})\wedge
m_{1}(\bar{x}))\vee(\psi^{\prime}(\bar{y})\wedge m_{2}(\bar{y}))$ où
$\varphi,\varphi^{\prime}$ des $L_{1}-$formules et $\psi,\psi^{\prime}$ des
$L_{2}-$formules positives et pour $i=1,2$,
$m_{i}(\bar{z})\equiv\bigwedge_{z_{j}\in\bar{z}}m_{i}(z_{j})$. Dans la suite
nous notons la formule $(\varphi(\bar{x})\wedge
m_{1}(\bar{x}))\wedge(\psi(\bar{y})\wedge m_{2}(\bar{y}))$ par
$(\varphi(\bar{x})\wedge\psi(\bar{y}))$ et la formule
$(\varphi^{\prime}(\bar{x})\wedge
m_{1}(\bar{x}))\vee(\psi^{\prime}(\bar{y})\wedge m_{2}(\bar{y}))$ par
$(\varphi^{\prime}(\bar{x})\vee\psi^{\prime}(\bar{y}))$.
Les modèles de la théorie $T$ sont de la forme $N_{1}\cup N_{2}$ avec
$N_{1}\models Tk(M_{1})$ et $N_{2}\models Tk(M_{2})$ et $N_{1}\cup
N_{2}\models\varphi(\bar{n}_{1})\wedge\psi(\bar{n}_{2})$ si et seulement si
$N_{1}\models\varphi(\bar{n}_{1})$ et $N_{2}\models\psi(\bar{n}_{2})$, et
$N_{1}\cup N_{2}\models\varphi(\bar{n}_{1})\vee\psi(\bar{n}_{2})$ si et
seulement si $N_{1}\models\varphi(\bar{n}_{1})$ ou
$N_{2}\models\psi(\bar{n}_{2})$. Ceci nous permet de déduire qu’une
application de $N_{1}\cup N_{2}$ dans un modèle de $T$ est un homomorphisme si
et seulement si les restrictions de $f$ sur $N_{1}$ et $N_{2}$ sont des
homomorphismes.
###### Lemme 12
Soit $N=N_{1}\cup N_{2}$ un modèle de $T$, alors $N$ est un pec de $T$ si et
seulement si $N_{1}$ et $N_{2}$ sont respectivement des pec de $Tk(M_{1})$ et
$Tk(M_{2})$
Preuve. Soit $N=N_{1}\cup N_{2}$ avec $N_{1}$ un pec de $Tk(M_{1})$ et $N_{2}$
un pec de $Tk(M_{2})$, et soit $f$ un homomorphisme de $N$ dans modèle
$M=N_{1}^{\prime}\cup N_{2}^{\prime}$ de $T$. Supposons que
$M\models\varphi(f(\bar{n}_{1}))\wedge\psi(f(\bar{n}_{2}))$, avec
$\bar{n}_{1}\in N_{1}$ et $\bar{n}_{2}\in N_{2}$. Comme $N_{1}$ et $N_{2}$
sont des pec, les restrictions de $f$ à $N_{1}$ et $N_{2}$ sont des
immersions. Ainsi $N\models\varphi(\bar{n}_{1})\wedge\psi(\bar{n}_{2})$. Ce
qui implique que $f$ est une immersion et par suite $N$ est un pec de $T$.
Inversement supposons que $N$ est un pec de $T$. Pour $i=1,2$, soit $f_{i}$ un
homomorphisme de $N_{i}$ dans $N_{i}^{\prime}$ un modèle de $Tk(M_{i})$ alors
$f=f_{1}\cup f_{2}$ est un homomorphisme de $N$ dans
$N^{\prime}=N_{1}^{\prime}\cup N_{2}^{\prime}$. Comme $N$ est un pec de $T$ on
déduit que $f$ est une immersion ce qui implique que $f_{1}$ et $f_{2}$ sont
des immersions. Ainsi $N_{1}$ est un pec de $Tk(M_{1})$ et $N_{2}$ un pec de
$Tk(M_{2})$.$\square$
###### Lemme 13
La théorie $T$ est complète et non bornée.
Preuve. En effet, comme la théorie $Tk(M_{2})$ n’est pas bornée, pour tout
cardinal $\alpha$ il existe $N_{2}$ un pec de $Tk(M_{2})$ de cardinal
$\geq\alpha$. Ainsi $M_{1}\cup N_{2}$ est un pec de $T$ est de cardinal
$\geq\alpha$, ce qui implique que $T$ est non bornée. $\square$
###### Proposition 1
La théorie $T$ est positivement stable.
Preuve. Supposons que la théorie $T$ est instable. D’après le corollaire 7 il
existe $N=N_{1}\cup N_{2}$ un pec de $T$, et une suite
$(\bar{n}_{i},\,i<\omega)$ de $N$ ordonnée par un couple de formules positives
$(\varphi,\psi)$ avec $\psi\in Res_{T}(\varphi)$.
Tout élément $\bar{n}_{i}$ est de la forme $(\bar{a}_{i},\bar{b}_{i})$ où
$\bar{a}_{i}\in N_{1}$ et $\bar{b}_{i}\in N_{2}$, et la formule positive
$\varphi$ vérifie
$\varphi((\bar{a}_{i},\bar{b}_{i}),(\bar{a}_{j},\bar{b}_{j}))\equiv\varphi_{1}(\bar{a}_{i},\bar{a}_{j})(C)\varphi_{2}(\bar{b}_{i},\bar{b}_{j})$
respectivement
$\psi((\bar{a}_{i},\bar{b}_{i}),(\bar{a}_{j},\bar{b}_{j}))\equiv\psi_{1}(\bar{a}_{i},\bar{a}_{j})(C)\psi_{2}(\bar{b}_{i},\bar{b}_{j})$
avec $\varphi_{i}$ (resp $\psi_{i}$) des formules positives dans le langage de
la théorie $Tk(M_{i})$ pour $i=1,2$ et $C\in\\{\wedge,\vee\\}$.
Supposons que $C=\vee$, alors
$\varphi((\bar{a}_{i},\bar{b}_{i}),(\bar{a}_{j},\bar{b}_{j}))\equiv\varphi_{1}(\bar{a}_{i},\bar{a}_{j})\vee\varphi_{2}(\bar{b}_{i},\bar{b}_{j})$,
et
$\psi((\bar{a}_{i},\bar{b}_{i}),(\bar{a}_{j},\bar{b}_{j}))\equiv\psi_{1}(\bar{a}_{i},\bar{a}_{j})\wedge\psi_{2}(\bar{b}_{i},\bar{b}_{j})$
avec $\psi_{i}\in Res_{T}(\varphi_{i})$ pour $i=1,2$. Comme d’une part, on
peut extraire de la suite de départ une sous suite
$((\bar{a}_{i},\bar{b}_{i}),\,i<\omega)$ telle que pour tous $i<j$ on a soit
$N_{1}\models\varphi_{1}(\bar{a}_{i},\bar{a}_{j})$ soit
$N_{2}\models\varphi_{2}(\bar{b}_{i},\bar{b}_{j})$. Et d’autre part, on a pour
tout $i<j$, $N_{1}\models\psi_{1}(\bar{a}_{j},\bar{a}_{i})$ et
$N_{2}\models\psi_{2}(\bar{b}_{j},\bar{b}_{i})$. On aboutit alors à l’une des
deux conclusions suivantes: soit on ordonne positivement définissablement un
ensemble infini de $N_{1}$; soit on ordonne positivement définissablement un
ensemble infini de $N_{2}$. Chaque possibilité mène à une contradiction car on
sait que $M_{1}$ et $M_{2}$ sont positivement stables.
On refait le même raisonement dans le cas ou $C=\wedge$ c’est à dire
$\varphi((\bar{a}_{i},\bar{b}_{i}),(\bar{a}_{j},\bar{b}_{j}))\equiv\varphi_{1}(\bar{a}_{i},\bar{a}_{j})\wedge\varphi_{2}(\bar{b}_{i},\bar{b}_{j})\
.$
Ainsi on déduit que la théorie $T$ est positivement stable. $\square$
On remarque que la théorie $Th(M_{1}\cup M_{2})$ dans le cadre de la théorie
des modèles usuelle est instable puisque la structure $M_{1}$ s’ordonne
définissablement.
### 3.6 Conséquences de la stabilité
Dans la première partie de cette section on étudie l’existence, le nombre des
extensions et des restrictions spéciales d’un types dans une théorie
h-inductive stable.
Dans la deuxième partie on propose un encadrement topoloqigue de la type-
définissabilité (définition 19) donné dans la caractérisation de la stabilité
(fait 9).
###### Définition 20
Soient $T$ une théorie h-inductive, $B$ un pec de $T$ de cardinal
$\geq\lambda$, et $p\in S(B)$. On dit que $p$ est $\lambda$-spécial, si et
seulement si il existe, $A\subset B$, de cardinal au plus $\lambda$ qui
vérifie la propriété suivante:
Pour tous $\bar{a},\bar{b}$, des uples de $B$, tels que $tp(\bar{a}\diagup
A)=tp(\bar{b}\diagup A)$, et toute formule positive $\varphi$
$p\vdash\varphi(x,\bar{a})\Leftrightarrow p\vdash\varphi(x,\bar{b}).$
Dans ce cas on dit que $p$ est $\lambda$-spécial sur $A$.
On dit que $p$ est $A$-spécial, si il est $|A|$-spécial sur $A$.
###### Lemme 14
Soient $T$ une théorie h-inductive, $M,N$ et $P$ trois pec de $T$ tels que
$M\subset N\subset P$ et $N$ réalise tous les types de $S(M)$. Soit $p\in
S_{n}(N)$ qui est $M$-spécial, alors il existe un unique un fils $q\in
S_{n}(P)$ de $p$ qui est $M$-spécial.
Preuve. Soit $\Gamma$ l’ensemble des formules positives,
$\varphi(\bar{x},\bar{d})$ avec $\bar{d}\in P$, telles qu’il existe
$\bar{a}\in N$ de même type sur $M$ que $\bar{d}$ (ie $tp(\bar{d}\diagup
M)=tp(\bar{a}\diagup M)$) et que $p\vdash\varphi(\bar{x},\bar{a})$. Montrons
que $\Gamma$ est un type de $S_{n}(P)$. Pour cela on va montrer que
$\Gamma\cup Tu(P)$ est consistant, ensuite on montrera que $\Gamma$ est une
famille maximale de formules positives.
$\Gamma\cup Tu(P)$ est consistant: En effet un fragment fini $\Sigma$ de cette
famille est de la forme
$\\{\neg\exists\bar{y}\varphi(\bar{y},\bar{d},\bar{m}),\psi(\bar{x},\bar{d},\bar{m})\\}$,
avec $\varphi,\psi$ des formules positives, $\bar{d}\in P$, $\bar{m}\in M$,
$\neg\exists\bar{y}\varphi(\bar{y},\bar{d},\bar{m})\in Tu(P)$, et
$\psi(\bar{x},\bar{d},\bar{m})\in\Gamma$. Soit $r$ le type de $\bar{d}$ sur
$M$. Comme $N$ satisfait tous les types à paramètres dans $M$, il existe
$\bar{n}\in N$ qui réalise $r$. Ainsi si on interprète $\bar{d}$ par
$\bar{n}$, alors il existe $\bar{a}\in N$ tel que
$N\models\psi(\bar{a},\bar{n},\bar{m})$ et
$N\vdash\neg\exists\varphi(\bar{y},\bar{n},\bar{m})$. Ainsi $N$ réalise
$\Sigma$. D’où la consistance de $\Gamma\cup Tu(P)$.
Montrons la maximalité de $\Gamma$. Soit $\varphi(\bar{x},\bar{d})$ une
formule positive, avec $\bar{d}\in P$, qui n’appartient pas à $\Gamma$. Soit
$\bar{n}\in N$ un uple qui a le même type sur $M$ que $\bar{d}$. Par
définition de $\Gamma$, on déduit que $p\vdash\neg\varphi(\bar{x},\bar{n})$,
ce qui implique l’existence d’une formule positive $\psi(\bar{x},\bar{y})\in
Res_{Tu(M)}(\varphi(\bar{x},\bar{y}))$ à paramètres dans $M$ telle que
$p\vdash\psi(\bar{x},\bar{n})$. Par définition de $\Gamma$ on déduit que
$\psi(\bar{x},\bar{d})\in\Gamma$, d’où la maximalité de $\Gamma$. Par
conséquence $\Gamma$ est un fils $M$-spécial de $p$.
D’autre part, on remarque que si $\Gamma^{\prime}$ est un autre fils
$M$-spécial de $p$, alors il contient par définition $\Gamma$, et comme
$\Gamma$ est maximal on a égalité. $\square$
###### Lemme 15
Soit $T$ une théorie h-inductive complète non bornée et $\lambda$-stable.
Soient $M,N$ deux pec de $T$ tels que $|M|\leq\lambda$ et $N$ réalise tous les
types de $S(M)$, alors
$|\\{p\in S(N)|\quad p$ est $M$-spécial $\\}|\leq\lambda$
Preuve. Comme $T$ est $\lambda$-stable, $|M|\leq\lambda$ et que $N$ réalise
tous les types de $S(M)$, il existe $N^{\star}$ un pec de $T$ de cardinal
$\lambda$ qui réalise tous les types de $S(M)$ et tel que $M\subset
N^{\star}\subset N$.
Soit $p\in S(N)$ un type $M$-spécial, alors $q=p\upharpoonright{N^{\star}}$
est $M$-spécial, et d’après le lemme 14, $p$ est l’unique fils de $p$ qui est
$M$-spécial. Comme $|N^{\star}|\leq\lambda$, on déduit que
$|\\{p\in S(N)|\quad p$ est $M$-spécial $\\}|\leq\lambda$
$\square$
Dans [17] Shelah introduit une notion de la stabilité dans les cas qui
s’adapte au contexte positif et permet d’obtenir des conditions nécessaires.
Le théorème suivant est un exemple dont la démonstration que nous proposons
n’utilise que des notions qui sont propre é la théorie des modèles positives
###### Théorème 4
Soient $T$ une théorie $h$-inductive $\mu$-stable, $M$ un pec de $T$ et $p\in
S(M)$. Il existe $N$ un pec de $T$ de cardinal $\leq\mu$ tel que
$N\prec_{+}M$, et $p$ est $N$-spécial.
Preuve. Soit $\chi$ le cardinal défini par
$\chi=min\\{\alpha\mid 2^{\alpha}>\mu,\alpha\leq\mu\\}$
Supposons que $p$ n’est $N$-spécial pour aucun pec $N\subset M$ de cardinal
$\leq\mu$. Par induction on construit une suite
$(M_{\alpha},N_{i\alpha};\,i=1,2;\alpha\leq\chi)$ telle que pour tout
$\alpha\leq\chi$ et $i=1,2$ on a:
* •
$M_{\alpha},N_{i\alpha}$ sont des pec de $T$ inclus dans $M$,
* •
$max(|M_{\alpha}|,|N_{i\alpha}|)\leq\mu$,
* •
$M_{\alpha}\subset N_{i\alpha}\subset M_{\alpha+1}$
Posons $M_{0}$ un pec de $T$ de cardinal au plus $\mu$ et qui est inclus dans
$M$. Comme par hypothèse $p$ n’est pas $M_{0}$-spécial, ils existent
$\bar{a}_{i},\,i=1,2$ deux uples de $M$ qui ont le même type sur $M_{0}$ et
tels que $p\vdash\varphi(\bar{x},\bar{a}_{1})$ et
$p\nvdash\varphi(\bar{x},\bar{a}_{2})$. Supposons que la suite est construite
jusqu’au $\alpha$, par amalgamation soit $M_{\alpha+1}$ l’amalgamé des
$N_{i\alpha};i=1,2$ sur la base $M_{\alpha}$ et qui est de cardinal $\leq\mu$.
(ie le diagramme commutatif suivant est donné par l’amalgamation des pec)
$\textstyle{M_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\rho}}$$\scriptstyle{i_{\alpha}}$$\textstyle{{N_{1\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{M_{2\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{h}_{\rho}}$$\textstyle{{P_{\alpha}}}$
Comme $p$ n’est pas $M_{\alpha+1}$-spécial, il existe
$\bar{a}_{i,\alpha+1};i=1,2$ des uples de $M$ tels que
$p\vdash\varphi(\bar{x},\bar{a}_{1,\alpha+1})$ et
$p\nvdash\varphi(\bar{x},\bar{a}_{2,\alpha+1})$. Soit $N_{i,\alpha+1}$ le pec
de $T$ engendré par $M_{\alpha+1}\cup\\{\bar{a}_{i\alpha+1}\\}$ (le lemme
LABEL:skolem), pour $i=1,2$. Ainsi on a construit la suite
$(M_{\alpha},N_{i,\alpha};\,i=1,2;\alpha\leq\chi)$
A partir de la suite $(M_{\alpha};N_{i\alpha};\alpha\leq\chi,i=1,2)$ on va
construire une deuxième suite
$(M^{\star}_{\alpha},h_{\rho},\alpha\leq\chi,\rho\in^{\alpha}2)$ qui vérifie
les propriétés suivantes:
* •
pour tout $\alpha\leq\chi$, $M^{\star}_{\alpha}$ est un pec de $T$ de cardinal
$\leq\mu$,
* •
$\alpha\leq\beta\Longrightarrow{M}^{\star}_{\alpha}\prec_{+}{M}^{\star}_{\beta}$,
* •
pour tout ordinal limite $\alpha\leq\chi$,
${M}^{\star}_{\alpha}=\bigcup_{\gamma\leq\alpha}{M}^{\star}_{\gamma}$,
* •
pour tout $\rho\in^{\alpha}2$,
$h_{\rho}:M_{\alpha}\longrightarrow{M}^{\star}_{\alpha}$ est un homomorphisme
(immersion),
* •
$\alpha<\beta\leq\chi$ et $\rho\in^{\beta}2\Longrightarrow
h_{\rho\mid\alpha}\subseteq h_{\rho}$,
* •
si $\beta=\alpha+1$ et $\rho\in^{\beta}2\Longrightarrow
h_{\rho\mbox{\textasciicircum}0}(N_{1\alpha})=h_{\rho\mbox{\textasciicircum}1}(N_{2\alpha})$
La construction de la suite $(M_{\alpha};N_{i\alpha};\alpha\leq\chi,i=1,2)$
est faite par induction comme suit: Posons ${M}^{\star}_{0}=M_{0}$ et
$h_{0}=id_{M_{0}}$. Supposons que la suite est contruite jusqu’au $\alpha$.
Soit $\rho\in^{\alpha}2$, et $h_{\rho}$ l’homomorphisme (immersion ) de
$M_{\alpha}$ dans $M^{\star}_{\alpha}$. Par l’amalgamation asymétrique on a le
digramme commutatif suivant avec $i_{\alpha}$ l’application inclusion et
$P_{\alpha}$ un pec de $T$ de cardinal au plus $\mu$
$\textstyle{M_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\rho}}$$\scriptstyle{i_{\alpha}}$$\textstyle{{M^{\star}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{M_{\alpha+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{h}_{\rho}}$$\textstyle{{P_{\alpha}}}$
Posons $h_{\rho\mbox{\textasciicircum}1}=\bar{h}_{\rho}$. Soit $f_{\alpha}$
l’homomorphisme qui fixe $M_{\alpha}$ point par point et qui envoie
$\bar{a}_{1\alpha}$ vers $\bar{a}_{2\alpha}$, donc on a
$f_{\alpha}(N_{1\alpha})=N_{2\alpha}$. Par amalgamation asymétrique on peut
trouver $Q_{\alpha}$ un pec de cardinal au plus $\mu$ et $g_{\alpha}$ un
homomorphisme (une immersion) de $M_{\alpha+1}$ dans $Q_{\alpha}$ tels que le
diagramme suivant est commutatif
$\textstyle{N_{1\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{h}_{\rho}\circ
f_{\alpha}}$$\scriptstyle{i_{\alpha}}$$\textstyle{{N_{2\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{M_{\alpha+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{\alpha}}$$\textstyle{{Q_{\alpha}}}$
Posons $h_{\rho\mbox{\textasciicircum}0}=g_{\alpha}$, alors on a
$h_{\rho\mbox{\textasciicircum}0}(N_{1\alpha})=h_{\rho\mbox{\textasciicircum}1}\circ
f_{\alpha}(N_{1\alpha})=h_{\rho\mbox{\textasciicircum}1}(N_{2\alpha})$
On définit $M^{\star}_{\alpha+1}$ comme étant l’amalgamé de tous les
$h_{\rho\mbox{\textasciicircum}0}(M_{\beta})$,
$h_{\rho\mbox{\textasciicircum}1}(M_{\beta})$ (où $\rho$ parcours l’ensemble
${}^{\alpha}2$) sur la base $M_{\alpha+1}$, on a
$|M^{\star}_{\alpha+1}|\leq\chi$.
Pour tout ordinal limite $\alpha$ soit:
${M}^{\star}_{\beta}=\bigcup_{\alpha<\beta}{M}^{\star}_{\alpha}\quad\text{et
pour tout }\quad\rho\in^{\beta}2,\quad
h_{\rho}=\bigcup_{\alpha<\beta}h_{\rho\upharpoonright\alpha}$
Une telle construction nous permet de déduire que pour tout $\alpha\leq\chi$,
${M}^{\star}_{\alpha}\prec_{+}{M}^{\star}_{\alpha+1}$. En effet par
construction on a
${M}^{\star}_{\beta}=<h_{\rho\mbox{\textasciicircum}0}(M_{\beta})\cup
h_{\rho\mbox{\textasciicircum}1}(M_{\beta})\mid\quad\rho\in^{\alpha}2>$
et
${M}^{\star}_{\alpha}=<h_{\rho}(M_{\alpha})\cup
h_{\rho}(M_{\alpha})\mid\quad\rho\in^{\alpha}2>$
Comme $h_{\rho\mbox{\textasciicircum}i\upharpoonright\alpha}=h_{\rho}$, pour
$i=0,1$ et $M_{\alpha}\prec_{+}M_{\beta}$, alors
${M}^{\star}_{\alpha}\prec_{+}{M}^{\star}_{\beta}$ Rappelons que
$|{M}^{\star}_{\chi}|\leq\chi$ car $\chi$ est un cardinal et par suite c’est
un ordinal limite, par contruction ${M}^{\star}_{\chi}$ est la limite
inductive d’une famille de pec tous de cardinal inférieur ou égale à $\mu$.
Soit $h_{\rho}\in^{\chi}2$, par amalgamation asymétrique il existe $H_{\rho}$
une immersion de $M$ dans un pec $N_{\rho}$ de $T$ telle que le diagramme
suivant est commutatif
$\textstyle{M_{\chi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\rho}}$$\scriptstyle{i}$$\textstyle{{M^{\star}_{\chi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{im}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H_{\rho}}$$\textstyle{{N_{\rho}}}$
Maintenant sous l’hypothèse de la non $\mu$-spécialité de $p$ on montrera que
la théorie $T$ est nécessairement non $\mu$-stable. Soient
$\eta,\nu\in^{\chi}2$ et $H_{\eta},H_{\nu}$ définit comme précédemment, et
soient
$\left\\{\begin{array}[]{rcl}p_{\eta}=H_{\eta}(p)_{\upharpoonright{M}^{\star}_{\chi}}\\\
p_{\nu}=H_{\nu}(p)_{\upharpoonright{M}^{\star}_{\chi}}.\end{array}\right.$
Alors on a:
$\eta\neq\nu\in^{\chi}2\Longrightarrow p_{\eta}\neq p_{\nu}$
En effet posons $\sigma=\eta\wedge\nu$ (l’intersection des deux suites) de
longueur $\gamma$ (ie $\sigma\in^{\gamma}2$ telle que
$h_{\sigma\mbox{\textasciicircum}0}\subseteq h_{\eta}$ et
$h_{\sigma\mbox{\textasciicircum}1}\subseteq h_{\nu}$). Par hypothèse de non
$\mu$-spécialité de $p$ il existe $\bar{a}\in N_{1\sigma}\subset M_{\chi}$ et
$\varphi$ une formule positive tels que
$(\ast)\left\\{\begin{array}[]{rcl}p\vdash\varphi(\bar{x},\bar{a})\\\
p\nvdash\varphi(\bar{x},f_{\sigma}(\bar{a})).\end{array}\right.$
Donc
$p_{\eta}\vdash\varphi(\bar{x},h_{\sigma\mbox{\textasciicircum}0}(\bar{a}))$,
par construction on a
$h_{\sigma\mbox{\textasciicircum}0}=\bar{h}_{\sigma}\circ f_{\sigma}$, ainsi
$p_{\eta}\vdash\varphi(\bar{x},\bar{h}_{\sigma}\circ f_{\sigma}(\bar{a}))$
D’autre part comme $p\nvdash\varphi(\bar{x},f_{\sigma}(\bar{a}))$ il existe
une formule positive $\psi\in Res_{T}(\varphi)$ telle que
$p\vdash\psi(\bar{x},f_{\sigma}(\bar{a}))$, et par le fait que
$h_{\sigma\mbox{\textasciicircum}1}=\bar{h}_{\sigma\upharpoonright
M_{\sigma+1}}=H_{\nu\upharpoonright M_{\sigma+1}}$ on déduit que
$p_{\nu}\vdash\psi(\bar{x},\bar{h}_{\sigma}\circ f_{\sigma}(\bar{a}))$ Par
conséquent
$\left\\{\begin{array}[]{rcl}p_{\eta}\vdash\varphi(\bar{x},\bar{h_{\sigma}}\circ
f_{\sigma}(\bar{a}))\\\ p_{\nu}\vdash\psi(\bar{x},\bar{h_{\sigma}}\circ
f_{\sigma}(\bar{a})).\end{array}\right.$
Ainsi $p_{\eta}\neq p_{\nu}$ car les deux formules $\varphi$ et $\psi$ sont
contradictoires, ce qui contredit la $\mu$-stabilité de $T$. $\square$
Dans le reste de cette section on propose un encadrement topologique du type
$q_{\varphi}$ (le type-définition ) associé à une formule positive $\varphi$.
Le théorème 5 nous donne cet encadrement, aucours de ce théorème on a adapté
positivement une technologie donnée dans le lemme 2.2 [11] pour arriver à
encadrer le fermé $F$ défini par le type partiel $q_{\varphi}$.
Rappelons que si $\varphi(\bar{x},\bar{a})$, $A$ un pec de $T$ et $\bar{a}$ un
uple de $A$. On note par $F_{\varphi(\bar{x},\bar{a})}$ (resp
$O_{\varphi(\bar{x},\bar{a})}$) le fermé (resp l’ouvert) défini par $\\{p\in
S(A);p\vdash\varphi(\bar{x},\bar{a})\\}$ (resp $\\{p\in
S(A);p\nvdash\varphi(\bar{x},\bar{a})\\}$). De même si $q$ est un type partiel
a paramètre dans $A$, on note $F_{q}$ par le fermé $\cup\\{F_{f};q\vdash f\\}$
###### Théorème 5
Soient $T$ une théorie h-inductive non bornée et stable , $A$ un pec de $T$,
et $p\in S_{n}(A)$. Alors pour toute formule positive
$\varphi(\bar{x},\bar{y})$, le fermé $F_{dp(\varphi)}$ associé au type
$dp(\varphi)$ vérifie
$\bigcap_{\psi\in Res_{T}(\varphi)}F_{\Phi^{0}_{\psi}(y)}\subset
F_{dp(\varphi)}\subset\bigcap_{\psi\in
Res_{T}(\varphi)}O_{\Phi^{1}_{\psi}(\bar{y})}$
où
$\Phi^{0}_{\psi}(\bar{y})=\bigvee_{w\subset\\{0,\cdots,2N_{\psi}+1\\}}\bigwedge_{i\in
w}\varphi_{0}(\bar{c}_{i},\bar{y})\ \ ,$
$\Phi^{1}_{\psi}(\bar{y})=\bigvee_{w\subset\\{0,\cdots,2N_{\psi}+1\\}}\bigwedge_{i\in
w}\psi(\bar{c}_{i},\bar{y})$
et les $\bar{c}_{i}$ des uplés de $A$ dependent de $\psi$
Preuve. Posons $\varphi_{0}=\varphi$, soit $\varphi_{1}\in
Res_{T}(\varphi_{0})$ telle qu’il existe $\bar{d},\bar{e}\in A$ qui vérifient
$p\vdash\varphi_{0}(\bar{x},\bar{d})$ et
$p\vdash\varphi_{1}(\bar{x},\bar{e})$. Comme par hypothèse la théorie $T$ est
stable alors elle n’a pas la propriété de l’ordre, par le lemme 6 il existe
$N\in\mbox{\Bbbb N}$ tel que:
* •
1: Il n’existe pas $\\{(\bar{c}_{i},\bar{a}_{i})\mid\quad i\leq N\\}$ dans $A$
tel que:
$\left\\{\begin{array}[]{rcl}A\models\varphi_{0}(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i<j\\\
A\models\varphi_{1}(\bar{c}_{i},\bar{a}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
* •
2: Il n’existe pas $\\{(\bar{c}_{i},\bar{b}_{i})\mid\quad i\leq N\\}$ dans $A$
tel que:
$\left\\{\begin{array}[]{rcl}A\models\varphi_{1}(\bar{c}_{i},\bar{b}_{j})\quad\text{si}\,\,i<j\\\
A\models\varphi_{0}(\bar{c}_{i},\bar{b}_{j})\quad\text{si}\,\,i>j.\end{array}\right.$
La première étape de cette preuve consiste en la construction de trois suites:
$\\{K(i),\bar{a}^{s}_{i+1}|K(i)\,\text{est une famille de parties de
}\\{0,\cdots,i\\},s\in K(i)\,\text{et}\,\bar{a}^{s}_{i+1}\in A\\}$
$\\{L(i),\bar{b}^{t}_{i+1}|L(i)\,\text{est une famille de parties de
}\\{0,\cdots,i\\},t\in L(i)\,\text{et}\,\bar{b}^{t}_{i+1}\in A\\}$
$(\bar{c}_{i},0\leq i<\omega)$ une suite de $A$ telle que pour tout $i<\omega$
$(\ast)\left\\{\begin{array}[]{ll}A\models\varphi_{1}(\bar{c}_{j+1},\bar{a}_{i+1}^{s})&\forall
i\leq j,s\in K(i)\\\
A\models\varphi_{0}(\bar{c}_{j+1},\bar{b}_{i+1}^{t})&\forall i\leq j,t\in
L(i)\end{array}\right.\ .$
La construction des trois suites est faite par induction comme suit:
Choisissons $\bar{c}_{0}$ arbitrairement dans $A$ et posons
$K(-1)=L(-1)=\emptyset$. Par hypothèse, supposons qu’on a
$(\bar{c}_{0},\cdots,\bar{c}_{n})$, $\\{K(i),\bar{a}^{s}_{i+1}|i\leq n-1,s\in
K(i)\\}$, $\\{L(i),\bar{b}^{t}_{i+1}|i\leq n-1,t\in L(i)\\}$. On définit
$K(n),L(n),\bar{c}_{n+1},\bar{a}^{s}_{n+1},\bar{b}^{t}_{n+1},s\in K(n),t\in
L(n)$ de la manière suivante:
$K(n)=\\{w\subset\\{0,\ldots,n\\}|\,\text{il existe}\,\bar{a}\in A\,\text{ tel
que pour tout}\,i\in w\ A\models\varphi_{0}(\bar{c}_{i},\bar{a}),\,\text{et}\
p\vdash\varphi_{1}(\bar{x},\bar{a})\\}$ Pour chaque $w\in K(n)$ on prend
$\bar{a}_{n+1}^{w}$ un élément de $A$ qui témoigne de l’existence de l’élément
$\bar{a}$ dans la definition de $w\in K(n)$ (ie
$A\models\varphi_{0}(\bar{c}_{i},\bar{a}_{n+1}^{w}),\forall i\in w$ et
$p\vdash\varphi_{1}(\bar{x},\bar{a}_{n+1}^{w})$.
De la même manière soit
$L(n)=\\{w\subset\\{0,\ldots,n\\}\mid\,\text{il existe}\,\bar{b}\in A\,\text{
tel que pour tout}\,i\in w\quad
A\models\varphi_{1}(\bar{c}_{i},\bar{b})\,\text{et}\
p\vdash\varphi_{0}(\bar{x},\bar{b})\\}$ Pour chaque $w\in L(n)$ on prend pour
$\bar{b}_{n+1}^{w}$ un élément de $A$ qui témoigne de cela.
Maintenant pour définir $\bar{c}_{n+1}$ on procède comme suit: Soit
$B_{n}=\\{\bar{a}_{i+1}^{s}\mid i\leq n,s\in
K(i)\\}\cup\\{\bar{b}_{i+1}^{t}\mid i\leq n,t\in L(i)\\}.$
Par définition
$\left\\{\begin{array}[]{rcl}\forall s\in K(i),\,\text{et}\,\,j\in
s:\,\,\,A\models\varphi_{0}(\bar{c}_{j},\bar{a}_{i+1}^{s})\quad\text{et
}\,\,p\vdash\varphi_{1}(\bar{x},\bar{a}_{i+1}^{s})\\\ \forall t\in
L(i),\,\text{et}\,\,k\in t:\quad
A\models\varphi_{1}(\bar{c}_{k},\bar{b}_{i+1}^{t})\quad\text{et}\,\,p\vdash\varphi_{0}(\bar{x},\bar{b}_{i+1}^{t}).\end{array}\right.$
Alors la famille finie $\Gamma_{n}$ définie par
$\Gamma_{n}=\\{\varphi_{1}(\bar{x},\bar{a}_{i+1}^{s}):\,i\leq n,s\in
K(i)\\}\cup\\{\varphi_{0}(\bar{x},\bar{b}_{i+1}^{t}):\,i\leq n,\,t\in
L(i)\\}\subset p$. Soit $\bar{c}_{n+1}$ une réalisation de $\Gamma_{n}$ dans
$A$. Alors on a
$(\ast\ast)\left\\{\begin{array}[]{rcl}A\models\varphi_{1}(\bar{c}_{n+1},\bar{a}_{i+1}^{s})\quad\forall
i\leq n,s\in K(i)\\\
A\models\varphi_{0}(\bar{c}_{n+1},\bar{b}_{i+1}^{t})\quad\forall i\leq n,t\in
L(i).\end{array}\right.$
La deuxième étape de la preuve du théorème consiste à montrer les deux
propriétés (a) et (b)
a- Soient $\\{i_{0}<i_{1}<\ldots<i_{n}<\omega\\}$ et $\bar{a}\in A$ tels que:
$\left\\{\begin{array}[]{cl}A\models\varphi_{1}(\bar{c}_{i_{k}},\bar{a})&\forall
0\leq k\leq n\\\ p\vdash\varphi_{0}(\bar{x},\bar{a})&\end{array}\right.$
Alors il existe $\\{\bar{d}_{i}:\,i=1,\cdots,n\\}$ des éléments de $A$ tels
que:
$(I)\left\\{\begin{array}[]{rcl}A\models\varphi_{1}(\bar{c}_{i_{k}},\bar{d}_{r})\quad\forall
k<r\\\ A\models\varphi_{0}(\bar{c}_{i_{k}},\bar{d}_{r})\quad\forall
k>r.\end{array}\right.$
En effet comme $i_{0}\in L(i_{0})$, on prend
$\bar{d}_{1}=\bar{b}_{i_{0}+1}^{i_{0}}$, alors
$A\models\varphi_{1}(\bar{c}_{i_{0}},\bar{d}_{0})$. D’après $(\ast)$ on a:
$\left\\{\begin{array}[]{ll}A\models\varphi_{1}(\bar{c}_{i_{0}},\bar{d}_{1})&\\\
A\models\varphi_{0}(\bar{c}_{i_{k}},\bar{d}_{1})&\forall
k=1,\cdots,n.\end{array}\right.$
Maintenant soit $0<k<n$ par hypothèses de (a) on a
$\left\\{\begin{array}[]{ll}A\models\varphi_{1}(\bar{c}_{i_{p}},\bar{a})&\forall
p\leq k\\\ p\vdash\varphi_{0}(\bar{x},\bar{a})&\end{array}\right.$
Par définition de $L(i_{k})$ on a $t=\\{i_{0},\cdots,i_{k}\\}\in L(i_{k})$.
Posons $\bar{d}_{k+1}=\bar{b}_{i_{k}+1}^{t}$ ainsi
$A\models\varphi_{1}(\bar{c}_{i_{p}},\bar{d}_{k+1})\quad p=0,\cdots,k$
Et par $(\ast)$
$A\models\varphi_{0}(c_{i_{p}},d_{k+1})\quad\forall p\geq k+1$
D’où la construction de la suite $\\{\bar{d}_{i}:\,i=1,\cdots,n\\}$ d’éléments
de $A$.
b- Soit $\\{i_{0}<i_{1}<\ldots<i_{n}<\omega\\}$ et $\bar{b}\in A$ tels que:
$\left\\{\begin{array}[]{ll}A\models\varphi_{0}(\bar{c}_{i_{k}},\bar{b})&\forall
0\leq k\leq n\\\
D\models\varphi_{1}(\bar{c}^{\star},\bar{b})&\end{array}\right.$
Alors il existe $\\{e_{i}:\,i=1,\cdots,n\\}$ des éléments de $A$ tels que:
$(II)\left\\{\begin{array}[]{rcl}A\models\varphi_{0}(\bar{c}_{i_{k}},\bar{e}_{r})\quad\forall
k<r\\\ A\models\varphi_{1}(\bar{c}_{i_{k}},\bar{e}_{r})\quad\forall
k>r.\end{array}\right.$
La démonstration de (b) est la même que celle de (a). Dans ce qui suit on va
conclure la preuve du théorème.
Soit $W$ un sous ensemble de $N+1$ élément de $\\{0,1,\cdots,2N+1\\}$.
Supposons qu’il existe $\bar{a}\in A$ tel que $A\models\bigwedge_{i\in
W}\varphi_{1}(\bar{c}_{i},\bar{a})$. Alors d’après $(2)+I$ on déduit que
$p\models\neg\varphi_{0}(\bar{x},\bar{a})$.
De même d’après s’il existe $\bar{a}\in A$ tel que $A\models\bigwedge_{i\in
W}\varphi_{0}(\bar{c}_{i},\bar{a})$. Par $(1)+II$ on déduit que
$p\vdash\varphi_{1}(\bar{x},\bar{a})$.
Maintenant pour toute formule positive $\psi\in
Res_{T}{\varphi_{0}}(\bar{x},\bar{y})$ on définit un couple de formules
positives $(\Phi^{0}_{\psi},\Phi^{1}_{\psi})$ telles que
$\left\\{\begin{array}[]{rcl}\Phi^{0}_{\psi}(\bar{y})=\bigvee_{w\subset\\{0,\cdots,2N_{\psi}+1\\}}\bigwedge_{i\in
w}\varphi_{0}(\bar{c}_{i},\bar{y})\\\
\Phi^{1}_{\psi}(\bar{y})=\bigvee_{w\subset\\{0,\cdots,2N_{\psi}+1\\}}\bigwedge_{i\in
w}\psi(\bar{c}_{i},\bar{y}).\end{array}\right.$
Avec $N_{\psi}$ est le nombre d’alternance du couple $(\varphi,\psi)$ donné
par le lemme 6 (rappelons aussi que les $\bar{c}_{i};i<\omega$ dépendent de
$\psi$). Posons
$\left\\{\begin{array}[]{rcl}P_{\varphi_{0}}(\bar{y})=\\{\Phi^{0}_{\psi}(\bar{y})\mid\,\psi\in
Res_{\varphi_{0}}(\bar{x},\bar{y})\\}.\\\
Q_{\varphi_{0}}(\bar{y})=\\{\Phi^{1}_{\psi}(\bar{y})\mid\,\psi\in
Res_{\varphi_{0}}(\bar{x},\bar{y})\\}\end{array}\right.$
On remarque que pour tout $\bar{a}\in A$ si $P_{\varphi_{0}}(\bar{a})$ alors
$p\models\neg\psi(\bar{x},\bar{a})$ pour toute $\psi$ dans
$Res_{\varphi_{0}}(\bar{x},\bar{y})$ donc
$p\vdash\varphi_{0}(\bar{x},\bar{a})$.
De même si $p\vdash\varphi_{0}(\bar{x},\bar{a})$, d’après $(a)+(2)$ on déduit
que que pour toute formule $\psi\in Res_{\varphi_{0}}(\bar{x},\bar{y})$ on a
$A\models\neg\Phi^{0}_{\psi}(a)$ autrement $a$ ne réalise aucune formule de
$Q_{\varphi_{0}}(y)$.
L’encadrement recherché en découle. $\square$
dans le reste de cette section on étudiera une forme faible de la stabilité
positive, on imposons d’autre conditions sur la type-définissabilité des
types. Un exemple de telle stabilité est donné par les théories h-inductives
modèle-complète.
###### Lemme 16
Soient $T$ une théorie h-inductive modéle-compléte stable, $A$ un modéle de
$T$, et $\varphi$ une formule positive. Alors pour tout type $p\in S(A)$ il
existe une famille finie $\\{\bar{c}_{i}|i<N_{\varphi}\\}$ d’uples de $A$ tels
que $dp(\varphi)$ est une combinaison boolèenne positive des formules
$\\{\varphi(\bar{c}_{i}|i<N_{\varphi})$
Preuve. Comme la théorie $T$ est modéle-compléte alors il existe une formule
positive $\psi(\bar{x},\bar{y})$ telle que $Res_{T}(\varphi)=\\{\psi\\}$. Par
le fait que $T$ est stable on a vu dans la preuve du théorème 5 après la
construction de la famille d’uples $\\{\bar{c}_{i}|i\leq 2N+1\\}$ ($N$ est le
nombre d’alternance associe au couple $(\varphi,\psi)$) de $A$. On a les deux
propositions (a) et (b) suivantes:
a- Soient $\\{i_{0}<i_{1}<\ldots<i_{n}<\omega\\}$ et $\bar{a}\in A$ tels que:
$\left\\{\begin{array}[]{cl}A\models\psi(\bar{c}_{i_{k}},\bar{a})&\forall
0\leq k\leq n\\\ p\vdash\varphi(\bar{x},\bar{a})&\end{array}\right.$
Alors il existe $\\{\bar{d}_{i}:\,i=1,\cdots,n\\}$ des éléments de $A$ tels
que:
$(I)\left\\{\begin{array}[]{rcl}A\models\psi(\bar{c}_{i_{k}},\bar{d}_{r})\quad\forall
k<r\\\ A\models\varphi(\bar{c}_{i_{k}},\bar{d}_{r})\quad\forall
k>r.\end{array}\right.$
b- Soit $\\{i_{0}<i_{1}<\ldots<i_{n}<\omega\\}$ et $\bar{b}\in A$ tels que:
$\left\\{\begin{array}[]{ll}A\models\varphi(\bar{c}_{i_{k}},\bar{b})&\forall
0\leq k\leq n\\\ p\models\psi(\bar{x},\bar{b})&\end{array}\right.$
Alors il existe $\\{e_{i}:\,i=1,\cdots,n\\}$ des éléments de $A$ tels que:
$(II)\left\\{\begin{array}[]{rcl}A\models\varphi(\bar{c}_{i_{k}},\bar{e}_{r})\quad\forall
k<r\\\ A\models\psi(\bar{c}_{i_{k}},\bar{e}_{r})\quad\forall
k>r.\end{array}\right.$
Supposons que $p\vdash\varphi(\bar{x},\bar{a})$ avec $\bar{a}\in A$ alors par
la propriété (a) on déduit que pour toute sous-famille $w$ de cardinal $N$ de
$\\{\bar{c}_{i}|i<2N+1\\}$ on a
$A\nvdash\bigwedge_{i\in w}\psi(\bar{c}_{i},\bar{a})$
car sinon le couple $(\varphi,\psi)$ ordonnera une famille de taille
supérieure à $N$. Ainsi $A\models\bigvee_{i\in w}$ pour tout $w$ de cardinal
$N$ extraite de la famille $\\{\bar{c}_{i}|i<2N+1\\}$.
Maintenant supposons que $p\nvdash\varphi(\bar{x},\bar{a})$, alors
$p\vdash\psi(\bar{x},\bar{a})$. Toujours par la même raisons on déduit à
partir de la propriété (b) que pour toute famille $w$ de cardinal $N$ de
$\\{\bar{c}_{i}|i<2N+1\\}$ on a $A\nvDash\bigwedge_{i\in
w}\varphi(\bar{c}_{i},\bar{a})$.
Ainsi on déduit que $dp(\varphi)(\bar{y})$ est donné par
$\bigwedge_{w}\bigvee_{i\in w}\varphi(\bar{c}_{i},\bar{y})$. $\square$
Une question naturelle à partir du lemme 16 sur la nature du type-
définissabilité dans le cas d’une théorie h-inductive, est ce qu’une
morliésation convenable peut nous donner des informations sur la nature du
type-définissabilité.
###### Définition 21
Une théorie h-inductive est dite stable faiblement si pour tout pec $A$ de $T$
et toutes type $p\in S(A)$ et $B$ une extension élémentaire positive de $A$.
La famille de formule positive
$\Gamma=\\{\varphi(\bar{x},\bar{b})|B\models dp(\varphi)(\bar{b})\\}$
est un type de $S(B)$.
Remarque: Sous les hypothèses de la définition 21, supposons que
$\Gamma\nvdash\varphi(\bar{x},\bar{b})$ d’une part ceci est équivalent à
$B\nVdash dp(\varphi)(\bar{b})$, d’autre part comme $B$ est un pec
$\Gamma\nvdash\varphi(\bar{x},\bar{b})$ implique l’existence d’une formule
positive $\psi$ dans $Res_{T(A)}(\varphi)$ telle que
$\Gamma\vdash\psi(\bar{x},\bar{b})$. Ainsi on déduit que si $B\nVdash
dp(\varphi)(\bar{b})$, alors il existe $\psi$ dans $Res_{T(A)}(\varphi)$ telle
que $\Gamma\vdash\psi(\bar{x},\bar{b})$
###### Corollaire 8
Toute théorie h-inductive modéle-compléte stable est stable fortemant.
## 4 De la stabilité à la simplicité positives
Dans [12] Pillay a défini les notions de division, déviation, et de simplicité
dans le cadre d’une théorie universelle au sens classique de la théorie des
modèles. La spécialité dans ce contexte est la définition de la déviation
qu’on peut traduire de la manière suivante: un type (existentiel) $p(\bar{x})$
dévie sur un ensemble $A$ si et seulement si le fermé défini par le type
$p(\bar{x})$ est recouvert par une famille de fermés $F_{\varphi_{i}}$, $i\in
I$ (avec $I$ un ensemble fini ou infini, et $F_{\varphi_{i}}$ est le fermé
défini par la formule $\varphi_{i}$) tels que pour tout $i\in I$, la formule
$\varphi_{i}$ divise sur $I$. Dans le cadre d’une théorie universelle ou dans
le cadre de la théorie des modèles positive, les fermés $F_{\varphi_{i}}$ ne
sont pas nécéssairement des ouverts. Par suite, on ne peut pas en extraire un
sous recouvrement fini. En se basant sur cette notion de déviation, il a
étudié les propriétés de la simplicité qu’il a définie par la non-déviation
sur un petit ensemble de paramètres. Nous nous référerons à cette notion de
simplicité comme “simplicité au sens de Pillay”.
Dans [3], Ben Yaacov a introduit une division positive qui est équivalente à
celle donnée dans [12] et qui témoigne la $k-$inconsistance par une formule
positive. Ensuite il a proposé une nouvelle définition de la simplicité
caractérisée par la non-division sur une petite partie de l’ensemble des
paramètres ou par la finitude du rang $D$. Cette simplicité sera dite “au sens
de Ben Yaacov”.
Il est clair que la simplicité au sens de Pillay implique celle au sens de Ben
Yaacov. En effet, l’implication est stricte comme le justifie l’exemple 4.3 de
[3]. Ce même exemple montre que la stabilité positive n’implique pas la
simplicité au sens de Pillay. En renforçant nos hypothèses sur les rangs des
types et leurs propriétés d’extension, et en reprenant les travaux de Pillay
[12], de Ben Yaacov [3], de Kim [8] et de Wagner [22], nous obtiendrons une
condition suffisante pour l’équivalence entre les deux notions de simplicité
(le corollaire 9).
### 4.1 Les simplicités positives
Nous utiliserons les appellations “simplicité au sens de Pillay” et
“simplicité au sens de Ben Yaacov” telles qu’elles étaient définies dans
l’introduction. Les notions de division et déviation seront introduites dans
les définitions 22 et 23 respectivement.
###### Définition 22
* •
Soient $\varphi(\bar{x},\bar{y})$ une L-formule positive, $k\in\mbox{\Bbbb
N}$,
$\psi(\bar{y}_{1},\cdots,\bar{y}_{k})\in
Res_{T}(\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{y}_{i}))$
et $(\bar{a}_{i}|i<\lambda)$ une suite d’uples de $M$. On dit que $\psi$ est
un témoin de k-inconsistance de $\\{\varphi(\bar{x},\bar{a}_{i})|i<\lambda\\}$
si $M\models\psi(\bar{a}_{i_{1}},\cdots,\bar{a}_{i_{k}})$, pour tout
$(\bar{a}_{i_{1}},\cdots,\bar{a}_{i_{k}})$ de taille $k$ de la suite
$(\bar{a}_{i}|i<\lambda)$. En parlant du même phénomène, on dira
alternativement que la famille $\\{\varphi(\bar{x},\bar{a}_{i})|i<\lambda\\}$
est $(\psi,k)$-inconsistante.
* •
Soient $p(\bar{x},B)$ un type partiel à paramètres dans $B$ et $A\subset B$.
On dit que $p(\bar{x},B)$ divise sur $A$ s’il existe une suite
$A$-indiscernable $(\bar{b}_{i}|i<\omega)$ telle que $tp(B\diagup
A)=tp(\bar{b}_{0}\diagup A)$ et la famille
$\\{p(\bar{x},\bar{b}_{i})|i<\omega\\}$ est inconsistante. On peut en fait
supposer que $b_{0}$ est une énumération de $B$.
* •
Soient $\varphi(\bar{x},\bar{y})$ une L-formule positive, $k\in\mbox{\Bbbb
N}$,
$\psi(\bar{y}_{1},\cdots,\bar{y}_{k})\in
Res_{T}(\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{y}_{i}))$
et $\bar{b}\in M$. On dit que $\varphi(\bar{x},\bar{b})$ $(\psi,k)$-divise sur
$A\subset M$ s’il existe une suite $A$-indiscernable $(\bar{b}_{i}|i<\omega)$
telle que $tp(\bar{b}\diagup A)=tp(\bar{b}_{0}\diagup A)$ et que la formule
$\psi$ est un témoin de k-inconsistance de la suite
$(\varphi(\bar{x},\bar{b}_{i})|i<\omega)$.
En utilisant la compacité positive, on vérifie le caractère fini de la
division.
###### Lemme 17
Un type partiel $p(\bar{x},B)$ divise sur $A$, si et seulement si il existe
$\varphi(\bar{x},\bar{b})$ une formule positive, $k\in\mbox{\Bbbb N}$ et
$\psi(\bar{y}_{1},\cdots,\bar{y}_{k})\in
Res_{T}(\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{y}_{i}))$ tels
que $p(\bar{x},B)\vdash\varphi(\bar{x},\bar{b})$ et $\varphi(\bar{x},\bar{b})$
$(\psi,k)$-divise sur $A$.
Preuve. Par définition de la division de $p(\bar{x},B)$ sur $A$, il existe
$(\bar{b}_{i}|i<\omega)$ une suite $A$-indiscernable telle que $tp(B\diagup
A)=tp(\bar{b}_{0}\diagup A)$ et que la famille
$\\{p(\bar{x},\bar{b}_{i})|i<\omega\\}$ est inconsistante. Par compacité
positive, il existe $\varphi(\bar{x},\bar{b})$ telle que
$p(\bar{x},B)\vdash\varphi(\bar{x},\bar{b})$ et la famille
$\\{\varphi(\bar{x},\bar{b}_{i})|i<\omega\\}$ est inconsistante. Ainsi, il
existe $k\in\mbox{\Bbbb N}$ telle que
$M\models\neg\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{b}_{i})$,
ce qui implique l’existence d’une formule positive
$\psi(\bar{y}_{1},\cdots,\bar{y}_{k})\in
Res_{T}(\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{y}_{i}))$ telle
que $M\models\psi(\bar{b}_{1},\cdots,\bar{b}_{k})$. Comme la suite
$(b_{i}|i<\omega)$ est $A$-indiscernable on déduit que
$\varphi(\bar{x},\bar{b})$ $(\psi,k)$-divise sur $A$. $\square$
Remarque: Le lemme 17 nous permet aussi de déduire l’équivalence entre
division et $(\psi,k)$-division pour un certain entier naturel $k$ et une
formule positive $\psi$.
###### Fait 10 ( [3])
Soit $T$ une théorie h-inductive, les deux propriétés suivantes sont
équivalentes
1. 1.
Pour tout couple de formules positives $(\varphi,\psi)$ et $k$ un entier
naturel tels que $\psi(\bar{y}_{1},\cdots,\bar{y}_{k})\in
Res_{T}(\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{y}_{i}))$, on a
$D(\bar{x}=\bar{x},\varphi,\psi,k)<\omega.$
2. 2.
Pour tout ensemble $A\subset M$, et $\bar{b}\in M$, il existe $A_{0}\subset A$
tel que $|A_{0}|\leq|T|+|\bar{b}|$ et $tp(\bar{b}\diagup A)$ ne k-divise pas
sur $A_{0}$.
###### Fait 11
Toute théorie h-inductive et stable positivement vérifier les deux propriétées
du fait 10
###### Définition 23 ([12], Definition 3.2)
Soient $p$ un type partiel à paramètres dans un ensemble $B$, et $A\subset B$.
On dit que $p$ dévie (resp k-dévie) sur $A$ s’il existe
$\\{\chi_{i}(\bar{x},\bar{m}_{i})|i\in I\\}$ une famille de formules positives
à paramètres dans $M$ telles que:
1. 1.
$p(\bar{x})\vdash\bigvee_{i\in I}\chi_{i}(\bar{x},\bar{m}_{i})$.
2. 2.
Pour tout $i\in I$, $\chi_{i}(\bar{x},\bar{m}_{i})$ divise (resp. k-divise)
sur $A$.
###### Définition 24 ([12], Definition 3.6)
Soient $A\subset M$ un pec et $p(\bar{x})$ un type maximal sur $A$ (ie
$p(\bar{x})\in S(A)$). Une suite $(\bar{a}_{i}|i<\omega)$ de réalisations de
$p(\bar{x})$ est dite une suite de Morley de $p(\bar{x})$, si elle est
$A-$indiscernable et $tp(\bar{a}_{i}\diagup{A\cup\\{\bar{a}_{j}|j<\omega\\}})$
ne divise pas sur $A$ pour tout $i<\omega$.
###### Fait 12 ([12] Lemma 3.5)
Soient $p(\bar{x})$ un type partiel à paramètres dans $C$, et $A,B$ deux
ensembles tels que $A\subset C\subset B$. Supposons que $p(\bar{x})$ ne dévie
pas sur $A$, alors il existe une réalisation $\bar{a}$ de $p$ telle que
$tp(\bar{a}\diagup B)$ ne divise pas sur $A$.
Preuve. Soit $\varphi(\bar{x})$ la famille des formules positives qui divisent
sur $A$. Comme $p(\bar{x})$ ne dévie pas sur $A$, il existe $\bar{a}$ une
réalisation de $p(\bar{x})$ telle que $M\models\neg\varphi(\bar{a})$ pour
toute formule $\varphi(\bar{x})$ de $\varphi(\bar{x})$.
Montrons que $tp(\bar{a}\diagup B)$ ne divise pas sur $A$. En effet sinon il
existe $\varphi(\bar{x})$ une formule positive à paramètres dans $B$ telle que
$tp(\bar{a}\diagup B)\vdash\varphi(\bar{x})$ et $\varphi(\bar{x})$ divise sur
$A$. C’est une contradiction puisque par définition de $\bar{a}$,
$M\models\neg\varphi(\bar{a})$. $\square$
###### Lemme 18
Soient $T$ une théorie h-inductive telle que tout type à paramètres dans un
pec ne dévie pas sur lui même, $A\subset M$ un pec de $T$ et $p(\bar{x})$ un
type maximal sur $A$ (ie $p\in S(A)$). Alors $p(\bar{x})$ admet une suite de
Morley.
Preuve. Par induction on construit une suite large $(\bar{a}_{i},i<\lambda)$
de réalisations de $p(\bar{x})$ telle que pour tout $i<\lambda$,
$tp(\bar{a}_{i}\diagup{A\cup\\{\bar{a}_{j}|j<i\\}})$ ne divise pas sur $A$.
Soit $\bar{a}_{0}$ une réalisation de $p(\bar{x})$. Supposons par induction
que la suite est construite jusqu’à $\alpha<\lambda$. Posons
$B=A\cup\\{\bar{a}_{i}|i\leq\alpha\\}$, Comme $p(\bar{x})$ ne dévie pas sur
$A$, ainsi par le fait 12 il existe $\bar{a}_{\alpha+1}$ une réalisation de
$p(\bar{x})$ telle que $tp(\bar{a}_{\alpha+1}\diagup B)$ ne divise pas sur
$A$. La construction de la suite s’ensuit.
Maintenant pour un $\lambda$ convenable et par le corollaire 8, on peut
extraire de la suite construite $(\bar{a}_{i}|i<\lambda)$ une sous suite
$(\bar{b}_{i}|i<\omega)$ $A-$indiscernable. Ainsi la suite
$(\bar{b}_{i}|i<\omega)$ est une suite de Morley de $p(\bar{x})$. $\square$
###### Lemme 19 ([12], Lemma 3.8)
Soient $T$ une théorie $h$-inductive, non bornée et complète. Soient
$I=(\bar{a}_{i}|i<\omega)$ une suite de Morley sur $A$,
$(\bar{a}^{i}|i<\omega)$ une suite $A-$indiscernable telle que
$\bar{a}^{0}=\bar{a}_{0}$. Alors il existe $J=(\bar{b}_{i}|1\leq i<\omega)$
une suite telle que pour tout $j<\omega$,
$tp(\bar{a}^{j},\bar{b}_{1},\bar{b}_{2},\cdots\diagup A)=tp(I)$ et la suite
$(\bar{a}^{j}|j<\omega)$ est $A\cup\\{J\\}-$indiscernable.
Preuve. Par induction supposons qu’il existe
$\\{\bar{b}_{1},\cdots,\bar{b}_{n}\\}$ tels que
$tp(\bar{a}^{0},\bar{b}_{1},\cdots,\bar{b}_{n}\diagup
A)=tp(\bar{a}_{0},\bar{a}_{1},\cdots,\bar{a}_{n}\diagup A)$
et que $(\bar{a}^{j}|j<\omega)$ est
$A\cup\\{\bar{b}_{1},\cdots,\bar{b}_{n}\\}-$ indiscernable. Par compacité
positive, pour tout ordinal $\lambda$ on peut prolonger la suite
$(\bar{a}^{j}|j<\omega)$ en une suite de taille $\lambda$ et qui est
$A\cup\\{\bar{b}_{1},\cdots,\bar{b}_{n}\\}-$ indiscernable. Soit $p(\bar{x})$
l’image de $tp(\bar{a}_{n+1}\diagup
A\cup\\{\bar{a}_{0},\cdots,\bar{a}_{n}\\})$ par un automorphisme de $M$ qui
fixe $A$ point par point et qui envoie $(\bar{a}_{0},\cdots,\bar{a}_{n})$ vers
$(\bar{a}^{0},\bar{b}_{1},\cdots,\bar{b}_{n})$. Comme $tp(\bar{a}_{n+1}\diagup
A\cup\\{\bar{a}_{0},\cdots,\bar{a}_{n}\\})$ ne divise pas sur $A$,
$p(\bar{x})$ ne divise pas sur $A$. Maintenant pour tout $j<\lambda$ notons
par $p^{j}(\bar{x})$ l’image de $p(\bar{x})$ par un automorphisme de $M$ qui
fixe $A\cup\\{\bar{b}_{1},\cdots,\bar{b}_{n}\\}$ point par point et qui envoie
$\bar{a}^{0}$ vers $\bar{a}^{j}$. Comme $p(\bar{x})$ ne divise pas sur $A$, on
conclut que $\bigcup\\{p^{j}(\bar{x})|j<\lambda\\}$ est consistante, et on en
fixe une réalisation $\bar{c}$. Pour un $\lambda$ large, en utilisant le lemme
8, on extrait de la suite $((\bar{a}^{j},\bar{c})|j<\lambda)$ une sous suite
$((\bar{b}^{j},\bar{c}),j<\omega)$ qui est
$A\cup\\{\bar{b}_{1},\cdots,\bar{b}_{n}\\}-$ indiscernable. Soit
$\bar{c}^{\prime}$ l’image de $\bar{c}$ par un automorphisme de $M$ qui fixe
$A\cup\\{\bar{b}_{1},\cdots,\bar{b}_{n}\\}$ point par point et qui envoie
$(\bar{b}^{j}|j<\omega)$ vers $(\bar{a}^{j}|j<\omega)$. Posons
$\bar{b}_{n+1}=\bar{c}^{\prime}$. La construction de la suite
$(\bar{b}_{i}|1\leq i<\omega)$ s’ensuit. $\square$
Un raisonnement basé sur le lemme 17 nous permet d’utiliser la méthode
originale de [8] pour aboutir à une caractérisation bien connue de la
division.
###### Proposition 2 ([8] Proposition 2.1)
Soient $T$ une théorie h-inductive complète non bornée, et qui vérifie les
deux propriétés suivantes:
1. 1.
$T$ est positivement stable;
2. 2.
tout type à paramètres dans un pec de $T$ ne dévie pas sur lui même,
$p(\bar{x})$ un type a paramètres dans $B$ et $A\subset B$. Alors les deux
propriétés suivantes sont équivalentes:
1. 1.
$p(\bar{x},B)$ divise sur $A$.
2. 2.
Pour toute suite de Morley $(\bar{b}_{i}|i<\omega)$ sur $A$, telle que
$tp(\bar{b}_{i}\diagup A)=tp(B\diagup A)$, la famille
$\\{p(\bar{x},\bar{b}_{i})|i<\omega\\}$ est inconsistante.
Preuve.
($1\Longrightarrow 2$) Soit $I=(\bar{b}_{i}|i<\omega)$ une suite de Morley de
$tp(B\diagup A)$. Supposons que $p(\bar{x},B)$ divise sur $A$, alors par la
définition 22 et le lemme 17 il existe $(\bar{a}_{i}|i<\omega)$ une suite
$A$-indiscernable de $tp(B\diagup A)$, un couple de formules positives
$(\varphi,\psi)$ et un entier naturel $k$, tels que:
* •
$\psi(\bar{y}_{1},\cdots,\bar{y}_{k})\in
Res_{T}(\exists\bar{x}\bigwedge_{i=1}^{k}\varphi(\bar{x},\bar{y}_{i}))$,
* •
$p\vdash\varphi(\bar{x},B)$,
* •
$(\varphi(\bar{x},\bar{a}_{i})|i<\omega)$ est $(\psi,k)-$inconsistente.
* •
$\bar{a}_{0}=\bar{b}_{0}$, pour cette propriété il suffit de prendre l’image
de la suite donnée par la définition 22 par un automorphisme qui fixe $A$ et
qui envoie $\bar{a}_{0}$ vers $\bar{b}_{0}$.
Ensuite par le fait 19, il existe une suite $J=(\bar{c}_{i}|1\leq i<\omega)$
telle que pour tout $j<\omega$
* •
$tp(\bar{a}_{j},J\diagup A)=tp(I\diagup A)$,
* •
la suite $(\bar{a}_{i}|j<\omega)$ est $A\cup J$-indiscernable.
Remarquons que la suite $J$ est $A$-indiscernable et que $tp(J\diagup
A)=tp(I\diagup A)$.
Maintenant nous allons montrer que la famille
$P_{J}=\\{p(\bar{x},\bar{c}_{j})|1\leq j<\omega\\}$ est inconsistante. Par
l’absurde supposons que $P_{J}$ est consistante, et posons
$\Phi_{J}=\\{\varphi(\bar{x},\bar{c}_{i})|1\leq i<\omega\\}$, alors $\Phi_{J}$
est consistante. Comme la théorie $T$ est supposée stable, par le fait 11 il
existe $n\in\mbox{\Bbbb N}$ tel que:
* •
$D(\Phi_{J},\varphi,\psi,k)=n$,
* •
$D(\Phi_{J}\cup\\{\varphi(\bar{x},\bar{a}_{j}),\varphi,\psi,k\\})=n$. En
effet, ceci découle directement du fait que pour tout $j<\omega$ on a
$tp(\bar{a}_{j},J\diagup A)=tp(I\diagup A)=tp(J\diagup A)$ et que le rang $D$
est invariant par automorphisme.
Or la famille $(\varphi(\bar{x},\bar{a}_{j})|j<\omega)$ est
$(\psi,k)-$inconsistante, ce qui implique par la définition du rang $D$
positive [3] que $D(\varphi_{J},\varphi,\psi,k)\geq n+1$, contradiction. Ainsi
$P_{J}$ est inconsistante. Comme $tp(I\diagup A)=tp(J\diagup A)$, on déduit
que la famille $P_{I}=\\{p(\bar{x},\bar{b}_{i})|i<\omega\\}$ est
inconsistante.
($2\Longrightarrow 1$) est immédiate. $\square$
###### Fait 13 ([12] Corollaire 3.10)
Soient $T$ une théorie h-inductive complète non bornée, et qui vérifie les
deux propriété suivantes:
1. 1.
$T$ est positivement stable;
2. 2.
tout type à paramètres dans un pec de $T$ ne dévie pas sur lui même.
Soient $p(\bar{x},B)$ un type sur $B$ et $A\subset B$. Alors les deux
propriétés suivantes sont équivalentes:
1. 1.
$p(\bar{x},B)$ divise sur $A$.
2. 2.
$p(\bar{x},B)$ dévie sur $A$.
Preuve. ($1\Longrightarrow 2$) est immédiat.
($2\Longrightarrow 1$) Supposons que $p(\bar{x},B)$ dévie sur $A$. Par
définition, il existe $\\{\varphi_{i}(\bar{x},C)|i\in I\\}$ une famille de
formules positives à paramètres dans $C$ telles que
$p(\bar{x})\vdash\bigvee_{i\in I}\varphi_{i}(\bar{x},C)$ et pour tout $i\in
I$, $\varphi_{i}(\bar{x},C)$ divise sur $A$. Ceci implique que le type partiel
$p(\bar{x},B)=p(\bar{x},B\cup C)$ dévie sur $A$. Posons $B\cup C=D$. Soit
$(\bar{c}_{j}|j<\omega)$ une suite de Morley de $tp(D\diagup A)$. En utilisant
la compacité positive, on prolonge cette suite à $(\bar{d}_{i}|i<\lambda)$,
$A-$indiscernable avec $\lambda>|I|$.
Montrons que la famille $\cup\\{p(\bar{x},\bar{d}_{j})|j<\lambda\\}$ n’est pas
consistante. Supposons par l’absurde que $\bar{e}$ est une réalisation de
cette famille dans $M$. Pour tout $j<\lambda$, il existe $i\in I$ tel que
$\bar{e}$ réalise $\varphi_{i}(\bar{x},\bar{d}_{i})$. Pour un choix de
$\lambda$ assez large justifié par le corollaire 8, il existe $i\in I$ et $E$
une partie de $\lambda$ tels que $\bar{e}$ réalise
$\varphi_{i}(\bar{x},\bar{d}_{j})$ pour tout $j\in E$. Comme
$(\bar{d}_{j}|j<\lambda)$ est $A$-indiscernable, soit $f$ un automorphisme de
$M$ qui fixe $A$ point par point et qui envoie $(\bar{d}_{j}|j<\omega)$ vers
$(\bar{c}_{j}|j<\omega)$. On déduit que $f(\bar{e})$ réalise la famille
$\\{\varphi_{i}(\bar{x},\bar{c}_{j});\ j<\omega\\}$. Comme
$\varphi_{i}(\bar{x},\bar{c})$ divise sur $A$, par la proposition 2 on aboutit
à une contradiction. $\square$
###### Corollaire 9
Soient $T$ une théorie h-inductive complète non bornée, et qui vérifie les
deux propriétés suivantes:
1. 1.
$T$ est positivement stable;
2. 2.
tout type à paramètres dans un pec de $T$ ne dévie pas sur lui même,
Alors la simplicité au sens Ben Yaacov est équivalente à la simplicité au sens
de Pillay.
Preuve. En général, la simplicité au sens de Pillay implique la simplicité au
sens de Ben Yaacov.
Maintenant soit $T$ une théorie simple au sens de Ben Yaacov et qui vérifie
les conditions (1) et (2). Par définition de la simplicité de Ben Yaacov et
par le fait 13, on a équivalence entre division et déviation. Ce qui implique
la simplicité au sens de Pillay. $\square$
###### Lemme 20
Toute théorie stable faiblement est simple au sens de Pillay.
Preuve. Il suffit de montrer que sous l’hypothèse de la stabilité faible tout
type ne dévie pas sur lui même. Supposons qu’il exite $A$ un pec de $T$ et
$p\in S(A)$ qui dévie. Alors il existe une famille de formule positive à
paramètres dans $M$ indexée par un ordinal $\alpha$,
$\\{\varphi_{i}(\bar{x},\bar{m}_{i})|i\alpha\\}$ telle que
$p\vdash\bigvee_{i<\alpha}\varphi(\bar{x},\bar{m}_{i})$, et pour tout
$i<\alpha$ la formule $\varphi_{i}(\bar{x},\bar{m}_{i})$ divise sur $A$.
Pour tout $i<\alpha$ soit $\\{\bar{m}_{ij}|j<\omega\\}$ une suite
$A$-indiscernable telle que $\bar{m}_{i0}=\bar{m}_{i}$ et la famille
$\\{\varphi(\bar{x},\bar{m}_{ij})|j<\omega\\}$ est inconsistante. Soit $B$ un
pec de $T$ qui contient $A\cup\\{\bar{m}_{ij}|i,j<\omega\\}$, et soit $q$ le
type de $S(B)$ associe à la définition $p$ sur $A$. Alors il existe $i<\alpha$
tel que $q\vdash\varphi_{i}(\bar{x},\bar{m}_{i})$, autrement $B\models
dp(\varphi_{i})(\bar{m}_{i})$, comme $dp(\varphi_{i})$ est à paramètres dans
$A$ alors pour tout $j<\omega$ $B\models dp(\varphi_{i})(\bar{m}_{ij})$.
Absurde, ainsi $p$ ne dévie pas sur lui même. $\square$
## References
* [1] Mohammed Belkasmi. Positive model theory and amalgamation. prépublication , 2011
* [2] Itaï Ben Yaacov. Positive model theory and compact abstract theories. Journal of Mathematical Logic, vol. 3, No. 1 (2003), 85–118
* [3] Itaï Ben Yaacov. Simplicity in compact abstract theories. Journal of Mathematical Logic, vol. 3, No 2 (2003), 163–191
* [4] Itaï Ben Yaacov, Bruno Poizat. Fondements de la logique positive. Journal of Symbolic Logic, 72, 4, 1141–1162, 2007.
* [5] Steven Buechler. Essential stability theory . Springer, 1996.
* [6] Wilfrid Hodges. Model theory. CUP, 1993.
* [7] Ehud Hrushovski Simplicity and the Lascar group. prépublication. 1997
* [8] Byunghan Kim. Forking in simple unstable theories . Journal of the London Mathematical Society.,1989, no. 2, 257–267.
* [9] Almaz Kungozhin Existentially closed and maximal models in positive logic. prepublication. 2011
* [10] Anand Pillay. An Introduction to Stability Theory. Oxford Science Publications, 1983.
* [11] Anand Pillay. Geometric Stability Theory. Oxford Science Publications, 1996.
* [12] Anand Pillay Forking in the category of existentially closed structures Quaderni di Matematica,6, Université de Naples.
* [13] Bruno Poizat. Cours de théorie des modèles. Nur Al-Mantiq Wal-Ma’rifah, 1985.
* [14] Bruno Poizat. Quelques effets pervers de la positivité. Preprint, 2008.
* [15] Bruno Poizat. Univers Positifs. Journal of Symbolic Logic, 71, 3, 969–976, 2006.
* [16] Saharon Shelah. The lazy model-theoretician’s guide to stability. Logique et Analyse, vol. 71-72, 241-308, 1975.
* [17] Saharon Shelah. Categoricity of abstract classes with amalgamation. Annals of Pure and Applied Logic, 98(1-3), pages 141–187, 1999.
* [18] Saharon Shelah Classification Theory and the number of non-isomorphic models North-Holland. Studies in logic and foundations of mathematic, volume 92, 1990.
* [19] Saharon Shelah Classification Theory for Elementary Abstract Classes Mathematical Logic and Foundations., 2009.
* [20] Saharon Shelah Simple unstable theories Annals of mathematical logic. 19 (1980) 177-203.
* [21] Thomas Jech, Karel Hrbacek. Introduction to set theory. CRC Press. Pure and applied mathematics, 1999.
* [22] Frank Olaf Wagner. Simple theories. Mathematics and its applications, v. 503 , 2000.
|
arxiv-papers
| 2012-04-19T22:32:11 |
2024-09-04T02:49:29.933968
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Mohammed Belkasmi",
"submitter": "Belkasmi Mohammed medkas",
"url": "https://arxiv.org/abs/1204.4488"
}
|
1204.4519
|
# Topological susceptibility and axial symmetry at finite temperature
JLQCD Collaboration: a, Sinya Aokib, Shoji Hashimotoa,c, Takashi Kanekoa,c,
Hideo Matsufurua, Jun-ichi Noakia, Eigo Shintanid
aTheory Center, IPNS, High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801, Japan
bGraduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba
305-8571, Japan
cSchool of High Energy Accelerator Science, The Graduate University for
Advanced Studies (Sokendai), Tsukuba 305-0801, Japan
dRIKEN-BNL Research Center, Upton, NY 11973-5000, USA E-mail:
cossu@post.kek.jp
###### Abstract:
We consider the simulation of finite temperature QCD with two flavors of
dynamical overlap fermions in order to study the suppression of the axial
$U(1)$ symmetry breaking at the chiral phase transition point. As a
preliminary study, pure gauge simulations are performed to investigate how
fixing the topology affects physical quantities like the topological
susceptibility, $\chi_{t}$, at finite temperature, showing that it is possible
to reconstruct known results from the fixed topology sector. First results on
the degeneracy of meson correlators in the high temperature QGP sector are
shown.
## 1 Introduction
It is an interesting and long standing problem to understand whether the
flavour-singlet axial $U(1)$ symmetry is restored or not above the finite
temperature transition in the chirally symmetric phase of QCD. At low
temperature it is well known that the chiral symmetry is spontaneously broken
while the axial $U(1)$ symmetry is broken at quantum level by the presence of
configurations with non-trivial topological structure. The semi-classical
configurations that contribute to the axial charge are called instantons and
it can be shown that zero modes of the Dirac operator are related to the
presence of these and other configurations that have non-trivial topological
charge. Hence, topology plays a major role in this subject. One clear example
is the Witten-Veneziano relation [1, 2] that connects the topological
susceptibility of pure gauge theory to the heavy mass of the
$\eta^{\prime}(958)$ particle, the candidate to be the would-be Nambu-
Goldstone boson of the axial $U(1)$ symmetry.
Our purpose is to study the behavior of meson correlators at high temperature,
looking for signals of (approximate) degeneracy in all the singlet and triplet
channels ($\sigma,\delta,\pi,\eta^{\prime}$) toward the chiral limit. This
would be an evidence of effective restoration of both chiral and axial
symmetries, several intermediate steps are to be considered to control the
systematic errors. The best available discretization of the Dirac operator is
the Overlap fermions [3]. Dynamical simulations with Overlap fermions are
possible with current machines and were performed by JLQCD collaboration in
the past years [4]. The price payed in order to be able to have a very precise
fulfillment of the Ginsparg-Wilson relation [5] on the lattice is to fix the
topology. Changing topology in a HMC trajectory requires generating rough
configurations associated with near-zero modes of the hermitian Wilson
operator ($H_{W}$). A discontinuity arises in the HMC when one of these modes
crosses zero. Treating this discontinuity is a costly task, although it could
be achieved using the reflection/refraction methods [6]. One solution to this
problem is to suppress near zero modes of $H_{W}$ and to avoid the zero
crossing, with the disadvantage of preventing topology change during the
simulation. The simulation is no more ergodic, and some technique has to be
developed to obtain the physical results at $\theta=0$. This program was
indeed developed by JLQCD collaboration (see [7] for details) and it works
nicely at zero temperature. In order to accomplish our main target, we need to
be sure that the same methods can be applied even in the finite temperature
case. The first step towards the understanding and controlling the effect of
fixing topology at finite temperature is to reproduce the known results of
pure gauge theory. In particular we focus on the most relevant quantity, i.e.
the topological susceptibility $\chi_{t}$, and see if it can be reproduced
with the same methods adopted at zero temperature.
The rest of the paper consists of two main sections. The first is devoted to
the discussion of our study of topological susceptibility in pure gauge theory
at finite temperature. The second is the measurements of meson correlators, in
flavor singlet and triplet channels, and the spectral density of the Overlap
Dirac operator, where we show some evidences that the axial $U(1)$ symmetry
breaking is suppressed after the chiral phase transition. In the last section
we will draw some conclusions.
## 2 Pure gauge simulations
In a previous paper [8], the JLQCD-TWQCD collaboration calculated the
topological susceptibility at zero temperature in the case of two flavors of
overlap fermions. They demonstrated that it is possible to obtain physical
quantities even with simulations at fixed topological sector. They checked the
prediction of $\chi$PT, i.e. $\chi_{t}=(m_{q}\Sigma)/N_{f}+O(m_{q}^{2})$,
finding the expected linear behavior of $\chi_{t}$ versus the sea quark mass.
The methods described in [7] were applied in that case, i.e. the topological
susceptibility is measured by the long distance value of disconnected
correlators of pseudoscalar operators (they are non-zero because in a fixed
topology environment clustering property of field theory is violated). We will
investigate numerically whether we can apply the same methodology at finite
temperature too, with a preliminary test in pure gauge simulations, in order
to be trained in dealing with this systematic error in the most interesting
full QCD case.
The correlators are calculated assuming that the major contribution comes
solely from the lowest 50 eigenmodes. We tested for one value of $\beta$ that
this is really the case by changing the number of eigenmodes to 30 and 40. The
result is that the long distance behavior saturates at 40-50 eigenvalues, the
short distance correlator has a worse approximation, as expected.
In the pure gauge theory,there is an anomalous contribution to the pseudo-
scalar meson correlator in the flavor singlet channel, which is called hairpin
diagram [9, 10]:
$H(p)=f_{P}\frac{1}{p^{2}+m_{\pi}^{2}}m_{0}^{2}\frac{1}{p^{2}+m_{\pi}^{2}}f_{P}.$
(1)
This is a zero temperature result, and we assume that it is valid at finite
temperature with all parameters depending on temperature. We actually find
that this fits our data very well. Also, the results are independent of the
valence quark mass, as it should be in the case of pure gauge theory. For
statistical purposes, results are obtained with a joint fit of connected and
disconnected part. The joint fit assumes an identical decay rate for the two
correlators, long distance of connected part going to zero, to constrain the
rest of parameters. Fits also discard the first and last three points of the
spatial separation range.
Simulations were performed using an Iwasaki lattice gauge action, on a lattice
of dimensions $24^{3}\times 6$ and $\beta\in[2.35,\dots,2.55]$ in order to be
in the range of temperatures $[249,\dots,347]$ MeV. The phase transition was
estimated by the Polyakov loop behavior to be at $T=288$ MeV. Except for one
run at $Q=1$ all configurations were generated at $Q=0$.
The outcome for the topological susceptibility measurements is shown in figure
1. Our results are shown as black dots. They follow with a good accuracy the
reference results by the Regensburg group [11], obtained with direct zero
eigenmode counting. In order to cross-check the methodology we also
accumulated configurations without fixing topology, by using the same HMC
algorithm. The parameters for this run were chosen so that it corresponds to
the same temperature as the $\beta=2.50$ run, $T/T_{C}\simeq 1.1$. We then
extracted two subsets of configurations with $Q=0$ and $Q=1$ and applied the
correlator methodology to extract topological susceptibility. The final
results agree among two sectors and moreover are in accordance with an
estimate using the Atiyah-Singer’s index theorem for all configurations
(points indicated by cross $Q=$all, blue dot $Q=0$ and star $Q=1$ in the
figure). This result indicates that the term in the action that forces fixed
topology is not preventing appearance of local topological fluctuations, like
pairs of caloron-anticalorons (so called topological molecules) which are
associated to very low eigenmodes. These topological fluctuations are
distributed in the same way as the ones generated without fixing topology.
Figure 1: (Topological susceptibility on the $24^{3}\times 6$ lattice.
Statistical errors only. Reference data points are diamonds and squares. The
black dots are our results from measurements at Q=0. Figure 2: Lowest
eigenvalues at several temperatures. On abscissas the HMC trajectory number,
one point every 100 trajectories except for RUN-II where we saved one
configuration every 50.
So far we demonstrated that our method works very well even in the finite
temperature case. However we discovered and reported at the Lattice conference
that topology fixing poses potential problems in the high temperature regime,
at finer lattice spacing. The lowest eigenmodes (main source of the
topological susceptibility signal) take longer time to thermalize than usual
thermodynamical quantities. This is a known problem but in our case the effect
appeared more strongly. The points at $\beta=2.50$ and $\beta=2.55$ were the
most problematic, exhibiting some dependence on the initial configuration, see
Figure 2. In particular the run tagged RUN-II, obtained by starting from a
thermalized configuration at $\beta=2.50$ and heated up to $\beta=2.55$, was
giving different results from a similar run starting from unit configuration.
This was a severe problem, mining the reliability of higher temperature runs.
We thus further investigated the issue by accumulating more configurations,
especially for the highest temperature point. Final result is shown in Figure
2. We recall that all runs started from a unit configuration, high
temperature, and then thermalized (except RUN-II). Thermalization is monitored
by plaquette and Polyakov Loop as standard observables. By looking at the
lowest eigenvalue history, the first three panels do not show any particular
problem in thermalization, except the expected increasing autocorrelation time
going to finer lattices. With newest data no anomalous behavior is detected
also for $\beta=2.55$. At Lattice 2011 the history was much shorter, less than
10k trajectories, and no jumps were present. Transition between two different
values of the lowest eigenmode is sharp but the tunneling rate is non-zero,
and no dependence on the starting configuration is observed. The issue was
just statistical and is not expected to affect full QCD results.
We thus conclude that physical quantities can be obtained from fixed topology
simulations at finite temperature.
## 3 Dynamical overlap simulations
In this section we will discuss some of the results from simulations of two
flavors of dynamical overlap fermions in the high temperature region.
We accumulated O(200-300) thermalized configurations per $\beta$ with several
masses and temperatures, see table 1. Accumulating the corresponding zero
temperature configurations would have been a really demanding task so we do
not have currently a measurement of the lattice spacing and pion mass for
every $\beta$. At $\beta=2.30$ and bare quark mass $am=0.015$ was 286 MeV, 360
MeV at $am=0.025$ [12]. By looking at the Dirac operator spectral density we
observe that a gap is already opened at a temperature of around 192 MeV
($\beta=2.25$) near the chiral limit, $am=0.01$, confirmed by higher
temperature simulations. This could be a signal of strong suppression of axial
symmetry breaking, because the splitting between flavor-singlet and non-
singlet correlators dominantly comes from the near-zero modes, as we
demonstrate below. We do not observe this behavior with current data at the
temperature of 177 MeV ($\beta=2.20$, $am>0.025$).
We report also our first observation of the meson correlators in this range of
temperatures (figure 4). We plot the singlet and non-singlet pseudo-scalars
($\eta^{\prime}$ and $\pi$) and singlet and non-singlet scalars ($\sigma$ and
$\delta$).
Data at $\beta=2.20$ do not show any degeneracy among the correlators of the
lightest mesons. This could be an effect of the high mass ($am=0.025$) but it
is also possible that this temperature is still in the chirally broken region
(see the spectral density plot). The number of configurations is not
sufficient to precisely determine the transition temperature, so we cannot
settle this question. The meson correlators at the highest temperature,
starting from $\beta=2.25$, show a clear tendency toward degeneracy in the
chiral limit, a signal that $U(1)_{A}$ axial symmetry is effectively restored.
We need to accumulate more data in order to be conclusive, but certainly we
observe robust signals of axial symmetry restoration. Further study around the
phase transition is on the way.
$\beta$ | $a$(fm) | $T$(MeV) | Masses ($am$)
---|---|---|---
2.18 | 0.144 | 171 | 0.05
2.20 | 0.139 | 177 | 0.05, 0.025
2.25 | 0.128 | 192 | 0.01
2.30 | 0.118 | 208 | 0.05, 0.025, 0.01
Table 1: Parameters of some of the collected data for two flavors of dynamical
overlap fermions, lattice dimensions $16^{3}\times 8$. The reported lattice
spacings are the extrapolations to the chiral limit. Figure 3: Spectral
density of the overlap Dirac operator for $N_{f}=2$. The several $\beta$s were
isolated to emphasize the mass dependence of the density. Zero counting of
eigenvalues is intended on the left when line stops.
Figure 4: Meson correlators at several temperatures and masses.
## 4 Conclusions and perspectives
We have shown that it is possible to perform finite temperature lattice
simulations with overlap fermions at fixed topology. It is tested in the case
of pure gauge theory where measurements of topological susceptibility can be
compared with known results.
Having under control the systematics of topology fixing we performed
simulations of two flavors of overlap fermions at several temperatures in the
range $[170\sim 208]$ MeV. We measured the correlators of meson operators in
all the (pseudo)scalar channels looking for their degeneracy in the chiral
limit, signal of effective restoration of both chiral and axial symmetry. We
found evidence of this restoration, corroborated also by the spectral density
analysis that exhibits a gap in the chiral limit at temperatures above $>192$
MeV, in the current data set.
We can currently fairly say that we have clear evidence of $U(1)_{A}$
effective restoration in a region just above the chiral phase transition in
two flavors QCD. The next step is narrowing the region of uncertainty about
the temperature when the gap starts opening. In a forthcoming paper in
preparation a complete analysis and the newly collected data will be
presented.
This work is supported in part by the HPCI Strategic Program of Ministry of
Education and in part by the Grant-in-Aid for Scientific Research on
Innovative Areas (No. 2004: 20105001, 20105003, 20105005, 21674002, 21684013).
## References
* [1] Edward Witten. Current Algebra Theorems for the U(1) Goldstone Boson. Nucl.Phys., B156:269, 1979.
* [2] G. Veneziano. U(1) Without Instantons. Nucl.Phys., B159:213–224, 1979.
* [3] Herbert Neuberger. Exactly massless quarks on the lattice. Phys.Lett., B417:141–144, 1998.
* [4] T. Kaneko et al. JLQCD’s dynamical overlap project. PoS, LAT2006:054, 2006.
* [5] Paul H. Ginsparg and Kenneth G. Wilson. A Remnant of Chiral Symmetry on the Lattice. Phys.Rev., D25:2649, 1982.
* [6] G.I. Egri, Z. Fodor, S.D. Katz, and K.K. Szabo. Topology with dynamical overlap fermions. JHEP, 0601:049, 2006.
* [7] Sinya Aoki, Hidenori Fukaya, Shoji Hashimoto, and Tetsuya Onogi. Finite volume QCD at fixed topological charge. Phys.Rev., D76:054508, 2007.
* [8] S. Aoki et al. Topological susceptibility in two-flavor lattice QCD with exact chiral symmetry. Phys.Lett., B665:294–297, 2008.
* [9] Thomas A. DeGrand and Urs M. Heller. Witten-Veneziano relation, quenched QCD, and overlap fermions. Phys.Rev., D65:114501, 2002.
* [10] William A. Bardeen, A. Duncan, E. Eichten, and H. Thacker. Anomalous chiral behavior in quenched lattice QCD. Phys.Rev., D62:114505, 2000.
* [11] Christof Gattringer, Roland Hoffmann, and Stefan Schaefer. The Topological susceptibility of SU(3) gauge theory near T(c). Phys.Lett., B535:358–362, 2002.
* [12] S. Aoki et al. Two-flavor QCD simulation with exact chiral symmetry. Phys.Rev., D78:014508, 2008.
|
arxiv-papers
| 2012-04-20T02:32:30 |
2024-09-04T02:49:29.950935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guido Cossu, Sinya Aoki, Shoji Hashimoto, Takashi Kaneko, Hideo\n Matsufuru, Jun-ichi Noaki, Eigo Shintani",
"submitter": "Guido Cossu",
"url": "https://arxiv.org/abs/1204.4519"
}
|
1204.4597
|
# Noise, Bifurcations, and Modeling of Interacting Particle Systems
Luis Mier-y-Teran-Romero, Eric Forgoston and Ira B. Schwartz This work is
supported by the Office of Naval ResearchLuis Mier-y-Teran-Romero is a joint
NIH postdoctoral fellow with the Johns Hopkins University and the US Naval
Research Laboratory, Nonlinear Systems Dynamics Section, Code 6792,
Washington, DC 20375, USA, luis@nlschaos.nrl.navy.milEric Forgoston is with
the Department of Mathematical Sciences, Montclair State University,
Montclair, NJ 07043, USA, eric.forgoston@montclair.eduIra B. Schwartz is with
the Nonlinear Systems Dynamics Section, Code 6792, US Naval Research
Laboratory, Washington, DC 20375, USA, ira.schwartz@nrl.navy.mil
###### Abstract
We consider the stochastic patterns of a system of communicating, or coupled,
self-propelled particles in the presence of noise and communication time
delay. For sufficiently large environmental noise, there exists a transition
between a translating state and a rotating state with stationary center of
mass. Time delayed communication creates a bifurcation pattern dependent on
the coupling amplitude between particles. Using a mean field model in the
large number limit, we show how the complete bifurcation unfolds in the
presence of communication delay and coupling amplitude. Relative to the center
of mass, the patterns can then be described as transitions between
translation, rotation about a stationary point, or a rotating swarm, where the
center of mass undergoes a Hopf bifurcation from steady state to a limit
cycle. Examples of some of the stochastic patterns will be given for large
numbers of particles.
## I INTRODUCTION
The collective motion of interacting multi-particle systems has been the
subject of many recent experimental and modeling studies. It is especially
astounding that numerous coherent states of great complexity can arise
spontaneously in spite of the absence of a particle acting as a leader. The
study of these swarming systems has proven useful in understanding the spatio-
temporal patterns formed by bacterial colonies, fish, birds, locusts, ants,
pedestrians, etc. [1, 2, 3, 4, 5, 6]. Moreover, these studies have provided
valuable information that may be exploited in the design of systems of
autonomous, inter-communicating robotic systems [7, 8, 9].
Investigators have used various mathematical approaches to study swarm
systems. Some studies have preserved the individual character of each agent in
the system, using ordinary or delay differential equations (ODEs/DDEs) to
describe their trajectories [10, 11, 12, 8]. Other researchers have proposed
continuum models written in terms of averaged velocity and particle density
fields that satisfy partial differential equations (PDEs) [2, 3, 5, 6]. In
addition, authors have also studied the effects of noise in the swarms and
shown the existence of noise-induced transitions [13, 14].
More recently, authors have begun to study the effects of communication time-
delays between particles. Time-delay models are common in many areas of
mathematical biology including population dynamics, neural networks, blood
cell maturation, virus dynamics and genetic networks [15, 16, 17, 18, 19, 20,
21, 22, 23]. In the context of swarming particles, it has been shown that the
introduction of a communication time-delay may induce transitions between
different coherent states [14]. The type of transition is dependent on the
coupling strength between particles and the noise intensity.
Here we make a more detailed study of the bifurcation structure of the mean
field approximation to the delay-coupled model proposed studied in [14] and
investigate how the bifurcations in the system are modified in the presence of
noise.
Figure 1: Snapshots of a swarm taken at (a) $t=50$, (b) $t=60$, (c) $t=62$,
(d) $t=64$, (e) $t=66$, (f) $t=68$, (g) $t=70$, (h) $t=72$, (i) $t=74$, and
(j) $t=76$, with $a=4$, $N=300$, and $D=0.08$. The swarm was in a rotational
state when the time delay of $\tau=1$ was switched on at $t=40$. For a movie,
see the relevant mpeg video. Figure reproduced with permission from [14].
## II The Swarm Model
We consider a two-dimensional swarm with $N$ self-propelling particles that
are mutually attracted in a symmetric fashion. Additionally, we consider the
case in which particles communicate with each other with a time delay. The
swarm is governed by the following system of ODEs:
$\displaystyle\dot{\mathbf{r}}_{i}=$ $\displaystyle\mathbf{v}_{i},$ (1a)
$\displaystyle\dot{\mathbf{v}}_{i}=$
$\displaystyle\left(1-|\mathbf{v}_{i}|^{2}\right)\mathbf{v}_{i}-\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau))+\boldsymbol{\eta}_{i}(t),$ (1b)
for $i=1,2\ldots,N$. Here $\mathbf{r}_{i}$ and $\mathbf{v}_{i}$ represent the
position and velocity of the $i$-th particle, respectively; the strength of
the attraction is measured by the coupling constant $a$ and the time delay is
uniform and given by $\tau$. The self- propulsion and frictional drag on each
particle is given by the term
$\left(1-|\mathbf{v}_{i}|^{2}\right)\mathbf{v}_{i}$. In the absence of
coupling, particles tend to move on a straight line with unit speed
$|\mathbf{v}_{i}|=1$ as time goes to infinity. The term
$\boldsymbol{\eta}_{i}(t)=(\eta_{i}^{(1)},\eta_{i}^{(2)})$ is a two-
dimensional vector of stochastic white noise with intensity equal to $D$ and
correlation functions $\langle\eta_{i}^{(\ell)}(t)\rangle=0$ and
$\langle\eta_{i}^{(\ell)}(t)\eta_{j}^{(k)}(t^{\prime})\rangle=2D\delta(t-t^{\prime})\delta_{ij}\delta_{\ell
k}$ for $i,j=1,2,\ldots N$ and $\ell,k=1,2$.
The coupling between particles arises from a time-delayed, spring-like
potential. Hence, our equations of motion may be considered to be the first
term in a Taylor expansion of other more general time-delayed potential
functions about an equilibrium point. The model described by Eqs. (1a)-(1b)
with $\tau=0$ (i.e. no time delay) possesses a noise-induced transition
whereby a large enough noise intensity causes a translating swarm of
individuals to transition to a rotating swarm with a stationary center of mass
[24, 14]. Regardless of which state the swarm is in (translating or rotating),
the addition of a communication time delay leads to another type of
transition. This transition occurs if the coupling parameter $a$, is large
enough. As an example, we consider a swarm that has already undergone a noise-
induced transition to a rotational state before switching on the communication
time delay.
Figures 1(a)-1(j) show snapshots of a swarm at $t=50$, $t=60$, $t=62$, $t=64$,
$t=66$, $t=68$, $t=70$, $t=72$, $t=74$, and $t=76$ respectively. For these
simulations, $N=300$, $\tau=1$, $D=0.08$, the noise was switched on at $t=10$
(causing the swarm to transition to a stationary, rotating state), and once in
this rotating state, the time delay was switched on at $t=40$. One can see
that with the evolution of time, the individual particles become aligned with
one another and the swarm becomes more compact. Additionally, the swarm is no
longer stationary, but has begun to oscillate [Figs. 1(g)-1(j)].
This compact, oscillating aligned swarm state looks similar to a single
“clump” that is described in [25]. However, where each “clump” of [25]
contains only some of the total number of swarming particles, our swarm
contains every particle. Additionally, while a deterministic model along with
global coupling is used to attain the “clumps” of [25], our oscillating
aligned swarm is attained with the use of noise and a time delay.
## III Mean Field Approximation
As we have shown, once the stochastic swarm is in the stationary, rotating
state, the addition of a time delay induces an instability. We investigate the
stability of the swarm by deriving the mean field equations and performing a
bifurcation analysis.
We carry out a mean field approximation of the swarming system by switching to
particle coordinates relative to the center of mass and disregarding the noise
terms. The center of mass of the swarming system is given by
$\displaystyle\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t).$ (2)
We decompose the position of each particle into
$\displaystyle\mathbf{r}_{i}=\mathbf{R}+\delta\mathbf{r}_{i},\qquad
i=1,2\ldots,N,$ (3)
where
$\displaystyle\sum_{i=1}^{N}\delta\mathbf{r}_{i}(t)=0.$ (4)
Inserting Eq. (3) into the second order system equivalent to Eqs. (1a)-(1b)
with $D=0$ and simplifying one obtains
$\displaystyle\ddot{\mathbf{R}}+\delta\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)(\dot{\mathbf{R}}+\delta\dot{\mathbf{r}_{i}})$
$\displaystyle-\frac{a(N-1)}{N}\bigg{(}\mathbf{R}(t)-\mathbf{R}(t-\tau)+\delta\mathbf{r}_{i}(t)\bigg{)}$
$\displaystyle-\frac{a}{N}\delta\mathbf{r}_{i}(t-\tau),$ (5)
where we used Eq. (4) in the form
$\delta\mathbf{r}_{i}(t-\tau)=-\sum_{j=1,\ i\neq
j}^{N}\delta\mathbf{r}_{j}(t-\tau).$ (6)
Summing Eq. (III) over $i$ and using Eq. (4), one arrives at
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}-\frac{1}{N}\sum_{i=1}^{N}|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{R}}$
$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\left(2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}+|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\delta\dot{\mathbf{r}_{i}}$
$\displaystyle-a\frac{N-1}{N}\left(\mathbf{R}(t)-\mathbf{R}(t-\tau)\right).$
(7)
Inserting Eq. (III) into Eq. (III) the following equation for
$\delta\ddot{\mathbf{r}}_{i}$ is obtained:
$\displaystyle\delta\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(\frac{1}{N}\sum_{j=1}^{N}|\delta\dot{\mathbf{r}}_{j}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{R}}$
$\displaystyle+\left(1-|\dot{\mathbf{R}}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\delta\dot{\mathbf{r}}_{i}$
$\displaystyle+\frac{1}{N}\sum_{j=1}^{N}\left(2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{j}+|\delta\dot{\mathbf{r}}_{j}|^{2}\right)\
\delta\dot{\mathbf{r}}_{j}-a\frac{N-1}{N}\delta\mathbf{r}_{i}$
$\displaystyle-\frac{a}{N}\delta\mathbf{r}_{i}(t-\tau),$ (8)
for $i=1,2\ldots,N$.
Equations (III) and (III) are fully equivalent to Eqs. (1a)-(1b) when $D=0$,
and simply consist of rewriting the original system using the relationship
between the particle coordinates $\mathbf{r}_{i}$, the center of mass
$\mathbf{R}$, and the coordinates relative to the center of mass
$\delta\mathbf{r}_{i}$. This mapping has transformed the original $2N$
differential equations into $2N+2$ differential equations. There is, however,
no inconsistency since in the transformed set of equations only $2N$ of them
are independent, because of the relation seen in Eq. (4).
We then obtain a mean field approximation by neglecting the fluctuation of the
swarm particles, $\delta\mathbf{r}_{i}$’s, from the center of mass:
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{R}(t)-\mathbf{R}(t-\tau)\right),$
(9)
where we made the approximation $a\frac{N-1}{N}\approx a$ since we consider
the thermodynamic limit.
## IV Bifurcations in the Mean Field Equation
The behavior of the system in the mean field approximation in different
regions of parameter space may be better understood by using bifurcation
analysis. Equation (9) may be written in component form using
$\mathbf{R}=(X,Y)$ and $\dot{\mathbf{R}}=(U,V)$ as
$\displaystyle\dot{X}$ $\displaystyle=U,$ (10a) $\displaystyle\dot{U}$
$\displaystyle=(1-U^{2}-V^{2})U-a(X-X(t-\tau)),$ (10b) $\displaystyle\dot{Y}$
$\displaystyle=V,$ (10c) $\displaystyle\dot{V}$
$\displaystyle=(1-U^{2}-V^{2})V-a(Y-Y(t-\tau)).$ (10d)
For all values of $a$ and $\tau$, Eqs. (10a)-(10d) have translationally
invariant stationary solutions
$\displaystyle X=X_{0},\quad U=0,\quad Y=Y_{0},\quad V=0,$ (11)
with two free parameters $X_{0}$ and $Y_{0}$. They also have a three parameter
family of uniformly translating solutions
$\displaystyle X=U_{0}t+X_{0},\quad U=U_{0},\quad Y=V_{0}t+Y_{0},\quad
V=V_{0},$ (12)
which requires
$\displaystyle U_{0}^{2}+V_{0}^{2}=1-a\tau,$ (13)
and thus exists only for $a\tau<1$. In the two-parameter space $(a,\tau)$, the
hyperbola $a\tau=1$ is in fact a pitchfork bifurcation line on which the
uniformly translating states are born from the stationary state
$(X_{0},0,Y_{0},0)$. The other branch of the pitchfork is an unphysical
solution with negative speed.
Linearizing Eqs. (10a)-(10d) about the stationary state, we obtain the
characteristic equation
$\displaystyle\left(a(1-e^{-\lambda\tau})-\lambda+\lambda^{2}\right)^{2}=0.$
(14)
It suffices to study the zeros of the function
$\displaystyle{\cal{D}}(\lambda)=a(1-e^{-\lambda\tau})-\lambda+\lambda^{2}=0,$
(15)
since the eigenvalues of Eqs. (10a)-(10d) are obtained by duplicating those of
Eq. (15).
We now search for Hopf bifurcations in the two parameter space $(a,\tau)$ by
letting $\lambda=i\omega$ in Eq. (15). Substitution leads to
$\displaystyle a-\omega^{2}-i\omega=ae^{-i\omega\tau}.$ (16)
Taking the modulus of Eq. (16), we find $a$ at the Hopf point, $a_{H}$, is
given by
$\displaystyle a_{H}^{2}=(a_{H}-\omega^{2})^{2}+\omega^{2},$ (17)
or, considering $\omega\neq 0$,
$\displaystyle a_{H}=\frac{1+\omega^{2}}{2}.$ (18)
We eliminate $a$ in Eq. (16) by using Eq. (18) and taking the complex
conjugate to obtain an equation for $\tau$ at the Hopf point
$\displaystyle\frac{1-\omega^{2}}{1+\omega^{2}}+i\frac{2\omega}{1+\omega^{2}}=e^{i\omega\tau}.$
(19)
We obtain $\tau$ by equating the arguments of both sides, being careful to use
the branch of $\tan\theta$ in $(0,\pi)$ since the left hand side of Eq. (19)
is on the upper complex plane for $\omega>0$. The result is a family of Hopf
bifurcation curves parameterized by $\omega$:
$\displaystyle a_{H}(\omega)$ $\displaystyle=\frac{1+\omega^{2}}{2},$ (20a)
$\displaystyle\tau_{Hn}(\omega)$
$\displaystyle=\frac{1}{\omega}\left(\arctan\left(\frac{2\omega}{1-\omega^{2}}\right)+2n\pi\right)$
$\displaystyle n$ $\displaystyle=0,1,\ldots.$ (20b)
These curves are shown in Fig. 2. We may eliminate the parameter $\omega$
between these two equations and obtain
Figure 2: (a) Hopf (blue) and pitchfork (red) bifurcation curves in $a$ and
$\tau$ space. (b) A zoom-in of the branches in the first panel displaying also
the saddle to node transition (dashed black); the number in each region
indicates the number of eigenvalues with a real part greater than zero with
the solid lines as boundaries. (Color online.)
$\displaystyle\tau_{Hn}(a)$
$\displaystyle=\frac{1}{\sqrt{2a-1}}\left(\arctan\left(\frac{\sqrt{2a-1}}{1-a}\right)+2n\pi\right)$
$\displaystyle n$ $\displaystyle=0,1,\ldots$ (21)
In spite of their appearance, the Hopf curves in Eqs. (20a)-(20) and (IV) are
in fact continuous at $\omega=1$ and $a=1$, respectively (with the correct
branch of $\tan\theta$ in $(0,\pi)$). From Eq. (20a)-(20), we see that the
Hopf frequency depends only on the value of $a$ for all members in the family;
it has the value one at $a=1$ and the frequency tends to infinity as $a$
grows. Interestingly, only the first Hopf curve of the family in Eq. (IV) is
defined at $a=1/2$; it has the value $\tau_{H0}|_{a=1/2}=2$. The point
($a=1/2$, $\tau=2$) which lies both on the first member of the family of Hopf
curves and on the pitchfork branch is in fact a Bogdanov-Takens (BT) point
[26], where $\omega=0$. None of the other Hopf branches meet the pitchfork
bifurcation line since they tend asymptotically to infinity as $a\rightarrow
1/2$.
Figure 3: Location of the dominating eigenvalues around the Bogdanov-Takens
point at $a=1/2$, $\tau=2$. Parameter values are (a) $a=0.60$, $\tau=2.0$, (b)
$a=0.48$, $\tau=2.09$, (c) $a=0.40$, $\tau=2.01$, (d) $a=0.53$, $\tau=1.90$,
and (e) $a=0.55$, $\tau=1.91$.
We used a numerical continuation method (DDE-Biftool) [27] to calculate the
pitchfork and Hopf branches in the $(a,\tau)$ parameter space; these results
are in perfect agreement with our analytical calculations (results not shown).
These numerical studies reveal that the number of eigenvalues with real part
greater than zero is as indicated in Fig. 2. In addition, our numerical
continuation analyses also reveal node to focus transitions of the steady
state. These occur at points where there are two real and equal eigenvalues,
i.e. where ${\cal{D}}(\lambda)=0$ and ${\cal{D}^{\prime}}(\lambda)=0$, for
$\lambda$ real. From ${\cal{D}^{\prime}}(\lambda)=0$ we obtain
$e^{-\tau\lambda}=\frac{1-2\lambda}{a\tau}$, which we can insert into
${\cal{D}}(\lambda)=0$ to obtain
$\displaystyle\lambda^{2}-\left(1-\frac{2}{\tau}\right)\lambda+a-\frac{1}{\tau}=0,$
(22)
with solutions
$\lambda=\frac{1}{2}\left[1-\frac{2}{\tau}\pm\sqrt{1+\frac{4}{\tau^{2}}-4a}\right]$.
For the roots to be repeated, we set the discriminant to zero and this gives
the curve where the node-focus transitions occur:
$\displaystyle\tau=\frac{1}{\sqrt{a-1/4}}.$ (23)
Moreover, from the solutions to Eq. (22) we see that the repeated eigenvalues
have positive real parts if $\tau>2$ and negative real parts if $\tau<2$. In
Fig. 2, we show the pitchfork and Hopf bifurcation curves overlaid with the
node-focus transition curve given by Eq. (23).
As seen in Fig. 2, the pitchfork and Hopf branches, together with the node-
focus transition curves split the area around the BT point into five different
regions. The behavior of the leading eigenvalues (excluding the one at the
origin) as one probes these five regions is shown in Figs. 3-3. At a point
directly to the right of the BT point in $(a,\tau)$ space, the stationary
solution has a pair of eigenvalues with positive real parts and non-zero
imaginary parts [Fig. 3]. Moving counter-clockwise in the $(a,\tau)$ plane,
the eigenvalue pair collapses onto the real line after crossing the upper
branch of the node-focus transition [Fig. 3]. Still moving in the same
direction in parameter space, we observe two different instances of
eigenvalues crossing the origin: first, the smaller of the two purely real and
positive eigenvalues does so on the upper part of the pitchfork bifurcation
line [Fig. 3] and then the remaining purely real and positive eigenvalue
crosses the origin on the lower part of the bifurcation line [Fig. 3].
Finally, at the node-focus transition line, the two purely real and negative
eigenvalues coincide on the real axis and acquire non-zero imaginary parts
[Fig. 3]. Continuing upwards in parameter space, the complex pair crosses the
imaginary axis on the Hopf bifurcation curve, giving birth to a stable limit
cycle.
### IV-A Numerical simulations
Figure 4: The limit cycle of the center of mass is shown through a comparison
of analytical (solid line) and numerical (“cross” markers) values of $a_{H}$
and $\omega$ for several choices of $\tau$. The analytical result is found
using Eqs. (20a)-(20), while the numerical result is found using a
continuation method [27] for Eq. (9). The inset shows the stochastic
trajectory of the center of mass of the swarm from $t=45$ to $t=90$.Figure
reproduced with permission from [14].
Figure 4 shows an excellent comparison of the analytical result given by Eqs.
(20a)-(20) with a numerical result which was found using a continuation method
[27] for the mean field model for several choices of $\tau$. Furthermore, for
$\tau=1$, the value of coupling $a$ at the bifurcation point is $a_{H}\approx
3.2$. This value of $a_{H}$ corresponds very well to the change in behavior of
the stochastic swarm (results not shown).
More evidence of the Hopf bifurcation is seen in the inset of Fig. 4. The
inset shows the stochastic trajectory of the center of mass of the swarm from
$t=45$ to $t=90$ for the example shown in Fig. 1. Once the time delay is
switched on at $t=40$ (with the swarm located at the center of the inset
figure), the swarm begins to oscillate. The swarm moves along an elliptical
path [the position of its center of mass is denoted at several times that
correspond to Figs. 1(b), 1(d), 1(f), 1(h), and 1(j)], until it eventually
converges to the circular limit cycle.
Figures 5 and 5 show a time series simulation of a swarm with $N=75$
particles. Figure 5 shows the position components, while Fig. 5 shows the
velocity components., One can see that the swarm follows a circular-like path
over time. A perturbation that is applied at $t=20$ shows that for the chosen
parameters, the pattern is stable in the presence of noise.
Figure 5: The limit cycle of the center of mass is shown through the (a)
position and (b) velocity time series of the swarm using $N=75$ particles,
$a=0.7$, $\tau=2.2$, and noise intensity $D=0.045$. A velocity perturbation is
applied at $t=20$.
Figure 6: Long time behavior of the mean particle alignment (defined in the
text) for different values of noise intensity ($D=\sigma^{2}/2$) and two
different initial conditions. In panel (a), all particles start off from the
origin with equal velocity vectors; in panel (b), all particles start from
rest, distributed uniformly over the unit square. For these simulations,
$N=150$, $a=2$ and $\tau=2$. The time-delay is turned on at $t=50$, and the
simulations run until $t=300$.
The presence of noise introduces interesting switching behavior that make the
initial conditions of the swarm critical in determining the long time behavior
of the system. To demonstrate this, we have performed a series of simulations
for different noise intensities and two different initial conditions: (i) all
particles start at the origin with unit $x$ and $y$ speeds [Fig. 6] and (ii)
all particles are distributed uniformly over the unit square and start from
rest [Fig. 6]. The simulations are run until $t=300$ using a coupling constant
$a=2$ and a time-delay $\tau=2$ which is turned on at $t=50$. Our simulations
reveal that in the long time limit and for small values of noise, the swarm
converges to either a compact state that rotates as a whole [case (i)] or to a
ring state with particles going both clockwise and counterclockwise [case
(ii)]. The asymptotic behavior of the system is readily visualized by
calculating the mean alignment of the swarm particles. We quantify this mean
alignment of the swarm by calculating the cosine between the directions of the
$i$-th particle and the center of mass,
$\cos\theta_{i}=(\dot{\mathbf{r}}_{i}\cdot\dot{\mathbf{R}})/(|\dot{\mathbf{r}}_{i}||\dot{\mathbf{R}}|)$,
and then averaging over all particles and over the last 100 time units of
simulation. Figure 6 shows that in case (i) the particles converge to the
compact, aligned state for low and moderate noise intensities. However, this
state is broken up at high noise levels ($\sigma\approx 0.8$). In contrast,
Fig. 6 shows that in case (ii) the particles converge to a ring for small
values of noise ($\sigma\lesssim 0.25$), evidenced by the low values of the
mean particle alignment in Fig. 6, but converge to the aligned case for higher
values of noise ($\sigma\gtrsim 0.25$). Observing the full simulation runs in
detail (not shown) reveals a switching behavior: for case (ii) with a noise
level $\sigma\gtrsim 0.25$, the particles first converge to a noisy ring and
then switch to the rotating state due to the effect of noise. The simulations
suggest that the transition to the rotating state occurs once the velocities
of the particles cross an alignment threshold. The system, in fact, displays
hysteresis: one can force the swarm to transition from the ring state with
$\sigma=0.2$ to the rotating state by raising the noise to $\sigma=0.25$;
however, it seems that the inverse transition, i.e. making the swarm
transition back to the ring state by lowering the noise level, is extremely
unlikely.
## V CONCLUSIONS
To summarize, we studied the dynamics of a self-propelling swarm in the
presence of noise and a constant communication time delay and prove that the
delay induces a transition that depends upon the size of the interaction
coupling coefficient. Although our analytical and numerical results were
obtained using a model with linear, attractive interactions, the analysis may
be applied to models with more general forms of social interaction.
We further uncovered a complete analytical description of the bifurcation
point which control the instabilities arising from noise induced transitions.
The analysis allows us to completely classify, using mean field
approximations, where the swarm exhibits a stable translation, stationary
center of mass, or rotation.
In general, our results provide insight into the stability of complex systems
comprised of individuals interacting with one another with a finite time delay
in a noisy environment. Furthermore, the results may prove to be useful in
controlling man-made vehicles where actuation and communication are delayed,
as well as in understanding swarm alignment in biological systems.
## VI ACKNOWLEDGMENTS
The authors gratefully acknowledge the Office of Naval Research for their
support. LMR and IBS are supported by Award Number R01GM090204 from the
National Institute Of General Medical Sciences. The content is solely the
responsibility of the authors and does not necessarily represent the official
views of the National Institute Of General Medical Sciences or the National
Institutes of Health. E.F. is supported by the Naval Research Laboratory
(Award No. N0017310-2-C007).
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|
arxiv-papers
| 2012-04-20T12:21:15 |
2024-09-04T02:49:29.958400
|
{
"license": "Public Domain",
"authors": "Luis Mier-y-Teran-Romero, Eric Forgoston and Ira B. Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1204.4597"
}
|
1204.4600
|
# Robotic Manifold Tracking of Coherent Structures in Flows
M. Ani Hsieh, Eric Forgoston, T. William Mather, and Ira B. Schwartz This work
was supported by the Office of Naval Research (ONR). MAH was an ONR Summer
Faculty Fellow and supported by ONR Contract No. N0001411WX20079. EF is
supported by the Naval Research Laboratory (Award No. N0017310-2-C007) TWM is
supported by the National Science Foundation under Grant No. DGE-0947936.M.
Ani Hsieh and T. William Mather are with the SAS Laboratory, Mechanical
Engineering & Mechanics Department, Drexel University, Philadelphia, PA 19104,
USA {mhsieh1,twm32}@drexel.eduEric Forgoston is with the Department of
Mathematical Sciences, Montclair State University, Montclair, New Jersey
07043, USA eric.forgoston@montclair.eduIra B. Schwartz is with the Nonlinear
Systems Dynamics Section, Plasma Physics Division, Code 6792, U.S. Naval
Research Laboratory, Washington, DC 20375, USA ira.schwartz@nrl.navy.mil
###### Abstract
Tracking Lagrangian coherent structures in dynamical systems is important for
many applications such as oceanography and weather prediction. In this paper,
we present a collaborative robotic control strategy designed to track stable
and unstable manifolds. The technique does not require global information
about the fluid dynamics, and is based on local sensing, prediction, and
correction. The collaborative control strategy is implemented on a team of
three robots to track coherent structures and manifolds on static flows as
well as a noisy time-dependent model of a wind-driven double-gyre often seen
in the ocean. We present simulation and experimental results and discuss
theoretical guarantees of the collaborative tracking strategy.
## I INTRODUCTION
In this paper, we present a collaborative control strategy for a class of
autonomous underwater vehicles (AUVs) to track the coherent structures and
manifolds on flows. In realistic ocean flows, these time-dependent coherent
structures, or Lagrangian coherent structures (LCS), are similar to
separatrices that divide the flow into dynamically distinct regions. LCS are
extensions of stable and unstable manifolds to general time-dependent flows
[1], and they carry a great deal of global information about the dynamics of
the flows. For two-dimensional (2D) flows, LCS are analogous to ridges defined
by local maximum instability, and quantified by local measures of Finite-Time
Lyapunov Exponents (FTLE) [2].
Recently, LCS have been shown to coincide with optimal trajectories in the
ocean which minimize the energy and the time needed to traverse from one point
to another [3, 4]. Furthermore, to improve weather and climate forecasting,
and to better understand various physical, chemical, and geophysical processes
in the ocean, there has been significant interest in the deployment of
autonomous sensors to measure a variety of quantities of interest. One
drawback to operating sensors in time-dependent and stochastic environments
like the ocean is that the sensors will tend to escape from their monitoring
region of interest. Since the LCS are inherently unstable and denote regions
of the flow where more escape events may occur [5], knowledge of the LCS are
of paramount importance in maintaining a sensor in a particular monitoring
region.
Existing work in cooperative boundary tracking for robotic teams that relies
on one-dimensional (1D) parameterizations include [6, 7] and [8, 9] for static
and time-dependent cases respectively. Formation control strategies for
distributed estimation of level surfaces and scalar fields in the ocean are
presented in [10, 11, 12] and pattern formation for surveillance and
monitoring by robot teams is discussed in [13, 14, 15]. Our work is
distinguished from existing work in that we use cooperative robots to find
coherent structures without requiring a global picture of the ocean dynamics.
We take inspiration from [16] and design a strategy to enable a team of robots
to track the stable/unstable manifolds of general 2D conservative flows
through local sensing alone. We verify the feasibility of our method through
simulations and experiments and show how the proposed strategy can be extended
to track coherent structures in time-dependent conservative flows with
measurement noise. To our knowledge, this is the first attempt in the
development of tracking strategies for mapping LCS in the ocean using AUVs.
The novelty of this work lies in the use of nonlinear dynamical and chaotic
system analysis techniques to derive a tracking strategy for a team of robots.
The cooperative control strategy leverages the spatio-temporal sensing
capabilities of a team of networked robots to track the boundaries separating
the regions in phase space that support distinct dynamical behavior.
Additionally, our boundary tracking relies solely on local measurements of the
velocity field. Our technique is quite general, and may be applied to any
conservative flow.
The paper is structured as follows: We formulate the problem and outline key
assumptions in Section II. The cooperative control strategy is presented in
Section III and its theoretical properties analyzed in Section IV. Section V
presents our simulation and experimental results. The extension of the
proposed strategy to a noisy time-dependent model of a wind-driven double-gyre
is presented in VI. We conclude with a discussion of our results and
directions for future work in Sections VII and VIII respectively.
## II PROBLEM FORMULATION
We consider the problem of controlling a team of $N$ planar AUVs to
collaboratively track the material lines that separate regions of flow with
distinct fluid dynamics. This is similar to the problem of tracking the stable
(and unstable) manifolds of a general nonlinear dynamical system where the
manifolds separate regions in phase space with distinct dynamical behaviors.
We assume the following 2D kinematic model for each of the AUVs:
$\dot{x}_{i}=V_{i}\cos\theta_{i}+u_{i},$ (1a)
$\dot{y}_{i}=V_{i}\sin\theta_{i}+v_{i},$ (1b)
where $\mathbf{x}_{i}=[x_{i},\,y_{i}]^{T}$ is the vehicle’s planar position,
$V_{i}$ and $\theta_{i}$ are the vehicle’s linear speed and heading, and
$\mathbf{u}_{i}=[u_{i},\,v_{i}]^{T}$ is the velocity of the fluid current
experienced/measured by the $i^{th}$ vehicle. Additionally, we assume each
agent can be circumscribed by a circle of radius $r$, i.e., each vehicle can
be equivalently described as a disk of radius $r$.
In this work, $\mathbf{u}_{i}$ is provided by a 2D planar conservative vector
field described by a differential equation of the form
$\mathbf{\dot{x}}=F(\mathbf{x}).$ (2)
In essence, $u_{i}=F_{x}(\mathbf{x}_{i})$ and $v_{i}=F_{y}(\mathbf{x}_{i})$.
Let $B_{S}$ and $B_{U}$ denote the stable and unstable manifolds of (2). In
general, $B_{S}$ and $B_{U}$ are the separating boundaries between regions in
phase space with distinct dynamics. For 2D flows, $B_{*}$ are simply one-
dimensional curves where $*$ denotes either stable ($S$) or unstable ($U$)
boundaries. For a small region centered about a point on $B_{*}$, the system
is unstable in one dimension. Finally, let $\rho(B_{*})$ denote the radius of
curvature of $B_{*}$ and assume that the minimum of the radius of curvature
$\rho_{min}(B_{*})>r$. This last assumption is needed to ensure the robots do
not lose track of the $B_{*}$ due to sharp turns.
The objective is to develop a collaborative strategy to enable a team of
robots to track $B_{*}$ in general 2D planar conservative flow fields through
local sampling of the velocity field. While the focus is on the development of
a tracking strategy for $B_{S}$, the proposed method can be easily extended to
track $B_{U}$ since $B_{U}$ are simply stable manifolds of (2) for $t<0$. We
present our methodology in the following section.
## III METHODOLOGY
Our methodology is inspired by the Proper Interior Maximum (PIM) Triple
Procedure [16] – a numerical technique designed to find stationary
trajectories in chaotic regions with no attractors. While the original
procedure was developed for chaotic dynamical systems, the approach can be
employed to reveal the stable set of a saddle point of a general nonlinear
dynamical system. The procedure consists of iteratively finding an appropriate
PIM Triple on a saddle straddling line segment and propagating the triple
forward in time. We briefly summarize the procedure in the following section
and refer the interested reader to [16] for further details.
### III-A The PIM Triple Procedure
Given the dynamical system described by (2), let ${\cal D}\in\mathbb{R}^{2}$
be a closed and bounded set such that ${\cal D}$ does not contain any
attractors of (2). Given a point $\mathbf{x}\in{\cal D}$, the escape time of
$\mathbf{x}$, denoted by $T_{E}(\mathbf{x})$, is the time $\mathbf{x}$ takes
to leave the region ${\cal D}$ under the differential map (2).
Let $J$ be a line segment that crosses the stable set $B_{S}$ in ${\cal D}$,
i.e., the endpoints of the $J$ are on opposite sides of $B_{S}$. Let
$\\{\mathbf{x}_{L},\mathbf{x}_{C},\mathbf{x}_{R}\\}$ denote a set of three
points in $J$ such that $\mathbf{x}_{C}$ denotes the interior point. Then
$\\{\mathbf{x}_{L},\mathbf{x}_{C},\mathbf{x}_{R}\\}$ is an Interior Maximum
triple if
$T_{E}(\mathbf{x}_{C})>\max\\{T_{E}(\mathbf{x}_{L}),T_{E}(\mathbf{x}_{R})\\}$.
Furthermore, $\\{\mathbf{x}_{L},\mathbf{x}_{C},\mathbf{x}_{R}\\}$ is a Proper
Interior Maximum (PIM) triple if it is an Interior Maximum triple and the
interval $[\mathbf{x}_{L},\mathbf{x}_{R}]$ in $J$ is a proper subset of $J$.
Then the numerical computation of any PIM triple can be obtained iteratively
starting with an initial saddle straddle line segment $J_{0}$. Let
$\mathbf{x}_{L_{0}}$ and $\mathbf{x}_{R_{0}}$ denote the endpoints of $J_{0}$
and apply an $\epsilon_{0}>0$ discretization of $J_{0}$ such that
$\mathbf{x}_{L_{0}}=\mathbf{q}_{0}<\mathbf{q}_{1}<\ldots<\mathbf{q}_{M}=\mathbf{x}_{R_{0}}$.
For every point $\mathbf{q}_{i}$, determine $T_{E}(\mathbf{q}_{i})$ by
propagating $\mathbf{q}_{i}$ forward in time using (2). Then the PIM triple in
$J_{0}$ is given by the the points
$\\{\mathbf{q}_{k-1},\mathbf{q}_{k},\mathbf{q}_{k+1}\\}$ where
$\mathbf{q}_{k}=\arg\max\limits_{i=1,\ldots,M}T_{E}(\mathbf{q}_{i})$. This PIM
triple can then be further refined by choosing $J_{1}$ to be the line segment
containing $\\{\mathbf{q}_{k-1},\mathbf{q}_{k},\mathbf{q}_{k+1}\\}$ and
reapplying the procedure with another $\epsilon_{1}>0$ discretization where
$\epsilon_{1}<\epsilon_{0}$.
Given an initial saddle straddling line segment $J_{0}$, it has been shown
that the line segment given by any subsequent PIM triple on $J_{0}$ is also a
saddle straddling line segment [16]. Furthermore, if we use a PIM triple
$\mathbf{x}(t)=\\{\mathbf{x}_{L},\mathbf{x}_{C},\mathbf{x}_{R}\\}$ as the
initial conditions for the dynamical system given by (2) and propagate the
system forward in time by $\Delta t$, then the line segment containing the set
$\mathbf{x}(t+\Delta t)$, $J_{t+\Delta t}$, remains a saddle straddle line
segment. As such, the same numerical procedure can be employed to determine an
appropriate PIM trip on $J_{t+\Delta t}$. This procedure can be repeated to
eventually reveal the entire stable set $B_{S}$ and unstable set $B_{U}$
within ${\cal D}$ if time was propagated forwards and backwards respectively.
Furthermore, since the procedure always begins with a valid saddle straddling
line segment, by construction, the procedure always results in a non-empty
set.
Inspired by the PIM Triple Procedure, we propose a cooperative saddle straddle
control strategy for a team of $N\geq 3$ robots to track the stable (and
unstable) manifolds of a general conservative time-independent flow field
$F(\mathbf{x})$. Different from the procedure, our robots will solely rely on
information that can be gathered via local sensing and shared through the
network. In contrast, a straight implementation of the PIM Triple Procedure
would require global knowledge of the structure of the system dynamics
throughout a given region given its reliance on computing escape times. We
describe our approach in the following section.
### III-B Controller Synthesis
Consider a team of three robots and identify them as robots $\\{L,C,R\\}$.
While the robots may be equipped with similar sensing and actuation
capabilities, we propose a heterogeneous cooperative control strategy.
Let
$\mathbf{x}(0)=[\mathbf{x}_{L}^{T}(0),\,\mathbf{x}_{C}^{T}(0),\,\mathbf{x}_{R}^{T}(0)]^{T}$
be the initial conditions for the three robots. Assume that $\mathbf{x}(0)$
lies on the line segment $J_{0}$ where $J_{0}$ is a saddle straddle line
segment and $\\{\mathbf{x}_{L}(0),\mathbf{x}_{C}(0),\mathbf{x}_{R}(0)\\}$
constitutes a PIM triple. Similar to the PIM Triple Procedure, the objective
is to enable the robots to maintain a formation such that a valid saddle
straddle line segment can be maintained between robots $L$ and $R$. Instead of
computing the escape times for points on $J_{0}$ as proposed by the PIM Triple
Procedure, robot $C$ must remain close to $B_{S}$ using only local
measurements of the velocity field provided by the rest of the team. As such,
we refer to robot $C$ as the tracker of the team while robots $L$ and $R$
maintains a straddle formation across the boundary at all times. Robots $L$
and $R$ may be thought of herding robots, since they control and determine the
actions of the tracking robot.
#### III-B1 Straddling Formation Control
The controller for the straddling robots consists of two discrete states: a
passive control state, $U_{P}$, and an active control state, $U_{A}$. The
robots initialize in the passive state $U_{P}$ where the objective is to
follow the flow of the ambient vector field. Therefore, $V_{i}=0$ for $i=L,R$.
Robots execute $U_{P}$ until they reach the maximum allowable separation
distance $d_{Max}$ from robot $C$. When
$\|\mathbf{x}_{i}-\mathbf{x}_{C}\|>d_{Max}$, robot $i$ switches to the active
control state, $U_{A}$, where the objective is to navigate to a point
$\mathbf{p}_{i}$ on the current projected saddle straddle line segment
$\hat{J}_{t}$ such that $\|\mathbf{p}_{i}-\mathbf{p}_{c}\|=d_{Min}$ and
$\mathbf{p}_{C}$ denotes the midpoint of $\hat{J}_{t}$. When robots execute
$U_{A}$ , $V_{i}=\|(\mathbf{p}_{i}-\mathbf{x}_{i})-\mathbf{u}_{i}\|$ and
$\theta_{i}(t)=\alpha_{i}(t)$ where $\alpha_{i}$ is the angle between the
desired, $(\mathbf{p}_{i}-\mathbf{x}_{i})$, and current heading,
$\mathbf{u}_{i}$, of robot $i$ as shown in Fig. 1. In summary, the straddling
control strategy for robots $L$ and $R$ is given by
$\displaystyle V_{i}$ $\displaystyle=\left\\{\begin{array}[]{ll}0&\textrm{if
}d_{Min}<\|\mathbf{x}_{i}-\mathbf{x}_{C}\|<d_{Max}\\\
\|(\mathbf{p}_{i}-\mathbf{x}_{i})-\mathbf{u}_{i}\|&\textrm{otherwise}\end{array}\right.,$
(3c) $\displaystyle\theta_{i}$
$\displaystyle=\left\\{\begin{array}[]{ll}0&\textrm{if
}d_{Min}<\|\mathbf{x}_{i}-\mathbf{x}_{C}\|<d_{Max}\\\
\alpha_{i}&\textrm{otherwise}\end{array}\right.~{}.$ (3f)
Figure 1: Three robots tracking $B_{S}$ in a conservative vector field. The
blue dash-dot lines represent the robot trajectories, the green dashed line
represents the saddle straddle line segment $J$, and $\mathbf{p}_{L}$ and
$\mathbf{p}_{R}$ denotes the target positions for $L$ and $R$ respectively
when executing $U_{P}$ and $U_{A}$.
We note that while the primary control objective for robots $L$ and $R$ is to
maintain a straddle formation across $B_{S}$, robots $L$ and $R$ are also
constantly sampling the velocity of the local vector field and communicating
these measurements and their relative positions to robot $C$. Robot $C$ is
then tasked to use these measurements to track the position of $B_{S}$.
#### III-B2 Manifold Tracking Control
Let $\mathbf{\hat{u}}_{L}(t)$, $\mathbf{\hat{u}}_{C}(t)$, and
$\mathbf{\hat{u}}_{C}(t)$ denote the current velocity measurements obtained by
robots $L$, $C$, and $R$ at their respective positions. Let $d(\cdot,\cdot)$
denote the Euclidean distance function and assume that
$d(\mathbf{x}_{C},B_{S})<\epsilon$ such that $\epsilon>0$ is small. Given the
straddle line segment $J_{t}$ such that $\mathbf{x}_{L}(k)$ and
$\mathbf{x}_{R}(k)$ are the endpoints of $J_{t}$, we consider an
$\epsilon_{t}<\epsilon$ discretization of $J_{t}$ such that
$\mathbf{x}_{L}=\mathbf{q}_{1}<\mathbf{q}_{2}<\ldots<\mathbf{q}_{M}=\mathbf{x}_{R}$.
The objective is to use the velocity measurements provided by the team to
interpolate the vector field at the points
$\mathbf{q}_{1},\ldots,\mathbf{q}_{M}$. Since (2) has ${\cal C}^{1}$
continuity and if $\mathbf{x}_{C}$ is $\epsilon$-close to $B_{S}$, then the
point
$\mathbf{q}_{B}=\arg\max\limits_{k=1,\ldots,M}\mathbf{u}(\mathbf{q}_{k})^{T}\mathbf{\hat{u}}_{C}(t)$
should be $\delta$-close to $B_{S}$ where $\epsilon<\delta<A$ and $A$ is a
small enough positive constant.
While there are numerous vector field interpolation techniques available [17,
18, 19], we employ the inverse distance weighting method described in [17].
For a given set of velocity measurements $\mathbf{\hat{u}}_{i}(t)$ and
corresponding position estimates $\mathbf{\hat{x}}_{i}(t)$, the velocity
vector at some point $\mathbf{q}_{k}$ is given by
$\displaystyle\mathbf{u}(\mathbf{q}_{k})=\sum_{j}\sum_{i=1}^{N}\frac{w_{ij}\mathbf{\hat{u}}_{i}(j)}{\sum_{j}\sum_{i=1}^{N}w_{ij}}$
where $w_{ij}=\|\mathbf{\hat{x}}_{i}(j)-\mathbf{q}_{i}\|^{-2}$. Rather than
rely solely on the current measurements provided by the three robots, it is
possible to include the recent history of $\mathbf{\hat{u}}_{i}(t)$ to improve
the estimate of $\mathbf{u}(\mathbf{q}_{k})$, i.e.,
$\mathbf{\hat{u}}_{i}(t-\Delta T)$, $\mathbf{\hat{u}}_{i}(t-2\Delta T)$, and
so on, where $\Delta T$ is the sampling period and $i=\\{L,C,R\\}$. Thus, the
control strategy for the tracking robot $C$ is given by
$\displaystyle V_{C}$
$\displaystyle=\|[(\mathbf{q}_{B}+b\mathbf{\hat{u}}_{B})-\mathbf{x}_{C}]-\mathbf{u}_{C}\|$
(4a) $\displaystyle\theta_{C}$ $\displaystyle=\beta_{C}$ (4b)
where $\beta_{C}$ denotes the difference in the heading of robot $C$ and the
vector $(\mathbf{q}_{B}-\mathbf{\hat{u}}_{B})$ and $b>r$ is a small number.
The term $b\mathbf{\hat{u}}_{B}$ is included to ensure that the control
strategy aims for a point in front of robot $C$ rather than behind it. As
such, the projected saddle straddle line segment $\hat{J}_{t}$ at each time
step is given by $\mathbf{p}_{c}=q_{C}+b\mathbf{u}_{C}$ with $\hat{J}_{t}$
orthogonal to $B_{S}$ at $q_{C}$ and $\|\hat{J}_{t}\|$ chosen to be in the
interval $[2d_{Min},2d_{Max}]$.
## IV ANALYSIS
In this section, we discuss the theoretical feasibility of the proposed saddle
straddle control strategy. We begin with the following key assumption on the
robots’ initial positions.
###### Assumption 1
Given a team of three robots $\\{L,C,R\\}$, assume that
$d(\mathbf{x}_{C}(0),B_{S})<\epsilon$ for a small value of $\epsilon>0$,
$\|\mathbf{x}_{L}-\mathbf{x}_{C}\|=\|\mathbf{x}_{R}-\mathbf{x}\|=d_{Min}$ with
$d_{Min}>2r$, and robots $L$ and $R$ are on opposite sides of $B_{S}$.
In other words, assume that the robots initialize in a valid PIM triple
formation and their positions form a saddle straddle line segment orthogonal
to $B_{S}$. Our main result concerns the validity of the saddle straddle
control strategy.
###### Theorem 1
Given a team of $3$ robots with kinematics given by (1) and $\mathbf{u}_{i}$
given by (2), the feedback control strategy (3) and (4) maintains a valid
saddle straddle line segment in the time interval $[t,t+\Delta t]$ if the
initial positions of the robots, $\mathbf{x}(t)$, is a valid PIM triple.
###### Proof:
To show this, we must show that at time $t+\Delta t$, robots $L$ and $R$
remain on opposite sides of $B_{S}$. Consider the rate of change of the
following function
$\displaystyle H(\mathbf{x}_{L},\mathbf{x}_{R})$
$\displaystyle=\frac{1}{2}(\mathbf{x}_{L}-\mathbf{x}_{R})^{T}(\mathbf{x}_{L}-\mathbf{x}_{R}).$
The above expression is simply one half the square of the distance between
robots $L$ and $R$. Let $J_{t}$ denote the saddle straddle line segment
defined by $\mathbf{x}_{L}(t)$ and $\mathbf{x}_{R}(t)$ at $t$ and let
$\mathbf{p}_{B}$ be the intersection of $J_{t}$ and $B_{S}$. By construction,
if we linearize (2) about the point $\mathbf{p}_{B}$, then the Jacobian of (2)
at $\mathbf{p}_{B}$ will have one positive eigenvalue. Furthermore, the
linearized system can be diagonalized such that the direction of instability
lies along $J_{t}$ [20]. Thus, $\frac{d}{dt}H>0$ in the time interval
$[t,t+\Delta t]$ when $V_{i}=0$ in (3).
When $V_{i}\neq 0$ in (3) for $i=L,R$, $\frac{d}{dt}H<0$ if the robots $L$ and
$R$ are moving closer to robot $C$ after reaching the maximum allowable
separation distance. Recall $\rho_{min}(B_{S})>r$, the smallest radius of
curvature of $B_{S}$, and $d_{Min}>2r$. Furthermore, robot $C$ initializes
$\epsilon$-close to the boundary and (4) steers $C$ towards $\mathbf{p}_{C}$
on $\hat{J}_{t}$ where $\hat{J}_{t}$ is orthogonal to $B_{S}$ at
$\mathbf{x}_{C}$. This ensures that the rate of the change of the radius of
curvature of the manifold $B_{S}$ is small enough such that $\hat{J}_{t}$
intersects with $B_{S}$ only once. Since $d_{Min}>2r$, this ensures that even
if $\frac{d}{dt}H<0$, the straddling robots never cross the boundary as they
move closer to the tracking robot. ∎
The above theorem guarantees that for any given time interval $[t,t+\Delta t]$
the team maintains a valid PIM triple formation. As such, the iterative
application of the proposed control strategy leads to the following
proposition.
###### Proposition 1
Given a team of $3$ robots with kinematics given by (1) and $\mathbf{u}_{i}$
given by (2), the feedback control strategy (3) and (4) results in an estimate
of $B_{S}$, denoted as $\hat{B}_{S}$, such that
$\left<B_{S},\hat{B}_{S}\right>_{L_{2}}<W$ for some $W>0$ where
$\left<\cdot,\cdot\right>_{L_{2}}$ denotes the inner product (which provides
an $L_{2}$ measure between the $B_{S}$ and $\hat{B}_{S}$ curves).
From Thm. 1, since the team is able to maintain a valid PIM triple formation
across $B_{S}$ for any given time interval $[t,t+\Delta t]$, this ensures that
an estimate of $B_{S}$ in the given time interval also exists. Applying this
reasoning in a recursive fashion, one can show that an estimate of $B_{S}$ can
be obtained for any arbitrary time interval. The challenge, however, lies in
determining the bound on $W$ such that $\hat{B}_{S}$ results in a good enough
approximation since $W$ depends on the sensor and actuation noise, the vector
interpolation routine, the sampling frequency, and the time scales of the flow
dynamics. This is a direction for future work.
## V RESULTS
### V-A Simulations
We illustrate the proposed control strategy given by (3) and (4) with some
simulation results. Fig. 2 shows the trajectories of three robots tracking a
sinusoidal boundary while Fig. 2 shows the team tracking a 1D star-shaped
boundary. We note that throughout the entire length of the simulation, the
team maintains a saddle straddle formation across the boundary.
In both examples, $\mathbf{u}=-a\nabla\varphi-b\nabla\times\psi$ where $a,b>0$
and $\varphi$ is an artificial potential function such that
$\varphi(\mathbf{x})=0$ for all $\mathbf{x}\in B_{*}$ and
$\varphi(\mathbf{x})<0$ for any $\mathbf{x}\in\mathbb{R}^{2}/B_{*}$. The
vector $\psi$ is a $3\times 1$ vector whose entries are given by
$[0,\,0,\,\gamma(x,y)]^{T}$ where $\gamma(x,y)$ is the curve describing the
desired boundary [15]. Lastly, the estimated position of the boundary is given
by the position of the tracking robot, i.e., robot $C$. In these examples, we
filtered the boundary position using a simple first-order low pass filter.
Figure 2: Trajectories of $3$ robots tracking a (a) sinusoidal boundary and a
(b) star-shaped boundary. The red dashed line is the estimated position of the
desired boundary shown in solid black. The start positions are shown by
$\triangle$ and the end positions are shown by the circle-enclosed blue
triangles.
### V-B Experiments
We also implemented the control strategy on our multi-robot testbed. The
testbed consisted of three mSRV-1 robots in a 4.8x5.4 meter workspace. The
mSRV-1 are differential-drive robots equipped with an embedded processor,
color camera, and 802.11 wireless capability. Localization for each robot was
provided via a network of overhead cameras. Fig. 3 shows the trajectories of
the robots tracking a star shaped boundary shown in black. Fig. 3 is a
snapshot of the experimental run. We refer the interested reader to the
attached multimedia file for a movie of the full simulation and experimental
runs.
Figure 3: Trajectories of the $3$ robot team tracking a (a) star shape. The
red dashed line is the estimated position of the desired boundary shown in
solid black. The start positions are shown by $\triangle$ and the end
positions are shown by $\bigcirc$. (b) Snapshot of the multi-robot experiment.
## VI EXTENSION TO PERIODIC BOUNDARIES
In this section, we consider the system of $3$ robots with kinematics given by
(1) where $\mathbf{u}_{i}$ is determined by the wind-driven double-gyre flow
model with noise
$\displaystyle\dot{x}$ $\displaystyle=-\pi
A\sin(\pi\frac{f(x,t)}{s})\cos(\pi\frac{y}{s})-\mu x+\eta_{1}(t),$ (5a)
$\displaystyle\dot{y}$ $\displaystyle=\pi
A\cos(\pi\frac{f(x,t)}{s})\sin(\pi\frac{y}{s})\frac{df}{dx}-\mu
y+\eta_{2}(t),$ (5b) $\displaystyle f(x,t)$
$\displaystyle=\varepsilon\sin(\omega t+\psi)x^{2}+(1-2\varepsilon\sin(\omega
t+\psi))x.$ (5c)
When $\varepsilon=0$, the double-gyre flow is time-independent, while for
$\varepsilon\neq 0$, the gyres undergo a periodic expansion and contraction in
the $x$ direction. In (5), $A$ approximately determines the amplitude of the
velocity vectors, $\omega/2\pi$ gives the oscillation frequency, $\varepsilon$
determines the amplitude of the left-right motion of the separatrix between
the gyres, $\psi$ is the phase, $\mu$ determines the dissipation, $s$ scales
the dimensions of the workspace, and $\eta_{i}(t)$ describes a stochastic
white noise with mean zero and standard deviation $\sigma=\sqrt{2I}$, for
noise intensity $I$. In this work, $\eta_{i}(t)$ can be viewed as either
measurement or environmental noise. Fig. 4 shows the phase portrait of the
time-independent double-gyre model.
Figure 4: Phase portrait of the model given by (5) with $A=10$, $\mu=0.005$,
$\varepsilon=0$, $\psi=0$, $I=0$, and $s=50$.
Fig. 5 shows the use of the control strategy (3) and (4) to track the
Lagrangian coherent structures of the periodic double-gyre model with noise.
As mentioned in Section I, LCS are extensions of stable and unstable manifolds
to non-autonomous dynamical systems [21]. We note that while the control
strategy was based on techniques developed for time-independent systems, the
method performs surprisingly well in tracking LCS for slow time-varying
systems in the presence of noise. Details regarding LCS computation can be
found in [5] and we refer the interested reader to the attached multimedia
file for a movie of the full simulation run. While the control strategy was
developed for static flows, the movie shows the robustness of the strategy for
tracking LCS in time-varying flows.
(a) t=0.8
(b) t=1.4
(c) t=1.8
(d) t=2.6
(e) t=3.0
(f) t=3.2
(g) t=3.6
(h) t=4.0
Figure 5: Trajectories of the team of $3$ robots tracking the Lagrangian
coherent structures of the system described by (5) with $A=10$, $\mu=0.005$,
$\varepsilon=0.1$, $\psi=0$, $I=0.01$, and $s=50$. The trajectories of the
straddling robots are shown in black and the estimated LCS is shown in white.
## VII DISCUSSION
In this paper, we have designed a control strategy that allows collaborating
robots to track coherent structures and manifolds on general static
conservative flows. In addition, we showed how the strategy can be used to
track LCS in time-dependent conservative flows with measurement noise. The
saddle straddle control strategy is based on the communication of local
velocity field measurements obtained by each robot. Using the local velocity
field information provided by the two straddling robots (the herders), one
robot (the tracker), is able to detect the coherent structures, a global
structure that delineates the phase space into different dynamical regions.
Our work is novel in that the robots are determining the location of a global
structure based solely on local information, and as far as we know, the
sensing of LCS in the ocean has never been performed using autonomous
vehicles. Moreover, only initial state knowledge of the LCS is required
locally to get an accurate prediction of the global structure.
While the cooperative control strategy was inspired by the PIM Triple
Procedure, a procedure that relies on the computation of escape times which is
a global property of the system, the controller itself only relies on
information provided by each robot’s onboard sensors. We also note that the
realization of the control strategy by the team of robots can be achieved
without the need for global localization information. As such, the strategy is
a purely local strategy. Furthermore, the cooperative control strategy was
derived to track the manifolds on a static flow, but performs surprisingly
well at tracking the LCS in the time-dependent double-gyre model in the
presence of noise.
Since realistic quasi-geostrophic ocean models exhibit double-gyre flow
solutions, our first attempt seems to suggest that our methods may be and
general enough to be applied to more complicated models, including multi-layer
PDE ocean models. As mentioned in Section IV the robustness of the control
strategy is dependent on numerous parameters in the system which includes
robots’ sensing and communication ranges, the bounds on the sensor and
actuation noise, the vector interpolation technique, the sampling frequency,
and the relative time scales of the AUV dynamics in relation to the
surrounding flow dynamics. While our initial results suggest that our approach
may be robust enough to measurement noise, a more thorough understanding of
the sensitivity of the proposed strategy to these various parameters is
instrumental in extending our approach to more realistic ocean models and for
field deployment.
## VIII FUTURE WORK
In recent years, there has been significant interest in the use of AUVs to
collect scientific data in the ocean to improve our ability to forecast
harmful algae blooms and weather and climate patterns. One drawback to
operating sensors in time-dependent and stochastic environments like the ocean
is that the sensors will tend to escape from their monitoring region of
interest. As such, the ability to identify and track Lagrangian coherent
structures (LCS) in these dynamic environments is paramount in maintaining
appropriate sensor coverage in regions of interest. Additionally, since LCS
have been shown to coincide with optimal trajectories in the ocean which
minimize the energy and navigation time [3, 4], real-time knowledge of these
“super-highways” is key in planning efficient AUVs paths.
Of particular interest is the extension of our method to more realistic ocean
models. Specifically, can we extend our current cooperative tracking strategy
to a swarm of heterogenous mobile and stationary sensors? By increasing the
team size and incorporating both stationary and mobile sensing devices, it is
possible to refine our tracking technique to reveal the coherent structures at
various spatial and time scales. One immediate direction for future work is to
investigate how the proposed strategy scales to larger team sizes. Second,
underwater environments pose unique challenges in terms of wireless
communications. In general, acoustic transmissions generally have low data
rates and acoustic wave propagation can be further affected by the surrounding
fluid dynamics [22]. As such, a second direction for future work is to
investigate how communication delays and missed transmissions impact the
overall accuracy of the tracking methodology. In this work, we assume an
initial state knowledge of the LCS is required. This initial formation may be
difficult to achieve without any prior global knowledge of the flows. By
considering a team of both stationary and mobile sensors, one can potentially
obtain an initial estimate of a local LCS through the stationary sensing
network which can then be tracked and further refined by the mobile nodes. A
third direction for future work is to determine how one can strategically
place a combination of mobile and stationary sensors to provide real-time
updates on the locations of LCS.
## References
* [1] G. Haller and G. Yuan, “Lagrangian coherent structures and mixing in two-dimensional turbulence,” _Phys. D_ , vol. 147, pp. 352–370, Dec 2000\.
* [2] S. C. Shadden, F. Lekien, and J. E. Marsden, “Definition and properties of lagrangian coherent structures from finite-time lyapunov exponents in two-dimensional aperiodic flows,” _Physica D: Nonlinear Phenomena_ , vol. 212, no. 3-4, pp. 271 – 304, 2005.
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|
arxiv-papers
| 2012-04-20T12:37:27 |
2024-09-04T02:49:29.966840
|
{
"license": "Public Domain",
"authors": "M. Ani Hsieh, Eric Forgoston, T. William Mather, and Ira B. Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1204.4600"
}
|
1204.4602
|
arxiv-papers
| 2012-04-20T12:41:58 |
2024-09-04T02:49:29.974259
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J.Q. Li, J. Li, H. X. Yang, H. F. Tian, S-W. Cheong, C. Ma, S. Zhang,\n and Y. G. Zhao",
"submitter": "Jun Li",
"url": "https://arxiv.org/abs/1204.4602"
}
|
|
1204.4606
|
# Randomly Distributed Delayed Communication and Coherent Swarm Patterns
Brandon Lindley and Luis Mier-y-Teran-Romero and Ira B. Schwartz This work was
supported by the Office of Naval Research and the National institutes of
Health.Brandon Lindley is an NRC postodctoral fellow at the US Naval Research
Labooratory, Code 6792, Washington, DC 20375 USA,
brandon.lindley.ctr@nrl.navy.milL. Mier-y-Teran-Romero is an NIH post doctoral
fellow at the Naval Research Laboratory. luis.miery@nrl.navy.milI. B. Schwartz
is at the US Naval Research Labooratory, Code 6792, Washington, DC 20375 USA
Ira.schwartz@nrl.navy.mil
###### Abstract
Previously we showed how delay communication between globally coupled self-
propelled agents causes new spatio-temporal patterns to arise when the delay
coupling is fixed among all agents [1]. In this paper, we show how discrete,
randomly distributed delays affect the dynamical patterns. In particular, we
investigate how the standard deviation of the time delay distribution affects
the stability of the different patterns as well as the switching probability
between coherent states.
## I INTRODUCTION
Numerous recent investigations have been devoted to the study of interacting
multi-agent or swarming systems in various natural and engineering fields of
study. Investigations of interacting systems have revealed the emergence of
highly complex dynamic behaviors in space and time which arise even though the
dynamics of a single agent is quite simple [2]. In particular, these multi-
agent swarms can self-organize in complicated spatio-temporal patterns that
depend on the details of the inter-agent interactions. These investigations
have been motivated by and had an impact on many diverse biological systems
such as bacterial colonies, schooling fish, flocking birds, swarming locusts,
ants, and pedestrians [3, 4, 5, 6, 7]. In this paper, we are interested in the
application that biological analogies have on the design of systems of
autonomous, inter-communicating robotic systems [8, 9, 10, 11] and mobile
sensor networks [12].
There is great interest to design agent-interaction protocols to carry out
robotic motion planning, consensus and cooperative control, and spatio-
temporal formation. One methodology is to combine inter-agent potentials with
external ones in order to achieve multi-agent cooperative motion in a manner
that is not too sensitive with respect the number of agents. Some important
applications making use of scalable numbers of agents are: obstacle avoidance
[10], boundary tracking [13, 14], environmental sensing [12, 15],
decentralized target tracking [16], environmental consensus estimation [12,
17] and task allocation [18].
Authors have employed very diverse approaches in the study of multi-agent
systems. Some authors have described the swarms at the individual level,
writing their models in terms of ordinary differential equations (ODEs) or
delay differential equations (DDEs) to describe their trajectories [19, 20,
9]. The addition of noise on the swarm’s dynamics introduces even richer
behavior, such as noise-induced transitions between different coherent
patterns [2, 1]. The study of noisy swarm dynamics has benefited from tools
from statistical physics applied to both first and second order phase
transitions that have been found in the formation of coherent states [21].
One important aspect of the understanding and design of space-time behavior in
communicating robotic systems is that of time delay. Time delay arises in
latent communication between agents, as well as actuation lag times due to
inertia. Time delays can have interesting and surprising dynamical
consequences in a system, such as large-scale synchronization [22, 23, 24],
and have been used successfully for control purposes [25, 26]. Many of the
initial time-delay studies focused on the case of one or a few discrete time
delays. Recently, more complex situations have been considered such as the
case of having several [27] and random time-delays [28, 29]. Another
interesting case is that of distributed time delays, i.e. when the dynamics of
the system depends on a continuous interval in its past instead of on a
discrete instant [30].
In the case of swarming systems in stochastic environments, it has been
observed that the introduction of a discrete communication time delay induces
a transition from one spatio-temporal pattern to another as the time delay
passes a certain threshold [1]. It was shown in [1] how the complex interplay
exists between the attractive coupling and the time delay in the transitions
between different spatio-temporal patterns [31, 32]. Time delays in robotic
systems have been also studied in the contexts of consensus estimation [17]
and task allocation [18]; in the latter, the time delays originate from the
period of time required to switch between different tasks.
In this paper, we consider a swarming model with discrete, randomly
distributed time delays. We explicitly show how a distribution of delay times
perturb the dynamics from the single discrete case delay case analytically. We
illustrate the dynamical effects of delay distributions with varying width and
show that the system is bistable, and very sensitive to choice of initial
starting conditions.
## II Swarm Model
We investigate the dynamics of a two-dimensional system of $N$ identical self-
propelling agents that are attracted to each other in a symmetric manner. We
consider the attraction between agents to occur in a time-delayed fashion, due
to the finite communication speeds and information-processing times.
Specifically, we focus on the situation in which the time-delay is nonuniform
across agents: there is one time delay for every pair of agents
$\tau_{ij}(=\tau_{ji})$, for particles $i$ and $j$. The time delays
$\tau_{ij}$’s are time-independent and are drawn independently from a random
distribution $\rho_{\tau}(\tau)$. The swarm dynamics are described by the
following governing equations:
$\displaystyle\dot{\mathbf{r}}_{i}=$ $\displaystyle\mathbf{v}_{i},$ (1a)
$\displaystyle\dot{\mathbf{v}}_{i}=$
$\displaystyle\left(1-|\mathbf{v}_{i}|^{2}\right)\mathbf{v}_{i}-\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau_{ij})),$ (1b)
for $i=1,2\ldots,N$. The position and velocity of the $i$th agent at time $t$
are denoted by $\mathbf{r}_{i}$ and $\mathbf{v}_{i}$, respectively. Each agent
has self-propulsion and frictional drag forces given by the expression term
$\left(1-|\mathbf{v}_{i}|^{2}\right)\mathbf{v}_{i}$. The coupling constant $a$
measures the strength of the attraction between agents and the communication
time delay between particles $i$ and $j$ is given by $\tau_{ij}$. Note that in
the absence of coupling agents tend to move in a straight line with unit speed
as time tends to infinity.
## III Mean Field Approximation
We carry out a mean field approximation of the swarming system by switching to
particle coordinates relative to the center of mass and disregarding the noise
terms. The center of mass of the swarming system is given by
$\displaystyle\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t).$ (2)
We can decompose the position of each particle into
$\displaystyle\mathbf{r}_{i}=\mathbf{R}+\delta\mathbf{r}_{i},\qquad
i=1,2\ldots,N,$ (3)
where we’ll have
$\displaystyle\sum_{i=1}^{N}\delta\mathbf{r}_{i}(t)=0.$ (4)
Inserting Eq. (3) into the second order system equivalent to Eq. (1) and
simplifying we get
$\displaystyle\ddot{\mathbf{R}}+\delta\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)(\dot{\mathbf{R}}+\delta\dot{\mathbf{r}_{i}})$
$\displaystyle-\frac{a(N-1)}{N}\bigg{(}\mathbf{R}(t)+\delta\mathbf{r}_{i}(t)\bigg{)}$
$\displaystyle+\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\left(\mathbf{R}(t-\tau_{ij})+\delta\mathbf{r}_{j}(t-\tau_{ij})\right),$ (5)
Summing Eq. (III) over $i$ and using Eq. (4), we get
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}-\frac{1}{N}\sum_{i=1}^{N}|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{R}}$
$\displaystyle-\frac{1}{N}\sum_{i=\
1}^{N}\left(2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}+|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\delta\dot{\mathbf{r}_{i}}$
$\displaystyle-a\frac{N-1}{N}\mathbf{R}(t)+\frac{a}{N^{2}}\sum_{i=1}^{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\left(\mathbf{R}(t-\tau_{ij})+\delta\mathbf{r}_{j}(t-\tau_{ij})\right).$ (6)
We now make some approximations on the terms with the double sums. For the
displacements from the center of mass, we have
$\displaystyle\frac{a}{N^{2}}$
$\displaystyle\sum_{i=1}^{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\delta\mathbf{r}_{j}(t-\tau_{ij})=\frac{a(N-1)}{N^{2}}\sum_{j=1}^{N}\frac{1}{N-1}\mathop{\sum_{i=1}^{N}}_{i\neq
j}\delta\mathbf{r}_{j}(t-\tau_{ij})$
$\displaystyle\approx\frac{a(N-1)}{N^{2}}\int_{0}^{\infty}\sum_{j=1}^{N}\delta\mathbf{r}_{j}(t-\tau)\rho_{\tau}(\tau)d\tau=0,$
(7)
since $\sum_{j=1}^{N}\delta\mathbf{r}_{j}(t-\tau)=0$ by Eq. (4). In passing
from the discrete to the continuous averaging above, we argue as follows. The
expression $\frac{1}{N-1}\mathop{\sum_{i=1}^{N}}_{i\neq
j}\delta\mathbf{r}_{j}(t-\tau_{ij})$ is the average of
$\delta\mathbf{r}_{j}(t)$ at the $N-1$ times $t-\tau_{ij}$. Since $N\gg 1$ and
the times $\tau_{ij}$ are distributed with density $\rho_{\tau}(\tau)$, this
is approximately equal to
$\int_{0}^{\infty}\delta\mathbf{r}_{j}(t-\tau)\rho_{\tau}(\tau)d\tau$.
Similarly,
$\displaystyle\frac{a}{N^{2}}\sum_{i=1}^{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\mathbf{R}(t-\tau_{ij})\approx\frac{a(N-1)}{N}\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho_{\tau}(\tau)d\tau.$
(8)
In a purely heuristic manner, we neglect all fluctuation terms
$\delta\mathbf{r}_{j}(t)$ in the dynamics of the center of mass and obtain the
mean field approximation:
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{\
R}(t)-\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho_{\tau}(\tau)d\tau\right).$ (9)
where we approximated $\frac{N-1}{N}\approx 1$, since we are considering large
numbers of agents.
## IV Bifurcations in the Mean Field Equation
The behavior of the system in the mean field approximation in different
regions of parameter space may be better understood by using bifurcation
analysis. This mathematical technique will allow us to show how the parameter
plane of coupling constant $a$ and mean time delay $\mu_{\tau}$ is divided
into regions with different dynamical behaviors.
First we show that Eq. (9) has a uniformly translating solution
$\mathbf{R}(t)=\mathbf{R}_{0}+\mathbf{V}_{0}\cdot t$, where $\mathbf{R}_{0}$
and $\mathbf{V}_{0}$ are constant, two-dimensional vectors. Inserting the
uniformly translating state into Eq. (9), we get
$\displaystyle 0=$
$\displaystyle\left(1-|\mathbf{V}_{0}|^{2}\right)\mathbf{V}_{0}-a\int_{0}^{\infty}\tau\rho_{\tau}(\tau)d\tau\mathbf{\
V}_{0},$ (10)
since $\int_{0}^{\infty}\rho_{\tau}(\tau)d\tau=1$. Hence, the speed
$|\mathbf{V}_{0}|$ of the uniformly translating state must satisfy
$\displaystyle|\mathbf{V}_{0}|^{2}=1-a\int_{0}^{\infty}\tau\rho_{\tau}(\tau)d\tau=1-a\mu_{\tau},$
(11)
where $\mu_{\tau}$ is the mean of the $\rho_{\tau}$ distribution. We note that
the direction of motion and starting point $\mathbf{R}_{0}$ are arbitrary.
The other state of interest is the stationary state
$\mathbf{R}(t)=\mathbf{R}_{0}$, for an arbitrary constant vector
$\mathbf{R}_{0}$. In the two-parameter space $(a,\mu_{\tau})$, the hyperbola
$a\mu_{\tau}=1$ is in fact a pitchfork bifurcation line on which the uniformly
translating states are born from the stationary state.
The linear stability of the stationary state is determined by the solutions to
the characteristic equation of Eq. (9):
$\displaystyle\mathcal{D}(\lambda)=a\left(1-\int_{0}^{\infty}\rho_{\tau}(\tau)e^{-\lambda\tau}d\tau\right)-\lambda+\lambda^{2}=0,$
(12)
and so involves the Laplace transform of the distribution $\rho_{\tau}$.
In our numerical simulations of system (1), we considered a truncated Gaussian
distribution:
$\displaystyle\rho_{\tau}=\begin{cases}{\cal
N}e^{\frac{(\tau-\tau_{0})^{2}}{2\tau_{1}^{2}}}&\text{if }\tau\geq 0\\\
0&\text{if }\tau<0,\end{cases}$ (13)
where ${\cal N}$ is the normalization constant. Note that because of the
truncation, $\tau_{0}$ and $\tau_{1}$ are only approximately equal to the mean
and standard deviation of $\rho_{\tau}$ and ${\cal N}$ is only approximately
$1/\sqrt{2\pi\tau_{1}^{2}}$.
We approximate the Laplace transform of the truncated Gaussian distribution by
extending the integration range to the whole real line and taking ${\cal
N}\approx\sqrt{2\pi\tau_{1}^{2}}$. In addition, we approximate the mean and
standard deviation of $\rho_{\tau}$ as $\mu_{\tau}\approx\tau_{0}$ and
$\sigma_{\tau}\approx\tau_{1}$, respectively. The result is
$\displaystyle\int_{0}^{\infty}\rho_{\tau}(\tau)e^{-\lambda\tau}d\tau\approx
e^{\lambda\mu_{\tau}+\lambda^{2}\sigma_{\tau}^{2}/2}.$ (14)
We use the above approximation to the Laplace transform of $\rho_{\tau}$ to
search for Hopf bifurcation curves in the $(a,\ \mu_{\tau})$ plane, by taking
$\lambda=i\omega$ in the characteristic equation (12). The equation
$\mathcal{D}(i\omega)=0$ is equivalent to:
$\displaystyle
a-\omega^{2}-i\omega=ae^{i\omega\mu_{\tau}-\omega^{2}\sigma_{\tau}^{2}/2},$
(15)
from which we obtain the Hopf curves parameterized by $\omega$:
$\displaystyle a_{H}(\omega)=$
$\displaystyle\frac{\omega\left(\omega\pm\sqrt{e^{-\sigma_{\tau}^{2}\omega^{2}}(1+\omega^{2})-1}\right)}{1-e^{-\sigma_{\tau}^{2}\omega^{2}}},$
(16a) $\displaystyle\mu_{\tau H}(\omega)=$
$\displaystyle\frac{1}{\omega}\left(\arctan\left(\frac{\omega}{a_{H}(\omega)-\omega^{2}}\right)+2n\pi\right),\
n=0,1,\ldots.$ (16b)
In the above expression for $\mu_{\tau H}(\omega)$, the branch of tan in
$(0,\pi)$ should be used, since the complex number on the left hand side of
Eq. (15) is always on the top half plane. This family of Hopf curves labeled
by $n$, together with the pitchfork bifurcation curve $a\mu_{\tau}=1$ are
shown in Figure 1, for various values of $\sigma_{\tau}$.
Figure 1: Hopf (blue) and pitchfork (red) branches in $a$ and $\mu_{\tau}$
space. The standard deviations of the time-delay distribution $\rho_{\tau}$
for the panels (a) through (d) are 0, 0.2, 0.4 and 0.6, respectively. Note the
change of scale in the abscissae.
When $\sigma_{\tau}=0$, the system exhibits a degenerate point at $a=1/2$,
$\mu_{\tau}=2$ (Fig. 1), where the Hopf bifurcation frequency becomes zero.
This is similar, but not equivalent to a Bogdanov-Takens bifurcation as is
known from previous work [31, 32]. Since the point on the Hopf curve in a two-
parameter bifurcation plane occurs when the Hopf frequency becomes zero, we
define this point as a Zero Frequency Hopf (ZFH) point.
For $\sigma_{\tau}>0$, this ZFH point shifts and a second ZFH point appears at
$a\rightarrow\infty$ and $\mu_{\tau}\rightarrow 0$. The location of the two
ZFH points in the $(a,\ \mu_{\tau})$ plane is given by
$(a_{{ZFH}}^{(\pm)},1/a_{{ZFH}}^{(\pm)})$, where
$a_{{ZFH}}^{(\pm)}=\frac{1}{\sigma_{\tau}^{2}}(1\pm\sqrt{1-\sigma_{\tau}^{2}})$.
When $\sigma_{\tau}=0$, the behavior of the mean field in the vicinity of the
ZFH point is relatively well understood [32, 31] and is as follows (see Fig.
1). In the region between the pitchfork and the first member of the Hopf
family, the stationary state is stable. A simulation of the full system (1)
with parameters in this area reveals that indeed the center of mass of the
agents comes to rest as time progresses and the particles spread themselves
along a ring with radius $1/\sqrt{a}$. Roughly half of the particles move
clockwise and the other half counterclockwise. Along the first Hopf curve, a
stable limit cycle is born and the center of mass begins to oscillate
periodically on a circular orbit. Below the pitchfork bifurcation curve
$a\mu_{\tau}=1$, the translating state is stable. Finally, we mention that
there is a region of bistability in the parameter region above the ZFH point
$(1/2,\ 2)$ between the pitchfork curve $a\mu_{\tau}=1$ and the curve
$a\mu_{\tau}^{2}=2$ (not shown), where the center of mass can either translate
or rotate. On the curve $a\mu_{\tau}^{2}=2$ there is a global bifurcation
where the radius of the orbit diverges and the limit cycle disappears.
The above discussion helps us understand the bifurcation planes in Figs. 1
through 1. Most significantly, we see that the parameter region where the
stationary state is stable decreases in size as the width of the time delay
distribution widens. Hence the system has a higher tendency to behave in an
oscillatory manner for wider time delay distributions. This effect has been
corroborated in numerical simulations (results not shown).
## V Numerical Simulations
We analyze the dynamics of system (1) by solving the system of DDEs
numerically. We use Heun’s method together with quadratic Lagrange
interpolation to evaluate the time-delayed terms of Eqs. (1). Overall, the
numerical method is second order with respect to the step-size $\Delta t$. For
all simulations we take the agents to be uniformly distributed in a random
fashion within the unit box $0\leq x\leq 1$ and $0\leq y\leq 1$, and each
particle is initially at rest $\mathbf{v}_{j}=0$. Moreover, since we are
interested in investigating the time-asymptotic behavior, for all numerical
experiments the time of integration is long enough to allow transients to
decay.
In [31, 32] it was shown that for the parameter set $a=2$, $\tau=2$ (fixed
delay) that the system exhibited a bistable set of solutions. In the rotating
state solution, all particles collapse to a point and that cluster of
particles rotates around a fixed center in a circular orbit. The other
possible stable solution is a ring state, in which all particles distribute
themselves uniformly along a circle and orbit around its center at unit speed.
Interestingly, not all particles traverse the ring in the same direction;
roughly half move clockwise and half move anti-clockwise. We will now examine
these two states, but with random delays given by the truncated Gaussian
distribution in Eq. (13).
Figure 2: Two stable attractors for the swarm dynamics. Here $a=2$,
$\mu_{\tau}=2$, and $\sigma_{\tau}=0.15$. The number of particles is set to be
$N=150$. The final state shown for both simulations is $t=300$. Panel (a)
depicts the rotating state at three snapshots at times $t=$297.6, 298.8, 300,
in red, green and blue, respectively. Panel (b) depicts the ring state.
Figure 2 shows the two final particle distributions after transients in a
simulation with an initial state of $N=150$ randomly placed particles, and
where $\mu_{\tau}=2$ and $\sigma_{\tau}=0.15$. In this case, depending upon
the random selection of delays, either stable solution (ring or rotating) is
possible. To understand the effects of increasing the standard deviation of
the random delays, we use a Monte Carlo method. At 100 different values of
$\sigma_{\tau}$ in the range $0\leq\sigma_{\tau}\leq 0.5$, we generate random
time delays from the distribution in Eq. (13), we then simulate the system
starting from the same initial condition and we determine what state is
acquired by the swarm in the long-time limit. To determine this, we first
measure the time-averaged distance of particle $j$ to the center of mass over
the interval $(t_{1},\ t_{2})$:
$\langle\delta\mathbf{r}_{j}\rangle_{(t_{1},t_{2})}=\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}|\delta\mathbf{r}_{j}(t)|dt$
(17)
where the size of the interval $(t_{1},\ t_{2})$ is long enough to include
several periods of oscillation. The ensemble average of Eq. (17) is then
$\langle\delta\mathbf{r}\rangle_{(t_{1},t_{2})}=\frac{1}{N}\sum_{j=1}^{N}\langle\delta\mathbf{r}_{j}\rangle_{(t_{1},t_{2})}.$
(18)
A value $\langle\delta\mathbf{r}\rangle_{(t_{1},t_{2})}\sim 1/\sqrt{a}$ will
indicate111When the delays are uniform, the ring state has a radius of
$1/\sqrt{a}$ [32]. that the system has converged to the ring state, while
$\langle\delta\mathbf{r}\rangle_{(t_{1},t_{2})}\ll 1/\sqrt{a}$ shows that the
rotating state has been adopted instead222This is true for the range of values
of $\sigma_{\tau}$ considered here..
Figure 3 demonstrates the effect of increasing $\sigma_{\tau}$ on the final
state. The blue circles show that for $\sigma_{\tau}$ small, this initial
condition converges to the rotating state. However, for $\sigma_{\tau}\gtrsim
0.2$ the same initial collection of particles will converge to the ring state
with high probability. In between, there is a transition region where both
states are commonly observed; the state that occurs depends on the random
choice of time delays. The black dashed lines of 3 show two simulations, one
which starts near the rotating state (the lower curve), and one which starts
near the ring state (the upper curve) as $\sigma_{\tau}$ is increased. These
curves demonstrate the stability of these steady states, and the effect of
random delays near these states.
Figure 3 shows the conditional probability of ending up in the ring state as a
function of $\sigma_{\tau}$. As expected, for this choice of initial
conditions, for $\sigma_{\tau}$ small enough, there is zero probability of
leaving the rotating state; however, as $\sigma_{\tau}$ is increased, the
probability increases to one.
The results of these numerical studies strongly suggest that even though there
is bi-stability between the ring and the rotating states, the size of their
respective basins of attraction is changing dramatically as the standard
deviation $\sigma_{\tau}$ increases.
Figure 3: As $\sigma_{\tau}$ is increased, we see a bifurcation from the
stable rotating state. Panel (a) captures the transition from the rotating
state to the ring state as the standard deviation of the random delay
increases. Panel (b) shows the probability of converging to the ring state for
a given $\sigma_{\tau}$ of the delays. These results were compiled using a
Monte Carlo simulation with 100 random distributions of delays for 100
uniformly-spaced values of $\sigma_{\tau}$ and for $N=50$ particles. See
accompanying online movie and Appendix to see the agents converge to each
stable pattern.
## VI Discussion
In this paper we studied the dynamics of a self-propelling swarm with time-
delayed inter-agent attraction. In contrast to the previously considered case
of uniform time delay across agents, we considered the situation in which the
time delay between every pair of agents is drawn randomly from a distribution
$\rho_{\tau}$.
Using a mean-field model of the swarm, we showed how the two parameter
bifurcation plane of coupling strength and mean time delay changes with
respect to the case in which all time delays are equal. The full implications
of these bifurcation results are the subject of our ongoing work. In
particular, it is unclear what the stable solutions are. Nevertheless, the
dramatic changes seen in the two parameter bifurcation plane as the standard
deviation $\sigma_{\tau}$ increases suggest that the basins of attraction of
each attractor undergo big changes as well.
Our numerical experiments show that the swarm displays bi-stable behavior
between the ring and rotating states, at the parameters considered.
Interestingly, however, our work suggests that the basin of attraction of the
ring state greatly expands as the distribution of time-delays $\rho_{\tau}$
widens. Thus, in a sense, widening the distribution of time-delays stabilizes
the stationary state of the swarm center of mass.
Even though in our model the attractive force among agents is linear, we
believe this work is useful since it represents a first approximation for
other, more general forms of attractive interaction. Here, we have limited our
focus to the case where the delays between agents are symmetric and constant.
However, one important generalization of this system involves incorporating
time dependent delays, including those which vary as a function of the
distance between the two agents. This particular refinement of our model is
the subject of ongoing work and beyond the scope of the current paper.
Finally, although we did not consider repulsion between agents, preliminary
research leads us to believe that the patterns observed in this investigation
persist when the characteristic repulsion strength between robots is small
compared to global attraction parameters. For these reasons, our results
indicate how to exploit time-delayed actuation when designing swarm robotic
systems with desired tasks and functionalities.
## VII Appendix-Video Description
The purpose of this research is to investigate the effects of randomized
communication delay on emerging patterns in swarming dynamics. This short
video captures the transition between two different stable patterns for a
swarm as a function of the standard distribution of the delays.
The two coordinate axes in the video show a scatter plot of the positions of
the particles animated in time. The initial positions are identically randomly
distributed particles in the unit box. The temporal state of the swarm is
updated in time using a numerical scheme called Heun’s Method, and a snapshot
is captured at every discrete time interval. Here, the left coordinate axis
uses a standard deviation of the delays $\sigma_{\tau}=0.1$ while the right
axis uses a standard deviation of $\sigma_{\tau}=0.3$. The mean delay for each
simulation is set to be $\mu_{\tau}=2$, and the number of particles for both
is $N=50$. The vectors at each particle give the velocity associated with that
particle. The two simulations are run side by side to demonstrate the dynamics
involved in converging to the “rotating” final state on the left, and the
“ring” final state on the right. The video demonstrates the dynamics over the
time interval from $t=0$ to $t=45$, and so includes transients.
## VIII ACKNOWLEDGMENTS
The authors gratefully acknowledge the Office of Naval Research for their
support. LMR and IBS are supported by Award Number R01GM090204 from the
National Institute Of General Medical Sciences. The content is solely the
responsibility of the authors and does not necessarily represent the official
views of the National Institute Of General Medical Sciences or the National
Institutes of Health. E.F. is supported by the Naval Research Laboratory
(Award No. N0017310-2-C007).
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|
arxiv-papers
| 2012-04-20T12:45:56 |
2024-09-04T02:49:29.978896
|
{
"license": "Public Domain",
"authors": "Brandon Lindley and Luis Mier-y-Teran-Romero and Ira B. Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1204.4606"
}
|
1204.4805
|
11institutetext: Chemoinformatics and Metabolism,
European Bioinformatics Institute, Cambridge, UK 22institutetext: Swiss Center
for Affective Sciences, University of Geneva, Switzerland 33institutetext:
Informatics, Royal Society of Chemistry, Cambridge, UK 44institutetext:
National Institute of Standards and Technology, Gaithersburg, MD, USA
55institutetext: University of Maryland Baltimore County, MD, USA
# What’s in an ‘is about’ link? Chemical diagrams and the Information Artifact
Ontology
Janna Hastings To whom correspondence should be addressed, email:
hastings@ebi.ac.uk1122 Colin Batchelor 33 Fabian Neuhaus 4455
Christoph Steinbeck 11
###### Abstract
The Information Artifact Ontology is an ontology in the domain of information
entities. Core to the definition of what it is to be an information entity is
the claim that an information entity must be ‘about’ something, which is
encoded in an axiom expressing that all information entities are about some
entity. This axiom comes into conflict with ontological realism, since many
information entities seem to be about non-existing entities, such as
hypothetical molecules. We discuss this problem in the context of diagrams of
molecules, a kind of information entity pervasively used throughout
computational chemistry. We then propose a solution that recognizes that
information entities such as diagrams are expressions of diagrammatic
languages. In so doing, we not only address the problem of classifying
diagrams that seem to be about non-existing entities but also allow a more
sophisticated categorisation of information entities.
## Introduction
As the importance of ontology in biomedicine grows, the attention of
ontologists is being pressed to the tasks of disambiguation of domain
terminology and clarification of underlying hierarchies and relationships in
an ever-wider network of interrelated domains [2, 10]. Some issues are
emerging as similarly problematic in many of these different domains. One such
is the clear definition and distinction of foundational types such as
processes and dispositions [1]. Another is the confusion between information
entities, such as computer simulations, models and diagrams, and the entities
that they are models and diagrams of. It is to this latter problem that we
turn in this paper.
Chemical graphs are the molecular models that are used throughout chemistry to
succinctly describe chemical entities and allow for computational
manipulations [12, 6]. Chemical graphs are typically depicted graphically as
schematic illustrations – chemical diagrams. Chemical graphs and chemical
diagrams are examples of information entities in the chemical domain, and
their use has become so pervasive that language used by chemists to refer to
chemicals regularly interchanges words for information (such as ‘graph’) with
words for actual chemicals [6].
The Information Artifact Ontology (IAO) [8] is an ontology being developed for
the domain of information entities of relevance in biomedicine. The
fundamental criterion by which information entities are defined and
categorised in the IAO is their aboutness, that is, the types of entities that
they are about. A diagram illustrating the chemical structure of caffeine
molecules, for example, is about the class of caffeine molecules. While in
this case the chemical diagram corresponds to something in reality (caffeine
molecules), there are many other useful and scientifically relevant chemical
diagrams that are not about something that exists. Thus, these chemical graphs
are not information entities as currently defined in IAO. A similar scenario
applies to many other models used in biomedicine, for example pathway diagrams
and the mathematical models used in quantitative systems biology. Using
chemical diagrams as examples, we will argue that information entities in IAO
are defined too narrowly. Since information entities may not necessarily be
about something, they cannot be categorized merely by what they are about.
But, as we will argue, they should rather be categorised by what sort of
information entities they are in their own right.
The remainder of this paper proceeds as follows. In the next section we
briefly describe the IAO and the theory of chemical graphs and their related
diagrams. Thereafter, we highlight the insufficiency of aboutness in defining
types of diagrams. We go on to introduce some semantics for the representation
relationship between chemical diagrams and chemical entities; and finally, we
propose a modified approach to information ontology that is free of the
problems with the current approach.
## 1 Background
### 1.1 The Information Artifact Ontology
The Information Artifact Ontology (IAO) [8] is an ontology of information
entities being developed in the context of the Open Biological and Biomedical
Ontologies (OBO) Foundry [9], beneath the upper level ontology Basic Formal
Ontology (BFO) [11, 5]. Within this context, information entities are defined
as:
###### Definition 1
An information content entity (_ICE_) is an entity that is generically
dependent on some artifact and stands in the relation of _aboutness_ to some
entity.
The generic dependence on an artifact (i.e., a human creation) in the above
definition restricts the scope of the domain to human-created information
entities. The ‘generic’ part of the dependence captures the intuition that
information can be copied, that is, reproduced in multiple bearers, in a way
that hair colour, for example, cannot. The textual definition also refers to a
relation of ‘aboutness’, which is further supplemented by the axiom:
ICE subClassOf is about some Entity (1)
The above is given in the Manchester Web Ontology Language (OWL) syntax, in
which the existential quantification ($\exists$) is expressed using the infix
some operator. This should not, however, obscure the strong existential
dependency claimed, namely: for every ICE, there exists some entity to which
the ICE is related by the is about relationship.
A hierarchical overview of the IAO together with some examples of information
content entities (ICEs) is illustrated in Figure 1.
Figure 1: An overview of the Information Artifact Ontology
### 1.2 Chemical graphs and diagrams
The principal object of graph theory is a graph, which consists of a set of
objects and the binary relations between them. Graph theory has found many
applications in chemistry and is used to represent molecular entities through
the molecular graph. These graphs represent the constitution of a molecule in
terms of nodes (usually atoms, but in some cases groups of atoms) and edges
(chemical bonds) [12].
For the purposes of this paper we define chemical graphs as follows111We
ignore additional complexity such as the representation of stereochemistry..
###### Definition 2
A _chemical graph_ , denoted _CG_ , is a tuple $(V,E)$ in which each vertex
$i\in V$ corresponds to an atom in a molecule; and each undirected edge
$\\{i,j\\}\in E$ corresponds to a chemical bond between the atoms $i$ and $j$.
These CGs are based on the valence bond model of quantum mechanics [7]. For
many of the molecules most relevant to the pharmaceutical industry this model
reasonably accurately represents (1) by atoms, those portions of the molecules
that chemists associate with particular atoms, and (2) by bonds, those
portions of the molecules that have high electron probability density.
Cheminformatics software uses these to make useful predictions about the
chemical properties of a molecule so represented and the physical properties
of an ensemble of those molecules. They also enable the schematic
representation of molecules in diagrams.
###### Definition 3
A _chemical diagram_ , denoted _CD_ , is a diagrammatic illustration of the
information encoded in a _CG_ , which follows an agreed _diagrammatic syntax_
for the representation of the graph information.
Some examples of CDs are illustrated in Figure 2. In the 2D wireframe
depiction, the diagrammatic syntax used specifies that the CD corresponds to
the CG in that, for each edge $\\{i,j\\}\in E$ there is a corresponding line,
and for each vertex $i\in V$ there is a corresponding corner or line ending in
the CD. In the 3D ball and stick diagram, edges are illustrated with lines
while vertices are illustrated with coloured, labelled spheres. In the 3D
spacefill diagram, vertices are illustrated with large coloured spheres. Both
the colours and the radii of the sphere are arbitrary—atoms are much too small
to have colours, but the radii are based on experimental averages and are an
approximation to the actual molecular structure.
Notice that there is not a one-to-one correspondence between CDs and CGs,
since the same CG can be illustrated in many different CDs, obeying different
syntaxes.
Figure 2: Some examples of CDs for the molecule caffeine
CDs, like maps, represent spatial information. Let us call spatial
representations such as street maps, chemical diagrams, and engineering design
models structural diagrams and, to a first approximation, assume that they
have a direct structural association with a portion of reality, which they are
intended to represent.
###### Definition 4
A structural diagram (_SD_) is a diagrammatic representation of spatial
aspects, such as position, topology and connection, of a structured portion of
reality.
This definition, however, does not suffice, for reasons that will be described
in the following section.
## 2 When ‘is about’ isn’t enough
The agreed syntax of CDs allows their informational content to be reliably
understood by all members of the community who use them for exchange of such
information.
The agreed syntax also allows for the depiction of molecules, which are
1. 1.
Planned, in that the representation is used as a precursor to a synthesis
procedure expected to produce a corresponding molecule instance.
2. 2.
Hypothesised, in that the representation corresponds to a molecule class for
which it is not known whether corresponding instances exist.
3. 3.
Chemically infeasible, in that it is known that the representation illustrates
a class of molecules for which no instances can exist for a measurable
duration of time under normal conditions.
4. 4.
Impossible, in that the representation cannot be the structure of any molecule
instances, since it violates the rules of molecular compositionality.
In the first two cases the CD might or might not be about molecules that
exist. In the third case chemists expect, and in the fourth case they are
certain that, the aboutness criterion of the IAO is violated. Nevertheless,
these CDs are used by chemists to communicate and exchange information in the
same ways as CDs that are known to correspond to something in reality. Thus,
the way CDs are used does not justify treating only a subset of them as
information entities. It also indicates that Definition 4 is not along the
right lines.
A conceptualist resolution to this issue might defend a view of ontology as
containing representations of concepts, and thereby not be required to
differentiate between chemical diagrams for real or impossible molecules, or
differentiate at the level of metadata only [4]. However, this seems to
overlook the fundamental distinction between these cases, one that chemists
recognise. Another strategy for addressing this problem is provided by
Ceusters and Smith [3] who distinguish between referring and non-referring
representational units in the context of a mental representation. The
application of this distinction to an ontology of SDs beneath IAO is
illustrated in Figure 3.
Figure 3: Referring and non-referring information entities in the IAO
One obvious problem with this approach is that it leads to a massive level of
parallel maintenance, since most types of ICE can appear twice in the
ontology. A more fundamental objection is that this approach violates the
fundamental design principles of BFO: categorization according to ontological
nature, which does not change. For example, it is impossible for a tree (an
independent continuant) to become a temporal region, or for a smile (a
dependent continuant) to become a soccer game (an occurrent). However,
according to the approach in [3] a CD might be a non-referring ICE now, but
become a referring ICE tomorrow, because somewhere in some lab somebody
accidentally synthesized the corresponding molecule. Thus, in contrast to the
other ontological categories in BFO, it would be possible for non-referring
ICEs to change their ontological nature. Even worse, the ontological nature of
CDs would be affected by events that had no causal connection to the CD and
did not change its structure in any way. Since the ontological nature of an
entity is not affected by Cambridge changes, that is to say changes only in
its description, we conclude that ‘non-referring ICE’ and ‘referring ICE’ are
not true ontological categories.
In summary, we agree with Ceusters and Smith that non-referring ICEs are ICEs.
However, we reject the idea that the distinction between referring and non-
referring should be the primary basis for classifying ICEs. There are some
ICEs that are necessarily about something (e.g., photographs). But structural
diagrams are information entities in virtue of the fact that they are well-
formed expressions in a diagrammatic language. For each type of SD, there is a
vocabulary (the symbols and icons that are used in diagrams of that type), a
grammar that regulates how the elements of the vocabulary can be combined, and
compositionality in the sense that the semantics of a complex expression is
determined by the semantics of its components and the way these components are
arranged.
The elements of the vocabulary of the diagrammatic language do need to
correspond to something existing, otherwise the diagrams will not be
scientifically relevant. However, not all combinations of the vocabulary that
are permissible by the grammar will correspond to something in reality. It
would seem strange indeed, on giving an ontological account of natural
language, to divide all sentences into those that are about facts and those
that are not. “Submariners love periscopes.” is a declarative sentence with a
transitive verb regardless of whether it is a fact that submariners love
periscopes. The same is true for expressions of diagrammatic languages.
## 3 The ontology of structural diagrams
Different types of CD (such as 2D wireframe, 3D ball and stick) obey different
diagrammatic syntaxes. What is essential to distinguish different types of
diagrams is thus to provide a definition for these syntaxes.
###### Definition 5
A _diagrammatic language_ $L_{D}=\langle V,G\rangle$ is an ordered pair that
consists of the vocabulary $V$ (a set of icons and symbols) and a syntax $G$
of composition rules.
###### Definition 6
An _interpreted diagrammatic language_ is a quadruple $IL_{D}=\langle
V,G,T,\phi\rangle$ such that $\langle V,G\rangle$ is a diagrammatic language,
$T$ is a set of types that is partitioned set of independent continuants $IT$
and dependent continuants $DT$, and $\phi$ is a function that maps the
elements from $V$ onto $T$.
###### Definition 7
Let $IL_{D}$ be an intepreted diagrammatic language as above, and let $D$ be a
well-formed expression in $L_{D}$ (i.e., a diagram). $D$ is a _structural
diagram_ that is about an entity $x$ iff there is some injective
interpretation function $\iota$ such that:
* •
for each element of $V$ and each token $t$ of $V$ that is part of $D$,
$\iota(t)$ is an instance of $\phi(V)$
* •
for two tokens $t_{1}$, $t_{2}$ that are part of $D$ and $\iota(t_{1})$,
$\iota(t_{2})$ are instances of elements of $IT$: $t_{1}$ is connected to
$t_{2}$ iff $\iota(t_{1})$ is connected to $\iota(t_{2})$
* •
for all tokens $t$, $t_{1}$, … $t_{n}$: if $\iota(t)$ is an instance of some
element of $DT$ and $t_{1}$ … $t_{n}$ are all connected to $t$, then
$\iota(t)$ inheres in $\iota(t_{1})$ … $\iota(t_{n})$.
* •
there is no part $y$ of $x$ such that $y$ is an instance of some type in $T$
and for all $t$ that are part of $D$ there is no $\iota(t)=y$.
Chemical diagrams of hypothetical molecules that do not exist are not about
anything, but they are still well-formed expressions of an interpreted
diagrammatic language. For example, the vocabulary $V$ of the 3D ball and
stick language consists of colored spheres and lines. The syntax $G$ describes
how these elements can be combined to diagrams. The set $IT$ consists of types
of atoms, the set $DP$ consists of the types of chemical bonds that connect
atoms within a molecule. The function $\phi$ maps the color-coded balls to
types of atoms and the links to types of bonds. The second diagram in Figure 2
is a structural diagram of a given instance of a caffeine molecule $x$, since
it is possible to map the spheres of the diagram to the atoms that are part of
$x$ and the links of the diagram to the chemical bonds of $x$ such that the
connections in the diagrams corresponds to the chemical reality in the
molecule. Conversely, if the diagram contains a link that does not correspond
to a bond in a given molecule $x$ or if it contains a sphere that is mapped to
a type of atoms that do not occur as part of $x$, then the diagram does not
represent $x$.222The second clause of definition 7 is irrelevant in the case
of CDs, because in CDs tokens of symbols for independent continuants (the
atoms) are always connected by tokens of symbols for dependent continuants
(the bonds). However, definition 7 is also intended to be applicable to
diagrams where symbols for independent continuants might be connected
directly; for example architectural drawings and engineering blueprints.
To place SDs (and therefore CDs) as subtypes of IAO’s ICE, we need to change
the fundamental aboutness criterion from Equation (1) to a value rather than
existential restriction:
ICE subClassOf is about only Entity (2)
This restriction no longer expresses an existential dependence. Rather, it now
has the effect that if there is some entity that the ICE is about, then it
must be of the required type to avoid a logical inconsitency. Note that this
formula expresses a schema, which will be made more precise for different
types of ICE. With the inclusion of conforms to axioms to relate the ICE to
the $L_{D}$, we are now in a position to provide a better definition for SDs
and CDs to replace Definition 4:
SD subClassOf ICE and is about only StructuredEntity and conforms to some
DiagrammaticLanguage CD subClassOf SD and is about only MolecularEntity
We can safely include in the resulting ontology, illustrated in Figure 4,
diagrams of planned, hypothetical, infeasible, and impossible molecules.
Figure 4: The ontology of chemical diagrams with distinctions for different
syntaxes
Now, we can define different types of chemical diagrams regardless of their
aboutness, and furthermore express the difference between different types of
diagrams that are about the same entity (such as 2D and 3D diagrams of
caffeine molecules). However, we can go one step further and define a
relationship between 2D and 3D depictions of the same molecule.
###### Definition 8
Let $L_{1}$, $L_{2}$ be two interpreted diagrammatic languages. Let
$\Theta_{1}$ be a non-empty set of all well-formed expressions of $L_{1}$,
such that there is at least one diagram $D$ in $\Theta_{1}$ and one entity
$x$, such that $D$ is about $x$ in $L_{1}$. Let $\Theta_{2}$ be a non-empty
set of all well-formed expressions of $L_{2}$, such that there is at least one
diagram $D$ in $\Theta_{2}$ and one entity $x$, such that $D$ is about $x$ in
$L_{2}$.
The function $m$ is a _coarsening_ from $\Theta_{1}$ (in $L_{1}$) to
$\Theta_{2}$ (in $L_{2}$) iff
* •
$m$ is a function from $\Theta_{1}$ onto $\Theta_{2}$; and
* •
for all diagrams $D$ in $\Theta_{1}$ and all entities $x$: if $D$ is about $x$
in $L_{1}$, then $m$($D$) is about $x$ in $L_{2}$; and
* •
for all diagrams $D_{2}$ in $\Theta_{2}$ and all entities $x$: if $D_{2}$ is
about $x$ in $L_{2}$, then there is a diagram $D$ such that $D$ is about $x$
and $m$($D$) = $D_{2}$.
Coarsening functions map between two different diagrammatic languages, such
that if a diagram in one language represents an entity, then it is possible to
construct a diagram in the other language that also represents the entity.
Typically, coarsening functions are directed from a greater to a lesser level
of detail; that is, it is possible to map diagrams in a more detailed language
to a diagram in a coarser language, but not the reverse. Coarsening functions
allow us to define a relationship coarser than between SDs.
###### Definition 9
Let $D_{1}$ and $D_{2}$ be diagrams conforming to languages $L_{1}$ and
$L_{2}$, respectively. $D_{2}$ is _coarser than_ $D_{1}$ iff
* •
there exists a function $m$ and sets of diagrams $\Theta_{1}$, $\Theta_{2}$ of
$L_{1}$ and $L_{2}$, respectively, such that $m$ is a coarsening from
$\Theta_{1}$ (in $L_{1}$) to $\Theta_{2}$ (in $L_{2}$) and $m$($D_{1}$) =
$D_{2}$; and
* •
there is no function $m^{\prime}$ such that $m^{\prime}$ is a coarsening from
$D_{2}$ (in $L_{2}$) to $D_{1}$ (in $L_{1}$).
This is illustrated in Figure 5.
Figure 5: Some examples of chemical diagrams and their relationships
## 4 Conclusion
We have argued that the is about relationship is not enough to define CDs, for
two reasons. Firstly, given the possibility of having several different CDs
corresponding to the same molecule, we see that distinguishing between
different types of diagrams, which obey different representational syntaxes,
is not possible using only distinctions in what the diagram is about.
Secondly, a challenge is posed in that CDs may be used validly to illustrate
classes of molecules for which no instances exist. The existential dependency
expressed in IAO means that the IAO cannot, in its present form, allow for the
inclusion of such non-referring information entities.
We evaluated an approach based on parallel maintenance of IAO hierarchies with
differing is about commitment. While such parallel maintenance may be a
scientifically-valid strategy in some scenarios, it is unable to express the
fact that the same representational formalism (i.e., diagrammatic syntax) is
used across the hierarchies. Of course, the diagrammatic syntax, if it is to
be scientifically valid, must typically represent entities which do exist. But
the syntax allows for compositionality and it would be absurd to require the
existence of instances for all the complex expressions obtained by composing
the elements of the representational vocabulary.
We therefore propose the definition of structural diagrams such as chemical
diagrams based on their syntaxes. Any diagram expressed in an interpreted
diagrammatic syntax is a valid information content entity regardless of the
existence of instances that the diagram is about; although the existence of
such an instance may be an interesting property depending on the application
scenario.
## References
* [1] Batchelor, C., Hastings, J., Steinbeck, C.: Ontological dependence, dispositions and institutional reality in chemistry. In: Galton, A., Mizoguchi, R. (eds.) Proceedings of the 6th Formal Ontology in Information Systems conference. Toronto, Canada (2010)
* [2] Bodenreider, O., Stevens, R.: Bio-ontologies: current trends and future directions. Briefings in Bioinformatics 7(3), 256–274 (2006)
* [3] Ceusters, W., Smith, B.: Foundations for a realist ontology of mental disease. Journal of Biomedical Semantics 1(1), 10 (2010)
* [4] Dumontier, M., Hoehndorf, R.: Scientific realism. In: Galton, A., Mizoguchi, R. (eds.) Proceedings of the 6th Formal Ontology in Information Systems conference. Toronto, Canada (2010)
* [5] Grenon, P., Smith, B., Goldberg, L.: Biodynamic ontology: Applying BFO in the biomedical domain. In: Stud. Health Technol. Inform. pp. 20–38. IOS Press (2004)
* [6] Hastings, J., Batchelor, C., Steinbeck, C., Schulz, S.: What are chemical structures and their relations? In: Galton, A., Mizoguchi, R. (eds.) Proceedings of the 6th Formal Ontology in Information Systems conference. Toronto, Canada (2010)
* [7] Pauling, L.: The shared-electron chemical bond. Proc. Natl. Acad. Sci. USA 14, 359–362 (1928)
* [8] Ruttenburg, A., Courtot, M., The IAO Community: The Information Artifact Ontology (2010), http://code.google.com/p/information-artifact-ontology/
* [9] Smith, B., Ashburner, M., Rosse, C., Bard, J., Bug, W., Ceusters, W., Goldberg, L.J., Eilbeck, K., Ireland, A., Mungall, C.J., The OBI Consortium, Leontis, N., Rocca-Serra, P., Ruttenberg, A., Sansone, S.A., Scheuermann, R.H., Shah, N., Whetzel, P.L., Lewis, S.: The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration. Nat Biotechnol 25(11), 1251–1255 (Nov 2007)
* [10] Smith, B., Ceusters, W.: Ontological realism as a methodology for coordinated evolution of scientific ontologies. Applied Ontology 5, 139–188 (2010)
* [11] Smith, B., Grenon, P.: The cornucopia of formal ontological relations. Dialectica 58, 279–296 (2004)
* [12] Trinajstic, N.: Chemical graph theory. CRC Press, Florida, USA (1992)
|
arxiv-papers
| 2012-04-21T12:00:16 |
2024-09-04T02:49:29.991040
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Janna Hastings and Colin Batchelor and Fabian Neuhaus and Christoph\n Steinbeck",
"submitter": "Janna Hastings",
"url": "https://arxiv.org/abs/1204.4805"
}
|
1204.4886
|
# The universal simplicial bundle is a simplicial group
David Michael Roberts School of Mathematical Sciences
University of Adelaide
Adelaide, SA 5005
Australia david.roberts@adelaide.edu.au
###### Abstract.
The classical universal bundle functor $W:s\textbf{{Grp}}(C)\to sC$ for
simplicial groups in a category $C$ with finite products lifts to a monad on
$s\textbf{{Grp}}(C)$. This result extends to simplicial algebras for any
Lawvere theory containing that of groups.
###### Key words and phrases:
simplicial group, universal bundle, Lawvere theory
###### 1991 Mathematics Subject Classification:
18G30 (Primary) 55R65, 55R15 (Secondary)
###### Contents
1. 1 Introduction
2. 2 Results
3. 3 Proof
4. 4 Postscript
## 1\. Introduction
The present note is motivated by two observations. The first, by Segal [8], is
that the total space $EG$ of the universal bundle for a (well-pointed)
topological group $G$ can be chosen to be a topological group. The easiest way
to see this is to pass through the simplicial construction Segal introduced,
which from the group $G$ gives a simplicial topological group. The geometric
realisation of this simplicial group is then a topological group (using the
product in the category of $k$-spaces). Additionally, the group $G$ is a
(closed) subgroup of $EG$ and the quotient is one of the standard
constructions of the classifying space of a topological group.
The second observation, appearing in [6], is that given a strict 2-group $G$
there is a natural construction of a universal bundle $INN(G)$ for $G$ which
is a group-like object in $2\textbf{{Gpd}}$. Again, there is an inclusion of
groups $G\hookrightarrow INN(G)$. This was proved for 2-groups in Set, but
also works for strict 2-groups internal to a finitely complete category.
Given a growing interest in higher gauge theory, derived geometry and higher
topos theory, it is natural to consider a generalisation of these results to
$\infty$-groups, at least in the first instance as presented by simplicial
groups. Because all the constructions involved are very simple, we can work
internal to an arbitrary category $C$ with finite products.
The functor $W:s\textbf{{Grp}}\to s\textbf{{Set}}$, introduced in [2], plays
the role in the simplicial world analogous to that
$E:\textbf{{Grp}}(\textbf{{Top}})\to\textbf{{Top}}$ does for universal bundles
for topological groups. One can easily see that the construction of $W$ works
for simplicial groups in a category $C$ with finite products. The main result
here is that $W$ lifts (up to isomorphism) through the forgetful functor
$s\textbf{{Grp}}(C)\to sC$. Furthermore, not only is this an endofunctor on
$s\textbf{{Grp}}(C)$, it is a monad, with the unit of the monad being the
subgroup inclusion. The result that $WG$ is a group, at least under the
assumption that $C$ has all finite limits, is proved in [7] in a more
conceptual manner.
We can say even more about the status of $WG$ as a universal bundle in
categories other than $s\textbf{{Set}}$, by recent joint work of Nikolaus,
Schreiber and Stevenson [5]. If $\mathcal{C}=Sh_{\infty}(S)$ is the
$\infty$-topos of $\infty$-sheaves on a site $S$ with a terminal object, then
any $\infty$-group object $\mathcal{G}$ in $\mathcal{C}$ is presented by a
simplicial group $G$ in $sSh(C)$, and moreover every principal $\infty$-bundle
is presented by a pullback (in $sSh(C)$) of the universal bundle
$WG\to\overline{W}G$ described here.
For background on simplicial objects and simplicial groups the reader may
consult the classic [4]. We shall describe simplicial objects in Set, using
elements, but all constructions here are possible in a category with finite
products, if we take the definition as using generalised elements.
A remark is perhaps necessary for the history of this result. The main theorem
of this note was proved around the time [6] was written, but the original
version of the notes languished, being referred to in one or two places,
themselves until now unpublished work. Thanks are due to Jim Stasheff for
encouraging a broader distribution. Urs Schreiber and Danny Stevenson made
useful suggestions on a draft.
## 2\. Results
We must first present the definitions of the objects we are considering. To
start with, we have the classical universal bundle for a simplicial group $G$.
###### Definition 1.
The _universal $G$-bundle $WG$_ has as its set of $n$-simplices
$(WG)_{n}=G_{n}\times\ldots\times G_{0}$
and face and degeneracy operators
$\displaystyle d_{0}(g_{n},\ldots,g_{0})$
$\displaystyle=(d_{0}g_{n}g_{n-1},g_{n-2},\ldots,g_{0}),$ $\displaystyle
d_{i}(g_{n},\ldots,g_{0})$
$\displaystyle=(d_{i}g_{n},\ldots,d_{1}g_{n-i+1},d_{0}g_{n-i}g_{n-i-1},g_{n-i-2},\ldots,g_{0}),\quad
i>0$ $\displaystyle s_{i}(g_{n},\ldots,g_{0})$
$\displaystyle=(s_{i}g_{n},\ldots,s_{0}g_{n-i},id_{G_{n-i}},g_{n-i-1},\ldots,g_{0})$
The simplicial group $G$ acts (on the left) on $WG$ by multiplication on the
first factor, and the quotient $(WG)/G$ is denoted $\overline{W}G$. We will
not need a detailed description for the present purposes, we only need to note
that this quotient exists even if we consider simplicial objects internal to
other cateories $C$ without assuming existence of colimits.
We now define a simplicial group $W_{gr}G$ for any simplicial group $G$.
###### Definition 2.
The set of $n$-simplices of $W_{gr}G$ is given by
$(W_{gr}G)_{n}=G_{n}\times\ldots\times G_{0}.$
The face and degeneracy maps are
$\displaystyle d_{0}(g_{n},\ldots,g_{0})$
$\displaystyle=(g_{n-1},g_{n-2},\ldots,g_{0}),$ $\displaystyle
d_{i}(g_{n},\ldots,g_{0})$
$\displaystyle=(d_{i}g_{n},\ldots,d_{1}g_{n-i+1},g_{n-i-1},\ldots,g_{0}),\quad
i>0$ $\displaystyle s_{i}(g_{n},\ldots,g_{0})$
$\displaystyle=(s_{i}g_{n},\ldots,s_{0}g_{n-i},g_{n-i},g_{n-i-1},\ldots,g_{0}).$
If we let the product on $(W_{gr}G)_{n}$ be componentwise, these face and
degeneracy maps are homomorphisms, because those of $G$ are. $W_{gr}G$ is then
a simplicial group. The construction is clearly functorial.
We state the main result of this note, and then prove it in the following
section after some observations. The last section contains some observations
which are more open-ended.
###### Theorem.
The endofunctor $W_{gr}\colon s\textbf{{Grp}}(C)\to s\textbf{{Grp}}(C)$ is a
lift, up to isomorphism, of the universal bundle functor $W$ through the
forgetful functor $s\textbf{{Grp}}(C)\to sC$. Moreover, $W_{gr}$ is a monad.
One immediate extension of this result is to any Lawvere theory $T$ extending
that of groups.111A Lawvere theory [1] roughly corresponds to any algebraic
structure that can be defined using maps between finite (including nullary)
products, for example monoids, groups, abelian groups, rings and so on. A
Lawvere theory is said to extend the theory of groups if every algebra for
that Lawvere theory has an underlying group. Given such a Lawvere theory,
there is a forgetful functor $T\textrm{-}Alg(C)\to\textbf{{Grp}}(C)$ from the
category of $T$-algebras in a category $C$ to the category of groups in $C$.
Similarly, we can consider simplicial $T$-algebras in $C$ (equivalently,
$T$-algebras in $sC$). We then have a composite functor
$sT\textrm{-}Alg(C)\to s\textbf{{Grp}}(C)\stackrel{{\scriptstyle W}}{{\to}}sC$
which by the theorem lifts to $s\textbf{{Grp}}(C)$. The construction of
$W_{gr}$ is such this composite lifts (on the nose) to a functor
$W_{T}\colon sT\textrm{-}Alg(C)\to sT\textrm{-}Alg(C).$
Given the monad structure maps for $W_{gr}$, one easily sees that $W_{T}$ is
also a monad.
One simple observation which is worth making, given our first motivating fact
(from Segal’s [8]), is that for $C$ some subcategory of Top, $G$ is a _closed_
sub-simplicial group of $W_{gr}G$. Similarly for other algebras in such a $C$
for a more general Lawvere theory.
Finally, given a finite-product-preserving homotopy colimit functor $sC\to C$,
it is clear that the object $\operatorname{hocolim}W_{gr}G\in C$ is actually a
group object (equiv. a $T$-algebra). We thus have come full circle and
recovered Segal’s result described above (using the fact geometric realisation
is a homotopy colimit for a well-pointed simplicial topological group).
## 3\. Proof
Proof of theorem: There is an isomorphism between $(WG)_{n}$ and (the
underlying set of) $(W_{gr}G)_{n}$, given by
$\displaystyle\Phi_{n}\colon(WG)_{n}$ $\displaystyle\to(W_{gr}G)_{n}$
$\displaystyle(g_{n},\ldots,g_{0})$
$\displaystyle\mapsto(k_{n},k_{n-1},\ldots,k_{0}),$
where the $k_{j}$ are defined recursively as
$k_{n}=g_{n},\qquad k_{j-1}=d_{0}k_{j}g_{j-1}\ \ (j<n).$
One can see that the maps $\Phi_{n}$ define a map $\Phi$ of simplicial sets by
the use of the standard identities for the boundary and degeneracy maps for
$G$. One can check the inverse map is
$\displaystyle\Phi^{-1}\colon W_{gr}G$ $\displaystyle\to WG$
$\displaystyle(h_{n},\ldots,h_{0})$
$\displaystyle\mapsto(h_{n},d_{0}h_{n}^{-1}h_{n-1},\ldots,d_{0}h_{1}^{-1}h_{0}).$
Thus $W_{gr}$ is an up-to-isomorphism lift of $W$. Note that since $W_{gr}G$
is isomorphic to $WG$ its underlying simplicial set is contractible.
We use the isomorphism $\Phi$ to see how $G$ includes into $W_{gr}G$:
$\displaystyle G_{n}\hookrightarrow(WG)_{n}$
$\displaystyle\stackrel{{\scriptstyle\Phi}}{{\to}}(W_{gr}G)_{n}$
$\displaystyle g_{n}\mapsto(g_{n},1,\ldots,1)$
$\displaystyle\mapsto(g_{n},d_{0}g_{n},d_{0}^{2}g_{n},\ldots,d_{0}^{n}g_{n})$
Call this homomorphism $\iota_{G}$. As an aside, from this we can see how
$G_{n}$ is closed in $(W_{gr}G)_{n}$ when we are working with a subcategory
$C\hookrightarrow\textbf{{Top}}$, as it is given by the conjunction of the
collection of equations $g_{n-i}=d_{0}^{i}g_{n}$ for $i=0,\ldots,n$.
Since the (left) action of $G$ on $W_{gr}G$ is defined via $\Phi$, it is
trivial to see that $\Phi$ is a $G$-equivariant isomorphism between free
$G$-spaces. This means that $\overline{W}G\simeq(W_{gr}G)/G$; this quotient
therefore exists in all categories $C$ with finite products.
Thus far we have an endofunctor
$W_{gr}\colon s\textbf{{Grp}}(C)\to s\textbf{{Grp}}(C),$
and a natural transformation
$\iota\colon 1_{s\textbf{{Grp}}}\to W_{gr}$
whose component at $G$ is given by the inclusion $\iota_{G}\colon
G\hookrightarrow W_{gr}G$.
We now have to prove that $W$ is a monad. For background, see for example [3],
chapter VI. Notice that
$\displaystyle(W_{gr}^{2}G)_{n}$ $\displaystyle=$
$\displaystyle(W_{gr}G)_{n}\times(W_{gr}G)_{n-1}\times\ldots(W_{gr}G)_{0}$
$\displaystyle=$ $\displaystyle(G_{n}\times\ldots\times
G_{0})\times(G_{n-1}\times\ldots\times G_{0})\times\ldots\times(G_{0}).$
If $\operatorname{pr}_{1}\colon(W_{gr}G)_{j}\to G_{j}$ denotes projection on
the first factor, define the maps
$(\mu_{G})_{n}=\operatorname{pr}_{1}\times\ldots\times\operatorname{pr}_{1}\colon(W_{gr}^{2}G)_{n}\to
G_{n}\times\ldots\times G_{0}=(W_{gr}G)_{n},$
which clearly assemble into a map of simplicial groups
$\mu_{G}\colon W_{gr}^{2}G\to W_{gr}G,$
and these form the components of a natural transformation
$\mu:W_{gr}^{2}\to W_{gr}.$
Now to show that $W_{gr}$ is a monad we need to check that the following
diagrams commute
|
---|---
$\textstyle{W_{gr}^{3}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{gr}(\mu_{G})}$$\scriptstyle{\mu_{W_{gr}G}}$$\textstyle{W_{gr}^{2}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{W_{gr}^{2}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{W_{gr}G}$
and
| | |
---|---|---|---
$\textstyle{W_{gr}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{gr}(\iota_{G})}$$\scriptstyle{=}$$\textstyle{W_{gr}^{2}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
W_{gr}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{W_{gr}G}}$$\scriptstyle{=}$$\textstyle{W_{gr}G}$
This can be done level-wise and is a fairly easy if tedious exercise in
indices. This completes the proof.
## 4\. Postscript
We can prove that $W_{gr}G$ (and more generally $W_{T}A$ for a $T$-algebra
$A$) is contractible in the category of simplicial groups in $C$ (resp. in
$sT\textrm{-}Alg(C)$) directly, rather than showing the underlying simplicial
object is contractible via the isomorphism
$WG\stackrel{{\scriptstyle\sim}}{{\to}}W_{gr}G$. The extra degeneracies
$\displaystyle s_{-1}\colon(W_{gr}G)_{n}$ $\displaystyle\to(W_{gr}G)_{n+1}$
$\displaystyle(g_{n},\ldots,g_{0})$
$\displaystyle\mapsto(1,g_{n},g_{n-1},\ldots,g_{0}),$
give rise to a contracting homotopy in $s\textbf{{Grp}}(C)$ (resp.
$sT\textrm{-}Alg(C)$). This allows us to consider the monad $W_{mon}$ on the
category of simplicial _monoids_ in $C$, which is defined in exactly the same
way as above. $W_{mon}$ lands in the subcategory of _contractible_ simplicial
monoids, but without the inversion operation we cannot display the isomorphism
with the usual construction of $W$ on the category of simplicial monoids. As
such the interpretation of the quotient $(W_{mon}M)/M$ is not straightforward,
especially in the level of generality of this paper, where we do not assume
the existence of colimits. Even when the required quotient exists, it does not
have a nice interpretation as does the universal bundle
$W_{gr}G\to\overline{W}G$.
Finally, we observe that the construction of $W_{mon}$ extends to a monad
$W_{T,m}$ on the category of simplicial $T$-algebras in a finite-product
category $C$ where $T$ is a Lawvere theory containing a specified222That we
need a _specified_ monoid operation is clear by the example of the theory of
associative, unital $k$-algebras for $k$ a field; such $k$-algebras have two
underlying monoids. monoid operation $m$ (i.e. an inclusion $m\colon
Th(monoids)\hookrightarrow T$ of the theory of monoids). This monad takes a
simplicial algebra for such a Lawvere theory and returns a contractible
simplicial $T$-algebra containing the original simplicial $T$-algebra.
## References
* [1] F. W. Lawvere, Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories, PhD thesis Columbia University, 1963. Available in Reprints in Theory and Applications of Categories, No. 5 (2004) pp. 1–121.
* [2] S. MacLane, Constructions Simpliciales Acycliques, Colloque Henri Poincaré, Paris 1954.
* [3] S. MacLane, _Categories for the working mathematician_ , Springer-Verlag (1971)
* [4] J. P. May, _Simplicial objects in algebraic topology_ , Van Nostrand Mathematical Studies, No. 11 (1967) Available from http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu
* [5] T. Nikolaus, U. Schreiber, D. Stevenson, Principal $\infty$-bundles – Presentations, preprint (2012). available from http://ncatlab.org/schreiber/files/bundles_presentation.pdf.
* [6] D. M. Roberts, U. Schreiber, The inner automorphism 3-group of a strict 2-group, J. Homotopy Relat. Struct. vol 3, no. 1 (2008) pp. 193–245. Available from http://arxiv.org/abs/0708.1741.
* [7] D. M. Roberts, D. Stevenson, Simplicial principal bundles in parameterized spaces, preprint (2012). Available from http://arxiv.org/abs/1203.2460.
* [8] G. Segal, Classifying spaces and spectral sequences, Pub. Math. IHES tome 34 (1968)
|
arxiv-papers
| 2012-04-22T13:18:29 |
2024-09-04T02:49:30.000599
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David M. Roberts",
"submitter": "David Roberts",
"url": "https://arxiv.org/abs/1204.4886"
}
|
1204.4952
|
# Sculptures in $S^{3}$
Saul Schleimer Henry Segerman111This work is in the public domain. Mathematics
Institute Department of Mathematics and Statistics University of Warwick
University of Melbourne Coventry CV4 7AL Parkville VIC 3010 United Kingdom
Australia s.schleimer@warwick.ac.uk segerman@unimelb.edu.au
###### Abstract
We construct a number of sculptures, each based on a geometric design native
to the three-dimensional sphere. Using stereographic projection we transfer
the design from the three-sphere to ordinary Euclidean space. All of the
sculptures are then fabricated by the 3D printing service Shapeways.
## 1 Introduction
The three-sphere, denoted $S^{3}$, is a three-dimensional analog of the
ordinary two-dimensional sphere, $S^{2}$. In general, the $n$–dimensional
sphere is a subset of Euclidean space, $\mathbb{R}^{n+1}$, as follows:
$S^{n}=\\{(x_{0},x_{1},\ldots,x_{n})\in\mathbb{R}^{n+1}\mid
x_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2}=1\\}.$
Thus $S^{2}$ can be seen as the usual unit sphere in $\mathbb{R}^{3}$.
Visualising objects in dimensions higher than three is non-trivial. However
for $S^{3}$ we can use stereographic projection to reduce the dimension from
four to three. Let $N=(0,\ldots,0,1)$ be the north pole of $S^{n}$. We define
stereographic projection $\rho:S^{n}-\\{N\\}\to\mathbb{R}^{n}$ by
$\rho(x_{0},x_{1},\ldots,x_{n})=\left(\frac{x_{0}}{1-x_{n}},\frac{x_{1}}{1-x_{n}},\ldots,\frac{x_{n-1}}{1-x_{n}}\right).$
See [1, page 27]. Figure 1a displays stereographic projection in dimension
one. For any point $(x,y)\in S^{1}-\\{N\\}$ draw the straight line $L$ through
$N$ and $(x,y)$. Then $L$ meets $\mathbb{R}^{1}$ at a single point; this is
$\rho(x,y)$. Notice that the figure is also a two-dimensional cross-section of
stereographic projection in any dimension. Additionally, there is nothing
special about the choice of $N=(0,\ldots,0,1)$. We may alter the formula so
that any point in $S^{3}$ becomes the projection point.
2pt $N$ at 135 213 $\frac{x}{1-y}$ at 102 86 $\mathbb{R}^{1}$ at -8 107
$(x,y)$ at 46 18
(a) Stereographic projection from $S^{1}-\\{N\\}$ to $\mathbb{R}^{1}$.
(b) Two-dimensional stereographic projection applied to the Earth. Notice that
features near the north pole are very large in the image.
Figure 1: Stereographic projection.
By adding in a point at infinity corresponding to the north pole,
stereographic projection extends to a homeomorphism from $S^{n}$ to
$\mathbb{R}^{n}\cup\\{\infty\\}$. So we may use stereographic projection to
represent, in $\mathbb{R}^{3}$, objects that live in $S^{3}$.
## 2 The geometry of $S^{3}$
A generic plane in $\mathbb{R}^{4}$, meeting $S^{3}$, meets $S^{3}$ in a
circle. The following circline property is fundamental: stereographic
projection maps any circle $C\subset S^{3}$ to a circle or line in
$\mathbb{R}^{3}$. Accordingly we use the word circline as a shorthand for
circles and lines in $\mathbb{R}^{3}$. See [1, Section 3.2] for a more general
discussion. Note that a circle $C$ of $S^{3}$ maps to a line in
$\mathbb{R}^{3}$ if and only if $C$ meets the projection point.
Any plane, meeting the origin in $\mathbb{R}^{4}$, cuts $S^{3}$ in a great
circle. The great circles are the geodesics, or locally shortest paths, in the
geometry on $S^{3}$. Just as for the usual sphere, $S^{2}$, two distinct great
circles meet at two points: say at $x\in\mathbb{R}^{4}$ and also at the
antipodal point $-x$.
Stereographic projection is conformal: if two circles in $S^{3}$ intersect at
a given angle then the corresponding circlines in $\mathbb{R}^{3}$ meet at the
same angle. So stereographic projection preserves angles [1, Section 3.2].
Note that lengths are not preserved; as shown in Figure 1b the distortion of
length becomes infinite as we approach the projection point. However, this
defect is unavoidable; there is no isometric embedding of any open subset of
the three-sphere into $\mathbb{R}^{3}$.
2pt $-j$ at 129 181 $i$ at 174 121 $1$ at 221 163 $k$ at 221 253 $-k$ at 245
47 $-i$ at 254 194 $j$ at 335 140
Figure 2: The unit quaternions in $S^{3}$ stereographically projected to
$\mathbb{R}^{3}$ from the projection point $-1$.
#### The quaternionic picture of $S^{3}$
In order to get a sense of the shape of $S^{3}$, it is useful to have some
landmarks. A good way to do this is to view $S^{3}$ in terms of the unit
quaternions [2]. The quaternions are an extension of the complex numbers, from
two dimensions to four. A quaternion is a formal sum $a+bi+cj+dk$ where
$a,b,c,d\in\mathbb{R}$ and where $i,j,k$ are non-commuting symbols satisfying
$i^{2}=j^{2}=k^{2}=ijk=-1.$
The set of quaternions is called $\mathbb{H}$ in honour of Hamilton, its
discoverer. There is a natural bijection between $\mathbb{R}^{4}$ and
$\mathbb{H}$ via $(a,b,c,d)\mapsto a+bi+cj+dk$. So we may view $S^{3}$ as the
set of unit quaternions: those with length
$|a+bi+cj+dk|=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}}$ equal to one. Once this is
established the points $\pm 1,\pm i,\pm j,\pm k$ serve as our landmarks. See
Figure 2. All of the circlines shown correspond to great circles in $S^{3}$
with particularly nice quaternionic expressions.
#### The isometries of $S^{3}$
The isometries of $S^{1}$ are the rigid motions of $\mathbb{R}^{2}$ that fix
the origin, namely rotations and reflections. Under composition, these form
the orthogonal group $O(2)$. Analogously, the isometries of $S^{n}$ form the
group $O(n)$. The unit quaternions can be realised as a subgroup of $O(4)$ in
the following manner. As above we identify $\mathbb{H}$ and $\mathbb{R}^{4}$.
For $q\in\mathbb{H}$ with $|q|=1$, the map $f_{q}:\mathbb{H}\to\mathbb{H}$
given by $f_{q}(x)=q\cdot x$ is an element of $O(4)$. So, if we want to move
the point $a$ to the point $b$ in $S^{3}$, then one way to achieve this is to
apply the isometry corresponding to the quaternion $b\cdot a^{-1}$.
An application of this technique is to adjust the stereographic projection of
a subset of $S^{3}$. If $F\subset S^{3}$ is a surface then, as $q$ varies, the
image of $q\cdot F$ in $\mathbb{R}^{3}$ changes dramatically. Equivalently we
can think of this as moving the projection point.
## 3 Designs in $S^{3}$
### 3.1 Four-dimensional polytopes
Suppose that $\sigma\subset\mathbb{R}^{n}$ is a finite set. Then
$P=P(\sigma)$, the convex hull of $\sigma$, is a polytope [11, page 4].
Suppose that $\tau\subset\sigma$. If $P(\tau)$ lies in the boundary of $P$ and
if for all $\tau\subsetneq\mu\subset\sigma$ we have $\dim P(\tau)<\dim P(\mu)$
then we say $P(\tau)$ is a face of $P$. Let $P^{(k)}$ be the $k$–skeleton:
the union of the $k$–dimensional faces of $P$. A maximal chain of faces
$P(\tau_{0})\subset P(\tau_{1})\subset\ldots\subset P(\tau_{n})=P$
is called a flag. Then $P$ is regular if for any two flags $F$ and $G$ of $P$
there is an isometry of $\mathbb{R}^{n}$ that preserves $P$ and sends $F$ to
$G$.
In dimensions one, two, and three the regular polytopes are known of old.
These are the interval, the regular $k$–gons, and the Platonic solids: the
tetrahedron (simplex), the cube, the octahedron (cross-polytope), the
dodecahedron, and the icosahedron. In all higher dimensions there are versions
of the simplex, cube, and cross-polytope. In dimension four these are known as
the $5$–cell, the $8$–cell, and the $16$–cell. Surprisingly, the only
remaining regular polytopes appear in dimension four! There are only three of
them: the $24$–cell, the $120$–cell, and the $600$–cell [3, page 136].
Suppose $P$ is a regular $n$–polytope. The extreme symmetry of $P$ implies
that we can move $P$ so that the vertices $P^{(0)}$ lie in the unit sphere,
$S^{n-1}$. Projecting radially from the origin transfers the one-skeleton
$P^{(1)}$ from $\mathbb{R}^{n}$ into $S^{n-1}$. Stereographic projection then
places $P^{(1)}$ in $\mathbb{R}^{n-1}$.
Applied to a $4$–polytope, these projections turn the Euclidean geometry of
$P^{(1)}$ first into a design of arcs of great circles in $S^{3}$ and then
into a design of segments of circlines in $\mathbb{R}^{3}$. If $P^{(1)}$ meets
the projection point then the design includes line segments running off to
infinity. Coincidentally, Figure 2 shows this for the $16$–cell. In order to
produce such a design as a physical object, we need to thicken the circline
segments to have non-zero volume. One possible approach uses the Euclidean
geometry of $\mathbb{R}^{3}$: we could thicken all segments of the design to
get tubular neighbourhoods of constant radius. However, the result is not
satisfactory; near the origin in $\mathbb{R}^{3}$ the tubes are much too thick
compared to their separation.
A better solution is to use tubular neighbourhoods in the intermediate $S^{3}$
geometry. For this we must parameterise the image of such a tube under
stereographic projection. Here the circline property is very useful. The
boundary of a tubular neighbourhood of a geodesic in $S^{3}$ can be made as a
union of small circles in $\mathbb{R}^{4}$. (These circles lie in $S^{3}$, but
are not great.) The small circles map to circlines in $\mathbb{R}^{3}$, which
can be directly parameterised. Computer visualisation of stereographic
projections of 4-polytopes, in this style, are beautifully rendered by the
program Jenn3d [8]. In Figure 3 we show four views of a 3D print of the
$24$–cell, with the projection point chosen to be at the center of one of the
cells. See also Ocneanu’s “Octacube” [9].
(a) A generic viewpoint.
(b) A 2-fold symmetry axis.
(c) A 3-fold symmetry axis.
(d) A 4-fold symmetry axis.
Figure 3: $24$–Cell, 2011, $9.0\times 9.0\times 9.0$ cm.
The sculpture in Figure 3 illustrates a problem inherent in 3D printing of
stereographic projections. Suppose that $P$ is a symmetric design in $S^{3}$
and $Q=\rho(P)$ is the stereographic projection from the north pole, $N$. In
this case the largest features of $Q$ will correspond to the parts of $P$
closest to $N$. These are the main contributers to volume and thus to cost.
The smallest features of $Q$ will be roughly half the size of the parts of $P$
nearest the south pole. The 3D printing process places a lower bound on the
size of the smallest printable feature: current technology allows around $1$
mm.
Of course we may scale $Q$ in $\mathbb{R}^{3}$; scaling up ensures
printability while scaling down reduces volume. Thus printability and cost are
in tension. For example, if we rotate the $120$–cell so that $N$ lies at the
center of a dodecahedral face, and stereographically project, then the largest
feature is around $29.4$ times larger than the smallest. So here scaling to
ensure printability also ensures unaffordability.
One solution to this problem, as employed by Hart [5], is use a projective
transformation instead of stereographic projection. This takes a $4$–polytope
to its Schlegel diagram [11, page 133]. This is typically much more compact.
However, conformality is lost; the resulting figure distorts both lengths and
angles.
Our alternative, shown in Figure 4, is to only print half of the object. We
cut $S^{3}$ along the equatorial $S^{2}$; the sphere equidistant from the
north and south poles. Choosing the north pole as the projection point, we
project the half of the design in the southern hyperhemisphere. The image is
contained in the unit ball $\mathbb{B}^{3}=\\{x\in\mathbb{R}^{3}:|x|\leq
1\\}$. This done, the thinnest and thickest parts differ only by a factor of
two, at the most. For stereographic projection, parts of the design near the
projection point are the real problem, in terms of size. Eliminating the half
nearest the projection point eliminates the problem.
(a) A generic viewpoint.
(b) A 2-fold symmetry axis.
(c) A 3-fold symmetry axis.
(d) A 5-fold symmetry axis.
Figure 4: Half of a 120-Cell, 2011, $9.9\times 9.9\times 9.9$ cm.
Note that half of the $120$–cell is still very complicated! However, one can
understand the whole of the $120$–cell by imagining reflecting the object
across the cutting two-sphere. Note as well, that printing only the southern
half of a design allows us to print objects that pass through the north pole,
which ordinarily would be infinitely expensive. For example, in Figure 5 we
show one-half of the stereographic projection of the vertex centered
$600$–cell. This version of the $600$–cell is positioned so as to be dual to
the facet-centered 120-cell shown in Figure 4. The other half of the vertex-
centered $600$–cell cannot be printed because the vertex antipodal to the
origin meets the projection point.
(a) A generic viewpoint.
(b) A 2-fold symmetry axis.
(c) A 3-fold symmetry axis.
(d) A 5-fold symmetry axis.
Figure 5: Half of a 600-Cell, 2011, $9.9\times 9.9\times 9.9$ cm.
### 3.2 Parameterisations of surfaces and torus knots
The geometry of $S^{3}$ lends itself particularly well to the representation
of tori and torus knots. There seem to be two reasons for this. First, in its
natural position certain geodesics in the torus are great circles in $S^{3}$.
Second, quaternionic multiplication and its relatives directly parametrise
torus knots.
When representing a surface as a 3D printed object, it is often a good idea to
drill holes in the surface, both to save on material used and so the viewer
can see, partly, through the surface to what is behind. In our approach, the
pattern of holes shows the parameterisation, by realising the surface as a
grid with grid-lines in the direction of the parameters.
#### Clifford torus
Recall that $e^{i\theta}=\cos(\theta)+i\sin(\theta)$ parametrises a great
circle $S^{1}$. The same formula holds replacing $i$ everywhere by $j$ or by
$k$. A Clifford torus is foremost a torus, and so can be parameterised as a
product [4, page 139] via
$\displaystyle\mathbb{T}=S^{1}\times S^{1}$
$\displaystyle=\left\\{\frac{1}{\sqrt{2}}\bigl{(}\cos(\alpha),\sin(\alpha),\cos(\beta),\sin(\beta)\bigr{)}\;\bigg{|}\;0\leq\alpha<2\pi,\,0\leq\beta<2\pi\right\\}$
$\displaystyle=\left\\{\frac{1}{\sqrt{2}}\bigl{(}e^{i\alpha}+e^{i\beta}\cdot
j\bigr{)}\;\bigg{|}\;0\leq\alpha<2\pi,\,0\leq\beta<2\pi\right\\}.$
The factor of $1/\sqrt{2}$ rescales the torus to lie inside of the unit sphere
$S^{3}\subset\mathbb{R}^{4}$. Note that if we vary $\alpha$ while fixing
$\beta$, then the point traces out a $(1,0)$ curve on $\mathbb{T}$. Conversely
varying $\beta$ while fixing $\alpha$ yields a $(0,1)$ curve. Unfortunately
none of these curves are great circles in $S^{3}$.
On the other hand, if we vary $\alpha$ and $\beta$ simultaneously, at the same
(respectively, opposite) velocity the the point traces out a $(1,1)$
(respectively $(1,-1)$) curve. As we shall see, these are great circles.
Note that $\mathbb{T}$ divides $S^{3}$ into a pair of isometric solid tori:
copies of $S^{1}\times D^{2}$. We want to rotate the torus $\mathbb{T}$ so
that it meets the projection point. This way, after stereographic projection
there is a pleasing symmetry; the two solid tori are interchangeable by an
isometry of $\mathbb{R}^{3}$.
We can use quaternions to fix the parameterisation, giving us the desired
$(1,1)$ and $(1,-1)$ curves, and to also move $\mathbb{T}$ to meet the
projection point $1\in S^{3}\subset\mathbb{H}$. Solving the second problem
first, note that $\frac{1}{\sqrt{2}}(1+j)$ lies in $\mathbb{T}$. If $q$ is the
quaternion satisfying $\frac{1}{\sqrt{2}}(1+j)q=1$, then
$q=\frac{1}{\sqrt{2}}(1-j)$. The new parameterisation of the torus is given by
post-multiplication by $q$:
$\frac{1}{\sqrt{2}}\bigl{(}e^{i\alpha}+e^{i\beta}\cdot
j\bigr{)}\cdot\frac{1}{\sqrt{2}}(1-j)=\frac{1}{2}\bigl{(}e^{i\alpha}+e^{i\beta}+(e^{i\beta}-e^{i\alpha})\cdot
j\bigr{)}.$
The torus meets the desired projection point when $\alpha=\beta=0$.
We now solve the second problem, by rotating the coordinates through
$45^{\circ}$. Take new coordinates $\theta,\phi$ where
$\theta=(\alpha+\beta)/2$ and $\phi=(\alpha-\beta)/2$. So $\alpha=\theta+\phi$
and $\beta=\theta-\phi$. Plugging in and simplifying, the above
parametrisation becomes $e^{i\theta}e^{-k\phi}$. Keeping $\phi$ fixed and
varying $\theta$ now gives a $(1,1)$ curve, which is also a great circle. Note
that we only need $0\leq\theta<2\pi,\,0\leq\phi<\pi$ to cover the whole torus.
We permute coordinates and change a sign to get a slightly neater form:
$e^{i\phi}e^{j\theta}=\bigl{(}\cos(\theta)\cos(\phi),\cos(\theta)\sin(\phi),\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi)\bigr{)}$
for $0\leq\theta<2\pi,0\leq\phi<\pi$. The operations of permuting the
coordinates and changing the sign are symmetries of $S^{3}$, so the geometry
is unchanged and the surface $\mathbb{T}$ still meets the desired projection
point, $1$. The resulting parametrization is Lawson’s minimal surface
$\tau_{1,1}$; see [6].
(a) A 2-fold symmetry axis.
(b) A generic viewpoint.
Figure 6: Clifford Torus, 2011, $10.8\times 10.8\times 3.4$ cm.
#### Finding the normal
After stereographic projection, we get a 2-dimensional surface in
$\mathbb{R}^{3}\cup\\{\infty\\}$. As in Section 3.1, for 3D printing we must
thicken our design to have positive volume. Our plan is to additionally
parametrise the normal (that is, perpendicular) to the surface, and then
thicken in that direction. As before, we do this thickening in $S^{3}$ rather
than $\mathbb{R}^{3}$.
Suppose that $F$ is any surface in $S^{3}$, with parametrisation
$p(\theta,\phi)\in S^{3}\subset\mathbb{R}^{4}$. Compute the tangent vectors
$\frac{\partial}{\partial\theta}p(\theta,\phi)$ and
$\frac{\partial}{\partial\phi}p(\theta,\phi)$ in $\mathbb{R}^{4}$. Since $F$
lies in $S^{3}$, these vectors are tangent to $S^{3}$ and so perpendicular to
$p(\theta,\phi)$, thought of as a vector from the origin. The desired normal
vector $n(\theta,\phi)$ is a unit vector that is perpendicular to the three
given vectors $p$, $\frac{\partial}{\partial\theta}p$, and
$\frac{\partial}{\partial\phi}p$. This determines $n$ up to sign. Thus finding
$n$ amounts to computing the kernel of the matrix with rows $p$,
$\frac{\partial}{\partial\theta}p$ and $\frac{\partial}{\partial\phi}p$. As
these vectors vary with the parameters $\theta$ and $\phi$ it is most
convenient to compute $n$ via an application of Cramer’s rule: $n$ is the
determinant of the matrix with first three rows $p$,
$\frac{\partial}{\partial\theta}p$, $\frac{\partial}{\partial\phi}p$, and
fourth row the vector $(1,i,j,k)$.
For the above parametrisation of the Clifford torus we find:
$\begin{array}[]{rccrrrrc}p(\theta,\phi)&=&\bigl{(}&\cos(\theta)\cos(\phi),&\cos(\theta)\sin(\phi),&\sin(\theta)\cos(\phi),&\sin(\theta)\sin(\phi)&\bigr{)}\\\
\frac{\partial}{\partial\theta}p(\theta,\phi)&=&\bigl{(}&-\sin(\theta)\cos(\phi),&-\sin(\theta)\sin(\phi),&\cos(\theta)\cos(\phi),&\cos(\theta)\sin(\phi)&\bigr{)}\\\
\frac{\partial}{\partial\phi}p(\theta,\phi)&=&\bigl{(}&-\cos(\theta)\sin(\phi),&\cos(\theta)\cos(\phi),&-\sin(\theta)\sin(\phi),&\sin(\theta)\cos(\phi)&\bigr{)}\\\
n(\theta,\phi)&=&\bigl{(}&-\sin(\theta)\sin(\phi),&\sin(\theta)\cos(\phi),&\cos(\theta)\sin(\phi),&-\cos(\theta)\cos(\phi)&\bigr{)}\end{array}$
We introduce the parameter $\psi$ for the thickness of the surface. We move a
distance $\psi$ along the geodesic from $p(\theta,\phi)$ to $n(\theta,\phi)$
to reach
$r(\theta,\phi,\psi)=\cos(\psi)p(\theta,\phi)+\sin(\psi)n(\theta,\phi)$. Let
$N_{\epsilon}(\mathbb{T})$ be the $\epsilon$–neighborhood of $\mathbb{T}$,
taken in $S^{3}$. This is the same as thickening $\mathbb{T}$ in the normal
direction, using $r$.
Since $N_{\epsilon}(\mathbb{T})$ contains the projection point, the sculpture
$\rho(N_{\epsilon}(\mathbb{T}))$ would have infinite volume. We therefore
remove a rectangular solid from $N_{\epsilon}(\mathbb{T})$; the boundary of
the removed material is visible around the outside of the sculpture shown in
Figure 6.
(a) A 2-fold symmetry axis.
(b) A generic viewpoint.
Figure 7: Round Möbius Strip, 2011, $15.2\times 10.9\times 6.2$ cm.
(a) The 4-fold symmetry axis.
(b) One of the 2-fold symmetry axes.
(c) The other 2-fold symmetry axis.
Figure 8: Round Klein Bottle, 2011, $15.2\times 15.2\times 10.9$ cm.
#### Möbius strip and Klein Bottle
A slight variant of the torus gives a Möbius strip:
$\left\\{\bigl{(}\cos(\theta)\cos(\phi),\cos(\theta)\sin(\phi),\sin(\theta)\cos(2\phi),\sin(\theta)\sin(2\phi)\bigr{)}\mid
0\leq\theta<\pi,0\leq\phi<\pi\right\\}$
This is a parameterisation of the “Sudanese Möbius strip” [7]. The border of
the Möbius strip is given by the points for which $\theta$ is $0$ or $\pi$.
Since these points form a geodesic in $S^{3}$, the boundary is a circline in
$\mathbb{R}^{3}$ by the circline property. With the given parameterisation,
stereographic projection from $(0,0,-1,0)$ gives a circular boundary as in
Figure 7. The normal vector is calculated analogously to the torus case, as
follows.
$\begin{array}[]{rcrrrrrc}p(\theta,\phi)&=&\bigl{(}&\cos(\theta)\cos(\phi),&\cos(\theta)\sin(\phi),&\sin(\theta)\cos(2\phi),&\sin(\theta)\sin(2\phi)&\bigr{)}\\\
\frac{\partial}{\partial\theta}p(\theta,\phi)&=&\bigl{(}&-\sin(\theta)\cos(\phi),&-\sin(\theta)\sin(\phi),&\cos(\theta)\cos(2\phi),&\cos(\theta)\sin(2\phi)&\bigr{)}\\\
\frac{\partial}{\partial\phi}p(\theta,\phi)&=&\bigl{(}&-\cos(\theta)\sin(\phi),&\cos(\theta)\cos(\phi),&-2\sin(\theta)\sin(2\phi),&2\sin(\theta)\cos(2\phi)&\bigr{)}\\\
n(\theta,\phi)&=&\frac{1}{\sqrt{1+3\sin^{2}(\theta)}}\bigl{(}&-2\sin(\theta)\sin(\phi),&2\sin(\theta)\cos(\phi),&\cos(\theta)\sin(2\phi),&-\cos(\theta)\cos(2\phi)&\bigr{)}\end{array}$
Again the surface is punctured at the projection point, with a rectangular
hole in the grid pattern. See Perry’s sculpture “Zero” [10] for a similar
design. If we extend the strip across its boundary, taking $0\leq\theta<2\pi$,
we get the union of two punctured Möbius strips, giving the twice-punctured
Klein bottle shown in Figure 8. This parameterisation of the (unpunctured)
Klein bottle is Lawson’s surface $\tau_{2,1}$.
#### Torus knot
A further variant gives a parameterisation of a torus knot, in this case the
trefoil knot:
$\left\\{\bigl{(}\cos(\theta)\cos(\phi),\cos(\theta)\sin(\phi),\sin(\theta)\cos((3/2)\phi),\sin(\theta)\sin((3/2)\phi)\bigr{)}\mid
0\leq\phi<4\pi\right\\}$
Here $\theta$ has a fixed value, greater than 0 and smaller than $\pi/2$.
Altering the fraction $3/2$ will produce other torus knots. The normal vector
may be found as before; however for this model we used an “alternative” to the
normal vector, namely
$n(\theta,\phi)=\bigl{(}-\sin(\theta)\sin(\phi),\sin(\theta)\cos(\phi),\cos(\theta)\sin((3/2)\phi),-\cos(\theta)\cos((3/2)\phi)\bigr{)}.$
Figure 9: Knotted Cog, 2011, $3.8\times 3.4\times 1.3$ cm.
Using the local coordinates $(\theta,\phi,\psi)$, we can add small features to
the sculpture, using any shape we could define in ordinary 3-dimensional
space. In the case shown in Figure 9, we add cog teeth, which are simply
truncated pyramids in $(\theta,\phi,\psi)$ coordinates. The alternative normal
vector adds a slight shear slope to the teeth, which we feel is aesthetically
preferable.
## 4 Future directions
Our sculptures are tangible representives of topological and geometric
abstractions. In order to do this, we naturally must construct designs that
occur in $\mathbb{R}^{3}$: that is, in actual space. In each case we attempted
to choose the most canonical such geometries available and then the most
faithful projections.
There is a wild array of further topological and combinatorial objects. For
example, there is a rich theory of knots and surfaces and their
interrelations. We have not yet found (or perhaps better, understood)
satisfactory geometric representations, or at least representatives which map
to $\mathbb{R}^{3}$ in satisfactory ways. An example of the latter problem
would be surfaces of genus at least two. These have nice hyperbolic
structures, but they cannot be mapped into $\mathbb{R}^{3}$ in a very
satisfying way.
## References
* [1] Alan F. Beardon, _The geometry of discrete groups_ , Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983.
* [2] John H. Conway and Derek A. Smith, _On quaternions and octonions: their geometry, arithmetic, and symmetry_ , A K Peters Ltd., Natick, MA, 2003.
* [3] H. S. M. Coxeter, _Regular polytopes_ , third ed., Dover Publications Inc., New York, 1973.
* [4] Manfredo Perdigão do Carmo, _Riemannian geometry_ , Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty.
* [5] George W. Hart, _4d polytope projection models by 3d printing_ , to appear in Hyperspace.
* [6] H. Blaine Lawson, _Complete minimal surfaces in $S^{3}$_, Annals of Mathematics 92 (1970), no. 3, 335–374.
* [7] D. Lerner and D. Asimov, _The Sudanese Möbius band (video)_ , In SIGGRAPH Video Review, 1984.
* [8] Fritz H. Obermeyer, _Jenn3d_ , a computer program for visualizing Coxeter polytopes, available from http://www.math.cmu.edu/f̃ho/jenn/.
* [9] Adrian Ocneanu, _Octacube_ , http://science.psu.edu/news-and-events/2005-news/math10-2005.htm.
* [10] Charles Perry, _Zero_ , http://www.charlesperry.com/sculpture/zero.
* [11] Günter M. Ziegler, _Lectures on polytopes_ , Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995.
|
arxiv-papers
| 2012-04-23T00:20:30 |
2024-09-04T02:49:30.010956
|
{
"license": "Public Domain",
"authors": "Saul Schleimer and Henry Segerman",
"submitter": "Henry Segerman",
"url": "https://arxiv.org/abs/1204.4952"
}
|
1204.4954
|
# THE KINEMATICS AND CHEMISTRY OF RED HORIZONTAL BRANCH
STARS IN THE SAGITTARIUS STREAMS
W.B.SHI11affiliationmark: 22affiliationmark: Y.Q.CHEN11affiliationmark:
K.CARRELL11affiliationmark: and G.ZHAO11affiliationmark: 22affiliationmark:
† 1 Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, China; swb@sdu.edu.cn;
cyq@bao.ac.cn; carrell@nao.cas.cn; gzhao@nao.cas.cn 2 Shandong Provincial Key
Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of
Space Science and Physics, Shandong University at Weihai, Weihai 264209, China
###### Abstract
We have selelcted 556 Red Horizontal Branch (RHB) stars along the streams of
the Sagittarius dwarf galaxy (Sgr) from SDSS DR7 spectroscopic data using a
theoretical model. The metallicity and $\alpha$-elements distributions are
investigated for stars in the Sgr streams and for Galactic stars at the same
locations. We find that the Sgr stars have two peaks in the metallicity
distribution while the Galactic stars have a more prominent metal-poor peak.
Meanwhile, [$\alpha$/Fe] ratios of the Sgr stars are lower than those of the
Galactic stars. Among the Sgr stars, we find a difference in the metallicity
distribution between the leading and trailing arms of the Sgr tidal tails. The
metallicity and [$\alpha$/Fe] distribution of the leading arm is similar to
that of the Galaxy. The trailing arm is composed mainly of a metal rich
component and [$\alpha$/Fe] is obviously lower than that of the Galactic
stars. The metallicity gradient is -(1.8 $\pm$ 0.3)$\times 10^{-3}$ dex
degree-1 in the first wrap of the trailing arm and -(1.5 $\pm$ 0.4)$\times
10^{-3}$ dex degree-1 in the first wrap of the leading arm. No significant
gradient exists along the second wraps of the leading or trailing arms. It
seems that the Sgr dwarf galaxy initially lost the metal poor component in the
second wrap (older) arms due to the tidal force of our Galaxy and then the
metal rich component is disrupted in the first wrap (younger) arms. Finally,
we found that the velocity dispersion of the trailing arm from
$88^{\circ}<\Lambda_{\odot}<112^{\circ}$ is $\sigma$ = 9.808 $\pm$ 1.0 km s-1,
which is consistent with previous work in the literature.
Galaxy: halo — galaxy: Sagittarius — stars: red horizontal-branch
## 1 Introduction
The Sagittarius dwarf galaxy is the second nearest galaxy to our Milky
Way(assuming the Canis Major dwarf galaxy is the nearest). The Sgr is
currently being disrupted under the strain of the Milky Way. Studying the
metallicity and kinematic distributions of Sgr stars has now become an
important issue. Many works on chemical abundances of the Sgr stars have been
done based on high resolution spectra. Bellazzini et al. (2008) selected 321
RGB stars in the Sgr nucleus and give the average [Fe/H]$\sim$ -0.45 dex from
the infrared Ca II triplet. Carretta et al. (2010) derived homogeneous
elemental abundances with 27 red giant stars belonging to the Sgr nucleus and
found on average [Fe/H] $\sim$ -0.61 dex – -0.74 dex. Keller et al. (2010)
observed 11 M giant stars with the Gemini South telescope which indicated the
[Fe/H] of stars decreases along the tidal stream. Chou et al. (2007) present a
reliable measurement on M giants with high resolution at different points
along the tidal stream and show a significant metallicity gradient. They found
a median [Fe/H]$\sim$-0.4 in the core that decreases to -1.1 dex over the
leading arm.
However, these works based on high resolution spectra have small samples of
stars. Based on low resolution spectra Yanny et al. (2009) traced the Sgr
tidal streams with red K/M-giants from the SDSS survey. They found an average
[Fe/H] in the range -0.8$\pm$0.2 with 33 K/M-giant stars in two areas. Carlin
et al. (2012) derived metallicity from low-resolution spectra of stars along a
stretch of the Sgr stream and find a constant [Fe/H]$\sim$-1.15. Thus far, the
metallicity and abundance studies of the Sgr tails have been less detailed in
large samples of stars and in various locations, which are the advantages of
the present work. We analyze the metallicity distribution at different points
along the tidal streams of the Sgr with low resolution data for a large sample
of stars.
The SDSS spectroscopic survey and the LAMOST project (Zhao et al., 2006), will
provide a large sample of Red Horizontal Branch (RHB) stars with low
resolution spectra in the Sgr. Currently, the spectroscopic data of the SDSS
survey provides stellar parameters, distances, radial velocities and
metallicities for many stars spread across a wide area. In this work, we
investigate the properties of Sgr RHB stars from SDSS and compare them with
stars in the Milky Way. We present the procedure for selecting the RHB stars
in Sect.2 and give the metallicity analysis in Sect.3. In Sect.4 we test the
theoretical model and sample selection, and a summary is given in Sect.5.
## 2 Sample selection
We obtained 8535 RHB Stars, 5391 with (U,V,W) and 3144 stars without, from
SDSS DR7 low resolution spectral data (Chen et al., 2010). We choose the Sgr
stars with the aid of a theoretical model by Law & Majewski (2010). Law &
Majewski (2010) provide a model of the Sgr orbiting in a triaxial Galactic
potential with $10^{5}$ points. The model divides the $10^{5}$ points into
four parts: leading arm 1 and 2 and trailing arm 1 and 2. The model is based
on observational data from 2MASS and SDSS (for more details please see Law &
Majewski (2010)).
We select our sample stars by using the Law & Majewski (2010) model as a
reference to provide cuts on the RHB stars. Firstly, we obtain 3512 stars from
the full 8535 RHB star sample using Ra-Dec positions. Secondly, we choose RHB
stars in the Sagittarius leading and trailing tidal tails using a
Distance-$\Lambda_{\odot}$ map of the Law & Majewski (2010) model. Here
$\Lambda_{\odot}$ is the Sgr longitude scale along the orbital plane. We
obtain 586 stars in leading arm 1, 585 stars in leading arm 2, 973 stars in
trailing arm 1 and 502 stars in trailing arm 2 from the 3512 stars. The first
and second wrap of the Law & Majewski (2010) model is denoted by arm 1 and 2
respectively. Third, we select stars to be likely members of the Sgr stream
based on their radial velocities, which are appropriate for the Sgr stream at
these positions based on Sgr debris models (Chou et al., 2010). Specifically,
we select stars with a $V_{gsr}$(the velocity in the Galactic standard of
rest)-$\Lambda_{\odot}$ map (Figure 1). A local standard of rest rotation
velocity of 220 km s-1 is adopted for the Sun, for consistency with Law &
Majewski (2010). We also calculate $V_{gsr}$ with the same equation as Law &
Majewski (2010) for consistency, i.e. $V_{gsr}=rv+9.0\cos b\cos l+232.0\cos
b\sin l+7.0\sin b$ km s-1. With the $V_{gsr}$ criteria, 118 stars satisfy the
cuts on leading arm 1 in both the Distance-$\Lambda_{\odot}$ and
$V_{gsr}$-$\Lambda_{\odot}$ maps from 586 stars. In a similar way, 80, 329 and
47 stars in the leading arm group 2, trailing arm group 1 and trailing arm
group 2, are selected from 585, 973 and 502 stars, respectively (see Figure
1). There are 18 RHB stars overlapped in the leading arm and trailing arm. We
omit these overlapping stars from the leading and trailing arm groups. Finally
there are 102, 78, 327 and 31 RHB stars in leading arm 1 and 2 and trailing
arm 1 and 2, respectively (shown in the $X_{GC}$-$Z_{GC}$ map of Figure 2). We
adopt 556 (including 18 overlapping stars) as the total number of RHB stars in
our Sgr samples. The table of our Sgr samples is provided in a electronic
version.
Leading arm 1
Leading arm 2
Trailing arm 1
Trailing arm 2
Figure 1: Selected RHB stars with the Law & Majewski (2010) model (red points)
in a $V_{gsr}$-$\Lambda_{\odot}$ map. We obtain 118 stars (green points) from
586 stars (green points + yellow points + black points) in leading arm 1 and
80 stars (green points) from 585 stars (green points + yellow points + black
points) in leading arm 2. We obtain 329 stars (blue points) from 973 stars
(blue points + yellow points \+ black points) in trailing arm 1 and 47 stars
(blue points) from 502 stars (blue points + yellow points + black points) in
trailing arm 2. The yellow points indicate the Galactic RHB stars. We replaced
the high density stars for a generalization which is shown as the black boxes.
For clearly dividing the stars into Sgr and Galaxy components, we omit the
stars (black points) located at the edge of the model.
Leading arm stars in Sgr
Trailing arm stars in Sgr
Leading arm stars in Galaxy
Trailing arm stars in Galaxy
Figure 2: Spatial distribution of target stars in the debris streams of Sgr.
Upper panels: A plot of Sgr RHB stars in the leading and trailing arms in an
$X_{GC}$-$Z_{GC}$ map. Green and blue points indicate arm 1 and arm 2
respectively in the leading and trailing arms. The large black point indicates
the location of the Galactic Center, while the asterisk indicates the location
of the Sun. Lower panels: A plot of the Galactic RHB stars in an
$X_{GC}$-$Z_{GC}$ map.
For comparison, we selected a sample of Galactic stars with the same positions
but different velocities from the Sgr tidal stars. That is, we select Galactic
RHB stars from the full 8535 star sample by finding stars that satisfy the Ra-
Dec criteria and the Distance-$\Lambda_{\odot}$ criteria of the Law & Majewski
(2010) model, but do not satisfy the $V_{gsr}$-$\Lambda_{\odot}$ criteria
mentioned above. In Figure 1 the yellow points indicate Galactic RHB stars. We
excluded the high density stars with a generalization which is shown as the
black boxes in order to reduce the potential effect from the undetected stream
stars in the Galaxy, see Figure 1. Again, 159 overlapping stars at the
positions of the leading and trailing arms are removed from the leading and
trailing sample. Finally, there are 202, 164, 347 and 129 Galactic stars at
the positions of leading arm 1 and 2 and trailing arm 1 and 2, respectively
(shown in the $X_{GC}$-$Z_{GC}$ map of Figure 2). We adopt 1001 (including 159
overlapping stars) as the total number of RHB stars in our Galaxy sample. The
table of Galactic samples is also provided in a electronic version.
## 3 Results and Discussions
### 3.1 Comparing the kinematics and chemistry of RHB stars in the Sgr and in
the Galaxy
In Figure 3, we plot the histograms of [Fe/H], $V_{r}$, and [$\alpha$/Fe] and
the distribution map of [Fe/H]-[$\alpha$/Fe] for RHB stars in the Sgr (556)
and Galaxy (1001). One sees that the value of $V_{r}$ for all RHB stars in the
Sgr have a sharp peak at -140 km s-1, while there is a large dispersion in the
distribution for the Galactic RHB stars. This is mainly due to the selection
effect. We set a dashed line at -90 km s-1 in the $V_{r}$ histogram to define
the lower velocity group and analyze the metallity distribution of those lower
velocity stars. Clearly, the lower velocity stars are more than half of all
the stars in the Sgr, but the lower velocity stars are only a small part of
all the Galactic stars. Meanwhile, the Galactic stars are dominated by a metal
poor component while the Sgr stars have a significant contribution from a more
metal rich component. The distribution of [Fe/H] in the Sgr stars has two
peaks, one at -1.3 dex and one at -0.8 dex. There are also two peaks in the
[Fe/H] distribution of Galactic RHB stars, which are at the same [Fe/H] value,
but the peak at -0.8 dex is less pronounced and could be due to Sgr stars near
the edges of our selection criteria. Yanny et al. (2009) show that the giant
branch in the Sgr leading tidal tail is consistent with those of globular
clusters with [Fe/H] of -1.0 $\pm$ 0.5. They also find that the 33 identified
Sgr K/M-giant stars have metallicities of -0.8 $\pm$ 0.2. Our results are
similar to the distribution of [Fe/H] in Yanny et al. (2009). We also show the
lower velocity stars with a dashed line in the histograms of [Fe/H] and
[$\alpha$/Fe]. For the Sgr stars, the dashed line shows two peaks in the
[Fe/H] histogram and the two components have equal contributions, while the
solid line shows a bigger contribution from the peak at -1.3 dex than that
from the peak at -0.8 dex. For Galactic RHB stars, the dashed line is similar
to the solid line.
All RHB stars in Sgr
All RHB stars in Galaxy
Figure 3: We compare all RHB stars (red solid lines) in the Sgr (556) and
Galaxy (1001). Blue dashed lines show the stars whose $V_{r}$ is less than -90
km s-1. Left (right) panels show RHB stars in the Sgr (Galaxy).
From the [Fe/H]-[$\alpha$/Fe] map in Figure 3, we can see that the
[$\alpha$/Fe] of most stars is lower than 0.2 dex in Sgr, while that of most
Galactic stars is larger than 0.2 dex. The low [$\alpha$/Fe] stars mainly come
from the metal rich component of the Sgr tidals at -0.8 dex. These results are
consistent with the results of early dwarf galaxy fragments. [$\alpha$/Fe]
deficiencies were found by Smecker-Hane & McWilliam (2002), McWilliam &
Smecker-Hane (2005), Sbordone et al. (2007) and Carretta et al. (2010) for the
more metal-rich stars in the Sgr (McWilliam (2010)). The existence of some
Galactic stars in our sample may lead to analysis error, but they could also
be real since there are plenty of examples of Galactic halo stars with low
[$\alpha$/Fe] (e.g. Nissen & Schuster (1997) and Brown et al. (1997)).
### 3.2 Comparing the properties of Sgr RHB stars in the leading and trailing
arms
Leading arm stars in Sgr
Leading arm stars in Galaxy
Figure 4: Same as Figure 3 but for comparing RHB stars in the leading arm of
the Sgr (180) and Galaxy (366).
Trailing arm stars in Sgr
Trailing arm stars in Galaxy
Figure 5: Same as Figure 3 but for comparing RHB stars in the trailing arm of
the Sgr (358) and Galaxy (476).
It is interesting to compare the properties of RHB stars between the leading
and trailing arms of the Sgr tidal tails. Firstly, the distribution of $V_{r}$
for the RHB stars have big differences between the leading and trailing arms
(Figures 4 \- 5). There are two peaks, -20 km s-1 and -100 km s-1, in the
$V_{r}$ histogram of leading arm stars. The distribution of the Sgr leading
arm stars is similiar with that of the Galactic RHB stars, which also presents
two peaks. Meanwhile, nearly all the trailing arm stars are centered around
one peak near -140 km s-1. The dashed line corresponds to low velocity stars
with $V_{r}$ less than -90 km s-1 in the $V_{r}$ histogram, the same as in
Figure 3. Again, this difference comes from the predictions of the Law &
Majewski (2010) model. We find that the metallicity distribution of the stars
also has large differences between the two arms (Figures 4 \- 5). The
metallicity distribution of RHB stars in the leading arm is similar to that of
the Galactic stars both in the solid and dashed lines. The metal rich peak is
not clear and the metal poor peak is prominent in the leading arm. The [Fe/H]
distribution of the trailing arm stars has two peaks and the metal rich peak
is significant, as shown in the solid line and even more clearly for low
velocity stars as shown in the dashed line.
From the [$\alpha$/Fe] histograms of Figures 4 \- 5, we can see that the
distributions of leading arm RHB stars is also similar with that of Galactic
stars both in the solid and dashed lines, while trailing arm stars show most
stars have lower values of [$\alpha$/Fe], which is different from the Galactic
stars. The properties of the trailing arm RHB stars are more consistent with
the core of the Sgr: [Fe/H] is more metal rich than that of the Galaxy and
[$\alpha$/Fe] is lower than that of Galactic halo stars. It is unexplained
that the leading arm stars do not follow the chemical history of the Sgr core.
Further work is necessary to investigate the leading arm of the Sgr tidals.
### 3.3 The metallicity gradient along the leading and trailing arms
Figure 6: [Fe/H] as a function of angular distance from the main body of Sgr
along the leading arm (left panels) and trailing arm (right panels). The upper
panels show the individual points. In the lower panels, the distribution of
[Fe/H] is displayed as the median. The solid line shows the result of a least-
squares linear fit to the median data. The metallicity gradient is -(1.5 $\pm$
0.4)$\times 10^{-3}$ dex degree-1 in leading arm 1 and -(1.8 $\pm$ 0.3)$\times
10^{-3}$ dex degree-1 in trailing arm 1. The fitted line shows that the
metallicity is nearly flat in leading arm 2 and trailing arm 2.
Figure 7: [$\alpha$/Fe] as a function of angular distance from the main body
of Sgr along the leading arm (left panels) and trailing arm (right panels).
The [$\alpha$/Fe] gradient is (0.67 $\pm$ 0.15)$\times 10^{-3}$ dex degree-1
in leading arm 1 and (0.86 $\pm$ 0.12) $\times 10^{-3}$ dex degree-1 in
trailing arm 1. There is no obvious trend in leading arm 2 or trailing arm 2
except for the fluctuations of individual points.
We would like to see if the metallicity is a function of orbital longitude
along the Sgr leading and trailing tidal streams. Figures 6 and 7 give the
distributions of $\Lambda_{\odot}$-[Fe/H] and $\Lambda_{\odot}$-[$\alpha$/Fe]
for RHB stars. There is a metallicity gradient in trailing arm 1 while there
is a lower one in leading arm 1\. We find that in the trailing arm, when
moving farther from the Sgr core along arm 1 and then to arm 2, the
metallicity shifts to more metal poor values, which suggests an evolution
toward more ancient stars since metal poor RHB stars must be older than metal
rich RHB stars. This is in agreement with dwarf galaxy formation theories
where the more metal rich core of the galaxy is surrounded by older and more
metal poor stars since it is this outer, older and metal poor population that
will be tidally stripped before the younger, inner component. Our results also
agree with the gradient found by Chou et al. (2007) and is similar to Figure
15 of Law & Majewski (2010), which gives the distribution of
$\Lambda_{\odot}$-[Fe/H]. However, Yanny et al. (2009) have studied the
metallicity of blue horizontal-branch (BHB) stars as a function of
$\Lambda_{\odot}$ and find that there is no significant trend in the BHB
metallicity. It is possible that the BHB stars in Yanny et al. (2009) are the
old and metal poor component of the Sgr, which is not easily distinguished
from the Galactic components with the same properties, in contrast with our
comparison sample.
### 3.4 Sgr RHB stars in the bright and faint streams
A recent paper by Koposov et al. (2012) suggests the tidal debris of the Sgr
is actually two separate streams of stars separated by $\sim 10^{\circ}$ in
the Sgr orbital coordinate system. Their work is an extension of the work of
Belokurov et al. (2006) who found two branches of the leading arm debris in
the north Galactic cap. The brighter and thicker stream is claimed to have
more than one stellar population with a large fraction being metal-rich. The
fainter and thinner stream is said to be primarily a single, metal-poor
population. No estimate of the metallicity of either stream is given by
Koposov et al. (2012), but using our RHB sample we can make a qualitative
comparison.
Figure 8: The metallicity distribution of stars belonging to the bright stream
of Koposov et al. (2012). The stars were selected based on the positions and
distances given in their Tables 1 and 2.
As we have shown, our Sgr RHB sample is somewhat metal-rich. Further, since
the brighter stream is also the more metal-rich one according to Koposov et
al. (2012), we expect our RHB sample to be composed primarily of stars from
this stream. To check if this is the case, we separate our sample into stars
belonging to the bright and faint streams using the positions and distances
given in Tables 1 and 2 of Koposov et al. (2012). There are 84 stars (80, 2
and 2 from trailing arm, leading arm and overlapping group, respectively) in
the bright stream and 5 stars (all from the trailing arm) in the faint stream.
The metallicity distribution of the stars in the bright stream is shown in
Figure 8. As expected, most of the stars are metal-rich.
We do not have enough stars in the faint stream to make a meaningful
comparison. This could be due to a couple of factors. One reason we may not
have many stars corresponding to the faint stream is because our selection
criteria are based on the model of Law & Majewski (2010). This selection may
preclude these stars simply based on positions and/or kinematics. Another
possibility is that the metal-poor faint stream has little or no RHB
component. A more detailed description of the two streams is necessary in
order to distinguish between these two possibilities.
## 4 Error Estimate and Model Test
### 4.1 Comparing with Besançon model
In order to estimate the level of contamination from halo RHB stars we use the
Besançon model of the Galaxy (Robin et al., 2003). We selected stars from all
possible Galactic components and applied our selection criteria mentioned
above.
We find that the possible contribution from the halo in our sample varies
greatly depending on the area. In particular, the leading arm areas we select
will suffer from more contamination than the trailing arm areas because of the
closer distances, lower velocities, and wider spread in velocities, all of
which will increase the number of expected halo stars. Further, in the second
wraps of the tidal tails the model constraints are not as strong and therefore
allow for more contamination. Our cleanest sample is that for trailing arm 1
in part because of the larger distances, but more importantly, from the narrow
range of velocities with large negative values. We also have the largest
sample of RHB stars in trailing arm 1 so we expect the results from this area
to be the best and most robust.
The fact that our metallicity gradients for trailing arm 1 and leading arm 1
are so similar and agree within errors means that contamination in our sample
is small and/or has little effect on our results. We also point out that halo
contamination is not unique to our RHB sample and disentangling the halo
component from the Sgr component is very difficult since the stars are at the
same distances and have the same velocities. Previous work using similar
selection criteria as our work will also suffer from the same problem (such as
Yanny et al. (2009); Monaco et al. (2007); Keller et al. (2010); Correnti et
al. (2010); Carlin et al. (2012); Koposov et al. (2012)).
### 4.2 Error Analysis
In the current models (especially in Law & Majewski (2010)), the younger
segments of tidal debris are constrained to match the 2MASS/SDSS observations
while the older segments are regarded as predictions for where tidal debris
might be expected if it extends beyond that which is currently traced by
2MASS/SDSS. The dynamical ’age’ of a particle in the Law & Majewski (2010)
model is given by the parameter ’Pcol’ where values of Pcol $<=$ 3 correspond
to tidal debris observed by 2MASS/SDSS. In our analysis, we use stars whose
’Pcol’ range from 1 to 7 in the model. For more accuracy we could only use the
tidal debris previously observed by 2MASS/SDSS. These parts nearly correspond
to the first wrap of the leading and trailing arms. This would restrict our
results to only arm 1 of the leading and trailing arms. With this sample, the
results become stronger and thus our results are reliable. In particular, the
metallicity and [$\alpha$/Fe] gradients are only detectable in arm 1 of the
leading and trailing arms.
We vary our ranges by 10% in distance and velocity for the sample selection to
obtain a larger or smaller sample and perform the same analysis procedure. The
results are very similar with the original ones. This indicates that the
sample selection criteria are reasonable and the results are robust.
### 4.3 Distance distributions for stars with $\Lambda_{\odot}<130^{\circ}$
as a model test
$70^{\circ}<\Lambda_{\odot}\leq 80^{\circ}$
$80^{\circ}<\Lambda_{\odot}\leq 90^{\circ}$
$90^{\circ}<\Lambda_{\odot}\leq 100^{\circ}$
$100^{\circ}<\Lambda_{\odot}\leq 110^{\circ}$
$110^{\circ}<\Lambda_{\odot}\leq 120^{\circ}$
$120^{\circ}<\Lambda_{\odot}\leq 130^{\circ}$
Figure 9: The distance distribution of RHB stars for
$\Lambda_{\odot}<130^{\circ}$. Each panel shows the stars in a $10^{\circ}$
bin. The dashed lines show the range of model distances in trailing arm 1.
In our sample there are a significant number of stars in trailing arm 1 with
$\Lambda_{\odot}<130^{\circ}$, but not enough stars for good statistics in
other areas. We thus investigate the distance distributions of our RHB stars
and compare them with model predictions since we expect that Galactic stars
have a broad distribution and there should be an overdensity when the Sgr
stream passes through the Galactic field. Figure 9 shows the distance
distributions of RHB stars and dashed lines show the distance range given in
the model for Sgr trailing arm 1 for $\Lambda_{\odot}<130^{\circ}$ with a bin
width of $10^{\circ}$. One sees that almost all bins show distance peaks
within the model predicted ranges despite the significant selection effect of
the SDSS spectroscopic survey. It seems that the distance prediction in the
Law & Majewski (2010) model is correct and our sample selection of RHB stars
based on this model is reasonable.
### 4.4 The velocity dispersion at $88^{\circ}<\Lambda_{\odot}<112^{\circ}$
as a model test
Figure 10: Left panel: radial velocity as a function of longitude of the Sgr
orbital plane for our RHB stars. The polynomial fit to the distribution is
also plotted. Right panel: distribution of the differences between the
polynomial fit and the data. A Gaussian fit is also shown.
Our sample has the largest number of stars at
$88^{\circ}<\Lambda_{\odot}<112^{\circ}$, which covers a similar area as
Majewski et al. (2004) and Monaco et al. (2007). Thus, we investigate the
velocity dispersion for this area so that we can compare our result to these
works and provide a test to the model prediction. In the left panel of Figure
10, we plot Vgsr as a function of the Sgr longitude $\Lambda_{\odot}$. The
solid line is a polynomial fit to the data and it describes a characteristic
trend of decreasing Vgsr with increasing $\Lambda_{\odot}$ along the Sgr
trailing tail, as already discussed by Majewski et al. (2004) and Monaco et
al. (2007). The fit is for $88^{\circ}<\Lambda_{\odot}<112^{\circ}$ because
for $\Lambda_{\odot}<90^{\circ}$ an increase of the velocity dispersion is
evident (see Majewski et al. (2004) and Monaco et al. (2007)). The right panel
shows residuals of our sample stars with respect to the polynomial fit. The
distribution is fit with a Gaussian of width $\sigma$=9.808$\pm$1.0 km s-1
using 119 stars. Monaco et al. (2007) give a velocity dispersion of
$\sigma$=8.3$\pm$0.9 km s-1 using 41 stars with high resolution spectroscopy
and Majewski et al. (2004) give $\sigma$=10.4$\pm$1.3 km s-1 for stars with
low resolution spectroscopy. These three values are consistent within errors.
The agreement indicates that the Law & Majewski (2010) model prediction is
reasonable, which is what our sample star selection is based on. These parts
of the trailing tail are dynamically colder than the Sgr core, which has
dispersions of 11.17 km s-1 and 11.4 km s-1 in Monaco et al. (2005).
## 5 Summary
In this paper we present the properties of the metallicity and
$\alpha$-abundance distributions for a large sample of RHB stars belonging to
the Sgr tidal streams. The Sgr stars have two components in [Fe/H] while the
Galactic stars have a more prominent metal-poor one. [$\alpha$/Fe] is lower
for the Sgr stars than for Milky Way stars, especially along the trailing arm.
There are metallicity gradients along the streams of Sgr, with a value of
-(1.8 $\pm$ 0.3)$\times 10^{-3}$ dex degree-1 in trailing arm 1 and of -(1.5
$\pm$ 0.4)$\times 10^{-3}$ dex degree-1 in leading arm 1. No significant
gradient exists along trailing arm 2 or leading arm 2. Stars belonging to more
ancient wraps of the streams in arm 2 are more metal-poor.
We test the model and sample selection in four aspects as follows. First, by
comparing with the Besançon model of the Galaxy we find that contamination
from the Galactic halo is small for the largest sample of RHB stars in
trailing arm 1. Then we change the selection range for the width of the
leading and trailing arms and find no significant difference. Third we
investigated the distance distribution of RHB stars in trailing arm 1
($\Lambda_{\odot}<130^{\circ}$) and the peaks fall within the model prediction
ranges. Fourth we test the velocity dispersion for the Sgr trailing tail at
$88^{\circ}<\Lambda_{\odot}<112^{\circ}$ and found a value of
$\sigma$=9.808$\pm$1.0 km s-1, which is consistent with the results of
Majewski et al. (2004) and Monaco et al. (2007).
With the upcoming LAMOST spectroscopic survey, we can expect to analyze RHB
stars in the Sgr for an even larger sample and in different Galactic locations
in order to further study the chemical history of the Sgr galaxy.
We thank the referees for their helpful comments which significantly improved
the paper. This work was supported by the National Natural Science Foundation
of China (Grant No.11178013, 11073026, 11150110135 and 10978015), and by the
Provincial Natural Science Foundation of ShanDong (Y2008A08 and ZR2010AM006).
## References
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* Carlin et al. (2012) Carlin Jeffrey L., Majewski Steven R., Casetti-Dinescu Dana I., et al. 2012, ApJ, 744, 25
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* Chou et al. (2007) Chou M. Y., Majewski S. R., Cunha K., et al. 2007, ApJ, 670, 346
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|
arxiv-papers
| 2012-04-23T00:51:55 |
2024-09-04T02:49:30.018313
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W.B. Shi, Y.Q. Chen, K. Carrell and G. Zhao",
"submitter": "Weibin Shi",
"url": "https://arxiv.org/abs/1204.4954"
}
|
1204.5048
|
arxiv-papers
| 2012-04-23T12:57:31 |
2024-09-04T02:49:30.030586
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaoqiang Deng",
"submitter": "Shaoqiang Deng",
"url": "https://arxiv.org/abs/1204.5048"
}
|
|
1204.5133
|
# Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures
Carlos Batista carlosbatistas@df.ufpe.br Departamento de Física, Universidade
Federal de Pernambuco, 50670-901 Recife - PE, Brazil
###### Abstract
It is well known that the classification of the Weyl tensor in Lorentzian
manifolds of dimension four, the so called Petrov classification, was a great
tool to the development of general relativity. Using the bivector approach it
is shown in this article a classification for the Weyl tensor in all four-
dimensional manifolds, including all signatures and the complex case, in an
unified and simple way. The important Petrov classification then emerges just
as a particular case in this scheme. The boost weight classification is also
extended here to all signatures as well to complex manifolds. For the Weyl
tensor in four dimensions it is established that this last approach produces a
classification equivalent to the one generated by the bivector method.
Weyl tensor, Petrov classification, General relativity.
## I Introduction
Weyl tensor algebraic classification in space-times of dimension four, the
Petrov classification, provides many useful techniques to deal with general
relativity. In particular it can be used in the search of new solutions to
Einstein’s equation, the main example being typeD , where it was found all
type D vacuum solutions. This classification is also related to important
geometric properties of space-times, as shows the celebrated Goldberg-Sachs
theorem Goldberg-Sachs .
There are many methods to state the Petrov classification Stephani ;
PlebanskiBook , the original one was worked out by A. Z. Petrov Petrov and is
based on the fact that the Weyl tensor can be seen as an operator on the
bivector space. An extensive review of this approach to the Petrov
classification and a thorough analysis of the bivector space in Lorentzian
four-dimensional manifolds can be found in reference HallBook . This bivector
method is the one adopted here to generalize this classification to all four-
dimensional manifolds endowed with a metric and a Levi-Civita connection. The
method presented in this article provides an unified local classification
scheme to the Weyl tensor in four-dimensional complex manifolds and real
manifolds of all signatures.
The classification of the Weyl tensor in complex manifolds of complex
dimension four was already done in Plebanski75 , but using a spinorial method.
Weyl tensor classification in Euclidean spaces of dimension four was also
developed before Hacyan ; Karlhede , but again using different methods than
the one adopted here. The (2,2) signature case was studied in Petrov-livro ;
Law1 ; Nurwoski2 using the bivector approach, but the classification schemes
in these references are different from the one obtained here, while in Law2
the spinorial method was used to produce a classification that has direct
relation with the one presented in this article. So the classification
presented here is not new, thus the originality of the present work comes only
on the approach used. It is also important to stress the unification achieved
in this article, the classification of the different signatures comes easily
from a common origin, this certainly helps to understand the meaning of the
Weyl tensor classification and can be useful in the physical study of Wick
rotated space-times. The advantages of the bivector approach are that it is
simple to understand and it is useful in the analysis of the integrability of
null structures on algebraically special manifolds art2 . The fact that the
Weyl tensor can be seen as an operator on the bivector space is valid in all
dimensions and reference ColeyBiv took advantage of this to refine the CMPP
classification of this tensor in higher-dimensional Lorentzian manifolds.
It is worth to mention that there are other forms to classify the curvature of
a manifold other than the Petrov classification and its generalization
presented here. The Ricci tensor, for example, can be seen as an operator on
the space of vectors and the Segre classification can be used to define the
different types that this operator can have Stephani . Also the CMPP approach,
which is based on the boost transformations, is a useful form to classify any
tensor in Lorentzian spaces of all dimensions CMPP . Another important method
to classify the curvature is by means of the scalar invariants
ColeyScInvariant .
In section II it will be reviewed some properties of the bivectors and of the
Weyl tensor that will be required for the development of the classification.
Section III treats the complex manifolds, this case being the paradigm to the
other classifications. In sections IV, V and VI the classification of
Lorentzian, Euclidean and (2,2) signatures are respectively done, they are
seen as particular cases of the complex one. Finally, in section VII the
classification scheme based on the boost weight obtained in ColeyPSEUD in
enhanced to include also Complex and Euclidean manifolds. It is also shown
that in four dimensions the Weyl tensor classification obtained by the boost
weight technique is equivalent to the one furnished by the bivector approach
in all cases.
## II Bivectors in 4 dimensions
Let $M$ be a differential manifold of dimension four endowed with a metric
$g_{\mu\nu}$ of signature $s$. In Euclidean case $s=4$, in Lorentzian $s=2$
and in the case (2,2), where the metric can be put in the form
$diag(+,+,-,-)$, the signature is $s=0$. The volume form,
$\epsilon_{\mu\nu\rho\sigma}$, is a completely skew-symmetric tensor whose
non-zero components in an orthonormal frame are $\pm 1$. This tensor obeys the
following equation:
$\epsilon^{\mu_{1}...\mu_{p}\nu_{1}...\nu_{4-p}}\,\epsilon_{\mu_{1}...\mu_{p}\sigma_{1}...\sigma_{4-p}}=(-1)^{\frac{4-s}{2}}\,p!(4-p)!\delta_{\sigma_{1}}^{[\nu_{1}}...\delta_{\sigma_{4-p}}^{\nu_{4-p}]}\,.$
(1)
A contravariant tensor of rank two which is skew-symmetric in its indices,
$B^{\mu\nu}=-B^{\nu\mu}$, is called a bivector. In what follows it will also
be used the term bivectors to denote the covariant and the mixed versions of
these tensors. This is not inconsistent because the metric provides a map
between these distinct kinds of tensors. In four dimensions the dual of a
bivector $B_{\mu\nu}$ is defined by
$\widetilde{B}_{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}B^{\rho\sigma}\,.$
(2)
It is easy to see that given any two bivectors, $B_{\mu\nu}$ and $F_{\mu\nu}$,
we have
$\widetilde{B}_{\mu\nu}F^{\mu\nu}=B_{\mu\nu}\widetilde{F}^{\mu\nu}.$ (3)
Taking the double dual of a bivector and using equation (1) we get:
$\widetilde{\widetilde{B}_{\mu\nu}}=\frac{1}{4}\epsilon_{\mu\nu\rho\sigma}\epsilon^{\rho\sigma\alpha\beta}B_{\alpha\beta}=(-1)^{\frac{s}{2}}B_{\mu\nu}\,.$
(4)
Let us concentrate at a specific point of the manifold, $p\in M$, and denote
by $T_{p}$M the tangent space at this point. From now on all considerations of
this paper are at this arbitrary point. In four dimensions the space of
bivectors constructed from $T_{p}M$ has 6 dimensions, denote this space and
its complexification by $\mathfrak{B}$ and $\mathfrak{B}_{\mathbb{C}}$
respectively. The action of the dual operation can be extended form
$\mathfrak{B}$ to $\mathfrak{B}_{\mathbb{C}}$ in the usual way. Let us
investigate the consequences of equation (4) in the various signatures and in
the complex case.
* •
Lorentzian signature
In this case we shall let the duality be an operation in the complexification
of the bivector space,
$\sim\,$:$\,\mathfrak{B}_{\mathbb{C}}\rightarrow\mathfrak{B}_{\mathbb{C}}$.
Since $\widetilde{\widetilde{B}_{\mu\nu}}=-B_{\mu\nu}$ when $s=2$ it follows
that the eigenvalues of $\sim$ are $\pm$ i. This permits to split the space
$\mathfrak{B}_{\mathbb{C}}$ into a direct sum of invariant subspaces under the
dual operation:
$\mathfrak{B}_{\mathbb{C}}=\mathfrak{D}^{+}\oplus\,\mathfrak{D}^{-},$ (5)
$\mathfrak{D}^{+}=\\{Z^{+}_{\mu\nu}\in\mathfrak{B}_{\mathbb{C}}|\widetilde{Z^{+}}_{\mu\nu}=iZ^{+}_{\mu\nu}\\}\;;\;\mathfrak{D}^{-}=\\{Z^{-}_{\mu\nu}\in\mathfrak{B}_{\mathbb{C}}|\widetilde{Z^{-}}_{\mu\nu}=-iZ^{-}_{\mu\nu}\\}.$
The complex dimension of both spaces $\mathfrak{D}^{+}$ and $\mathfrak{D}^{-}$
is three. We call the former the space of self-dual bivectors and the later
the space of anti-self-dual bivectors.
* •
Euclidean and (2,2) signatures
In these cases we do not need to complexify $\mathfrak{B}$ in order to split
it into the sum of invariant subspaces by the dual operation. This happens
because by (4) we have $\widetilde{\widetilde{B}_{\mu\nu}}=B_{\mu\nu}$ when
$s=4$ or 0, so that the eigenvalues of the duality operator,
$\sim\,$:$\,\mathfrak{B}\rightarrow\mathfrak{B}$, are real, $\pm$1\. This
enables us to split $\mathfrak{B}$ into a direct sum of two three-dimensional
invariant subspaces under the dual operation:
$\mathfrak{B}=\mathfrak{D}^{+}\oplus\,\mathfrak{D}^{-},$
$\mathfrak{D}^{+}=\\{Z^{+}_{\mu\nu}\in\mathfrak{B}|\widetilde{Z^{+}}_{\mu\nu}=Z^{+}_{\mu\nu}\\}\;;\;\mathfrak{D}^{-}=\\{Z^{-}_{\mu\nu}\in\mathfrak{B}|\widetilde{Z^{-}}_{\mu\nu}=-Z^{-}_{\mu\nu}\\}.$
* •
Complex case
In complex manifolds of complex dimension four the duality operator is such
that its square can give both results, the identity or minus the identity.
This happens because in this kind of manifold vectors of an orthonormal frame
can be multiplied by a factor of $i$, changing the apparent signature of the
metric. The important thing is that once a choice of volume form is made, the
bivector space $\mathfrak{B}_{\mathbb{C}}$, can be split in a direct sum of
invariant subspaces, under the duality operation, of complex dimension three,
as in equation (5). Here it will be assumed, without loss of generality, that
the volume form is conveniently chosen is such a way that the dual operation
squared gives the identity map.
Note that in the various signatures the same symbols, $\mathfrak{D}^{+}$ and
$\mathfrak{D}^{-}$, were used to denote different spaces. This shall make no
confusion since it will be clear in the context which space it is meant. It is
worth to keep in mind that there is nothing intrinsic to the manifold which
distinguishes these two spaces, it is just a choice of orientation sign. If we
change the sign of $\epsilon_{\mu\nu\rho\sigma}$ the spaces $\mathfrak{D}^{+}$
and $\mathfrak{D}^{-}$ are interchanged. Note also that (3) implies that if
$Z^{+}_{\mu\nu}\in\mathfrak{D}^{+}$ and $Z^{-}_{\mu\nu}\in\mathfrak{D}^{-}$
then $Z^{+}_{\mu\nu}Z^{-\mu\nu}=0$. Until the end of this section the
calculations are valid to all signatures and also to the complex case.
The Weyl tensor of the manifold $(M,g_{\mu\nu})$ has the following
symmetries111It is assumed that the connection on the manifold is torsion-free
and compatible with the metric, the so called Levi-Civita connection.:
$C_{\mu\nu\rho\sigma}=C_{[\mu\nu]\rho\sigma}=C_{\mu\nu[\rho\sigma]}=C_{\rho\sigma\mu\nu}\;;\;C^{\mu}_{\phantom{\mu}\nu\mu\sigma}=0\;;\;C_{\mu[\nu\rho\sigma]}=0.$
(6)
Because of the skew-symmetry in the first and second pairs of the Weyl tensor
indices it is natural to define
$C_{\mu\nu\widetilde{\rho\sigma}}\equiv\frac{1}{2}\epsilon_{\rho\sigma\alpha\beta}C_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\;;\;C_{\widetilde{\mu\nu}\rho\sigma}\equiv\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}C^{\alpha\beta}_{\phantom{\alpha\beta}\rho\sigma}.$
(7)
It is then trivial to see that
$C_{\mu\nu\widetilde{\rho\sigma}}=C_{\widetilde{\rho\sigma}\mu\nu}$, but it
less obvious that
$C_{\mu\nu\widetilde{\rho\sigma}}=C_{\widetilde{\mu\nu}\rho\sigma}$, let us
prove this.
$\textrm{Define},\;\;T_{\mu\nu\rho\sigma}\equiv
C_{\mu\nu\widetilde{\rho\sigma}}-C_{\widetilde{\mu\nu}\rho\sigma}.$
Note that
$T_{\mu\nu\rho\sigma}=T_{[\mu\nu]\rho\sigma}=T_{\mu\nu[\rho\sigma]}=-T_{\rho\sigma\mu\nu}$.
$\epsilon^{\mu\rho\alpha\beta}T_{\mu\nu\rho\sigma}=\epsilon^{\mu\rho\alpha\beta}[\frac{1}{2}\epsilon_{\rho\sigma\theta\gamma}C_{\mu\nu}^{\phantom{\mu\nu}\theta\gamma}-\frac{1}{2}\epsilon_{\mu\nu\theta\gamma}C^{\theta\gamma}_{\phantom{\theta\gamma}\rho\sigma}]=$
$=-3(-1)^{\frac{s}{2}}(\delta_{\sigma}^{[\mu}\delta_{\theta}^{\alpha}\delta_{\gamma}^{\beta]}C_{\mu\nu}^{\phantom{\mu\nu}\theta\gamma}+\delta_{\nu}^{[\rho}\delta_{\theta}^{\alpha}\delta_{\gamma}^{\beta]}C^{\theta\gamma}_{\phantom{\theta\gamma}\rho\sigma})=-(-1)^{\frac{s}{2}}(C_{\sigma\nu}^{\phantom{{\sigma\nu}}\alpha\beta}+C^{\alpha\beta}_{\phantom{\alpha\beta}\nu\sigma})=0\,,$
(8)
where in the second equality it was used (1) and in the third (6). In
particular, equation (8) means that
$T_{\mu\nu\rho\sigma}=T_{\rho\nu\mu\sigma}$. Further if we contract $\alpha$
and $\nu$ in equation (8) we get $T_{[\mu\nu\rho]\sigma}=0$. Using the
symmetries of $T_{\mu\nu\rho\sigma}$ in this equation it is easily seen that
this tensor vanishes, which implies the wanted relation. Summing-up we have,
$C_{\mu\nu\widetilde{\rho\sigma}}=C_{\widetilde{\rho\sigma}\mu\nu}\;\;;\;\;C_{\mu\nu\widetilde{\rho\sigma}}=C_{\widetilde{\mu\nu}\rho\sigma}.$
(9)
The above identities are the most important relations of this paper, since
from them it is trivial to deduce that the Weyl operator, to be defined next
section, has the interesting property of sending elements of
$\mathfrak{D}^{\pm}$ into elements of $\mathfrak{D}^{\pm}$.
## III Complex Weyl Tensor
The intent of this section is to produce an algebraic classification for the
Weyl tensor in complex manifolds. This was already done in Plebanski75 by
means of Penrose’s spinor techniques, while here the bivector approach is
taken, but at the end the possible algebraic types are the same. The complex
case will be used in forthcoming sections to extract the Weyl tensor
classification in real manifolds of arbitrary signature in an easy and quick
way.
Let us assume that $(M,g_{\mu\nu})$ is a complex manifold of complex dimension
four. The metric can be complex, so that the Weyl tensor is, in general, not
real. The classification scheme done in this article is based on the
possibility of seeing the Weyl tensor as an operator on the complex bivector
space, $C:\,\mathfrak{B}_{\mathbb{C}}\rightarrow\mathfrak{B}_{\mathbb{C}}$,
with action given by:
$C(B)=F\;\;\textrm{where}\;\;F_{\mu\nu}\equiv
C_{\mu\nu\rho\sigma}B^{\rho\sigma}.$ (10)
By making an analysis of the non-trivial solutions of the eigenvalue equation
$C(B)=\lambda B$ it is possible to classify the Weyl tensor. But this approach
may be laborious if the Weyl operator action is studied in the whole
$\mathfrak{B}_{\mathbb{C}}$, as A. Z. Petrov did in the Lorentzian case Petrov
. However a shortcut can be taken using the restrictions of this operator to
the subspaces of self-dual and anti-self-dual bivectors. This makes possible
to attack this eigenvalue problem working with two $3\times 3$ matrices
instead of the $6\times 6$ matrix which represents the Weyl operator on the
full bivector space.
Many calculations in this section will be done in such a way to include not
only the complex manifold case, but also real manifolds of all signatures,
this shall be clear in the context. Let $Z^{+}_{\mu\nu}$ be a self-dual
bivector, this means that $\widetilde{Z^{+}}_{\mu\nu}=\varepsilon
Z^{+}_{\mu\nu}$, where $\varepsilon$ is $+i$ in Lorentzian signature and $+1$
in the other two signatures as well in the complex case. Then if $T_{\mu\nu}$
is the image of this bivector under the Weyl operator,
$T_{\mu\nu}=C_{\mu\nu\rho\sigma}Z^{+\rho\sigma}$, it must be self-dual:
$\widetilde{T}_{\mu\nu}=C_{\widetilde{\mu\nu}\rho\sigma}Z^{+\rho\sigma}=C_{\mu\nu\widetilde{\rho\sigma}}Z^{+\rho\sigma}=C_{\mu\nu\rho\sigma}\widetilde{Z}^{+\rho\sigma}=\varepsilon
C_{\mu\nu\rho\sigma}Z^{+\rho\sigma}\equiv\varepsilon T_{\mu\nu}.$ (11)
Where in the second equality it was used equation (9) , in the third eq. (3)
and in the fourth it was used the self-duality of $Z^{+}_{\mu\nu}$. Equation
(11) implies that $T_{\mu\nu}$ is self-dual. In an analogous way this operator
sends anti-self-dual bivectors into anti-self-dual bivectors. This together
with (5) guarantees that the Weyl operator is the direct sum of its
restrictions to $\mathfrak{D}^{+}$ and $\mathfrak{D}^{-}$,
$C_{\mu\nu\rho\sigma}=C^{+}_{\mu\nu\rho\sigma}+C^{-}_{\mu\nu\rho\sigma}\;\;;\;C^{\pm}_{\mu\nu\rho\sigma}=\frac{1}{2}(C_{\mu\nu\rho\sigma}\pm\varepsilon^{3}C_{\mu\nu\widetilde{\rho\sigma}}).$
(12)
Where $C^{+}(C^{-})$ is called the self-dual(anti-self-dual) part of the Weyl
tensor, its action in $\mathfrak{D}^{-}$ ($\mathfrak{D}^{+}$) gives zero. For
example, if $Z^{-}_{\mu\nu}$ is an element of $\mathfrak{D}^{-}$ then using
(3) and the relation $\widetilde{Z^{-}}=-\varepsilon Z^{-}$, we have:
$C^{+}_{\mu\nu\rho\sigma}Z^{-\rho\sigma}=\frac{1}{2}(C_{\mu\nu\rho\sigma}Z^{-\rho\sigma}+\varepsilon^{3}C_{\mu\nu\rho\sigma}\widetilde{Z}^{-\rho\sigma})=\frac{1}{2}(C_{\mu\nu\rho\sigma}+\varepsilon^{3}(-\varepsilon)C_{\mu\nu\rho\sigma})Z^{-\rho\sigma}=0\,.$
Thus to classify the Weyl tensor is equivalent to classify the operators
$C^{+}:\mathfrak{D}^{+}\rightarrow\mathfrak{D}^{+}$ and
$C^{-}:\mathfrak{D}^{-}\rightarrow\mathfrak{D}^{-}$. Now let us see that the
these two operators have vanishing trace. Using equations (6), (7) and (12) we
see that:
$C^{\pm\mu}_{\phantom{\pm\mu}\nu\mu\sigma}=\frac{1}{2}(C^{\mu}_{\phantom{\mu}\nu\mu\sigma}\pm\varepsilon^{3}\frac{1}{2}\epsilon_{\mu\sigma\alpha\beta}C^{\mu\phantom{\nu}\alpha\beta}_{\phantom{\mu}\nu})=\mp\varepsilon^{3}\frac{1}{4}\epsilon_{\mu\sigma\alpha\beta}C_{\nu}^{\phantom{\nu}[\mu\alpha\beta]}=0.$
In particular this implies that $C^{\pm\mu\nu}_{\phantom{\pm\mu\nu}\mu\nu}=0$,
proving that the operators $C^{\pm}$ have zero trace. Below it will be
explicitly shown that in complex manifolds these two operators are independent
of each other and have in general no simplifying property other than the
vanishing trace. Thus the Weyl tensor classification in complex manifolds
reduces to the classification of two trace-free operators acting on three-
dimensional spaces. The three eigenvalues,
$(\lambda_{1},\lambda_{2},\lambda_{3})$, of a trace-free operator acting on a
three-dimensional vector space can have the following features: (a) All
eigenvalues are different,
$(\lambda_{1},\lambda_{2},-\lambda_{1}-\lambda_{2})$; (b) There is a non-zero
repeated eigenvalue, $(\lambda,\lambda,-2\lambda)$ (c) All eigenvalues are
zero, $(0,0,0)$. In the case (a) the operator is called type I, in the case
(b) there are two algebraically distinct possibilities, if the operator can be
put in a diagonal form it is type D, otherwise it is type II, finally the case
(c) allows three distinctions, when the operator vanishes it is called type O,
when it is non-zero but with vanishing square it is type N and when its square
is different from zero it is said to be type III. These types are summarized
by the bellow algebraic characteristics:
$\left\\{\begin{array}[]{ll}\textbf{Type O}^{\pm}\rightarrow\;C^{\pm}=0\\\
\textbf{Type I}^{\pm}\;\,\rightarrow\;C^{\pm}$ allows 3 distinct eigenvalues
$\\\ \textbf{Type D}^{\pm}\rightarrow C^{\pm}$ is diagonalizable with a
repeated non-zero eigenvalue $\\\ \textbf{Type II}^{\pm}\rightarrow C^{\pm}$
is non-diagonalizable with a repeated non-zero eigenvalue$\\\ \textbf{Type
III}^{\pm}\rightarrow\,(C^{\pm})^{3}=0\,$and$\,(C^{+})^{2}\neq 0$\,(all
eigenvalues are zero)$\\\ \textbf{Type
N}^{\pm}\rightarrow\,(C^{\pm})^{2}=0\,$and$\,C^{+}\neq 0$ \,(all eigenvalues
are zero)$\end{array}\right.$
The classification of the full Weyl tensor is then made by a composition of
these algebraic types. For example, we say that this tensor is type (I,O) if
$C^{+}$ is type I+ and $C^{-}$ is type O-. But it is important to note that
type (I,O) is intrinsically equivalent to type (O,I), since the operators
$C^{+}$ and $C^{-}$ are interchanged by a simple change of orientation sign,
and from an intrinsic point of view this sign choice is arbitrary. So at the
end there are 21 possible classifications:
(O,O); (O,I); (O,D);(O,II); (O,III); (O,N); (I;I); (I;D); (I,II); (I,III);
(I,N);
(D;D); (D,II); (D,III); (D,N); (II,II); (II,III); (II,N); (III,III); (III,N);
(N,N).
It is worth to note that such classification was made at a specific point of
the manifold, $p\in M$, and in general the Weyl tensor can change its type
from point to point. Now to understand better this classification let us find
explicit representations for the operators $C^{\pm}$. In the complex tangent
space it is always possible to construct a null tetrad frame,
$\\{l,n,m_{1},m_{2}\\}$, defined to be such that the only non-zero
contractions are
$l^{\mu}n_{\mu}=1\;\;\textrm{and}\;\;m_{1}^{\mu}m_{2\mu}=-1\,.$ (13)
In particular all basis vectors are null. For example, if
$\\{e_{(1)},e_{(2)},e_{(3)},e_{(4)}\\}$ is an orthonormal frame such that
$e_{(a)}^{\mu}e_{(b)\mu}=\delta_{ab}$, then the vectors
$l=\frac{1}{\sqrt{2}}(e_{(1)}+ie_{(2)})\,,\,n=\frac{1}{\sqrt{2}}(e_{(1)}-ie_{(2)})\,,\,m_{1}=\frac{1}{\sqrt{2}}(e_{(3)}+ie_{(4)})\;\textrm{and}\;m_{2}=\frac{-1}{\sqrt{2}}(e_{(3)}-ie_{(4)})\,$
form a null tetrad frame.
The Weyl tensor in a real four-dimensional manifold has 10 independent real
components, because of the symmetries shown in equation (6), while in complex
manifolds it has 10 independent complex components. In the complex case it is
convenient to choose these independent components to be
$\Psi^{+}_{0}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m_{1}^{\nu}l^{\rho}m_{1}^{\sigma}\;;\;\Psi^{+}_{1}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}l^{\rho}m_{1}^{\sigma}\;;\;\Psi^{+}_{2}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m_{1}^{\nu}m_{2}^{\rho}n^{\sigma}$
$\Psi^{+}_{3}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}m_{2}^{\rho}n^{\sigma}\;;\;\Psi^{+}_{4}\equiv
C_{\mu\nu\rho\sigma}n^{\mu}m_{2}^{\nu}n^{\rho}m_{2}^{\sigma}$ (14)
$\Psi^{-}_{0}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m_{2}^{\nu}l^{\rho}m_{2}^{\sigma}\;;\;\Psi_{1}^{-}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}l^{\rho}m_{2}^{\sigma}\;;\;\Psi_{2}^{-}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m_{2}^{\nu}m_{1}^{\rho}n^{\sigma}$
$\Psi_{3}^{-}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}m_{1}^{\rho}n^{\sigma}\;;\;\Psi_{4}^{-}\equiv
C_{\mu\nu\rho\sigma}n^{\mu}m_{1}^{\nu}n^{\rho}m_{1}^{\sigma}\,.$
These scalars are called the Weyl scalars. The main advantage of using a null
tetrad frame when dealing with the Weyl tensor is that the trace-free property
and the Bianchi identity, (6), are easily written. Equations below are the
explicit forms of these two properties of the Weyl tensor in this kind of
frame:
$\displaystyle C_{1l1n}$ $\displaystyle=$ $\displaystyle
C_{2l2n}=C_{l1l2}=C_{n1n2}=0\;;$ $\displaystyle C_{1l12}$ $\displaystyle=$
$\displaystyle\Psi^{+}_{1}\;;\;C_{2l21}=\Psi_{1}^{-}\;;\;C_{1n12}=\Psi_{3}^{-}\;;\;C_{2n21}=\Psi^{+}_{3}\;;$
(15) $\displaystyle C_{1212}$ $\displaystyle=$ $\displaystyle
C_{lnln}=\Psi^{+}_{2}+\Psi_{2}^{-}\;;\;C_{12ln}=\Psi_{2}^{-}-\Psi^{+}_{2}.$
Where, for example, $C_{12ln}$ means
$C_{\mu\nu\rho\sigma}m^{\mu}_{1}m^{\nu}_{2}l^{\rho}n^{\sigma}$. In particular,
from equation (III) we see that the Weyl scalars, defined in (14), are indeed
independent of each other in complex manifolds, so that they can be used to
represent the Weyl tensor degrees of freedom. Choosing the manifold
orientation to be such that
$\epsilon_{\mu\nu\rho\sigma}e^{\mu}_{(1)}e^{\nu}_{(2)}e^{\rho}_{(3)}e^{\sigma}_{(4)}=1,$
then convenient bases for spaces $\mathfrak{D}^{+}$ and $\mathfrak{D}^{-}$ are
respectively given by:
$Z^{+1}_{\mu\nu}=2l_{[\mu}m_{1\nu]}\;;\;Z^{+2}_{\mu\nu}=2m_{2[\mu}n_{\nu]}\;;\;Z^{+3}_{\mu\nu}=\sqrt{2}n_{[\mu}l_{\nu]}+\sqrt{2}m_{1[\mu}m_{2\nu]}$
(16)
$Z^{-1}_{\mu\nu}=2l_{[\mu}m_{2\nu]}\;;\;Z^{-2}_{\mu\nu}=2m_{1[\mu}n_{\nu]}\;;\;Z^{-3}_{\mu\nu}=\sqrt{2}n_{[\mu}l_{\nu]}+\sqrt{2}m_{2[\mu}m_{1\nu]}.$
(17)
The only non-zero full contractions of the elements of these bases are:
$Z^{+1\mu\nu}Z^{+2}_{\mu\nu}=2\;;\;Z^{+3\mu\nu}Z^{+3}_{\mu\nu}=-2\;;\;Z^{-1\mu\nu}Z^{-2}_{\mu\nu}=2\;;\;Z^{-3\mu\nu}Z^{-3}_{\mu\nu}=-2\,.$
(18)
Using equations (14), (III), (16) and (17) it is straightforward to see that
the Weyl scalars can be written in the form
$\displaystyle\Psi^{\pm}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}C_{\mu\nu\rho\sigma}Z^{\pm 1\mu\nu}Z^{\pm
1\rho\sigma}\;;\;\Psi^{\pm}_{1}=\frac{-\sqrt{2}}{8}C_{\mu\nu\rho\sigma}Z^{\pm
1\mu\nu}Z^{\pm 3\rho\sigma}$ $\displaystyle\Psi^{\pm}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}C_{\mu\nu\rho\sigma}Z^{\pm 1\mu\nu}Z^{\pm
2\rho\sigma}=\frac{1}{8}C_{\mu\nu\rho\sigma}Z^{\pm 3\mu\nu}Z^{\pm
3\rho\sigma}$ (19) $\displaystyle\Psi^{\pm}_{3}$ $\displaystyle=$
$\displaystyle\frac{-\sqrt{2}}{8}C_{\mu\nu\rho\sigma}Z^{\pm 2\mu\nu}Z^{\pm
3\rho\sigma}\;;\;\Psi^{\pm}_{4}=\frac{1}{4}C_{\mu\nu\rho\sigma}Z^{\pm
2\mu\nu}Z^{\pm 2\rho\sigma}.$
The matrix representations of operators $C^{\pm}$ in these bivector bases are
$\mathcal{C}^{\pm\phantom{j}i}_{\;\;j}$, defined to be such that
$C^{\pm}(Z^{\pm i})=\mathcal{C}^{\pm\phantom{j}i}_{\;\;j}Z^{\pm j}$ or, in
component form, $\mathcal{C}^{\pm\phantom{j}i}_{\;\;j}Z^{\pm
j}_{\mu\nu}=C_{\mu\nu\rho\sigma}Z^{\pm i\,\rho\sigma}$. Where it was used the
fact that $C^{\pm}_{\mu\nu\rho\sigma}Z^{\pm
i\,\rho\sigma}=C_{\mu\nu\rho\sigma}Z^{\pm i\,\rho\sigma}$, which stems from
(12) and from the triviality of $C^{\pm}$ action on the spaces of opposite
duality. Now using these definitions and equations (18) and (III) we get that
the matrix representations of operators $C^{\pm}$ are:
$\mathcal{C}^{\pm\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}\Psi^{\pm}_{2}&\Psi^{\pm}_{4}&-\sqrt{2}\Psi^{\pm}_{3}\\\
\Psi^{\pm}_{0}&\Psi^{\pm}_{2}&-\sqrt{2}\Psi^{\pm}_{1}\\\
\sqrt{2}\Psi^{\pm}_{1}&\sqrt{2}\Psi^{\pm}_{3}&-2\Psi^{\pm}_{2}\\\
\end{array}\right].$ (20)
Note that these matrices have vanishing trace, as it should be. In next
section these matrix representations will be used to find a canonical form for
each classification type seen above. Complex results will be used to generate
a Weyl tensor classification on real four-dimensional manifolds in the
forthcoming sections.
## IV Lorentzian Case, The Petrov Classification
Real Lorentzian manifolds are characterized by the local existence of a real
frame {$e_{t},e_{x},e_{y},e_{z}$} such that the only non-zero contractions are
$e^{\mu}_{t}e_{t\mu}=1$ and
$e^{\mu}_{x}e_{x\mu}=e^{\mu}_{y}e_{y\mu}=e^{\mu}_{z}e_{z\mu}=-1$. This kind of
frame is called a Lorentz frame or a tetrad. From these real vectors it is
possible to construct a null tetrad frame in the complexified tangent space,
$\mathbb{C}\otimes T_{p}M$, given by
$l=\frac{1}{\sqrt{2}}(e_{t}+e_{z})\,,\,n=\frac{1}{\sqrt{2}}(e_{t}-e_{z})\,,\,m_{1}=\frac{1}{\sqrt{2}}(e_{x}+ie_{y})\;\textrm{and}\;m_{2}=\frac{1}{\sqrt{2}}(e_{x}-ie_{y})$.
Since this null tetrad frame has exactly the same inner products as the null
tetrad frame seen in the previous section it follows that basically the whole
formalism is the same in the present case. The differences that must be
stressed are that now the Weyl tensor is assumed to be real,
$\overline{C}_{\mu\nu\rho\sigma}=C_{\mu\nu\rho\sigma}$, and the basis vectors
$l$ and $n$ are real while $m_{1}$ and $m_{2}$ are complex and conjugates to
each other, $\overline{m_{1}}=m_{2}$. These simple observations make huge
restrictions on the possible classifications of the Weyl tensor, since now the
Weyl scalars $\Psi^{+}_{a}$ and $\Psi^{-}_{a}$ are complex conjugates to each
other, $\overline{\Psi^{+}_{a}}=\Psi^{-}_{a}$, which stems from their
definitions, (14). This reduces the number of independent components of the
Weyl tensor from 10 complex numbers to 5 complex numbers, or 10 real
parameters, as it should be in a real manifold.
So by equation (20) we have that $\mathcal{C}^{+\phantom{j}i}_{\;j}$ and
$\mathcal{C}^{-\phantom{j}i}_{\;j}$ are complex conjugates to each other222In
particular this implies that if $\lambda$ is an eigenvalue of the full Weyl
operator then its complex conjugate, $\overline{\lambda}$, is also an
eigenvalue. This result was proved by A. Z. Petrov in Petrov . . This implies
that if the operator $C^{+}$ has certain algebraic type then the operator
$C^{-}$ will have the same type. Thus the only allowable classifications are
(O,O); (I,I);(D,D); (II,II); (III,III) and (N,N). The notation can now be
condensed and we say that the Weyl tensor in a real Lorentzian manifold can
have just six types of classification: O, I, D, II, III and N, where type D,
for example, means (D,D). These types are the well known Petrov types, which
are widely studied since its discovery in 1954 Petrov 333When created by A. Z.
Petrov the classification consisted only of three non-trivial types, called
types 1, 2 and 3. Type 1 later was split into types I and D, type 2 was
refined to give types II and N, while type 3 was the now called type III Bel
..
There are various ways to approach the Petrov classification Stephani ,
probably the most elegant is Penrose’s spinor method Penrose . It is worth to
make a connection between the classification as presented in this text and
some of the other ones. In spinorial approach it is easy to see that the
different Petrov types are featured by the possibility of choosing convenient
bases where some Weyl scalars are zero. The final result of this analysis is
summarized in the table below.
Weyl Scalars that Can be Made to Vanish by a Suitable Choice of Basis
Type O - All | Type I - $\Psi^{\pm}_{0},\Psi^{\pm}_{4}$ | Type D - $\Psi^{\pm}_{0},\Psi^{\pm}_{1},\Psi^{\pm}_{3},\Psi^{\pm}_{4}$
---|---|---
Type II -$\Psi^{\pm}_{0},\Psi^{\pm}_{1},\Psi^{\pm}_{4}$ | Type III - $\Psi^{\pm}_{0},\Psi^{\pm}_{1},\Psi^{\pm}_{2},\Psi^{\pm}_{4}\>$ | Type N - $\Psi^{\pm}_{0},\Psi^{\pm}_{1},\Psi^{\pm}_{2},\Psi^{\pm}_{3}$
This approach to the characterization of Petrov types, which identify the many
types with the possibility to annihilate different Weyl tensor components, is
the origin of the main extension of this classification to higher dimensions,
the so called CMPP classification CMPP . Now it is easy matter to see that the
classification obtained here using the algebraic properties of the operators
$C^{\pm}$ is equivalent to the classification depicted in the above table. For
example, if the Weyl tensor is Petrov type N according to the above table it
is possible, using equation (20), to find a basis such that the
representations of $C^{+}$ and $C^{-}$ are respectively:
$\mathcal{C}^{+\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}0&\Psi^{+}_{4}&0\\\
0&0&0\\\ 0&0&0\\\
\end{array}\right]\;\;\textrm{and}\;\;\mathcal{C}^{-\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}0&\overline{\Psi^{+}_{4}}&0\\\
0&0&0\\\ 0&0&0\\\ \end{array}\right].$
Where $\Psi^{+}_{4}\neq 0$ and it was used the fact that
$\overline{\Psi^{+}_{a}}=\Psi^{-}_{a}$ in Lorentzian signature. It is now
trivial to see that $(C^{+})^{2}=0=(C^{-})^{2}$, which means that the Weyl
tensor, according to classification defined in section III, is type (N,N), or
type N in the condensed notation. This illustrates the equality of type N
Petrov classification in both approaches. It is also easy to verify the
equivalence of the other types.
As a final comment, note that the above table enables to determine the
canonical forms of operators $C^{\pm}$ matrix representations in the various
classification types. For example, type (II,II) is characterized by the
following canonical forms:
$\mathcal{C}^{+\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}\Psi^{+}_{2}&0&-\sqrt{2}\Psi^{+}_{3}\\\
0&\Psi^{+}_{2}&0\\\ 0&\sqrt{2}\Psi^{+}_{3}&-2\Psi^{+}_{2}\\\
\end{array}\right]\;\;\textrm{and}\;\;\mathcal{C}^{-\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}\overline{\Psi^{+}_{2}}&0&-\sqrt{2}\overline{\Psi^{+}_{3}}\\\
0&\overline{\Psi^{+}_{2}}&0\\\
0&\sqrt{2}\overline{\Psi^{+}_{3}}&-2\overline{\Psi^{+}_{2}}\\\
\end{array}\right],$
with $\Psi^{+}_{2}\neq 0\neq\Psi^{+}_{3}$. Since a complex manifold of complex
dimension four can always be seen locally as a complexified Lorentzian
manifold of real dimension four, it follows that canonical forms to the
complex case can also be easily found by using the above table. For example,
if the Weyl tensor on a complex manifold has type (D,III) then it can be found
a basis for space $\mathfrak{D}^{+}$ and an independent basis for
$\mathfrak{D}^{-}$ (the bases of these two spaces may be not related to each
other as they are in equations (16) and (17)) where the representations of the
operators $C^{\pm}$ are:
$\mathcal{C}^{+\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}\Psi^{+}_{2}&0&0\\\
0&\Psi^{+}_{2}&0\\\ 0&0&-2\Psi^{+}_{2}\\\
\end{array}\right]\;\;\textrm{and}\;\;\mathcal{C}^{-\phantom{j}i}_{\;\;j}=2\left[\begin{array}[]{ccc}0&0&-\sqrt{2}\Psi^{-}_{3}\\\
0&0&0\\\ 0&\sqrt{2}\Psi^{-}_{3}&0\\\ \end{array}\right].$
As a last comment note also that by definition of types I± in the preceding
section, the matrix representations of the operators $C^{\pm}$ with these
types can always be put in a diagonal form with three different eigenvalues
whose sum is zero. So
$diag(\lambda^{\pm}_{1},\lambda^{\pm}_{2},\lambda^{\pm}_{3})$, with
$\lambda^{\pm}_{i}\neq\lambda^{\pm}_{j}$ if $i\neq j$ and
$\lambda^{\pm}_{1}+\lambda^{\pm}_{2}+\lambda^{\pm}_{3}=0$, may be convenient
choices of canonical forms for the types I±.
## V Euclidean Signature
Now suppose that $(M,g_{\mu\nu})$ is an Euclidean four-dimensional real
manifold. In this case it is possible to find an orthonormal frame
$\\{e_{(1)},e_{(2)},e_{(3)},e_{(4)}\\}$ made of real vectors such that
$e_{(a)}^{\mu}e_{(b)\mu}=\delta_{ab}$. Thus a null tetrad frame,
$\\{l,n,m_{1},m_{2}\\}$, can be constructed in the complexified tangent space
by defining the vectors
$l=\frac{1}{\sqrt{2}}(e_{(1)}+ie_{(2)})\,,\,n=\frac{1}{\sqrt{2}}(e_{(1)}-ie_{(2)})\,,\,m_{1}=\frac{1}{\sqrt{2}}(e_{(3)}+ie_{(4)})\;\textrm{and}\;m_{2}=\frac{-1}{\sqrt{2}}(e_{(3)}-ie_{(4)})\,$.
Since $\\{e_{(a)},\small{a=1,2,3,4}\\}$ are real vectors it follows that $l$
is the complex conjugate of $n$, while $m_{1}$ is minus the complex conjugate
of $m_{2}$. Also the Weyl tensor is real, because it is being assumed that the
metric is real. Using these relations it is easy to see that the Weyl scalars,
(14), are such that: $\overline{\Psi^{\pm}_{0}}=\Psi^{\pm}_{4}$ ,
$\overline{\Psi^{\pm}_{1}}=-\Psi^{\pm}_{3}$ and
$\overline{\Psi^{\pm}_{2}}=\Psi^{\pm}_{2}$. These equalities together with
equation (20) implies that the matrix representations of $C^{+}$ and $C^{-}$
are hermitian and independent of each other. Since these matrices have
vanishing trace and every hermitian matrix can be put in a diagonal form by a
suitable choice of basis, it follows that there are just three algebraically
distinct canonical forms for the matrix representations of $C^{+}$ and
$C^{-}$, they are:
$diag(\lambda^{\pm}_{1},\lambda^{\pm}_{2},-\lambda^{\pm}_{1}-\lambda^{\pm}_{2})$
with all eigenvalues different,
$diag(\lambda^{\pm},\lambda^{\pm},-2\lambda^{\pm})$ with $\lambda^{\pm}\neq 0$
and $diag(0,0,0)$. This means that $C^{\pm}$ can be just of types I±, D± or
O±. Since the operators $C^{+}$ and $C^{-}$ are independent of each other,
there are also just six realizable classification types in the Euclidean case,
they are: (O, O); (O, I); (O, D); (I, I); (I, D) and (D, D). This
classification was already obtained in Hacyan , using a mixture of null tetrad
and spinorial formalisms, and in Karlhede , by means of splitting the four-
dimensional Weyl tensor into two three-dimensional independent tensors of rank
two and using the $SO(4|\mathbb{R})$ group to find canonical forms for these
tensors.
## VI (2,2) Signature
In a real four-dimensional manifold of signature (2,2) it is possible to find
a real frame $\\{e_{1},e_{2},e_{3},e_{4}\\}$ such that the only non-zero inner
products between the basis vectors are
$e_{1}^{\mu}e_{1\mu}=e_{2}^{\mu}e_{2\mu}=1$ and
$e_{3}^{\mu}e_{3\mu}=e_{4}^{\mu}e_{4\mu}=-1$. Because of this, in such
signature it is possible to construct a null tetrad frame made only of real
vectors:
$l=\frac{1}{\sqrt{2}}(e_{1}+e_{3})\,,\,n=\frac{1}{\sqrt{2}}(e_{1}-e_{3})\,,\,m_{1}=\frac{1}{\sqrt{2}}(e_{2}+e_{4})\,,\,m_{2}=\frac{-1}{\sqrt{2}}(e_{2}-e_{4})\,$.
From the reality of these vectors and the reality of Weyl tensor it follows
that all the Weyl scalars, (14), are real in this case. These scalars form a
total of ten real independent components of the Weyl tensor, as it should be
in a real manifold of dimension four.
But, differently from the other two signatures, there are no relations
connecting the different Weyl scalars, so that all classification types are
possible in (2,2) case. There are thus 21 possible algebraic types for the
Weyl tensor in a four-dimensional real manifold of zero signature, they are
(O,O); (O,I); (O,D);(O,II); (O,III); (O,N); (I;I); (I;D); $\ldots$, just as in
the complex case. Where, as commented before, type (X,Y) is equivalent to type
(Y,X). The canonical forms of these types are given in the same way as
described at the end of section IV444It is possible that the bivector basis
where the operators $C^{\pm}$ take the canonical forms is formed by complex
bivectors. For example, when $C^{+}$ is type I+ the null tetrad frame where
$\Psi_{0}^{+}=\Psi_{4}^{+}=0$ may be composed by complex vectors, in
particular in such a frame the Weyl scalars assume, in general, non-real
values..
A classification in signature (2,2) directly related to this one was obtained
before in Law2 , by means of spinors. The relation of the present
classification for $C^{+}$ and the one defined in Law2 is the following:
I+$\rightarrow\\{1111\\}$, $\\{1\overline{1}11\\}$ or
$\\{1\overline{1}1\overline{1}\\}$; II+$\rightarrow\\{211\\}$ or
$\\{1\overline{1}2\\}$; D+$\rightarrow\\{22\\}$ or $\\{2\overline{2}\\}$;
III+$\rightarrow\\{31\\}$ and N+$\rightarrow\\{4\\}$. There are more Weyl
tensor types in Law2 because there the reality of principal spinors is used
to refine the classification. An analogous refinement could be done in the
bivector approach, but probably this refinement do not bring any useful
geometric information.
## VII Boost Weight Method
The most famous and, up to now, fruitful tensor classification scheme for
higher-dimensional Lorentzian manifolds is the so called CMPP classification
CMPP , in particular when used to classify the Weyl tensor in four dimensions
it reduces to the Petrov classification. This classification is based on the
so called boost weight decomposition. In reference ColeyPSEUD S. Hervik and
A. Coley extended the boost weight classification to pseudo-Riemannian
manifolds of arbitrary dimension, but as depicted in ColeyPSEUD this
classification can not be applied to Euclidian nor complex manifolds. It will
be argued in the present section that this classification can be easily
extended to complex manifolds and the results on real manifolds, including
Euclidean signature, can be extracted from the complex case by imposition of
suitable reality conditions. In particular, the boost weight classification
for the four-dimensional Weyl tensor will be shown to be equivalent to the
classification obtained here by the bivector approach.
When the metric of a real manifold has signature $(p,q)$ the isometry group on
each point of the manifold (gauge group of the null frame) is
$SO(p,q|\mathbb{R})$. But if the manifold is complexified the isometry group
is enhanced to $SO(p+q|\mathbb{C})$. In ColeyPSEUD the analysis is restricted
to the real manifolds with real isometry groups, so that the boost weight
classification as treated there does not applies to the complex manifolds nor
to the Euclidean signature, since in this last case real null directions are
not allowed. But this treatment can be easily generalized to include complex
and Euclidean manifolds. In complex manifolds there is nothing special to do,
just apply directly the boost weight decomposition while keeping in mind that
the isometry group is $SO(n|\mathbb{C})$, where $n$ is the complex dimension
of the manifold. In the Euclidean case the tangent bundle must be complexified
so that the isometry passes from $SO(n|\mathbb{R})$ to $SO(n|\mathbb{C})$,
after this the boost weight classification is done and at the end the reality
condition must be imposed. As an example it will be shown below the
classification of the Weyl tensor in Euclidean four-dimensional manifolds.
In the previous sections it was shown that in four dimensions it is always
possible to locally find a null tetrad frame $\\{l,n,m_{1},m_{2}\\}$, although
it may be necessary to complexify the tangent bundle. In order to agree with
the notation of ColeyPSEUD let us define the following null basis:
$l^{1}=n,n^{1}=l,l^{2}=-m_{2}$ and $n^{2}=m_{1}$. When the component of a
tensor, $\Upsilon_{A}$, transforms as
$\Upsilon_{A}\mapsto\lambda^{-r}\tau^{-s}\Upsilon_{A}$ under the boost
$l^{1}\rightarrow\lambda
l^{1},n^{1}\rightarrow\lambda^{-1}n^{1},l^{2}\rightarrow\tau l^{2}$ and
$n^{2}\rightarrow\tau^{-1}n^{2}$ this component is said to have boost weight
[$r$,$s$]. Now looking to the definition of the Weyl scalars $\Psi^{+}_{a}$,
(14), it is easily seen that they have the following boost weights:
Boost Weights of the Weyl Scalars
$\Psi^{+}_{0}$ $\rightarrow$ [2,2] | $\Psi^{+}_{1}$ $\rightarrow$ [1,1] | $\Psi^{+}_{2}$ $\rightarrow$ [0,0] | $\Psi^{+}_{3}$ $\rightarrow$ [-1,-1] | $\Psi^{+}_{4}$ $\rightarrow$ [-2,-2]
---|---|---|---|---
The important thing to be observed in the above table is that the components
of $C^{+}$ are such that the boost weights relative to the null vectors
$\\{l^{1},n^{1}\\}$ are always equal to the boost weights relative to
$\\{l^{2},n^{2}\\}$. So in the language of CMPP ; ColeyPSEUD the only allowed
types for $C^{+}$ are [O,O], [I,I], [D,D], [II,II], [III,III] and [N,N]. These
are respectively what in the present article was called types O+, I+, D+, II+,
III+ and N+. Obviously the operator $C^{-}$ can have only the same six types.
Finally, to apply the boost weight classification to the whole Weyl tensor,
the classifications of $C^{+}$ and $C^{-}$ should be composed. It is then
clear that in the Lorentzian and (2,2) signatures the boost weight
classification for the Weyl tensor furnishes the same types as the bivector
approach presented here. Obviously it must be considered, as was shown in
section IV, that in the Lorentzian case $C^{+}$ is the complex conjugate of
$C^{-}$, so that the type of the first operator is the same of the second one.
Analogously the boost weight classification for the Weyl tensor for the
complex and Euclidean cases, in the form explained above, follows easily and
produces the same types as the bivector approach presented here. As an example
let us treat explicitly the Euclidean signature.
As explained in the preceding paragraph, the operator $C^{+}$ can have only
six types of boost weight: [O,O], [I,I], … [N,N]. Analogously the operator
$C^{-}$ can have only the same six types. In the particular case of the
Euclidean signature, as shown in section V, the Weyl scalars are such that
$\overline{\Psi^{\pm}_{0}}=\Psi^{\pm}_{4}$ and
$\overline{\Psi^{\pm}_{1}}=-\Psi^{\pm}_{3}$. This implies that for both
operators $C^{+}$ and $C^{-}$ the type [II,II] collapses to type [D,D] while
types [III,III] and [N,N] collapse to type [O,O]555For example, if the
operator $C^{+}$ is type [II,II] then, by definition, there exists a basis in
which $\Psi^{+}_{0}=\Psi^{+}_{1}=0$. Now because of the identities
$\overline{\Psi^{+}_{0}}=\Psi^{+}_{4}$ and
$\overline{\Psi^{+}_{1}}=-\Psi^{+}_{3}$ that are valid in the Euclidean
signature it follows that in this basis $\Psi^{+}_{3}$ and $\Psi^{+}_{4}$ must
also vanish. So that the type of $C^{+}$ is indeed [D,D] instead of [II,II]..
So that in this signature both operators $C^{+}$ and $C^{-}$ must have one of
the following three types: [O,O], [I,I] or [D,D]. This agrees perfectly with
the result obtained by means of the bivector approach.
## VIII Conclusion
A local classification for the Weyl tensor, based on the map of bivectors into
bivectors that this tensor provides, was developed in all four-dimensional
manifolds endowed with a metric and a Levi-Civita connection. The
classification was presented in an unified way, so that the classification on
real spaces can be seen as particular cases of the complex classification,
where different signatures correspond to different choices of reality
condition on the complex manifold. In the complex case and in the real one of
signature (2,2) there are 21 possible classifications. In the Lorentzian and
Euclidean signatures special relations appear when the reality condition is
imposed so that just 6 types of classifications can be realized. It was also
discussed the canonical forms of the Weyl tensor in the various classification
types. Finally it was proved that trivially extending the boost weight
classification of ColeyPSEUD to include complex and Euclidean manifolds we
get a classification scheme for the Weyl tensor that is equivalent to the
bivector approach presented here in all signatures as well in complex
manifolds.
## Acknowledgments
I am indebted to Professor Bruno G. Carneiro da Cunha for the patience in the
revision of the manuscript and for the valuable proposals. I also want to
thank Fábio M. Novaes Santos for the LaTeX lessons, to Marcello Ortaggio for
suggesting many references and to the referee for important suggestions. This
work had CNPq666Conselho Nacional de Desenvolvimento Científico e Tecnológico
financial support. The final publication is available at link.springer.com .
## References
* (1) W. Kinnersley, Type D vacuum Metrics, Journal of Mathematical Physics 10 (1969), 1195
* (2) J. Goldberg and R. Sachs, A theorem on Petrov types, General Relativity and Gravitation 41 (2009), 433. This is a republication of original 1962 paper.
* (3) H. Stephani et. al., Exact solutions of Einstein’s field equations, Cambridge University Press (2009)
* (4) J. Plebański and A. Krasinski, An introduction to general relativity and cosmology, Cambridge University Press (2006)
* (5) A. Z. Petrov, The classification of spaces definig gravitational fields, General Relativity and Gravitation 32 (2000), 1665. This is a translated republication of original 1954 paper.
* (6) J. Plebański, Some solutions of complex Einstein equations, Journal of Mathematical Physics 16 (1975), 2395
* (7) G. S. Hall, Lecture notes on symmetries and curvature structure in general relativity, World Scientific (2004)
* (8) S. Hacyan, Gravitational instantons in H-spaces, Physics Letters 75A (1979), 23
* (9) A. Karlhede, Classification of Euclidean metrics, Classical and Quantum Gravity 3 (1986),L1 (Letter to the Editor).
* (10) A. Z. Petrov, Einstein Spaces, Pergamon Press (1969), pages 99-101.
* (11) P. R. Law, Neutral Einstein metrics in four dimensions, Journal of Mathematical Physics 32 (1991), 3039
* (12) A. Gover, C. Hill and P. Nurowski, Sharp version of the Goldberg-Sachs theorem, Annali di Matematica Pura ed Applicata 190 Number 2 (2011), 295. Availabe at arXiv:0911.3364.
* (13) P. R. Law, Classification of the Weyl curvature spinors of neutral metrics in four dimensions, Journal of Geometry and Physics 56 (2006), 2093
* (14) C. Batista, A generalization of the Goldberg-Sachs theorem and its consequences. Availabe at arXiv:1205.4666
* (15) A. Coley and S. Hervik, Higher dimensional bivectors and classification of the Weyl operator, Classical and Quantum Gravity 27 (2010), 015002. Availabe at arXiv:0909.1160
* (16) A. Coley, R. Milson, V.Pravda and A. Pravdová, Classification of the Weyl tensor in higher dimensions, Classical and Quantum Gravity 21 (2004), L35. Availabe at arXiv:gr-qc/0401008v2.
* (17) A. Coley, S. Hervik and N. Pelavas, Spacetimes characterized by their scalar curvature invariants, Classical and Quantum Gravity 26 (2009), 025013. Availabe at arXiv:0901.0791
* (18) S. Hervik and A. Coley, On the algebraic classification of pseudo-Riemannian spaces. Availabe at arXiv:1008.3021
* (19) L. Bel, Radiation states and the problem of energy in general relativity, General Relativity and Gravitation 32 (2000), 2047. This is a republication of original 1962 paper.
* (20) R. Penrose and W. Rindler, Spinors and space-time vol.1 and 2, Cambridge University Press (1984 and 1986)
|
arxiv-papers
| 2012-04-23T18:26:56 |
2024-09-04T02:49:30.040502
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Batista",
"submitter": "Carlos A. Batista da S. Filho",
"url": "https://arxiv.org/abs/1204.5133"
}
|
1204.5232
|
# Clifford-Wolf translations of Homogeneous Randers spheres111Supported by
NSFC (no, 10671096, 10971104) and SRFDP of China
Shaoqiang Deng1 and Ming Xu2
1School of Mathematical Sciences and LPMC
Nankai University
Tianjin 300071, P. R. China
2Department of Mathematical Sciences
Tsinghua University
Beijing 100084, P. R. China Corresponding author. E-mail:
mgxu@math.tsinghua.edu.cn
###### Abstract
In this paper, we study Clifford-Wolf translations of homogeneous Randers
metrics on spheres. It turns out that we can present a complete description of
all the Clifford-Wolf translations of all the homogeneous Randers metrics on
spheres. The most important point of this paper is that a new phenomena
surfaces. Namely, we find that there are some CW-homogeneous Randers spaces
which are essentially not symmetric. This is a great difference compared to
Riemannian geometry, where any CW-homogeneous Riemannian manifold must be
locally symmetric.
Mathematics Subject Classification (2000): 22E46, 53C30.
Key words: Finsler spaces, Clifford-Wolf translations, Killing vector fields,
homogeneous Randers manifolds.
## 1 Introduction
In this paper we continue our study concerning Clifford-Wolf translations of
Finsler spaces in our previous article ([DXP]). Our main goal here is to give
a complete description of Clifford-Wolf translations of homogeneous Randers
metrics on spheres. Recall that a Clifford-Wolf of a locally compact connected
metric space is an isometry of the space which moves all the point in the same
distance. Although generically the distance function of a Finsler space is not
reversible, one can similarly define the Clifford-Wolf translation of a
Finsler space (see Definition 2.2 below).
A connected Riemannian manifold $(M,Q)$ is called Clifford-Wolf homogeneous
(CW-homogeneous) if for any $x,y\in M$, there is a Clifford-Wolf translation
$\sigma$ such that $\sigma(x)=y$. It is called restrictively CW-homogeneous if
for ant $x\in M$, there is a open neighborhood $V$ of $x$ such that for any
$x^{\prime}\in V$ there is an isometry $\sigma^{\prime}$ of $(M,Q)$ such that
$\sigma^{\prime}(x)=x^{\prime}$. CW-homogeneous Riemannian manifolds were
thoroughly studied by V. N. Berestovskii and Yu. G. Nikonorov in [BN09]. It
was proved in [BN09] that any restrictively CW-homogeneous Riemannian manifold
must be locally symmetric. Based on this, the authors of [BN09] obtained a
complete classification of all connected simply connected CW-homogeneous
Riemannian manifolds. The complete list consists of compact Lie groups with
bi-invariant Riemannian metrics, the odd-dimensional spheres with standard
metrics and the symmetric space $SU(2n+2)/Sp(n)$ with the standard symmetric
metrics.
The notion of CW-homogeneous and restrictively CW-homogeneous Riemannian
manifold can be generalized to the Finsler case (see Definition 2.5 below). It
is therefore a natural problem to find out whether the above conclusions still
hold for Finsler spaces and to give a complete classification of all the CW-
homogeneous Finsler spaces. This problem is much more difficult compared to
the Riemannian case. To begin with we first consider homogeneous Randers
metrics on spheres. The main results of this paper is that there are some
homogeneous Randers metrics on the spheres which are restrictively CW-
homogeneous but which are essentially not locally symmetric, in the following
senses: The underlying Riemannian metrics of such Randers metrics are not
locally symmetric and; Such Randers metrics are not of Berwald type. Note that
a locally symmetric Finsler space must be of Berwald type, this fact was
conjectured by the first author and Z. Hou in [DH07] and was recently proved
by V. S. Matveev and M. Troyanov in [MTP]. Meanwhile, we will give a complete
list of all the Clifford-Wolf translations of the homogeneous Randers metrics
on spheres.
## 2 Preliminary
Finsler geometry is introduced by Riemann in 1854 in his celebrated lecture on
the foundations of geometry, and revived in 1918 by Finsler in his doctoral
dissertation.
###### Definition 2.1
A Finsler metric on a manifold $M$ is a continuous function
$F:TM\rightarrow\mathbb{R}$, which is smooth on $TM\backslash 0$, and
satisfies the following conditions:
(1) (Positivity) $F(x,y)>0$ if $y\neq 0$.
(2) (Positive homogeneousness) $F(x,\lambda y)=\lambda F(x,y)$ for
$\lambda>0$.
(3) (Convexity) The Hessian matrices of $F^{2}$ for $y$, i.e.
$g_{ij}=\frac{1}{2}[F^{2}]_{y^{i}y^{j}}$, are positively definite on
$TM\backslash 0$.
The most familiar examples of Finsler metric is the Riemannian metrics, when
$F=\sqrt{g_{ij}(x)y^{i}y^{j}}$ is a quadratic function of $y$ for any $x$ on
the manifold. Similarly as in the Riemannian case, any Finsler metric $F$
gives the length for any tangent vector, and this gives arc length for any
piecewise smooth path. We can then define “distance” as the minimum of the arc
lengthes among all the piece-wise smooth curves from one point to another
[BCS00]. The distance of a Finsler metric does not satisfy the reversibility
of a metric space, unless $F$ is absolutely homogeneous, i.e.
$F(x,y)=F(x,-y)$, $\forall x\in M,y\in TM_{x}$. For simplicity, we will still
call it the distance and denote it as $d(\cdot,\cdot)$.
Among the non-Riemannian examples of Finsler metrics, Randers metrics are
well-known for its simplicity and importance in geometry and physics. A
Randers metric $F$ is a sum $F=\alpha+\beta$, where $\alpha$ is a Riemannian
metric and $\beta$ is an one-form whose length is everywhere less than $1$.
In [DXP] we have studied the Clifford Wolf translations in Finsler geometry.
We now recall the definitions.
###### Definition 2.2
A Clifford Wolf translation (or simply a CW-translation) $\rho$ of a Finsler
manifold $(M,F)$ is an isometry of $(M,F)$ such that $d(x,\rho(x))$ is a
constant function.
The interrelation between CW translations and Killing vector fields of
constant lengths in the Riemannian case, due to V. N. Berestovskii and Yu. G.
Nikonorov [BN09], was generalized to the Finslerian case in our previous paper
[DXP]. We have
###### Theorem 2.3
Let $(M,F)$ be a complete Finsler manifold with positive injective radius. If
$X$ is a Killing vector field of constant length, then the flow $\phi_{t}$
generated by $X$ is a CW-translation for all sufficiently small $t>0$.
###### Theorem 2.4
Let $(M,F)$ be a compact Finsler manifold. Then there is a $\delta>0$, such
that any CW-translation $\rho$ with $d(x,\rho(x))<\delta$ is generated by a
Killing vector field of constant length.
There are some other concepts related to CW-translations which has been
studied extensively in the Riemannian case. For example, Clifford-Wolf
homogeneous space and restrictively Clifford-Wolf homogeneous space.
###### Definition 2.5
A Finsler manifold $(M,F)$ is called Clifford-Wolf homogeneous if for any two
points $x_{1},x_{2}\in M$, there is a CW-translation $\sigma$ such that
$\sigma(x_{1})=x_{2}$. It is called restrictively Clifford-Wolf homogeneous if
for any point $x\in M$ there is a neighborhood $V$ of $x$, such that for any
two points $x_{1},x_{2}\in V$ there is a CW-translation $\sigma$ such that
$\sigma(x_{1})=x_{2}$.
We will simply call such a spce CW-homogeneous or restrictively CW-
homogeneous. As we will only deal with compact manifolds in this work, the
definition of restrictively CW-homogeneous can be simplified as the following
one.
###### Definition 2.6
A compact Finsler manifold $(M,F)$ is called restrictively CW-homogeneous if
there is a constant $\delta>0$, such that for any pair of points $x$ and
$x^{\prime}$ with $d(x,x^{\prime})<\delta$ (or equivalently,
$d(x^{\prime},x)<\delta$), there is a CW-translation $\rho$ such that
$\rho(x)=x^{\prime}$.
Obviously the CW-homogeneity or the restrictive CW-homogeneity of a Finsler
space $(M,F)$ implies the homogeneity of the space. Therefore to understand
CW-homogeneous Finsler space it is natural to start with CW-translations of
homogeneous Finler spaces. In this case, both the metric data and the
conditions for CW-translations can be reduced to the Lie algebra level, which
greatly reduces the complexity of the problem.
In [DXP] we have studied examples of CW translations on some compact Lie
groups, with left invariant non-Riemannian Randers metrics. In this work we
will see more examples of CW-translations of homogeneous Randers metrics on
spheres.
## 3 Homogeneous Randers metrics on spheres
Let $(M,F)$ be a connected compact Finsler space. It is called a homogeneous
space, or $F$ is called a homogeneous metric, if its full connected isometry
group $G_{0}=I_{0}(M,F)$ acts transitively on $M$. It has been proven that
$G_{0}$ is a compact Lie group [DH02]. Let $H_{0}\subset G_{0}$ be the
isotropic subgroup of a point of $M$. Then the manifold is naturally
diffeomorphic to $G_{0}/H_{0}$. In general there are more than one way to
express $M$ as a homogeneous space. In fact, any connected closed subgroup
$G\subset I_{0}(M,F)$ which acts transitively on $M$, with the isotropy
subgroup $H\subset G$ fixing the same point, gives a homogeneous space $G/H$
for $M$. No matter which $G$ is used, the quotient vector space of the Lie
algebras $\mathfrak{m}=\mathfrak{g}/\mathfrak{h}$ is the same. It can be
identified with the tangent space at the chosen point.
The Finsler metric $F$ is totally determined by its restriction to
$\mathfrak{m}$, which is an $Ad_{H}$-invariant Minkowski norm. This Minkowski
norm is transposed to other points by the left translations of $G$. On the
other hand, if $K$ is an effective transitive Lie transformation group on $M$
and $K_{1}$ is the isotropy subgroup of $K$ at a fixed point $x\in M$. Then
for any $Ad(K_{1})$-invariant Minkowski norm on the quotient space
$\mathfrak{k}/\mathfrak{k}_{1}$, one can construct a $K$-invariant Finsler
metric on $M$ using the above method.
Let us give an explicit example. Suppose $F=\alpha+\beta$ is a homogeneous
Randers metric on $M=G/H$, with $\mathfrak{m}=\mathfrak{g}/\mathfrak{h}$. Then
$\alpha$ is determined by an $\mbox{Ad}_{H}$-invariant inner product on
$\mathfrak{m}$, and $\beta$ is determined by an $\mbox{Ad}_{H}$ invariant
element of $\mathfrak{m}^{*}$. Equivalently, $\beta$ can be determined by its
dual with respect to the inner product, which is an $\mbox{Ad}_{H}$ invariant
vector $V\in\mathfrak{m}$.
The following lemma is useful for determining Killing vector fields of
constant length, which can generate CW-translations for a homogeneous space,
see [DXP].
###### Lemma 3.1
Let $(M,F)$ be a homogeneous Finsler space and $G$ be its full group of
isometries, Lie $G=\mathfrak{g}$. Then a Killing vector field generated by
$X\in\mathfrak{g}$ is of constant length $1$ if and only if the projection of
the $Ad_{G}$-orbit of $X$ to $\mathfrak{m}$ is contained in the indicatrix.
By studying the projections of the orbits, we can find the wanted homogeneous
Finsler metric from its indicatrix at the chosen point.
Now we turn to the main subject of this paper, namely, spheres with
homogeneous Randers metrics. Suppose $G$ is an effective transitive Lie
transformation group of $S^{n}$ and $H$ is the isotropy subgroup of $G$ at a
fixed point. If $F=\alpha+\beta$ is a $G$-invariant Randers metric on $S^{n}$,
then so is $\alpha$ ([DE08]). A complete list of Lie groups which admit an
effective transitive action on $S^{n}$ was obtained by Montgomery and Samelson
([MS43]). The list results in the following:
###### Lemma 3.2
The following list of Riemannian homogeneous spaces $G/H$ for spheres is
complete, in any case $G$ is a connected subgroup of the full isometry group
of a $G$-invariant Riemannian metric $\alpha$ on $S^{n}$.
(1) $S^{n}=SO(n+1)/SO(n)$, $n\geq 1$,
(2) $S^{2n+1}=SU(n+1)/SU(n)$, $n\geq 1$,
(3) $S^{2n+1}=U(n+1)/U(n)$, $n\geq 1$,
(4) $S^{4n+3}=Sp(n+1)/Sp(n)$, $n\geq 1$,
(5) $S^{4n+3}=Sp(n+1)U(1)/Sp(n)U(1)$, $n\geq 1$,
(6) $S^{4n+3}=Sp(n+1)Sp(1)/Sp(n)Sp(1)$, $n\geq 1$,
(7) $S^{6}=G_{2}/SU(3)$,
(8) $S^{7}=Spin(7)/G_{2}$,
(9) $S^{16}=Spin(9)/Spin(7)$.
The list gives all the possible $G\subset I_{0}(M,F)$ which acts transitively
on spheres, for all possible homogeneous Finsler metrics $F$ on them. In the
special case of Randers metrics, to produce a non-Riemannian metric on $G/H$,
we must have a non-zero vector in $\mathfrak{m}$ which is fixed under $Ad(h)$,
for any $h$ in the isotropy subgroup. This is equivalent to the condition that
the isotropy representation of $H$ on $\mathfrak{m}$ has a non-zero trivial
subrepresentation. From the above list it is obvious that this is the case
only in (2)-(5). Therefore we only need to deal with the cases of (2)-(5).
In (2) and (3), where $G=U(n)$ or $SU(n)$, the full isometry group of any
$G$-invariant non-Riemannian Randers metric must be $U(n)$. We will study this
case in more detail in Section 5. In (4) and (5), the full isometry group can
be $Sp(n)$, $Sp(n)U(1)$ or $U(2n)$, in this case our main focus will be on the
group $Sp(n)U(1)$.
## 4 CW-translations of left invariant Randers metrics on $SU(2)$
This work is motivated by the particular example $S^{3}$ which can be regarded
as $SU(2)=SU(2)/SU(1)$, $Sp(1)=Sp(1)/Sp(0)$, $U(2)/U(1)$ or $Spin(1)U(1)/U(1)$
appearing in each case of (2)-(5) in Lemma 3.2.
Let $F$ be a non-Riemannian homogeneous Randers metric on $S^{3}=SU(2)/{e}$.
There is no Killing vector field of non-zero constant length generated by the
elements of $\mathfrak{g}=\mathfrak{su}(2)$ (see [DXP]). Now let us see if we
can find a Killing vector field of constant length from the Lie algebra of the
full connected isometry group $U(2)=Sp(1)U(1)$. In $U(2)$, the center vectors
generate some special Killing vector fields. These Killing vector fields
generate CW-translations of the symmetric metric on $S^{3}$. Moreover, they
have constant length with respect to any $U(2)$-invariant Finsler metric on
$S^{3}$. This case is uninteresting and we just ignore them. So let us try to
find those Killing vector fields of constant length generated by non-central
elements of $\mathfrak{u}(2)$.
Denoting $\mathfrak{g}=\mathfrak{u}(2)$ and $\mathfrak{h}=\mathbb{R}$ the Lie
algebras of $G$ and $H$ respectively, we have a Lie algebra decomposition
$\mathfrak{g}=\mathfrak{u}(2)=\mathfrak{su}(2)\oplus\mathbb{R}$. The
subalgebra $\mathfrak{h}$ is generated by an element of the form
$(V,1)\in\mathfrak{g}$ with $V\neq 0$. On $\mathfrak{u}(2)$, there is a
standard inner product, i.e., $\langle A,B\rangle_{eq}=-trAB$. The above
decomposition of $\mathfrak{u}(2)$ is orthogonal with respect to this inner
product. Moreover, the restriction of this inner product to
$\mathfrak{m}\cong\mathfrak{su}(2)$ induces the the standard Riemannian metric
on $S^{3}$. For any $X$ in $\mathfrak{su}(2)$ with $|X|>|V|$, the ${\rm
Ad}_{G}$-orbit of $(X,1)$ is a $2$-dimensional round sphere centered at
$(0,1)$ with respect to $\langle\,,\,\rangle_{eq}$. The projection of this
orbit to $\mathfrak{m}=\mathfrak{g}/\mathfrak{h}=\mathfrak{su}(2)$ is a sphere
of the same radius, with center shifted to $-V$. By the assumption, this
sphere still surrounds the origin. This observation gives a method to find
homogeneous Randers metrics with the prescribed indicatrix, such that $(X,1)$
generates a Killing vector field of constant length. In fact, up to a constant
scalar, $\beta$ is the dual of $V$ with respect to $\langle\,,\,\rangle_{eq}$,
$\alpha$ has an ellipsoid indicatrix, which is a round sphere with respect to
the metric $langle\,,\,\rangle_{eq}$, with center stretched in the direction
of $V$ (?). Both $\alpha$ and $\beta$ are $Ad_{G}$-invariant(?), so they are
$Ad_{V}$-action as well.
Once we have found a non-vanishing Killing vector field of constant length of
a homogeneous Randers metric $F$, we can find an $Ad_{G}$-orbit of Killing
vector fields of the same constant length. It is easily seen that these
Killing vector fields exhaust all the tangent directions at the origin. By
Theorem 2.3, the homogeneous non-Riemannian Randers metrics constructed above
on $S^{3}$ are restrictively CW-homogeneous.
As a by-product, this construction can be generalized to other connected
compact Lie groups.
###### Proposition 4.1
On any connected compact Lie group $G$, there is a left invariant non-
Riemannian Randers metric $F$ which makes $(M,F)$ a restrictively CW-
homogeneous Finsler space.
Proof. Let $G^{\prime}=G\times S^{1}$, whose Lie algebras is
$\mathfrak{g}^{\prime}=\mathfrak{g}\oplus\mathbb{R}$. Select a nonzero $V$ in
${\mathfrak{g}}$ such that the one-parameter subgroup $\exp t(V)$ is
isomorphic to $S^{1}$. Let $H^{\prime}$ be the subgroup of $G^{\prime}$
generated by $(V,1)$. Obviously $H^{\prime}\cong S^{1}$ and
$G^{\prime}/H^{\prime}$ is a homogeneous space for $G$. Choose any bi-
invariant metric on $G$ and denote by $\langle\,,\,\rangle_{eq}$ the inner
production induced on $\mathfrak{g}$. We can assume $\langle
V,V\rangle_{eq}<1$. The sphere $S=\\{(X,1)|\langle X,X\rangle_{eq}^{1/2}=1\\}$
is the union of some ${\rm Ad}_{G^{\prime}}$-orbits. Projected to
$\mathfrak{m}\cong\mathfrak{g}$, its center is shifted to $-V$. Using the
above argument one can similarly find a $G^{\prime}$-invariant Randers metric
on $G\simeq G^{\prime}/H^{\prime}$ with the above sphere in $\mathfrak{g}$ as
the indicatrix. Then any vector $(X,1)\in S$ generates a Killing vector field
of constant length $1$, and these vectors exhaust all the tangent directions.
Therefore this Randers metric makes $G$ a restrictively CW-homogeneous Finsler
spaces.
For general compact Lie groups, it is still unknown whether the word
“restrictively” can be removed. But for the special case $SU(2)$, the answer
is positive.
###### Proposition 4.2
On $S^{3}$, there are non-Riemannian homogeneous Randers metrics which makes
it CW-homogeneous.
Proof. The homogeneous Randers metric we choose is the one constructed as
above. The proof is carried out by a closer observation of the geometry of the
Randers metric and Killing vector fields in the construction. Choose a bi-
invariant inner product on $\mathfrak{su}(2)$ such that the induced metric
makes $SU(2)$ the standard unit sphere. Without losing generality, we can
assume that $X$ has length $1$ and $V$ has length $l<1$ with respect to this
metric. Suppose $(X,1)$ generates the Killing vector field (of constant
length) of the Randers metric $F$. The flow of isometries generated by $(X,1)$
are $\phi_{t}(g)=\mbox{exp}(tX)g\mbox{exp}(-tV)$ which gives a geodesic at
$g$. Notice that $\exp(\pi X)=-\mbox{id}$ for each $X$ with length $1$, since
the length of $X$ is 1 implies that the eigenvalues of $X$ are $\pm\sqrt{-1}$.
Thus $\exp(\pi X)$ is a unitary conjugation of
$\exp(\mbox{diag}(\pi\sqrt{-1},-\pi\sqrt{-1}))=-\mbox{id}$.
Since these $X$ can exhaust all the unit vectors of $\mathfrak{su}(2)$, the
geodesics in all directions starting from $g$ with $t=0$ will end at
$-g\mbox{exp}(\pi V)$ with $t=\pi$. This means that all those geodesics from
$g$ to $-g\exp(\pi V)$ have the same length $\pi$.
Any point can be reached by a geodesic from $g$ for $t\in[0,\pi]$. Otherwise
we can choose a shortest geodesic from $g$ to it, passing $-g\mbox{exp}(\pi
V)$ in the midway. Then the geodesic from $g$ to $-g\mbox{exp}(\pi V)$ can be
changed to another one which turns a angle at $-g\mbox{exp}(\pi V)$, and the
new path is still a shortest path. This is a contradiction because the new
path is not a smooth geodesic.(?)
Those geodesics do not intersect each other when $t$ is restricted to
$(0,\pi)$. Otherwise, there will be a pair of unit vectors $X_{1}$ and $X_{2}$
in $\mathfrak{su}(2)$, and $t_{1},t_{2}\in(0,\pi)$, such that
$\exp(t_{1}X_{1})g\exp(-t_{1}V)=\exp(t_{2}X_{2})g\exp(-t_{2}V).$ (4.1)
If $t_{1}=t_{2}$, then we have $X_{1}={\rm Ad}_{g}X_{2}$. Thus $X_{1}=X_{2}$.
If $t_{1}\neq t_{2}$, say $t_{2}<t_{1}$, then we have
$\exp(t_{1}X_{1})=\exp(t_{2}X_{2})\exp((t_{1}-t_{2}){\rm Ad}_{g}V).$ (4.2)
The left side gives a point on a geodesic sphere with radius $t_{1}$ centered
at the point $e_{0}$ representing the identity matrix. The right side gives a
point on a geodesic sphere with radius $(t_{1}-t_{2})l$ which is centered at
the point $\exp(t_{2}X_{2})$ on geodesic sphere centered at $e_{0}$ with
radius $t_{2}$. Therefore there is a path from $e_{0}$ to the point given by
the left side that has a length smaller than $t_{1}$, which is a
contradiction.
So all those geodesic flow curves from $g$ to $-g\mbox{exp}(\pi V)$ are the
shortest ones. This property only depends on the length of the $X$’s and the
length of $V$. Change of $g$ only results in a change of unitary conjugation,
without changing the lengthes. Therefore given any two points $g_{1}$, $g_{2}$
in $SU(2)$, one can find a CW-translation $\phi_{t}$, with $t\in[0,\pi]$,
where $\phi_{t}$ is the flow of a Killing vector field generated by certain
$(X,1)$, such that $\phi_{t}(g_{1})=g_{2}$. This completes the proof of the
proposition.
## 5 Randers Spheres with unitary isometry groups
Let $F=\alpha+\beta$ be a non-Riemannian homogeneous Randers metric on
$S^{2n+1}$, such that $I_{0}(S^{2n+1},F)=U(n+1)$. Then the sphere can be
presented as $G/H$, with $G=U(n)$ and $H=U(n-1)\subset G$. Their Lie algebras
are $\mathfrak{g}=\mathfrak{u}(n)$ and $\mathfrak{h}=\mathfrak{u}(n-1)$
respectively. The tangent space at the origin is the quotient space
$\mathfrak{m}=\mathfrak{m}_{0}\oplus\mathfrak{m}_{1}$, where
$\mathfrak{m}_{0}=\mathbb{R}$ and $\mathfrak{m}_{1}=\mathbb{C}^{n}$. The
isotropy subgroup acts trivially on $\mathfrak{m}_{0}$ and acts on
$\mathfrak{m}_{1}$ by left multiplication. The projection of
$X\in\mathfrak{g}$ to $\mathfrak{m}$ is equal to
$X(0,\ldots,0,\sqrt{-1})^{*}$.
Since the underlying Reiamnnian metric $\alpha$ is also invariant under $G$,
it induces an ${\rm Ad}_{H}$-invariant linear metric (still denoted by
$\alpha$) on $\mathfrak{m}$, which must have the form
$\alpha^{2}(q,u)=a|q|^{2}+bu^{*}u$, $\forall q\in\mathbb{R}$ and
$u\in\mathbb{C}^{n}$, with positive constant $a$ and $b$. The standard inner
product, i.e., the one with $a=b=1$, is induced by the symmetric standard
Riemannian metric. The corresponding inner product on $\mathfrak{m}$ will be
denoted by $\langle\,,\,\rangle_{eq}$.
The non-vanishing $1$-form $\beta$ is also $G$-invariant, so it is induced by
an $\mbox{\rm Ad}(H)$-invariant vector $V\in{\mathfrak{m}}$. This means that
$V$ must be contained in ${\mathfrak{m}}_{0}$. Thus $V$ has the form
$\mbox{\rm diag}(0,\ldots,0,c)^{T}$, with $|c|<\sqrt{a}$.
Suppose there is a vector $X\in\mathfrak{u}(n+1)$ which generates a Killing
vector field of constant length $L>0$ with respect to $F=\alpha+\beta$. For
simplicity, we assume that $X$ is not in the center of $\mathfrak{u}(n+1)$,
since any vector in the center of $\mathfrak{u}(n+1)$ generates a Killing
vector field of constant length for any homogeneous Finsler metric on
$S^{2n+1}=U(n+1)/U(n)$. Moreover, it is also a CW-translation of the CW-
homogeneous Riemannian metric on $S^{2n+1}$ (i.e., the standard metric).
Up to a unitary conjugation, we can assume $X$ to be diagonal. By the action
of the Weyl group, each eigenvalue of $X$ can appear at the down right corner.
The projection of those diagonal matrices in the orbit of $X$ to
$\mathfrak{m}$ can be denoted as $(0,\ldots,0,a_{i})$, in which
$a_{1}\sqrt{-1},\ldots,a_{n+1}\sqrt{-1}$ are all the eigenvalues of $X$. Then
these $a_{i}$’s must be the solution of the equation
$\sqrt{a}|x|+cx=L.$ (5.3)
By the assumption, $X$ has at least two distinct eigenvalues. Then from (5.3)
it is easily seen that $X$ has exactly two distinct eigenvalues with opposite
signs. So by a suitable unitary conjugation, we can assume that
$X=\sqrt{-1}(x_{1}\mbox{I}+x_{2}\mbox{diag}(-m\mbox{I}_{l},l\mbox{I}_{m}))$,
where $l$ and $m$ are natural numbers satisfying $l+m=n+1$, $x_{2}\neq 0$ and
$(x_{1}-mx_{2})(x_{1}+lx_{2})<0.$ (5.4)
Suppose $U\in U(n+1)$. Denote its last row denoted by $(u^{*},v^{*})$. Then
the value of the metric $\alpha^{2}$ at the projection of $UXU^{*}$ in
$\mathfrak{m}$ is
$\displaystyle
bx_{2}^{2}(m^{2}(|u|^{2}-|u|^{4}))+l^{2}(|v|^{2}-|v|^{4})+2ml|u|^{2}|v|^{2})$
$\displaystyle+a(x_{2}(-m|u|^{2}+l|v|^{2})+x_{1})^{2}.$ (5.5)
Denote $t=l|v|^{2}-m|u|^{2}\in[-m,l]$. Since $|u|^{2}+|v|^{2}=1$, we have
$|u|^{2}=\frac{1-t}{n+1},\\\ |v|^{2}=\frac{t+m}{n+1}.\\\ $ (5.6)
Then the above $\alpha^{2}$ term can be summarized as
$f(t)=(a-b)x_{2}^{2}t^{2}+[(l-m)x_{2}^{2}b+2ax_{1}x_{2}]t+(x_{2}^{2}bml+ax_{1}^{2}).$
(5.7)
The $\mathfrak{m}_{0}$-coordinate of the projection of $UXU^{*}$ is
$x_{2}t+x_{1}$, so its $\beta$ value is $c(x_{2}t+x_{1})$.
The Killing vector field generated by $X$ having a constant length $L>0$ is
equivalent to the equation
$f(t)\equiv(-c(x_{2}t+x_{1})+L)^{2},\forall t\in[-m,l].$ (5.8)
For any $m$,$l$, $x_{1}$, $x_{2}$ and $L$, we can uniquely solve the triple
$(a,b,c)$,
$\displaystyle a$ $\displaystyle=$ $\displaystyle b+c^{2},$ (5.9)
$\displaystyle b$ $\displaystyle=$ $\displaystyle
L^{2}[(\frac{(n+1)x_{2}}{2})^{2}-(\frac{l-m}{2}x_{2}+x_{1})^{2}]^{-1},$ (5.10)
$\displaystyle c$ $\displaystyle=$
$\displaystyle-\frac{b}{L}(\frac{l-m}{2}x_{2}+x_{1}).$ (5.11)
To summarize, we have proved the following theorem.
###### Theorem 5.1
For any $X\in u(n+1)$ with exactly two eigenvalues of different signs and
different absolute values, there is a non-Riemannian homogeneous Randers
metric on $S^{2n+1}=U(n+1)/U(n)$, which is unique up to a scalar, such that
$X$ generates a CW-translation.
This theorem can also be stated as the following.
###### Theorem 5.2
Let $F$ be a non-Riemannian homogeneous Randers metric on
$S^{2n+1}=U(n+1)/U(n)$, determined by the parameters $(a,b,c)$ defined by
(5.10),(5.11) and (5.11). Then there exists a non-vanishing Killing vector
field of constant length which is not in the center of ${\mathfrak{u}}(n+1)$
if and only if $a=b+c^{2}$. The Killing vector fields of constant length $L>0$
generated by elements of the center of $\mathfrak{u}(n+1)$ are in one-to-one
correspondence with the unitary matrices with exactly two eigenvalues
$(-\frac{Lc}{b}\pm\sqrt{\frac{L^{2}}{b}+\frac{L^{2}c^{2}}{b^{2}}})\sqrt{-1}$.
It should be noted that the Riemannian CW-translations of the symmetric space
$S^{2n+1}$ can also be derived from the above discussion. In fact, when $c=0$,
we get those $X\in u(2n+1)$ whose eigenvalues have the same absolute values.
They are all the Killing vector fields of constant length of the symmetric
sphere, which commute with the given one, i.e.,
$\sqrt{-1}I\in\mathfrak{u}(2n+1)$. The matrix $X$ in Theorem 5.1 and Theorem
5.2 can be written as
$X=\sqrt{-1}(x_{1}+\frac{l-m}{2}x_{2})I+\sqrt{-1}\frac{l+m}{2}x_{2}\mbox{diag}(-I_{l},I_{m}).$
(5.12)
This means that there is a natural correspondence between the Killing vector
fields of constant length of a homogeneous non-Riemannian Randers metric and
the pairs of Killing vector fields of constant length of a symmetric metric,
with the later pair commuting with each other and having different lengthes.
Accordingly we have the correspondence for CW-translations.
The condition $a=b+c^{2}$ implies that the indicatrix of the Randers metric is
a sphere in $\mathfrak{m}$ with respect to the inner product
$\langle\cdot,\cdot\rangle_{eq}$ (in general not centered at $0$). Therefore
the projection of the $\mbox{Ad}_{G}$-orbit of $X$ is contained in a sphere.
In fact it is projected onto that sphere. If we choose a unitary matrix $U$ so
that its last row is $(u^{*},v^{*})$, with
$u\in\mathbb{C}^{l},v\in\mathbb{C}^{m}$ satisfying $v^{*}=(0,\ldots,0,s)$,
$s\in[-1,1]$, and its last column is of the form
$(\sqrt{1-s^{2}}w^{*},s)^{*}$, where $w$ is a unit vector in $\mathbb{C}^{n}$,
then for any $s\in[-1,1]$, and any unit vector $w$, this $U$ can be found by
the process of choosing a unitary basis. First choose
$(\sqrt{1-s^{2}}w^{*},s)$, then choose the next $m-1$ vectors from
$(\sqrt{1-s^{2}}w^{*},s)^{\perp}\cap(0,\ldots,0,1)^{\perp}$, then the others,
and rearrange the order at the end.(?)
When calculating $UXU^{*}$, we only need to consider the cases of $s\in[0,1]$.
The last column of
$UXU^{*}=\sqrt{-1}U(x_{1}I+x_{2}\mbox{diag}(-m\mbox{I}_{l},l\mbox{I}_{m}))U^{*}$
(5.13)
is
$\sqrt{-1}((n+1)x_{2}\sqrt{1-s^{2}}sw^{*},(n+1)x_{2}s^{2}-mx_{2}+x_{1})^{*}$.
They can give all the points on the sphere for
$\langle\cdot,\cdot\rangle_{eq}$, which is centered at
$(0,(l-m)x_{2}/2+x_{1})$ with radius $(n+1)|x_{2}|/2$. So it gives all
directions in $\mathfrak{m}$. The resulting non-Riemannian homogeneous Randers
metrics on $S^{2n+1}$ is restrictively CW-homogeneous. We Shall prove in the
following that they are in fact CW-homogeneous.
If $a\neq b+c^{2}$, then the only Killing vector fields of constant lengths of
the homogeneous Randers metric constructed by the triple are in the center
${\mathfrak{u}}(1)$. They only gives a flow of CW-translations on the sphere.
Obviously they are not restrictively Clifford-Wolf homogeneous.
###### Theorem 5.3
On $S^{2n+1}=U(n+1)/U(n)$, any non-Riemannian homogeneous Randers metrics
determined by a triple $(a,b,c)$ with $a=b+c^{2}$, are Clifford-Wolf
homogeneous.
Proof. Up to a scalar multiple, we can write $X$ as
$X=\sqrt{-1}(x\mbox{I}+\mbox{diag}(-I_{l},I_{m}))$, with $0<|x|<1$. With
respect to the homogeneous Randers metrics constructed above, any $X^{\prime}$
in the $\mbox{Ad}_{U(2m)}$-orbit of $X$ generates a flow of isometries
$\phi_{t}(v)=\exp(tX^{\prime})v$, $v\in S^{2n+1}\in\mathbf{C}^{2n+2}$ on the
sphere. Up to a scalar constant, $t$ parameterizes the length of geodesic flow
curves. For any $X^{\prime}$ in the same orbit, the flow curves $\phi_{t}(v)$
gives all the geodesic starting at $v$ with $t=0$, and all reach
$-\mbox{exp}(x\pi\sqrt{-1})v$ when $t=\pi$. For any other $v^{\prime}$ on the
sphere, the shortest geodesic from $v$ to $v^{\prime}$ must reach $v^{\prime}$
when $t\leq\pi$. Otherwise we can change it by choosing another geodesic for
the segment $t\in[0,\pi]$, then the path is not geodesic but gives the same
shortest distance from $v$ to $v^{\prime}$. This is a contradiction.
The study on those geodesics with $t\in(0,\pi)$ needs the following lemma
about the estimate of the eigenvalues of the product of two unitary matrices,
whose proof will be provided in the appendix.
###### Lemma 5.4
Suppose $P$ and $Q$ are two unitary matrix in $U(n)$, and denote the
eigenvalues of $P$ as $e^{a_{i}\sqrt{-1}}$, $a_{i}\in(-\pi,\pi)$, the
eigenvalues of $Q$ as $e^{b_{i}\sqrt{-1}}$, $b_{i}\in(-\pi,\pi)$,
$i=1,2,\ldots,n$. Let $m_{1}$ and $m_{2}$ be the maximum and minimum of the
$b_{i}$’s, respectively. Then the eigenvalues of $PQ$ must be of the form
$e^{c_{i}\sqrt{-1}}$, with $c_{i}\in[a_{i}+m,a_{i}+M]$.
If any two of those geodesics intersect within $t\in(0,\pi)$, i.e., there are
$X_{1}$ and $X_{2}$ which are unitary conjugate to $X$, and $t_{1}$ and
$t_{2}$ in $(0,\pi)$, such that $\exp(t_{1}X_{1})v=\exp(t_{2}X_{2})v$, then
$\exp(t_{1}X_{1})\cdot\exp(-t_{2}X_{2})$ has an eigenvalue $1$. If $t_{1}\neq
t_{2}$, then we may assume $t_{1}>t_{2}$. The eigenvalues of
$\mbox{exp}(-t_{2}X_{2})$ have the form $e^{b_{i}\sqrt{-1}}$,
$b_{i}\in[-t_{2}x-t_{2},-t_{2}x+t_{2}]$. By the lemma, the eigenvalues of
$\exp(t_{1}X_{1})v=\exp(t_{2}X_{2})v$ have the form $e^{c_{i}\sqrt{-1}}$,
$c_{i}\in[t_{1}x+t_{1}-t_{2}x-t_{2},t_{1}x+t_{1}-t_{2}x+t_{2}]$ or
$c_{i}\in[t_{1}x-t_{1}-t_{2}x-t_{2},t_{1}x-t_{1}-t_{2}x+t_{2}]$. In both cases
$c_{i}\in(-2\pi,0)\cup(0,2\pi)$, and $1$ can not be an eigenvalue of the
product. So if any two geodesics starting from $v$ with $t=0$ intersect in the
midway, then they have the same length between the two common points.
Therefore it suffices to prove that all the geodesics from $v$ to
$-\mbox{exp}(x\pi\sqrt{-1})v$ are the shortest pathes. In fact, when
$t=t_{1}=t_{2}\in(0,\pi)$, $\exp(-tX_{1})\exp(tX_{2})v=v$, we can prove that
these two geodesic coincides for all $t\in[0,\pi]$. The essential steps are
left in the Appendix. The flows generated by the Killing vector fields in the
orbit of $X$ are CW-translations for $t\in(0,\pi]$. This completes the proof
of the theorem.
## 6 Randers metrics on $S^{4n+3}$ with $I_{0}(S^{4n+3},F)\subset
Sp(n+1)U(1)$
Now we start the discussion on the cases (4) and (5) in the list of
homogeneous spaces for spheres. Let $F$ be a non-Riemannian homogeneous
Randers metric on $S^{4n+3}$, $n>0$, such that its connected isometry group
contains $Sp(n+1)$. Then $I_{0}(M,F)$ must be among $U(2n+2)$, $Sp(n+1)U(1)$
or $Sp(n+1)$. The unitary case has already been discussed. So we only need to
consider the case $I_{0}(M,F)=Sp(n+1)U(1)$ or $I_{0}(M,F)=Sp(n+1)$.
For $G=Sp(n+1)$ or $Sp(n+1)U(1)$, $\mathfrak{m}$ can be decomposed as
$\mathfrak{m}_{0}\oplus\mathfrak{m}_{1}$, where
$\mathfrak{m}_{0}=\mbox{Im}\,\mathbb{H}$ is the $3$-dimensional trivial
representation of $Sp(n-1)$, and $\mathfrak{m}_{1}=\mathbb{H}^{n}$ with the
action of $Sp(n)$ by left multiplication. When regarded as a $Sp(n)U(1)$
representation, $\mathfrak{m}_{1}$ also has the action of $U(1)$-scalar
multiplication from the right, and $\mathfrak{m}_{0}$ is further decomposed
into the sum of the $1$-dimensional trivial representation of $U(1)$,
generated by $\mathbf{i}\in\mathbb{H}$ (which is identified with $\sqrt{-1}\in
U(1)$), and the 2-dimensional space spanned by $\mathbf{j}$ and $\mathbf{k}$,
on which $U(1)$ acts as the rotation group. The projection from the Lie
algebra of $G=Sp(n+1)$ or $Sp(n+1)U(1)$ to $\mathfrak{m}$ is just the
differentiation of the group action on
$(0,\ldots,0,1)^{*}\in\mathfrak{m}\subset\mathbb{H}^{n+1}$ at $I$.
We have a standard inner product on $\mathfrak{m}$ induced by a symmetric
Riemannian metric on the sphere. It will be denoted as
$\langle\,,\,\rangle_{eq}$. Any $\mbox{Ad}_{Sp(n)Sp(1)}$-invariant linear
metric on $\mathfrak{m}$ can be written as
$\alpha^{2}(u,q)=\mbox{Re}(aq^{*}q+bu^{*}u)$, $q\in\mbox{Im}\,\mathbb{H}$,
$u\in\mathbb{H}^{n}$. The standard inner product $\langle\,,\,\rangle_{eq}$ is
corresponding to the case $a=b=1$. A non-Riemannian Randers metric $F$ can
written as
$F=\alpha+\langle\cdot,V\rangle_{eq},$ (6.14)
where $V\in\mathfrak{m}_{0}$ can be any non-zero vector if the isometry group
is $Sp(n+1)$, or generated by $\mathbf{i}$ if the isometry group is
$Sp(n+1)U(1)$.
We now prove
###### Proposition 6.1
If $X\in\mathfrak{sp}(n+1)$ generates a Killing vector field of constant
length with respect to a non-Riemannian homogeneous Randers metric $F$ on
$S^{4n+3}$, then $X=0$.
Proof. Up to a $Sp(n+1)$ conjugation, one can assume that $X$ is a diagonal
matrix in $\mathfrak{gl}(n+1,\mathbb{H})$. If a diagonal entry of $X$ is not
$0$, say $q\in{\mathbb{H}}$, then there is an element $\sigma$ in the Weyl
group such that the down right corner of $\sigma(X)$ is $q$. Moreover, up to a
$Sp(n+1)$ conjugation, we can further assume that the down right corner of
$\sigma(X)$ is real proportional to $V$ in (6.14). There are two choices for
this down right corner, namely $\frac{|q|}{|V|}V$ and $-\frac{|q|}{|V|}V$. The
projections of the corresponding matrix to $\mathfrak{m}$ form an opposite
pair of vectors, denoted by $q_{1}$ and $q_{2}$. Then we have
$\alpha(q_{1})=\alpha(q_{2})$, $F(q_{1})=F(q_{2})$ and
$\beta(q_{1})=-\beta(q_{2})$. This means that $\beta(q_{1})=\beta(q_{2})=0$.
Therefore we have $\langle V,V\rangle_{eq}=0$. This is a contradiction with
$V\neq 0$.
From now on, we assume that $G=I_{0}(S^{4n+3},F)=Sp(n+1)U(1)$. The vector $V$
in (6.14) will be denoted as $(0,\ldots,0,c\mathbf{i})^{T}$, where $c$ is a
nonzero real number. The metric $\alpha$ induced on $\mathfrak{m}$ can be
written as
$\alpha^{2}=\mbox{Re}(a_{1}\lambda_{1}^{2}+a_{2}\lambda_{2}^{2}+a_{2}\lambda_{3}^{2}+bu^{*}u)$,
$q=\lambda_{1}\mathbf{i}+\lambda_{2}\mathbf{j}+\lambda_{3}\mathbf{k}$,
$u\in\mathbb{H}^{n}$, for some positive parameter $a_{1}$, $a_{2}$ and $b$.
Moreover, we have $a_{2}\neq b$, since otherwise $I_{0}(S^{4n+3},F)$ will be
$U(2n+2)$.
As in the last section, the center of $G$ generates CW-translations for any
homogeneous Finsler metric on the sphere, and they are not the ones we are
searching for. Assume there is a non-central vector in $\mathfrak{g}$, which
generates a Killing vector field of constant length with respect to a non-
Riemannian homogeneous Randers metric $F$. Then it can be written as
$(X,x)\in\mathfrak{sp}(n+1)\oplus\mathbb{R}$, with $X\neq 0$. Then Proposition
6.1 asserts that $x$ is nonzero either.
Up to a suitable $Sp(n+1)$ conjugation, we can assume
$X=\mbox{diag}(x_{1}\mathbf{i},\ldots,x_{n+1}\mathbf{i}),x_{i}\in\mathbf{R},i=1,\ldots,n+1.$
(6.15)
Using the action of the Weyl group, we can reorder the $x_{i}$’s freely, and
change $x_{n+1}$ to $-x_{n+1}$. In this way, we get a set of elements in
$\mathfrak{g}$. Their projections to $\mathfrak{m}$ have the form
$(0,\ldots,0,(\pm x_{i}+x)\mathbf{i})^{T}$, $i=1,\ldots,n+1$. They all have
the same $F$ values. Similar discussions shows that $\\{\pm
x_{i}+x,i=1,\ldots,n+1\\}$ must take exactly $2$ values with opposite signs.
So the $|x_{i}|$’s must be equal to each other. Using actions of the Weyl
group, we can change all the $x_{i}$’s to the same positive number. Then we
can write $X=x^{\prime}\mathbf{i}I\in sp(n+1)\subset gl(n+1,\mathbb{H})$, with
$x^{\prime}>|x|>0$.
Now we need to calculate the projection of the
$\mbox{Ad}_{Sp(n+1)U(1)}$-orbit, which is also the $\mbox{Ad}_{Sp(n+1)}$-orbit
of $(X,x)$. Suppose $Q=Q_{1}+Q_{2}\mathbf{j}\in Sp(n+1)$, where $Q_{1}$ and
$Q_{2}$ are complex matrices, and $\sqrt{-1}$ is identified with $\mathbf{i}$.
The condition $Q\in Sp(n+1)$ implies
$\displaystyle Q_{1}^{*}Q_{2}-Q_{2}^{T}\bar{Q_{1}}$ $\displaystyle=$
$\displaystyle 0,$ (6.16) $\displaystyle Q_{1}^{*}Q_{1}+Q_{2}^{T}\bar{Q_{2}}$
$\displaystyle=$ $\displaystyle I.$ (6.17)
Then
$Q^{*}XQ=-\mathbf{i}I+2Q_{1}^{*}Q_{1}\mathbf{i}+2Q_{1}^{*}Q_{2}\mathbf{k}=\mathbf{i}(-I+2Q_{1}^{*}(Q_{1}+Q_{2}\mathbf{j}))$.
As we will project it to $\mathfrak{m}$, we only need to see its last column.
The last row (?) of $Q$ can be denoted as $(\sqrt{1-|q|^{2}}w,q)$, where
$w^{*}\in\mathbb{H}^{n-1}$ is a unit vector, and $q=q_{1}+q_{2}\mathbf{j}$,
$q_{i}\in\mathbb{C}$. We first assume that the last row of $Q_{1}^{*}$ is
$(0,\ldots,0,\bar{q_{1}})$. Then the projection of $(Q^{*}XQ,x)$ in
$\mathfrak{m}$ is
$-((2x^{\prime}(1-|q_{1}|^{2}-|q_{2}|^{2}))\mathbf{i}q_{1}w,(x^{\prime}(2|q_{1}|^{2}-1)+x)\mathbf{i}+2x^{\prime}\bar{q_{1}}q_{2}\mathbf{k})^{*}.$
(6.18)
It gives all the points on the sphere $S$ for $\langle\,,\,\rangle_{eq}$,
which is centered at $x$ and has a radius $x^{\prime}$ if we can find the
suitable $w$ and $q$. Note that this is the same sphere appearing in the last
section with $m=l$. With $Sp(n)$ changed by $SU(2n)$ or $U(2n)$, the
$\mbox{Ad}_{U(2n)}$-orbit ( containing the $\mbox{Ad}_{Sp(n)}$-orbit ) will be
mapped onto $S$. If the $\mbox{Ad}_{Sp(n)}$-orbit is also mapped onto $S$,
then the Randers metric must be the one constructed in the last section, and
in this case we have $I(S^{4n+3},F)=U(2n+2)$.
To see the $\mbox{Ad}_{Sp(n)}$-orbit is also mapped onto $S$, we need the
following lemma.
###### Lemma 6.2
For any unit vector in $\mathbb{H}^{n}$, there is $Q\in Sp(n)$, such that the
last row of $Q$ is the given vector, and the last row of $Q^{*}$ has the form
$(q^{\prime},0\ldots,0,q)$, where the imaginary part of $q^{\prime}$ is
contained in the real span of $\mathbf{j}$ and $\mathbf{k}$.
The proof can be given by the process of choosing an orthogonal basis for the
inner product $\langle x,y\rangle=x^{*}y\in\mathbb{H}$. First we choose an
arbitrary unit vector for the last row of $Q$, then choose the next $n-2$ in
orthogonal complement of the first one, at the same time perpendicular to
$(0,\ldots,0,1)$. For the last basis vector, we can use a suitable unit scalar
multiplication to make its last term only contain $\mathbf{j}$ and
$\mathbf{k}$. (?)
We conclude this paper with the following theorem.
###### Theorem 6.3
Let $F$ be a homogeneous Randers metric on $S^{4n+3}$ with
$I_{0}(S^{4n+3},F)=Sp(n+1)U(1)$. Then any Killing vector field of constant
length for $F$ is generated by a central vector in the Lie algebra of the
isometry group.
## 7 Appendix: estimates of the eigenvalues of unitary matrices
In this section we give a proof of Lemma 5.4.
Up to a suitable unitary conjugation, we can assume $Q=\exp B$, where
$B=\mbox{diag}(b_{1}\sqrt{-1},\ldots,b_{n}\sqrt{-1})$, $b_{i}\in(-\pi,\pi)$
for $i=1,\ldots,n$.
First we consider a special case such that $P\exp(tB)$ has no multiple
eigenvalues for each $t\in[0,1]$. The eigenvalues of $P\mbox{exp}(tB)$ can
then be presented as smooth functions $\lambda_{1}(t),\ldots,\lambda_{n}(t)$
of $t\in[0,1]$. We can also find the corresponding unit eigenvectors
$v_{1}(t),\ldots,v_{n}(t)$, respectively, which are vector-valued smooth
function of $t$. Differentiating the equation
$P\exp(tB)v_{1}(t)=\lambda_{1}(t)v_{1}(t)$ with respect to $t$ and taking
$t=0$, we get
$Pv^{\prime}_{1}(0)+Bv_{1}(0)=\lambda^{\prime}_{1}(0)v_{1}(0)+\lambda_{1}(0)v^{\prime}_{1}(0).$
(7.19)
Taking the inner product with $v_{1}(0)$ for both sides of (7.19), and
noticing that $Pv^{\prime}_{1}(0)$ and $v^{\prime}_{1}(0)$ are orthogonal to
$v_{1}(0)$, we have $Bv_{1}(0)=\lambda^{\prime}_{1}(0)$. So
$\lambda^{\prime}_{1}(0)/\sqrt{-1}$ is bounded between the minimum and the
maximum of all the $b_{i}$’s.
This calculation is valid for all $i=1,\ldots,n$ and all $t\in[0,1]$ with $P$
replaced by $P\exp(tB)$. If we write
$\lambda_{i}(t)=\mbox{exp}(c_{i}(t)\sqrt{-1})$ with smooth $c_{i}(t)$
satisfying $c_{i}(0)=a_{i}$, then
$c^{\prime}_{i}(t)=\lambda^{\prime}_{i}(t)/\sqrt{-1}$ is bounded between the
minimum $m_{1}$ and maximum $m_{2}$ for all the $b_{i}$’s.
When the matrix changes continuously, the eigenvalues also vary continuously.
To prove the lemma for general $P$ and $Q$, we only need to notice the fact
that generically the unitary matrices have no multiple eigenvalues. In fact,
the set of those unitary matrices with multiple eigenvalues form a real
subvariety of codimension $3$. Therefore for generically chosen $P$ and $O$,
then $1$-parameter curve $P\exp(tB)$ has no intersection with it. If they
intersect at finite points, it does not matter either. Though the eigenvalue
functions are not globally smoothly defined, in each small closed interval,
they can be continuous defined, and in each small open interval, they are
smooth. The argument can still be carried out for each interval, and one can
get the estimate of the eigenvalues for $t=1$.
The following lemma is the essential technique to prove the claim in section 5
that the Clifford Wolf homogeneous Randers metrics we constructed on
$S^{2n+1}$ satisfying the property that the geodesics starting from one point
will all pass another point, and they do not intersect each other in the
midway.
###### Lemma 7.1
Let $U\in U(m+l)$ and $X=\sqrt{-1}\mbox{diag}(-I_{l},I_{m})$. If the
commutator $\exp(tX)U\exp(-tX)U^{*}$ has an eigenvalue $1$ for some
$t\in(0,\pi)$, then for each $t\in(0,\pi)$ it has an eigenvalue $1$ with the
same eigenvector.
Proof. We denote $\mbox{exp}(tX)$ as $\left(\begin{array}[]{cc}\lambda&0\\\
0&\bar{\lambda}\\\ \end{array}\right)$, $\lambda\neq 1$ or $-1$, and $U$ as
$\left(\begin{array}[]{cc}U_{1}&U_{2}\\\ U_{3}&U_{4}\\\ \end{array}\right)$.
Then
$\mbox{exp}(tX)U\mbox{exp}(-tX)U^{*}=I+\left(\begin{array}[]{cc}(\lambda^{2}-1)U_{2}&0\\\
0&(\bar{\lambda^{2}}-1)U_{3}\\\
\end{array}\right)\left(\begin{array}[]{cc}U^{*}_{2}&U^{*}_{4}\\\
U^{*}_{1}&U^{*}_{3}\\\ \end{array}\right).$
It is not hard to see that if $1$ is an eigenvalue, then $U_{2}$ or $U_{3}$
must be singular. The eigenspace of $1$ is the direct sum of the kernel of
$U_{2}$ and $U_{3}$, multiplied by an invertible matrix irrelative to $t$.
Obviously the change of $\lambda\in S^{1}\backslash\\{\pm 1\\}$ or $t\in(0,1)$
does not affect the eigenspace for $1$.
Acknowledgement. We are grateful to Dr. Libing Huang and Dr. Zhiguang Hu for
useful discussions. This work was finished during the second author’s visit to
the Chern institute of Mathematics. He would like to express his deep
gratitude to the members of the institute for their hospitality.
## References
* [AW76] R. Azencott, E. Wilson, Homogeneous manifolds with negative curvature I, Trans. Amer. Math. Soc., 215 (1976), 323-362.
* [BCS00] D. Bao, S. S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, New York, 2000.
* [BN081] V. N. Berestovskii, Yu. G. Nikonorov, Killing vector fields of constant length on locally symmetric Riemannian manifolds, Transformation Groups, 13 (2008), 25 C45.
* [BN082] V. N. Berestovskii, Yu.G. Nikonorov, On $\delta$-homogeneous Riemannian manifolds, Diff. Geom. Appl., 26 2008, 514 C535.
* [BN09] V. N. Berestovskii, Yu.G. Nikonorov, Clifford-Wolf homogeneous Riemannian manifolds, Jour. Differ. Geom., 82 (2009), 467-500.
* [CS04] S. S. Chern, Z. Shen, Riemann-Finsler Geometry, World Scientific Publishers, 2004.
* [DE08] S. Deng, The S-curvature of homogeneous Randers spaces, Differ. Geom. Appl., 27 (2009) 75-84.
* [DH02] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific J. Math, 207 (2002), 149-157.
* [DH07] S. Deng and Z. Hou, On symmetric Finsler spaces, Israel J. Math., 162 (2007), 197-219.
* [DP] S. Deng, Clifford-Wolf translations of Finsler spaces of Finsler spaces of negative flag curvature, preprint.
* [DXP] S. Deng, M. Xu, Clifford-Wolf translations of Finsler spaces, preprint.
* [HE74] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann., 211 (1974), 23-34.
* [MS43] D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. Math., 44 (1943), 454-470.
* [MTP] V. S. Matveev, M. Troyanov, The Binet-Legendre elliposoid in Finsler geometry, preprint, ArXiv: 1104.1647v1.
* [RA04] H. B. Rademacher, Nonreversible Finsler spaces of positive flag curvature, In: A smaple of Finsler geometry, eds: D. Bao, R. Bryant, S. S. Chern, Z. Shen, MSRI Publ. 50, 2004, 261-302.
* [SH01] Z. Shen, Differential Geometry of Sprays and Finsler Spaces, Kluwer, Dordrent, 2001.
* [WO64] J. A. Wolf, Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math., 8 (1964), 14-18.
|
arxiv-papers
| 2012-04-23T23:18:36 |
2024-09-04T02:49:30.053035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaoqiang Deng and Ming Xu",
"submitter": "Shaoqiang Deng",
"url": "https://arxiv.org/abs/1204.5232"
}
|
1204.5233
|
# Clifford-Wolf translations of left invariant Randers metrics on compact Lie
groups111Supported by NSFC(no.10671096 and 10971104) and SRFDP of China.
Shaoqiang Deng1 and Ming Xu2
1School of Mathematical Sciences and LPMC
Nankai University
Tianjin 300071, P. R. China
2Department of Mathematical Sciences
Tsinghua University
Beijing 100084, P. R. China Corresponding author. E-mail:
mgxu@math.tsinghua.edu.cn
###### Abstract
A Clifford-Wolf translation of a connected Finsler space is an isometry which
moves each point the sam distance. A Finsler space $(M,F)$ is called Clifford-
Wolf homogeneous if for any two point $x_{1},x_{2}\in M$ there is a Clifford-
Wolf translation $\rho$ such that $\rho(x_{1})=x_{2}$. In this paper, we study
Clifford-Wolf translations of left invariant Randers metrics on compact Lie
groups. The mian result is that a left invariant Randers metric on a connected
compact simple Lie group is Clifford-Wolf homogeneous if and only if the
indicatrix of the metric is a round sphere with respect to a bi-invariant
Riemannian metric. This presents a large number of examples of non-reversible
Finsler metrics which are Clifford-Wolf homogeneous.
Mathematics Subject Classification (2000): 22E46, 53C30.
Key words: Finsler spaces, Clifford-Wolf translations, Killing vector fields,
compact Lie groups, left invariant Randers metrics.
## 1 Introduction
Let $(M,g)$ be a homogeneous Riemannian manifold. An isometry of the
Riemannian manifold $(M,g)$ is called a Clifford-Wolf translation (or simply
CW-transformation) if it moves each point of $M$ the same distance. If for any
two points $x_{1},x_{2}\in M$, there is a CW-translation $\rho$ such that
$\rho(x_{1})=x_{2}$, then $(M,g)$ is called Clifford-Wolf homogeneous (or
simply CW-homogeneous).
The general merit of the study of CW-translations lies in the fact that they
are closely related to the study of homogeneous Riemannian manifolds. Let
$\Gamma$ be a properly discontinuous subgroup of the full group of isometries
of $(M,g)$, which acts freely on $M$. It is a very important problem in
Riemannina geometry to find out the conditions that the quotient manifold
$M/\Gamma$ is again a homogeneous Riemannian manifolds. Through the work of a
number of researchers, we can say that at least in the cases that $(M,g)$ is a
Riemannian symmetric space, or a homogeneous Riemannian manifold of non-
positive curvature, or a homogeneous Riemannian manifold admitting a
transitive group of isometries which is semisimple of the noncompact type,
$M/\Gamma$ is a homogeneous Riemannian manifold, provided that $\Gamma$
consists of Cw-translations. Moreover, in the first two cases it is proved
that the above condition is also sufficient; see [WO60], [WO62], [WO64] and
[DMW86] for the details.
In view of the the above results, it is an important problem to determine all
the CW-transformations of a given Riemannian manifold, particularly for the
homogeneous ones. For Riemannian symmetric spaces this problem was first
completely settled by J. A. Wolf in [WO62]; see also [FR63, OZ69, OZ74] for
other proofs of the results. The related results has important application in
the study of homogeneous Riemannian manifolds of negative curvature; see for
example [HE74, AW76].
Recently the study on CW-translation has become active. In [BN08-1, BN08-2,
BN09], Berestovskii and Nikonorov studied the local one-parameter group of CW-
translations of general Riemannian manifolds. The main result is that there is
a correspondence between local one-parameter groups of CW-translations and
Killing vector fields of constant length. They also obtained a classification
of connected simply connected CW-homogeneous Riemannian manifolds. The list
consists of the euclidean spaces, the odd-dimensional spheres with constant
curvature, connected simply connected compact simple Lie groups with bi-
invariant Riemannian metrics and the direct products of the above manifolds.
Notice that there are also a local version of the related notion of CW-
homogeneous Riemannian manifolds; see the above cited papers above.
In our recent papers [DE12] and [DM12], we initiated the study of CW-
translations of Finsler spaces. The notions of CW-translations and CW-
homogeneity have been generalized to the Finslerian setting. It is proved that
the correspondence between the local one-parameter subgroup of CW-translations
and the Killing vector fields of constant length is also valid for a Finsler
space; see the next section for the details. We also studied this problem for
a special type of Finsler spaces-Randers spaces and obtained a necessary and
sufficient condition for a smooth vector field $X$ on a homogeneous Randers
space $(G/H,F)$, with $H$ connected, to be a Killing vector field of constant
length. This result gives a complete classification for all the local one-
parameter subgroup of CW-translations of a homogeneous Randers space.
In this paper we continue our study on this topic. Our main idea is to study
the condition for a homogeneous Randers space to be CW-homogeneous, and
futhermore, to classify all the CW-homogeneous Randers spaces. However, due to
the complexity of the problem, we will only deal with left invariant Randers
metrics on a compact simple Lie group. The main result of this paper is the
following
###### Theorem 1.1
Let $G$ be a compact connected simple Lie group and $F$ be a left invariant
Randers metric. Then $(G,F)$ is CW-homogeneous if and only if the indicatrix
of $F$ in $\mathfrak{g}$ is a round sphere with respect to a bi-invariant
metric.
Recall that the indicatrix of a Finsler space $(M,F)$ at a point $x\in M$ is
defined to be
$\mathcal{I}_{x}=\\{y\in T_{x}(M)|F(x,y)=1\\}.$
Notice that in the above theorem, we have identified $\mathfrak{g}$ with
$T_{e}G$. On the other hand, since $G$ is simple, up to a positive scalar the
bi-invariant Riemannian metric on $G$ is unique. Therefore the statement of
the theorem have no independence on the specific bi-invariant metric. The
above theorem says that a left invariant Randers metric on a compact simple is
CW-homogeneous if and only if it solves the Zermelo’s navigation problem of a
bi-invariant Riemannian metric under the influence of certain external vector
field. Put another way, this theorem presents many examples of non-reversible
Finsler spaces which are CW-homogeneous. It would be an interesting problem to
classify CW-homogeneous Finsler spaces, either reversible or non-reversible.
The arrangement of this paper is as the following. In Section 2, we recall
some notions and known results on Finsler geometry and CW-translations of
Finsler spaces. In Section 3, we study Killing vector fields of left invariant
Randers metrics and obtain a complete description of those with constant
length. In Section 4, we first state a theorem (Theorem 4.1) and then use this
theorem to give a proof of Theorem 1.1. Finally, in Section 5, we give a case
by case proof of Theorem 4.1.
## 2 Preliminary
Finsler geometry is introduced by Riemann in his celebrated lecture on the
foundations of geometry, addressed in 1854. Due to the complexity of the
problem, the study of Finsler geometry was dormant for a rather long period.
In 1918, Finsler studied the variation problem of Finsler spaces in his
doctoral dissertation, which initiated the systematic study of Finsler
geometry. For fundamental properties of Finsler spaces, we refer the readers
to [BCS00], [CS05] and [SH01].
###### Definition 2.1
A Finsler metric on a manifold $M$ is a continuous function
$F:TM\rightarrow\mathbb{R}^{+}$, which is smooth on the slit tangent bundle
$TM\backslash 0$. In any local standard coordinates $(x^{i},y^{j})$ for $TM$,
$F$ satisfies the following conditions:
(1) $F(x,y)>0$ for any $y\neq 0$.
(2) $F(x,\lambda y)=\lambda F(x,y)$ for any $y\in TM_{x}$, and $\lambda>0$.
(3) The Hessian matrix defined by $g_{ij}=\frac{1}{2}[F^{2}]_{y^{i}y^{j}}$ is
positive definite.
An important example is a Finsler metric with the form $F=\alpha+\beta$, where
$\alpha$ is a Riemannian metric and $\beta$ is a 1-form. This kind of Finsler
metrics are called Randers metrics. Notice that the norm of the $1$-form
$\beta$ with respect to the metric $\alpha$ must be smaller than $1$ and they
reduce to Riemannian metrics if and only if the $\beta$-terms vanish. Randers
metrics are among the most important examples of non-Riemannian Finsler
metrics in the study of geometry and physics.
Using the integration of $F$ along a path, we can define the arc length of
piece-wise smooth curves. The “distance” of two points is then defined to be
the infimum of the arc length of all the piece-wise smooth curves connecting
them. We call it the distance function and denote as $d(\cdot,\cdot)$. In
general, the distance function of a Finsler space is not reversible (i.e.,
$d(x,y)=d(y,x)$ for any $x,y$) unless $F(x,y)=F(x,-y)$, for all $x\in M$ and
$y\in TM_{x}$.
The notion of CW-translations can be generalized to Finsler geometry; see
[DM12].
###### Definition 2.2
A CW-translation $\rho$ of a Finsler manifold $(M,F)$ is an isometry of
$(M,F)$ such that $d(x,\rho(x))$ is a constant function.
In the Riemannian case, the interrelation between CW-translations and Killing
vector fields of constant lengths is studied by V.N.Berestovskii and
Yu.G.Nikonorov. The key observation is that the flow curves of a Killing
vector field of constant length are geodesics, so Killing vector fields of
constant length can be used to generate one-parameter groups of CW-tranlations
when the parameter var1able is close to $0$. This observation is still valid
in the Finslerian case, except that the Killing vector fields of constant
lengths can only generate CW-translations with positive small parameter
variables when $F$ is not absolutely homogeneous. In fact, we have the
following interrelation theorems ([DM12]).
###### Theorem 2.3
Let $(M,F)$ be a complete Finsler manifold with a positive injective radius.
If $X$ is a Killing vector field of constant length, then the flow $\phi_{t}$
generated by $X$ is a CW-translation for all sufficiently small $t>0$.
###### Theorem 2.4
Let $(M,F)$ be a compact Finsler manifold. Then there is a $\delta>0$, such
that any CW-translation $\rho$ with $d(x,\rho(x))<\delta$ is generated by a
Killing vector field of constant length.
We have also generalized the concept of (restrictively) CW-homogeneous space
in Finsler geometry.
###### Definition 2.5
A Finsler manifold $(M,F)$ is called CW-homogeneous if for any two points
$x_{1},x_{2}\in M$, there is a CW-translation $\rho$ such that
$\rho(x_{1})=x_{2}$. It is called restrictively CW-homogeneous if for any
point $x\in M$, there is a neighborhood $V$ of $x$, such that for any two
points $x_{1},x_{2}\in V$ there is a CW-translation $\rho$ mapping $x_{1}$ to
$x_{2}$.
Notice that the set of CW-translations of a Finsler space is generally not a
subgroup of the full group of isometries. Therefore, restrictive CW-
homogeneity is a weaker notion than CW-homogeneity. In the case that $(M,F)$
is a compact Finsler manifold, the interrelation between CW-translations and
Killing vector fields of constant length gives an equivalent description of
the restrictive CW-homogeneity, namely, the Finsler space $(M,F)$ is
restrictively CW-homogeneous if and only if any non-vanishing tangent vector
can be extended to a Killing vector field of constant length.
## 3 Left invariant Randers metrics and Killing vector fields of constant
length
Let $G$ be a connect compact Lie group with Lie algebra $\mathfrak{g}$. A
Finsler metric $F$ on $G$ is called left invariant if the left translation
group $L(G)$ is contained in the isometry group of $F$. A left invariant
Finsler metric is completely determined by its restriction to
$T_{e}G\cong\mathfrak{g}$. A left invariant Randers $F=\alpha+\beta$ is then
determined by an inner product on $\mathfrak{g}$ and an element of
$\mathfrak{g}^{*}$, which define the $\alpha$ and $\beta$ terms respectively.
Let $\langle\,,\,\rangle$ be the inner product induced by $\alpha$ and
$\langle\,,\,\rangle_{\mathrm{bi}}$ be a fixed $\mathrm{Ad}(G)$-invariant
inner product on $\mathfrak{g}$ induced by a bi-invariant metric on $G$, then
the restriction of $F$ to $T_{e}G$ can be written as $F(y)=\sqrt{\langle
y,y\rangle}+\langle y,V\rangle_{\mathrm{bi}}$ for some $V\in\mathfrak{g}$.
The Killing vector fields with constant length of a left invariant Randers
metric is the key subject for our considerations. In [DM12], we have already
discussed some techniques for determining Killing vector fields
$X\in\mathfrak{g}$ with constant length, and found some examples in which the
CW-translations are different from those Riemannian ones. Any Killing vector
field $X$ belongs to the Lie algebra of the full isometry group of $F$, which
contains $\mathfrak{g}$ as a Lie subalgebra corresponding to the Lie subgroup
$L(G)$. Therefore it is important to compute the full isometries of $(G,F)$.
In general, it is rather difficult to get a complete classification of all the
isometry groups of left invariant Randers metrics on a compact Lie group.
However, the following lemma tells us that its identity component is contained
in $L(G)R(G)$ when $G$ is simple (i.e., $\mathfrak{g}$ is simple).
###### Lemma 3.1
Let $G$ be a compact connected simple Lie group and $F$ a left invariant
Randers metric on $G$. Then the identity component $I_{0}(G,F)$ of the full
group $I(M,F)$ of isometries of $(G,F)$ is contained in $L(G)R(G)$.
Proof. Let $F=\alpha+\beta$ be the standard decomposition for the left
invariant Randers metric. Then we have $L(G)\subset I(G,F)\subset
I(G,\alpha)$, i.e., $\alpha$ is a left invariant Riemannian metric. Since $G$
is compact connected simple, $I_{0}(G,\alpha)$ is contained in $L(G)R(G)$
([OT76]). Thus $I_{0}(G,F)\subset I_{0}(G,\alpha)$ is also contained in
$L(G)R(G)$.
Let $G^{\prime}$ be the subgroup of $G$ such that $R(G^{\prime})$ is the
connected component of the group of all isometric right translations, and
denote $\mathrm{Lie}(G^{\prime})=\mathfrak{g}^{\prime}$. Then obviously
$I_{0}(G,F)=L(G)R(G^{\prime})$. In the Lie algebra level, the Lie algebra of
$I_{0}(G,F)$, i.e., the space of all Killing vector fields, is the direct sum
of $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$.
The following theorem gives a criterion for a Killing vector field
$(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ of the left
invariant Randers metric $F$ to have constant length.
###### Theorem 3.2
If $(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ generates a
Killing vector field of constant length of a left invariant Randers metric
$F$, then either $X=0$ or $X^{\prime}$ belongs to the center
$\mathfrak{c}(\mathfrak{g}^{\prime})$ of $\mathfrak{g}^{\prime}$.
Proof. Let $F(y)=\sqrt{\langle y,y\rangle}+\langle y,V\rangle_{\mathrm{bi}}$
be the restriction of $F$ to $T_{e}G$. If
$(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ generates a Killing
vector field of constant length for $F$, then the projection of the
$\mathrm{Ad}(L(G)R(G^{\prime}))$-orbit of $(X,X^{\prime})$ in $\mathfrak{g}$
has the same $F$ values, i.e.,
$\displaystyle F(\mathrm{Ad}((g,g^{\prime}))(X,X^{\prime}))$ $\displaystyle=$
$\displaystyle\alpha(\mathrm{Ad}(g)X-\mathrm{Ad}(g^{\prime})X^{\prime})+\langle\mathrm{Ad}(g)X-\mathrm{Ad}(g^{\prime})X^{\prime},V\rangle_{\mathrm{bi}}$
(3.1) $\displaystyle\equiv$ $\displaystyle\mbox{constant},\forall g\in
G,g^{\prime}\in G^{\prime}.$
Since $\langle\mathrm{Ad}(g^{\prime})X^{\prime},V\rangle_{\mathrm{bi}}=\langle
X^{\prime},V\rangle$, we see that for any fixed $g\in G$,
$\alpha(\mathrm{Ad}(g)X-\mathrm{Ad}(g^{\prime})X^{\prime})$ is a constant
function of $g^{\prime}\in G^{\prime}$. So
$\displaystyle\langle\mathrm{Ad}(g)X,\mathrm{Ad}(g^{\prime})X^{\prime}\rangle_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\alpha(\mathrm{Ad}(g)X)+\alpha(\mathrm{Ad}(g^{\prime})X^{\prime})-\alpha(\mathrm{Ad}(g)X-\mathrm{Ad}(g^{\prime})X^{\prime}))$
(3.2) $\displaystyle=$
$\displaystyle\frac{1}{2}(\alpha(\mathrm{Ad}(g)X)+\alpha(X^{\prime})-\alpha(\mathrm{Ad}(g)X-\mathrm{Ad}(g^{\prime})X^{\prime}))$
is a constant function of $g^{\prime}\in G^{\prime}$, for any fixed $g\in G$.
Now Select
$g^{\prime}=\exp(t_{1}Y_{1})\cdots\exp(t_{n}Y_{n}).$
Taking the derivative with respect to all the $t_{i}$’s and evaluating at $0$,
we easily deduce that for any $g\in G$, the vector $\mathrm{Ad}(g)X$ is
orthogonal to the ideal generated by $[X^{\prime},\mathfrak{g}^{\prime}]$ in
$\mathfrak{g}^{\prime}$ with respect to $\alpha$. Since the above assertion
holds for any $g\in G$, we see that the ideal generated by $X$ in
$\mathfrak{g}$ is orthogonal to the ideal generated by
$[X^{\prime},\mathfrak{g}^{\prime}]$ in $\mathfrak{g}^{\prime}$. Since
$\mathfrak{g}$ is simple, this implies that either $X=0$ generates the $0$
ideal, or $X\neq 0$ and it generates the ideal $\mathfrak{g}$. Notice that in
the later case we have $[X^{\prime},\mathfrak{g}^{\prime}]=0$, i.e.,
$X^{\prime}\in\mathfrak{c}(\mathfrak{g^{\prime}})$. This completes the proof
of the theorem.
Using a similar argument we can prove the following modification of Theorem
3.2, which allows more general $G^{\prime}$.
###### Theorem 3.3
Let $G$ be a compact connected simple Lie group, and $G^{\prime}$ a closed
connected subgroup of $G$ with Lie algebra $\mathfrak{g}^{\prime}$. Let $F$ be
a $L(G)R(G^{\prime})$-invariant Randers metric on $G$. If
$(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ generates a Killing
vector field of constant length, then either $X=0$ or $X^{\prime}$ belongs to
the center $\mathfrak{c}(\mathfrak{g}^{\prime})$ of $\mathfrak{g}^{\prime}$.
In Theorem 3.3, we treat $(G,F)$ as the homogeneous Randers space, in which is
viewed as the coset space $G=(L(G)R(G^{\prime}))/H$ and $H$ can be identified
with the quotient of $G^{\prime}$ by a discrete normal subgroup. The Lie
algebra $\mathfrak{h}$ of $H$ is isomorphic to $\mathfrak{g}^{\prime}$ and it
is a “diagonal” in $\mathfrak{g}\oplus\mathfrak{g}^{\prime}$. The projection
from $(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ to
$\mathfrak{m}\cong\mathfrak{g}$ is just $X-X^{\prime}$.
For simplicity, we assume that
$(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{c}(\mathfrak{g}^{\prime})$, with
$X\neq 0$, generates a Killing vector field of constant length $1$ for the
$L(G)R(G^{\prime})$-invariant Randers metric $F$ on $G$. Then the projection
from the orbit to $\mathfrak{g}$ is just a shift by $-X^{\prime}$. Its
projection in $\mathfrak{g}$ is contained in an indicatrix ellipsoid $S$ for
$F$ if and only if the $\mathrm{Ad}(G)$-orbit of $X$ is contained in another
ellipsoid $S^{\prime}$, which is the indicatrix for another left invariant
Randers metric $F^{\prime}$. Then $X\in\mathfrak{g}$ generates a Killing
vector field of constant length $1$ of $F^{\prime}$, and $X^{\prime}$ must be
inside the indicatrix ellipsoid of $F^{\prime}$. The Randers metric $F$ is
invariant under the right multiplications in $G^{\prime}$ if and only its
indicatrix $S$ in $\mathfrak{g}$ is $\mathrm{Ad}(G^{\prime})$-invariant. By
Theorem 3.3, the shifting from $S$ to $S^{\prime}$ by $X^{\prime}$ is
$\mathrm{Ad}(G^{\prime})$-invariant, so $F^{\prime}$ is also
$L(G)R(G^{\prime})$-invariant. The correspondence from $F^{\prime}$ to $F$ is
similar. We have thus proved the following theorem.
###### Theorem 3.4
Let $G$ be a compact Lie group with simple Lie algebra $\mathfrak{g}$,
$G^{\prime}$ a closed connected subgroup of $G$ with Lie algebra
$\mathfrak{g}^{\prime}$. Then for any
$(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{c}(\mathfrak{g}^{\prime})$,
$X\neq 0$, and $l>0$, there is an one-to-one correspondence between the
following two sets:
(1) The set of all $L(G)R(G^{\prime})$-invariant metrics such that
$(X,X^{\prime})$ generates a Killing vector field of constant length $l$.
(2) The set of all $L(G)R(G^{\prime})$-invariant metrics such that $X$
generates a Killing vector field of constant length $l$ and the length of
$X^{\prime}$ with respect to the Randers metric is less than $l$.
Theorem 3.4 provides a theoretical machinery to find left invariant Randers
metrics with von-vanishing Killing vector fields of constant length. First we
fix a nonzero $X\in\mathfrak{g}$ and find the left invariant Randers metrics
such that $X$ generates a Killing vector field of constant length for this
metric (as we did in the previous work [DM12]). If its isometry group is the
product of $L(G)R(G^{\prime})$ as given above, then we can freely choose any
$X^{\prime}$ from $\mathfrak{c}(\mathfrak{g}^{\prime})$ which is shorter than
$X$. In this way we get the a Randers metric by requiring its indicatrix to be
the parallel shifting of the former one by $-X^{\prime}$. Then it is easily
seen that the above Randers metric has a Killing vector field of constant
length generated by $(X,X^{\prime})$.
Besides the Killing vector fields generated by
$(X,X^{\prime})\in\mathfrak{g}\oplus\mathfrak{c}(\mathfrak{g})$ with $X\neq
0$, there are Killing vector fields of constant lengths generated by elements
of the form $(0,X^{\prime})$. It is easy to see that in this case $X^{\prime}$
can be any vector in $\mathfrak{g}^{\prime}$. Since the subgroup
$R(G^{\prime})\in I(G,F)$ commute with $L(G)$, and $L(G)$ acts transitively on
$G$, $R(G^{\prime})$ gives a group of CW-translations which is neither new nor
interesting for our consideration (see [DM12]).
## 4 Left invariant CW-homogeneous Randers metrics on simple compact Lie
groups
We now apply Theorem 3.4 to give a proof of Theorem 1.1. The proof contains
two steps. We first prove the statement of the theorem with CW-homogeneity
replaced by restrictive CW-homogeneity. Then we prove that, for a left
invariant Randers metric on a connected simply connected compact simple Lie
group, the CW-homogeneity and restrictive homogeneity are equivalent.
The first step of the proof needs the following theorem, which will be proved
in the next section.
###### Theorem 4.1
Let $G$ be a compact connected simple Lie group with Lie algebra
$\mathfrak{g}$. Then a generic $X\in\mathfrak{g}$ can not generate a Killing
vector field of constant length when the left invariant Randers metric $F$ on
$G$ is not Riemannian. The complement of all those generic elements is a
subvariety with a codimension at least $\mathrm{rk}(\mathfrak{g})+1$.
We remark here the generic condition is some condition for the eigenvalue
multiplicities when $\mathfrak{g}$ is realized as a Lie algebra of matrices,
and the exact sense may depend on the explicit Lie algebras; see the
interruption in Section 5 for the Lie algebras of $A_{n}$, $\geq 1$, $D_{n}$
with $n$ odd and $>2$, and $E_{6}$. Hence apparently the generic condition has
not been precisely defined. However, what is important in the above theorem is
the assertion on the codimension of the set of generic elements. In fact, in
the proof of Theorem 4.1, we will only use the generic condition in the above
mentioned three cases, in which the condition can be precisely described.
Proof of Theorem 1.1. Let $I_{0}(G,F)$ be the product $L(G)R(G^{\prime})$,
with $G^{\prime}\subset G$ and $R(G^{\prime})$ the connected subgroup of all
right translation isometries. Denote the Lie algebra of $I_{0}(G,F)$ by
$\mathfrak{g}\oplus\mathfrak{g^{\prime}}$. If the indicatrix of the
$L(G)R(G^{\prime})$-invariant metric $F$ in $T_{e}G\cong\mathfrak{g}$ is a
round sphere, centered at $-X^{\prime}$, with radius $r>0$ with respect to the
bi-invariant metric, then $X^{\prime}$ is $\mathrm{Ad}(G^{\prime})$-invariant,
i.e., $X^{\prime}$ lies in $\mathfrak{c}(\mathfrak{g})$. Since the indicatrix
of $F$ contains $0$, the length of $X^{\prime}$ with respect to the bi-
invariant metric satisfies $|X^{\prime}|_{\mathrm{bi}}<r$. Any vector
$(X,X^{\prime})$, with $|X|_{bi}=r$, generates a Killing vector field of
constant length, and these vectors exhaust all tangent directions in $T_{e}G$.
Through left translations, the Killing vector fields of constant lengths can
exhaust all tangent directions at any point. Thus $(G,F)$ is restrictively CW-
homogeneous.
Conversely, if $(G,F)$ is restrictively CW-homogeneous, then any tangent
vector $X^{\prime\prime}\in\mathfrak{g}\cong T_{e}G$, of length $1$ with
respect to $F$ can be extended to a Killing vector field of constant length
$1$. Such a Killing vector field $(X,X^{\prime})$ has either of the following
two forms:
(1) $X\in\mathfrak{g}$, $X^{\prime}\in\mathfrak{c}(\mathfrak{g}^{\prime})$,
and $X=X^{\prime}+X^{\prime\prime}$;
(2) $(0,X^{\prime})$, with
$X^{\prime}=-X^{\prime\prime}\in\mathfrak{g^{\prime}}$.
Notice that if $F$ is Riemannian, then it must be a bi-invariant metric
([DM12]). Now suppose $F$ is not Riemannian and write $F$ as
$F(y)=\sqrt{\langle y,y\rangle}+\langle y,V\rangle_{\mathrm{bi}}$, where $y\in
T_{e}G$ and $V$ is a non-vanishing vector in $\mathfrak{g}$. Then
$[X^{\prime},V]=[X^{\prime\prime},V]=0$ for a Killing vector field
$(X^{\prime},X^{\prime\prime})$ of constant length with the second form. In
this case $X^{\prime\prime}$ is contained in a subspace with a lower
dimension. So when $X^{\prime\prime}$ is generic, the corresponding Killing
vector field $(X,X^{\prime})$ is of the first form, i.e.,
$X^{\prime}\in\mathfrak{c}(\mathfrak{g}^{\prime})$. Theorem 3.4 implies that
$X$ generates a Killing vector field of constant length for another
$L(G)R(G^{\prime})$-invariant Randers metric $F^{\prime}$, whose indicatrix is
just a shift of that of $F$. Notice that all the possible $X^{\prime}$’s are
contained in $\mathfrak{c}(\mathfrak{g}^{\prime})$ which has a dimension at
most $\mathrm{rk}(\mathfrak{g})$. For generic $X^{\prime\prime}$, the
corresponding $X=X^{\prime}+X^{\prime\prime}$ must be generic in the sense of
Theorem 4.1, otherwise $X$ belongs to a subvariety with a codimension at least
$rk(\mathfrak{g})+1$, and this implies that $X^{\prime\prime}=X-X^{\prime}$
belongs to subvariety with a codimension at least $1$, which conflict with the
assumption that $X^{\prime\prime}$ is generic. So the corresponding
$F^{\prime}$ for $X$ must be Riemannian, or equivalently, its indicatrix must
be the only shifting of that of $F$ with its center shifted back to $0$. Thus
for a generic vector $X^{\prime\prime}$ with length $1$ for $F$, the
corresponding Killing vector field $(X,X^{\prime})$ of constant length has the
same $X^{\prime}$ factor, and the corresponding $F^{\prime}$ for the $X$ term
is also the same. As the set of the Killing vector fields with a fixed
constant length is a closed subset of the set of all the Killing vector
fields, the above assertion is true for all $X^{\prime\prime}$’s with length
$1$ for $F$. So $F^{\prime}$ is a restrictively CW-homogeneous Riemannian
metric, i.e., it is a bi-invariant Riemannian metric. Consequently the
indicatrix of $F^{\prime}$ in $\mathfrak{g}$ is a round sphere. Hence its
shifting, the indicatrix of $F$, is also a round sphere.
Up to now we have completed the first step of the proof. For the second step
we only need to prove that for a left invariant Randers metric on a connected
simply connected simple Lie group, the restrictive CW-homogeneity implies the
CW-homogeneity.
By suitable scalar changes, we can assume that
$\\{(X,V)||X|_{\mathrm{bi}}=1\\}$ generates all the Killing vector fields of
constant length $1$ with respect to $F$, in which the fixed $V$ satisfies
$|V|_{\mathrm{bi}}<1$. Then after a constant re-scaling of the parameter, any
geodesic of $(G,F)$ starting from $g$ can be written as $\exp(tX)g\exp(-tV)$
with $t\geq 0$, for some $X\in\mathfrak{g}$ with $|X|_{\mathrm{bi}}=1$. The
geodesic $\exp(tX)g\exp(-tV)$, $t\in[0,t_{0}]$ from $g$ to $g^{\prime}$ is not
minimizing if and only if there is another geodesic from $g$ to $g^{\prime}$
with the form $\exp(tX^{\prime})g\exp(-tV)$, $|X^{\prime}|_{\mathrm{bi}}=1$
and $t\in[0,t^{\prime}]$ with $t^{\prime}<t_{0}$. This implies that
$\exp(t_{0}X)g\exp(-t_{0}V)=\exp(t^{\prime}X^{\prime})g\exp(-t^{\prime}V),$
(4.3)
i.e.,
$\exp(t_{0}X)=\exp(t^{\prime}X^{\prime})\exp((t_{0}-t^{\prime})\mathrm{Ad}(g)V)$.
For the bi-invariant metric, the geodesic $\exp(tX)$, $t\in[0,t_{0}]$ from the
unit element $e$ to $g^{\prime}g^{-1}$ has a length $t_{0}$. The right side of
(4.3) gives a path from $e$ to
$\exp(t^{\prime}X^{\prime})\exp((t_{0}-t^{\prime})\mathrm{Ad}(g)V)$, which is
a geodesic from $e$ to $\exp((t_{0}-t^{\prime})V)$ with length
$(t_{0}-t^{\prime})|V|_{bi}$, and a geodesic from $\exp((t_{0}-t^{\prime})V)$
to $\exp(t^{\prime}X^{\prime})\exp((t_{0}-t^{\prime})\mathrm{Ad}(g)V)$ with
length $t^{\prime}$. The total length is less than $t_{0}$, so the geodesic
$\exp(tX)$, $t\in[0,t_{0}]$ is not minimizing with respect to the bi-invariant
metric. The next lemma shows the converse statement is also true.
###### Lemma 4.2
Let $X$ be a unit vector and $\exp(tX)$, $t\in[0,t_{0}]$, a geodesic from $e$
to $g=\exp(t_{0}X)$ which is not minimizing with respect to the bi-invariant
metric. Then for any $V\in\mathfrak{g}$ with $|V|_{bi}<1$, there is
$t^{\prime}\in[0,t_{0})$ and a unit vector $X^{\prime}\in\mathfrak{g}$, such
that
$g=\exp(t_{0}X)=\exp(t^{\prime}X^{\prime})\exp((t_{0}-t^{\prime})V).$ (4.4)
Proof. We construct a sequence $t_{n}\in[0,t_{0}]$, and a sequence of unit
vectors $X_{n}\in\mathfrak{g}$ with respect to the bi-invariant metric,
inductively as follows. Let $t_{1}$ and $X_{1}$ be the pair such that the
geodesic $\exp(tX_{1})$, $t\in[0,t_{1}]$, is minimizing from $e$ to $g$.
Suppose we have defined $t_{i}$ and $X_{i}$. Then we choose $t_{i+1}$ and
$X_{i+1}$ to be the pair such that the geodesic $\exp(tX_{i+1})$,
$t\in[0,t_{i+1}]$, is the shortest from $e$ to $g\exp((t_{i}-t_{0})V)$. We
have a sequence of equalities $g=\exp(t_{i+1}X_{i+1})\exp((t_{0}-t_{i})V)$,
$\forall i\geq 0$, which imply that
$\exp(t_{i+1}X_{i+1})\exp(-t_{i}X_{i})=\exp((t_{i}-t_{i-1})V)$. Using the
triangle inequality, we have
$|t_{i+1}-t_{i}|\leq|V|_{\mathrm{bi}}|t_{i}-t_{i-1}|$. So the sequence $t_{n}$
converges to some $t^{\prime}\geq 0$. Using a suitable subsequence if
necessary, we can assume that $X_{n}$ converge to some unit vector
$X^{\prime}$ satisfying (4.4). Since all the geodesics $\exp(tX_{n})$,
$t\in[0,t_{n}]$, are minimizing, the limit geodesic $\exp(tX^{\prime})$,
$t\in[0,t^{\prime}]$ is also a minimizing geodesic from $e$ to
$g\exp((t^{\prime}-t_{0})V)$. If $t^{\prime}>t_{0}$, then the path given by
the geodesic from $e$ to $\exp((t^{\prime}-t_{0})V)$ and the geodesic from
$\exp((t^{\prime}-t_{0})V)$ to
$g\exp((t^{\prime}-t_{0})V)=\exp(t_{0}X)\exp((t^{\prime}-t_{0})V)$ has a
length $t_{0}+|V|_{eq}(t^{\prime}-t_{0})<t^{\prime}$, which is a
contradiction. On the other hand, the condition that $\exp(tX_{0})$,
$t\in[0,t_{0}]$, is not minimizing, implies that $t^{\prime}\neq t_{0}$.
Therefore $t^{\prime}\in[0,t_{0})$. This completes the proof of the lemma.
For any $g_{1}\neq g_{2}$, we can find $X\in\mathfrak{g}$ with
$|X|_{\mathrm{bi}}=1$, such that $(X,V)$ is the Killing vector field of
constant length $1$, and its flow curve, $\exp(tX)g_{1}\exp(-tV)$,
$t\in[0,t_{0}]$, generates a minimizing geodesic from $g_{1}$ to $g_{2}$ for
the Randers metric $F$. Denote the local one-parameter group of
diffeomorphisms generated by $(X,V)$ by $\phi_{t}$. Given
$t^{\prime}\in[0,t_{0}]$, if $\phi_{t^{\prime}}$ is not a Clifford-Wolf
translation, then there is some $g^{\prime}\in G$, such that the geodesic
$\exp(tX)g^{\prime}$, $t\in[0,t^{\prime}]$, is not minimizing. Then with
respect to the bi-invariant metric, the geodesic $\exp(tX)$,
$t\in[0,t^{\prime}]$, does not minimizes the distance from $e$ to
$\exp(t^{\prime}X)$. By Lemma 4.2, there is $X^{\prime\prime}$ with
$|X^{\prime\prime}|_{\mathrm{bi}}=1$ and $t^{\prime\prime}\in[0,t^{\prime})$,
such that
$\exp(t^{\prime}X)=\exp(t^{\prime\prime}X^{\prime\prime})\exp((t^{\prime}-t^{\prime\prime})\mathrm{Ad}(g_{1})V)$.
This means that the geodesic $\exp(tX^{\prime\prime})g_{1}\exp(-tV)$,
$t\in[0,t^{\prime\prime}]$, gives a path with length $t^{\prime\prime}$. It is
shorter than the geodesic $\exp(tX)g_{1}\exp(-tV)$, $t\in[0,t^{\prime}]$, from
$g_{1}$ to $\exp(t^{\prime}X)g_{1}\exp(-t^{\prime}V)$. This is a contradiction
with the fact that for $t\in[0,t_{0}]$, $(X,V)$ generates a minimizing
geodesic from $g_{1}$ to $g_{2}$. Therefore $(G,F)$ is CW-homogeneous. This
completes the proof of Theorem 1.1.
## 5 Proof of Theorem 4.1
### 5.1 The theme of the proof
Let $G$ be a compact connected simple Lie group with Lie algebra
$\mathfrak{g}$, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ with
$\dim_{\mathbb{R}}\mathfrak{h}=\mathrm{rk}(\mathfrak{g})=n$, and $W$ the
corresponding Weyl group. The bi-invariant metric on $G$ induces a
$W$-invariant linear metric on $\mathfrak{h}$. The condition in Theorem 4.1,
that the vector field generated by $X\in\mathfrak{g}$ is not a Killing vector
field of constant length of the left invariant Randers metric $F$ on $G$ when
$F$ is not Riemannian, is equivalent to the condition that any ellipsoid
containing the $\mathrm{Ad}(G)$-orbit $X$ must be centered at $0$.
If $-\mbox{Id}$ is contained in the Weyl group, the proof goes easily by
applying the next lemma.
###### Lemma 5.1
If $-\mbox{Id}$ belongs to the Weyl group of a simple compact Lie algebra
$\mathfrak{g}$, then any ellipsoid containing the $\mathrm{Ad}(G)$-orbit of a
non-zero element $X\in\mathfrak{g}$ must be centered at $0$.
Proof. We only need to prove that for any non-zero $X$ in a Cartan subalgebra,
an ellipsoid containing its Weyl group orbit must be centered at $0$. Denote
the equation of the ellipsoid $E$ containing the Weyl group orbit of $X\neq 0$
as
$x^{T}Ax+b^{T}x+c=0.$ (5.5)
Then the ellipsoid defined by the equation
$x^{T}Ax-b^{T}x+c=0$ (5.6)
also contains the Weyl group orbit of $X$. So the Weyl group orbit of $X$ is
contained in the subspace $b^{T}x=0$ if $b\neq 0$, or equivalently, the
ellipsoid $E$ is not centered at $0$. This conflicts with the fact that the
representation of the Weyl group is irreducible on the Cartan subalgebra of a
simple compact $\mathfrak{g}$.
It is well known that, the Weyl groups of all simple compact Lie algebras
except $A_{n}$ with $n>1$, $D_{n}$ with $n$ odd, and $E_{6}$, contain the
endomorphism $-\mbox{Id}$ on the Cartan subalgebras. For the last three cases,
we need another lemma.
Let $\\{v_{1},\ldots,v_{n}\\}$ be a basis of the vector space $\mathfrak{h}$
given by roots, with corresponding reflections $\rho_{1},\ldots,\rho_{n}$ in
the Weyl group. Denote the linear subspace spanned by $\rho_{i}$ as
$v_{i}^{\perp}$. Then we have
###### Lemma 5.2
Suppose for any $i=1,\ldots,n$, we can find $n$ points on the Weyl group orbit
of $X$ outside $v_{i}^{\perp}$, such that their orthogonal projections in
$v_{i}^{\perp}$ form an affine basis of $v_{i}^{\perp}$. Then any ellipsoid
containing the Weyl group orbit of $X$ must be centered at the origin.
Proof. Let $E$ be the ellipsoid containing the Weyl group orbit of $X$.
Suppose $l$ is a line which has two intersectional points with $E$ and denote
by $L$ the set of all the lines which is parallel to $l$. Define $D$ to be the
set of the middle points of the intersectional points of the lines in $L$ with
$E$. Then $D$ is contained in a hyperplane. Notice that the center of the
ellipsoid must be contained in this hyperplane. For each $i=1,\ldots,n$, the
hyperplane containing all middle points with the lines parallel to $v_{i}$
contains a set of $n$ projections from the orbit of $X$, which gives an affine
basis. Thus it is identical with $v_{i}^{\perp}$. Since the $v_{i}^{\perp}$’s
have the unique common point (the origin), the center of the ellipsoid must be
the origin.
The generic condition for $X$ are certain conditions for the eigenvalue
multiplicities for the cases of $A_{n}$ and $D_{n}$, and more generally the
isomorphic type of its isotropy group for $E_{6}$. They are invariant under
$\mathrm{Ad}$-actions, namely, if $X$ is generic, then each element of the
$\mathrm{Ad}$-orbit of $X$ is also generic. If a generic element $X$ generates
a Killing vector field of constant length for some Randers metric $F$, then
the restriction of $F$ to each Cartan subalgebra must be Riemannian, since the
indicatrix ellipsoid is centered at $0$. This implies that $F$ must be
Riemannian. Therefore we are left to calculate the codimension of the
complement of the set of all generic elements in $\mathfrak{g}$. Next we deal
with this problem case by case.
### 5.2 The case of $A_{n}$ with $n>1$
The diagonal matrices in $\mathfrak{su}(n+1)$ with $n>0$ form a Cartan
subalgebra. Any matrix in it can be identified with a vector
$X=(a_{0},\ldots,a_{n})\in\mathbb{R}^{n+1}$, in which $\sum_{i=0}^{n}a_{i}=0$,
and the eigenvalues are $a_{i}\sqrt{-1}$, $i=0,\ldots,n$. The weyl group is
the full permutation group $S_{n+1}$ for all the entries of $X$.
Take $v_{i}=(1,0,\ldots,0,-1,0,\ldots,0)$ with $-1$ in the $i$th-entry. The
reflection $\rho_{i}$ is the permutation which interchanges $a_{0}$ and
$a_{i}$, and fixes all the other entries.
Let $X$ be a vector with the first entry equal to $a$, the last entry equal to
$b$ such that $b<a$ and suppose all the other entries $c_{i}$ lies in
$\mathbb{R}\backslash\\{a,b\\}$, $i=1,\ldots,n-1$. Then using the action of
$\rho_{i}$, $i=0,1,\ldots,n$, we get the following $n$ points in the Weyl
group orbit of $X$:
$\displaystyle X_{0}$ $\displaystyle=$
$\displaystyle(a,b,c_{1},\ldots,c_{n-1}),$ $\displaystyle X_{i}$
$\displaystyle=$
$\displaystyle(a,c_{i},c_{1},\ldots,c_{i-1},b,c_{i+1},\ldots,c_{n-1}),$
for $i=1,\ldots,n-1$. It is easy to see that neither of the above points is
contained in $v_{1}^{\perp}$. To see that their orthogonal projections form an
affine basis for $v_{1}^{\perp}$, we only need to notice the fact that the set
of vectors
$\mathrm{pr}(X_{i})-pr(X_{0})=((c_{i}-b)/2,0,\ldots,0,c_{i}-b,0,\ldots,0),\forall
i=1,\ldots,n-1,$ (5.7)
are linearly independent, which is obvious. The same technique can also be
applied to other $v_{i}^{\perp}$ with $i>1$. Therefore the condition of Lemma
5.2 is satisfied. Hence the ellipsoid containing this $X$ must be centered at
$0$. Thus the left-invariant Randers metric with Killing vector fields of
constant length given by this $X$ must be Riemannian on each Cartan
subalgebra. Therefore $F$ is Riemannian.
Any matrix in $\mathfrak{su}(n+1)$ have $n+1$ imaginary eigenvalues, with
their multiplicities denoted as $\\{n_{1},\ldots,n_{m}\\}$,
$\sum_{i=1}^{m}n_{i}=n+1$. The generic condition discussed above is in fact
that there are two $1$’s among all $n_{i}$’s. The matrix space
$\mathfrak{su}(n+1)$ can be naturally stratified as a finite union of
subvarieties with respect to the multiplicities. Notice that the isotropy
group for the ${\mathrm{A}d}(SU(n))$-action at $X\in\mathfrak{su}(n+1)$ with
eigenvalue multiplicities $\\{n_{1},\ldots,n_{m}\\}$ is
$S(U(n_{1})\times\cdots\times U(n_{m}))$. So the set of the matrices with the
same eigenvalue multiplicities $\\{n_{1},\ldots,n_{m}\\}$ is a subvariety with
codimension
$\sum_{i=1}^{m}n_{i}^{2}-m=\sum_{i=1}^{m}(n_{i}+1)(n_{i}-1)\geq n+m-2l,$ (5.8)
in which $l$ is the number of $1$’s among the $n_{i}$’s. Obviously, when
$m\geq 3$ and $l\leq 1$, the codimension is $n+m-2l\geq n+1$, and when $m=2$
and $l\leq 1$, it is also easy to see that the inequality in (5.8) is
restrict. Thus each stratified subset in the complement of the set of the
generic elements has a codimension at least $n+1$, and so does their union.
This completes the proof of Theorem 4.1 for the case of $A_{n}$.
### 5.3 The case of $D_{n}$ with odd $n>2$
In a Cartan subalgebra of $\mathfrak{so}(2n)$ with $n>2$, the matrix can be
identified with a vector $X=(a_{1},\ldots,a_{n})\in\mathbb{R}^{n}$, such that
the eigenvalues of the matrix are $\pm a_{i}\sqrt{-1}$ for $i=1,\ldots,n$. The
Weyl group actions permute the entries and change even numbers of signs of the
entries arbitrarily. The eigenvalue multiplicities for a matrix in
$\mathfrak{so}(2n)$ can be denoted as $\\{n_{0},n_{1},\ldots,n_{m}\\}$, where
$n_{0}$ is the even multiplicity of the eigenvalue $0$, and the $n_{i}$’s are
eigenvalue multiplicities for positive multiples of $\sqrt{-1}$. Then we have
$n_{0}+2\sum_{i=1}^{m}n_{i}=2n$. Suppose the eigenvalue multiplicities
$X\in\mathbb{R}^{n}$ satisfies the conditions that $m\geq 2$ and there is an
$i>0$ such that $n_{i}=1$. After a suitable change of the order and adjustment
of the signs, we can suppose that $X$ can be represented as
$(a,b,c_{1},\ldots,c_{n})$ such that $a$ and $b$ are non-zero, and $a$ has
different absolute value from the others. If there is a $c_{i}$ such that
$c_{i}=b$, then we change the sign for that $c_{i}$ and $a$ simultaneously
using a Weyl group element. Therefore we can assume further that $b\neq
c_{i}$, for $i=1,\ldots,n-2$. Let $v_{i}=(1,0,\ldots,-1,0,\ldots,0)$ be the
root with a $-1$ in the $(i+1)$-th entry, and denote $v_{0}=(1,1,0,\ldots,0)$.
One can find the following $n$ points from the Weyl group orbit of $X$ outside
$v_{1}^{\perp}$: $(a,b,c_{1},\ldots,c_{n-2})$,
$(a,c_{i},c_{1},\ldots,c_{i-1},b,c_{i+1},\ldots,c_{n-2})$ with
$i=1,\ldots,n-2$, and $(-a,-b,c_{1},\ldots,c_{n-2})$. Neither of the $n$
points is contained in $v_{1}^{\perp}$, and their orthogonal projections in
$v_{1}^{\perp}$ form an affine basis. This argument can also applied to
$v_{0}$ and the other $v_{i}$’s. Thus $X$ is a generic vector in the sense of
Lemma 5.2.
In this case the generic condition for $X$ can be stated as follows: the
eigenvalue multiplicities $\\{n_{0},\ldots,n_{m}\\}$ of $X$ satisfy the
condition that $m\geq 2$, and there is a $n_{i}=1$ for some $i>0$. The
isotropy group of the $\mathrm{Ad}(\mathrm{SO}(2n))$-action at $X$ with
eigenvalue multiplicities $(n_{0},\ldots,n_{m})$ is isomorphic to
$\mathrm{SO}(n_{0})\times\mathrm{U}(n_{1})\times\mathrm{U}(n_{m})$. Therefore
the subvariety of the elements $X$ whose eigenvalue multiplicities are all the
same, has codimension $n_{0}(n_{0}-1)/2+\sum_{i=1}^{m}n_{i}^{2}-m$. If $m=1$
and $n_{1}=1$, then the codimension equals $(n-1)(2n-3)>n+1$ when $n>2$. If
for all $i>0$ we have $n_{i}>1$, then the codimension is
$\displaystyle\frac{n_{0}(n_{0}-1)}{2}+\sum_{i=1}^{m}n_{i}^{2}-m$
$\displaystyle=$
$\displaystyle\frac{n_{0}(n_{0}-1)}{2}+\sum_{i=1}^{m}(n_{i}-1)(n_{i}+1)$ (5.9)
$\displaystyle\geq$ $\displaystyle n_{0}-1+\sum_{i=1}^{m}n_{i}+m$
$\displaystyle=$ $\displaystyle n+(m+\frac{n_{0}}{2}-1)\geq n$
in which the equality holds only when $n_{0}/2+m=1$. However, one can easily
check that the equality can not hold. So in the complement of the set of the
generic elements, any stratified subvariety of the fixed type of eigenvalue
multiplicities has a codimension at least $n+1$. This completes the proof of
Theorem 4.1 for the case of $D_{n}$.
### 5.4 The case of $E_{6}$
Now we consider the last case of $E_{6}$, which is also the most difficult
one. The Cartan subalgebra of $E_{6}$ can be identified with $\mathbb{R}^{6}$.
The root system consists of $\alpha=(\pm 1,\pm 1,0,0,0,0)$ together with all
permutations of $\alpha$ keeping the last entry $0$ fixed, and all the vectors
$(\pm 1/2,\pm 1/2,\pm 1/2,\pm 1/2$, $\pm 1/2,\pm\sqrt{3}/2)$ with odd positive
signs. It is easy to observe that the set $\Pi$ of roots which are
perpendicular to $(1/2,\ldots,1/2,-\sqrt{3}/2)$ consists of the permutations
of $(1,-1,0,0,0,0)$, together with the permutations of the vectors
$\pm(1/2,\ldots$, $1/2,-1/2,\sqrt{3}/2)$, both keeping the last entry fixed.
It is also easily checked that the set set $\Pi$ forms the root system of
$A_{5}$. We can use an orthogonal automorphism of $\mathbb{R}^{6}$ to map them
to the standard root system of $A_{5}$, i.e., all permutations of
$(1,-1,0,0,0,0)$. More precisely, the orthogonal automorphism keeps all
permutations of $(1,-1,0,0,0,0)$ fixing the last entry $0$, and maps
$(-1/2,\ldots,-1/2,1/2,-\sqrt{3}/2)$ to $(0,\ldots,0,1,-1)$. We can also
assume that it maps the roots $\pm(1/2,1/2,1/2,1/2,1/2,-\sqrt{3}/2)$ to
$\pm(3^{-1/2},\ldots,3^{-1/2})$, respectively. It is not hard to check that it
maps the remaining forty roots to $\pm((3^{-1/2}\pm 1)/2,\ldots,(3^{-1/2}\pm
1)/2)$ with three positive signs and three negative signs among the six
entries. To summarize, we have the following lemma.
###### Lemma 5.3
The root system of $E_{6}$ can be represented as the union of the standard
root system of $A_{5}\times A_{1}$, i.e., all permutations of
$(1,-1,0,\ldots,0)$ and $\pm(3^{-1/2},\ldots,3^{-1/2})$, and all the vectors
$\pm((3^{-1/2}\pm 1)/2,\ldots,(3^{-1/2}\pm 1)/2)$ with three positive signs
and three negative signs among the six entries.
For any vector $X\in\mathbb{R}^{6}$, or the Cartan subalgebra of $E_{6}$, the
multiplicities of its different entries can be denoted as a set of positive
integers $\\{n_{1},\ldots,n_{m}\\}$, with $\sum_{i=1}^{m}n_{m}=6$. For
simplicity, we will call it the multiplicity type of $X$. Suppose $X$ is a
vector in the Cartan subalgebra whose all entries sum to $0$, i.e., a vector
in the Cartan subalgebra of $A_{5}$. From Subsection 5.2, we see that if the
multiplicity type of $X$ contains two $1$’s, then any ellipsoid in the Cartan
subalgebra of $A_{5}$ containing the $A_{5}$-Weyl group orbit of $X$ is
centered at $0$. In fact it is true for more generic elements.
###### Lemma 5.4
If $X$ has three different entries and the sum of all its entries is equal to
$0$, then any ellipsoid in the Cartan subalgebra of $A_{5}$ which contains the
$A_{5}$-Weyl group orbit of $X$ must be centered at 0.
Proof. We only need to consider the case with multiplicity types $\\{1,2,3\\}$
and $\\{2,2,2\\}$ and repeat the arguments in Subsection 5.2.
Consider $X=(a,b,b,c,c,c)$ where $a$, $b$ and $c$ are three different numbers
with $a+2b+3c=0$. Let the roots $v_{i}=(1,0,\ldots,0,-1,0,\ldots,0)$ with a
$-1$ in the $(i+1)$-th entry, $i=1,\ldots,5$. For $v_{1}=(1,-1,0,\ldots,0)$,
the $A_{5}$-Weyl group orbit of $X$ contains the points $(a,b,b,c,c,c)$,
$(c,b,b,a,c,c)$, $(c,b,b,c,a,c)$, $(c,b,b,c,c,a)$ and $(a,b,c,b,c,c)$. They
are vectors outside $v_{1}^{\perp}$ and their orthogonal projections form an
affine basis for $v_{1}^{\perp}$. For other $v_{i}$’s, the arguments are
similar. By Lemma 5.2, any ellipsoid containing the $A_{5}$-Weyl group orbit
of $X$ is centered at $0$.
Consider $X=(a,a,b,b,c,c)$ where $a$, $b$ and $c$ are three different numbers
with $a+b+c=0$. Then for $v_{1}$, the $A_{5}$-Weyl group orbit of $X$ contains
the points $(a,b,a,b,c,c)$, $(a,b,b,a,c,c)$, $(a,b,a,c,b,c)$, $(a,b,a,c,c,b)$
and $(a,c,a,b,c,b)$. They are vectors outside $v_{1}^{\perp}$ and their
projections form an affine basis for $v_{1}^{\perp}$. Similar arguments also
apply to other $v_{i}$’s. This completes the proof of the lemma.
Finally, we prove that any ellipsoid containing the $E_{6}$-Weyl group orbit
of $X$ must be centered at $0$, provided that $X$ has three different entries.
First we consider the case that the sum of all entries of $X$ is not $0$. Let
$E$ be any ellipsoid containing the $A_{5}\times A_{1}$-Weyl group orbit of
$X$. Notice that the Cartan subalgebra of the $A_{1}$ factor is generated by
$v_{0}=(3^{-1/2},\ldots,3^{-1/2})$, and the Cartan subalgebra of $A_{5}$ is
the orthogonal complement $v_{0}^{\perp}$. The $A_{5}\times A_{1}$-Weyl group
orbit of $X$ has no intersection with $v_{0}^{\perp}$, and Lemma 5.4 implies
that their orthogonal projection to $v_{0}^{\perp}$ contains an affine basis
for $v_{0}^{\perp}$. So the middle points of the intersection between $E$ and
the lines parallel to $v_{0}$ are contained in $v_{0}^{\perp}$. Similarly, the
centers of the intersection ellipsoid between $E$ and the hyperplanes parallel
to $v_{0}^{\perp}$ is contained in $\mathbb{R}v_{0}$. Since the center of $E$
must be contained in both $v_{0}^{\perp}$ and $\mathbb{R}v_{0}$, it must be
$0$.
Next we consider the case that the sum of all entries of $X$ is $0$. Let $E$
be any ellipsoid containing the $E_{6}$-Weyl group orbit of $X$. Then Lemma
5.4 implies that the intersection of $E$ with $v_{0}^{\perp}$ is an ellipsoid
containing the $A_{5}$-Weyl group orbit of $X$, which is centered at $0$. On
the other hand, any element $\rho$ in the $E_{6}$-Weyl group maps
$v_{0}^{\perp}$ to another subspace. Thus the intersection between $E$ and
$\rho(v_{0}^{\perp})$ is also an ellipsoid containing the $\rho$-images of the
$A_{5}$-Weyl group orbit of $X$, which is also centered at $0$. Therefore the
center of $E$ must be $0$, since it is contained in both the lines generated
by $v_{0}$ and $\rho(v_{0})$ which intersect at $0$. This proves the
assertion.
For non-zero vector $X$ with other multiplicity types, namely, $\\{6\\}$,
$\\{5,1\\}$, $\\{4,2\\}$ and $\\{3,3\\}$, if we can find a vector $X^{\prime}$
in its $E_{6}$-Weyl group orbit with three different entries, the same
statement is still true. If we assume further that any vector in its Weyl
group orbit does not have three different entries, then additional linear
relations will be required between the entries of $X$. So up to nonzero
constant scalars, we can only find finite $X$’s in the Cartan subalgebra
$\mathfrak{h}$, even if there does exist such $X$. The centralizer of such a
$X$ is the Lie algebra of the isotropy group for the $\mathrm{Ad}(G)$-action.
It contains $\mathfrak{h}$ and is obviously not abelian. Hence its dimension
is at least $n+2$. So the union of all the $\mathrm{Ad}(G)$-orbits of the
generic elements in the above sense has a codimension at least $n+1$ ( the
dimension of the centralizers minus the one dimension for scalar changes).
Thus the assertion of Theorem 4.1 holds for $E_{6}$. This completes the proof
of Theorem 4.1.
Acknowledgements. We would like to thank Dr. Libing Huang and Dr. Zhiguang Hu
for useful discussions. This work was finished during the second author’s
visit to the Chern institute of Mathematics. He would like to express his deep
gratitude to the members of the institute for their hospitality.
## References
* [AW76] R. Azencott, E. Wilson, Homogeneous manifolds with negative curvature I, Trans. Amer. Math. Soc., 215 (1976), 323-362.
* [BCS00] D. Bao, S. S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, New York, 2000.
* [BN08-1] V. N. Berestovskii, Yu. G. Nikonorov, Killing vector fields of constant length on locally symmetric Riemannian manifolds, Transformation Groups, 13 (2008), 25 C45.
* [BN08-2] V. N. Berestovskii, Yu.G. Nikonorov, On $\delta$-homogeneous Riemannian manifolds, Diff. Geom. Appl., 26 (2008), 514 C535.
* [BN09] V. N. Berestovskii, Yu.G. Nikonorov, Clifford-Wolf homogeneous Riemannian manifolds, Jour. Differ. Geom., 82 (2009), 467-500.
* [CS05] S. S. Chern, Z. Shen, Riemann-Finsler Geometry, World Scientific Publishers, 2004.
* [DH02] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific J. Math, 207 (2002), 149-157.
* [DMW86] I. Dotti-Miatello, R. Miatello and J. A. Wolf, Bounded isometries and homogeneous Riemannian quotient manifolds, Geom. Dedicata, 21 (1986), 21 C27.
* [DE12] S. Deng, Clifford-Wolf translations of Finsler spaces of negative flag curvature, preprint.
* [DM12] S. Deng and M. Xu, Clifford-Wolf translations of Finsler spaces, to appear in Forum Math., [Math.DG] arXiv:1201.3714v1.
* [DR83] M. J. Druetta, Clifford translations in manifolds without focal points, Geom. Dedicata, 14 (1983), 95 C103.
* [FR63] H. Freudenthal, Clifford-Wolf-isometrien symmetrischer raüme, Math. Ann., 150 (1963), 136-149.
* [HE74] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann., 211 (1974), 23-34.
* [OT76] T. Ochiai and T. Takahashi, The group of isometries of a left invariant metric on a Lie group, Math. Ann., 223 (1976), 91-96.
* [OZ69] V. Ozols, Critical points of the displacement function of an isometry, J. Differential Geometry, 3 (1969), 411 C432.
* [OZ74] V. Ozols, Clifford translations of symmetric spaces, Proc. Amer. Math. Soc., 44 (1974), 169 C175.
* [SH01] Z. Shen, Differential Geometry of Sprays and Finsler Spaces, Kluwer, Dordrent, 2001.
* [WO60] J. A. Wolf, Sur la classification des varietes riemanniennes homogenes a courbure constante, C. R. Math. Acad. Sci. Paris, 250 (1960), 3443-3445.
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|
arxiv-papers
| 2012-04-23T23:21:49 |
2024-09-04T02:49:30.062935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaoqiang Deng and Ming Xu",
"submitter": "Shaoqiang Deng",
"url": "https://arxiv.org/abs/1204.5233"
}
|
1204.5259
|
# Numerical calculations of a high brilliance synchrotron source and on issues
with characterizing strong radiation damping effects in non-linear
Thomson/Compton backscattering experiments
A. G. R. Thomas1,2, C. P. Ridgers3, S. S. Bulanov4, B. J. Griffin2, S. P. D.
Mangles5 1Centre for Ultrafast Optical Science, University of Michigan, Ann
Arbor, MI 48109, US. 2Department of Nuclear Engineering and Radiological
Sciences, University of Michigan, Ann Arbor, MI 48109, USA. 3Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK. 4University
California Berkeley, Berkeley, CA 94720 USA. 5Blackett Laboratory, Imperial
College London, London SW7 2AZ, UK
###### Abstract
A number of theoretical calculations have studied the effect of radiation
reaction forces on radiation distributions in strong field counter-propagating
electron beam-laser interactions, but could these effects – including quantum
corrections – be observed in interactions with realistic bunches and focusing
fields, as is hoped in a number of soon to be proposed experiments? We present
numerical calculations of the angularly resolved radiation spectrum from an
electron bunch with parameters similar to those produced in laser wakefield
acceleration experiments, interacting with an intense, ultrashort laser pulse.
For our parameters, the effects of radiation damping on the angular
distribution and energy distribution of _photons_ is not easily discernible
for a “realistic” moderate emittance electron beam. However, experiments using
such a counter-propagating beam-laser geometry should be able to measure such
effects using current laser systems through measurement of the _electron beam_
properties. In addition, the brilliance of this source is very high, with peak
spectral brilliance exceeding $10^{29}$ photons s-1mm-2mrad${}^{-2}(0.1$%
bandwidth$)^{-1}$ with approximately 2% efficiency and with a peak energy of
10 MeV.
###### pacs:
41.75.Jv,52.38.Ph,41.60.-m
## I Introduction
The recent development of ultra-high intensity laser systems has generated a
great amount of interest in a class of well known theoretical problems
involving the interaction of strong fields with relativistic electron beams
that have not been experimentally demonstrated. Relativistic electron beams
are regularly measured in experiments by laser wakefield acceleration
Tajima_PRL_1979 ; mangles ; geddes ; faure and are characterized by being of
relatively high current density in short bunches. In laser wakefield
acceleration, oscillations of the electrons in the electromagnetic fields of
electron plasma cavities created by laser driven ponderomotive expulsion have
been shown to result in extremely bright sources of x-rays Rousse_PRL_2004 ;
ISI:000242538700029 ; ISI:000254024500038 ; ISI:000253724500017 ;
Thomas_POP_2009 ; Kneip_NP_2010 .
Another proposed source of radiation using the wakefield accelerated electron
beam is Thomson or Compton backscattering from a second laser Esarey_PRE_1993
; Hartemann_PRL_1996 ; Salamin_JPA_1998 ; Ueshima_LPB_1999 ;
Avetissian_PRE_2002 ; Lee_OE_2003 ; Brown_PRSTAB_2004 ; Hartemann_PRE_2005 ;
Koga_POP_2005 ; Albert_POP_2011 . In this scheme a counter-propagating laser
is used as a short wavelength undulator for producing high brightness,
monochromatic gamma rays. An undulator in a conventional synchrotron is
characterized by a strength parameter $K$ that characterizes the oscillation
amplitude relative to wavelength. For small $K$, the radiation is
monochromatic. For large $K$ the radiation is characterized by a synchrotron-
like spectrum Jacksonbook . In the counter-propagating laser scheme, the field
strength parameter (normalized peak vector potential)
$a_{0}=|eF_{0}|/m_{e}c\omega_{0}$ is analogous to $K$. $F_{0}$ is the peak
electric field strength of a laser with central angular frequency
$\omega_{0}$. For a laser with $a_{0}\ll 1$ ($I\lambda^{2}\ll 10^{18}$ Wcm-2),
the radiation is monochromatic. For $a_{0}>1$ harmonics in the radiation
spectrum start to appear, and for $a_{0}\gg 1$ the spectrum becomes broad. For
linear polarization of the laser there is also longitudinal motion due to the
Lorentz force, and therefore downshifting of the fundamental frequency occurs
He_PRL_2003 ; ISI:000182450200081 . The monochromatic regime using a laser
wakefield accelerated electron bunch has been proposed as a good source for
applications He_PRL_2003 ; Hartemann_PRSTAB_2007 ; Albert_PRSTAB_2010 ;
Albert_PRSTAB_2011 . In addition, experiments using this counter propagating
geometry with a very high intensity laser (Fig 1) should be an interesting
testbed for studying radiation reaction forces and non-linear quantum
electrodynamics Bulanov_NIMPR_2011 , due to the high field strength in the
electron rest frame.
Figure 1: Schematic of counter-propagating laser-beam interaction geometry
using laser wakefield accelerated electrons.
The transverse component of the laser vector potential is Lorentz invariant,
so the radiation emission of an $a_{0}\leq 1$ interaction is very different to
an $a_{0}\gg 1$ interaction independent of the reference frame (and therefore
electron energy, in the colliding geometry). The emission of photons in such
processes clearly indicates that a force should be applied to the electron to
conserve momentum. Conversely, the electric field strength is not a Lorentz
invariant, and hence the electron energy in this geometry may be crucial to
determining whether the field is quantum electrodynamically strong or not.
In this paper, radiation damping effects on the full angular and energy
distribution of photons produced in the counter-propagating geometry
interaction between a tightly focused $\sim 10^{22}$ Wcm-2 ultrashort pulse
with a electron beam is studied by solving modified classical equations of
motion numerically and generating spectra with a numerical radiation
spectrometer Thomas_PRSTAB_2010 . The layout of the manuscript is as follows:
First we parameterize the interplay between the field strength $a_{0}$ and
electron energy $\gamma m_{e}c^{2}$ in the colliding pulse geometry, and
identify the regime relevant to near term experiments where radiation damping
is strong but quantum electrodynamic effects are relatively small. Next we
introduce the numerical model for calculating both the electron dynamics and
radiation spectra. We then proceed to calculate the $\gamma$-ray spectrum with
“realistic” conditions, then examine the effect of radiation reaction on the
photon and electron phase-spaces. Finally, we show that semi-classical
corrections to the radiation reaction force may be observable in experiments.
## II Parameterizing strong field interactions
### II.1 Radiation reaction force effects
Although properly described by quantum electrodynamics, the radiation force
has a classical form that is self-consistent within the limits that the
acceleration timescale is much larger than $\tau_{0}=2e^{2}/3mc^{3}=6.4\times
10^{-24}$ s Dirac_PRSA_1938 ; Rohrlich_PRE_2008 . It is principally a damping
of motion due to loss of momentum to the radiation. The Lorentz-Abraham-Dirac
equation is a third order differential equation of motion for a charged
particle in the presence of accelerating forces, and includes the change of
momentum due to the radiation generated by the charge. The force on an
electron is given in covariant form by:
$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}\left(F^{\mu\nu}v_{\nu}+\tau_{0}D^{\mu}\right)\;,$
(1)
where $D^{\mu}$ is the radiation reaction (damping) force,
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the
electromagnetic field tensor and $v_{\nu}={dx^{\nu}}/{d\tau}=\\{\gamma
c,-\gamma\mathbf{v}\\}$ is the particle four-velocity.
The radiation reaction force, according to the Lorentz-Abraham-Dirac model, is
a source of much controversy precisely because it is a third order
differential equation, which allows for self-accelerating solutions that do
not conserve energy, for example. Various authors have reformulated the
equation to eliminate the third order term (See Sokolov Sokolov_POP_2009 ,
Hammond Hammond_PRA_2010 and references within). These are generally
identical to first order in $\tau_{0}$ (and are therefore basically all
equivalent to the Landau-Lifshitz form of the radiation reaction force
Landau_Lifshitz ), but are otherwise not identical. The modified force can be
written in the form Rohrlich_PRE_2008 :
$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}\left[F^{\mu\nu}v_{\nu}-\tau_{0}P^{\mu\alpha}\frac{d}{d\tau}\left(F_{\alpha}^{\nu}v_{\nu}\right)\right]\;.$
(2)
where $P^{\mu\alpha}=\eta^{\mu\alpha}+{v^{\mu}v^{\alpha}}/{c^{2}}$ and
$\eta^{\mu\nu}$ is the Minkowski metric tensor with trace -2. In Ref.
Bulanov_PRE_2011 , several examples were given, which show that the solutions
of the Lorentz-Abraham-Dirac model and equation 2 are identical in the
classical regime.
One of the interesting phenomena arising from this laser-electron interaction
is that the radiation damping is theoretically predicted to be so extreme that
for a sufficiently intense laser, the electron beam may lose almost all its
energy in the interaction time. In particular, Koga et al. studied the effect
of radiation damping on the radiation spectrum Koga_POP_2005 . Di Piazza et
al. also studied the effect of radiation damping on the angular distribution
of radiation Piazza_PRL_2009 . The effects of ‘real world’ conditions on the
radiation spectrum emitted have also been somewhat previously studied, for
example the effects of higher-order field corrections for tightly focused
pulses Hartemann_AJS_2000 ; Lee_EPL_2010 .
Radiation damping can be parameterized by considering the energy loss of the
electron due to the most significant damping term Koga_POP_2005 ;
Bulanov_NIMPR_2011 . Here we proceed from equation 2, where ignoring terms of
$\tau_{0}^{2}$ and higher and the Schott term, the damping contribution can be
written in the form Rohrlich_PRE_2008 :
$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}F_{\alpha}^{\nu}v_{\nu}\left[\eta^{\alpha\mu}-\tau_{0}\frac{e}{m_{e}c^{2}}v_{\nu}v^{\mu}F^{\alpha\nu}\right]\;,$
(3)
The electromagnetic four-force can be written in the form:
$\displaystyle
F^{\alpha\nu}v_{\nu}=-\frac{dA^{\alpha}}{d\tau}+v_{\nu}\partial^{\alpha}A^{\nu}\;.$
(4)
For the case of a linearly polarized plane wave,
$A^{\mu}=\Re\left[\\{A_{0}\\}^{\mu}e^{i\kappa_{\alpha}x^{\alpha}}f(\kappa_{\beta}x^{\beta}/\omega_{0}t_{L})\right]$
where $\kappa_{\alpha}$ is the four-wave-vector
$\kappa_{\alpha}=\omega_{0}\\{1,-\mathbf{\hat{k}}/c\\}$,
$f(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})$ is a function describing the
temporal envelope and $t_{L}$ is the pulse duration, interacting with a
counter-propagating electron with initial Lorentz factor $\gamma_{0}$ obeying
$a_{0}\ll\gamma_{0}\ll(a_{0}\omega_{0}\tau_{0})^{-1/2}$, the zeroth component
is well approximated by:
$\frac{d\gamma}{d\tau}=-\gamma\tau_{0}\frac{da^{\mu}}{d\tau}\frac{da_{\mu}}{d\tau}\;,$
(5)
where $a^{\mu}=eA^{\mu}/m_{e}c$. The condition on $\gamma$ is so that the
longitudinal Lorentz force is minimized but radiation damping does not affect
the transverse oscillations of the electron. For a slowly varying gaussian
envelope, i.e. $(1/f)df/d\tau\ll\kappa_{\mu}v^{\mu}$ with
$f=\exp(-(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})^{2})$, and averaging over
the fast oscillations, we can integrate to obtain the total energy loss by the
particle:
$\frac{\Delta\gamma_{\infty}}{\gamma_{0}}=\frac{\sqrt{\frac{\pi}{2}}\tau_{0}t_{L}\omega_{0}^{2}\gamma_{0}a_{0}^{2}}{1+\sqrt{\frac{\pi}{2}}\tau_{0}t_{L}\omega_{0}^{2}\gamma_{0}a_{0}^{2}}\;,$
(6)
This is similar to the result in Ref. Bulanov_PRE_2011 , but with a different
definition for the pulse duration because here $t_{L}$ is close to the full-
width-at-half-maximum duration commonly used in experiments. From this
expression, we can define a parameter
$\psi=10\sqrt{\frac{\pi}{2}}\tau_{0}t_{L}\omega_{0}^{2}\gamma_{0}a_{0}^{2}t_{rad}$,
for a particular characteristic timescale for radiation damping $t_{rad}$,
such that:
$\frac{\Delta\gamma_{\infty}}{\gamma_{0}}=\frac{0.1\psi(t/t_{rad})}{1+0.1\psi(t/t_{rad})}\;.$
(7)
which clearly defines strong radiation damping for $\psi\geq 1$ and weak
radiation damping for $\psi\ll 1$. Here we choose $t_{rad}=2\pi/\omega_{0}$ –
that is to say a laser period – which is slightly different from the choice of
Koga et al. Koga_POP_2005 , who chose the pulse duration for $t_{rad}$.
However, we have also added a factor of 10 into $\psi$, which is such that
$\psi=1$ corresponds to a 10% energy loss in a single cycle, which therefore
results in a condition similar to that of Koga et al. Koga_POP_2005 , since
they considered an approximately 10 cycle pulse. In addition, a 10% loss in a
single cycle can reasonably be defined as the threshold of “significant”
damping. Hence:
$\psi=10\sqrt{2\pi^{3}}\omega_{0}\tau_{0}\gamma_{0}a_{0}^{2}\;.$ (8)
For a 800 nm laser, $\psi=1.2\times 10^{-6}\gamma_{0}a_{0}^{2}$. The condition
$\psi=1$ leads to the condition for the laser pulse vector potential, i.e. the
strong radiation damping regime is realized for
$a_{0}>a_{rad}=\left(10\sqrt{2\pi^{3}}\omega_{0}\tau_{0}\gamma_{0}\right)^{-1/2}.$
(9)
### II.2 Quantum electrodynamics effects
Quantum electrodynamically strong interactions are parameterized by a
relativistically and gauge invariant parameters
$\chi_{e}=||F_{\mu\nu}v^{\nu}||/(cE_{cr})$ and
$\chi_{\gamma}=||F_{\mu\nu}\hbar k^{\nu}||/(m_{e}cE_{cr})$ Ritus_1979 , where
$\hbar k^{\nu}$ is the four-momentum of a photon and
$E_{cr}=m_{e}^{2}c^{3}/e\hbar=1.32\times 10^{18}$ Vm-1 is the Schwinger or
critical field of quantum electrodynamics. These parameters determine the
rates of photon creation by an electron or an electron-positron pair creation
by high-energy photon in a strong electromagnetic field, the latter being the
Breit-Wheeler process Breit_PR_1934 . The photon emission probability for
$\chi_{e}\ll 1$ is $\approx\left(5\alpha
m_{e}^{2}/2\sqrt{3}p_{0}\right)\chi_{e}$ and for $\chi_{e}\gg 1$ is
$\approx\left[14\Gamma(2/3)\alpha
m_{e}^{2}/27p_{0}\right]\left(3\chi_{e}\right)^{2/3}$, where $p_{0}$ is the
electron energy and $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt$ is the Euler
gamma function, and $\alpha=e^{2}/4\pi\epsilon_{0}\hbar c=1/137$ is the fine
structure constant Ritus_1979 . The pair production probability by a photon
for $\chi_{\gamma}\ll 1$ is $\approx\left(3\sqrt{3}\alpha
m_{e}^{2}/16\sqrt{2}k_{0}\right)\chi_{\gamma}\exp\left(-8/3\chi_{\gamma}\right)$
and for $\chi_{\gamma}\gg 1$ is $\approx\left[15\Gamma^{4}(2/3)\alpha
m_{e}^{2}/28\pi k_{0}\right]\left(3\chi_{\gamma}\right)^{2/3}$, where $k_{0}$
is the photon energy Ritus_1979 . Previously it was shown that extremely high
intensity counter propagating laser pulses could lead to prolific pair
production Bell_PRL_2008 ; Fedotov_PRL_2010 ; Bulanov_PRL_2010 .
For multi-100 TW lasers, such as the Hercules Yanovsky:2008 or Astra Gemini
Hooker_JPIV_2006 lasers, with focused field strength $|E|\sim 10^{-3}E_{cr}$,
interaction with GeV energy electron beams should be sufficient to achieve
$\chi_{e}\sim 1$ Schwinger_PR_1951 ; Sokolov_PRL_2010 ; Sokolov_PRE_2010 .
However the conversion of emitted photons into electron-positron pairs will be
suppressed due to the $\exp\left(-8/3\chi_{\gamma}\right)$ in the expression
for the probability for $\chi_{\gamma}\ll 1$.
A notable experiment in a similar geometry, using the 46 GeV electron beam
from the the Stanford Linear Accelerator (SLAC) colliding with a laser with
intensity of $I_{0}\sim 10^{18}$ Wcm-2, was an important demonstration of non-
linear quantum electrodynamics (multi photon Breit-Wheeler pair production)
Burke_PRL_1997 . A simplified version of the parameter $\chi_{e}$ for the
situation of an electron beam with energy $E=\gamma_{0}m_{e}c^{2}$ colliding
with a laser field with field strength parameter $a_{0}$ can be written as
Sokolov_PRE_2010
$\chi_{e}=\frac{2\hbar}{m_{e}c^{2}}\omega_{0}\gamma_{0}a_{0}\;.$ (10)
For an 800 nm laser system, this gives $\chi_{e}=6\times
10^{-6}\gamma_{0}a_{0}$. For the SLAC experiment (using a 527 nm laser), the
small $a_{0}$ ($a_{0}<1$) was compensated by the high beam energy
($\gamma_{0}\sim 10^{5}$), so that $\chi_{e}\approx 0.4$.
Quantum electrodynamics effects may be considered significant when the energy
of the emitted photons becomes of the order of the electron energy,
$\hbar\omega\gtrsim\gamma_{0}m_{e}c^{2}$. For a head-on collision of an
electron and a laser pulse, a characteristic emitted photon energy is
$\hbar\omega\approx\hbar\omega_{0}a_{0}\gamma_{0}^{2}$ Bulanov_NIMPR_2011 ,
which corresponds to the condition $\chi_{e}\sim 1$. Hence, quantum
electrodynamics effects may be considered to be strong for a field strength
of:
$a_{0}>a_{Q}=\frac{m_{e}c^{2}}{2\hbar\omega_{0}\gamma_{0}}.$ (11)
However, quantum effects in the radiation damping of electrons becomes
noticeable for much lower laser field strengths. It is well known Ritus_1979 ;
Erber_RMP_1966 ; Kirk_PPCF_2009 ; Sokolov_PRE_2010 , that the classical
description of an electron radiating in a strong electromagnetic field
overestimates the total emitted power. It is connected with the fact that in
the quantum description the emitted photon energy may not exceed the electron
energy, whereas the classical approach does not have such a restriction. This
effect can be approximately taken into account by introducing a function
$g(\chi_{e})$ into the expression for the total power of emitted radiation
Erber_RMP_1966 ; Kirk_PPCF_2009 ; Sokolov_PRE_2010 . $g(\chi_{e})$ enters the
equation of motion by modifying the expression for the radiation reaction
force as:
$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}F_{\alpha}^{\nu}v_{\nu}\left[\eta^{\alpha\mu}-g(\chi_{e})\tau_{0}\frac{e}{m_{e}c^{2}}v_{\nu}v^{\mu}F^{\alpha\nu}\right]\;,$
(12)
The strong damping parameter $\psi$ can be modified to include this quantum
effect to obtain a parameter $\psi_{Q}=\langle g(\chi_{e})\rangle\psi$, where
$\langle g(\chi_{e})\rangle$ is the time average of the $g$ factor. To do
this, we make use of a polynomial fraction fit to data for $g(\chi_{e})$ given
in Ref Kirk_PPCF_2009 :
$\displaystyle
g(\chi_{e})=\left(3.7\chi_{e}^{3}+31\chi_{e}^{2}+12\chi_{e}+1\right)^{-4/9}\;,$
(13)
which for $\chi_{e}\rightarrow 0$, $g\rightarrow 1$. The condition
$\chi_{e}=1$ corresponds to $g(\chi_{e})=0.18$, but even for $\chi_{e}=0.1$
this factor has a value of $g(\chi_{e})=0.66$. The time averaged field
strength parameter $a_{0}/\sqrt{2}$ (for linear polarization) is used to
approximate $\langle g(\chi_{e})\rangle\approx g(\langle\chi_{e}\rangle)$,
which is valid for $\chi_{e}\ll 1$.
Figure 2: The function $\psi_{Q}$ as a function of $a_{0}$ and $\gamma_{0}$
for an 800 nm central wavelength laser. The solid line indicates the threshold
between classical and quantum radiation reaction forces, and the dotted line
indicates the threshold where $g(\chi_{e})$ begins to be significant.
The modified strong damping parameter $\psi_{Q}$ for an 800 nm wavelength
laser is shown as a function of $a_{0}$ and $\gamma_{0}$ in figure 2. As shown
in Refs Kirk_PPCF_2009 ; Sokolov_PRE_2010 , for $\chi_{e}\sim 0.1$, the
spectrum emitted should not be changed significantly in shape, but $\langle
g(\chi_{e})\rangle\approx g(\langle\chi_{e}\rangle)$ indicates that the energy
loss of the electron beam due to radiation damping should be changed by a
measurable amount. This is also consistent with what we observe with our
model.
### II.3 Parameter regimes involving $\chi_{e}$ and $\psi_{Q}$
The counter-propagating geometry laser-electron beam experiment is an
excellent testbed for studying quantum electrodynamics and strong radiation
damping effects. This is because of the ability to choose between strongly
radiation damped behavior ($\psi_{Q}\gtrsim 1$), or fields that are quantum
electrodynamically strong ($\chi_{e}\gtrsim 1$), _or_ a situation where both
$\psi_{Q}\gtrsim 1$ and $\chi_{e}\gtrsim 1$ simultaneously; conditions where
even more exotic effects may occur. These are controlled through variation of
the laser field strength, $a_{0}$, central frequency, $\omega_{0}$, and the
electron beam energy, $\gamma_{0}m_{e}c^{2}$. We can compare the requirements
for $a_{0}$, $\omega_{0}$ and $\gamma_{0}$ for the interaction to be in the
strong radiation damping regime or quantum electrodynamics dominant regime
relevant to experiments using 30 fs class lasers.
Table 1 shows parameters for different scenarios for strongly radiation damped
($\psi_{Q}\gtrsim 1$) and quantum electrodynamically strong ($\chi_{e}\gtrsim
1$) physics in a non-linear Thomson/Compton scattering geometry for an 800 nm
laser pulse with intensity $I_{L}$ colliding with an electron beam with energy
$E_{b}$. (a) corresponds to the SLAC experiment Burke_PRL_1997 . (b)
corresponds to near term experiments using intense 30 fs lasers such as
Hercules Yanovsky:2008 or Astra Gemini Hooker_JPIV_2006 and a laser
wakefield generated electron beam. (c) corresponds to an ‘ideal’ experiment
using two laser beam lines (as Astra Gemini has) with the current maximum
experimentally demonstrated laser intensity Bahk_OL_2004 and laser-wakefield
accelerated electron beam energy Leemans_NP_2006 . The SLAC experiment is
shown for comparative purposes only since the quasi-static field approximation
is not valid for this case Hu_PRL_2010 .
Table 1: Different scenarios for strong radiation damping ($\psi_{Q}\gtrsim 1$) and QED strong ($\chi_{e}\gtrsim 1$) physics in a non-linear Thomson scattering geometry for 800 nm wavelength laser pulses with intensity $I_{L}$ colliding with an electron beam with energy $E_{b}$. | $E_{b}$ / GeV | $I_{L}$ / Wcm-2 | $a_{0}$ | $a_{rad}$ | $a_{Q}$ | $\chi_{e}$ | $\psi$ | $\psi_{Q}$
---|---|---|---|---|---|---|---|---
(a) | 46.6 | $1\times 10^{18}$ | 0.5 | 2.5 | 1.2 | 0.43 | 0.045 | 0.018
(b) | 0.2 | $5\times 10^{21}$ | 50 | 46 | 420 | 0.12 | 1.2 | 0.74
(c) | 1 | $2\times 10^{22}$ | 100 | 21 | 84 | 1.2 | 23 | 3.7
In the first case (a) the laser vector-potential is of the order of both
$a_{rad}$ and $a_{Q}$. Case (b) corresponds to a situation where there will be
strong radiation damping but quantum effects will be weak,
$a_{Q}>a_{0}>a_{rad}$. In case (c) the laser is sufficiently intense for both
radiation damping and quantum recoil to be manifest, $a_{0}>a_{Q}>a_{rad}$.
Since our model is classical – that is to say involving equations of motion
only – it is restricted to the parameter range where $\chi^{2}\ll 1$
Sokolov_PRE_2010 . For the parameters described here, $\chi^{2}=0.014$ and so
the classical approach is reasonable. This also motivates the use of the
description of this process as “non-linear Thomson scattering” rather than
“Compton” scattering. We also calculate the electron spectrum after the
interaction in the presence of radiation damping with and without the $g$
factor, showing that quantum modifications to radiation losses may be
measurable.
## III The model and numerical methods
The spectral intensity of radiation emitted by a number $N_{P}$ of
accelerating point charges can be expressed, in the far-field, as Jacksonbook
:
$\frac{d^{2}I}{d\omega
d\Omega}=\frac{\mu_{0}e^{2}c}{16\pi^{3}}\omega^{2}\Bigg{|}{\int_{-\infty}^{\infty}\sum_{j=1}^{N_{P}}\mathbf{\hat{s}}\times\mathbf{\beta}_{j}e^{i\omega(t-\mathbf{n}\cdot\mathbf{r}_{j}/c)}}dt\Bigg{|}^{2}\;,$
(14)
where the unit vector $\mathbf{\hat{s}}$ is in the direction of observation,
at a distance far compared with the scale of the emission region. This can be
written alternatively in terms of proper time, $\tau$:
$\frac{d^{2}I}{d\omega
d\Omega}=\frac{\mu_{0}e^{2}c}{16\pi^{3}}\omega^{2}\Bigg{|}\sum_{j=1}^{N_{P}}{\int_{-\infty}^{\infty}\mathbf{\hat{s}}\times\mathbf{v}_{j}e^{i\kappa_{\alpha}x_{j}^{\alpha}}}d\tau\Bigg{|}^{2}\;,$
(15)
where $\mathbf{v}_{j}$ is the momentum part of the $j$th particle’s four-
velocity defined as:
$v_{j}^{\alpha}=\frac{dx_{j}^{\alpha}}{d\tau}\;.$ (16)
To numerically integrate the equations of motion for charged particles, both
$x^{\alpha}$ and $v^{\alpha}$ have to be recorded at a number of discrete
points. To then perform the spectral integration numerically, equation 15 can
be reduced to the summation:
$\frac{d^{2}I}{d\omega
d\Omega}=\frac{\mu_{0}e^{2}c}{16\pi^{3}}\omega^{2}\Bigg{|}{\sum_{j=1}^{N_{P}}\sum_{n=0}^{N_{\tau}}\mathbf{\hat{s}}\times\mathbf{v}_{j}^{n}e^{i\kappa_{\alpha}x_{j}^{\alpha,n}}}\Delta\tau\Bigg{|}^{2}\;,$
(17)
One of the advantages of using proper time rather than ‘laboratory’ time for
numerical calculations is that the time resolution is effectively adaptive; as
the particle gains inertia and is therefore accelerated at a decreased rate
for a similar force, the time step-size increases. Numerically calculating
this integral by ‘brute force’ has the problem that the exponent is a fast
oscillating function, and therefore without resolving $\omega$ the integral
will in general not converge without a numerical timestep of $\Delta\tau\ll
1/\omega$ Filon_PRSE_1928 ; Nyquist ; ISI:A1949um84500004 . Since we are
interested in $\gamma$-ray photons in excess of an MeV energy generated from a
few fs interaction, the ratio of the necessary time step to the integration
timescale is computationally unfeasibly large. Recently, methods for
overcoming this limit by using interpolation have been developed
Martins_SPIE_2009 ; Thomas_PRSTAB_2010 ; Martins_AAC_2010 . Here we use the
method we previously developed Thomas_PRSTAB_2010 , and the reader is directed
towards that paper for further details of the numerical calculation.
The particle trajectories were calculated in the presence of four-potentials,
$A^{\mu}=\\{A^{0}=\phi/c,A^{1},A^{2},A^{3}\\}$, representative of a spatio-
temporally Gaussian laser pulse with no interaction between electrons. The
laser pulse propagated in the $+\mathbf{\hat{x}}_{3}$ direction with four-
potential described by:
$A^{\mu}=\Re\left[\\{A_{0}\\}^{\mu}(x^{\alpha})e^{i\kappa_{\alpha}x^{\alpha}}f(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})\right]\;,$
(18)
where $\\{A_{0}\\}^{\mu}(x^{\alpha})$ is the spatial distribution of four-
potential, in this case $\kappa_{\alpha}=\\{\omega_{0},0,0,\omega_{0}/c\\}$ is
the laser four wavevector, $f(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})$ is a
function of time describing the temporal envelope. The spatial-temporal
distribution of a tightly focused pulse that satisfies the vacuum Maxwell’s
equations is in general very complicated, but is easier to formulate in terms
of potentials than fields. That is because it is possible to have a purely
transverse (to propagation) vector potential and satisfy the vacuum Maxwell’s
equations, something that is not possible with fields. These can be formulated
by using the Lorentz-invariant Lorenz gauge condition
$\partial_{\mu}A^{\mu}=0$. Using a slowly varying envelope approximation, and
using a transverse vector potential linearly polarized in the
$\hat{\mathbf{x}}_{1}$ direction with propagation in the
$\hat{\mathbf{x}}_{3}$ direction, this can be approximated as
$\\{A_{0}\\}^{0}=-(ic/\omega_{0})\partial\\{A_{0}\\}^{1}/\partial x_{1}$
Davis_PRA_1979 . Here, vector and scalar potentials with corrections to the
basic Gaussian optics formulation were introduced up to order
${\theta_{0}}^{2}$, where $\theta_{0}=2c/\omega_{0}w_{0}$ is the asymptotic
divergence angle of a Gaussian laser beam with a waist of $w_{0}$. This yields
potentials:
$\\{A_{0}\\}^{1}=\left[1+\frac{\theta_{0}^{2}}{2}\left(\frac{1-i\zeta}{1+\zeta^{2}}\right)\left(1-\left(\frac{1-\zeta^{2}}{2\left(1+\zeta^{2}\right)}\right)\rho^{2}\right)\right]\Psi_{0}\;,$
(19)
$\\{A_{0}\\}^{0}=i\theta_{0}\xi_{1}e^{-i\tanh^{-1}\zeta}\left[\\{A_{0}\\}^{1}-{\theta_{0}}^{2}\left(\frac{1-\zeta^{2}}{1+\zeta^{2}}\right)\Psi_{0}\right]\;,$
(20)
and $\\{A_{0}\\}^{2}=\\{A_{0}\\}^{3}=0$, where
$\Psi_{0}=\frac{e^{-i\tanh^{-1}\zeta-(1+i\zeta)\rho^{2}}}{\sqrt{1+\zeta^{2}}}\;,$
(21)
$\rho=\sqrt{{x_{1}}^{2}+{x_{2}}^{2}}/w_{0}$, and
$\zeta=x_{3}\theta_{0}/w_{0}$. $w_{0}/\theta_{0}$ is the Rayleigh range of the
laser. Higher order corrections to the field structure could be employed to
account for extremely tight focusing, but here we restrict our numerical
calculations to foci with $w_{0}>\lambda$, where $\lambda$ is the laser
wavelength; these corrections in ${\theta_{0}}^{2}$ are of magnitude
$(1/\pi^{2})\lambda^{2}/w_{0}^{2}$, so these corrections are up to 10 % of the
zero order fields and can’t be considered negligible, but the next order
corrections are ${\theta_{0}}^{4}$ and therefore of less importance.
An electron beam was modeled using $N_{P}$ particles initiated with a momentum
$p_{0}$ in the $-\mathbf{\hat{x}}_{3}$ direction in front of the laser. In
order to simulate a more realistic beam, rejection sampling against a Gaussian
probability distribution function was used to generate a beam with a spread in
momentum, $\mathbf{\sigma_{p}}$, and position, $\mathbf{\sigma_{x}}$ which
statistically approximated the phase space distribution:
$\displaystyle f_{e}(\mathbf{x},\mathbf{p},t)=$
$\displaystyle\exp\left[-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{p^{2}}{2\sigma_{p}^{2}}\right]\;,$
(22)
where
$x^{2}/\sigma_{x}^{2}=x_{1}^{2}/\sigma_{x_{1}}^{2}+x_{2}^{2}/\sigma_{x_{2}}^{2}+x_{3}^{2}/\sigma_{x_{3}}^{2}$
and
$p^{2}/\sigma_{p}^{2}=p_{1}^{2}/\sigma_{p_{1}}^{2}+p_{2}^{2}/\sigma_{p_{2}}^{2}+\left(p_{3}-p_{0}\right)^{2}/\sigma_{p_{3}}^{2}$
The root-mean-square normalized emittance of the bunch is therefore given by
$\epsilon={\sigma_{p_{1}}}{\sigma_{p_{2}}}{\sigma_{p_{3}}}{\sigma_{x_{1}}}{\sigma_{x_{2}}}{\sigma_{x_{3}}}$.
Although the particle tracking routine could easily calculate a much larger
bunch, due to the computational demands of the numerical spectrometer for a
full angular sweep, the number of electrons in the bunch was limited to
$N_{P}=500$. Radiation from individual electrons was summed incoherently.
A gaussian temporal envelope is used in all cases,
$f=e^{-(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})^{2}}$. The pulse duration
is $t_{L}=65\;\omega_{0}$, where $\omega_{0}$ is the laser angular frequency,
which in the case of a typical $0.8$ $\mu$m laser is $2.36\times 10^{15}$ s-1,
yielding $t_{L}=27.5$ fs at $1/e^{2}$ radius, or 32 fs full-width-at-half-
maximum, of intensity. The electron beam parameters were varied, with $p_{0}$
corresponding to a beam energy typically of 204 MeV ($\gamma_{0}=400$). The
linearly polarized laser, with normalized vector potential of $a_{0}=50$
corresponding to a peak laser intensity of $5.3\times 10^{21}$ Wcm-2 was
focused to a spot with waist $w_{0}=2.55$ $\mu$m, or $w_{0}=20c/\omega_{0}$.
The electron beam parameters are comparable those routinely achieved in laser
wakefield acceleration experiments and are summarized in table 2.
Table 2: Emittance parameters of the electron beam used in the numerical model. (a) Finite momentum spread case, (b) Zero momentum spread case. | $\sigma_{x_{1}}$ | $\sigma_{x_{2}}$ | $\sigma_{x_{3}}$ | $\sigma_{p_{1}}$ | $\sigma_{p_{2}}$ | $\sigma_{p_{3}}$
---|---|---|---|---|---|---
A | $3c/\omega_{0}$ | $3c/\omega_{0}$ | $9c/\omega_{0}$ | $m_{e}c$ | $m_{e}c$ | $10m_{e}c$
B | $3c/\omega_{0}$ | $3c/\omega_{0}$ | $9c/\omega_{0}$ | $0$ | $0$ | $0$
## IV Numerical results
In this section we detail ‘real world’ numerical calculations of a
backscattering experiment applicable to near term experiments using current
laser systems and laser wakefield accelerated electrons. By ‘real world’, we
mean that the calculation of the radiation spectrum includes the effect of a
Gaussian shaped bunch of electrons with normalized emittance (longitudinal and
transverse) comparable to that produced in laser-wakefield acceleration
interacting with a tightly focused laser, that radiation reaction forces are
included, and that the radiation spectrum is calculated directly from the
electron trajectories. However, the self-consistent absorption of laser pulse
photons is not included. The energy radiated by a $10^{9}$ electron beam will
later be shown to be 0.3 J, which is a non-negligible 2% of the pulse energy
of the laser considered here. Including the depletion of laser energy would
modify the spectrum of photons slightly, but is likely to be less important
than the other effects we consider here.
Figure 3: The angularly resolved spectral intensity ($d^{2}I/d\omega d\Omega$)
due to a 500 electron bunch with $\gamma=400$ and emittance given in table 2
scattering from a laser pulse with $a_{0}=50$ with higher order field
contributions included as in equations 19-21. In (a) the radiation reaction
force is not included, and in (b) the radiation reaction force is included.
The contours are taken at identical spectral intensity levels for both cases,
normalized to the peak which is 1.6517$\times 10^{-26}$ Js-1 at 0.2, 0.3, 0.4,
0.5, 0.6, 0.7 and 0.8.
The spectral intensity $d^{2}I/d\omega d\Omega$, where differential solid
angle $d\Omega=\sin\theta d\theta d\phi$, was calculated on a grid consisting
of 150 cells in $\omega$ over the range
$10^{4}\omega_{0}<\omega<10^{8}\omega_{0}$, with $\Delta\omega$ exponentially
increasing with $\omega$, 117 cells in $\theta$ over the range $0<\theta<30$
mrad and 26 cells in $\phi$ over the range $0<\phi<\pi/2$ rad. For clarity in
the figures, symmetry is assumed and therefore the full range $-30<\theta<30$
mrad and $0<\phi<\pi$ is displayed.
### IV.1 The high brilliance synchrotron source
The properties of radiation from backscattering of an electron with a laser
pulse have been extensively studied, and we can therefore use analytic
formulae to predict that in the interaction of a electron beam with
$\gamma=400$ with a laser with field strength $a_{0}=50$, the synchrotron-like
spectrum will peak in energy at
$\hbar\omega_{peak}=2.56a_{0}\gamma^{2}\hbar\omega_{0}=30$ MeV
ISI:000182450200081 . Using the numerical model, we can more accurately model
the properties of the radiation produced in a high intensity laser interaction
with a laser wakefield accelerated electron beam as a source for applications,
in particular including the effects of radiation damping Koga_POP_2005 , non-
plane wave laser fields Hartemann_AJS_2000 and calculate the full angular
distribution of radiation.
Figure 3 shows the spectral intensity of radiation produced by a 500 electron
bunch with $\gamma=400$ scattering from a laser pulse with $a_{0}=50$. In this
example, the higher order field contributions are included, as in equations
19-21, as well as beam emittance as given in table 2 case A. This represents
reasonably “realistic” modeling of an experiment and results in a well
collimated, smooth, synchrotron-like radiation emission extending up to very
high energies, with a broad peak at approximately $10$ MeV, which is a factor
of 3 smaller than the analytic prediction due to the radiation reaction and
finite spot effects. Because the laser pulse is linearly polarized, as
expected, the radiation is strongly polarized, but also the angular intensity
distribution has a pronounced ellipticity, with the major axis in the
direction of polarization. Linear polarization also leads to higher photon
energies compared with a circularly polarized pulse with the same pulse
energy.
One other notable effect is that of the higher order terms in the laser
fields. These do not significantly change the spectral shape, but do change
the magnitude non-negligibly. Without the field contributions, the peak
spectral intensity is 1.57$\times 10^{-26}$ Js-1, but with them it is
1.65$\times 10^{-26}$ Js-1, which is a 5% difference. Although the order
$\theta^{2}$ pulse potential corrections have been simply added to the first
order potential – so that the energy in the corrected pulse is higher than the
uncorrected – because the additional potential are $\sim 10$% of the first
order potential, adding the corrections only represents a $\sim$1% increase in
pulse energy. The slight increase in pulse energy alone is too small to
account for the increased radiation output alone. Instead it is the additional
longitudinal motion due to these potentials that increases the spectral
output.
Figure 4: The on-axis peak spectral brilliance ($d^{2}I/d\omega
d\Omega/(1000\hbar t_{L}\pi(w_{0}/2)^{2})$ in standard light-source units of
${\rm photons}\cdot{\rm s}^{-1}{\rm mm}^{-2}{\rm mrad}^{-2}/0.1\%\;{\rm
bandwidth}$) due to a 100 pC electron bunch with $\gamma=400$ and momentum
spread given by table 2 scattering from a 30 fs laser pulse with $a_{0}=50$.
To compare this to other synchrotron light sources, it is also useful to plot
the on-axis spectrum in terms of the standard units of the synchrotron
community, ${\rm photons}\cdot{\rm s}^{-1}{\rm mm}^{-2}{\rm
mrad}^{-2}/0.1\%\;{\rm bandwidth}$. To this, the spectral intensity is
multiplied by a numerical factor that assumes the 500 electrons are a
reasonable statistical representation of a 100 pC electron bunch that is
typical of laser wakefield experiments mangles ; geddes ; faure . Also
necessary for this calculation, the source size of the radiation is taken to
be the laser spot area within the radius of half the pulse waist,
$\pi(w_{0}/2)^{2}$. The on-axis radiation spectrum is shown in figure 4. As
well as peaking at high energies, the peak spectral brilliance is also
extremely high, comparable to the FLASH free electron laser but at
significantly higher photon energies Robinson_NJP_2010 and significantly more
brilliant than conventional synchrotrons. The effect of the high intensity
dramatically increases the brilliance of the source, at the expense of the
band-width which at lower intensity can be extremely narrow, which may be of
more utility for some applications Hartemann_PRSTAB_2007 ; Albert_PRSTAB_2010
; Albert_PRSTAB_2011 .
In figure 5 the cumulative photon number,
$\int_{0}^{E}(dN/dE^{\prime})dE^{\prime}$, _per electron_ is shown for this
spectrum, showing that on average each electron interacting with the laser
field emits approximately 200 photons. When integrated numerically, the total
photon energy emitted by each electron is $3.5\times 10^{-10}$ J. This is 10
times more than the energy of a 200 MeV electron. This result may
superficially appear not to conserve energy, however, the radiated photon
energy is predominantly drawn from the laser pulse. For a bunch of $10^{9}$
electrons, which is of the order 100 pC of charge, the total energy output
would be 0.35 J. For a bunch of this size, depletion of the laser fields – if
treated self-consistently – would modify the electron dynamics and radiation
output, but should only be a small perturbation (the pulse energy used here is
19.0 J) and hence would not be expected to modify this output energy
significantly. Ignoring this correction, the conversion efficiency of laser
pulse energy into $\gamma$-rays is 1.8%.
Figure 5: The cumulative photon number
($\int_{0}^{E}dN/dE^{\prime}dE^{\prime}$) per electron due to a 500 electron
bunch with $\gamma=400$ and momentum spread scattering given by table 2 from a
laser pulse with $a_{0}=50$. Figure 6: The angularly resolved spectral
intensity ($d^{2}I/d\omega d\Omega$) due to a zero emittance 500 electron
bunch with $\gamma=400$ and no scattering from a laser pulse with $a_{0}=50$.
In (a) the radiation reaction force is not included, and in (b) the radiation
reaction force is included. The contours are taken at identical spectral
intensity levels for both cases, normalized to the peak which is 2.6872$\times
10^{-26}$ Js-1 at 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8.
### IV.2 On the observation of radiation reaction effects in the photon
distribution
It has been suggested that signatures of the radiation reaction forces may be
observed in the photon distribution emitted in a counter propagating
experiment Koga_POP_2005 ; Piazza_PRL_2009 . The numerical calculations
performed here suggest this may be difficult due to the momentum spread of the
electron beam. Figure 6 shows the spectral intensity of radiation emitted
under identical conditions to those of figure 3 except that here the electron
beam has zero momentum spread, as in table 2 case B. The distribution has fine
features that are smoothed out when the electron beam has a momentum spread,
as would be expected. To see more clearly the effect of momentum spread on the
radiation distribution, figures 7 and 8 show two dimensional slices through
the radiation intensity distribution, in the planes parallel and perpendicular
to the laser polarization. In addition, the spectral intensity has been
converted into a photon distribution _per electron_ ,
$\omega_{0}d^{2}N/d\omega d\Omega$, which is more likely to be the form of
data obtained in an experiment (i.e. a histogram of photon hits on an array of
single photon counting detectors).
Figure 7: The photon distribution (normalized to the laser frequency,
$\omega_{0}d^{2}N/d\omega d\Omega$) per electron due to a 500 electron bunch
with $\gamma=400$ and zero momentum spread scattering from a laser pulse with
$a_{0}=50$. In (a) and (c) radiation reaction force is not included, and in
(b) and (d) radiation reaction force is included. (a) and (b) show the photon
distribution in the plane perpendicular to the laser polarization and (c) and
(d) show the photon distribution in the plane parallel to the laser
polarization.
Figure 7 shows the photon distribution from a zero momentum spread electron
beam interaction. (a) and (c) _radiation reaction force is not included_ , and
in (b) and (d) _radiation reaction force is included_. (a) and (b) show the
photon distribution in the plane perpendicular to the laser polarization and
(c) and (d) show the photon distribution in the plane parallel to the laser
polarization. The angular distribution of photons shows pronounced differences
with and without radiation reaction forces, and the energy distribution is
also dramatically changed, in particular resulting in a large number of low
energy photons in the damped case compared to no damping. Another feature is
slow oscillations in the spectral intensity with frequency/energy. These may
be due to the short truncated electron bunch and laser pulse in the time
domain, which will result in long wavelength oscillations in the frequency
domain.
Figure 8: The photon distribution (normalized to the laser frequency,
$\omega_{0}d^{2}N/d\omega d\Omega$) per electron due to a 500 electron bunch
with $\gamma=400$ and momentum spread given by table 2 scattering from a laser
pulse with $a_{0}=50$. In (a) and (c) radiation reaction force is not
included, and in (b) and (d) radiation reaction force is included. (a) and (b)
show the photon distribution in the plane perpendicular to the laser
polarization and (c) and (d) show the photon distribution in the plane
parallel to the laser polarization.
When the electron bunch is given the momentum spread of table 2 case A, the
distinction between the cases with and without radiation reaction force
becomes significantly less distinct. Figure 8 shows the photon distribution
from this interaction. There is little difference in the spectral intensity
distribution with and without radiation reaction force effects, except that
the overall magnitude is reduced, and the peak energy is reduced. Differences
in the angular distribution, however, are small and are likely to be much
smaller than expected shot-to-shot fluctuations in electron beam emittance.
Coupling this to the intrinsic difficulty of measuring high energy photons in
a collimated beam, it appears to be unfeasible to suggest that radiation
reaction effects will be discernible in experimental measurements in this
configuration in the near term.
### IV.3 On the observation of radiation reaction effects in the electron
phase-space distribution
In contrast to the photon measurements, it should be very easy to observe
radiation reaction effects in the electrons as measured using a standard
scintillating screen configuration. It is typical in laser wakefield
accelerator experiments to measure either the electron beam profile using a
scintillating screen, or electron forward momentum spectrum using a deflecting
magnet and a scintillating screen geddes ; faure . These diagnostics
effectively correspond to the $p_{1}$-$p_{2}$ and $p_{1}$-$p_{3}$ electron
phase-space densities respectively – with a spectrometer, the deflection by
the magnetic field disperses the electrons by $p_{3}$, but the projection in
$p_{1}$ is maintained.
Figure 9: Two dimensional histograms of the $p_{1}$-$p_{2}$ phase space
distribution of a 500 electron bunch with momentum spread according to table 2
case B before (top) and after (bottom) interaction with the high intensity
($a_{0}=50$) pulse. In (a) and (c) there is no radiation reaction, and in (b)
and (d) a radiation reaction model included according to equation 2.
Figure 9 shows the $p_{1}$-$p_{2}$ phase-space density for the electron bunch
before and after the interaction as two dimensional histogram plots. The
electrons are deflected by the laser fields so that the transverse momentum
spread is increased in both cases, consistent with a ponderomotive deflection.
However, there is little difference between the cases with and without
radiation reaction forces. This is because the radiation damping effect
reduces both transverse and longitudinal momenta proportionally (to lowest
order the radiation force in equation 2 is
$dp^{\mu}/d\tau|_{fric}=-\tau_{0}\omega_{0}^{2}\gamma^{2}a^{2}p^{\mu}$), and
hence in general the exit angle of a particular electron $\theta_{exit}\simeq
p_{\bot}/p_{3}$ is not expected to change significantly.
Figure 10: Two dimensional histograms of the $p_{3}$-$p_{1}$ phase space
distribution of a 500 electron bunch with momentum spread according to table 2
case B before (top) and after (bottom) interaction with the high intensity
($a_{0}=50$) pulse. In (a) and (c) there is no radiation reaction, and in (b)
and (d) a radiation reaction model included according to equation 2. _Note
that the horizontal momentum scale is negative and not the same for each phase
space_.
The effect on the electron spectrum is dramatic however, as was also
previously shown by Koga et al. Koga_POP_2005 . In figure 10, two dimensional
histogram plots of the $p_{3}$-$p_{1}$ phase space density of the electron
bunch is shown with and without radiation reaction forces. Under the
conditions modeled here, the electron beam loses almost half its energy when
radiation damping is included (and as expected, it experiences little change
in energy without radiation damping).
Figure 11: Two dimensional histograms of the $p_{3}$-$p_{1}$ phase space
distribution of a 500 electron bunch with momentum spread according to table 2
case B before (top) and after (bottom) interaction with the high intensity
($a_{0}=50$) pulse with radiation damping. (a) and (c) use the purely
classical expression 2 and in (b) and (d) the radiation reaction four-force is
modified by multiplying by the instantaneous $g$ factor given by equation 13.
_Note that the horizontal momentum scale is negative and not the same for each
phase space._
Finally, figure 11 shows two dimensional histogram plots of the
$p_{3}$-$p_{1}$ phase space density of the electron bunch similar to figure
10, but this time the radiation reaction is shown with and without the factor
$g(\chi_{e})$ given by equation 13 included. It can be seen by the plot that
under these conditions the _electron_ spectrum after the interaction with the
$g$ factor differs from the purely classical result by $\sim 10$% relative to
the overall energy loss. There is a smaller energy loss due to the fact that
the expected radiation spectrum is less energetic than the purely classical
result would suggest. This difference between classical and quantum corrected
radiation reaction may be sufficiently large to be distinguishable over
experimental fluctuations if well characterized. The effect of the addition of
$g(\chi_{e})$ on the _photon_ spectrum calculated with classical radiation
reaction forces under these conditions is negligible.
## V Conclusions
The counter-propagating electron beam, ultra-high-intensity laser interaction
is likely to be attempted by numerous experimental groups in the near future.
In addition to ultimately studying quantum electrodynamic effects, initial
experiments with lower electron beam energies and laser intensities are likely
to be concerned with the brilliant high energy photon output and classical
forms of radiation forces. From these numerical calculations, we predict a
large flux of photons with energy in excess of 1 MeV, in a beam collimated
within a 10 mrad divergence angle, and with an elliptical angular distribution
due to the linear polarization of the laser pulse. Each electron should emit
$\sim$100 photons above 1 MeV for a 200 MeV Gaussian electron beam colliding
with a pulse of intensity $5\times 10^{21}$ Wcm-2. For a typical laser
wakefield accelerated electron bunch with 100 pC charge mangles ; geddes ;
faure , this should result in $\sim 10^{11}$ photons in a broad synchrotron-
like spectrum peaking at 10 MeV with approximately 2% conversion efficiency of
laser energy into $\gamma$-rays, in a beam collimated to less than 10 mrad
divergence and with a peak brightness exceeding $10^{29}$ photons
s-1mm-2mrad${}^{-2}(0.1$% bandwidth$)^{-1}$.
In addition we show that measurements of the _radiation_ will be unlikely to
be able to indicate signatures of radiation reaction forces, and in particular
the ability to distinguish between different classical or quantum formulations
of the radiation force, due to the effects of beam emittance and tight laser
focusing. However, it should still be easy to observe radiation reaction
effects in the electron spectrum, where differences compared with a no-
radiation force model is dramatic, even with moderate beam emittance.
Including quantum effects using the $g(\chi_{e})$ factor under these
parameters causes a sufficiently reduced damping effect on the electron energy
spectrum to be measurable.
Whether signatures of different classical radiation reaction force models can
be observed in experiment is not addressed by the results of this paper.
However, it is unlikely, since the primary measurable effect on the electrons
is energy loss, which is to low order similar for all formulations of the
radiation reaction force. It is likely that the differences between models
will be hidden by the effects of beam emittance and laser focusing conditions
also.
## VI Acknowledgements
This work was funded by the NSF under contracts 1054164 and PHY-0935197, DARPA
under contract N66001-11-1-4208, EPSRC (Grant numbers EP/I014462/1 and
EP/G055165/1) and the Royal Society.
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|
arxiv-papers
| 2012-04-24T03:11:05 |
2024-09-04T02:49:30.073895
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. G. R. Thomas, C. P. Ridgers, S. S. Bulanov, B. J. Griffin, S. P. D.\n Mangles",
"submitter": "Alexander Thomas",
"url": "https://arxiv.org/abs/1204.5259"
}
|
1204.5273
|
∎
11institutetext: M. Gersabeck 22institutetext: CERN, 1211 Geneva, Switzerland
22email: marco.gersabeck@cern.ch 33institutetext: V.V.
Gligorov44institutetext: CERN, 1211 Geneva, Switzerland
44email: vladimir.gligorov@cern.ch 55institutetext: N. Serra66institutetext:
University of Zuerich, 8006 Zuerich, Switzerland
66email: nicola.serra@cern.ch
# Experimental constraints from flavour changing processes and physics beyond
the Standard Model††thanks: MG and VVG are supported by a Marie Curie Action:
“Cofunding of the CERN Fellowship Programme (COFUND-CERN)” of the European
Community’s Seventh Framework Programme under contract number (PCOFUND-
GA-2008-229600).
M. Gersabeck V.V. Gligorov N. Serra
(Received: date / Accepted: date)
###### Abstract
Flavour physics has a long tradition of paving the way for direct discoveries
of new particles and interactions. Results over the last decade have placed
stringent bounds on the parameter space of physics beyond the Standard Model.
Early results from the LHC, and its dedicated flavour factory LHCb, have
further tightened these constraints and reiterate the ongoing relevance of
flavour studies. The experimental status of flavour observables in the charm
and beauty sectors is reviewed in measurements of $C\\!P$ violation, neutral
meson mixing, and measurements of rare decays.
###### Keywords:
Flavour physics CP violation Meson mixing Rare decays Charm mesons Beauty
mesons
###### pacs:
13.20.Fc13.20.He13.25.Ft 13.25.Hw13.35.Bv13.35.Dx14.40.Lb14.40.Nd
## 1 Introduction
Flavour physics has given key contributions to the understanding of
fundamental particles. The kaon system is an excellent example how the
interplay of meson anti-meson mixing Lande:1956pf ; Jackson:1957zzb ;
Niebergall:1974wh , and the search for rare decays BottBodenhausen1967194 ;
Foeth:1969hi led to the prediction of the charm quark and indeed charm mesons
GellMann1964214 ; Tarjanne:1963zz ; Hara:1963gw ; Bjorken:1964gz ;
Glashow:1970gm . Furthermore, the observation of $C\\!P$ violation in neutral
kaons Christenson:1964fg led to the prediction of a third generation of
quarks Kobayashi:1973fv . At the LHC, precision measurements of flavour
physics are sensitive to new particles contributing to quantum loops up to
scales of about $200\mathrm{\,Te\kern-1.00006ptV}$ Buras:2009if which,
according to the Heisenberg uncertainty principle Heisenberg:1927zz ,
correspond to distance scales of the order of $10^{-21}~{}{\rm m}$. This
exceeds the reach for direct production of particles by roughly two orders of
magnitude.
This review covers flavour changing processes of charm and beauty mesons;
recent results on lepton flavour violating decays are also briefly discussed.
These provide complementary access to effects from Physics Beyond the Standard
Model (PBSM). This complementarity will eventually help to identify the nature
of signs of new dynamics, should they be generated by a common source.
Sections 2 to 4 cover the status of mixing and $C\\!P$ violation measurements
while section 5 reviews measurements of rare decays.
## 2 $C\\!P$ violation in heavy flavour mesons
The mass eigenstates of neutral mesons, $|M_{1,2}\rangle$, with masses
$m_{1,2}$ and widths $\Gamma_{1,2}$, are linear combinations of the flavour
eigenstates, $|M^{0}\rangle$ and $|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$, as
$|M_{1,2}\rangle=p|M^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$ with complex coefficients
satisfying $|p|^{2}+|q|^{2}=1$. This allows the definition of the averages
$m\equiv(m_{1}+m_{2})/2$ and $\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$. The
phase convention of $p$ and $q$ is chosen such that
$C\\!P|M^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$.
Following the notation of Kagan:2009gb , the time dependent decay rates of
$M^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptM}{}^{0}$ decays to the
final state $f$ can be expressed as
$\displaystyle\Gamma$ $\displaystyle(M^{0}(t)\rightarrow
f)=\frac{1}{2}e^{-\tau}\left|A_{f}\right|^{2}$
$\displaystyle\times\Big{\\{}\left(1+|\lambda_{f}|^{2}\right)\cosh(y\tau)+\left(1-|\lambda_{f}|^{2}\right)\cos(x\tau)$
$\displaystyle\quad+2\Re(\lambda_{f})\sinh(y\tau)-2\Im(\lambda_{f})\sin(x\tau)\Big{\\}},$
$\displaystyle\Gamma$ $\displaystyle(\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}(t)\rightarrow
f)=\frac{1}{2}e^{-\tau}\left|\bar{A}_{f}\right|^{2}$
$\displaystyle\times\Big{\\{}\left(1+|\lambda^{-1}_{f}|^{2}\right)\cosh(y\tau)+\left(1-|\lambda^{-1}_{f}|^{2}\right)\cos(x\tau)$
$\displaystyle\quad+2\Re(\lambda^{-1}_{f})\sinh(y\tau)-2\Im(\lambda^{-1}_{f})\sin(x\tau)\Big{\\}},$
(1)
where $\tau\equiv\Gamma t$, $\kern-1.99997pt\stackrel{{\scriptstyle\kern
1.39998pt\textsf{(---)}}}{{A}}_{\kern-2.10002ptf}\kern-3.00003pt$ are the
decay amplitudes and $\lambda_{f}$ is given by
$\lambda_{f}\equiv\frac{q\bar{A}_{f}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|e^{i\phi},$
(2)
where the right-hand expression is valid for a $C\\!P$ eigenstate $f$ with
eigenvalue $\eta_{C\\!P}$ and $\phi$ is the $C\\!P$ violating relative phase
between $q/p$ and $\bar{A}_{f}/A_{f}$.
In general, $C\\!P$ symmetry is violated if $\lambda_{f}$, as defined in
Equation 2, deviates from $1$. This can have different origins: the case
$|q/p|\neq 1$ is called $C\\!P$ violation in mixing, $|\bar{A}_{f}/A_{f}|\neq
1$ is $C\\!P$ violation in the decay, and a non-zero phase $\phi$ between
$q/p$ and $\bar{A}_{f}/A_{f}$ causes $C\\!P$ violation in the interference
between mixing and decay. Mixing is common to all decay modes and hence
$C\\!P$ violation originating in this process is universal which is called
indirect $C\\!P$ violation. Decay-specific $C\\!P$ violation is called direct
$C\\!P$ violation. An excellent discussion on the different types of $C\\!P$
violation can be found in section 7.2.1 of Sozzi:2008zza . As opposed to the
strange and the beauty system, $C\\!P$ violation has not yet been discovered
in the charm system, though the LHCb collaboration has recently found first
evidence for $C\\!P$ violation in two-body $D^{0}$ decays Aaij:2011in .
In the charm system one defines the differences
$\Delta{}m_{D}\equiv{}m_{2}-m_{1}$ and
$\Delta\Gamma_{D}\equiv\Gamma_{2}-\Gamma_{1}$. Furthermore, the mixing
parameters are defined as $x\equiv\Delta{}m/\Gamma$ and
$y\equiv\Delta\Gamma/(2\Gamma)$. Analogously, in the beauty system one defines
the differences $\Delta{}m_{d,s}\equiv{}m_{2}-m_{1}$ and
$\Delta\Gamma_{d,s}\equiv\Gamma_{1}-\Gamma_{2}$, where the subscripts denote
the $B^{0}_{d}$ and $B^{0}_{s}$ systems, respectively.
Within the Standard Model (SM), quark mixing is described by the CKM matrix
$\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\ V_{cd}&V_{cs}&V_{cb}\\\
V_{td}&V_{ts}&V_{tb}\end{array}\right)=\\\
\left(\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}\left(\rho-i\eta\right)\\\
-\lambda&1-\frac{1}{2}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}\left(1-\rho+i\eta\right)&-A\lambda^{2}&1\end{array}\right),$ (3)
given on the right in the Wolfenstein parametrization where $\lambda\approx
0.22$ is the sine of the Cabibbo angle. $C\\!P$ violation then arises solely
from the imaginary term in this matrix. Since the matrix is unitary, it can be
represented by six triangles in the complex plane, defined by unitarity
conditions such as
$V^{*}_{ub}V_{ud}+V^{*}_{cb}V_{cd}+V^{*}_{tb}V_{td}=0,$ (4)
which is known as the “Unitarity Triangle”. This particular unitarity
condition is chosen because the three terms, corresponding to the sides of the
triangle, are of approximately equal size. The fact that the SM predicts
$O(10\%)$ $C\\!P$ violating effects in many $B$ decays, while the predictions
for $D$ decays are generally at least two orders of magnitude smaller, has led
to differing experimental approaches. In the case of $B$ decays, the focus has
been on precise measurements of mixing and $C\\!P$ violation in order to
overconstrain the sides and angles of the Unitarity Triangle, in particular
its apex, as illustrated in Fig. 1.
Figure 1: The current constraints on the Unitarity Triangle. These meet at the
overconstrained apex, and the shaded ellipse indicates the allowed region for
the apex when all measurements are taken together. Reproduced from
Charles:2004jd .
In $D$ decays the focus has been on searches for $C\\!P$ violation and a
precise understanding of the mixing parameters.
## 3 Charm mixing and $C\\!P$ violation
The studies of charm mesons have gained in momentum with the measurements of
first evidence for meson anti-meson mixing in neutral charm mesons in 2007
Aubert:2007wf ; Staric:2007dt . Mixing of $D^{0}$ mesons is the only mixing
process where down-type quarks contribute to the box diagram. Unlike $B$-meson
mixing where the top-quark contribution dominates, the third generation quark
is of similar mass to the other down-type quarks. This leads to a combination
of GIM cancellation Glashow:1970gm and CKM suppression Cabibbo:1963yz ;
Kobayashi:1973fv , which results in a strongly suppressed mixing process
Bobrowski:2010xg .
Since experimental evidence has shown that quantum-loop effects are accessible
in the charm sector, measurements of $D$ mesons provide access to effects from
particles beyond the SM, complementary to measurements in the $B$ sector. It
was discussed whether the measured size of the mixing parameters could be
interpreted as a hint for new physics Hou:2006mx ; Ciuchini:2007cw ;
Nir:2007ac ; Blanke:2007ee ; He:2007iu ; Chen:2007yn ; Golowich:2007ka . New
physics effects were also searched for in numerous $C\\!P$-violation
measurements, which are covered in the remainder of this section, and searches
for rare decays as discussed in section 5.
### 3.1 Charm mixing
Mixing of $D^{0}$ mesons can be measured in several different modes. All
require identifying the flavour of the $D^{0}$ at production as well as at the
time of the decay. Tagging the flavour at production usually exploits the
strong decay $D^{*+}\\!\rightarrow D^{0}\pi^{+}$, where the charge of the pion
determines the flavour of the $D^{0}$. Charge conjugate decays are implicitly
included here and henceforth. The small amount of free energy in this decay
leads to the difference in the reconstructed invariant mass of the $D^{*+}$
and the $D^{0}$, $\delta m\equiv{}m_{D^{*+}}-m_{D^{0}}$, exhibiting a sharply
peaking structure over a threshold function as background. An alternative to
using this decay mode is tagging the $D^{0}$ flavour by reconstructing a
flavour-specific decay of a $B$ meson. This method has not yet been used in a
measurement as it did not yet yield competitive quantities of tagged $D^{0}$
mesons. At LHCb this approach may be of interest due to differences in trigger
efficiencies partly compensating for lower production rates. Another option
available, particularly at $e^{+}e^{-}$ colliders, is the reconstruction of
the opposite side charm meson in a flavour specific decay.
Theoretically, the most straight-forward mixing measurement is that of the
rate of the forbidden decay $D^{0}\\!\rightarrow
K^{+}\mu^{-}\overline{\nu}_{\mu}$ which is only accessible through
$D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing. The ratio
of the time-integrated rate of these forbidden decays to their allowed
counterparts, $D^{0}\\!\rightarrow K^{-}\mu^{+}\nu_{\mu}$, determines $R_{\rm
m}\equiv(x^{2}+y^{2})/2$. As this requires very large samples of $D^{0}$
mesons no measurement has thus far reached sufficient sensitivity to see
evidence for $D^{0}$ mixing. The most sensitive measurement to date has been
made by the Belle collaboration to $R_{\rm m}=(1.3\pm 2.2\pm 2.0)\times
10^{-4}$ Bitenc:2008bk , where the first uncertainty is of statistical and the
second is of systematic nature. This notation is applied to all results where
two uncertainties are quoted.
Related to the semileptonic decay is the suppressed decay $D^{0}\\!\rightarrow
K^{+}\pi^{-}$, called the wrong-sign (WS) decay. For this decay, a doubly
Cabibbo-suppressed (DCS) amplitude interferes with the decay through a mixing
process followed by the Cabibbo-favoured (CF) decay $D^{0}\\!\rightarrow
K^{-}\pi^{+}$. Following from equation 2 the time-dependent decay rate of the
WS decay is, in the limit of $C\\!P$ conservation, proportional to
Bergmann:2000id
$\frac{\Gamma(D^{0}(t)\rightarrow K^{+}\pi^{-})}{e^{-\Gamma
t}}\propto\left(R_{\rm D}+\sqrt{R_{\rm D}}y^{\prime}\Gamma{}t+R^{2}_{\rm
m}(\Gamma{}t)^{2}\right),$ (5)
where the mixing parameters are rotated by the strong phase between the DCS
and the CF amplitude, leading to the observable
$y^{\prime}=y\cos\delta_{K\pi}-x\sin\delta_{K\pi}$. The parameter $R_{\rm D}$
is the ratio of the DCS to the CF rate. Measurements with sufficient
sensitivity to unveil evidence for $D^{0}$ mixing have been performed by the
BaBar and CDF collaborations, leading to $x^{\prime 2}=(-0.22\pm 0.30\pm
0.20)\times 10^{-3}$ and $y^{\prime}=(9.7\pm 4.4\pm 3.1)\times 10^{-3}$
Aubert:2007wf , and $x^{\prime 2}=(-0.12\pm 0.35)\times 10^{-3}$ and
$y^{\prime}=(8.5\pm 7.6)\times 10^{-3}$ Aaltonen:2007uc , respectively.
Similarly, the CF and DCS amplitudes can also lead to higher mass states of
the same quark content. The decay $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{0}$ is
the final state of several such resonances. Thus, by studying the decay-time
dependence of the various resonances a mixing measurement can be obtained. The
BaBar collaboration achieved a measurement showing evidence for $D^{0}$ mixing
with central values of $x^{\prime\prime}=(26.1^{+5.7}_{-6.8}\pm 3.9)\times
10^{-3}$ and $y^{\prime\prime}=(-0.6^{+5.5}_{-6.4}\pm 3.4)\times 10^{-3}$
Aubert:2008zh , where the rotation between the observables and the system of
mixing parameters is given by a strong phase as
$x^{\prime\prime}=x\cos\delta_{K^{-}\pi^{+}\pi^{0}}+y\sin\delta_{K^{-}\pi^{+}\pi^{0}}$
and
$y^{\prime\prime}=y\cos\delta_{K^{-}\pi^{+}\pi^{0}}-x\sin\delta_{K^{-}\pi^{+}\pi^{0}}$.
The strong phases are not accessible in these measurements but have to come
from measurements performed using quantum-correlated $D^{0}$-$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ pairs produced at threshold. These
are available from CLEO PhysRevD.73.034024 ; PhysRevD.77.019901 ;
Rosner:2008fq ; Asner:2008ft and can be further improved by BESIII.
By the time of this review no single experiment observation of mixing in
$D^{0}$ mesons with a significance exceeding $5\sigma$ has been possible.
However, the combination of the numerous measurements by the Heavy Flavor
Averaging Group (HFAG) excludes the no-mixing hypothesis by about $10\sigma$
Asner:2010qj . Under the assumption of no $C\\!P$ violation the world average
of the mixing parameters is $x=(6.5^{+1.8}_{-1.9})\times 10^{-3}$ and
$y=(7.3\pm 1.2)\times 10^{-3}$.
### 3.2 Charm $C\\!P$ violation
Indirect $C\\!P$ violation is often measured in conjunction with mixing
parameters. One example is the measurement of effective inverse lifetimes in
decays of $D^{0}$ ($\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) mesons
into final states which are $C\\!P$ eigenstates, $\hat{\Gamma}$
($\hat{\bar{\Gamma}}$). The comparison of these lifetimes to that of a
Cabibbo-favoured flavour eigenstate ($\Gamma$) leads to the observable
$\displaystyle y_{C\\!P}$
$\displaystyle=\frac{\hat{\Gamma}+\hat{\bar{\Gamma}}}{2\Gamma}-1$
$\displaystyle\approx\eta_{C\\!P}\left[\left(1-\frac{A_{m}^{2}}{8}\right)y\cos\phi-\frac{A_{m}}{2}x\sin\phi\right],$
(6)
where $A_{m}$ is the $C\\!P$ violation in mixing defined alongside the direct
$C\\!P$ violation $A_{d}$ by $|\lambda_{f}^{\pm 1}|^{2}\approx(1\pm
A_{m})(1\pm A_{d})$ Gersabeck:2011xj . In the limit of $C\\!P$ conservation
$y_{C\\!P}$ equals the mixing parameter $y$.
Comparing the $C\\!P$ eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ to the
Cabibbo-favoured mode $K^{-}\pi^{+}$, the Belle and BaBar collaborations have
measured $y_{C\\!P}=(13.1\pm 3.2\pm 2.5)\times 10^{-3}$ Staric:2007dt and
$y_{C\\!P}=(11.6\pm 2.2\pm 1.8)\times 10^{-3}$ PhysRevD.80.071103 ,
respectively. The Belle collaboration has also published a measurement using
only the decay $D^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{-}K^{+}$
in which they compare the effective lifetime around the $\phi$ resonance with
that measured in sidebands of the $K^{-}K^{+}$ invariant mass. The effective
$C\\!P$ eigenstate content in these regions is determined with two different
models. Their result is $y_{C\\!P}=(1.1\pm 6.1\pm 5.2)\times 10^{-3}$
Zupanc:2009sy . Provided measurements of sufficient precision, the comparison
of $y_{C\\!P}$ with the mixing parameter $y$ is a test of $C\\!P$ violation.
However, while one would expect $y_{C\\!P}<y$ in the presence of $C\\!P$
violation, the experimental results currently favour $y_{C\\!P}>y$, i.e. no
sign of $C\\!P$ violation is observed.
A second, more sensitive, way of measuring indirect $C\\!P$ violation is
through the comparison of effective lifetimes of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to $C\\!P$ eigenstates.
This leads to the observable
$A_{\Gamma}=\frac{\hat{\Gamma}-\hat{\bar{\Gamma}}}{\hat{\Gamma}+\hat{\bar{\Gamma}}}\approx\eta_{C\\!P}\left[\frac{1}{2}\left(A_{m}+A_{d}\right)y\cos\phi-x\sin\phi\right],$
(7)
which has contributions from both direct and indirect $C\\!P$ violation
Kagan:2009gb ; Gersabeck:2011xj . Currently there are three measurements of
$A_{\Gamma}$, which are all compatible with zero. The Belle, BaBar, and LHCb
collaborations have measured $A_{\Gamma}=(0.1\pm 3.0\pm 1.5)\times 10^{-3}$
Staric:2007dt , $A_{\Gamma}=(2.6\pm 3.6\pm 0.8)\times 10^{-3}$
PhysRevD.78.011105 , and $A_{\Gamma}=(-5.9\pm 5.9\pm 2.1)\times 10^{-3}$
Aaij:2011ad , respectively. With the LHCb result being based only on a small
fraction of the data recorded so far, significant improvements in sensitivity
may be expected in the near future. Using current experimental bounds values
of $A_{\Gamma}$ up to $\mathcal{O}(10^{-4})$ are expected Kagan:2009gb ;
Bigi:2011re . It has however been shown that enhancements up to about one
order of magnitude are possible for example in the presence of a fourth
generation of quarks Bobrowski:2010xg or in a Little Higgs Model with
T-Parity Bigi:2011re . This would bring $A_{\Gamma}$ close to the current
experimental limits.
Eventually, the interpretation of $C\\!P$ violation results requires precise
knowledge of both mixing and $C\\!P$ violation parameters. The analysis of the
decays $D^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{-}\pi^{+}$ and
$D^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{-}K^{+}$ offers
separate access to the parameters $x$, $y$, $|q/p|$ and $\arg(q/p)$. This
require the decay-time dependence of the phase space structure of these
decays, which is possible in two ways: using Dalitz plot models or based on a
measurement of the strong phase difference across the Dalitz plot by the CLEO
collaboration Libby:2010eq . One measurement made by the Belle collaboration
has determined these parameters based on a Dalitz plot model Abe:2007rd .
Other measurements were performed by the CLEO Asner:2005sz and BaBar
delAmoSanchez:2010xz collaborations assuming $C\\!P$ conservation and thus
extracting only $x$ and $y$. With the data samples available and being
recorded at LHCb and those expected at future flavour factories, these
measurements will be very important to understand charm mixing and $C\\!P$
violation. However, in order to avoid systematic limitations it will be
important to reduce model uncertainties or to improve model-independent strong
phase difference measurements, which is possible at BESIII.
Direct $C\\!P$ violation is searched for in decay-time integrated
measurements. However, the decay-time distribution of the data has to be taken
into account to estimate the contribution from indirect $C\\!P$ violation.
Currently, the most striking measurements have been made in decays of $D^{0}$
mesons into two charged pions or kaons. While early measurements of BaBar
Aubert:2007if and Belle Staric:2008rx had not shown significant deviations
from zero, LHCb recently reported first evidence for $C\\!P$ violation in the
charm sector Aaij:2011in
$\displaystyle\Delta A_{C\\!P}$
$\displaystyle\equiv{}A_{C\\!P}(K^{-}K^{+})-A_{C\\!P}(\pi^{-}\pi^{+})$
$\displaystyle=(-8.1\pm 2.1\pm 1.1)\times 10^{-3}.$
Meanwhile, CDF has released a preliminary measurement of $\Delta
A_{C\\!P}=(-6.2\pm 2.1\pm 1.0)\times 10^{-3}$ CDF10784 which shows a hint of
a deviation from zero, in support of the LHCb result. The observable $\Delta
A_{C\\!P}$ exploits the cancellation of systematic uncertainties in the
difference of asymmetries. It gives access to the difference in direct $C\\!P$
violation of the two decay modes through
$\Delta A_{C\\!P}=\Delta{}a_{C\\!P}^{\rm
dir}\left(1+y_{C\\!P}\frac{\overline{\langle{}t\rangle}}{\tau}\right)+\overline{A}_{\Gamma}\frac{\Delta\langle{}t\rangle}{\tau},$
(8)
where $\tau$ is the nominal $D^{0}$ lifetime,
$\overline{X}\equiv(X(K^{-}K^{+})+X(\pi^{-}\pi^{+}))/2$, and
$\Delta{}X\equiv{}X(K^{-}K^{+})-X(\pi^{-}\pi^{+})$ Gersabeck:2011xj . With the
current precision on $A_{\Gamma}$ the influence of direct $C\\!P$ violation on
$A_{\Gamma}$ can be neglected as it is known to be $\leq 10^{-4}$ and hence
$A_{\Gamma}=-a_{C\\!P}^{\rm ind}$ is assumed. Thus, the world average leads to
central values of $\Delta{}a_{C\\!P}^{\rm dir}=(-6.6\pm 1.5)\times 10^{-3}$
and $a_{C\\!P}^{\rm ind}=(-0.3\pm 2.3)\times 10^{-3}$ which has a confidence
level of being in agreement with the no $C\\!P$ violation hypothesis of
$6.1\times 10^{-5}$ Asner:2010qj (see Fig. 2).
Figure 2: HFAG combination of measurements of $\Delta A_{C\\!P}$ and
$A_{\Gamma}$. Shown are the experimental results as bands indicating their
$\pm 1\sigma$ uncertainties, the best fit value with one-dimensional
uncertainties as a cross, and the $1\sigma$, $2\sigma$, and $3\sigma$
ellipses. The dot marks the point of no $C\\!P$ violation. Reproduced from
Asner:2010qj .
While it was commonly stated in literature that $C\\!P$ violation effects in
these channels were not expected to exceed $10^{-3}$, this statement has been
revisited in numerous recent publications. To date, no clear understanding of
whether Brod:2011re ; Feldmann:2012js ; Bhattacharya:2012ah ; Franco:2012ck
or not Bigi:2011re ; Rozanov:2011gj ; Cheng:2012wr ; Li:2012cf $C\\!P$
violation of this level can be accommodated within the SM has emerged. In
parallel with attempts to improve the SM calculations, many estimates of
potential effects of PBSM have been made Bigi:2011re ; Feldmann:2012js ;
Rozanov:2011gj ; Grossman:2006jg ; Isidori:2011qw ; Wang:2011uu ;
Hochberg:2011ru ; Pirtskhalava:2011va ; Chang:2012gn ; Giudice:2012qq ;
Altmannshofer:2012ur ; Chen:2012am ; Gedalia:2012pi . To complement
theoretical calculations, measurements in related modes have been and will be
performed in order to single out effects from particular amplitudes.
A related way of searching for $C\\!P$ violation is using decays of charged
$D$ mesons. One group of measurements studies decays of $D^{+}$ and
$D^{+}_{s}$ mesons into three charged hadrons, namely pions or kaons. Here,
$C\\!P$ violation can occur in two-body resonances contributing to these decay
amplitudes. Asymmetries in the Dalitz-plot substructure can be measured using
an amplitude model or using model-independent statistical analyses
Bediaga:2009tr ; Williams:2011cd . The latter allow $C\\!P$ asymmetries to be
discovered while eventually a model-dependent analysis is required to identify
its source. Neither phase-space integrated asymmetry measurements
Aitala:1996sh ; Link:2000aw ; Aubert:2005gj ; Alexander:2008aa ; Rubin:2008aa
; Mendez:2009aa , nor searches for local asymmetries in the Dalitz plot
Aubert:2005gj ; Rubin:2008aa ; Staric:2011en ; PhysRevD.84.112008 ;
PhysRevLett.108.071801 have shown any evidence for $C\\!P$ violation. The
largest signal is the recently reported measurement of $C\\!P$ violation in
$D^{+}\\!\rightarrow\phi\pi^{+}$ of $A_{C\\!P}^{\phi\pi^{+}}=(5.1\pm 2.8\pm
0.5)\times 10^{-3}$ by the Belle collaboration PhysRevLett.108.071801 , which
exploits cancellation of uncertainties through a comparison of asymmetries in
the decays of $D^{+}$ and $D^{+}_{s}$ mesons into the final state
$\phi\pi^{+}$.
Decays of $D^{+}$ and $D^{+}_{s}$ into a $K^{0}_{\rm\scriptscriptstyle S}$ and
either a $K^{+}$ or a $\pi^{+}$ are closely related to their $D^{0}$
counterparts. Measurements of time-integrated asymmetries in these decays are
expected to exhibit a contribution from $C\\!P$ violation in the kaon system.
As pointed out recently Grossman:2011to this contribution depends on the
decay-time acceptance of the $K^{0}_{\rm\scriptscriptstyle S}$. This can lead
to different expected values for different experiments which so far has not
been taken into account. Measurements of asymmetries in the decays
$D^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$
PhysRevD.80.071103 ; Mendez:2009aa ; PhysRevLett.104.181602 and
$D^{+}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ Mendez:2009aa
; PhysRevLett.104.181602 show significant asymmetries. Future, more precise
measurements will reveal whether or not these are in agreement with the
expected contribution from the kaon system.
In the light of the recent measurements it is evident that there are four
directions to pursue: more precise measurements of $\Delta A_{C\\!P}$ and the
individual asymmetries are required to establish the effect; further searches
for time-integrated $C\\!P$ violation need to be carried out in a large range
of modes that allow to identify the source of the $C\\!P$ asymmetry; searches
for time-dependent $C\\!P$ asymmetries, particularly via more precise
measurements of $A_{\Gamma}$; and finally a more precise determination of the
mixing parameters is required. Complementary to this are searches for rare
charm decays, studies of the top quark Wang:2011uu ; Hochberg:2011ru ,
measurements of nuclear electric dipole moments Giudice:2012qq , and many
other flavour observables which are beyond the scope of this review.
## 4 Beauty mixing and $C\\!P$ violation
The existence of $B^{0}$ and $B^{0}_{s}$ meson mixing is well established, and
the mass difference between the light and heavy eigenstates has been measured
to high precision in both systems. In addition, evidence exists for $C\\!P$
violation in $B^{0}$, $B^{+}$, and $B^{0}_{s}$ decays. The interpretation of
the experimental data focuses on the compatibility of the various measurements
with each other, and their compatibility with the SM description of $C\\!P$
violation as arising from a single weak phase in the CKM matrix. Two tensions
stand out at present: the discrepancy between the large mixing-induced $C\\!P$
asymmetry measured in semileptonic $B^{0}$ and $B^{0}_{s}$ decays
Abazov:2011yk and the small $C\\!P$ violating phase in $B^{0}_{s}$ mixing
LHCb-CONF-2012-002 on the one hand, and the discrepancy between
$\sin\left(2\beta\right)$ and $|V_{ub}|$ measured from the branching ratio of
$B^{+}\rightarrow\tau^{+}\nu$ Asner:2010qj on the other hand.
### 4.1 $B^{0}_{s}$ mixing
The mixing of $B^{0}_{s}$ mesons is described by the width difference between
the light and heavy mass eigenstates, $\Delta\Gamma_{s}$, the mass difference
$\Delta m_{s}$, and a single $C\\!P$ violating phase $\phi_{s}$. Within the SM
the width difference is substantial,
$\Delta\Gamma_{s}=\Gamma_{\textrm{L}}-\Gamma_{\textrm{H}}=0.087\pm 0.021$ ps-1
Lenz:2011ti , while the $C\\!P$ violating phase, as determined from indirect
fits to experimental data, is small $\phi_{s}=-0.036\pm 0.002$ rad Lenz:2011ti
; Lenz:2006hd ; Charles:2011va . Both can deviate substantially from these
predictions in other models.
The first observation of $B^{0}_{s}$ mixing was made by CDF Abulencia:2006ze ,
while the most precise measurement of the mass difference $\Delta m_{s}$ comes
from the recent LHCb measurement Aaij:2011qx . The most precise measurements
of both the width difference and phase come from the measurement of the time-
dependent $C\\!P$ asymmetry in
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
LHCb-CONF-2012-002 ; CDF-CONF-10778 ; PhysRevD.85.032006
$\displaystyle\phi_{s}$ $\displaystyle=$ $\displaystyle-0.001\pm 0.101\pm
0.027\;\textrm{rad}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb-
CONF-2012-002}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\Delta\Gamma_{s}$ $\displaystyle=$ $\displaystyle 0.116\pm
0.018\pm 0.006\;\textrm{ps}^{-1}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb-
CONF-2012-002}{\@@citephrase{(}}{\@@citephrase{)}}}},$ $\displaystyle\phi_{s}$
$\displaystyle\in$
$\displaystyle[\frac{\pi}{2},-1.51]\cup[-0.06,0.30]\cup[1.26,\frac{\pi}{2}]\;\textrm{rad}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{CDF-CONF-10778}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\phi_{s}$ $\displaystyle=$
$\displaystyle-0.55^{+0.38}_{-0.36}\;\textrm{rad}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PhysRevD.85.032006}{\@@citephrase{(}}{\@@citephrase{)}}}}.$
All these measurements are in good agreement with the SM, and it is notable
that a non-zero $\Delta\Gamma_{s}$ has been directly measured for the first
time at $5\sigma$. In addition, the sign of $\Delta\Gamma_{s}$ has been
unambiguously determined to be positive through the study of S-wave and P-wave
contributions to the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
decay amplitude LHCb-PAPER-2011-028 .
The measurement of $\phi_{s}$ from
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ is
complicated by the vector-vector final state, which necessitates a time-
dependent angular analysis, whereas it was proposed Stone:2008ak to study the
vector-pseudoscalar decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$
in which no such analysis is required. This measurement has recently been
performed by the LHCb collaboration, which, combined with the LHCb
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
measurement leads to
$\phi_{s}=-0.002\pm 0.083\pm
0.027\;\textrm{rad}\;\textrm{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb-
CONF-2012-002}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$
in good agreement with the SM prediction.
As noted in Fleischer:2010ib , the interplay of $\Delta\Gamma_{s}$ and
$\phi_{s}$ leads to predictions for the effective lifetimes of $B^{0}_{s}$
mesons decaying into $C\\!P$ eigenstates. In the specific case of
$B^{0}_{s}\rightarrow K^{+}K^{-}$, the lifetime has already been measured
Aaij:2011kn ; LHCb-CONF-2012-001 to be $1.468\pm 0.046\pm 0.006\textrm{ps}$.
Using the latest measurement of $\Gamma_{s}$ and $\Delta\Gamma_{s}$ by LHCb
LHCb-CONF-2012-002 , as well as the $B^{0}_{s}$ lifetime
$\tau_{B^{0}_{s}}=1.472\pm 0.025$ ps Asner:2010qj , the SM prediction from
Fleischer:2010ib can be updated to $\tau_{K^{+}K^{-}}=1.40\pm 0.02$.
Moreover, recent first observations of $B^{0}_{s}\rightarrow D^{0}D^{0}$ and
$B^{0}_{s}\rightarrow D^{+}D^{-}$ LHCb-CONF-2012-009 by LHCb indicate that it
will be possible in the near future to measure effective lifetimes in many
different $B^{0}_{s}$ decays to $C\\!P$ eignestates, and further constrain
$(\phi_{s},\Delta\Gamma_{s})$ in this manner.
The decay $B^{0}_{s}\rightarrow K^{+}K^{-}$ is not only a decay to a $C\\!P$
eigenstate, but is one example of a $b\rightarrow s$ penguin transition in the
decays of $B^{0}_{s}$ mesons. One of the experimentally most interesting modes
of this type is $B^{0}_{s}\rightarrow\phi\phi$ where, because of a
cancellation of $C\\!P$ violating effects from decay and mixing, the SM
predicts an upper limit of $0.02$ for $C\\!P$ violation Raidal:2002ph .
Although the time-dependent analysis is yet to be performed, time-integrated
analyses based on measuring triple products have been performed, and have
found no significant asymmetries Aaltonen:2011rs ; LHCb-PAPER-2012-004 , in
agreement with SM predictions.
Another interesting Ciuchini:2007hx decay is $B^{0}_{s}\rightarrow
K^{*0}K^{*0}$, which has recently been observed for the first time by LHCb
LHCb-PAPER-2011-012 . Because of the $V-A$ structure of the weak interaction,
the $C\\!P$-even longitudinal polarization component was expected to be
dominant Beneke:2006hg ; Cheng:2009mu ; Ali:2007ff in both this decay and
$B^{0}_{s}\rightarrow\phi\phi$. However, both B-factory measurements in
$b\rightarrow s$ penguin modes Abe:2004mq ; Chen:2005zv ; Aubert:2006uk ;
Aubert:2006fs ; Aubert:2008zza ; delAmoSanchez:2010mz , as well as the recent
LHCb measurements of $B^{0}_{s}\rightarrow\phi\phi$ and $B^{0}_{s}\rightarrow
K^{*0}K^{*0}$, find roughly equal longitudinal and $C\\!P$-odd transverse
polarization components. Proposed explanations have included large penguin
annihilation contributions Kagan:2004uw or final state interactions
Datta:2007qb . The time dependent $C\\!P$ violation measurements in both these
modes should become experimentally accessible in the near future, further
constraining PBSM.
### 4.2 $B^{0}$ mixing
The mixing of $B^{0}$ mesons can be described within the same formalism as
that of $B^{0}_{s}$ mesons, but now it is the width difference
$\Delta\Gamma_{d}$ which is small in the SM while the mixing phase $\phi_{d}$
is large. The most precise measurements of $\Delta m_{d}$ were made by BaBar
Aubert:2005kf and Belle Abe:2004mz , leading to the current world average
$\Delta m_{d}=0.505\pm 0.004$ Nakamura:2010zzi . The mixing phase can also be
expressed as the angle $\beta$ of the Unitarity Triangle, whose most precise
measurement comes from the study of time-dependent $C\\!P$ violation in the
“golden mode” $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and related decays
$\displaystyle\sin(2\beta)$ $\displaystyle=$ $\displaystyle 0.687\pm 0.028\pm
0.012\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Asner:2010qj,Aubert:2009aw}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\sin(2\beta)$ $\displaystyle=$ $\displaystyle 0.667\pm 0.023\pm
0.012\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Adachi:2012et}{\@@citephrase{(}}{\@@citephrase{)}}}},$
The measurement of this angle can be related to the CKM matrix element
$|V_{ub}|$ through the unitarity relation in equation 4, and can be compared
to the value of $\sin(2\beta)$ as determined from a fit to the other
parameters of the Unitarity Triangle Charles:2004jd ; Ciuchini:2000de of
$0.830^{+0.013}_{-0.033}$ and $0.80\pm 0.05$ from the CKMFitter and UTFit
collaborations respectively. This tension is driven by the branching fraction
of the decay $B^{+}\rightarrow\tau^{+}\nu$
$\displaystyle B(B\rightarrow\tau\nu)$ $\displaystyle=$
$\displaystyle(1.80^{+.57}_{-.54}\pm 0.26)\times
10^{-4}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{delAmoSanchez:2010ab}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle B(B\rightarrow\tau\nu)$ $\displaystyle=$
$\displaystyle(1.54^{+.38}_{-.37}{}^{+.29}_{-.31})\times
10^{-4}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hara:2010dk}{\@@citephrase{(}}{\@@citephrase{)}}}},$
which can be transformed into a measurement of $|V_{ub}|$ and hence a
constraint on the apex of the Unitarity Triangle.
Resolving this tension will require a precise understanding of the size of
doubly Cabibbo-suppressed penguin topologies in
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ Faller:2008zc . In this respect it is
interesting to note the observation of the U-spin partner decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ at LHCb LHCb-CONF-2011-048 , which has
been proposed Fleischer:1999nz ; DeBruyn:2010hh as one way of measuring these
effects.
An important additional null-test of the SM comes from the measurement of
$\Delta\Gamma_{d}$. As noted in Gershon:2010wx , the fact that the SM
prediction for $\Delta\Gamma_{d}/\Gamma_{d}$ is so small,
$40.9^{+8.9}_{-9.9}\times 10^{-4}$ Lenz:2006hd , while plausible scenarios of
PBMS exist in which this value is enhanced Lenz:2012az , means that any non-
zero measurement with current experimental sensitivity would be a clear sign
of new physics effects. Indeed, such effects are needed to explain the
anomalous dimuon asymmetry observed by DØ, as discussed in the following
section. Both BaBar and Belle have measured $\Delta\Gamma_{d}$ Aubert:2003hd ;
Aubert:2004xga ; Higuchi:2012kx through fits to the time dependent decay
rates in $B^{0}\rightarrow D^{(*)-}(\pi,\rho,a_{1})^{+}$ and $B^{0}\rightarrow
c\bar{c}K^{0}_{\textrm{S,L}}$ modes. The average is dominated by the recent
Belle result of $\Delta\Gamma_{d}/\Gamma_{d}=[-1.7\pm 1.8\pm 1.1]\times
10^{-2}$. As the uncertainty on this measurement is still an order of
magnitude larger than the SM prediction, it remains to be seen if the
systematic uncertainties can be kept under control in the era of the next
generation flavour factories.
### 4.3 Semileptonic asymmetries
The mixing induced semileptonic asymmetry $A_{\textrm{sl}}$ is predicted to be
$O(10^{-4})$ in the SM within both the $B^{0}$ $(a^{d}_{\textrm{sl}})$ and
$B^{0}_{s}$ $(a^{s}_{\textrm{sl}})$ meson systems Lenz:2011ti . The most
precise experimental measurement to date was made by the DØ Collaboration
Abazov:2011yk , which found a percent-level $C\\!P$ asymmetry
$\displaystyle A_{\textrm{sl}}$ $\displaystyle\approx$ $\displaystyle
0.6\times a^{d}_{\textrm{sl}}+0.4\times a^{s}_{\textrm{sl}},$ $\displaystyle
A_{\textrm{sl}}$ $\displaystyle=$ $\displaystyle(-0.787\pm 0.172\pm
0.093)\%\;.$
Because DØ cannot distinguish between dimuon pairs coming from $B^{0}$ and
$B^{0}_{s}$ decays, it measures a combination of the two semileptonic
asymmetries. In the same paper, the collaboration attempts to separate effects
caused by $B^{0}_{s}$ oscillations from those caused by $B^{0}$ oscillations
by indirectly studying the lifetime of the decaying $B$ meson, and concludes
that the asymmetry is largest at short lifetimes. The authors take this as a
hint that the asymmetry is dominated by $B^{0}_{s}$ decays because the
$B^{0}_{s}$ meson oscillates much more quickly than the $B^{0}$.
When interpreting this result, it is important to keep in mind that the
background levels are also highest at short lifetimes; for this reason, it is
critical that $(a^{d}_{\textrm{sl}})$ and $(a^{s}_{\textrm{sl}})$ are measured
separately in a low background environment where the decaying $B$ meson can be
unambiguously tagged as a $B^{0}$ or $B^{0}_{s}$. Nevertheless, taking the DØ
result at face value, it is not trivial to reconcile it with the measurements
of $B^{0}_{s}$ and $B^{0}$ mixing mentioned earlier. An easy way of seeing
this is to consider why, if the dimuon asymmetry is driven by $B^{0}_{s}$
mixing, the mixing phase in
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ is
so close to the SM value while the direct and indirect measurements of
$\sin\left(2\beta\right)$ are in tension. One proposed explanation requires
Lenz:2012az contributions from PBSM to both $M^{d,s}_{\textrm{12}}$ and
$\Gamma^{d,s}_{\textrm{12}}$
### 4.4 $B\rightarrow h\gamma$ decays
CP asymmetry measurements of $b\rightarrow s\gamma$ transitions are sensitive
to PBSM, for instance through measurements of the photon polarisation which
probes models involving right-handed currents Gershon:2006mt ;
DescotesGenon:2011yn ; Mahmoudi:2006wq ; Altmannshofer:2011gn . $C\\!P$
asymmetries in $b\rightarrow s\gamma$ transitions have been measured by BaBar
Aubert:2008gy ; Aubert:2008be ; Aubert:2009ak , Belle Nishida:2003yw ;
Ushiroda:2006fi , and LHCb LHCb-CONF-2012-004 and the results are consistent
with SM expectations and statistically limited. In this context, it has been
recently noted Benzke:2010tq that the difference in $C\\!P$ asymmetries
between the inclusive processes $X_{s}^{+}\gamma$ and $X_{s}^{0}\gamma$ offers
a cleaner probe of PBSM than either measurement taken on its own.
Thanks to the large value of $\Delta\Gamma_{s}$, the $B_{s}$ system is
particularly promising for measuring the photon polarisation by studying time
dependent $C\\!P$ violation in the decay $B_{s}\rightarrow\phi\gamma$
Muheim:2008vu . This mode was first observed at Belle Wicht:2007ni , while the
LHCb collaboration has recently measured LHCb:2012ab the ratio of the
branching ratios $\frac{{\cal B}(B^{0}\rightarrow
K^{*}\gamma)}{B_{s}\rightarrow\phi\gamma)}=1.12\pm 0.08^{+0.11}_{-0.09}$.
### 4.5 The CKM angle $\gamma$
A precise determination of the angle $\gamma$ of the CKM unitarity triangle is
important in order to further overconstrain the position of the triangle’s
apex, in particular with respect to the previously discussed measurements of
$\sin\left(2\beta\right)$ and $V_{ub}$. In this respect $\gamma$ can be
measured either from tree-level or loop-mediated processes, and a comparison
of the two kinds of measurements provides another opportunity for PBSM to
manifest itself. In either case, $\gamma$ is experimentally determined from a
measurement of $C\\!P$ violation in those $B$ meson decays where diagrams
involving $|V_{ub}|$ and $|V_{cb}|$ result in the same final state.
The determination of $\gamma$ from tree-level decays is one of the most
sensitive tests of the SM precisely because the associated theoretical
uncertainties are confined to electroweak corrections associated with box-
diagram decays, and are at the level of $\delta\gamma/\gamma\approx 10^{-6}$
Zupan:2011mn . Experimentally the challenge is that the sensitivity to
$\gamma$ comes from the interference of $|V_{ub}|$ and $|V_{cb}|$ diagrams,
which means that the final state must be carefully chosen in order to make the
amplitudes of similar size and hence maximize the interference. Unfortunately
those modes which have the highest interference also have the biggest
associated experimental difficulties, whether it be low overall branching
ratios, difficult to reconstruct final state particles, or the requirement for
a time-dependent analysis. This means that the ultimate precision on $\gamma$
can only be achieved by combining several different measurements.
The current sensitivity on $\gamma$ is dominated by measurements of $C\\!P$
violation and partial widths in $B^{+}\rightarrow D^{0}K^{+}$ decays, in which
the $D^{0}$ then decays to either a $C\\!P$-eigenstate Gronau1991483 ;
Gronau1991172 , a doubly-Cabibbo suppressed decay mode Atwood:1996ci ;
Atwood:2000ck , or a multibody decay whose Dalitz distribution gives rise to
interference effects Giri:2003bs . These are known as the GLW, ADS, and GGSZ
methods respectively after their inventors.
In the first two cases the charge-averaged partial width ratios of the
$D^{0}K^{+}$ and $D^{0}\pi^{+}$ decays are measured,
$R^{f}_{K/\pi}=\frac{\Gamma(B\rightarrow[f]_{D}K)}{\Gamma(B\rightarrow[f]_{D}\pi)},$
(9)
where $f$ represents the $C\\!P$-eigenstate $\pi\pi$ and $KK$ decays and the
Cabibbo-favoured $K\pi$ decay mode; the $C\\!P$ asymmetries
$A^{f}_{h}=\frac{\Gamma(B^{+}\rightarrow[f]_{D}h^{+})-\Gamma(B^{-}\rightarrow[f]_{D}h^{-})}{\Gamma(B^{+}\rightarrow[f]_{D}h^{+})+\Gamma(B^{-}\rightarrow[f]_{D}h^{-})},$
(10)
where $h$ is a pion or a kaon; and the charge-separated partial width ratios
of the Cabibbo-favoured and doubly Cabibbo-suppressed $B^{+}\rightarrow
D^{0}K^{+}$ decay modes
$R^{\pm}_{h}=\frac{\Gamma(B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}h^{\pm})}.$
(11)
As these are the most experimentally accessible modes for measuring $\gamma$,
they have been studied at BaBar delAmoSanchez:2010dz ; delAmoSanchez:2010ji ,
Belle Abe:2006hc ; Belle:2011ac ,CDF Aaltonen:2009hz ; Aaltonen:2011uu , and
recently at LHCb Aaij:2012kz . In particular, LHCb has observed the doubly
Cabibbo-suppressed decay $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$
with $10\sigma$ significance, and has made a $5.8\sigma$ observation of
$C\\!P$-violation in $B^{+}\rightarrow D^{0}K^{+}$ decays. It is worth
highlighting the cleanliness of the LHCb signals, as seen in Fig. 4, as well
as the intriguing hint of $C\\!P$-violation in the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}\pi^{\pm}$ which can be seen in the
same picture.
Figure 3: Invariant mass distribution of selected
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$ candidates. The left plots
are $B^{-}$, the right plots are $B^{+}$. Top are $h=K$ and bottom are
$h=\pi$. The red curve is the signal, the shaded area, green, and magenta
curves are backgrounds. Reproduced from Aaij:2012kz .
In the third case, what is measured are the different Dalitz plot
distributions of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}hh$ in
$B^{+}\rightarrow D^{0}K^{+}$ and $B^{-}\rightarrow D^{0}K^{-}$ decays, and
measurements have been made with delAmoSanchez:2010rq ; Poluektov:2010wz or
without Belle:2011ab assuming an amplitude model for the $D^{0}$ decay. The
advantage of this method is that it only suffers from a two-fold ambiguity in
the measured value of $\gamma$, as opposed to the eightfold ambiguity in e.g.
the GLW method.
The average value of $\gamma$ from these decay modes, as computed by the
CKMFittter collaboration, is shown in Fig. 4, from which it is apparent that
while direct measurements of $\gamma$ agree well with its indirect
determination from other Unitarity Triangle parameters, they are not yet
strongly constraining the apex of the triangle. A historical tension exists
between the frequentist (CKMFitter) and Bayesian (UTFit) averages of $\gamma$,
driven by the different treatment of the nuissance parameters which
parametarize the size of the interference in each decay mode. The most up-to-
date averages from the two collaborations are
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle(66\pm 12)^{\circ}$
CKMFitter $\displaystyle\;\textrm{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Charles:2004jd}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle(76\pm 9)^{\circ}$ UTFit
$\displaystyle\;\textrm{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ciuchini:2000de}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$
where the CKMFitter average includes the most recent ADS/GLW results from LHCb
and the UTFit average does not. The larger uncertainty in the CKMFitter
average comes from the treatment of the nuissance parameters, while there is
an interesting discrepancy developing in the central values which is not
understood at present.
Figure 4: Averaged constraints on $\gamma$ from direct measurements.
Reproduced from Charles:2004jd .
Many other tree-level determinations of $\gamma$ are possible, for example
from $B^{0}\rightarrow D^{0}h^{+}h^{-}$ decays Gronau1991483 ; Gronau1991172 ;
Atwood:1996ci ; Atwood:2000ck ; Gershon:2008pe ; Gershon:2009qc whether in a
quasi-two-body approach, selecting the $h^{+}h^{-}$ mass to lie at a
particular resonance, or through an amplitude analysis. An important milestone
on this road to $\gamma$ is the first observation of the decay mode
$B^{0}_{s}\rightarrow D^{0}K^{*0}$ at LHCb Aaij:2011tz . It is also possible
to make an unambiguous measurement of $\gamma$ through the study of $C\\!P$
violation in the interference of $B^{0}_{s}$ mixing and the decay
$B^{0}_{s}\rightarrow D^{\pm}_{s}K^{\mp}$ FleischerDsK , whose branching ratio
has recently been precisely measured LHCb-CONF-2011-057 . Within measurements
of $\gamma$ from loop-mediated processes, the study of two body
$B_{s,d}\rightarrow h^{+}h^{-}$ decays stands out. The U-spin partner decays
$B^{0}_{s}\rightarrow K^{+}K^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$ are
able to extract $\gamma$ unambiguously in a combined analysis Fleischer:1999pa
; Fleischer:2007hj , and recently the time-dependent $C\\!P$ asymmetry in
$B^{0}_{s}\rightarrow K^{+}K^{-}$ has been measured LHCb-CONF-2012-007 for
the first time
$\displaystyle A^{\textrm{dir}}_{KK}$ $\displaystyle=$ $\displaystyle 0.02\pm
0.18\pm 0.04\;,$ $\displaystyle A^{\textrm{mix}}_{KK}$ $\displaystyle=$
$\displaystyle 0.17\pm 0.18\pm 0.05\;,$
to add to the existing Aubert:2008sb ; Ishino:2006if measurements in
$B^{0}\rightarrow\pi^{+}\pi^{-}$.
## 5 Rare Decays
Rare decays which proceed via Flavour Changing Neutral Currents (FCNC) are
induced by one-loop diagrams in the SM and are excellent probes for PBSM. New
particles can enter in competing loop-order diagrams, resulting in large
deviations from SM predictions. In general, an effective hamiltonian formalism
is used to describe the amplitudes of FCNC processes, according to the
formula:
$H_{eff}=\frac{G_{F}}{\sqrt{2}}\sum_{i}V_{CKM}^{i}C_{i}(\mu)Q_{i}\;,$ (12)
where $V^{i}_{CKM}$ are the relevant factors of the CKM matrix; $Q_{i}$ are
local operators; $C_{i}$ are the corresponding couplings (Wilson
coefficients); and $\mu$ is the QCD renormalization scale. The correlation of
different channels, where common Wilson coefficients contribute, is a powerful
tool for searching and understanding the structure of PBSM.
This approach is complementary to direct searches for PBSM. Moreover indirect
searches often allow to set more stringent constraints than direct ones. For
instance, strong lower bounds on the mass of the charged Higgs in Two-Higgs-
Doublets-Models of type II have been obtained from the analysis of
$\overline{B}\rightarrow X\gamma$ decays, where the SM prediction
Misiak:2006zs is found in agreement with inclusive measurements performed by
the experiments BaBar Aubert:2006gg ; Aubert:2005cba ; Aubert:2007my , Belle
Abe:2001hk ; Limosani:2009qg and CLEO Chen:2001fja (other bounds from
$B\rightarrow X_{s}\gamma$ are discussed in Buras:2011zb and the references
therein).
As a result of the many measurements performed by the B-factories and more
recently by the CDF experiment,our knowledge of suppressed processes has
considerably improved in the last decade. Consequently, constraints on PBSM
have become much stronger.
While inclusive measurements are challenging at hadron colliders, studies of
exclusive decays are competitive with $e^{+}e^{-}$ machines. Moreover, hadron
colliders have the advantage that all B-hadron species are produced. With the
start-up of the LHCb experiment a new round in the precision measurements of
rare decays has begun.
### 5.1 $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ decays
Purely leptonic decays of B-mesons are a key ingredient in the search for
PBSM, since the prediction of their branching fractions is largely free from
hadronic uncertainties. The two decays $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ have
a clear experimental signature and are easier to reconstruct and identify than
the other leptonic decays of B-mesons. Their branching fractions are predicted
to be ${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})=(3.2\pm 0.2)\times 10^{-9}$
and ${\cal B}(B_{d}\rightarrow\mu^{+}\mu^{-})=(1.0\pm 0.1)\times 10^{-10}$ in
the SM Buras:2010mh ; Buras:2010wr . Contributions from PBSM, especially in
models with an extended Higgs sector, can enhance these branching fractions.
For instance, in the Minimal Supersymmetric extension of the SM the branching
fraction of the decay $B_{s}\rightarrow\mu^{+}\mu^{-}$ is proportional to the
sixth power of $\tan\beta$ (the ratio of the vacuum expectation values of the
neutral components of the Higgs fields $H_{u}$ and $H_{d}$) Hurth:2008jc .
This fact makes this observable particularly sensitive to supersymmetric
models with large $\tan\beta$. More generally measurements of this branching
fractions probe the Wilson coefficients $C_{s}$ and $C_{p}$, which are
negligibly small in the SM. Present measurements of ${\cal
B}(B_{s}\rightarrow\mu^{+}\mu^{-})$ are shown in Fig. 5.
Figure 5: Present limits on ${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})$ at
$95\%$ CL set by the experiments D0 Abazov:2010fs CDF CDF-Bsmumu , ATLAS
ATLAS-CONF-2012-010 CMS Chatrchyan:2012rg and LHCb Aaij:2012ac . The SM
prediction is indicated by the blue-dashed line.
Presently, the most stringent upper limits on ${\cal
B}(B_{s,d}\rightarrow\mu^{+}\mu^{-})$ are set by the LHCb experiment
Aaij:2012ac . This analysis profits from the good momentum resolution and the
good particle identification performances of LHCb to reject the different
sources of background. The branching fraction of the signal was extracted by
using the three normalization channels: $B^{+}\rightarrow J/\psi K^{+}$,
$B^{0}\rightarrow K^{+}\pi^{-}$ and $B_{s}\rightarrow J/\psi\phi$. For the
first two of these channels, the ratio of the hadronization fractions
$\frac{f_{s}}{f_{d}}$ is needed111Isospin symmetry, i.e. $f_{u}=f_{d}$, has
been assumed.. This variable was measured at LHCb by combining measurements
with semi-leptonic and hadronic decays Fleischer:2010ay :
$f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$ Aaij:2011jp ; Aaij:2011hi . The
uncertainty on this parameter is, in the long run, a limiting systematic
uncertainty for discriminating between SM and BSM contributions in the
$B_{s}\rightarrow\mu^{+}\mu^{-}$ decay, as well as for the measurement of the
golden ratio $\frac{{\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})}{{\cal
B}(B_{d}\rightarrow\mu^{+}\mu^{-})}$ Buras:2010pi . The correlation between
the branching fractions of the decays $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ is
shown in Fig. 6 for several beyond SM scenarios.
Figure 6: Correlation for the branching fractions of the decays
$B_{s}\rightarrow\mu^{+}\mu^{-}$ and $B_{d}\rightarrow\mu^{+}\mu^{-}$ for
several models of PBSM. Details on the models can be found in Straub:2010ih .
The recent upper limits by LHCb are shown by the shaded region. Reproduced
from Straubb-Moriond .
The upper limits set by LHCb for the $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ decays
are: ${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})<4.5\times 10^{-9}$ and ${\cal
B}(B_{d}\rightarrow\mu^{+}\mu^{-})<1.05\times 10^{-9}$ at $95\%$ CL and are
illustrated in Fig. 6 by the shaded region. These measurements are in
agreement with SM expectations and give additional constraints for PBSM with
respect to those provided by $b\rightarrow s\gamma$ and other $b\rightarrow
sl^{+}l^{-}$ transitions.
### 5.2 $B\rightarrow h\mu^{+}\mu^{-}$ decays
In the decay $B_{d}\rightarrow K^{*}\mu^{+}\mu^{-}$ several angular
observables can be built which are sensitive to PBSM, and for which form
factor uncertainties are theoretically under control, (see for example
Ali:1991is ; Altmannshofer:2008dz and references therein). These observables
include the forward-backward asymmetry of the dimuon system,
$A_{\mathrm{FB}}$, the fraction of $K^{*}$ longitudinal polarization, $F_{L}$,
the transverse asymmetry, $S_{3}$ Altmannshofer:2008dz (often referred to as
$\frac{1}{2}(1-F_{L})A_{T}^{2}$ in the literature Kruger:2005ep ), and the
T-odd CP asymmetry $A_{\textrm{Im}}$ Bobeth:2008ij . They can be extracted by
performing an angular analysis as a function of the dimuon invariant mass
squared, $q^{2}$, with respect to the following angles: the angle $\theta_{l}$
between the $\mu^{+}$ ($\mu^{-}$) and the $B^{0}$ ($\overline{B}^{0}$) in the
dimuon rest frame; the angle $\theta_{K}$ between the kaon and $B^{0}$ in the
$K^{*}$ rest frame; and the angle $\phi$ between the planes of the dimuon
system and the plane of the $K^{*}$. A formal definition of these angles can
be found in Egede:1048970 . Present measurements of the observables
$A_{\mathrm{FB}}$, $F_{L}$, $S_{3}$ and $A_{\textrm{Im}}$ are shown in Fig. 7.
These measurements provide information about the Wilson coefficients $C_{7}$,
$C_{9}$ and $C_{10}$ and on their right-handed counterparts.
The LHCb experiment has recently made the world’s best measurements on these
angular observables LHCb-CONF-2012-008 . The physics parameters were extracted
by fitting the partial decay rate as a function of the three angles for
different bins in $q^{2}$. In order to reduce the number of parameters in the
fit, due to the small size of the data sample, the angle $\phi$ was folded by
taking $\phi\rightarrow\phi+\pi$ when $\phi<0$. This transformation cancels
out the terms containing $cos\phi$ and $sin\phi$ in the differential decay
rate. This strategy is different from that followed by other experiments,
where only projections of the angular distributions were used.
Figure 7: The $A_{\mathrm{FB}}$, $F_{L}$, $S_{3}$ and $A_{\textrm{Im}}$
measured by the experiments BaBar BaBarLakeLouise , Belle Wei:2009zv , CDF
Aaltonen:2011ja and LHCb LHCb-CONF-2012-008 . The comparison with the SM
prediction, taken from Bobeth:2011gi is also shown. Reproduced from LHCb-
CONF-2012-008 . Figure 8: The $A_{\mathrm{FB}}$ as a function of $q^{2}$
extracted from an unbinned counting experiment. The shaded region correspond
to the $68\%$ CL of the zero-crossing point. Comparison with the SM prediction
Bobeth:2011gi is shown. Reproduced from LHCb-CONF-2012-008 .
The so called zero-crossing point, where $A_{\mathrm{FB}}$ changes sign, is
largely free from form factor uncertainties and sensitive to PBSM Ali:1991is .
The SM predicts this point to be in the range 4.0-4.3 GeV2/c4 Bobeth:2011nj ;
Beneke:2004dp ; Ali:2006ew . The zero-crossing point of $A_{\mathrm{FB}}$ was
measured for the first time by LHCb to be $q_{0}^{2}=4.9^{+1.1}_{-1.3}$GeV2/c4
LHCb-CONF-2012-008 . This observable was extracted in an unbinned counting
experiment with respect to $q^{2}$, integrating the angular distributions with
respect to the three angles Jansen:1156131 . The result is shown in Fig. 8.
Other exclusive $b\rightarrow sll$ processes have been measured by the
B-factories, CDF and LHCb. The measurements of the differential branching
fractions of the decays $\Lambda_{b}\rightarrow\Lambda\mu^{+}\mu^{-}$
Aaltonen:2011qs , $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ Aaltonen:2011qs ;
Wei:2009zv ; BaBarLakeLouise , $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$
Aaltonen:2011qs , $B^{0}\rightarrow K_{S}\mu^{+}\mu^{-}$ Aaltonen:2011qs ;
Wei:2009zv ; BaBarLakeLouise and $B^{+}\rightarrow K^{*+}\mu^{+}\mu^{-}$
Aaltonen:2011qs ; Wei:2009zv ; BaBarLakeLouise and the $A_{FB}$ for the
decays $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ Aaltonen:2011ja ; Wei:2009zv ;
BaBarLakeLouise and $B^{+}\rightarrow K^{*+}\mu^{+}\mu^{-}$ Aaltonen:2011ja ;
Wei:2009zv ; BaBarLakeLouise were found to be in agreement with SM
predictions.
Another observable which is potentially sensitive to PBSM is the isospin
asymmetry, $A_{I}$, defined as:
$A_{I}=\frac{{\cal B}(B^{0}\rightarrow
K^{(*)0}l^{+}l^{-})-\frac{\tau_{0}}{\tau_{+}}{\cal B}(B^{\pm}\rightarrow
K^{(*)\pm}l^{+}l^{-})}{{\cal B}(B^{0}\rightarrow
K^{(*)0}l^{+}l^{-})+\frac{\tau_{0}}{\tau_{+}}{\cal B}(B^{\pm}\rightarrow
K^{(*)\pm}l^{+}l^{-})},$ (13)
where $l=(e,\mu)$. This observable was measured by the experiments Belle
Wei:2009zv , BaBar BaBarLakeLouise and CDF Aaltonen:2011ja . The results are
shown in Fig. 9.
Figure 9: Isospin asymmetry for the decays $B\rightarrow K^{(*)}l^{+}l^{-}$,
measured by the experiments BaBar BaBarLakeLouise , Belle Wei:2009zv (with
electrons and muons) and CDF Aaltonen:2011ja (with muons).
The SM predicts a small asymmetry in all $q^{2}$ bins. Present results seem to
hint at a non-zero $A_{I}$ in some $q^{2}$ bins. Measurements with larger data
samples and good control of the theoretical uncertainties are important; a
measurement of $A_{I}$ from LHCb can be expected in the near future.
Recently, the LHCb collaboration reported the first observation of a
$b\rightarrow dll$ transition, by measuring the branching fraction ${\cal
B}(B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-})=(2.4\pm 0.6\pm 0.2)\times 10^{-8}$
LHCb-CONF-2012-006 . The invariant mass distribution of
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates is shown in Fig. 10.
Figure 10: The invariant mass distribution of
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates. Reproduced from LHCb-
CONF-2012-006 .
This process is further suppressed by the factor $|V_{td}/V_{ts}|$, with
respect to the $b\rightarrow sll$ transitions. The measured branching fraction
is in good agreement with the SM expectation.
### 5.3 Search for Lepton Flavour Violating and very rare decays
The search for Lepton Flavour Violating (LFV) decays is a crucial way to test
the SM flavour structure. These searches have been performed by several
experiments. A complete discussion of LFV searches goes beyond the scope of
this review.
Decays of the type $B^{+}\rightarrow h^{-}l^{+}l^{+}$, where $h^{-}$ is a
meson, can be considered the analogues of neutrinoless double $\beta$ decays
and can be used to search for heavy Majorana neutrinos Atre:2009rg ;
Zhang:2010um ; Boyarsky:2006fg ; Ilakovac:1999md . These searches have been
performed by the LHCb Aaij:2012zr ; Aaij:2011ex , BaBar BABAR:2012aa , Belle
Seon:2011ni and CLEO Edwards:2002kq experiments. Upper limits for these
decays are summarised in Table 1. Heavy Majorana neutrinos can also be
searched for by using the corresponding charm decays $D^{+}\rightarrow
h^{-}l^{+}l^{+}$. Constraints on these decays are expected to improve
substantially with measurements from LHCb.
LFV decays of charged leptons are allowed in several extensions of the SM, for
instance supersymmetric models Dedes:2002rh ; Ciuchini:2007ha ; Arganda:2005ji
, left-right symmetric models Akeroyd:2006bb and models with heavy neutrinos
Atre:2009rg ; Zhang:2010um ; Boyarsky:2006fg ; Ilakovac:1999md . Stringent
upper limits on the decay $\mu^{-}\rightarrow e^{-}\gamma$ have been set by
the MEG experiment Adam:2011ch , while the most stringent upper limits on
$\tau^{-}\rightarrow l^{-}l^{+}l^{-}$ were set by the Belle experiment
Hayasaka:2010np .
In addition, searches for exotic very rare decays have been carried out at
LHCb. Upper limits for the decays
$B_{d,s}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ and
$D^{0}\rightarrow\mu^{+}\mu^{-}$ were recently set LHCb-CONF-2012-010 ; LHCb-
CONF-2012-005 and are listed in Table 1.
For the moment no hint of the existence of any of such processes has been
observed, and all searches are statistically limited at present.
Channel | Upper Limit (CL) | Reference
---|---|---
${\cal B}(B^{+}\rightarrow K^{-}\mu^{+}\mu^{+})$ | $5.4\times 10^{-8}$ ($95\%$) | LHCb Aaij:2011ex
${\cal B}(B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+})$ | $1.3\times 10^{-8}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow\pi^{-}e^{+}e^{+})$ | $2.3\times 10^{-8}$ ($90\%$) | BaBar BABAR:2012aa
${\cal B}(B^{+}\rightarrow K^{-}e^{+}e^{+})$ | $3.0\times 10^{-8}$ ($90\%$) | BaBar BABAR:2012aa
${\cal B}(B^{+}\rightarrow D^{-}\mu^{+}\mu^{+})$ | $6.9\times 10^{-7}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow D^{*-}\mu^{+}\mu^{+})$ | $2.8\times 10^{-6}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow D^{-}e^{+}e^{+})$ | $2.6\times 10^{-6}$ ($90\%$) | Belle Seon:2011ni
${\cal B}(B^{+}\rightarrow D^{-}\mu^{+}e^{+})$ | $1.8\times 10^{-6}$ ($90\%$) | Belle Seon:2011ni
${\cal B}(B^{+}\rightarrow D_{s}^{-}\mu^{+}\mu^{+})$ | $5.8\times 10^{-7}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow D^{0}\pi^{-}\mu^{+}\mu^{+})$ | $1.5\times 10^{-6}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(D^{0}\rightarrow\mu^{+}\mu^{+})$ | $1.3\times 10^{-8}$ ($95\%$) | LHCb LHCb-CONF-2012-005
${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-})$ | $1.3\times 10^{-8}$ ($95\%$) | LHCb LHCb-CONF-2012-010
${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-})$ | $5.4\times 10^{-9}$ ($95\%$) | LHCb LHCb-CONF-2012-010
${\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$ | $2.1\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow e^{-}e^{+}e^{-})$ | $2.7\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow e^{-}\mu^{+}\mu^{-})$ | $2.7\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow e^{+}\mu^{-}\mu^{-})$ | $1.7\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow\mu^{+}e^{-}e^{-})$ | $1.5\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow\mu^{-}e^{+}e^{-})$ | $1.8\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\mu^{-}\rightarrow e^{-}\gamma)$ | $2.4\times 10^{-12}$ ($90\%$) | MEG Adam:2011ch
Table 1: Upper Limit for several very rare or forbidden decays.
## 6 Conclusion
Despite the ongoing lack of a direct discovery of particles beyond the
Standard Model, recent results in flavour physics are giving ever stronger
hints of effects beyond the Standard Model. In particular, the observation of
permille-level $C\\!P$ violation in $D^{0}$ decays, the large dimuon asymmetry
in $B^{0}$ and $B^{0}_{s}$ decays, as well as the values of
$\sin\left(2\beta\right)$ and the branching ratio of
$B^{+}\rightarrow\tau^{+}\nu$, are difficult to simultaneously interpret
within the Standard Model framework. At the same time, measurements of rare
decays such as $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ which are in good agreement with the Standard Model have
placed the most stringent limits yet on many Standard Model extensions. What
this contradiction highlights is the ongoing relevance of flavour physics as
key tool not only for the indirect discovery of new particles and processes,
but also for discriminating between the many proposed theories of physics
beyond the Standard Model. With the excellent performance of the LHC, and the
wealth of precision flavour measurements coming from its detectors, it is
reasonable to hope for a deepening, and eventual resolution, of these
contradictions in the years to come.
###### Acknowledgements.
The authors would like to thank the CKMFitter, HFAG, and UTFit collaborations
for preparing updated world averages of many flavour parameters in time for
this review, and Mitesh Patel for useful comments on the draft.
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|
arxiv-papers
| 2012-04-24T06:24:34 |
2024-09-04T02:49:30.085369
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Gersabeck and Vladimir V. Gligorov and Nicola Serra",
"submitter": "Vladimir Vava Gligorov",
"url": "https://arxiv.org/abs/1204.5273"
}
|
1204.5326
|
# Rare B decays in the $\cal F$-$SU(5)$ Model
Tianjun Li State Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China
George P. and Cynthia W. Mitchell Institute for Fundamental Physics and
Astronomy, Texas A$\&$M University, College Station, TX 77843, USA Dimitri V.
Nanopoulos George P. and Cynthia W. Mitchell Institute for Fundamental
Physics and Astronomy, Texas A$\&$M University, College Station, TX 77843, USA
Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell
Campus, Woodlands, TX 77381, USA Academy of Athens, Division of Natural
Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece Wenyu Wang Institute
of Theoretical Physics, Beijing University of Technology, Beijing 100124,
China Xiao-Chuan Wang Institute of Theoretical Physics, Beijing University
of Technology, Beijing 100124, China Zhao-Hua Xiong Institute of Theoretical
Physics, Beijing University of Technology, Beijing 100124, China
###### Abstract
In the testable Flipped $SU(5)\times U(1)_{X}$ model with TeV-scale vector-
like particles from F-theory model building dubbed as the $\cal F$-$SU(5)$
model, we study the vector-like quark contributions to B physics processes,
including the quark mass spectra, Feynman rules, new operators and Wilson
coefficients, etc. We focus on the implications of the vector-like quark mass
scale on B physics. We find that there exists the $\bar{s}bZ$ interaction at
tree level, and the Yukawa interactions are changed. Interestingly, different
from many previous models, the effects of vector-like quarks on rare B decays
such as $B\to X_{s}\gamma$ and $B\to X_{s}\ell^{+}\ell^{-}$ do not decouple in
some viable parameter space, especially when the vector-like quark masses are
comparable to the charged Higgs boson mass. Under the constraints from $B\to
X_{s}\gamma$ and $\ B\to X_{s}\ell^{+}\ell^{-}$, the latest measurement for
$B_{s}\to\mu^{+}\mu^{-}$ can be explained naturally, and the branching ratio
of $B_{s}\to\ell^{+}\ell^{-}\gamma$ can be up to $(4\sim 5)\times 10^{-8}$.
The non-decouling effects are much more predictable and thus the $\cal
F$-$SU(5)$ model may be tested in the near future experiments.
###### pacs:
12.15.-g, 12.15.Lk, 12,15.Ff, 14.20.Mr, 12.39.-x
††preprint: ACT-05-12, MIFPA-12-16
## I Introduction
Supersymmetry provides a natural solution to the gauge hierarchy problem in
the Standard Model (SM). In the supersymmetric SM (SSM) with R-parity under
which the SM particles are even while the supersymmetric particles
(sparticles) are odd, the $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ gauge
couplings can be unified around $2\times 10^{16}$ GeV Langacker:1991an , the
lightest supersymmetric particle (LSP) such as the neutralino can be a cold
dark matter candidate Ellis:1983ew ; Goldberg:1983nd , and the electroweak
(EW) precision constraints can be evaded, etc. Especially, the gauge coupling
unification strongly suggests Grand Unified Theories (GUTs). However, in the
supersymmetric $SU(5)$ models, there exist the doublet-triplet splitting
problem and dimension-five proton decay problem. Interestingly, these problems
can be solved elegantly in the Flipped $SU(5)\times U(1)_{X}$ models smbarr ;
dimitri ; AEHN-0 via missing partner mechanism AEHN-0 . On the other hand,
string theory is the most promising candidate for quantum gravity, and it can
unify all the fundamental interactions in the Nature. However, the string
scale is at least one-order larger than the conventional GUT scale.
To solve the little hierarchy problem between the traditional GUT scale and
string scale, two of us (TL and DVN) with Jing Jiang have proposed the
testable Flipped $SU(5)\times U(1)_{X}$ models, where the TeV-scale vector-
like particles are introduced Jiang:2006hf . Such kind of models can be
constructed from the free fermionic string constructions at the Kac-Moody
level one Antoniadis:1988tt ; Lopez:1992kg and locally from the F-theory
model building Beasley:2008dc ; Jiang:2009zza , and is dubbed as ${\cal
F}$-$SU(5)$ Jiang:2009zza . In particular, these models are very interesting
from the phenomenological point of view Jiang:2009zza : the vector-like
particles can be observed at the Large Hadron Collider (LHC), proton decay is
within the reach of the future Hyper-Kamiokande Nakamura:2003hk and Deep
Underground Science and Engineering Laboratory (DUSEL) DUSEL experiments
Li:2009fq ; Li:2010dp , the hybrid inflation can be naturally realized, the
correct cosmic primodial density fluctuations can be generated Kyae:2005nv ,
and the lightest CP-even Higgs boson mass can be lifted Huo:2011zt ; Li:2011ab
. With no-scale boundary conditions at $SU(5)\times U(1)_{X}$ unification
scale Cremmer:1983bf , two of us (TL and DVN) with James Maxin and Joel Walker
have described an extraordinarily constrained “golden point” Li:2010ws and
“golden strip” Li:2010mi that satisfied all the latest experimental
constraints and has an imminently observable proton decay rate Li:2009fq . For
a review of the recent progresses, see Ref. Li:2012uj .
Interestingly, the vector-like quarks in the $\cal F$-$SU(5)$ model predict
rich phenomenology on low energy processes. If the model is treated seriously,
constraints from electroweak parameters such as $U,\ S,T$ and $R_{b},\ R_{c}$
and B processes should be taken into account. We also would like to point out
that the $\cal F$-$SU(5)$ model has no Landau pole problem and then is very
different from the other simple SM extensions in quark sector (also see the
next Section) BNPfourth , and the $3\times 3$ SM-like quark mixing matrix is
now replaced by a $5\times 5$ one and then is no longer unitary, and there
exists the tree-level $\bar{s}bZ$ interaction, which will play an important,
even dominant, role in some parameter space for rare B decays.
Thanks to the efforts of the B factories and LHC, the exploration of quark-
flavor mixing is now entering a new interesting era. It is well known that the
rare B decays induced by the flavor changing neutral current (FCNC) only occur
at loop level in the SM and then are sensitive to new physics. Thus, the rare
radiative, leptonic and semi-leptonic B meson decays are valuable in testing
the SM at loop level and probe new physics. On the theoretical side, the rare
B inclusive radiative decays $B\to X_{s}\gamma$ and $B\to
X_{s}\ell^{+}\ell^{-}(\ell=e,\mu)$ as well as the exclusive decays
$B_{s}\to\mu^{+}\mu^{-}$ and $B_{s}\to\ell^{+}\ell^{-}\gamma$ have been
studied extensively at the leading logarithm order (LO) BLOSM and high order
in the SM BHOSM and various new physics models BNPfourth ; BNPMSSM ; BNP2HDM
. On the experimental side, $B\to X_{s}\gamma$ and $B\to
X_{s}\ell^{+}\ell^{-}(\ell=e,\mu)$ have been measured and the latest upper
bound on $B_{s}\to\mu^{+}\mu^{-}$ is achieved Bmeasured . By comparing the
predictions with experimental measurements, we will present some constraints
on the parameter space in the $\cal F$-$SU(5)$ model.
The first task of this work will be deriving the quark mass spectra and
Feynman rules. We stress that the Feynman rules which not be presented in
previous studies are used not only in B physics but also in research of all
low energy processes. B physics constraints on the model is the second task of
this work, we will concentrate our attention on the vector-like quark
contributions to B physics, in particular, the contributions from the new
operators induced by tree-level FCNC. We will show that the $\bar{s}bZ$
interaction can be generated at tree level, and the Yukawa interactions are
changed, new operators $O_{9}^{\prime}$ and $O_{10}^{\prime}$ in effective
Hamiltonian should be introduced. We will demonstrate that different from many
previous models, the effects of vector-like quarks on rare B decays such as
$B\to X_{s}\gamma$ and $B\to X_{s}\ell^{+}\ell^{-}$ do not decouple in some
allowed parameter space, especially when the vector-like quark masses are
comparable to the charged Higgs boson mass. Within the constraints from $B\to
X_{s}\gamma$ and $\ B\to X_{s}\ell^{+}\ell^{-}$, and the latest measurement
for $B_{s}\to\mu^{+}\mu^{-}$ will be explained naturally, and the branching
ratio of $B_{s}\to\ell^{+}\ell^{-}\gamma$ can be up to $(4\sim 5)\times
10^{-8}$. Because the non-decouling effects are very predictable, the $\cal
F$-$SU(5)$ model may be tested in the near future experiments.
This paper is organized as follows. We present a brief description for the
TeV-scale $\cal F$-$SU(5)$ model and derive all the Feynman rules for our
calculations in Section II. We discuss the implications of vector-like quarks
on B physics in Section III. Our numerical results are presented in Section
IV, and Section V is the summary.
## II The $\cal F$-$SU(5)$ Model around the TeV Scale
To achieve the string-scale gauge coupling unification in the $\cal F$-$SU(5)$
model, we introduce the vector-like particles which from complete Flipped
$SU(5)\times U(1)_{X}$ multiplets. The quantum numbers for these additional
vector-like particles under the $SU(5)\times U(1)_{X}$ gauge symmetry are
Jiang:2006hf
$\displaystyle
XF={\mathbf{(10,1)}}~{},~{}{\overline{YF}}={\mathbf{({\overline{10}},-1)}}~{},~{}$
$\displaystyle
Xf={\mathbf{(5,3)}}~{},~{}{\overline{Yf}}={\mathbf{({\overline{5}},-3)}}~{},~{}$
$\displaystyle
Xl={\mathbf{(1,-5)}}~{},~{}{\overline{Yl}}={\mathbf{(1,5)}}~{}.~{}$ (1)
To avoid the confusion in the following discussions, we change the convention
in Ref. Jiang:2006hf a little bit. It is obvious that $XF$,
${\overline{YF}}$, $Xf$, ${\overline{Yf}}$, $Xl$, and ${\overline{Yl}}$ are
standard vector-like particles with contents as follows
$\displaystyle
XF=(XQ,XD^{c},XN^{c})~{},~{}{\overline{YF}}=(YQ^{c},YD,YN)~{},~{}$
$\displaystyle Xf=(XU,XL^{c})~{},~{}{\overline{Yf}}=(YU^{c},YL)~{},~{}$
$\displaystyle Xl=XE~{},~{}{\overline{Yl}}=YE^{c}~{}.~{}$ (2)
Under the $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ gauge symmetry, the
quantum numbers for the extra vector-like particles are
$\displaystyle XQ={\mathbf{(3,2,{1\over
6})}}~{},~{}YQ^{c}={\mathbf{({\bar{3}},2,-{1\over 6})}}~{},~{}$ $\displaystyle
XU={\mathbf{({3},1,{2\over 3})}}~{},~{}YU^{c}={\mathbf{({\bar{3}},1,-{2\over
3})}}~{},~{}$ $\displaystyle XD={\mathbf{({3},1,-{1\over
3})}}~{},~{}YD^{c}={\mathbf{({\bar{3}},1,{1\over 3})}}~{},~{}$ $\displaystyle
XL={\mathbf{({1},2,-{1\over 2})}}~{},~{}YL^{c}={\mathbf{(1,2,{1\over
2})}}~{},~{}$ $\displaystyle
XE={\mathbf{({1},1,{-1})}}~{},~{}YE^{c}={\mathbf{({1},1,{1})}}~{},~{}$
$\displaystyle XN={\mathbf{({1},1,{0})}}~{},~{}YN^{c}={\mathbf{({1},1,{0})}}.$
(3)
At the GUT scale the superpotential is given by
$\displaystyle W_{GUT}$ $\displaystyle=$ $\displaystyle
Y_{ij}^{D}F_{i}F_{j}h+Y_{ij}^{U\nu}F_{i}\bar{f_{j}}\bar{h}+Y_{ij}^{E}\bar{l_{i}}\bar{f_{j}}h+\mu
h\bar{h}+Y_{kj}^{N}\phi_{k}\bar{H}F_{j}$ (4) $\displaystyle+$ $\displaystyle
Y^{\prime}{}_{j}^{D}XFF_{j}h+Y^{\prime}{}_{j}^{U\nu}XF\bar{f}_{j}\bar{h}+Y"{}_{i}^{U\nu}F_{i}\overline{Xf}\bar{h}+Y^{\prime}{}_{j}^{E}\overline{Xl}\bar{f}_{j}h$
$\displaystyle+$ $\displaystyle
Y"{}_{j}^{E}\bar{l}_{j}\overline{Xf}h+Y^{\prime}{}_{k}^{N}\phi_{k}\bar{H}XF+Y^{2D}XFXFh+Y^{\prime
2D}\overline{YF}\overline{YF}\bar{h}$ $\displaystyle+$ $\displaystyle
Y^{2U\nu}XF\overline{Xf}\bar{h}+Y^{\prime
2U\nu}\overline{YF}Yfh+Y{}^{2E}\overline{Xl}\overline{Xf}h+Y^{\prime
2E}YlYf\bar{h}$ $\displaystyle+$ $\displaystyle
M_{j}^{1}F_{j}\overline{YF}+M_{j}^{2}\bar{f}_{j}Yf+M_{j}^{3}\bar{l}_{j}Yl$
$\displaystyle+$ $\displaystyle
M^{4}XF\overline{YF}+M^{5}\overline{Xf}Yf+M^{6}\overline{Xl}Yl~{},$
where $i$ is the generation indices. The first line is the SSM superpotential,
the second line is the Yukawa mixing terms between the SM fermions and vector-
like particles, the third and fourth lines are the SM-like superpotential for
vector-like multiplets, and the fifth and sixth lines are bilinear mass terms.
After the $SU(5)\times U(1)_{X}$ gauge symmetry breaking down to the SM gauge
symmetry, we obtain the superpotential as follows
$\displaystyle W_{EW}$ $\displaystyle=$
$\displaystyle(Y_{ij}^{D}-Y_{ji}^{D})(D^{c})_{i}Q_{j}\cdot
H_{d}+Y_{ij}^{U\nu}U_{j}^{c}Q_{i}\cdot H_{u}-Y_{ij}^{U\nu}N_{i}^{c}L_{j}\cdot
H_{u}$ (5) $\displaystyle-$ $\displaystyle Y_{ij}^{E}E_{i}^{c}L\cdot
H_{d}-Y_{j}^{{}^{\prime}D}(XD^{c}Q_{j}\cdot H_{d}+D_{j}^{c}XQ\cdot
H_{d})+Y_{{}^{\prime}j}^{U\nu}U_{j}^{c}XQ\cdot H_{u}$ $\displaystyle-$
$\displaystyle Y_{j}^{{}^{\prime}U\nu}XN^{c}L\cdot
H_{u}+Y_{i}^{"U\nu}XU^{c}Q\cdot H_{u}-Y_{i}^{"U\nu}N_{i}^{c}XL\cdot
H_{u}-Y_{j}^{{}^{\prime}E}XE^{c}L\cdot H_{d}$ $\displaystyle-$ $\displaystyle
Y_{j}^{"E}E_{j}^{c}XL\cdot H_{d}-2Y^{2D}XD^{c}XQ\cdot
H_{d}-2Y^{{}^{\prime}2D}YDYQ^{c}\cdot H_{u}$ $\displaystyle+$ $\displaystyle
Y^{2U\nu}XU^{c}XQ\cdot H_{u}-Y^{2U\nu}XN^{c}XL\cdot
H_{u}-Y^{{}^{\prime}2U\nu}YUYQ^{c}\cdot H_{d}$ $\displaystyle+$ $\displaystyle
Y^{{}^{\prime}2U\nu}YNYL^{c}\cdot H_{d}-Y^{2E}XE^{c}XL\cdot
H_{d}-Y^{{}^{\prime}2E}YEYL^{c}\cdot H_{u}$ $\displaystyle-$ $\displaystyle
2M_{j}^{1}\left[D_{j}^{c}YD+Q\cdot
YQ^{c}+N_{j}^{c}YN\right]+M_{j}^{2}\left[U^{c}YU+L\cdot
YL^{c}\right]+M_{j}^{3}E_{j}^{c}YE$ $\displaystyle-$ $\displaystyle
2M^{4}\left[XD^{c}YD+XQ\cdot
YQ^{c}+XN^{c}YN\right]+M^{5}\left[XU^{c}YU+XL\cdot YL^{c}\right]$
$\displaystyle+$ $\displaystyle M^{6}XE^{c}YE~{}.~{}\,$
At low energy, the sparticles decouple rapidly when $M_{S}$ increases. Note
that the LHC already put strong constraints on squark masses around 1500 GeV,
we will concentrate on the contributions from new vector-like quark multiplets
$XU,~{}YU^{c},~{}XD,~{}{\rm and}~{}YD^{c}$ for simplicity. At first glance
these multiplets seem to be similar to the fourth and fifth generation quarks,
but indeed $(XU,~{}YU^{c})$ and $(XD,~{}YD^{c})$ are vector-like. This makes
them very different from the fourth and fifth generation quarks. The down-type
quark mass matrix is
$\displaystyle M_{D}=$
$\displaystyle\left(\begin{array}[]{ccccc}(Y_{11}^{D}+Y_{11}^{D})v_{d}&(Y_{12}^{D}+Y_{21}^{D})v_{d}&(Y_{13}^{D}+Y_{31}^{D})v_{d}&Y^{\prime}{}_{1}^{D}v_{d}&-2M_{1}^{1}\\\
(Y_{21}^{D}+Y_{12}^{D})v_{d}&(Y_{22}^{D}+Y_{22}^{D})v_{d}&(Y_{23}^{D}+Y_{32}^{D})v_{d}&Y^{\prime}{}_{2}^{D}v_{d}&-2M_{2}^{1}\\\
(Y_{31}^{D}+Y_{13}^{D})v_{d}&(Y_{32}^{D}+Y_{23}^{D})v_{d}&(Y_{33}^{D}+Y_{33}^{D})v_{d}&Y^{\prime}{}_{3}^{D}v_{d}&-2M_{3}^{1}\\\
Y^{\prime}{}_{1}^{D}v_{d}&Y^{\prime}{}_{2}^{D}v_{d}&Y^{\prime}{}_{3}^{D}v_{d}&2Y^{2D}v_{d}&-2M^{4}\\\
2M_{1}^{1}&2M_{2}^{1}&2M_{3}^{1}&2M^{4}&-2Y^{\prime
2D}v_{u}\end{array}\right)~{},~{}$ (11)
and the up-type quark matrix is
$\displaystyle
M_{U}=\left(\begin{array}[]{ccccc}Y_{11}^{U\nu}v_{u}&Y_{21}^{U\nu}v_{u}&Y_{31}^{U\nu}v_{u}&Y^{\prime}{}_{1}^{U\nu}v_{u}&M_{1}^{2}\\\
Y_{12}^{U\nu}v_{u}&Y_{22}^{U\nu}v_{u}&Y_{32}^{U\nu}v_{u}&Y^{\prime}{}_{2}^{U\nu}v_{u}&M_{2}^{2}\\\
Y_{13}^{U\nu}v_{u}&Y_{23}^{U\nu}v_{u}&Y_{33}^{U\nu}v_{u}&Y^{\prime}{}_{3}^{U\nu}v_{u}&M_{3}^{2}\\\
Y"{}_{1}^{U\nu}v_{u}&Y"{}_{2}^{U\nu}v_{u}&Y"{}_{3}^{U\nu}v_{u}&Y^{2U\nu}v_{u}&M^{5}\\\
-2M_{1}^{1}&-2M_{2}^{1}&-2M_{3}^{1}&-2M^{4}&Y^{\prime
2U\nu}v_{d}\end{array}\right)~{},~{}$ (17)
where $v_{u}$ and $v_{d}$ are the vacuum expectation values (VEVs) for $H_{u}$
and $H_{d}$. These two matrixes can be diagonalized by unitary matrices $U$
and $V$,
$\displaystyle V_{d}^{\dagger}M_{D}U_{d}={\rm
diag.}[m_{d},m_{s},m_{b},m_{d_{x}},m_{d_{y}}],$ $\displaystyle
V_{u}^{\dagger}M_{U}U_{u}={\rm diag.}[m_{u},m_{c},m_{t},m_{u_{x}},m_{u_{y}}].$
(18)
Thus, the quark mixings are described by a matrix $V=U_{u}^{\dagger}U_{d}$.
From Eqs. (11) and (17), we can see that the mass matrices of the down-type
quarks and up-type quarks are related to each other, implying that the Yukawa
couplings are different from those in the SM. In the Feynman gauge the Feynman
rules for charged $W$ boson, Goldstone boson, and charged Higgs boson with
quarks $\overline{u_{l}}d_{j}\chi^{+}(\chi=W,~{}G,~{}h)$ and for $Z$ boson
$\overline{d_{j}}d_{l}Z$ needed in our calculations are given as follows
$\displaystyle
i\frac{g}{\sqrt{2}}\gamma^{\mu}\left[g^{\chi}_{L}(l,j)P_{L}+g^{\chi}_{R}(l,j)P_{R}\right],~{}~{}(\chi=W,~{}Z)~{},$
(19) $\displaystyle
i\frac{g}{\sqrt{2}}\left[g^{\chi}_{L}(l,j)P_{L}+g^{\chi}_{R}(l,j)P_{R}\right],~{}~{}(\chi=G,~{}h)~{},$
(20)
where
$\displaystyle g^{W}_{L}(i,j)$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{4}U_{u}^{*mi}U_{d}^{m,j},\ \ \
g^{W}_{R}(i,j)=V_{u}^{*5i}V_{d}^{5j},$ (21) $\displaystyle g^{Z}_{L}(i,j)$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}\cos\theta_{W}}\left[\left(1-\frac{2}{3}\sin^{2}\theta_{W}\right)\delta^{ij}-U_{d}^{*5i}U_{d}^{5j}\right],$
$\displaystyle g^{Z}_{R}(i,j)$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}\cos\theta_{W}}\left[-\frac{2}{3}\sin^{2}\theta_{W}\delta^{ij}+V_{d}^{*5i}V_{d}^{5j}\right],$
(22) $\displaystyle g^{G}_{L}(i,j)$ $\displaystyle=$
$\displaystyle\left(\sum_{k,m=1}^{4}Y^{U\nu}_{km}V_{u}^{*ki}U_{d}^{mj}+2Y^{\prime
2D}V_{u}^{*5i}U_{d}^{5j}\right)\frac{v_{u}}{m_{W}},$ $\displaystyle
g^{G}_{R}(i,j)$ $\displaystyle=$
$\displaystyle-\left(\sum_{k,m=1}^{4}(Y^{D}_{mk}+Y^{D}_{km})V_{d}^{*kj}U_{u}^{mi}-2Y^{\prime
U\nu}V_{d}^{*5j}U_{d}^{5i}\right)\frac{v_{d}}{m_{W}},$ (23) $\displaystyle
g^{h}_{L}(i,j)$ $\displaystyle=$
$\displaystyle\left(\sum_{k,m=1}^{4}Y^{U\nu}_{km}V_{u}^{*ki}U_{d}^{mj}+2Y^{\prime
2D}V_{u}^{*5i}U_{d}^{5j}\right)\frac{v_{d}}{m_{W}},$ $\displaystyle
g^{h}_{R}(i,j)$ $\displaystyle=$
$\displaystyle\left(\sum_{k,m=1}^{4}(Y^{D}_{mk}+Y^{D}_{km})V_{d}^{*kj}U_{u}^{mi}-2Y^{\prime
U\nu}V_{d}^{*5j}U_{d}^{5i}\right)\frac{v_{u}}{m_{W}}.$ (24)
Because the vector-like particles do not change $U(1)_{EM}$ interaction, the
interactions of photon and quarks are still the same as those in the SM. From
the above mass matrices we can see that the TeV-scale $\cal F$-$SU(5)$ model
has two points for rich physics to be explored:
* •
Since the quark mass matrices are not the same as two Higgs doublet model
(2HDM) BNP2HDM or the Minimal Supersymmetric Standard Model (MSSM) BNPMSSM ,
the loop-level FCNC will be changed by the Yukawa interactions, and then may
change the prediction of process $b\to s\gamma$ significantly.
* •
The last terms in Eqs.(21)-(24), which we call the “tail terms”, will cause
the tree-level FCNC processes induced by $b\to s\ell^{+}\ell^{-}$ and then the
stringent constraints on the model parameter space will be expected.
## III Implications on B physics
Apart from the directly search for the light vector-like quarks at the LHC,
another way to test the $\cal F$-$SU(5)$ model is to measure their effects on
low energy processes such as rare B decays.
### III.1 Effective Hamiltonian
The starting point for rare B decays $B\to X_{s}\gamma$, $B\to
X_{s}\ell^{+}\ell^{-}$, $B_{s}\to\ell^{+}\ell^{-}$ and
$B_{s}\to\ell^{+}\ell^{-}\gamma$ is the determination of a low-energy
effective Hamiltonian obtained by integrating out the heavy degrees of freedom
in the theory. For $b\to s$ transition, this can be written as
${\cal H}_{\rm
eff}=-\frac{G_{F}}{\sqrt{2}}V_{ts}^{*}V_{tb}\sum_{i=1}^{10}[C_{i}(\mu)O_{i}(\mu)+C^{{}^{\prime}}_{i}(\mu)O^{{}^{\prime}}_{i}(\mu)]~{},~{}\,$
(25)
where the effective operators $O_{i}$ are same as those in the SM defined in
Ref. BLOSM . The chirality-flipped operators $O^{\prime}_{i}$ are obtained
from $O_{i}$ by the replacement $\gamma_{5}\to-\gamma_{5}$ in quark current.
It is obvious that $O^{\prime}_{9,10}$ can be got directly from the tail terms
in the Feynman rules of the $\cal F$-$SU(5)$ model. A few remarks follow on
the operators and Wilson coefficients:
* •
As mentioned in introduction, the three generation quark mixing matrix is
replaced by a $5\times 5$ matrix $U_{u}^{\dagger}U_{d}$ and then is non-
unitary. In our analyses we take a reasonable assumption that the deviation
from unitary is not large. Otherwise, the tree-level FCNC will modify
significantly the low energy processes such as $Z\to b\overline{b}$ and
$B_{s}\to\mu^{+}\mu^{-}$.
* •
Since the Wilson coefficient
$C_{2}(m_{W})=-\frac{V_{cb}V_{cs}^{*}}{V_{tb}V_{ts}^{*}}\simeq 1$ is always a
good approximation in $\cal F$-$SU(5)$ model, and the coefficients of four
quark operators $C_{i}(\mu_{b})\ (i=1,3-6)$ depend actually on the value
$C_{2}(m_{W})$, the contributions from the four-quark operator matrix elements
to effective coefficient $C_{9}^{eff}(\mu_{b})$ can not be ignored and have
the same expressions as the SM.
* •
The coefficient of operator
$O_{2}^{\prime}=(\overline{s}c)_{V+A}(\overline{c}b)_{V-A}$, for example, is
proportional to the elements of quark mixing matrix $V^{5j}_{u}$ or
$U^{5i}_{d}$ denoted the mixings between the ordinary quarks and vector-like
quarks. Thus, it can be reasonably set to be much smaller than
$\mathcal{O}(1)$, and the contributions from the four-quark primed operators
to $C_{9}^{eff}(\mu_{b})$ and $C_{9}^{{}^{\prime},eff}(\mu_{b})$ can be
neglected safely. This means
$C_{9,10}^{{}^{\prime},eff}(\mu_{b})=C_{9,10}^{{}^{\prime}}(m_{W})~{},~{}\,$
(26)
which receive contributions mainly from the tree-level diagrams, loop diagrams
for $b\to s\gamma$, and box diagrams. We also neglect the operator
$O_{7}^{\prime}$ contribution.
* •
For $b\to s\gamma$, the new contributions mainly come from the new type Yukawa
interactions, and for $b\to s\ell^{+}\ell^{-}$, the new contributions mainly
arise from the new operators $O_{9,10}^{\prime}$.
### III.2 Analyses in B Physics Calculations
In the $\cal F$-$SU(5)$ model the contributions to operators $O_{i}\ (i=1-10)$
and $O^{\prime}_{9,10}$ can be encoded by the values of the coefficients
$C_{i}$ and $C^{\prime}_{i}$ at the matching scale $m_{W}$. In this Section,
we will present the Wilson coefficients at the matching scale and decay widths
for some rare B decays. We keep both new physics contributions and the SM
results at the LO for consistency.
* •
The Wilson coefficient $C_{7}$ at the matching scale is
$\displaystyle C_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{tb}V_{ts}^{\ast}}\sum_{i=1}^{5}\\{A(x_{i})g_{L}^{W\ast}(i,2)g_{L}^{W}(i,3)-B(x_{i})\frac{m_{W}}{m_{b}}g_{L}^{W\ast}(i,2)g_{R}^{G}(i,3)$
(27) $\displaystyle+$ $\displaystyle
g_{L}^{G\ast}(i,2)[C(x_{i})g_{L}^{G}(i,3)-\frac{m_{u_{i}}}{m_{b}}D(x_{i})g_{R}^{G}(i,3)]$
$\displaystyle+$
$\displaystyle\frac{x_{i}}{y_{i}}g_{L}^{{h}\ast}(i,2)[C(y_{i})g_{L}^{{h}}(i,3)-\frac{m_{u_{i}}}{m_{b}}D(y_{i})g_{R}^{{h}}(i,3)]\\},$
where $x_{i}=m_{u_{i}}^{2}/m_{W}^{2}$ and $y_{i}=m_{u_{i}}^{2}/m_{h^{+}}^{2}$.
For cross check, using the loop functions given in the appendix and the CKM
matrix unitarity condition, one can easily obtain the predication
$C_{7}^{SM}(m_{W})=A(x_{t})+B(x_{t})+x_{t}[C(x_{t})+D(x_{t})]$ which is
consistent with that in Ref. BLOSM . Furthermore, $C_{7}$ receives a large
non-decoupling contribution not only from top quark as in the SM but also from
the up-type vector-like quark loops at the electroweak scale. The non-
decoupling effects are unique and will be demonstrated in next Section.
The Wilson coefficient $C_{9}$ at the matching scale is
$\displaystyle C_{9}$ $\displaystyle=$
$\displaystyle\frac{P(x_{t})-Q(x_{t})}{\sin^{2}\theta_{W}}+4Q(x_{t})$ (28)
$\displaystyle-$
$\displaystyle\frac{2\pi}{\alpha_{em}}\frac{U_{d}^{*52}U_{d}^{53}}{V_{tb}V_{ts}^{\ast}}(\frac{1}{4}-\sin^{2}\theta_{W})$
$\displaystyle+$
$\displaystyle\frac{1}{V_{tb}V_{ts}^{\ast}}\left\\{\sum_{i=3}^{5}\left[R(x_{i})g_{L}^{W\ast}(i,2)g_{L}^{W}(i,3)+S(x_{i})g_{R}^{G\ast}(i,2)g_{L}^{G}(i,3)\right]\right.$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{5}\frac{m_{W}}{m_{u_{i}}}T(x_{i})\left[g_{L}^{W\ast}(i,2)g_{L}^{G}(i,3)+g_{R}^{G\ast}(i,2)g_{L}^{W}(i,3)\right]$
$\displaystyle+$
$\displaystyle\left.\frac{x_{i}}{y_{i}}S(y_{i})g_{R}^{{h}\ast}(i,2)g_{L}^{{h}}(i,3)]\right\\}+\frac{4}{9}.$
Note the first part related to $P(x_{t})$ and $Q(x_{t})$ from the box diagrams
and the effective vertex $b\to sZ^{*}$ at loop level have the same expression
as those in the SM, while the second part denotes the interaction at tree
level enhanced by a large factor $\frac{2\pi}{\alpha_{em}}$. The last part
comes from the effective vertex $b\to s\gamma^{*}$ at loop level for
consistency. The contribution from one-loop matrix element of the operator
$O_{2}$ is also included as in the SM BLOSM . Moreover, the Wilson
coefficients $C_{10}$, $C^{\prime}_{9}$, and $C^{\prime}_{10}$ at the matching
scale are
$\displaystyle C_{10}$ $\displaystyle=$
$\displaystyle-\frac{P(x_{t})-Q(x_{t})}{\sin^{2}\theta_{W}}+\frac{2\pi}{\alpha_{em}}\frac{1}{4}\frac{U_{d}^{*52}U_{d}^{53}}{V_{tb}V_{ts}^{\ast}}~{},~{}$
(29) $\displaystyle C_{9}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}-\sin^{2}\theta_{W})\frac{2\pi}{\alpha_{em}}\frac{V_{d}^{*52}V_{d}^{53}}{V_{tb}V_{ts}^{\ast}}~{},~{}$
(30) $\displaystyle C_{10}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{\alpha_{em}}\frac{1}{4}\frac{V_{d}^{*52}V_{d}^{53}}{V_{tb}V_{ts}^{\ast}}~{}.$
(31)
The contributions from loop diagrams to $C^{\prime}_{9,10}$ can be neglected
safely.
* •
Branching Ratios
Considering that the Wilson coefficients do not separate into the SM and new
physics parts easily and new operators are introduced, we need to list some
explicit expressions for the branching ratios of B decays as follows
1. 1.
$B\to X_{s}\gamma$
The inclusive $B\to X_{s}\gamma$ rate is the most precise and clean short-
distance information that we have, at present, on $\Delta B=1$ FCNCs. The new
contributions mainly come from the new type Yukawa interactions to operator
$O_{7}$. The calculation of the branching ratio is usually normalized by the
process $B\rightarrow X_{c}e\overline{\nu_{e}}$, so we get
${\rm Br}(B\rightarrow X_{s}\gamma)={\rm Br}^{ex}(B\rightarrow
X_{c}e\overline{\nu_{e}})\frac{|V_{ts}^{\ast}V_{tb}|^{2}}{|V_{cb}|^{2}}\frac{6\alpha}{\pi
f(z)}|C^{eff}_{7}(\mu_{b})|^{2}.$ (32)
Here $z=\frac{m_{c}}{m_{b}}$, and $f(z)=1-8z^{2}+8z^{6}-z^{8}-24z^{4}\ln z$ is
the phase-space factor in the semi-leptonic B-decay. From the formula of
$C^{eff}_{7}$ in Eq.(27) and the corresponding coefficients in Eqs. (21)-(24),
we can see that if we sum the flavor indices from 1 to 5 in Eqs. (21)-(24),
$C_{7}$ will be exactly the same as the five generation 2HDM. In our numerical
calculation we will compare both results in these two models, since it will
show clearly the implications of the new type Yukawa interactions in the $\cal
F$-$SU(5)$ model.
2. 2.
$B\to X_{s}\ell^{+}\ell^{-}$
Since the new operators $O_{9}^{\prime}$ and $O_{10}^{\prime}$ contribute to
$B\to X_{s}\ell^{+}\ell^{-}$ and the exclusive decays, the analytical
expression of invariant dileptonic mass distribution is found to be similar to
the SM as follows
$\displaystyle\frac{d\Gamma(B\to X_{s}\ell^{+}\ell^{-})}{ds}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{b}^{5}}{768\pi^{5}}\alpha_{em}^{2}|V_{tb}V_{ts}^{*}|^{2}(1-s)^{2}(1-\frac{4r}{s})^{1/2}$
(33) $\displaystyle\times$
$\displaystyle\left\\{4|C_{7}^{eff}|^{2}(1+\frac{2}{s})+(|C_{9}^{eff}|^{2}+|C_{9}^{\prime}|^{2})(1+2s)\right.$
$\displaystyle+$
$\displaystyle\left.(|C_{10}|^{2}+|C^{\prime}_{10}|^{2})(1+2s)+12Re(C_{7}^{eff}C_{9}^{eff*})\right\\}~{},$
where $s=(p_{\ell^{+}}+p_{\ell^{-}})^{2}/m_{b}^{2}$. Also, we use the
normalization process $B\rightarrow X_{c}e\overline{\nu_{e}}$ to get rid of
large uncertainties due to $m_{b}^{5}$ and CKM elements as in Eq. (32).
3. 3.
$B_{s}\to\mu^{+}\mu^{-}$
The purely leptonic decays constitute a special case among exclusive
transitions. It is strongly helicity suppressed and only receives
contributions from two axial-current operators $O_{10}$ and $O^{\prime}_{10}$
in the models we studied. The decay width is given by
$\displaystyle\Gamma(B_{s}\to\mu^{+}\mu^{-})=\kappa\frac{\alpha_{em}^{2}G_{F}^{2}}{16\pi^{3}}\left|V_{tb}V_{ts}^{*}\right|^{2}f_{B_{s}}^{2}m_{B_{s}}m_{\mu}^{2}|C_{10}-C^{\prime}_{10}|^{2}~{},~{}\,$
(34)
where $f_{B_{s}}$ is the decay constant for $B_{s}$ determined by $\langle
0|\overline{q}\gamma_{\mu}\gamma_{5}b|B_{q}\rangle=-if_{B_{q}}p_{\mu}.$ The
factor $\kappa$ denotes the non-zero width difference of the $B_{s}$-meson
system effect on the branching ratio of the $B_{s}\to\mu^{+}\mu^{-}$ decay and
it reads deBruyn:2012wk
$\displaystyle\kappa=\frac{1+\frac{1}{2}\tau_{B_{s}}{\cal
A}_{\Delta\Gamma}\Delta\Gamma_{s}}{1-\frac{1}{4}\tau^{2}_{B_{s}}(\Delta\Gamma_{s})^{2}},$
(35)
where $\Delta\Gamma_{s}$ is the difference between the decay widths of the
light and heavy $B_{s}$ mass eigenstates and $\tau_{B_{s}}$ is the $B_{s}$
mean lifetime. The parameters ${\cal A}_{\Delta\Gamma}$ is related to the
effective $B_{s}\to\mu^{+}\mu^{-}$ lifetime $\tau_{\mu^{+}\mu^{-}}$ and
depends sensitively on new physics.
4. 4.
$B_{s}\to\ell^{+}\ell^{-}\gamma$
The exclusive decay can be obtained from the inclusive decay $b\to
s\ell^{+}\ell^{-}\gamma$, and further, from $b\to s\ell^{+}\ell^{-}$. To
achieve this, for $\ell=e,\mu$ we just attach photons to any external quark
lines in the Feynman diagrams of $b\to s\ell^{+}\ell^{-}$ xiong08 . The decay
rate is
$\displaystyle\frac{d\Gamma}{ds}$ $\displaystyle=$
$\displaystyle\left|\frac{\alpha_{em}^{3/2}G_{F}}{4\sqrt{6\pi}}V_{tb}V_{ts}^{*}\right|^{2}\frac{m^{7}_{B_{s}}}{(2\pi)^{3}}s(1-s)^{3}\left[|K|^{2}+|L|^{2}+|M|^{2}+|N|^{2}\right]~{},~{}\,$
(36)
where $s=p^{2}/m_{B_{s}}^{2}$ is normalized dileptonic mass squared, and
$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{1}{m_{B_{s}}^{2}}\left\\{[C_{9}^{eff}(\mu_{b})+C_{9}^{\prime}]G_{1}(p^{2})-2C_{7}^{eff}(\mu_{b})\frac{m_{b}}{p^{2}}G_{2}(p^{2})\right\\},$
$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{m_{B_{s}}^{2}}\left\\{[C_{9}^{eff}(\mu_{b})-C_{9}^{\prime}]F_{1}(p^{2})-2C_{7}^{eff}(\mu_{b})\frac{m_{b}}{p^{2}}F_{2}(p^{2})\right]~{},~{}\,$
$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{C_{10}+C^{\prime}_{10}}{m_{B_{s}}^{2}}G_{1}(p^{2}),\ \ \ \
N=\frac{C_{10}-C^{\prime}_{10}}{m_{B_{s}}^{2}}F_{1}(p^{2})~{},~{}\,$ (37)
with $G_{i}$ and $\ F_{i}$ being the form factors Eilam95 .
## IV Numerical Results
Since additional vector like quark introduced in the model, there are many new
input parameters appear in Wilson coefficients $C_{7},\ C_{9},\ C_{10},\
C_{9}^{\prime},\ C_{10}^{\prime}$. These parameters are not independent and
constrained by conditions Eq. (18). As the first study on B physics in the
model, we will not scan the parameter space completely, but focus on the
implication of mass scale of the vector-like quark on B physics, this will
give us the most important information of the model. Thus in the numerical
study we scan the mass $m_{u_{x}}$ in the range $180\ {\rm GeV}\sim 2000\ {\rm
GeV}$, and $m_{u_{y}}$ in the range $40\sim 60$ GeV heavier than $m_{u_{x}}$.
As for other parameters, we use the shooting method to randomly generate
$5\times 5$ unitary matrix $V_{u}$ and $U_{u}$, then use the CKM matrix to get
the $V_{d}$, $U_{d}$ to let mass of down-type quark matrix satify the Eq.
(18). Note that to take in account impact of the non-zero width difference of
$B_{s}$ system Aaij:2012kn on the branching ratio of
$B_{s}\to\mu^{+}\mu^{-}$, we use $y_{s}=0.088\pm 0.014$ deBruyn:2012wk . We
also use the following experimental constraints from B physics:
1. 1.
In the model with three generation quarks, the CKM matrix unitarity is already
used in the calculations of the loop-level FCNC induced rare B decays.
Therefore for consistency, in the model we study the constraints on CKM matrix
element measurements are not from rare B decays but from tree-level B decays
CKM as shown in Table 1.
Table 1: The CKM matrix elements constrained by the tree-level B decays. | absolute value | relative error | direct measurement from
---|---|---|---
$V_{ud}$ | $0.97418\pm 0.00027$ | $0.028\%$ | nuclear beta decay
$V_{us}$ | $0.2255\pm 0.0019$ | $0.84\%$ | semi-leptonic K-decay
$V_{ub}$ | $0.00393\pm 0.00036$ | $9.2\%$ | semi-leptonic B-decay
$V_{cd}$ | $0.230\pm 0.011$ | $4.8\%$ | semi-leptonic D-decay
$V_{cb}$ | $0.0412\pm 0.0011$ | $2.7\%$ | semi-leptonic B-decay
$V_{tb}$ | $>0.74$ | | (single) top-production
2. 2.
To see the implications of the vector-like quark multiplets, we use the
following bounds on the rare B decays Bmeasured ; Aaij:2012kn
$\displaystyle Br(b\to ce\overline{\nu}_{e})=(10.74\pm 0.16)\times
10^{-2}~{},~{}$ $\displaystyle Br({\overline{B}}\to X_{s}\gamma)=(3.06\pm
0.23)\times 10^{-4}~{},~{}$ $\displaystyle Br(B\to
X_{s}\ell^{+}\ell^{-})=(4.5\pm 1)\times 10^{-6}~{},~{}$ $\displaystyle
Br(B_{s}\to\mu^{+}\mu^{-})<4.5\times 10^{-9}\ \ (95\%C.L.)~{}.$ (38)
3. 3.
Other input parameters are the same as those in the SM, except for $\tan\beta$
and the charged Higgs boson mass $m_{h^{+}}$. In our numerical calculations we
scan the two parameters randomly and choose two typical points
($\tan\beta=2,~{}~{}m_{h^{+}}={3000\ {\rm GeV}}$) and
($\tan\beta=40,~{}~{}m_{h^{+}}={500\ {\rm GeV}}$) for the demonstration.
Figure 1: Comparison of $B\to X_{s}\gamma$ versus $m_{u_{x}}$ in the $\cal
F$-$SU(5)$ model (red cross) and 2HDM (green triangle).
The numerical results of $B\to X_{s}\gamma$ as a function of the vector-like
quark mass are displayed in Fig. 1. For the comparison, Fig. 1 also shows the
results of the five-generation 2HDM. From this figure one can see some
features clearly: (i) The new physics effects decouple when the charged Higgs
boson is very heavy. However, for a much heavier charged Higgs, the branching
ratio of $B\to X_{s}\gamma$ increases with $m_{u_{x}}$ in the $\cal F$-$SU(5)$
model while is almost independent on the extra quark mass in 2HDM, indicating
the large non-decoupling effects; (ii) Unlike the 2HDM where the large
$\tan\beta$ is preferred if the charged Higgs boson mass is at the EW scale,
the small $\tan\beta$, which is excluded in 2HDM, is still survived in the
$\cal F$-$SU(5)$ model; (iii) It is clear from the left plot of this figure
that the branching ratio can be much bigger than the detection result when
$m_{u_{x}}$ getting close to the charged Higgs boson mass. So the detection
results of $B\to X_{s}\gamma$ can give stringent constraints on the $\cal
F$-$SU(5)$ model. The tendency of the figure can be understood as following:
* •
$C_{7}$ determined by Eq. (27) in both $\cal F$-$SU(5)$ model and 2HDM BNP2HDM
will approach to the SM value when the charged Higgs boson is much heavier
than EW scale. Nevertheless, the contributions from the fourth and fifth
generation up-type vector-like quarks in 2HDM can be suppressed by small
$V^{5i}$ and $V^{4i}$ due to the unitarity condition of $5\times 5$ matrix;
* •
Because the summed indices are only from 1 to 4 in the $\cal F$-$SU(5)$ model,
the unitary condition of the CKM matrix can not be maintained. When the
vector-like particle mass approaches to the charged Higgs boson mass, the
suppression from $5\times 5$ CKM mixing matrix will be released and then the
non-decoupling effects will be sizable. In fact, the non-decoupling effects
are a very special part of the $\cal F$-$SU(5)$ model at EW scale and can be
tested at the LHC and other B physics detectors.
Figure 2: Branching ratio of $B\to X_{s}\ell^{+}\ell^{-}$ versus $B\to
X_{s}\gamma$ in the $\cal F$-$SU(5)$ model.
Fig. 2 shows the branching ratio of $B\to X_{s}\ell^{+}\ell^{-}$ versus $B\to
X_{s}\gamma$ in the $\cal F$-$SU(5)$ model. Clearly, both processes will give
stringent constraints on our model. Especially, most part of the points are
excluded when the charged Higgs boson is several hundred GeV, leaving a narrow
part in the parameter space. Similar phenomenology can be seen in Fig. 3 which
shows branching ratios of $B_{s}\to\mu^{+}\mu^{-}$ versus $B\to
X_{s}\ell^{+}\ell^{-}$. The non-decoupling effects can be stringently
constrained by the experiments as expected. Here we should emphasize that the
upper bounds from the Tevatron and the first LHCb constraints Aaij:2012kn ,
which are about one order of magnitude above the SM expectation, as well as
the recent CDF results of $B_{s}\to\mu^{+}\mu^{-}$ detection Aaltonen:2011fi
can be explained naturally. It is interesting to see that there is an
approximate linear relation between branching ratios of
$B_{s}\to\mu^{+}\mu^{-}$ and $B\to X_{s}\ell^{+}\ell^{-}$. In fact, we find
that in the allowed parameter space with $U_{d}\simeq V_{d}^{\dagger}$, the
dominant contributions to both processes come from $C_{i}$ and
$C_{i}^{{}^{\prime}}(i=9,10)$. From Eqs. (28) to (31), we can easily draw the
conclusion that the branching ratios are nearly proportional to
$|C_{10}^{{}^{\prime}}|^{2}$.
Figure 3: Branching ratios of $B_{s}\to\mu^{+}\mu^{-}$ versus $B\to
X_{s}\ell^{+}\ell^{-}$ in the $\cal F$-$SU(5)$ model.
To see whether there are solutions simultaneously satisfied with the allowed
ranges for these data, we can offer now some predications for
$B_{s}\to\ell^{+}\ell^{-}\gamma$, which might be measured at the LHCb and B
factories. The numerical results are illustrated in Fig. 4. We can see clearly
that under the constraints from the inclusive decays $B\to X_{s}\gamma$ and
$B\to X_{s}\ell^{+}\ell^{-}$, exclusive decays $B_{s}\to\mu^{+}\mu^{-}$, as
well as CKM measurements extracted by the tree-level B decays, the branching
ratio, which is very sensitive to $\tan\beta$ and charged Higgs boson mass,
can still be up to $(4\sim 5)\times 10^{-8}$. Thus, it may be tested by the
LHCb soon.
Figure 4: Branching ratio of $B_{s}\to\ell^{+}\ell^{-}\gamma$ with the
combained constraints from $B\to X_{s}\gamma$, $B\to X_{s}\ell^{+}\ell^{-}$
and $B_{s}\to\mu^{+}\mu^{-}$. Red cross stands for the type inputs
($\tan\beta=2,~{}~{}m_{h^{+}}={3000\rm GeV}$) and green triangle for
($\tan\beta=40,~{}~{}m_{h^{+}}={500\rm GeV}$) in the $\cal F$-$SU(5)$ model,
respectively.
Rare B decays continue to be the valuable probes of physics beyond the SM. In
the current early phase of the LHC era, the exclusive modes with muons in the
final states are among the most promising decays. The decay
$B_{s}\to\mu^{+}\mu^{-}$ is likely to be confirmed before the end of 2012 LHCb
. If an enhancement beyond $10^{-8}$ and further non-decoupling effects are
observed, we will have an indication of the $\cal F$-$SU(5)$ model. Although
there are some theoretical challenges including calculation of the hadronic
form factors and non-factorable corrections, $B_{s}\to\ell^{+}\ell^{-}\gamma$
can be expected as the next goal once $B_{s}\to\mu^{+}\mu^{-}$ measurement is
finished since the final states can be identified easily and branching ratios
are large. Our predictions for such processes can be tested in the near
future.
## V Summary
In this paper, we studied the vector-like quark contributions to B physics
processes in the $\cal F$-$SU(5)$ model, including the quark mass spectra,
Feynman rules, the new operators in low energy effective theory and the
correspondence Wilson coefficients, etc. As for the first time study, we focus
on the implication of mass scale of vector like quark. The main conclusions we
obtained are the following:
1. 1.
There exists the $\overline{s}bZ$ interaction at tree level, and the Yukawa
interactions are changed. The new operators $O_{9}^{\prime}$ and
$O_{10}^{\prime}$ must be introduced in effective Hamiltonian, and the Wilson
coefficients are changed due to the violation of the unitarity condition.
2. 2.
Different from many previous models, the effects of vector-like quarks on rare
B decays such as $B\to X_{s}\gamma$ and $B\to X_{s}\ell^{+}\ell^{-}$ do not
decouple in some allowed parameter space, especially when the vector-like
quark mass is comparable to the charged Higgs boson mass.
3. 3.
Under the constraints from $B\to X_{s}\gamma$ and $B\to
X_{s}\ell^{+}\ell^{-}$, there exist scenarios in the model the latest
measurement for $B_{s}\to\mu^{+}\mu^{-}$ can be explained naturally, and the
branching ratio of $B_{s}\to\ell^{+}\ell^{-}\gamma$ can be up to $(4\sim
5)\times 10^{-8}$.
All in all, due to the participation of vector-like particles, the $\cal
F$-$SU(5)$ model is different from the ordinary models such as 2HDM. In
particular, the non-decouling effects are much more predictable and may be
tested in the near future experiments. Finally, we should note that the large
input parameter space and the sparticle effects in the $\cal F$-$SU(5)$ model
needs further work.
###### Acknowledgements.
This research was supported in part by the Natural Science Foundation of China
under grant numbers 11005006, 11172008, 10821504, 11075194, and 11135003, by
the DOE grant DE-FG03-95-Er-40917, and by the Doctor Foundation of BJUT No.
X0006015201102.
## Appendix
The loop functions for calculating the Wilson coefficients at the matching
scale are the following
$\displaystyle A(x)$ $\displaystyle=$
$\displaystyle\frac{5x+38x^{2}-55x^{2}}{36(x-1)^{3}}+\frac{4x-17x^{2}+15x^{3}}{6(x-1)^{4}}\ln
x,$ $\displaystyle B(x)$ $\displaystyle=$
$\displaystyle\frac{x+x^{2}}{4(x-1)^{2}}-\frac{x^{2}}{2(x-1)^{3}}\ln x,$
$\displaystyle C(x)$ $\displaystyle=$
$\displaystyle\frac{20-19x+5x^{2}}{18(x-1)^{3}}+\frac{-2+x}{3(x-1)^{4}}\ln x$
$\displaystyle D(x)$ $\displaystyle=$
$\displaystyle\frac{-5-5x+4x^{2}}{12(x-1)^{3}}+\frac{2x-x^{2}}{2(x-1)^{4}}\ln
x,$ $\displaystyle P(x)$ $\displaystyle=$
$\displaystyle\frac{-x}{4(x-1)}+\frac{x}{4(x-1)^{2}}\ln x~{},$ $\displaystyle
Q(x)$ $\displaystyle=$
$\displaystyle\frac{x^{2}-6x}{8(x-1)}+\frac{3x^{2}+6x}{8(x-1)^{2}}\ln x~{},$
$\displaystyle R(x)$ $\displaystyle=$
$\displaystyle\frac{31x^{2}+20x^{3}}{9(x-1)^{3}}+\frac{-4+18x-30x^{2}+6x^{3}}{9(x-1)^{4}}\ln
x,$ $\displaystyle S(x)$ $\displaystyle=$
$\displaystyle\frac{38-79x+47x^{2}}{108(x-1)^{3}}+\frac{-4x+6x^{2}-3x^{4}}{18(x-1)^{4}}\ln
x,$ $\displaystyle T(x)$ $\displaystyle=$
$\displaystyle\frac{x-5x^{2}-2x^{3}}{12(x-1)^{3}}+\frac{x^{3}}{2(x-1)^{4}}\ln
x.$ (39)
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|
arxiv-papers
| 2012-04-24T10:30:52 |
2024-09-04T02:49:30.098622
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tianjun Li, Dimitri V. Nanopoulos, Wenyu Wang, Xiao-Chuan Wang and\n Zhao-Hua Xiong",
"submitter": "Wenyu Wang",
"url": "https://arxiv.org/abs/1204.5326"
}
|
1204.5523
|
# Anomalous Magnetoresistance in Fibonacci Multilayers
L.D. Machado1, C.G. Bezerra1, M.A. Correa1, C. Chesman1,2, J.E. Pearson2 and
A. Hoffmann2 1Departamento de Física Teórica e Experimental, Universidade
Federal do Rio Grande do Norte, Natal-RN 59072-970, Brazil 2Materials Science
Division, Argonne National Laboratory, Argonne, IL, 60439, USA
###### Abstract
The present paper theoretically investigates magnetoresistance curves in
quasiperiodic magnetic multilayers for two different growth directions, namely
[110] and [100]. We considered identical ferromagnetic layers separated by
non-magnetic layers with two different thicknesses chosen based on the
Fibonacci sequence. Using parameters for Fe/Cr multilayers, four terms were
included in our description of the magnetic energy: Zeeman, cubic anisotropy,
bilinear and biquadratic couplings. The minimum energy was determined by the
gradient method and the equilibrium magnetization directions found were used
to calculate magnetoresistance curves. By choosing spacers with a thickness
such that biquadratic coupling is stronger than bilinear coupling, unusual
behaviors for the magnetoresistance were observed: (i) for the [110] case
there is a different behavior for structures based on even and odd Fibonacci
generations; and more interesting, (ii) for the [100] case we found magnetic
field ranges for which the magnetoresistance increases with magnetic field.
A. Nanometric Structures; B. Multilayers; C. Quasiperiodic; D.
Magnetoresistance
## I Introduction
The study of magnetic multilayers has been the focus of much attention since
the discovery of antiferromagnetic bilinear coupling between magnetic Fe
layers separated by nonmagnetic Cr layersGrunberg . The subsequent discovery
of giant magnetoresistance (GMR)Fert , which allowed the electrical resistance
in these systems to be controlled through external magnetic fields, led to
several applications, particularly in the field of information storage. In
1990 Parkin et al. Parkin showed that, depending on spacer thickness,
bilinear coupling between magnetic films oscillated between being
ferromagnetic and antiferromagnetic. One year later, Rührig et al. Ruhrig
discovered a novel form of coupling (later called biquadratic coupling) in
which, for certain spacer thickness, non-collinear coupling existed between
the magnetic films, resulting in a $90^{\circ}$ angle between magnetization of
adjacent films.
Around the same time, other important breakthroughs were being made in what
was then an unrelated field. The discovery of quasicrystals by Shechtman et
al. Shechtman in 1984 confirmed the existence of an intermediate phase
between ordered crystals and disordered solids. A year later, Merlin et al.
Merlin reported performing the first quasiperiodic superlattice following the
Fibonacci sequence. More recently, first quasiperiodic Fe/Cr magnetic
multilayers with biquadratic coupling were grown experimentally Thatiara ,
illustrating the development of crystal growth techniques, which allow
substantial thickness control for each layer.
The magnetic properties of multilayers can depend significantly on the
stacking pattern of their layers, which can now be tailored in unusual
stacking arrangements. For instance, a quasiperiodic stacking pattern in Fe/Cr
magnetic multilayers induces new magnetic phases which would not be observed
in a periodic arrangement. The consequences of these new phases are observed
in the static PRB99 and dynamic properties JPCM2002 of these magnetic
structures. As previously mentioned, the spacer thickness greatly influences
the property of these multilayers; a relevant question that naturally arises
is: what are the consequences of a quasiperiodic stacking pattern of the non-
magnetic spacers? This paper investigates a new stacking pattern with varying
spacer thickness. In our model the spacer can have one out of three different
thicknesses, which results in variations of the relative strength of bilinear
and biquadratic couplings. The non-magnetic layers are arranged in a Fibonacci
quasiperiodic sequence, and interesting properties emerge for specific
combinations of spacers. Furthermore, results were obtained for two different
growth directions - [100] and [110].
This paper is organized as follows. In Sec. II we discuss the theoretical
model, with emphasis on the description of the quasiperiodic sequences and the
crystallographic orientations considered here. The numerical method, used to
obtain the equilibrium configurations, is described in Sec. II, as well. The
numerical results are described in Sec. III and our findings are summarized in
Sec. IV.
## II Theory
A quasiperiodic multilayer can be built by juxtaposing two building blocks
($A,B$) following a quasiperiodic sequence. The Fibonacci sequence is widely
used, with building blocks transforming according to the following rule:
$A\rightarrow AB$, $B\rightarrow A$. The first Fibonacci sequence is
$S_{1}=A$, the second is $S_{2}=AB$, the third is $S_{3}=ABA$ and so on. A
more detailed description of quasiperiodic sequences can be found in the Ref.
[10].
In the present study, non-magnetic Cr layers, between ferromagnetic Fe layers,
were chosen with thicknesses following the Fibonacci sequence. $A$ is a Cr
layer with thickness $t_{1}$ and $B$ is a Cr layer with thickness $t_{2}$. For
instance, the multilayer $Fe/Cr(t_{1})/Fe/Cr(t_{2})/Fe/Cr(t_{1})/Fe$,
corresponds to $Fe/A/Fe/B/Fe/A/Fe$. Illustrations of multilayers with non-
magnetic layers following sequences $S_{1}$, $S_{2}$, $S_{3}$ and $S_{4}$, are
shown in Fig. 1.
In order to describe the magnetic behavior of these multilayer systems, we
considered four terms in the magnetic energy: the Zeeman term (owing to
interaction between the magnetization of the ferromagnetic films and the
applied external magnetic field), the cubic anisotropy term (due to
interaction between the crystalline structure and electronic spins) and the
two aforementioned terms that couple the magnetization of Fe layers separated
by Cr layers, namely bilinear and biquadratic couplings. Considering these
terms, the total magnetic energy can be written as Mariz ,
$\frac{E_{T}}{dM_{S}}=\displaystyle\sum_{i=1}^{n}\left[-H_{0}\cos(\theta_{i}-\theta_{H})+\frac{H_{ac}}{8}\sin^{2}(2\theta_{i})\right]$
$+\displaystyle\sum_{i=1}^{n-1}\left[-H_{bl_{i}}\cos(\theta_{i}-\theta_{i+1})+H_{bq_{i}}\cos^{2}(\theta_{i}-\theta_{i+1})\right]$
(1)
for the [100] direction, and
$\frac{E_{T}}{dM_{S}}=\displaystyle\sum_{i=1}^{n}\left[-H_{0}\cos(\theta_{i}-\theta_{H})+\frac{H_{ac}}{8}\left(\cos^{4}\theta_{i}+\sin^{2}2\theta_{i}\right)\right]$
$+\displaystyle\sum_{i=1}^{n-1}\left[-H_{bl_{i}}\cos(\theta_{i}-\theta_{i+1})+H_{bq_{i}}\cos^{2}(\theta_{i}-\theta_{i+1})\right]$
(2)
for the [110] direction. A comparison of the two equations shows that the
cubic anisotropy terms depends on the growth direction. A thorough description
of how this term is calculated for both growth directions can be found in Ref.
[11]. In these equations $d$ represents the thickness of the Fe layers (which
in our model is constant), $M_{S}$ is the saturation magnetization, $n$ is the
total number of ferromagnetic films, $H_{0}$ is the external magnetic field
that we consider to be maintained within the plane of the films (in our case
the x-z plane, see Fig. 1), $\theta_{H}$ is the angle between the external
magnetic field and the z axis, $H_{bl}$ is the bilinear coupling term that
gives rise to parallel (anti-parallel) magnetization alignment of adjacent Fe
films if positive (negative), and $H_{bq}$ is the biquadratic coupling term
aligning the magnetization of adjacent Fe films in a perpendicular manner.
$H_{ca}$ measures the strength of the cubic anisotropy field and, for the
[100] case, tends to align magnetization of the films parallel to the
crystalline axis (either x or z), whereas in the [110] case the magnetization
tends to be aligned parallel to the x direction (although there is a local
minimum along the z direction, and a maximum at $\theta\approx 35^{\circ}$).
In accordance with the values given by Refs. [12,13], we used the numerical
value of $H_{ca}=0.5$ kOe and selected $\theta_{H}=0$ for both cases (this
means the field is applied in the easy axis for the [100] case and the
intermediate axis in the [110] case).
Another important aspect of these equations is that the bilinear and
biquadratic fields change from one pair of layers to the next. This is due to
the varying spacer thicknesses since, as previously mentioned, the values of
these coupling terms strongly depend on this thickness. We performed
calculations for three different values of spacer thickness:
1. 1.
$t=1.0$ nm for which $H_{bq}=0.1|H_{bl}|$ with $H_{bl}=-1.0$ kOe;
2. 2.
$t=1.5$ nm for which $H_{bq}=0.3|H_{bl}|$ with $H_{bl}=-0.15$ kOe;
3. 3.
$t=3.0$ nm for which $H_{bq}=|H_{bl}|$ with $H_{bl}=-0.035$ kOe.
These values are the same as those found in Refs. [12,13]. In general, if we
choose the second set for Cr layers that correspond to $A$ and the third set
for Cr layers that correspond to $B$, we obtain different results from those
we would have obtained if we had chosen the third set for $A$ and the second
set for $B$. This means there is a total of six sets of parameters. We found
more interesting results for the case where the biquadratic is relatively
strong.
In order to calculate magnetoresistance for these multilayer systems, we need
the set $\\{\theta_{i}\\}$ of equilibrium angles that minimize equation 1 (or
2). As the number of ferromagnetic films rises, the computational cost of
numerically minimizing these equations increases, requiring an efficient
method of calculating this minimum. As such, we applied the gradient method,
which takes into account the gradient of $E_{T}$ in relation to the set
$\\{\theta_{i}\\}$,
$\vec{\nabla}E_{T}=\sum_{i=1}^{n}{\partial
E_{T}\over\partial\theta_{i}}\hat{\theta}_{i}.$ (3)
A brief description of this algorithm is:
1. (i)
An initial set of angles was randomly chosen, $\\{\theta_{i}\\}_{0}$. These
were used to calculate an initial energy $E_{0}$;
2. (ii)
The gradient of the magnetic energy was calculated, $\vec{\nabla}E_{T}$, and
the set $\\{\theta_{i}\\}_{0}$ was employed to find its numerical value
$\vec{\nabla}E_{T}(\\{\theta_{i}\\}_{0})$;
3. (iii)
We applied the calculated energy and gradient to find the next set of angles
$\\{\theta_{i}\\}_{1}$, using
$\\{\theta_{i}\\}_{1}=\\{\theta_{i}\\}_{0}-\alpha\vec{\nabla}_{i}E_{T}(\\{\theta_{i}\\}_{0})$
for each ferromagnetic film;
4. (iv)
The energy $E_{1}$ was then calculated based on this new set of angles. If
$E_{1}<E_{0}$ this energy and the new set of angles were stored, otherwise we
halved the value of $\alpha$ and repeated step (iii);
5. (v)
This process was repeated until $\alpha$ was smaller than a given tolerance.
A complete discussion of this method can be found in Ref. [10].
Theoretically, it is well known that spin-dependent scattering is responsible
for the magnetoresistance ($M_{R}$) effect in these multilayers Gallagher . It
was also shown that $M_{R}$ varies linearly with $\cos(\Delta\theta)$ when
electrons form a free-electron gas, i.e., there are no barriers between
adjacent films Vedy . Here, $\cos(\Delta\theta)$ is the angular difference
between adjacent film magnetizations. In metallic systems such as Fe/Cr this
angular dependence is valid and once the set $\\{\theta_{i}\\}$ of equilibrium
angles is determined, we obtain normalized values for magnetoresistance PRB99
, i.e.,
$M_{R}(H_{0})=R(H_{0})/R(0)=\frac{\displaystyle\sum_{i=1}^{n-1}\left[1-\cos(\theta_{i}-\theta_{i+1})\right]}{2(n-1)},$
(4)
where $R(0)$ is the electric resistance at zero field.
## III Numerical Results
Although calculations were performed with several different sets of
parameters, the remainder of this paper focuses on only one of these, since we
determined this is sufficient to illustrate our system’s most relevant
properties. We selected the second set of parameters for Cr films associated
with $A$ letters of the quasiperiodic sequence and the third set of parameters
for Cr films associated with $B$ letters of the quasiperiodic sequence. From
now on, we label them as Cr($A$) and Cr($B$), respectively.
### III.1 [110] cubic anisotropy
Let us discuss our numerical results for the magnetoresistance in the case of
the [110] growth direction. These results are illustrated in Figs. 2 and 3. In
Fig. 2 we present the magnetoresistance considering the Cr layers following
the fourth and sixth Fibonacci generations, which means 6 and 14 Fe films,
respectively. As we can see, all transitions are first-order type,
characterized by discontinuous jumps in the magnetoresistance. For the fourth
generation of the Fibonacci sequence ($S_{4}=ABAAB$), which is illustrated in
Fig. 2(a), in the small field region, the magnetoresistance value is $1$
because all magnetizations are antiparallel to each other at zero field. As
the external magnetic field increases ($\sim 80$ Oe), a transition takes place
to a magnetic phase in which the magnetization of the bottom layer is aligned
with the field. We can observe that, increasing the magnetic field, more
transitions take place and the saturated phase emerges when $H_{0}\geq 570$
Oe. A similar behavior is observed for the sixth generation of the Fibonacci
sequence ($S_{6}=ABAABABAABAAB$) which is shown in Fig. 2(b). As in the fourth
generation case, in the low field region the magnetizations are in the
antiferromagnetic configuration. As the field increases, ten different
transitions are observed, from the antiferromagnetic configuration ($H_{0}<90$
Oe) to the saturated regime ($H_{0}\geq 570$ Oe). It is easy to note the self-
similar pattern, which is the basic signature of a quasiperiodic system,
present in Fig. 2, i.e., the magnetoresistance profile of the fourth
generation is reproduced in the magnetoresistance profile of the sixth
generation. Let us now take a look at the results for the magnetoresistance
considering the Cr layers following the fifth ($S_{5}=ABAABABA$) and seventh
($S_{7}=ABAABABAABAABABAABABA$) Fibonacci generations, which means 9 and 22 Fe
films, respectively. These results are illustrated in Fig. 3. Once again,
there is a clear self-similar pattern which is shown Fig. 3, i.e., the
magnetoresistance profile of the fifth generation is reproduced in the
magnetoresistance profile of the seventh generation. One can observe that in
the low field region the central magnetoresistance step is much larger than
the case of even generations. This is because the even generations of the
Fibonacci sequence are terminated by $B$. This letter is associated with the
third set of parameters (lower values of $H_{bl}$ and $H_{bq}$). As a
consequence, the Fe film at the bottom of the multilayer is weakly coupled to
its only adjacent Fe film, and a lower magnetic field is enough to induce a
transition. Therefore, if we compare Figs. 2 and 3, we can see that structures
built using even and odd Fibonacci generations present different profiles for
the magnetoresistance. Moreover, we can also remark that there are two self-
similar patterns: one for the even generations and another for the odd
generations. As explained above, this is also a consequence of the subtle
difference between even and odd Fibonacci generations. Such behavior had been
observed previously in the specific heat of quasiperiodic magnetic
superlattices bezerra1 .
### III.2 [100] cubic anisotropy
Fig. 4 depicts the (a) fourth and (b) fifth Fibonacci generations obtained for
growth direction [100]. It illustrates a number of interesting properties. As
in the [110] case, there are various first-order phase transitions, which are
proportional to the number of ferromagnetic layers. Much more interesting,
however, is the behavior of the magnetoresistance in the low magnetic field
region. Fig. 4 shows, for both fourth and fifth generations, a region where
one can see a positive change of the magnetoresistance, i.e., a region where
an increase in the magnetic field leads to a rise in magnetoresistance, that
is, $\Delta M_{R}/\Delta H>0$. In order to understand these positive changes
in magnetoresistance, it is necessary to analyze the magnetization behavior of
the various films. Fig. 5 shows a diagram of the fourth Fibonacci generation,
illustrating the magnetization direction of each ferromagnetic layer. The
numbered arrows indicate the magnetization direction of different layers. For
example, number 1 represents the first layer, on the top, and number 6
represents the last layer, on the bottom, of the multilayer. Once the cubic
anisotropy is dominant, all magnetizations remain close to a crystalline axis,
as observed in the diagram. For low magnetic field, the Zeeman term is not
important and it can be ignored. As a consequence the film magnetizations tend
to form a configuration that minimizes the bilinear and biquadratic energies.
For Cr($A$), the sum of the two terms is minimized when
$\theta_{i}-\theta_{i+1}=180^{\circ}$, whereas for Cr($B$), the sum of the two
terms is minimized when $\theta_{i}-\theta_{i+1}=90^{\circ}$. In Fig. 5 one
can observe that in the low field region the film magnetizations are not in
the anti-parallel configuration because of two Cr($B$) spacers in the
multilayer. As the magnetic field increases, the Zeeman energy plays a more
important role. A transition takes place for $H\sim 46$ Oe. For this
configuration, all magnetizations, except for the bottom film magnetization,
are in the anti-parallel configuration. Thus, the magnetoresistance increases,
resulting in a transition with $\Delta M_{R}/\Delta H>0$. When the magnetic
field reaches $90$ Oe, a second transition takes place. In this configuration
only the fourth Fe film changes its magnetization anti-parallel to the
magnetic field. Therefore, the magnetoresistance drops to $\sim 0.64$. With
further increase of the magnetic field, it becomes energetically too costly
for the film magnetizations to be opposite to the external field. This implies
that the next transition, which takes place for $H\sim 130$ Oe, leads to a
configuration for which there is no film magnetization anti-parallel to the
external magnetic field. However, most of magnetizations are orthogonal to
each other. As a consequence, according to Eq. 4, the magnetoresistance of
this configuration is higher than the previous one. Once again, we observe a
transition with $\Delta M_{R}/\Delta H>0$. As the magnetic field increases,
the film magnetizations gradually become aligned with the field, and the
magnetoresistance monotonically decreases with the magnetic field. Saturation
is reached for $H_{S}\sim 450$ Oe. An analogous analysis applies to the fifth
generation of the Fibonacci sequence depicted in Fig. 4b.
## IV Conclusion
In summary, we studied quasiperiodic magnetic multilayers, composed by
ferromagnetic Fe layers separated by non-magnetic Cr layers. The non-magnetic
Cr layers were arranged according to the Fibonacci quasiperiodic sequence,
such that the letters A and B in the sequence correspond to Cr layers with
thicknesses $t_{1}$ and $t_{2}$, respectively. The Fe layers are between Cr
layers as well as on the top and bottom of the multilayer structure. The
calculation is based on a phenomenological model which includes the following
contributions to the magnetic energy: Zeeman, cubic anisotropy, bilinear and
biquadratic exchanges. The magnetic energy was minimized using the gradient
method and the resulting equilibrium angles were used to calculate
magnetoresistance curves for the system. We selected a particular set of
parameters such that the thickness of Cr($A$) layer corresponds to
$H_{bq}=0.3|H_{bl}|$ and the thickness of Cr($B$) layer corresponds to
$H_{bq}=|H_{bl}|$. These two sets of exchange couplings are responsible for
the exchange energies between two adjacent Fe films. We numerically calculated
the magnetoresistance curves assuming two possible crystallographic
orientations namely, [110] and [100]. Our results show that quasiperiodic
magnetic multilayers exhibit a rich variety of configurations induced by the
external magnetic field. In particular, two points may be emphasized: (i) the
well-defined even and odd parity observed in the behavior of the
magnetoresistance curves and (ii) the positive change of the magnetoresistance
with $\Delta M_{R}/\Delta H>0$.
As illustrated in Figs. 2 and 3, magnetoresistance curves for odd and even
Fibonacci generations show different profiles. This is a consequence of the
quasiperiodic sequence itself, since even generations of the sequence
terminate with $B$, while odd generations start and end with $A$. This subtle
difference is responsible for the well-defined even and odd parity related to
the generation number of the Fibonacci structure. A similar parity had been
observed previously in the specific heat of quasiperiodic magnetic super-
lattices bezerra1 . On the other hand, a much more interesting and novel
behavior is the positive change of magnetoresistance characterized by $\Delta
M_{R}/\Delta H>0$, illustrated in Figs. 4 and 5. Our numerical results showed
that in the low field region, the transitions, induced by the increase of the
magnetic field, may lead to a magnetic configuration with a higher
magnetoresistance. This is a direct consequence of the quasiperiodic
distribution of the Cr layers in the multilayer structure.
We would like to thank the Brazilian Research Agencies CNPq, CAPES, FINEP and
FAPERN for financial support. Work at Argonne was supported by the US
Department of Energy, Basic Energy Sciences under contract No. DE-
AC02-06CH1135.
## References
* (1) P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky and H. Sowers, Phys. Rev. Lett. 57, 2442 (1986).
* (2) M. N. Baibich, J.M. Broto, A. Fert, F. Nguyen van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988); G. Binasch, P. Grünberg, F. Saurenbach and W. Zinn, Phys. Rev. B 39, 4828 (1989).
* (3) S. S. P. Parkin, N. More and K. P. Roche, Phys. Rev. Lett. 64, 2304 (1990).
* (4) M. Rührig, R. Schäfer, A. Hubert, R. Mosler, J. A. Wolf, S. Demokritov and P. Grünberg, Phys. Stat. Sol. (a) 125, 635 (1991).
* (5) D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
* (6) R. Merlin, K. Bajema, R. Clarke, F. Y. Juang and P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
* (7) T. Freire, C. Salvador, M.A. Correa, C.G. Bezerra, C. Chesman, A.B. Oliveira and F. Bohn, Solid State Commun. 151 (2011) 337.
* (8) C. G. Bezerra, J. M. de Araújo, C. Chesman, and E. L. Albuquerque, Phys. Rev. B 60, 9264 (1999); C.G. Bezerra and M.G. Cottam, J. Magn. Magn. Mater. 240, 52 (2002).
* (9) P. W. Mauriz, E. L. Albuquerque and C. G. Bezerra, J. Phys.: Condens. Matter 14, 1785 (2002); C.G. Bezerra and M.G. Cottam, Phys. Rev. B 65, 054412 (2002).
* (10) C.G. Bezerra, J. M. de Araújo, C. Chesman and E. L. Albuquerque, J. Appl. Phys. 89, 2286 (2001).
* (11) C. Chesman, C.G. Bezerra, E.L. Albuquerque and A.M. Mariz, Phys. Lett. A 354, 221 (2006).
* (12) C. Chesman, M. A. Lucena, M. C. de Moura, A. Azevedo, F. M. de Aguiar, and S. M. Rezende, Phys.Rev. B 58, 101( 1998).
* (13) S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, F. M. de Aguiar and S. S. P. Parkin, J. Appl. Phys. 84, 958 (1998).
* (14) W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, 1992.
* (15) W.J. Gallagher, S.S.P. Parkin, Yu Lu, X.P. Bian, A. Marley, K.P. Roche, R.A. Altman, S.A. Rishton, C. Jahnes, T.M. Shaw and Gang Xiao, J. Appl. Phys. 81, 3741 (1997).
* (16) A. Vedyayev, B. Dieny, N. Ryzhanova, J.B. Genin and C. Cowache, Europhys. Lett. 25, 465 (1994).
* (17) C.G. Bezerra, E.L. Albuquerque, A.M. Mariz, L.R. da Silva and C. Tsallis, Physica A 294, 415 (2001); C.G. Bezerra, E.L. Albuquerque and M.G. Cottam, Physica A 301, 341 (2001).
Figure 1: Schematic multilayers constructed following the Fibonacci sequence.
(a) and (b) correspond to $S_{1}$, one representing Cr thickness equal to
$t_{1}=3.0$ nm and the other for Cr thickness equal to $t_{2}=1.5$ nm. (c)
$S_{2}$ and (d) $S_{3}$ depict the magnetic counterpart of the second and
third Fibonacci sequence, respectively.
Figure 2: Normalized magnetoresistance curves for growth direction [110] for
the (a) fourth and (b) sixth Fibonacci generations.
Figure 3: Normalized magnetoresistance curves for growth direction [110] for
the (a) fifth and (b) seventh Fibonacci generations.
Figure 4: Normalized magnetoresistance curves for the (a) fourth and (b) fifth
Fibonacci generations obtained for the growth direction [100]. Positive
magnetoresistance changes are evident.
Figure 5: Diagram of the fourth Fibonacci generation (growth direction [100]).
The magnetization of each ferromagnetic layer is represented schematically by
an arrow.
|
arxiv-papers
| 2012-04-25T00:55:01 |
2024-09-04T02:49:30.118795
|
{
"license": "Public Domain",
"authors": "L. D. Machado, C. G. Bezerra, M. A. Correa, C. Chesman, J. E. Pearson\n and A. Hoffmann",
"submitter": "Claudionor Bezerra",
"url": "https://arxiv.org/abs/1204.5523"
}
|
1204.5564
|
# Segmentor3IsBack: an R package for the fast and exact segmentation of Seq-
data
A. Cleynen1; Michel Koskas1; Emilie Lebarbier1; Guillem Rigaill2 and Stéphane
Robin1
###### Abstract
## Background:
Genome annotation is an important issue in biology which has long been
addressed with gene prediction methods and manual experiments requiring
biological expertise. The expanding Next Generation Sequencing technologies
and their enhanced precision allow a new approach to the domain: the
segmentation of RNA-Seq data to determine gene boundaries.
## Results:
Because of its almost linear complexity, we propose to use the Pruned Dynamic
Programming Algorithm, which performances had been acknowledged for CGH
arrays, for Seq-experiment outputs. This requires the adaptation of the
algorithm to the negative binomial distribution with which we model the data.
We show that if the dispersion in the signal is known, the PDP algorithm can
be used and we provide an estimator for this dispersion. We then propose to
estimate the number of segments, which can be associated to coding or non-
coding regions of the genome, using an oracle penalty.
## Conclusions:
We illustrate the results of our approach on a real data-set and show its good
performance. Our algorithm is available as an R package on the CRAN
repository.
1AgroParisTech, UMR 518 MIA, 16, rue Claude Bernard, 75005 Paris, France
INRA, UMR 518 MIA, 16, rue Claude Bernard, 75005 Paris, France.
2URGV INRA-CNRS-Université d’Évry Val d’Essonne, 2 Rue Gaston Crémieux,
91057 Evry Cedex, France
## Keywords
segmentation algorithm, exact, fast, RNA-Seq data, count data
## 1 Background
Change-point detection methods have long been used in the analysis of genetic
data, for instance they proved a useful tool in the study of DNA sequences
with various purposes. Braun and Muller (1998); Durot et al. (2009) have
developed segmentation methods for categorical variables with the aim of
identifying patterns for gene predictions, while Bockhorst and Jojic (2007)
uses the sequence segmentation for the detection of SNPs. In the last two
decades, with the large spread of micro-arrays, change-point methods have been
widely used for the analysis of DNA copy number variations and the
identification of amplification or deletion of genomic regions in pathologies
such as cancers Zhang et al. (2012); Erdman and Emerson (2008); Olshen et al.
(2004); Picard et al. (2005, 2011).
The recent development of Next-Generation Sequencing technologies gives rise
to new applications along with new difficulties: ($a$) the increased size of
profiles (up to $10^{8}$ data-points when micro-arrays signals were closer to
$10^{5}$), and ($b$) the discrete nature of the output (number of reads
starting at each position of the genome). Yet applying segmentation methods to
DNA-Seq data and its greater resolution should lead to the analysis of copy-
number variation with a much improved precision than CGH arrays. Moreover, in
the case of poly-(A) RNA-Seq data on lower organisms, since coding regions of
the genome are well separated from non-coding regions with lower activity,
segmentation methods should allow the identification of transcribed genes as
well as address the issue of new transcript discovery. Our objective is
therefore to develop a segmentation method to tackle both ($a$) and ($b$) with
some specific requirements: the amount of reads falling in a segment should be
representative of the biological information associated (relative copy-number
of the region, relative level of expression of the gene) and comparison to
neighboring regions should be sufficient to label the segment (for instance
normal or deleted region of the chromosome in DNA-Seq data, exon or non-coding
region in RNA-Seq), so that no comparison profile should be needed. This also
suppresses the need for normalization, and thus we wish to analyze the raw
count-profile.
So far, most methods addressing the analysis of these datasets require some
normalization process to allow the use of algorithms relying on Gaussian-
distributed data or previously developed for micro-arrays Chiang et al.
(2009); Xie and Tammi (2009); Yoon et al. (2009); Boeva et al. (2011). Indeed,
methods adapted to count data-sets are not many, and highly focused on Poisson
distribution. Shen and Zhang (2012) proposes a method based on the comparison
of Poisson processes associated with the read counts of a case and a control
sample, allowing for the detection of alteration of genomic sequences but not
for expressed genes in a normal condition. Rivera and Walther (2012) developed
a likelihood ratio statistic for the localization of a shift in the intensity
of a Poisson process while Franke et al. (2012) developed a test statistic for
the existence of a change-point in the Poisson autoregression of order 1.
Those two latter methods do not require a comparison profile but they only
allow for the detection of a single change-point and have too high a time-
complexity to be applied to RNA-Seq profiles. Binary Segmentation, a fast
heuristic Olshen et al. (2004) and Pruned Exact Linear Time (PELT), Killick et
al. (2012) an exact algorithm for optimal segmentation with respect to the
likelihood, are both implemented for the Poisson distribution in package
changepoint. Even though both are extremely fast, do not require a comparison
profile and analyse count-data, the Poisson distribution is in-adapted to our
kind of data-sets.
A recent study of Hocking has compared 13 segmentation methods for the
analysis of chromosomal copy number profiles and has shown the excellent
performances of the Pruned Dynamic Programming (PDP) algorithm proposed by
Rigaill (2010) in its initial implementation for the analysis of Gaussian data
in the R package cghseg. We propose to use the PDP algorithm which we have
implemented for the Poisson and negative binomial distributions.
In the next section we recall the general segmentation framework and the
definition and requirements of the PDP algorithm. Our contributions are given
in the third section where we define the negative binomial model and show that
it satisfies the PDP algorithm requirements. We also give a model selection
criterion with theoretical guaranties, which makes the whole approach
complete. We conclude with a simulation study, which illustrates the
performances of the proposed method.
## 2 Segmentation model and algorithm
### 2.1 General segmentation model
The general segmentation problem consists in partitioning a signal of $n$
data-points $\\{y_{t}\\}_{t\in[\\![1,n]\\!]}$ into $K$ pieces or segments. The
model can be written as follows: the observed data
$\\{y_{t}\\}_{t=1,\ldots,n}$ are supposed to be a realization of an
independent random process $Y=\\{Y_{t}\\}_{t=1,\ldots,n}$. This process is
drawn from a probability distribution $\mathcal{G}$ which depends on a set of
parameters among which one parameter $\theta$ is assumed to be affected by
$K-1$ abrupt changes, called change-points, so that
$Y_{t}\sim\mathcal{G}(\theta_{r},\phi)\qquad\mbox{if }t\ \in\
r\quad\text{and}\quad r\in m$
where $m$ is a partition of $[\\![1,n]\\!]$ into segments $r$, $\theta_{r}$
stands for the parameter of segment $r$ and $\phi$ is constant. The objective
is to estimate the change-points or the positions of the segments and the
parameters $\theta_{r}$ both resulting from the segmentation. More precisely,
we define $\mathcal{M}_{k,t}$ the set of all possible partitions in $k>0$
regions of the sequence up to point $t$. We remind that the number of possible
partitions is
$\text{card}(\mathcal{M}_{K,t})={t-1\choose K-1}.$
We aim at choosing the partition in $\mathcal{M}_{K,n}$ of minimal loss
$\gamma$, where the loss is usually taken as the negative log-likelihood of
the model. We define the loss of a segment with given parameter $\theta$ as
$c(r,\theta)=\sum_{i\ \in\ r}\gamma(y_{i},\theta)$, so its optimal cost is
$c(r)=\min_{\theta}\left\\{c(r,\theta)\right\\}$. This allows us to define the
cost of a segmentation $m$ as $\sum_{r\ \in\ m}c(r)$ and our goal is to
recover the optimal segmentation $M_{K,n}$ and its cost $C_{K,n}$ where :
$\displaystyle M_{k,t}$ $\displaystyle=$ $\displaystyle{\arg\min}_{\\{m\ \in\
\mathcal{M}_{k,t}\\}}\left\\{\sum_{r\ \in\ m}c(r)\right\\}$
$\displaystyle\text{and}\quad C_{k,t}$ $\displaystyle=$
$\displaystyle{\min}_{\\{m\ \in\ \mathcal{M}_{k,t}\\}}\left\\{\sum_{r\ \in\
m}c(r)\right\\}.$
### 2.2 Quick overview of the pruned DPA
The pruned DPA relies on the function $H_{k,t}(\theta)$ which is the cost of
the best partition in $k$ regions up to $t$, the parameter of the last segment
being $\theta$:
$\displaystyle H_{k,t}(\theta)=\min_{k-1\leq\tau\leq t}\\{\ C_{k-1,\tau}+\
c([\tau+1,t],\theta)\ \\},$
and from there gets $C_{k,t}$ as $min_{\theta}\\{H_{k,t}(\theta)\\}$. More
precisely, for each total number of regions, $k$, from $2$ to $K$, the pruned
DPA works on a list of last change-point candidates:
$\text{ListCandidate}_{k}$. For each of these candidate change-points, $\tau$,
the algorithm stores a cost function and a set of optimal-cost intervals. To
be more specific, we define:
* •
$H_{k,t}^{\tau}(\theta)=C_{k,\tau}+\sum_{j=\tau+1}^{t}\gamma(y_{j},\theta)$:
the optimal cost if the last change is $\tau$;
* •
$S_{k,t}^{\tau}=\\{\theta\ |\ H_{k,t}^{\tau}(\theta)\ \leq\ H_{k,t}(\theta)\
\\}$: the set of $\theta$ such that $\tau$ is optimal;
* •
$I_{k,t}^{\tau}=\\{\theta\ |\ H_{k,n}^{\tau}(\theta)\ \leq\
H_{k,n}^{t}(\theta)\ \\}$: the set of $\theta$ such that $\tau$ is better than
$t$ in terms of cost, with $\tau<t$.
We have $H_{k,t}(\theta)=\min_{\tau\leq t}\\{H_{k,t}^{\tau}(\theta)\\}$.
The PDP algorithm rely on four basic properties of these quantities:
1. $(i)$
if all $\sum_{j=\tau+1}^{t+1}\gamma(y_{j},\theta)$ are unimodal in $\theta$
then $I_{k,t}^{\tau}$ are intervals;
2. $(ii)$
$H_{k,t+1}^{\tau}(\theta)$ is obtained from $H_{k,t}^{\tau}(\theta)$ using:
$\displaystyle H_{k,t+1}^{\tau}(\theta)=$ $\displaystyle
H_{k,t}^{\tau}(\theta)+\gamma(y_{t+1},\theta);$
3. $(iii)$
it is easy to update $S_{k,t+1}^{\tau}$ using:
$\displaystyle S_{k,t+1}^{\tau}$ $\displaystyle=$ $\displaystyle
S_{k,t}^{\tau}\ \cap I_{k,t+1}^{\tau}$ $\displaystyle S_{k,t}^{t}$
$\displaystyle=$
$\displaystyle\complement_{\mathbb{R}}(\cup_{\tau\in[\\![k-1,t-1]\\!]}I_{k,t}^{\tau});$
4. $(iv)$
once it has been determined that $S_{k,t}^{\tau}$ is empty, the region-border
$\tau$ can be discarded from the list of candidates $ListCandidate_{k}$:
$\displaystyle S_{k,t}^{\tau}=\emptyset$
$\displaystyle\Rightarrow\qquad\forall\ t^{\prime}\geq t\quad
S_{k,t^{\prime}}^{\tau}=\emptyset.$
#### Requirements of the pruned dynamic programming algorithm.
###### Proposition 2.1.
Properties ($i$) to ($iv$) are satisfied as soon as the following conditions
on the loss $c(r,\theta)$ are met:
1. (a)
it is point additive,
2. (b)
it is convex with respect to its parameter $\theta$,
3. (c)
it can be stored and updated efficiently.
It is possible to include an additional penalty term in the loss function. For
example, in the case of RNA-seq data one could add a lasso ($\lambda|\theta|$)
or ridge penalty ($\lambda\theta^{2}$) to encode that a priori the coverage in
most regions should be close to 0. Our C++ implementation of the pruned DPA
includes the possibility to add such a penalty term, however we do not provide
an R interface to this functionality in our R package. One of the reason for
this choice is that choosing an appropriate value for $\lambda$ is not a
simple problem.
## 3 Contribution
### 3.1 Pruned dynamic programming algorithm for count data
We now show that the PDP algorithm can be applied to the segmentation of RNA-
Seq data using a negative binomial model, and propose a criterion for the
choice of $K$. Though not discussed here, our results also hold for the
Poisson segmentation model.
#### Negative binomial model.
We consider that in each segment $r$ all $y_{t}$ are the realization of random
variables $Y_{t}$ which are independent and follow the same negative binomial
distribution. Assuming the dispersion parameter $\phi$ to be known, we will
use the natural parametrization from the exponential family (also classically
used in R) so that parameter $\theta_{r}$ will be the probability of success.
In this framework, $\theta_{r}$ is specific to segment $r$ whereas $\phi$ is
common to all segments.
We have $E(Y_{t})=\phi(1-\theta)/\theta$ and
$Var(Y_{t})=\phi(1-\theta)/\theta^{2}$. We choose the loss as the negative
log-likelihood associated to data-point $t$ belonging to segment $r$ :
$-\phi\log(\theta_{r})-y_{t}\log(1-\theta_{r})+A(\phi,y_{t})$, or more simply
$\gamma(y_{t},\theta_{r})=-\phi\log(\theta_{r})-y_{t}\log(1-\theta_{r})$ since
$A$ is a function that does not depend on $\theta_{r}$.
#### Validity of the pruned dynamic programming algorithm for the negative
binomial model
###### Proposition 3.1.
Assuming parameter $\phi$ to be known, the negative binomial model satisfies
(a), (b) and (c):
1. (a)
As we assume that $Y_{t}$ are independent we indeed have that the loss is
point additive : $c(r,\theta)=\sum_{t\ \in\ r}\gamma(y_{t},\theta).$
2. (b)
As $\gamma(y_{t},\theta)=-\phi\log(\theta)-y_{t}\log(1-\theta)$ is convex with
respect to $\theta$, $c(r,\theta)$ is also convex as the sum of convex
functions.
3. (c)
Finally, we have $c(r,\theta)=-n_{r}\phi\log(\theta)+\sum_{t\ \in\
r}y_{t}\log(1-\theta)$. This function can be stored and updated using only two
doubles: one for $-n_{r}\phi$, and the other for $\sum_{t\ \in\ r}y_{t}$.
#### Estimation of the overdispersion parameter.
We propose to estimate $\phi$ using a modified version of the estimator
proposed by Johnson et al. (2005): compute the moment estimator of $\phi$ on
each sliding window of size $h$ using the formulae
$\phi=\mathbb{E}(Y)^{2}/(Var(Y)-\mathbb{E}(Y))$ and keep the median
$\widehat{\phi}$.
### 3.2 C++ implementation of the pruned DPA
We implemented the pruned DPA in C++ with in mind the possibility of adding
new loss functions in potential future applications. The difficulties we had
to come through were the versatility of the program to design and the design
of the objects it could work on. Indeed, the use of full templates implied
that we used stable sets of objects for the operations that were to be
performed on.
Namely:
* •
The sets were to be chosen in a _tribe_. This means that they all belong to a
set ${\cal S}$ of sets such that any set $s\in{\cal S}$ can be conveniently
handled and stored into the computer. A set of sets ${\cal S}$ is said
_acceptable_ if it satisfies:
1. 1.
if $s$ belongs to $s$, $\mathbb{R}\setminus s\in{\cal S}$
2. 2.
if $s_{1},s_{2}\in{\cal S},\,\,s_{1}\cap s_{2}\in{\cal S}$
3. 3.
if $s_{1},s_{2}\in{\cal S},\,\,s_{1}\cup s_{2}\in{\cal S}$
* •
The cost functions were chosen in a set ${\cal F}$ such that
1. 1.
each function may be conveniently handled and stored by the software
2. 2.
for any $f\in{\cal F}$, $f(x)=0$ can be easily solved and the set of solutions
belongs to an acceptable set of sets
3. 3.
for any $f\in{\cal F}$ and any constant $c$, $f(x)\leq c$ can be easily solved
and the set of solutions belongs to an acceptable set of sets
4. 4.
for any $f,g\in{\cal F},\,f+g\in{\cal F}$.
Thus we defined two collections for the sets of sets, intervals and
parallelepipeds, and implemented the loss functions corresponding to negative
binomial, Poisson or normal distributions. The program is thus designed in a
way that any user can add his own cost function or acceptable set of
probability function and use it without rewriting a line in the code.
### 3.3 Model Selection
The last issue concerns the estimate of the number of segments $K$. This model
selection issue can be solved using penalized $\log$-likelihood criterion
where the choice of a good penalty function is crucial. This kind of procedure
requires the visit of the optimal segmentations in $k=1,\ldots,K_{\max}$
segments where $K_{\max}$ is generally chosen smaller than $n$. The most
popular criteria (AIC Akaike (1974) and BIC Yao (1984)) failed in the
segmentation context due to the discrete nature of the segmentation parameter.
In a non-asymptotic point of view and for the negative binomial model, Cleynen
and Lebarbier (2013) proposed to choose the number of segments as follows:
denoting $\hat{m}_{K}$ the optimal segmentation of the data in $K$ segments,
$\displaystyle\hat{K}=\arg\min_{K\in
1:K_{\max}}\left\\{\sum_{r\in\hat{m}_{K}}\sum_{t\in
r}\left[-\phi\log\dfrac{\phi}{\phi+\bar{y}_{r}}-Y_{t}\log(1-\dfrac{\phi}{\phi+\bar{y}_{r}})\right]+\beta
K\left(1+4\sqrt{1.1+\log{\left(\frac{n}{K}\right)}}\right)^{2}\right\\},$ (1)
where $\bar{y}_{r}=\dfrac{\sum_{t\in r}y_{t}}{\hat{n}_{r}}$ and $\hat{n}_{r}$
is the size of segment $r$. The first term corresponds to the cost of the
optimal segmentation while the second is a penalty term which depends on the
dimension $K$ and of a constant $\beta$ that has to be tuned according to the
data (see the next section). With this choice of penalty, so-called oracle
penalty, the resulting estimator satisfies an oracle-type inequality. A more
complete performance study is done in Cleynen and Lebarbier (2013) and showed
that the proposed criterion outperforms the existing ones.
### 3.4 R package
The Pruned Dynamic Programming algorithm is available in the function
Segmentor of the R package Segmentor3IsBack. The user can choose the
distribution with the slot model (1 for Poisson, 2 for Gaussian homoscedastic,
3 for negative binomial and 4 for segmentation of the variance). It returns an
S4 object of class Segmentor which can later be processed for other purposes.
The function SelectModel provides four criteria for choosing the optimal
number of segments: AIC Akaike (1974), BIC Yao (1984), the modified BIC Zhang
and Siegmund (2007) (available for Gaussian and Poisson distribution) and
oracle penalties (available for the Gaussian distribution Lebarbier (2005) and
for the Poisson and negative binomial Cleynen and Lebarbier (2013) as
described previously). This latter kind of penalties require tuning a constant
according to the data, which is done using the slope heuristic Arlot and
Massart (2009).
Figure 4 (which is detailed in the Results and discussion section) was
obtained with the following $4$ lines of code (assuming the data was contained
in vector x):
> Seg<-Segmentor(x,model=3,Kmax=200)
> Kchoose<-SelectModel(Seg, penalty="oracle")
> plot(sqrt(x),col=’dark red’)
> abline(v=getBreaks(Seg)[Kchoose, 1:Kchoose],col=’blue’)
The function BestSegmentation allows, for a given $K$, to find the optimal
segmentation with a change-point at location $t$ (slot $bestSeg). It also
provides, through the slot $bestCost, the cost of the optimal segmentation
with $t$ for $j^{th}$ change-point. Figure 1(b) illustrates this result for
the optimal segmentations in $4$ segments of a signal simulated with only $3$
segments. We can see for instance that any choice of first change-point
location between $1$ and $2000$ yields almost the same cost (the minimum is
obtained for $t=1481$), thus the optimal segmentation is not clearly better
than the second or third. On the contrary, the same function with $3$ segments
shows that the optimal segmentation outperforms all other segmentations in $3$
segments (Figure 1(a)).
(a) 4 segments
(b) 3 segments
Figure 1: Cost of optimal segmentation in $4$ and $3$ segments. Cost of
optimal segmentation depending on the location of the $j^{th}$ change-point
when the number of segments is $4$ (figure 1(a)) and $3$ (figure 1(b)) and the
signal was simulated with $3$ segments. Illustration of the output of function
BestSegmentation.
## 4 Results and discussion
### 4.1 Performance study
We designed a simulation study on the negative binomial distribution to assess
the performance of the PDP algorithm in terms of computational efficiency,
while studying the impact of the overdispersion parameter $\phi$ by comparing
the results for two different values of this parameter. After running
different estimators (median on sliding windows of maximum, quasi-maximum
likelihood and moment estimators) on several real RNA-Seq data (whole
chromosome and genes of various sizes) we fixed $\phi_{1}=0.3$ as a typical
value for highly dispersed data as observed in real RNA-Seq data, and chose
$\phi_{2}=2.3$ for comparison with a reasonably dispersed data-set. For each
value, we simulated data-sets of size $n$ with various densities of number of
segments $K$, and only two possible values for the parameter $p_{J}$: $0.8$ on
even segments (corresponding to low signal) and $0.2$ on odd segments for a
higher signal. We had $n$ vary on a logarithmic scale between $10^{3}$ and
$10^{6}$ and $K$ between $\sqrt{n}/6$ and $\sqrt{n}/3$. For each
configuration, we segmented the signal up to $K_{\max}=\sqrt{n}$ twice: once
with the known value of $\phi$ and once with our estimator $\widehat{\phi}$ as
described above. We started with a window width $h=15$. When the estimate was
negative, we doubled $h$ and repeated the experience until the median is
positive.
Each configuration was simulated $100$ times.
For our analysis we checked the run-time on a standard laptop, and assessed
the quality of the segmentation using the Rand Index $\mathcal{I}$.
Specifically, let $C_{t}$ be the true index of the segment to which base $t$
belongs and let $\hat{C}_{t}$ be the index estimated by the method, then
$\mathcal{I}=\frac{2\sum_{t>s}\left[\mathbf{1}_{C_{t}=C_{s}}\mathbf{1}_{\hat{C}_{t}=\hat{C}_{s}}+\mathbf{1}_{C_{t}\neq
C_{s}}\mathbf{1}_{\hat{C}_{t}\neq\hat{C}_{s}}\right]}{(n-1)(n-2)}.$
Figure 2 shows, for the particular case of $K=\sqrt{n}/3$, the almost linear
complexity of the algorithm in the size $n$ of the signal. As the maximal
number of segments $K_{\max}$ considered increased with $n$, we normalized the
run-time to allow comparison. This underlines an empirical complexity smaller
than $\mathcal{O}(K_{\max}n\log n)$, and independent on the value of $\phi$ or
its knowledge. Moreover, the algorithm, and therefore the pruning, is faster
when the overdispersion is high, phenomenon already encountered with the
$L^{2}$ loss when the distribution of the errors is Cauchy. However, the
knowledge of $\phi$ does not affect the run-time of the algorithm. Figure 3
illustrates through the Rand Index the quality of the proposed segmentation
for a few values of $n$. Even though the indexes are slightly lower for
$\phi_{1}$ than for $\phi_{2}$ (see left panel), they range between $0.94$ and
$1$ showing a great quality in the results. Moreover, the knowledge of $\phi$
does not increase the quality (see right panel), which validates the use of
our estimator.
Figure 2: Run-time analysis for segmentation with negative binomial
distribution.This figure displays the normalized (by $K_{\max}$) run-time in
seconds of the Segmentor3IsBack package for the segmentation of signals with
increasing length $n$, for two values of the dispersion $\phi$, and with
separate analysis when its value is known or estimated. While the algorithm is
faster for more over-dispersed data, the estimation of the parameter does not
slow the processing.
Figure 3: Rand-Index for the quality of the segmentation. This figure displays
the boxplot of the Rand-index computed for each of the hundred simulations
performed in the following situations: comparing the values with $\phi_{1}$
and $\phi_{2}$ when estimated (left figure), and comparing the impact of
estimating $\phi_{1}$ (right figure). While the estimation does not decrease
the quality of the segmentation, the value of the dispersion affects the
recovery of the true change-points.
### 4.2 Yeast RNAseq experiment
We applied our algorithm to the segmentation of chromosome $1$ of the S.
Cerevisiae (yeast) using RNA-Seq data from the Sherlock Laboratory at Stanford
University Risso et al. (2011) and publicly available from the NCBI’s Sequence
Read Archive (SRA, http://www.ncbi.nlm.nih.gov/sra, accession number
SRA048710). We selected the number of segments using our oracle penalty
described in the previous section. An existing annotation is available on the
Saccharomyces Genome Database (SGD) at http://www.yeastgenome.org, which
allows us to validate our results.
With a run-time of $25$ minutes (for a signal length of $230218$), we selected
$103$ segments with the negative binomial distribution, most of which (all but
$3$) were found to surround known genes from the SGD. Figure 4 illustrates the
result.
Figure 4: Segmentation of the yeast chromosome 1 using the negative binomial
loss. The model selection procedure chooses $K=103$ segments, most of which
surround genes given by the SGD annotation.
## 5 Conclusion
Segmentation has been a useful tool for the analysis of biological data-sets
for a few decades. We propose to extend its application with the use of the
Pruned Dynamic Programming algorithm for count data-sets such as outputs of
sequencing experiments. We show that the negative binomial distribution can be
used to model such data-sets on the condition that the overdispersion
parameter is known, and proposed an estimator of this parameter that performs
well in our segmentation framework.
We propose to choose the number of segments using our oracle penalty
criterion, which makes the package fully operational. This package also allows
the use of other criteria such as AIC or BIC. Similarly, the algorithm is not
restricted to the negative binomial distribution but also allows the use of
Poisson and Gaussian losses for instance, and could easily be adapted to other
convex one-parameter losses.
With its empirical complexity of $\mathcal{O}(K_{\max}n\log n)$, it can be
applied to large signals such as read-alignment of whole chromosomes, and we
illustrated its result on a real-data sets from the yeast genomes. Moreover,
this algorithm can be used as a base for further analysis. For example, Luong
et al. (2012) use it to initialize their Hidden Markov Model to compute
change-point location probabilities.
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|
arxiv-papers
| 2012-04-25T05:28:32 |
2024-09-04T02:49:30.131748
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alice Cleynen, Michel Koskas, Emilie Lebarbier, Guillem Rigaill,\n Stephane Robin",
"submitter": "Alice Cleynen",
"url": "https://arxiv.org/abs/1204.5564"
}
|
1204.5643
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-111 LHCb-PAPER-2012-005 April 25, 2012
Analysis of the resonant components in $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The decay $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ can be exploited to study $C\\!P$ violation. A detailed
understanding of its structure is imperative in order to optimize its
usefulness. An analysis of this three-body final state is performed using a
1.0 fb-1 sample of data produced in 7 TeV $pp$ collisions at the LHC and
collected by the LHCb experiment. A modified Dalitz plot analysis of the final
state is performed using both the invariant mass spectra and the decay angular
distributions. The $\pi^{+}\pi^{-}$ system is shown to be dominantly in an
S-wave state, and the $C\\!P$-odd fraction in this $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay is shown to be greater
than 0.977 at 95% confidence level. In addition, we report the first
measurement of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ branching fraction relative to $J/\psi\phi$ of
$(19.79\pm 0.47\pm 0.52)$%.
Submitted to Physics Review D
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
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Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
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Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
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Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Measurement of mixing-induced $C\\!P$ violation in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays is of prime importance
in probing physics beyond the Standard Model. Final states that are $C\\!P$
eigenstates with large rates and high detection efficiencies are very useful
for such studies. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$,
$f_{0}(980)\rightarrow\pi^{+}\pi^{-}$ decay mode, a $C\\!P$-odd eigenstate,
was discovered by the LHCb collaboration [1] and subsequently confirmed by
several experiments [2, *Abazov:2011hv, *Aaltonen:2011nk]. As we use the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, the final state has four charged
tracks, and has high detection efficiency. LHCb has used this mode to measure
the $C\\!P$ violating phase $\phi_{s}$ [5], which complements measurements in
the $J/\psi\phi$ final state [6, 7, *Abazov:2011ry]. It is possible that a
larger $\pi^{+}\pi^{-}$ mass range could also be used for such studies.
Therefore, to fully exploit the $J/\psi\pi^{+}\pi^{-}$ final state for
measuring $C\\!P$ violation, it is important to determine its resonant and
$C\\!P$ content. The tree-level Feynman diagram for the process is shown in
Fig. 1.
Figure 1: Leading order diagram for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays into
$J/\psi\pi^{+}\pi^{-}$.
In this paper the $J/\psi\pi^{+}$ and $\pi^{+}\pi^{-}$ mass spectra, and decay
angular distributions are used to study the resonant and non-resonant
structures. This differs from a classical “Dalitz plot” analysis [9] because
one of the particles in the final state, the $J/\psi$, has spin-1 and its
three decay amplitudes must be considered. We first show that there are no
evident structures in the $J/\psi\pi^{+}$ invariant mass, and then model the
$\pi^{+}\pi^{-}$ invariant mass with a series of resonant and non-resonant
amplitudes. The data are then fitted with the coherent sum of these
amplitudes. We report on the resonant structure and the $C\\!P$ content of the
final state.
## 2 Data sample and analysis requirements
The data sample contains 1.0 fb-1 of integrated luminosity collected with the
LHCb detector [10] using $pp$ collisions at a center-of-mass energy of 7 TeV.
The detector is a single-arm forward spectrometer covering the pseudorapidity
range $2<\eta<5$, designed for the study of particles containing $b$ or $c$
quarks. Components include a high precision tracking system consisting of a
silicon-strip vertex detector surrounding the $pp$ interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet with a
bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip
detectors and straw drift-tubes placed downstream. The combined tracking
system has a momentum resolution $\Delta p/p$ that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at 100$\mathrm{\,Ge\kern-1.00006ptV}$
(we work in units where $c=1$), and an impact parameter resolution of
20$\,\upmu\rm m$ for tracks with large transverse momentum with respect to the
proton beam direction. Charged hadrons are identified using two ring-imaging
Cherenkov (RICH) detectors. Photon, electron and hadron candidates are
identified by a calorimeter system consisting of scintillating-pad and pre-
shower detectors, an electromagnetic calorimeter and a hadronic calorimeter.
Muons are identified by a muon system composed of alternating layers of iron
and multiwire proportional chambers. The trigger consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by a
software stage which applies a full event reconstruction.
Events selected for this analysis are triggered by a
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay. Muon candidates are selected at the
hardware level using their penetration through iron and detection in a series
of tracking chambers. They are also required in the software level to be
consistent with coming from the decay of a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson into a $J/\psi$. Only
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays that are triggered on
are used.
## 3 Selection requirements
The selection requirements discussed here are imposed to isolate $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates with high signal
yield and minimum background. This is accomplished by first selecting
candidate $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays, selecting a pair of pion
candidates of opposite charge, and then testing if all four tracks form a
common decay vertex. To be considered a $J/\psi\rightarrow\mu^{+}\mu^{-}$
candidate particles of opposite charge are required to have transverse
momentum, $p_{\rm T}$, greater than 500 MeV, be identified as muons, and form
a vertex with fit $\chi^{2}$ per number of degrees of freedom (ndf) less than
11. After applying these requirements, there is a large
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal over a small background
[1]. Only candidates with dimuon invariant mass between $-$48 MeV to +43 MeV
relative to the observed $J/\psi$ mass peak are selected. The requirement is
asymmetric because of final state electromagnetic radiation. The two muons
subsequently are kinematically constrained to the known $J/\psi$ mass [11].
Pion and kaon candidates are positively identified using the RICH system.
Cherenkov photons are matched to charged tracks, the emission angles of the
photons compared with those expected if the particle is an electron, pion,
kaon or proton, and a likelihood is then computed. The particle identification
is done by using the logarithm of the likelihood ratio comparing two particle
hypotheses (DLL). For pion selection we require DLL$(\pi-K)>-10$.
Candidate $\pi^{+}\pi^{-}$ combinations are selected if each particle is
inconsistent with having been produced at the primary vertex. This is done by
use of the impact parameter (IP) defined as the minimum distance of approach
of the track with respect to the primary vertex. We require that the
$\chi^{2}$ formed by using the hypothesis that the IP is zero be greater than
9 for each track. Furthermore, each pion candidate must have $p_{\rm T}>250$
MeV and the scalar sum of the two pion candidate momentum, $p_{\rm
T}(\pi^{+})+p_{\rm T}(\pi^{-})$, must be greater than 900 MeV. To select
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates we further
require that the two pion candidates form a vertex with a $\chi^{2}<10$, that
they form a candidate $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
vertex with the $J/\psi$ where the vertex fit $\chi^{2}$/ndf $<5$, that this
vertex is greater than $1.5$ mm from the primary vertex and the angle between
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ momentum vector and
the vector from the primary vertex to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex must be less than 11.8
mrad
We use the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$,
$\phi\rightarrow K^{+}K^{-}$ as a normalization and control channel in this
paper. The selection criteria are identical to the ones used for
$J/\psi\pi^{+}\pi^{-}$ except for the particle identification requirement.
Kaon candidates are selected requiring that DLL($K-\pi)>0$. Figure 2(a) shows
the $J/\psi K^{+}K^{-}$ mass for all events with $m(K^{+}K^{-})<1050$ MeV. The
$K^{+}K^{-}$ combination is not, however, pure $\phi$ due to the presence of
an S-wave contribution [12]. We determine the $\phi$ yield by fitting the data
to a relativistic P-wave Breit-Wigner function that is convolved with a
Gaussian function to account for the experimental mass resolution and a
straight line for the S-wave. We use the ${{}_{S}Plot}$ method to subtract the
background [13]. This involves fitting the $J/\psi K^{+}K^{-}$ mass spectrum,
determining the signal and background weights and then plotting the resulting
weighted mass spectrum, shown in Fig. 2(b). There is a large peak at the
$\phi$ meson mass with a small S-wave component. The mass fit gives 20,934$\pm
150$ events of which $(95.5\pm 0.3)$% are $\phi$ and the remainder is the
S-wave contribution.
Figure 2: (a) Invariant mass spectrum of $J/\psi K^{+}K^{-}$ for candidates
with $m(K^{+}K^{-})<1050$ MeV. The data has been fitted with a double-Gaussian
signal and linear background functions shown as a dashed line. The solid curve
shows the sum. (b) Background subtracted invariant mass spectrum of
$K^{+}K^{-}$ for events with $m(K^{+}K^{-})<1050$ MeV. The dashed line (barely
visible along the $x$-axis) shows the S-wave contribution and the solid curve
is the sum of the S-wave and a P-wave Breit-Wigner functions, fitted to the
data.
The invariant mass of the selected $J/\psi\pi^{+}\pi^{-}$ combinations, where
the dimuon candidate pair is constrained to have the $J/\psi$ mass, is shown
in Fig. 3. There is a large peak at the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass and a smaller one at the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass on top of a
background. A double-Gaussian function is used to fit the signal, the core
Gaussian mean and width are allowed to vary, and the fraction and width ratio
for the second Gaussian are fixed to that obtained in the fit of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$. Other
components in the fit model take into account contributions from
$B^{-}\rightarrow J/\psi K^{-}(\pi^{-})$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\eta^{\prime},\eta^{\prime}\rightarrow\rho\gamma$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi,\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$
backgrounds and a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
J/\psi K^{-}\pi^{+}$ reflection. Here and elsewhere charged conjugated modes
are used when appropriate. The shape of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$
signal is taken to be the same as that of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$. The exponential combinatorial
background shape is taken from wrong-sign combinations, that are the sum of
$\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ candidates. The shapes of the other
components are taken from the Monte Carlo simulation with their normalizations
allowed to vary (see Sect. 4.2). The mass fit gives $7598\pm 120$ signal and
$5825\pm 54$ background candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.
Figure 3: Invariant mass of $J/\psi\pi^{+}\pi^{-}$ candidate combinations. The
data have been fitted with double-Gaussian signal and several background
functions. The (red) solid line shows the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal, the (brown) dotted
line shows the combinatorial background, the (green) short-dashed shows the
$B^{-}$ background, the (purple) dot-dashed is $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$,
the (black) dot-long dashed is the sum of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\eta^{\prime}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi$ when $\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$ backgrounds, the
(light blue) long-dashed is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
reflection, and the (blue) solid line is the total.
## 4 Analysis formalism
The decay of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ with the $J/\psi\rightarrow\mu^{+}\mu^{-}$ can be
described by four variables. These are taken to be the invariant mass squared
of $J/\psi\pi^{+}$ ($s_{12}\equiv m^{2}(J/\psi\pi^{+})$), the invariant mass
squared of $\pi^{+}\pi^{-}$ ($s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$), the
$J/\psi$ helicity angle ($\theta_{J/\psi}$), which is the angle of the
$\mu^{+}$ in the $J/\psi$ rest frame with respect to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and the angle
between the $J/\psi$ and $\pi^{+}\pi^{-}$ decay planes ($\chi$) in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame. To improve the
resolution of these variables we perform a kinematic fit constraining the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $J/\psi$ masses to
their PDG mass values [11], and recompute the final state momenta. To simplify
the probability density function (PDF), we analyze the decay process after
integrating over $\chi$, that eliminates several interference terms. The
$\chi$ distribution is shown in Fig. 4 after background subtraction using
wrong-sign events. The distribution has little structure, and thus the $\chi$
acceptance can be integrated over without biasing the other variables.
Figure 4: Background subtracted $\chi$ distribution from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ candidates.
### 4.1 The decay model for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$
One of the main challenges in performing a Dalitz plot angular analysis is to
construct a realistic probability density function (PDF), where both the
kinematic and dynamical properties are modeled accurately. The overall PDF
given by the sum of signal, $S$, and background, $B$, functions is
$F(s_{12},s_{23},\theta_{J/\psi})=\frac{f_{\rm sig}}{{\cal{N}}_{\rm
sig}}\varepsilon(s_{12},s_{23},\theta_{J/\psi})S(s_{12},s_{23},\theta_{J/\psi})+\frac{(1-f_{\rm
sig})}{{\cal{N}}_{\rm bkg}}B(s_{12},s_{23},\theta_{J/\psi}),$ (1)
where $f_{\rm sig}$ is the fraction of the signal in the fitted region and
$\varepsilon$ is the detection efficiency. The normalization factors are given
by
$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(s_{12},s_{23},\theta_{J/\psi})S(s_{12},s_{23},\theta_{J/\psi})\,ds_{12}ds_{23}d\cos\theta_{J/\psi},$
$\displaystyle{\cal{N}}_{\rm bkg}$ $\displaystyle=$
$\displaystyle\int\\!B(s_{12},s_{23},\theta_{J/\psi})\,ds_{12}ds_{23}d\cos\theta_{J/\psi}.$
(2)
In this analysis we apply a formalism similar to that used in Belle’s analysis
of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
K^{-}\pi^{+}\chi_{c1}$ decays [14].
To investigate if there are visible exotic structures in the $J/\psi\pi^{+}$
system as claimed in similar decays [15], we examine the $J/\psi\pi^{+}$ mass
distribution shown in Fig. 5. No resonant effects are evident.
Figure 5: Distribution of $m(J/\psi\pi^{+})$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ candidate decays within $\pm 20$ MeV of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass shown with the (blue)
solid line; $m(J/\psi\pi^{+})$ for wrong-sign $J/\psi\pi^{+}\pi^{+}$
combinations is shown with the (red) dashed line, as an estimate of the
background.
Examination of the event distribution for $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}(J/\psi\pi^{+})$ in Fig. 6 shows obvious structure in
$m^{2}(\pi^{+}\pi^{-})$ that we wish to understand.
Figure 6: Distribution of $s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$ versus
$s_{12}\equiv m^{2}(J/\psi\pi^{+})$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate decays within $\pm
20$ MeV of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass.
#### 4.1.1 The signal function
The signal function is taken to be the sum over resonant states that can decay
into $\pi^{+}\pi^{-}$, plus a possible non-resonant S-wave contribution
$S(s_{12},s_{23},\theta_{J/\psi})=\sum_{\lambda=0,\pm
1}\left|\sum_{i}a^{R_{i}}_{\lambda}e^{i\phi^{R_{i}}_{\lambda}}\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{J/\psi})\right|^{2},$
(3)
where $\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{J/\psi})$ is the
amplitude of the decay via an intermediate resonance $R_{i}$ with helicity
$\lambda$. Each $R_{i}$ has an associated amplitude strength
$a_{\lambda}^{R_{i}}$ for each helicity state $\lambda$ and a phase
$\phi_{\lambda}^{R_{i}}$. The amplitudes are defined as
$\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi})=F_{B}^{(L_{B})}\;A_{R}(s_{23})\;F_{R}^{(L_{R})}\;T_{\lambda}\Big{(}\frac{P_{B}}{m_{B}}\Big{)}^{L_{B}}\;\Big{(}\frac{P_{R}}{\sqrt{s_{23}}}\Big{)}^{L_{R}}\;\Theta_{\lambda}(\theta_{J/\psi}),$
(4)
where $P_{B}$ is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum
in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame and
$P_{R}$ is the momentum of either of the two pions in the dipion rest frame,
$m_{B}$ is the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass,
$F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson and $R_{i}$ resonance
decay form factors, $L_{B}$ is the orbital angular momentum between the
$J/\psi$ and $\pi^{+}\pi^{-}$ system, and $L_{R}$ the orbital angular momentum
in the $\pi^{+}\pi^{-}$ decay, and thus is the same as the spin of the
$\pi^{+}\pi^{-}$. Since the parent $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ has spin-0 and the $J/\psi$ is
a vector, when the $\pi^{+}\pi^{-}$ system forms a spin-0 resonance, $L_{B}=1$
and $L_{R}=0$. For $\pi^{+}\pi^{-}$ resonances with non-zero spin, $L_{B}$ can
be 0, 1 or 2 (1, 2 or 3) for $L_{R}=1(2)$ and so on. We take the lowest
$L_{B}$ as the default.
The Blatt-Weisskopf barrier factors $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$
[16] are
$\displaystyle F^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
F^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{1+z_{0}}}{\sqrt{1+z}},$
(5) $\displaystyle F^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{z_{0}^{2}+3z_{0}+9}}{\sqrt{z^{2}+3z+9}}.$
For the $B$ meson $z=r^{2}P_{B}^{2}$, where $r$, the hadron scale, is taken as
5.0 GeV-1; for the $R$ resonance $z=r^{2}P_{R}^{2}$, and $r$ is taken as 1.5
GeV-1. In both cases $z_{0}=r^{2}P_{0}^{2}$ where $P_{0}$ is the decay
daughter momentum at the pole mass, different for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and the resonance decay.
The angular term, $T_{\lambda}$, is obtained using the helicity formalism and
is defined as
$T_{\lambda}=d^{J}_{\lambda 0}(\theta_{\pi\pi}),$ (6)
where $d$ is the Wigner d-function [11], $J$ is the resonance spin,
$\theta_{\pi\pi}$ is the $\pi^{+}\pi^{-}$ resonance helicity angle which is
defined as the angle of $\pi^{+}$ in the $\pi^{+}\pi^{-}$ rest frame with
respect to the $\pi^{+}\pi^{-}$direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame and calculated from
the other variables as
$\cos\theta_{\pi\pi}=\frac{\left[m^{2}(J/\psi\pi^{+})-m^{2}(J/\psi\pi^{-})\right]m(\pi^{+}\pi^{-})}{4P_{R}P_{B}m_{B}}.$
(7)
The $J/\psi$ helicity dependent term $\Theta_{\lambda}(\theta_{J/\psi})$ is
defined as
$\displaystyle\Theta_{\lambda}(\theta_{J/\psi})$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}\theta_{J/\psi}}\;\;\;\;\;\;\;\;\;\text{for}\;\;\text{helicity}=0$
(8) $\displaystyle=$
$\displaystyle\sqrt{\frac{1+\cos^{2}\theta_{J/\psi}}{2}}\;\;\text{for}\;\;\text{helicity}=\pm
1.$
The function $A_{R}(s_{23})$ describes the mass squared shape of the resonance
$R$, that in most cases is a Breit-Wigner (BW) amplitude. Complications arise,
however, when a new decay channel opens close to the resonant mass. The
proximity of a second threshold distorts the line shape of the amplitude. This
happens for the $f_{0}(980)$ because the $K^{+}K^{-}$ decay channel opens.
Here we use a Flatté model [17]. For non-resonant processes, the amplitude
$A_{R}(s_{23})$ is constant over the variables $s_{12}$ and $s_{23}$, and has
an angular dependence due to the $J/\psi$ decay.
The BW amplitude for a resonance decaying into two spin-0 particles, labeled
as 2 and 3, is
$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma(s_{23})}~{},$ (9)
where $m_{R}$ is the resonance mass, $\Gamma(s_{23})$ is its energy-dependent
width that is parametrized as
$\Gamma(s_{23})=\Gamma_{0}\left(\frac{P_{R}}{P_{R_{0}}}\right)^{2L_{R}+1}\left(\frac{m_{R}}{\sqrt{s_{23}}}\right)F^{2}_{R}~{}.$
(10)
Here $\Gamma_{0}$ is the decay width when the invariant mass of the daughter
combinations is equal to $m_{R}$.
The Flatté model is parametrized as
$A_{R}(s_{23})=\frac{1}{m_{R}^{2}-s_{23}-im_{R}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK})}.$
(11)
The constants $g_{\pi\pi}$ and $g_{KK}$ are the $f_{0}(980)$ couplings to
$\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ final states respectively. The $\rho$
factors are given by Lorentz-invariant phase space
$\displaystyle\rho_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sqrt{1-\frac{4m^{2}_{\pi^{\pm}}}{m^{2}(\pi^{+}\pi^{-})}}+\frac{1}{3}\sqrt{1-\frac{4m^{2}_{\pi^{0}}}{m^{2}(\pi^{+}\pi^{-})}},$
(12) $\displaystyle\rho_{KK}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{\pm}}}{m^{2}(\pi^{+}\pi^{-})}}+\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{0}}}{m^{2}(\pi^{+}\pi^{-})}}.$
(13)
The non-resonant amplitude is parametrized as
$\mathcal{A}(s_{12},s_{23},\theta_{J/\psi})=\frac{P_{B}}{m_{B}}\sqrt{\sin^{2}\theta_{J/\psi}}.$
(14)
### 4.2 Detection efficiency
The detection efficiency is determined from a sample of one million $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ Monte Carlo (MC) events that are generated flat in phase
space with $J/\psi\rightarrow\mu^{+}\mu^{-}$, using Pythia [18] with a special
LHCb parameter tune [19], and the LHCb detector simulation based on Geant4
[20] described in Ref [21]. After the final selections the MC has 78,470
signal events, reflecting an overall efficiency of $7.8\%$. The acceptance in
$\cos\theta_{J/\psi}$ is uniform.
Next we describe the acceptance in terms of the mass squared variables. Both
$s_{12}$ and $s_{13}$ range from $10.2\;\;\rm GeV^{2}$ to $27.6\;\;\rm
GeV^{2}$, where $s_{13}$ is defined below, and thus are centered at 18.9 GeV2.
We model the detection efficiency using the symmetric Dalitz plot observables
$x=s_{12}-18.9~{}{\rm GeV}^{2},~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}y=s_{13}-18.9~{}{\rm GeV}^{2}.$ (15)
These variables are related to $s_{23}$ as
$s_{12}+s_{13}+s_{23}=m^{2}_{B}+m^{2}_{J/\psi}+m^{2}_{\pi^{+}}+m^{2}_{\pi^{-}}~{}.$
(16)
The detection efficiency is parametrized as a symmetric 4th order polynomial
function given by
$\displaystyle\varepsilon(s_{12},s_{23})$ $\displaystyle=$ $\displaystyle
1+\varepsilon_{1}(x+y)+\varepsilon_{2}(x+y)^{2}+\varepsilon_{3}xy+\varepsilon_{4}(x+y)^{3}+\varepsilon_{5}xy(x+y)$
(17)
$\displaystyle+\varepsilon_{6}(x+y)^{4}+\varepsilon_{7}xy(x+y)^{2}+\varepsilon_{8}x^{2}y^{2},$
where the $\varepsilon_{i}$ are the fit parameters.
The fitted polynomial function is shown in Fig. 7.
Figure 7: Parametrized detection efficiency as a function of $s_{23}\equiv
m^{2}(\pi^{+}\pi^{-})$ versus $s_{12}\equiv m^{2}(J/\psi\pi^{+})$. The scale
is arbitrary.
The projections of the fit used to measure the efficiency parameters are shown
in Fig. 8. The efficiency shapes are well described by the parametrization.
Figure 8: Projections of invariant mass squared of (a) $s_{12}\equiv
m^{2}(J/\psi\pi^{+})$ and (b) $s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$ of the MC
Dalitz plot used to measure the efficiency parameters. The points represent
the MC generated event distributions and the curves the polynomial fit.
To check the detection efficiency we compare our simulated $J/\psi\phi$ events
with our measured $J/\psi\phi$ helicity distributions. The events are
generated in the same manner as for $J/\psi\pi^{+}\pi^{-}$. Here we use the
measured helicity amplitudes of $\left|A_{||}(0)\right|^{2}=0.231$ and
$\left|A_{0}(0)\right|^{2}=0.524$ [7]. The background subtracted $J/\psi\phi$
angular distributions, $\cos\theta_{J/\psi}$ and $\cos\theta_{KK}$, defined in
the same manner as for the $J/\psi\pi^{+}\pi^{-}$ decay, are compared in Fig.
9 with the MC simulation. The $\chi^{2}$/ndf =389/400 is determined by binning
the angular distributions in two dimensions. The p-value is 64.1%. The
excellent agreement gives us confidence that the simulation accurately
predicts the acceptance.
Figure 9: Distributions of (a) $\cos\theta_{J/\psi}$, (b) $\cos\theta_{KK}$
for $J/\psi\phi$ background subtracted data (points) compared with the MC
simulation (histogram).
### 4.3 Background composition
The main background source is taken from the wrong-sign combinations within
$\pm 20$ MeV of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass peak. In addition, an extra 4.5% contribution from combinatorial
background formed by ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and random
$\rho(770)$, which cannot be present in wrong-sign combinations, is included
using a MC sample. The level is determined by measuring the background yield
as a function of $\pi^{+}\pi^{-}$ mass. The background model is parametrized
as
$B(s_{12},s_{23},\theta_{J/\psi})=B_{1}(s_{12},s_{23})\times\left(1+\alpha\cos\theta_{J/\psi}+\beta\cos^{2}\theta_{J/\psi}\right),$
(18)
where the first part $B_{1}(s_{12},s_{23})$ is modeled using the technique of
multiquadric radial basis functions [22]. These functions provide a useful way
to parametrize multi-dimensional data giving sensible non-erratic behaviour
and yet they follow significant variations in a smooth and faithful way. They
are useful in this analysis in providing a modeling of the decay angular
distributions in the resonance regions. Figure 10 shows the mass squared
projections from the fit. The $\chi^{2}/{\rm ndf}$ of the fit is 182/145. We
also used such functions with half the number of parameters and the changes
were insignificant. The second part
$\left(1+\alpha\cos\theta_{J/\psi}+\beta\cos^{2}\theta_{J/\psi}\right)$ is a
function of $J/\psi$ helicity angle. The $\cos\theta_{J/\psi}$ distribution of
background is shown in Fig. 11, fit with the function
$1+\alpha\cos\theta_{J/\psi}+\beta\cos^{2}\theta_{J/\psi}$ that determines the
parameters $\alpha=-0.0050\pm 0.0201$ and $\beta=-0.2308\pm 0.0036$.
Figure 10: Projections of invariant mass squared of (a) $s_{12}\equiv
m^{2}(J/\psi\pi^{+})$ and (b) $s_{23}\equiv m^{2}(\pi^{\pm}\pi^{\pm})$ of the
background Dalitz plot. Figure 11: The $\cos\theta_{J/\psi}$ distribution of
the background and the fitted function
$1+\alpha\cos\theta_{J/\psi}+\beta\cos^{2}\theta_{J/\psi}$.
## 5 Final state composition
### 5.1 Resonance models
To study the resonant structures of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ we use 13,424 candidates with invariant mass within $\pm
20$ MeV of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass
peak. This includes both signal and background. Possible resonance candidates
in the decay $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ are listed in Table 1.
Table 1: Possible resonance candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$ decay mode. Resonance | Spin | Helicity | Resonance
---|---|---|---
| | | formalism
$f_{0}(600)$ | 0 | 0 | BW
$\rho(770)$ | 1 | $0,\pm 1$ | BW
$f_{0}(980)$ | 0 | 0 | Flatté
$f_{2}(1270)$ | 2 | $0,\pm 1$ | BW
$f_{0}(1370)$ | 0 | 0 | BW
$f_{0}(1500)$ | 0 | 0 | BW
To understand what resonances are likely to contribute, it is important to
realize that the $s\bar{s}$ system in Fig. 1 is isoscalar ($I=0$) so when it
produces a single meson it must have zero isospin, resulting in a symmetric
isospin wavefunction for the two-pion system. Since the two-pions must be in
an overall symmetric state, they must have even total angular momentum. In
fact we only need to consider spin-0 and spin-2 particles as there are no
known spin-4 particles in the kinematically accessible mass range below 1600
MeV. The particles that could appear are spin-0 $f_{0}(600)$, spin-0
$f_{0}(980)$, spin-2 $f_{2}(1270)$, spin-0 $f_{0}(1370)$ and spin-0
$f_{0}(1500)$. Diagrams of higher order than the one shown in Fig. 1 could
result in the production of isospin-one $\pi^{+}\pi^{-}$ resonances, thus we
use the $\rho(770)$ as a test of the presence of these higher order processes.
We proceed by fitting with a single $f_{0}(980)$, established from earlier
measurements [1], and adding single resonant components until acceptable fits
are found. Subsequently, we try the addition of other resonances. The models
used are listed in Table 2.
Table 2: Models used in data fit. Name | Components
---|---
Single R | $f_{0}(980)$
2R | $f_{0}(980)+f_{0}(1370)$
3R | $f_{0}(980)+f_{0}(1370)+f_{2}(1270)$
3R+NR | $f_{0}(980)+f_{0}(1370)+f_{2}(1270)+~{}$non-resonant
3R+NR + $\rho(770)$ | $f_{0}(980)+f_{0}(1370)+f_{2}(1270)+~{}$non-resonant $+\rho(770)$
3R+NR +$f_{0}(1500)$ | $f_{0}(980)+f_{0}(1370)+f_{2}(1270)+~{}$non-resonant $+f_{0}(1500)$
3R+NR +$f_{0}(600)$ | $f_{0}(980)+f_{0}(1370)+f_{2}(1270)+~{}$non-resonant $+f_{0}(600)$
The masses and widths of the BW resonances are listed in Table 3. When used in
the fit they are fixed to these values, except for the $f_{0}(1370)$, for
which they are not well measured, and thus are allowed to vary using their
quoted errors as constraints in the fits, taking the errors as being Gaussian.
Besides the mass and width, the Flatté resonance shape has two additional
parameters $g_{\pi\pi}$ and $g_{KK}$, which are also allowed to vary in the
fit. Parameters of the non-resonant amplitude are also allowed to vary. One
magnitude and one phase in each helicity grouping have to be fixed, since the
overall normalization is related to the signal yield, and only relative phases
are physically meaningful. The normalization and phase of $f_{0}(980)$ are
fixed to 1 and 0 respectively. The phase of $f_{2}(1270)$, with helicity $=\pm
1$ is also fixed to zero when it is included. All background and efficiency
parameters are held static in the fit.
Table 3: Breit-Wigner resonance parameters. Resonance | Mass (MeV) | Width (MeV) | Source
---|---|---|---
$f_{0}(600)$ | $513\pm 32$ | $335\pm 67$ | CLEO [23]
$\rho(770)$ | $775.5\pm 0.3$ | $149.1\pm 0.8$ | PDG [11]
$f_{2}(1270)$ | $1275\pm 1$ | $185\pm 3$ | PDG [11]
$f_{0}(1370)$ | $1434\pm 20$ | $172\pm 33$ | E791 [24]
$f_{0}(1500)$ | $1505\pm 6$ | $109\pm$7 | PDG [11]
To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as
${\cal{L}}=\prod_{i=1}^{N}F\left(s_{12}^{i},s_{23}^{i},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}^{i}\right),$ (19)
where $N$ is the total number of events, and $F$ is the total PDF defined in
Eq. 1. The PDF is constructed from the signal fraction $f_{\rm sig}$,
efficiency model $\varepsilon(s_{12},s_{23})$, background model
$B(s_{12},s_{23},\theta_{J/\psi})$ and the signal model
$S(s_{12},s_{23},\theta_{J/\psi})$. The PDF needs to be normalized. This is
accomplished by first normalizing the $J/\psi$ helicity dependent part by
analytical integration, and then for the mass dependent part using numerical
integration over 500$\times$500 bins.
### 5.2 Fit results
In order to compare the different models quantitatively an estimate of the
goodness of fit is calculated from 3D partitions of the one angular and two
mass-squared variables. We use the Poisson likelihood $\chi^{2}$ [25] defined
as
$\chi^{2}=2\sum_{i=1}^{N_{\rm
bin}}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(20)
where $n_{i}$ is the number of events in the three dimensional bin $i$ and
$x_{i}$ is the expected number of events in that bin according to the fitted
likelihood function. A total of $N_{\rm bin}=1356$ bins are used to calculate
the $\chi^{2}$, using the variables $m^{2}(J/\psi\pi^{+})$,
$m^{2}(\pi^{+}\pi^{-})$, and $\cos\theta_{J/\psi}$. The $\chi^{2}/\text{ndf}$
and the negative of the logarithm of the likelihood, $\rm-ln\mathcal{L}$, of
the fits are given in Table 4. There are two solutions of almost equal
likelihood for the 3R+NR model. Based on a detailed study of angular
distributions (see Section 5.3) we choose one of these solutions and label it
as “preferred”. The other solution is called “alternate.” We will use the
differences between these to assign systematic uncertainties to the resonance
fractions.
Table 4: $\chi^{2}/\text{ndf}$ and $\rm-ln\mathcal{L}$ of different resonance models. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$ | Probability (%)
---|---|---|---
Single R | 59269 | 1956/1352 | 0
2R | 59001 | 1498/1348 | 0.25
3R | 58973 | 1455/1345 | 1.88
3R+NR (preferred) | 58945 | 1415/1343 | 8.41
3R+NR (alternate) | 58946 | 1414/1343 | 8.70
3R+NR + $\rho(770)$ (preferred) | 58945 | 1418/1341 | 7.05
3R+NR + $\rho(770)$ (alternate) | 58944 | 1416/1341 | 7.57
3R+NR + $f_{0}(1500)$ (preferred) | 58943 | 1416/1341 | 7.57
3R+NR + $f_{0}(1500)$ (alternate) | 58941 | 1407/1341 | 10.26
3R+NR + $f_{0}(600)$ (preferred) | 58935 | 1409/1341 | 9.60
3R+NR + $f_{0}(600)$ (alternate) | 58937 | 1412/1341 | 8.69
The probability is improved noticeably adding components up to 3R+NR. Figure
12 shows the preferred model projections of $m^{2}(\pi^{+}\pi^{-})$ for the
preferred model including only the 3R+NR components. The projections for the
other considered models are indiscernible.
Figure 12: Dalitz fit projections of $m^{2}(\pi^{+}\pi^{-})$ fit with 3R+NR
for the preferred model. The points with error bars are data, the signal fit
is shown with a (red) dashed line, the background with a (black) dotted line,
and the (blue) solid line represents the total. The normalized residuals in
each bin are shown below, defined as the difference between the data and the
fit divided by the error on the data.
The preferred model projections of $m^{2}(J/\psi\pi^{+})$ and
$\cos\theta_{J/\psi}$ are shown in Fig. 13 for the preferred model 3R+NR fit.
The projections of the other preferred model fits including the additional
resonances are almost identical.
Figure 13: Dalitz fit projections of (a) $s_{12}\equiv m^{2}(J/\psi\pi^{+})$
and (b) $\cos\theta_{J/\psi}$ fit with the 3R+NR preferred model. The points
with error bars are data, the signal fit is shown with a (red) dashed line,
the background with a (black) dotted line, and the (blue) solid line
represents the total.
While a complete description of the decay is given in terms of the fitted
amplitudes and phases, knowledge of the contribution of each component can be
summarized by defining a fit fraction, ${\cal{F}}^{R}_{\lambda}$. To determine
${\cal{F}}^{R}_{\lambda}$ we integrate the squared amplitude of $R$ over the
Dalitz plot. The yield is then normalized by integrating the entire signal
function over the same area. Specifically,
${\cal{F}}^{R}_{\lambda}=\frac{\int\left|a^{R}_{\lambda}e^{i\phi^{R}_{\lambda}}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi})\right|^{2}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}{\int
S(s_{12},s_{23},\theta_{J/\psi})~{}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}.$
(21)
Note that the sum of the fit fractions is not necessarily unity due to the
potential presence of interference between two resonances. Interference term
fractions are given by
${\cal{F}}^{RR^{\prime}}_{\lambda}=2\mathcal{R}e\left(\frac{\int
a^{R}_{\lambda}\;a^{R^{\prime}}_{\lambda}e^{i(\phi^{R}_{\lambda}-\phi^{R^{\prime}}_{\lambda})}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi}){\mathcal{A}_{\lambda}^{R^{\prime}}}^{*}(s_{12},s_{23},\theta_{J/\psi})ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}{\int
S(s_{12},s_{23},\theta_{J/\psi})~{}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}\right),$
(22)
and
$\sum_{\lambda}\left(\sum_{R}{\cal{F}}^{R}_{\lambda}+\sum_{RR^{\prime}}{\cal{F}}^{RR^{\prime}}_{\lambda}\right)=1.$
(23)
If the Dalitz plot has more destructive interference than constructive
interference, the total fit fraction will be greater than one. Note that,
interference between different spin-$J$ states vanishes because the
$d^{J}_{\lambda 0}$ angular functions in $\mathcal{A}^{R}_{\lambda}$ are
orthogonal.
The determination of the statistical errors of the fit fractions is difficult
because they depend on the statistical errors of every fitted magnitude and
phase. A toy Monte Carlo approach is used. We perform 500 toy experiments:
each sample is generated according to the model PDF, input parameters are
taken from the fit to the data. The correlations of fitted parameters are also
taken into account. For each toy experiment the fit fractions are calculated.
The distributions of the obtained fit fractions are described by Gaussian
functions. The r.m.s. widths of the Gaussians are taken as the statistical
errors on the corresponding parameters. The fit fractions are listed in Table
5.
Table 5: Fit fractions (%) of contributing components for the preferred model. For P- and D-waves $\lambda$ represents the final state helicity. Here $\rho$ refers to the $\rho(770)$ meson. Components | 3R+NR | 3R+NR+$\rho$ | 3R+NR+$f_{0}(1500)$ | 3R+NR+$f_{0}(600)$
---|---|---|---|---
$f_{0}(980)$ | $107.1\pm 3.5$ | $104.8\pm 3.9$ | $73.0\pm 5.8$ | $115.2\pm 5.3$
$f_{0}(1370)$ | $32.6\pm 4.1$ | $32.3\pm 3.7$ | $114\pm 14$ | $34.5\pm 4.0$
$f_{0}(1500)$ | - | - | $15.0\pm 5.1$ | -
$f_{0}(600)$ | - | - | - | $4.7\pm 2.5$
NR | $12.84\pm 2.32$ | $12.2\pm 2.2$ | $10.7\pm 2.1$ | $23.7\pm 3.6$
$f_{2}(1270)$, $\lambda=0$ | $0.76\pm 0.25$ | $0.77\pm 0.25$ | $1.07\pm 0.37$ | $0.90\pm 0.31$
$f_{2}(1270)$, $|\lambda|=1$ | $0.33\pm 1.00$ | $0.26\pm 1.12$ | $1.02\pm 0.83$ | $0.61\pm 0.87$
$\rho$, $\lambda=0$ | - | $0.66\pm 0.53$ | - | -
$\rho$, $|\lambda|=1$ | - | $0.11\pm 0.78$ | - | -
Sum | $153.6\pm 6.0$ | $151.1\pm 6.0$ | $214.4\pm 15.7$ | $179.6\pm 8.0$
$\rm-ln\mathcal{L}$ | 58945 | 58944 | 58943 | 58935
$\chi^{2}$/ndf | 1415/1343 | 1418/1341 | 1416/1341 | 1409/1341
Probability(%) | 8.41 | 7.05 | 7.57 | 9.61
The 3R+NR fit describes the data well. For models adding more resonances, the
additional components never have more than 3 standard deviation ($\sigma$)
significance, and the fit likelihoods are only slightly improved. In the 3R+NR
solution all the components have more than $3\sigma$ significance, except for
the $f_{2}(1270)$ where we allow the helicity $\pm$1 components since the
helicity 0 component is significant. In all cases, we find the dominant
contribution is S-wave which agrees with our previous less sophisticated
analysis [5]. The D-wave contribution is small. The P-wave contribution is
consistent with zero, as expected. The fit fractions from the alternate model
are listed in Table 6. There are only small changes in the $f_{2}(1270)$ and
$\rho(770)$ components.
Table 6: Fit fractions (%) of contributing components from different models for the alternate solution. For P- and D-waves $\lambda$ represents the final state helicity. Here $\rho$ refers to the $\rho(770)$ meson. Components | 3R+NR | 3R+NR+$\rho$ | 3R+NR+$f_{0}(1500)$ | 3R+NR+$f_{0}(600)$
---|---|---|---|---
$f_{0}(980)$ | $100.8\pm 2.9$ | $99.2\pm 4.2$ | $96.9\pm 3.8$ | $111\pm 15$
$f_{0}(1370)$ | $7.0\pm 0.9$ | $6.9\pm 0.9$ | $3.0\pm 1.7$ | $8.0\pm 1.1$
$f_{0}(1500)$ | - | - | $4.7\pm 1.7$ | -
$f_{0}(600)$ | - | - | - | $4.3\pm 2.3$
NR | $13.8\pm 2.3$ | $13.4\pm 2.7$ | $13.4\pm 2.4$ | $24.7\pm 3.9$
$f_{2}(1270)$, $\lambda=0$ | $0.51\pm 0.14$ | $0.52\pm 0.14$ | $0.50\pm 0.14$ | $0.51\pm 0.14$
$f_{2}(1270)$, $|\lambda|=1$ | $0.24\pm 1.11$ | $0.19\pm 1.38$ | $0.63\pm 0.84$ | $0.48\pm 0.89$
$\rho$, $\lambda=0$ | - | $0.43\pm 0.55$ | - | -
$\rho$, $|\lambda|=1$ | - | $0.14\pm 0.78$ | - | -
Sum | $122.4\pm 4.0$ | $120.8\pm 5.3$ | $119.2\pm 5.2$ | $148.7\pm 15.5$
$\rm-ln\mathcal{L}$ | 58946 | 58945 | 58941 | 58937
$\chi^{2}$/ndf | 1414/1343 | 1416/1341 | 1407/1341 | 1412/1341
Probability(%) | 8.70 | 7.57 | 10.26 | 8.69
The fit fractions of the interference terms for the preferred and alternate
models are computed using Eq. 22 and listed in Table 7.
Table 7: Fit fractions (%) of interference terms for both solutions of the 3R+NR model. Components | Preferred | Alternate
---|---|---
$f_{0}(980)$ \+ $f_{0}(1370)$ | $-36.6\pm 4.6$ | $-5.4\pm 2.3$
$f_{0}(980)$ \+ NR | $-16.1\pm 2.7$ | $-23.6\pm 2.6$
$f_{0}(1370)$ \+ NR | $~{}0.8\pm 1.0$ | $~{}~{}6.6\pm 0.8$
Sum | $-53.6\pm 5.5$ | $-22.4\pm 3.6$
### 5.3 Helicity distributions
Only S and D waves contribute to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ final state in the $m(\pi^{+}\pi^{-})$ region below 1550
MeV. Helicity information is already included in the signal model via Eqs. 7
and 8. For a spin-0 $\pi^{+}\pi^{-}$ system $\cos\theta_{J/\psi}$ should be
distributed as $1-\cos^{2}\theta_{J/\psi}$ and $\cos\theta_{\pi\pi}$ should be
flat. To test our fits we examine the $\cos\theta_{J/\psi}$ and
$\cos\theta_{\pi\pi}$ distribution in different regions of $\pi^{+}\pi^{-}$
mass. The decay rate with respect to the cosine of the helicity angles is
given by [5]
$\displaystyle\frac{d\Gamma}{\textbullet
d\cos\theta_{J/\psi}d\cos\theta_{\pi\pi}}=$
$\displaystyle\left|A_{00}+\frac{1}{2}A_{20}e^{i\phi}\sqrt{5}(3\cos^{2}\theta_{\pi\pi}-1)\right|^{2}\sin^{2}\theta_{J/\psi}$
$\displaystyle+\frac{1}{4}\left(\left|A_{21}\right|^{2}+\left|A_{2-1}\right|^{2}\right)\left(15\sin^{2}\theta_{\pi\pi}\cos^{2}\theta_{\pi\pi}\right)\left(1+\cos^{2}\theta_{J/\psi}\right),$
where $A_{00}$ is the S-wave amplitude, $A_{2i},\;i=-1,0,1$, the three D-wave
amplitudes, and $\phi$ is the strong phase between $A_{00}$ and $A_{20}$
amplitudes. Non-flat distributions in $\cos\theta_{\pi\pi}$ would indicate
interference between the S-wave and D-wave amplitudes.
To investigate the angular structure we then split the helicity distributions
into three different $\pi^{+}\pi^{-}$ mass regions: one is the $f_{0}(980)$
region defined within $\pm 90$ MeV of the $f_{0}(980)$ mass and the others are
defined within one full width of the $f_{2}(1270)$ and $f_{0}(1370)$ masses,
respectively (the width values are given in Table 3). The
$\cos\theta_{J/\psi}$ and $\cos\theta_{\pi\pi}$ background-subtracted
efficiency corrected distributions for these three different mass regions are
presented in Figs. 14 and 15. The distributions are in good agreement with the
3R+NR preferred signal model. Furthermore, splitting into two bins, $[-90,0]$
and $[0,90]$ MeV, we see different shapes, because across the pole mass of
$f_{0}(980)$, the $f_{0}(980)$’s phase changes by $\pi$. Hence the relative
phase between $f_{0}(980)$ and the small D-wave in the two regions changes
very sharply. This feature is reproduced well by the “preferred” model and
shown in Fig. 16. The “alternate” model gives an acceptable, but poorer
description.
Figure 14: Background subtracted and acceptance corrected
$\cos\theta_{J/\psi}$ helicity distributions fit with the preferred model: (a)
in $f_{0}(980)$ mass region defined within $\pm 90$ MeV of 980 MeV
($\chi^{2}$/ndf =39/40), (b) in $f_{2}(1270)$ mass region defined within one
full width of $f_{2}(1270)$ mass ($\chi^{2}$/ndf =25/40), (c) in $f_{0}(1370)$
mass region defined within one full width of $f_{2}(1370)$ mass
($\chi^{2}$/ndf = 24/40). The points with error bars are data and the solid
blue lines show the fit from the 3R+NR model. Figure 15: Background subtracted
and acceptance corrected $\cos\theta_{\pi\pi}$ helicity distributions fit the
preferred model: (a) in $f_{0}(980)$ mass region defined within $\pm 90$ MeV
of 980 MeV ($\chi^{2}$/ndf =38/40), (b) in $f_{2}(1270)$ mass region defined
within one full width of $f_{2}(1270)$ mass ($\chi^{2}$/ndf = 32/40), (c) in
$f_{0}(1370)$ mass region defined within one full width of $f_{2}(1370)$ mass
($\chi^{2}$/ndf =37/40). The points with error bars are data and the solid
blue lines show the fit from the 3R+NR model. Figure 16: Background subtracted
and acceptance corrected $\cos\theta_{\pi\pi}$ helicity distributions fit the
preferred model: (a) in $[-90,0]$ MeV of 980 MeV ($\chi^{2}$/ndf =41/40), (b)
in $[0,90]$ MeV of 980 MeV ($\chi^{2}$/ndf =31/40)
### 5.4 Resonance parameters
The fit results from the four-component best fit are listed in Table 8 for
both the preferred and alternate solutions. The table summarizes the
$f_{0}(980)$ mass, the Flatté resonances parameters $g_{\pi\pi}$,
$g_{KK}/g_{\pi\pi}$, $f_{0}(1370)$ mass and width and the phases of the
contributing resonances.
Table 8: Fit results from the 3R+NR model for both the preferred and alternate solutions. $\phi$ indicates the phase with respect to the $f_{0}(980)$. For the $f_{2}(1270)$, $\lambda$ represents the final state helicity. The parameters | Preferred | Alternate
---|---|---
$m_{f_{0}(980)}$(MeV) | $939.9\pm 6.3$ | $939.2\pm 6.5$
$g_{\pi\pi}$(MeV) | $199\pm 30$ | $197\pm 25$
$g_{KK}/g_{\pi\pi}$ | $3.0\pm 0.3$ | $3.1\pm 0.2$
$m_{f_{0}(1370)}$(MeV) | $1475.1\pm 6.3$ | $1474.4\pm 6.0$
$\Gamma_{f_{0}(1370)}$(MeV) | $113\pm 11$ | $108\pm 11$
$\phi_{980}$ | 0 (fixed) | 0 (fixed)
$\phi_{1370}$ | $241.5\pm 6.3$ | $181.7\pm 8.4$
$\phi_{\rm NR}$ | $217.0\pm 3.7$ | $232.2\pm 3.7$
$\phi_{1270}$, $\lambda=0$ | $165\pm 15$ | $118\pm 15$
$\phi_{1270}$, $|\lambda|=1$ | 0 (fixed) | 0 (fixed)
The mass and resonance parameters depend strongly on the final state in which
they are measured, and the form of the resonance fitting function. Thus we do
not quote systematic errors on these values. The value found for the
$f_{0}(980)$ mass in the Flatté function $939.9\pm 6.3$ MeV is lower than most
determinations, although the observed peak value is close to 980 MeV, the
estimated PDG value [11]. This is due to the interference from other
resonances. The BES collaboration using the same functional form found a mass
value of 965$\pm 8\pm$6 MeV in the $J/\psi\rightarrow\phi\pi^{+}\pi^{-}$ final
state [26]. They also found roughly similar values of the coupling constants
as ours, $g_{\pi\pi}=165\pm 10\pm 15$ MeV, and $g_{KK}/g_{\pi\pi}=4.21\pm
0.25\pm 0.21$. The PDG provides only estimated values for the $f_{0}(1370)$
mass of 1200$-$1500 MeV and width 200$-$500 MeV, respectively [11]. Our result
is within both of these ranges.
### 5.5 Angular moments
The angular moment distributions provide an additional way of visualizing the
effects of different resonances and their interferences, similar to a partial
wave analysis. This technique has been used in previous studies [27,
*delAmoSanchez:2010yp].
We define the angular moments $\langle Y_{l}^{0}\rangle$ as the efficiency
corrected and background subtracted $\pi^{+}\pi^{-}$ invariant mass
distributions, weighted by spherical harmonic functions
$\langle
Y_{l}^{0}\rangle=\int_{-1}^{1}d\Gamma(m_{\pi\pi},\cos\theta_{\pi\pi})Y_{l}^{0}(\cos\theta_{\pi\pi})d\cos\theta_{\pi\pi}.$
(25)
The spherical harmonic functions satisfy
$\int_{-1}^{1}Y_{i}^{0}(\cos\theta_{\pi\pi})Y_{j}^{0}(\cos\theta_{\pi\pi})d\cos\theta_{\pi\pi}=\frac{\delta_{ij}}{2\pi}.$
(26)
If we assume that no $\pi^{+}\pi^{-}$ partial-waves of a higher order than
D-wave contribute, then we can express the differential decay rate ($d\Gamma$)
derived from Eq. (3) in terms of S-, P-, and D-waves including helcity 0 and
$\pm 1$ components as
$\displaystyle d\Gamma(m_{\pi\pi},\cos\theta_{\pi\pi})$ $\displaystyle=$
$\displaystyle 2\pi\left|{\cal A}_{S_{0}}Y_{0}^{0}(\cos\theta_{\pi\pi})+{\cal
A}_{P_{0}}e^{i\phi_{P_{0}}}Y_{1}^{0}(\cos\theta_{\pi\pi})+{\cal
A}_{D_{0}}e^{i\phi_{D_{0}}}Y_{2}^{0}(\cos\theta_{\pi\pi})\right|^{2}$ (27)
$\displaystyle+$ $\displaystyle 2\pi\left|{\cal A}_{P_{\pm 1}}e^{i\phi_{P_{\pm
1}}}\sqrt{\frac{3}{8\pi}}\sin\theta_{\pi\pi}+{\cal A}_{D_{\pm
1}}e^{i\phi_{D_{\pm
1}}}\sqrt{\frac{15}{8\pi}}\sin\theta_{\pi\pi}\cos\theta_{\pi\pi}\right|^{2},$
where ${\cal A}_{k_{\lambda}}$ and $\phi_{k_{\lambda}}$ are real-valued
functions of $m_{\pi\pi}$, and we have factored out the S-wave phase. We then
calculate the angular moments
$\displaystyle\sqrt{4\pi}\langle Y_{0}^{0}\rangle$ $\displaystyle=$
$\displaystyle{\cal A}_{S_{0}}^{2}+{\cal A}_{P_{0}}^{2}+{\cal
A}_{D_{0}}^{2}+{\cal A}_{P_{\pm 1}}^{2}+{\cal A}_{D_{\pm 1}}^{2},$
$\displaystyle\sqrt{4\pi}\langle Y_{1}^{0}\rangle$ $\displaystyle=$
$\displaystyle 2{\cal A}_{S_{0}}{\cal
A}_{P_{0}}\cos\phi_{P_{0}}+\frac{4}{\sqrt{5}}{\cal A}_{P_{0}}{\cal
A}_{D_{0}}\cos(\phi_{P_{0}}-\phi_{D_{0}})+8\sqrt{\frac{3}{5}}{\cal A}_{P_{\pm
1}}{\cal A}_{D_{\pm 1}}\cos(\phi_{P_{\pm 1}}-\phi_{D_{\pm 1}}),$
$\displaystyle\sqrt{4\pi}\langle Y_{2}^{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{5}}{\cal A}_{P_{0}}^{2}+2{\cal A}_{S_{0}}{\cal
A}_{D_{0}}\cos\phi_{D_{0}}+\frac{2\sqrt{5}}{7}{\cal
A}_{D_{0}}^{2}-\frac{1}{\sqrt{5}}{\cal A}_{P_{\pm
1}}^{2}+\frac{\sqrt{5}}{7}{\cal A}_{D_{\pm 1}}^{2},$
$\displaystyle\sqrt{4\pi}\langle Y_{3}^{0}\rangle$ $\displaystyle=$
$\displaystyle 6\sqrt{\frac{3}{35}}{\cal A}_{P_{0}}{\cal
A}_{D_{0}}\cos(\phi_{P_{0}}-\phi_{D_{0}})+\frac{6}{\sqrt{35}}{\cal A}_{P_{\pm
1}}{\cal A}_{D_{\pm 1}}\cos(\phi_{P_{\pm 1}}-\phi_{D_{\pm 1}}),$
$\displaystyle\sqrt{4\pi}\langle Y_{4}^{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{6}{7}{\cal A}_{D_{0}}^{2}-\frac{4}{7}{\cal A}_{D_{\pm
1}}^{2}.$ (28)
Figure 17 shows the distributions of the angular moments for the preferred
solution. In general the interpretation of these moments is that $\langle
Y^{0}_{0}\rangle$ is the efficiency corrected and background subtracted event
distribution, $\langle Y^{0}_{1}\rangle$ the interference of the sum of S-wave
and P-wave and P-wave and D-wave amplitudes, $\langle Y^{0}_{2}\rangle$ the
sum of the P-wave, D-wave and the interference of S-wave and D-wave
amplitudes, $\langle Y^{0}_{3}\rangle$ the interference between P-wave and
D-wave, and $\langle Y^{0}_{4}\rangle$ the D-wave.
In our data the $\langle Y^{0}_{1}\rangle$ distribution is consistent with
zero, confirming the absence of any P-wave. We do observe the effects of the
$f_{2}(1270)$ in the $\langle Y^{0}_{2}\rangle$ distribution including the
interferences with the S-waves. The other moments are consistent with the
absence of any structure, as expected.
Figure 17: The $\pi^{+}\pi^{-}$ mass dependence of the spherical harmonic
moments of $\cos\theta_{\pi\pi}$ after efficiency corrections and background
subtraction: (a) $\langle Y^{0}_{0}\rangle$, (b) $\langle Y^{0}_{1}\rangle$,
(c) $\langle Y^{0}_{2}\rangle$, (d) $\langle Y^{0}_{3}\rangle$, (e) $\langle
Y^{0}_{4}\rangle$, (f) $\langle Y^{0}_{5}\rangle$, (g) $\langle
Y^{0}_{6}\rangle$, and (h) $\langle Y^{0}_{7}\rangle$. The points with error
bars are the data points and the solid curves are derived from the 3R+NR
preferred model.
## 6 Results
### 6.1 $C\\!P$ content
The main result in this paper is that $C\\!P$-odd final states dominate. The
$f_{2}(1270)$ helicity $\pm 1$ yield is ($0.21\pm 0.65$)%. As this represents
a mixed $C\\!P$ state, the upper limit on the $C\\!P$-even fraction due to
this state is $<1.3$ % at 95% confidence level (CL). Adding the $\rho(770)$
amplitude and repeating the fit shows that only an insignificant amount of
$\rho(770)$ can be tolerated; in fact, the isospin violating $J/\psi\rho(770)$
final state is limited to $<$ 1.5% at 95% CL. The sum of $f_{2}(1270)$
helicity $\pm 1$ and $\rho(770)$ is limited to $<$ 2.3% at 95% CL. In the
$\pi^{+}\pi^{-}$ mass region within $\pm$90 MeV of 980 MeV, this limit
improves to $<$ 0.6% at 95% CL.
### 6.2 Total branching fraction ratio
To avoid the uncertainties associated with absolute branching fraction
measurements, we quote branching fractions relative to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ channel.
The detection efficiency for this channel from Monte Carlo simulation is
$(1.07\pm 0.01)$%, where the error is due to the limited Monte Carlo sample
size.
The simulated detection efficiency for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ as a function of the $m^{2}(\pi^{+}\pi^{-})$ is shown in
Fig. 18. The simulation does not model the pion and kaon identification
efficiencies with sufficient accuracy for our purposes. Therefore, we measure
the kaon identification efficiency with respect to the Monte Carlo simulation.
We use samples of $D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow
K^{-}\pi^{+}$ events selected without kaon identification to measure the kaon
and pion efficiencies with respect to the simulation, and an additional sample
of $K_{s}^{0}\rightarrow\pi^{+}\pi^{-}$ decay for pions. The identification
efficiency is measured in bins of $p_{\rm T}$ and $\eta$ and then the averages
are weighted using the event distributions in the data. We find the correction
to the $J/\psi\phi$ efficiency is 0.970 (two kaons) and that to the $J/\psi
f_{0}$ efficiency is 0.973 (two pions). The additional correction due to
particle identification then is 0.997$\pm$0.010. In addition, we re-weight the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ $p$ and $p_{\rm T}$
distributions in the simulation which lowers the $\pi^{+}\pi^{-}$ efficiency
by 1.01% with respect to the $K^{+}K^{-}$ efficiency.
Dividing the number of the $J/\psi\pi^{+}\pi^{-}$ signal events by the $J/\psi
K^{+}K^{-}$ yield, applying the additional corrections as described above, and
taking into account ${\cal{B}}\left(\phi\rightarrow K^{+}K^{-}\right)=(48.9\pm
0.5)$% [11], we find
$\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi\right)}=(19.79\pm 0.47\pm 0.52)\%.$
Whenever two uncertainties are quoted the first is statistical and the second
systematic. The latter will be discussed later in Section 7. This branching
fraction ratio has not been previously measured.
Figure 18: Detection efficiency of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ as a function of $s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$.
### 6.3 Relative resonance yields
Next we evaluate the relative yields for the 3R+NR fit to the
$J/\psi\pi^{+}\pi^{-}$ final state from the preferred solution. We normalize
the individual fit fractions reported in Table 5 by the sum. These normalized
fit fractions are listed in Table 9 along with the branching fraction relative
to $J/\psi\phi$, $\phi\rightarrow K^{+}K^{-}$, defined as $R_{r}$, where $r$
refers to the particular final state under consideration. Thus
$R_{r}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
r\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi\right)}~{}.$ (29)
We use the difference between the preferred and alternate solutions found for
the 3R+NR fit to assign a systematic uncertainty. Other systematic
uncertainties are described in Section 7.
The value found for $R_{r}$ for the $f_{0}(980)$, $0.139\pm
0.006^{+0.025}_{-0.012}$, is consistent with the prediction of Ref. [12], and
consistent with the our first observation using 33 pb-1 of integrated
luminosity [1], after multiplying by ${\cal{B}}\left(\phi\rightarrow
K^{+}K^{-}\right)$. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(1370)$
is now established. Previously both LHCb [1] and Belle [2] had seen evidence
for this final state. The normalized $f_{2}(1270)$ helicity zero rate is
(0.49$\pm$0.16)% in the preferred model and (0.42$\pm$0.11)% for the alternate
solution.
Table 9: Normalized fit fractions (%) for alternate and preferred 3R+NR models and the ratio $R$ (%) relative to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$. The numbers for the $f_{2}(1270)$ refer only to the $\lambda=0$ state. State | Preferred | Alternate | $R$ preferred | $R$ alternate | Final $R$
---|---|---|---|---|---
$f_{0}(980)$ | $69.7\pm 2.3$ | $82.4\pm 2.3$ | $13.9\pm 0.6$ | $16.3\pm 0.6$ | $13.9\pm 0.6^{+2.5}_{-1.2}$
$f_{0}(1370)$ | $21.2\pm 2.7$ | $5.7\pm 0.7$ | $4.19\pm 0.53$ | $1.13\pm 0.15$ | $4.19\pm 0.53^{+0.12}_{-3.70}$
NR | $8.4\pm 1.5$ | $11.3\pm 1.9$ | $1.66\pm 0.31$ | $2.23\pm 0.39$ | $1.66\pm 0.31^{+0.96}_{-0.08}$
$f_{2}(1270)$ | $0.49\pm 0.16$ | $0.42\pm 0.11$ | $0.098\pm 0.033$ | $0.083\pm 0.022$ | $0.098\pm 0.033^{+0.006}_{-0.015}$
## 7 Systematic uncertainties
Systematic uncertainties on the $C\\!P$-odd fraction are negligible. In fact,
using any of the alternate fits with different additional components does not
introduce any significant fractions of $C\\!P$-odd final states.
The systematic uncertainties on the branching fraction ratios have several
contributions listed in Table 10. Since $R_{r}$ is measured relative to
$J/\psi\phi$ there is no systematic uncertainty due to differences in the
tracking performance between data and simulation. The $J/\psi\phi$ P-wave
yield is fully correlated with the S-wave yield whose uncertainty we estimate
as 0.7% by changing the signal PDF, and the background shape. By far the
largest uncertainty in every rate, except the total, is caused by our choice
of the preferred versus the alternate solutions. Using the difference between
these fit results for the systematic uncertainty causes relatively large and
asymmetric values. We also include systematic uncertainties due to the
possible presence of the $\rho(770)$, the $f_{0}(1500)$, or the $f_{0}(600)$
resonances by taking the maximum difference between the fit including one of
these resonances and our preferred solution, if the difference is larger than
the one between the preferred and alternate 3R+NR fit. In the case of the
$f_{0}(1500)$ the preferred solution is pathological in that it produces an
unacceptably large $f_{0}(1370)$ component along with a 214% component sum;
therefore here we use the alternate solution that is much better behaved.
The uncertainty from Monte Carlo sample size for the mass dependent
$\pi^{+}\pi^{-}$ efficiencies are accounted for in the statistical errors, a
residual systematic uncertainty is included that results from allowed changes
in the shape due to the distribution of the events. The size of these
differences depends on the mass range for the particular component multiplied
by the possible efficiency variation across this mass range. This is estimated
as 1% for the entire mass range and is smaller for individual resonances.
Small uncertainties are introduced if the simulation does not have the correct
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ kinematic
distributions. We are relatively insensitive to any these differences in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ $p$ and $p_{\rm T}$
distributions since we are measuring relative rates. These distributions are
varied by changing the weights in each bin by plus and minus the statistical
error in that bin. We see at most a 0.5% change. There is a 2% systematic
uncertainty assigned for the relative particle identification efficiencies.
These efficiencies have been corrected from those predicted in the simulation
by using pion data from $K_{s}^{0}\rightarrow\pi^{+}\pi^{-}$ decays and kaon
and pion data from $D^{*\pm}\rightarrow\pi^{\pm}D^{0}(\overline{D}^{0})$,
$D^{0}(\overline{D}^{0})\rightarrow K^{\mp}\pi^{\pm}$ decays. The uncertainty
on the corrections is 0.5% per track. The background modeling was changed by
using a second-order polynomial shape in the $J/\psi\pi^{+}\pi^{-}$ mass fit
giving a 0.6% change in the signal yield. Since the input $f_{0}(1370)$ mass
and width parameters were allowed to vary within Gaussian constraints, there
is no additional uncertainty to account for.
Table 10: Relative systematic uncertainties on $R$(%). Parameter | Total | $f_{0}(980)$ | $f_{0}(1370)$ | NR | $f_{2}(1270)$, $\lambda=0$
---|---|---|---|---|---
$m(\pi^{+}\pi^{-})$ dependent effic. | 1.0 | 0.2 | 0.2 | 1.0 | 0.2
PID efficiency | 2.0 | 2.0 | 2.0 | 2.0 | 2.0
$J/\psi\phi$ S-wave | 0.7 | 0.7 | 0.7 | 0.7 | 0.7
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ $p$ and $p_{\rm T}$ distributions | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
Acceptance function | 0 | 0.1 | 1.3 | 1.4 | 3.9
${\cal{B}}\left(\phi\rightarrow K^{+}K^{-}\right)$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Background | 0.6 | 0.6 | 0.6 | 0.6 | 0.6
Resonance fit | $-$ | ${}^{+18.2}_{-~{}8.0}$ | ${}^{+~{}0.8}_{-88.1}$ | ${}^{+57.6}_{-~{}3.7}$ | ${}^{+~{}3.0}_{-15.8}$
Total | $\pm$2.7 | ${}^{+18.3}_{-~{}8.4}$ | ${}^{+~{}2.9}_{-88.2}$ | ${}^{+57.7}_{-~{}4.8}$ | ${}^{+~{}5.5}_{-16.4}$
The effect on the fit fractions of changing the acceptance function is also
evaluated. Since the acceptance model was tested by its agreement with the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ data in Fig. 9, we vary the data so that the model does not fit as
well. This is accomplished by increasing the minimum IP $\chi^{2}$ requirement
from 9 to 12.25 on both of the kaon candidates, which has the effect of
increasing the $\chi^{2}$/ndf of the fit to angular distributions by 1 unit.
The Monte Carlo simulation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ with the changed requirement is then fitted to get an
acceptance function. This acceptance function is then applied to the data with
the original minimum IP $\chi^{2}$ cut of 9, and the likelihood fit is redone.
The resulting fitted values from the preferred solution are compared with the
original values in Table 11. The changes are small and well within the
statistical uncertainties.
Table 11: Changes due to modified acceptance function. Values | Original | After change | Variation(%)
---|---|---|---
Fit fractions
$f_{0}(980)$ | (107.1$\pm$3.5)% | 107.2% | 0.1
$f_{2}(1270)$ $\lambda=0$ | (0.76$\pm$0.25)% | 0.79% | 3.9
$f_{2}(1270)$ $|\lambda|=1$ | (0.33$\pm$1.00)% | 0.26% | 21.2
$f_{0}(1370)$ | (32.6$\pm$4.1)% | 31.2% | 1.3
NR | (12.8$\pm$2.3)% | 12.7% | 1.4
$f_{0}(980)$ parameters
$m_{f_{0}}$ (MeV) | 939.9$\pm$6.3 | 938.4 | 0.16
$g_{\pi\pi}$(MeV) | 199$\pm$30 | 205 | 2.7
$g_{KK}/g_{\pi\pi}$ | 3.01$\pm$0.25 | 3.05 | 1.3
$f_{0}(1370)$ parameters
$m_{f_{0}}$ (MeV) | 1475.1$\pm$6.3 | 1476.4 | 0.09
$\Gamma$ (MeV) | 112.7$\pm$11.1 | 113.0 | 0.27
## 8 Conclusions
We have studied the resonance structure of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ using a modified Dalitz plot analysis where we also
include the decay angle of the $J/\psi$. The decay distributions are formed
from a series of final states described by individual $\pi^{+}\pi^{-}$
interfering decay amplitudes. The largest component is the $f_{0}(980)$ that
is described by a Flatté function. The data are best described by adding
Breit-Wigner amplitudes for the $f_{0}(1370)$, the $f_{2}(1270)$ resonances
and a non-resonance contribution. Adding a $\rho(770)$ into the fit does not
improve the overall likelihood. Inclusion of $f_{0}(600)$ or $f_{0}(1500)$
does not result in significant signals for these resonances.
Our three resonance plus non-resonance best fit is dominantly $C\\!P$-odd
S-wave over the entire signal region. We also have a D-wave component arising
from the $f_{2}(1270)$ resonance. Part of this corresponds to the $A_{20}$
amplitude which is also pure $C\\!P$-odd and is $(0.49\pm
0.16^{+0.02}_{-0.08})\%$ of the total rate. A mixed $C\\!P$ part corresponding
to the $A_{2\pm 1}$ amplitude is $(0.2\pm 0.7)\%$ of the total. Adding this to
the amount of allowed $\rho(770)$, less than $1.5$% at 95% CL, we find that
the $C\\!P$-odd fraction is greater than 0.977 at 95% CL. Thus, the entire
mass range can be used to study $C\\!P$ violation with this almost pure
$C\\!P$-odd final state.
The measured relative branching ratio is
$\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi\right)}=(19.79\pm 0.47\pm 0.52)\%,$
where the first uncertainty is statistical and the second systematic. The
largest component is the $f_{0}(980)$ resonance. We also determine
$\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}\right){\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi\right)}=(13.9\pm 0.6^{+2.5}_{-1.2})\%,$
This state was predicted to exist and have a branching fraction about 10% that
of $J/\psi\phi$ [12]. Our new measurement is consistent with and somewhat
larger than this prediction. Other models give somewhat higher rates [29,
*Colangelo:2010bg, *Fleischer:2011au, *ElBennich:2011gm, *Leitner:2010fq]. We
also have firmly established the existence of the $J/\psi f_{0}(1370)$ final
state in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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arxiv-papers
| 2012-04-25T12:54:32 |
2024-09-04T02:49:30.143596
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn, S. De\n Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi,\n S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra\n Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.\n A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli,\n C. Haen, S. C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew,\n J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M.\n Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y. M. Kim, M. Knecht, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N.\n La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O.\n Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn,\n B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March,\n H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F.\n Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B.\n Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J.\n Merkel, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller,\n R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, B. K. Pal, J.\n Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel, S. K. Paterson,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid,\n A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P.\n Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci,\n M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B.\n Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, R. Waldi, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams,\n F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1204.5643"
}
|
1204.5675
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-107 LHCb-PAPER-2012-006 April 25, 2012
Measurement of the $C\\!P$-violating phase $\phi_{s}$ in $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
Measurement of the mixing-induced $C\\!P$-violating phase $\phi_{s}$ in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays is of prime importance
in probing new physics. Here 7421$\pm$105 signal events from the dominantly
$C\\!P$-odd final state ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ are selected in 1 fb-1 of $pp$ collision data collected
at $\sqrt{s}=7$ TeV with the LHCb detector. A time-dependent fit to the data
yields a value of $\phi_{s}=-0.019^{+0.173+0.004}_{-0.174-0.003}$ rad,
consistent with the Standard Model expectation. No evidence of direct $C\\!P$
violation is found.
Submitted to Physics Letters B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-Joergensen52, N. Torr52, E.
Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38,
M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J.
Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, H.
Voss10, R. Waldi55, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42,
A.D. Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Current knowledge of the Cabibbo-Kobayashi-Maskawa (CKM) matrix leads to the
Standard Model (SM) expectation that the mixing-induced $C\\!P$ violation
phase in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays
proceeding via the $b\rightarrow c\overline{c}s$ transition is small and
accurately predicted [1]. Therefore, new physics can be decisively revealed by
its measurement. This phase denoted by $\phi_{s}$ is given in the SM by
$-2\arg\left[{V_{ts}V_{tb}^{*}}/{V_{cs}V_{cb}^{*}}\right]$, where the $V_{ij}$
are elements of the CKM matrix. Motivated by a prediction in Ref. [2], the
LHCb collaboration made the first observation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$, $f_{0}(980)\rightarrow\pi^{+}\pi^{-}$ [3], which was
subsequently confirmed by others [4, *Abazov:2011hv, 6]. This mode is a
$C\\!P$-odd eigenstate and its use obviates the need to perform an angular
analysis in order to determine $\phi_{s}$ [7], as is required in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ final state [8, 9,
*CDF:2011af]. In this Letter we measure $\phi_{s}$ using the final state
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ over a large
range of $\pi^{+}\pi^{-}$ masses, 775$-$1550 MeV,111We work in units where
$c=\hbar=1$. which has been shown to be an almost pure $C\\!P$-odd eigenstate
[11]. We designate events in this region as $f_{\rm odd}$. This phase is the
same as that measured in ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decays, ignoring contributions from suppressed processes [12,
*Fleischer:2011au].
The decay time evolutions for initial $B_{s}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decaying into a $C\\!P$-odd
eigenstate, $f_{-}$, assuming only one CKM phase, are [14, *Bigi:2000yz]
$\Gamma\left(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{-}\scalebox{0.4}{)}}{B}_{s}^{0}\rightarrow
f_{-}\right)={\cal
N}e^{-\Gamma_{s}t}\,\Bigg{\\{}\frac{e^{\Delta\Gamma_{s}t/2}}{2}(1+\cos\phi_{s})+\frac{e^{-\Delta\Gamma_{s}t/2}}{2}(1-\cos\phi_{s})\pm\sin\phi_{s}\sin\left(\Delta
m_{s}\,t\right)\Bigg{\\}}\,,$ (1)
where $\Delta\Gamma_{s}=\Gamma_{\rm L}-\Gamma_{\rm H}$ is the decay width
difference between light and heavy mass eigenstates, $\Gamma_{s}=(\Gamma_{\rm
L}+\Gamma_{\rm H})/2$ is the average decay width, $\Delta m_{s}=m_{\rm
H}-m_{\rm L}$ is the mass difference, and ${\cal N}$ is a time-independent
normalization factor. The plus sign in front of the $\sin\phi_{s}$ term
applies to an initial $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and the minus sign to an initial $B^{0}_{s}$ meson. The time evolution of the
untagged rate is then
$\Gamma\left(B_{s}^{0}\rightarrow f_{-}\right)+\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow f_{-}\right)={\cal
N}e^{-\Gamma_{s}t}\,\Bigg{\\{}e^{\Delta\Gamma_{s}t/2}(1+\cos\phi_{s})+e^{-\Delta\Gamma_{s}t/2}(1-\cos\phi_{s})\Bigg{\\}}\,.$
(2)
Note that there is information in the shape of the lifetime distribution that
correlates $\Delta\Gamma_{s}$ and $\phi_{s}$. In this analysis we will use
samples of both flavour tagged and untagged decays. Both Eqs. 1 and 2 are
invariant under the change $\phi_{s}\rightarrow\pi-\phi_{s}$ when
$\Delta\Gamma_{s}\rightarrow-\Delta\Gamma_{s}$, which gives an inherent
ambiguity. Recently this ambiguity has been resolved [16], so only the allowed
solution with $\Delta\Gamma_{s}>0$ will be considered.
## 2 Data sample and selection requirements
The data sample consists of 1 fb-1 of integrated luminosity collected with the
LHCb detector [17] at 7 TeV centre-of-mass energy in $pp$ collisions at the
LHC. The detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. Components include a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\delta p/p$ that varies
from 0.4% at 5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$, and an impact parameter (IP) resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov (RICH)
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a muon system composed of alternating layers of iron and
multiwire proportional chambers. The trigger consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by a
software stage which applies a full event reconstruction.
Events were triggered by detecting two muons with an invariant mass within 120
MeV of the nominal ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [18].
To be considered a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate,
particles of opposite charge are required to have $p_{\rm T}$ greater than 500
MeV, be identified as muons, and form a vertex with fit $\chi^{2}$ per number
of degrees of freedom less than 16. Only candidates with a dimuon invariant
mass between $-$48 MeV and +43 MeV of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass peak are selected. For
further analysis the four-momenta of the dimuons are constrained to yield the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass.
For this analysis we use a Boosted Decision Tree (BDT) [19] to set the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ selection
requirements. We first implement a preselection that preserves a large
fraction of the signal events, including the requirements that the pions have
$p_{\rm T}$ $>$ 250 MeV and be identified by the RICH. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate decay tracks must
form a common vertex that is detached from the primary vertex. The angle
between the combined momentum vector of the decay products and the vector
formed from the positions of the primary and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay vertices (pointing
angle) is required to be consistent with zero. If more than one primary vertex
is found the one corresponding to the smallest IP significance of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate is chosen.
The variables used in the BDT are the muon identification quality, the
probability that the $\pi^{\pm}$ come from the primary vertex (implemented in
terms of the IP $\chi^{2}$), the $p_{\rm T}$ of each pion, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex $\chi^{2}$, the
pointing angle and the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
flight distance from production to decay vertex. For various calibrations we
also analyze samples of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow\pi^{+}K^{-}$, and its
charge-conjugate. The same selections are used as for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ except for
particle identification.
The BDT is trained with $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ Monte Carlo events generated using Pythia [20] and the LHCb
detector simulation based on Geant4 [21]. The following two data samples are
used to study the background. The first contains
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}\pi^{-}$ events with
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}\pi^{\pm})$ within
$\pm$50 MeV of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass, called the like-sign sample. The second consists of events in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ sideband having
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-})$ between 200
and 250 MeV above the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass peak. In both cases we require $775<m(\pi\pi)<$ 1550 MeV.
Separate samples are used to train and test the BDT. Training samples consist
of 74,230 signal and 31,508 background events, while the testing samples
contain 74,100 signal and 21,100 background events. Figure 1 shows the signal
and background BDT distributions of the training and test samples. The
training and test samples are in excellent agreement. We select $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidates with BDT $>$ 0 to maximize signal
significance for further analysis.
Figure 1: Distributions of the BDT variable for both training and test samples
of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi\pi$ signal and background
events. The signal samples are from simulation and the background samples
derived from data.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ mass
distribution is shown in Fig. 2 for the $f_{\rm odd}$ region. In the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal region, defined as
$\pm$20 MeV around the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass peak, there are 7421$\pm$105 signal events, 1717$\pm$38 combinatorial
background events, and 66$\pm$9 $\eta^{\prime}$ background events,
corresponding to an 81% signal purity. The $\pi^{+}\pi^{-}$ mass distribution
is shown in Fig. 3. The most prominent feature is the $f_{0}(980)$, containing
52% of the events within $\pm$90 MeV of 980 MeV, called the $f_{0}$ region.
The rest of the $f_{\rm odd}$ region is denoted as $\tilde{f}_{0}$.
Figure 2: Mass distribution of the selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ combinations in
the $f_{\rm odd}$ region. The blue solid curve shows the result of a fit with
a double Gaussian signal (red solid curve) and several background components:
combinatorial background (brown dotted line), background from
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}$ (green short-dashed
line), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ (purple dot-dashed), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ when $\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$ (black dot-long-
dashed), and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ (light-blue long-dashed). Figure 3: Mass distribution of
selected $\pi^{+}\pi^{-}$ combinations shown as the (solid black) histogram
for events in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
signal region. The (dashed red) line shows the background determined by
fitting the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ mass
in bins of $\pi^{+}\pi^{-}$ mass. The arrows designate the limits of the
$f_{\rm odd}$ region.
## 3 Resonance structure in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ final state
The resonance structure in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays has been studied using a modified Dalitz plot
analysis including the decay angular distribution of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson [11]. A fit is performed
to the decay distributions of several $\pi^{+}\pi^{-}$ resonant states
described by interfering decay amplitudes. The largest component is the
$f_{0}(980)$ that is described by a Flatté function [22]. The data are best
described by adding Breit-Wigner amplitudes for the $f_{0}(1370)$ and
$f_{2}(1270)$ resonances and a non-resonant amplitude. The components and
fractions of the best fit are given in Table 1.
Table 1: Resonance fractions in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ over the full mass range [11]. The final-state helicity of the D-wave is denoted by $\Lambda$. Only statistical uncertainties are quoted. Resonance | Normalized fraction (%)
---|---
$f_{0}(980)$ | $69.7\pm 2.3$
$f_{0}(1370)$ | $21.2\pm 2.7$
non-resonant $\pi^{+}\pi^{-}$ | $8.4\pm 1.5$
$f_{2}(1270)$, $\Lambda=0$ | $0.49\pm 0.16$
$f_{2}(1270)$, $|\Lambda|=1$ | $0.21\pm 0.65$
The final state is dominated by $C\\!P$-odd S-wave over the entire $f_{\rm
odd}$ region. We also have a small D-wave component associated with the
$f_{2}(1270)$ resonance. Its zero helicity ($\Lambda=0$) part is also pure
$C\\!P$-odd and corresponds to $(0.49\pm 0.16^{+0.02}_{-0.08})\%$ of the total
rate.222In this Letter whenever two uncertainties are given, the first is
statistical and the second systematic. The $|\Lambda|=1$ part, which is of
mixed $C\\!P$, corresponds to $(0.21\pm 0.65^{+0.01}_{-0.03})$% of the total.
Performing a separate fit, we find that a possible $\rho$ contribution is
smaller than 1.5% at 95% confidence level (CL). Summing the $f_{2}(1270)$
$|\Lambda|=1$ and $\rho$ rates, we find that the $C\\!P$-odd fraction is
larger than 0.977 at 95% CL. Thus the entire mass range can be used to study
$C\\!P$ violation in this almost pure $C\\!P$-odd final state.
## 4 Flavour tagging
Knowledge of the initial $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ flavour is necessary in order
to use Eq. 1. This is realized by tagging the flavour of the other $b$ hadron
in the event, exploiting information from four sources: the charges of muons,
electrons, kaons with significant IP, and inclusively reconstructed secondary
vertices. The decisions of the four tagging algorithms are individually
calibrated using $B^{\mp}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\mp}$ decays and combined using a neural network as described in Ref.
[23]. The tagging performance is characterized by $\varepsilon_{\rm
tag}D^{2}$, where $\varepsilon_{\rm tag}$ is the efficiency and $D$ the
dilution, defined as $D\equiv(1-2\omega)$, where $\omega$ is the probability
of an incorrect tagging decision.
We use both the information of the tag decision and of the predicted per-event
mistag probability. The calibration procedure assumes a linear dependence
between the predicted mistag probability $\eta_{i}$ for each event and the
actual mistag probability $\omega_{i}$ given by
$\omega_{i}=p_{0}+p_{1}\cdot\left(\eta_{i}-\langle\eta\rangle\right)$, where
$p_{0}$ and $p_{1}$ are calibration parameters and $\langle\eta\rangle$ the
average estimated mistag probability as determined from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\mp}$ calibration sample. The
values are $p_{0}=0.392\pm 0.002\pm 0.009$, $p_{1}=1.035\pm 0.021\pm 0.012$,
and $\langle\eta\rangle=0.391$. Systematic uncertainties are evaluated by
using ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ separately from
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$, performing the
calibration with $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{*+}\mu^{-}\overline{\nu}_{\mu}$ plus charge-conjugate channels, and viewing
the dependence on different data taking periods. We find $\varepsilon_{\rm
tag}=(32.9\pm 0.6)$% providing us with 2445 tagged signal events. The dilution
is measured as $D=0.272\pm 0.004\pm 0.015$, leading to $\varepsilon_{\rm
tag}D^{2}=(2.43\pm 0.08\pm 0.26)$%.
## 5 Decay time resolution
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay time is
defined here as $t=m{\vec{d}\cdot\vec{p}}/{|p|^{2}}$, where $m$ is the
reconstructed invariant mass, $\vec{p}$ the momentum and $\vec{d}$ the vector
from the primary to the secondary vertex. The time resolution for signal
increases by about 20% for decay times from 0 to 10 ps, according to both the
simulation and the estimate of the resolution from the reconstruction. To take
this dependence into account, we use a double-Gaussian resolution function
with widths proportional to the event-by-event estimated resolution,
$T(t-\hat{t};\sigma_{t})=\sum_{i=1}^{2}f^{T}_{i}\frac{1}{\sqrt{2\pi}S_{i}\sigma_{t}}e^{-\frac{(t-\hat{t}-\mu_{t})^{2}}{2(S_{i}\sigma_{t})^{2}}}\,,$
(3)
where $\hat{t}$ is the true time, $\sigma_{t}$ the estimated time resolution,
$\mu_{t}$ is the bias on the time measurement, $f^{T}_{1}+f^{T}_{2}=1$ are the
fractions of each Gaussian, and $S_{1}$ and $S_{2}$ are scale factors.
To determine the parameters of $T$ we use events containing a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, found using a dimuon trigger
without track impact parameter requirements, plus two opposite-sign charged
tracks with similar selection criteria as for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ events including
that the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ mass be
within $\pm$20 MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass. Figure 4 shows the decay
time distribution for this ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ prompt data sample for the $f_{0}$ region; the
$\tilde{f}_{0}$ data are very similar. The data are fitted with the time
dependence given by
$P^{\rm
prompt}(t)=(1-f_{1}-f_{2})T(t;\sigma_{t})+\left[\frac{f_{1}}{\tau_{1}}e^{{-\hat{t}}/{\tau_{1}}}+\frac{f_{2}}{\tau_{2}}e^{{-\hat{t}}/{\tau_{2}}}\right]\otimes
T(t-\hat{t};\sigma_{t})\,,$ (4)
where $f_{1}$ and $f_{2}$ are long-lived background fractions with lifetimes
$\tau_{1}$ and $\tau_{2}$, respectively.
Figure 4: Decay time distribution of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ candidates in the
$f_{0}$ region. The dashed (red) line shows the long-lived component, and the
solid curve the total.
The resulting parameter values of the function $T$ are given in Table 2.
Table 2: Parameters of the decay time resolution function determined from fits to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ prompt data samples. Parameter | $f_{0}$ region | $\tilde{f}_{0}^{\Large\vphantom{X}}$ region
---|---|---
$\mu_{t}$ (fs) | $-$3.32(12) | $-$2.91(7)
$S_{1}$ | 1.362(4) | 1.329(2)
$S_{2}$ | 12.969(3) | 9.108(3)
$f^{T}_{2}$ | 0.0193(7) | 0.0226(5)
Figure 5 shows the $\sigma_{t}$ distributions used in Eq. 3 for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ events in the
$f_{\rm odd}$ region after background subtraction, and for like-sign
background. Taking into account the calibration parameters of Table 2, the
average effective decay time resolution for the signal is 40.2 fs and 39.3 fs
for the $f_{0}$ and $\tilde{f}_{0}$ regions, respectively. The average of the
two samples is 39.8 fs.
Figure 5: Distribution of the estimated time resolution $\sigma_{t}$ for
opposite-sign ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
signal events after background subtraction, and for like-sign background.
## 6 Decay time acceptance
The decay time acceptance function is written as
$A(t;a,n,t_{0})=C\frac{\left[a\left(t-t_{0}\right)\right]^{n}}{1+\left[a\left(t-t_{0}\right)\right]^{n}}\,,$
(5)
where $C$ is a normalization constant. The other parameters are determined by
fitting the lifetime distribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ events, where $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow K^{-}\pi^{+}$. Figure
6(a) shows the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass when the $K^{-}\pi^{+}$
invariant mass is within $\pm$300 MeV of 892 MeV. There are 155,743$\pm$434
signal events. The sideband-subtracted decay time distribution is shown in
Fig. 6(b) together with a lifetime fit taking into account the acceptance and
resolution. This fit yields $a=2.11\pm 0.04$ ps-1, $n=1.82\pm 0.06$,
$t_{0}=0.105\pm 0.006$ ps and a lifetime of 1.516$\pm$0.008 ps, in good
agreement with the PDG average of 1.519$\pm$0.007 ps [18].
Figure 6: (a) Mass distribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates. The
dashed (red) line shows the background, and the solid (blue) curve the total.
(b) Decay time distribution, where the small background has been subtracted
using the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass sidebands.
The (blue) curve shows the lifetime fit.
We check our lifetime acceptance by comparing with a CDF measurement of the
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}$ effective lifetime of $\tau^{\rm eff}=1.70^{+0.12}_{-0.11}\pm
0.03$ ps [6] obtained from a single exponential fit.333This corresponds to the
lifetime of the $C\\!P$-odd eigenstate if $\phi_{s}$ is zero (see Eq. 2). A
fit of the $f_{0}$ sample (see Fig. 7) yields $\tau^{\rm eff}=1.71\pm 0.03$
ps, while we find $\tau^{\rm eff}=1.67\pm 0.03$ ps in the $\tilde{f}_{0}$
sample. These two values are consistent with each other, within the quoted
statistical errors, and with the CDF result.
Figure 7: Decay time distribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}$ candidates fitted with a single exponential function multiplied
by the acceptance and convolved with the resolution. The dashed line is signal
and the shaded area background.
## 7 Likelihood function definition
To determine $\phi_{s}$ an extended likelihood function is maximized using
candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
signal region
${\cal L}(\phi_{s})=e^{-(N_{\rm sig}+N_{\rm bkg})}\prod_{i=1}^{N_{\rm
obs}}P(m_{i},t_{i},{\sigma_{t}}_{i},q_{i},\eta_{i})\,,$ (6)
where the signal yield, $N_{\rm sig}$, and background yield, $N_{\rm bkg}$,
are fixed from the fit of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ mass distribution in the $f_{\rm odd}$ region (see Fig.
2). $N_{\rm obs}$ is the number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates, $m_{i}$ the
reconstructed mass, $t_{i}$ the reconstructed decay time, and
${\sigma_{t}}_{i}$ the estimated decay time uncertainty. The flavour tag,
$q_{i}$, takes values of +1, $-1$ or 0, respectively, if the signal meson is
tagged as $B_{s}^{0}$, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$,
or untagged, and $\eta_{i}$ is the estimated mistag probability. Backgrounds
are caused largely by mis-reconstructed $b$-hadron decays, so it is necessary
to include a long-lived background probability density function (PDF). The
likelihood function includes distinct contributions from the signal and the
background. For tagged events we have
$\displaystyle P(m_{i},t_{i},{\sigma_{t}}_{i},q_{i},\eta_{i})$
$\displaystyle=$ $\displaystyle N_{\rm sig}\varepsilon_{\rm tag}P_{m}^{\rm
sig}(m_{i})P_{t}^{\rm
sig}(t_{i},q_{i},\eta_{i}|{\sigma_{t}}_{i})P_{\sigma_{t}}^{\rm
sig}({\sigma_{t}}_{i})$ (7) $\displaystyle+N_{\rm bkg}\varepsilon^{\rm
bkg}_{\rm tag}P_{m}^{\rm bkg}(m_{i})P_{t}^{\rm
bkg}(t_{i}|{\sigma_{t}}_{i})P_{\sigma_{t}}^{\rm bkg}({\sigma_{t}}_{i})\,,$
where $\varepsilon^{\rm bkg}_{\rm tag}$ refers to the flavour tagging
efficiency of the background. The signal mass PDF, $P_{m}^{\rm sig}(m)$, is a
double Gaussian function, while the background mass PDF, $P_{m}^{\rm bkg}(m)$,
is proportional to $e^{-\alpha m}$ together with a very small contribution
from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}$, $N_{\eta^{\prime}}$, that is fixed in the $\phi_{s}$ fit
to 66 events obtained from the fit shown in Fig. 2.
The PDF used to describe the signal decay rate, $P_{t}^{\rm sig}$, depends on
the tagging results $q$ and $\eta$. It is modelled by a PDF of the true time
$\hat{t}$, $R(\hat{t},q,\eta)$, convolved with the decay time resolution and
multiplied by the decay time acceptance function found for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ events. From Eq. 1,
it can be expressed as
$R(\hat{t},q,\eta)\propto
e^{-\Gamma_{s}\hat{t}}\left\\{\cosh\frac{\Delta\Gamma_{s}\hat{t}}{2}+\cos\phi_{s}\sinh\frac{\Delta\Gamma_{s}\hat{t}}{2}-q[1-2\omega(\eta)]\sin\phi_{s}\sin(\Delta
m_{s}\hat{t})\right\\}\,,$ (8)
where $\omega(\eta)$ is the calibrated mistag probability. Thus the PDF of
reconstructed time is
$P_{t}^{\rm sig}(t,q,\eta|{\sigma_{t}})=R(\hat{t},q,\eta)\otimes
T(t-\hat{t};\sigma_{t})\cdot A(t;a,n,t_{0})\,.$ (9)
For untagged events we use
$\displaystyle P(m_{i},t_{i},{\sigma_{t}}_{i},q_{i}=0,\eta_{i})$
$\displaystyle=$ $\displaystyle N_{\rm sig}(1-\varepsilon_{\rm tag})P_{m}^{\rm
sig}(m_{i})P_{t}^{\rm
sig}(t_{i},0,\eta_{i}|{\sigma_{t}}_{i})P_{\sigma_{t}}^{\rm
sig}({\sigma_{t}}_{i})$ (10) $\displaystyle+N_{\rm bkg}(1-\varepsilon^{\rm
bkg}_{\rm tag})P_{m}^{\rm bkg}(m_{i})P_{t}^{\rm
bkg}(t_{i}|{\sigma_{t}}_{i})P_{\sigma_{t}}^{\rm bkg}({\sigma_{t}}_{i})\,.$
The PDF describing the long-lived background decay rate is
$P^{\rm bkg}_{t}(t|\sigma_{t})=\left[\frac{1-f^{\rm bkg}_{2}}{\tau^{\rm
bkg}_{1}}e^{-\frac{\hat{t}}{\tau^{\rm bkg}_{1}}}+\frac{f^{\rm
bkg}_{2}}{\tau^{\rm bkg}_{2}}e^{-\frac{\hat{t}}{\tau^{\rm
bkg}_{2}}}\right]\otimes T(t-\hat{t};\sigma_{t})\cdot A(t;a^{\rm bkg},n^{\rm
bkg},t_{0}^{\rm bkg})\,,$ (11)
where $\tau^{\rm bkg}_{1}$, $\tau^{\rm bkg}_{2}$ and $f^{\rm bkg}_{2}$
parameterize the underlying double exponential function. The same functional
form is used to describe the background decay time acceptance as for signal
(Eq. 5) with different parameters that are determined by fitting the like-sign
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}\pi^{\pm}$ events in an
interval $\pm$200 MeV around the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass. The $P^{\rm
sig}_{\sigma_{t}}({\sigma_{t}}_{i})$ and $P^{\rm
bkg}_{\sigma_{t}}({\sigma_{t}}_{i})$ functions are shown in Fig. 5. The
parameters that are fixed in the likelihood fit are listed in Table 3.
Table 3: Parameters used in the functions for the invariant mass and decay time describing the signal and background. These parameters are fixed to their central values in the fit for $\phi_{s}$. Function | Parameters
---|---
| $N_{\rm sig}=7421$, $N_{\rm bkg}=1717\pm 38$, $N_{\eta^{\prime}}=66\pm 9$
$P_{m}^{\rm sig}(m)$ | $m_{0}$= 5368.2(1) MeV, $\sigma^{m}_{1}$=8.1(1) MeV, $\sigma^{m}_{2}$=18.0(2) MeV, $f^{m}_{2}$= 0.196(2)
$P_{m}^{\rm bkg}(m)$ | $\alpha=(-5.35\pm 1.15)\times 10^{-4}$ MeV-1
$P^{\rm bkg}_{t}(t|\sigma_{t})$ | $\tau^{\rm bkg}_{1}=0.65(5)$ ps, $\tau^{\rm bkg}_{2}=2.0(8)$ ps, $f^{\rm bkg}_{2}=0.06(2)$
| $a^{\rm bkg}=3.22(10)$ ps-1, $n^{\rm bkg}=3.31(14)$, $t_{0}^{\rm bkg}=0$
ps,
$T(t-\hat{t};\sigma_{t})$ | see Table 2
## 8 Results
The likelihood of Eq. 6 is multiplied by Gaussian constraints on several of
the model parameters. These are the LHCb measured value of $\Delta
m_{s}=17.63\pm 0.11\pm 0.02$ ps-1 [24], the tagging parameters $p_{0}$ and
$p_{1}$, the decay time acceptance parameters $t_{0}$, $a$, and $n$, and both
$\Gamma_{s}=0.657\pm 0.009\pm 0.008$ ps-1 and $\Delta\Gamma_{s}=0.123\pm
0.029\pm 0.011$ ps-1 given by the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ analysis [8]. The fit has been validated with full Monte Carlo
simulations.
Figure 8 shows the difference of log-likelihood value, $\Delta\ln(\cal{L})$,
compared to the one at the point with the best fit, as a function of
$\phi_{s}$. At each value, the likelihood function is maximized with respect
to all other parameters. The best fit value is
$\phi_{s}=-0.019^{+0.173+0.004}_{-0.174-0.003}$ rad. (The systematic
uncertainty will be discussed subsequently.) Values for $\phi_{s}$ in the
$f_{0}$ and $\tilde{f}_{0}$ regions are $-0.26\pm 0.23$ rad and $0.29\pm 0.28$
rad, respectively, consistent within the uncertainties. The decay time
distribution is shown in Fig. 9.
Figure 8: Log-likelihood difference as a function of $\phi_{s}$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{\rm odd}$ events. Figure 9: Decay time distribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{\rm odd}$ candidates. The solid line shows the result of the fit, the
dashed line shows the signal, and the shaded region the background.
The presence of a $\sin\phi_{s}$ contribution in Eq. 1 can, in principle, be
viewed by plotting the asymmetry $\left[N(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})-N(B^{0}_{s})\right]/\left[N(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})+N(B^{0}_{s})\right]$ of the
background-subtracted tagged yields as a function of decay time modulo
$2\pi/\Delta m_{s}$, as shown in Fig. 10. The asymmetry is consistent with the
value of $\phi_{s}$ determined from the full fit and does not show any
significant structure.
Figure 10: $C\\!P$ asymmetry as a function of decay time modulo $2\pi/\Delta
m_{s}$. The curve shows the expectation for $\phi_{s}=-0.019$ rad.
The data have also been analyzed allowing for the possibility of direct
$C\\!P$ violation. In this case Eq. 8 must be replaced with
$\displaystyle R(\hat{t},q,\eta)$ $\displaystyle\propto$ $\displaystyle
e^{-\Gamma_{s}\hat{t}}\left\\{\cosh\frac{\Delta\Gamma_{s}\hat{t}}{2}+\frac{2|\lambda|}{1+|\lambda|^{2}}\cos\phi_{s}\sinh\frac{\Delta\Gamma_{s}\hat{t}}{2}\right.$
(12)
$\displaystyle\left.-\frac{q[1-2\omega(\eta)]}{1+|\lambda|^{2}}\left[2|\lambda|\sin\phi_{s}\sin(\Delta
m_{s}\hat{t})-(1-|\lambda|^{2})\cos(\Delta m_{s}\hat{t})\right]\right\\}\,.$
The fit gives $|\lambda|=0.89\pm 0.13$, consistent with no direct $C\\!P$
violation ($|\lambda|=1$). The value of $\phi_{s}$ changes only by $-0.002$
rad, and the uncertainty stays the same.
The systematic uncertainties are small compared to the statistical one. No
additional uncertainty is introduced by the acceptance parameters, $\Delta
m_{s}$, $\Gamma_{s}$, $\Delta\Gamma_{s}$ or flavour tagging, since Gaussian
constraints are applied in the fit. The uncertainties associated with the
fixed parameters are evaluated by changing them by $\pm$1 standard deviation
from their nominal values and determining the change in the fitted value of
$\phi_{s}$. These are listed in Table 4. The uncertainty due to a change in
the signal time acceptance function is evaluated by multiplying
$A(t;a,n,t_{0})$ with a factor $(1+\beta t)$, and redoing the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ fit with the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime fixed to the PDG value.
The resulting value of $\beta=(1\pm 3\pm 3)\times 10^{-3}$ is then varied by
$\pm 4.4\times 10^{-3}$ to estimate the uncertainty in $\phi_{s}$. An
additional uncertainty is included due to a possible $C\\!P$-even component.
This has been limited to 2.3% of the total $f_{\rm odd}$ rate at 95% CL, and
contributes an uncertainty to $\phi_{s}$ as determined by repeating the fit
with an additional multiplicative dilution of 0.954. The asymmetry between
$B^{0}_{s}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
production is believed to be small, and similar to the asymmetry between
$B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production
which has been measured by LHCb to be about 1% [25]. The effect of neglecting
this production asymmetry is the same as making a relative 1% change in the
tagging efficiencies, up for $B^{0}_{s}$ and down for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, which has a negligible effect
on $\phi_{s}$.
Table 4: Summary of systematic uncertainties on $\phi_{s}$. Quantities fixed in the fit that are not included here give negligible uncertainties. The total uncertainty is found by adding in quadrature all the positive and negative contributions separately. Quantity (Q) | $\pm\Delta$Q | $+$Change | $-$Change
---|---|---|---
| | in $\phi_{s}$ (rad) | in $\phi_{s}$ (rad)
$\beta$ | $4.4\times 10^{-3}$ | 0.0008 | $-0.0007$
$\tau^{\rm bkg}_{1}$ (ps) | 0.046 | $-0.0006$ | 0.0014
$\tau^{\rm bkg}_{2}$ (ps) | 0.8 | $-0.0014$ | 0.0014
$f^{\rm bkg}_{2}$ | 0.02 | $-0.0006$ | 0.0012
$N_{\rm bkg}$ | 38 | 0.0009 | $-0.0001$
$N_{\eta^{\prime}}$ | 9 | 0.0006 | 0.0001
$m_{0}$ (MeV) | 0.12 | 0.0012 | $-0.0004$
$\sigma^{m}_{1}$ (MeV) | 0.1 | $-0.0002$ | 0.0008
$\alpha$ | $1.1\times 10^{-4}$ | $0.0003$ | $0.0003$
$T$ function | 5% | $0.0005$ | 0.0005
$C\\!P$-even | multiply dilution by 0.954 | $-0.0008$ | $-$
Direct $C\\!P$ | free in fit | $-0.0020$ | $-$
Total systematic uncertainty on $\phi_{s}$ | ${}^{+0.004}_{-0.003}$
## 9 Conclusions
Using 1 fb-1 of data collected with the LHCb detector, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays are selected and used to measure the $C\\!P$
violating phase $\phi_{s}$. The signal events have an effective decay time
resolution of 39.8 fs. The flavour tagging is based on properties of the decay
of the other $b$ hadron in the event and has an efficiency times dilution-
squared of 2.4%. We perform a fit of the time dependent rates with the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime and the difference in
widths of the heavy and light eigenstates used as input. We measure a value of
$\phi_{s}=-0.019^{+0.173+0.004}_{-0.174-0.003}$ rad. This result subsumes our
previous measurement obtained with 0.41 $\mbox{\,fb}^{-1}$ of data [7].
Combining this result with our previous result from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays [8] by performing a joint fit to the data gives a combined
LHCb value of $\phi_{s}=+0.06\pm 0.12\pm 0.06$ rad. Our result is consistent
with the SM prediction of $-0.0363^{+0.0016}_{-0.0015}$ rad [1]. In addition,
we find no evidence for direct $C\\!P$ violation.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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arxiv-papers
| 2012-04-25T15:01:57 |
2024-09-04T02:49:30.158098
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, V. Balagura,\n W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C.\n Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M. -O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N.\n Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De\n Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi,\n L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F.\n Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D.\n Esperante Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez,\n V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H.\n Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E.\n Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison,\n P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A.\n Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E.\n Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R.\n Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J.\n Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim,\n M. Knecht, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi,\n D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti,\n G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J. -P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy,\n T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B.\n Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H.\n Lu, J. Luisier, A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc,\n O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, S.\n Miglioranzi, D. A. Milanes, M. -N. Minard, J. Molina Rodriguez, S. Monteil,\n D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R.\n Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, B. K. Pal, J.\n Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel, S. K. Paterson,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid,\n A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M. -H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P.\n Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci,\n M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B.\n Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, R. Waldi, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams,\n F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1204.5675"
}
|
1204.5701
|
# Orbital linearization of smooth completely integrable vector fields
Nguyen Tien Zung Institut de Mathématiques de Toulouse, UMR5219, Université
Toulouse 3 tienzung.nguyen@math.univ-toulouse.fr
(Date: Version 1, April 2012)
###### Abstract.
The main purpose of this paper is to prove the smooth local orbital
linearization theorem for smooth vector fields which admit a complete set of
first integrals near a nondegenerate singular point. The main tools used in
the proof of this theorem are the formal orbital linearization theorem for
formal integrable vector fields, the blowing-up method, and the Sternberg–Chen
isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.
###### Key words and phrases:
integrable system, normal form, linearization, nondegenerate singularity
###### 1991 Mathematics Subject Classification:
37G05, 58K50,37J35
## 1\. Introduction
The main purpose of this paper is to show the following _orbital linearization
theorem_ for smooth ($C^{\infty}$) vector fields which admit a complete set of
first integrals near a nondegenerate singular point:
###### Theorem 1.1.
Let $X$ be a smooth vector field in a neighborhood of $O=(0,\ldots,0)$ in
$\mathbb{R}^{n},$ which vanishes at $O$ and satisfies the following
conditions:
i) (Complete integrability): $X$ admits $n-1$ functionally independent smooth
first integrals $F_{1},\ldots,F_{n-1},$ i.e. $X(F_{1})=\ldots=X(F_{n-1})=0$
and $dF_{1}\wedge\ldots\wedge dF_{n-1}\neq 0$ almost everywhere.
ii) (Nondegeneracy 1): The semisimple part of the linear part of $X$ at $O$ is
non-zero, and the $\infty$-jets of $F_{1},\ldots,F_{n-1}$ at $O$ are
funtionally independent.
iii) (Nondegeneracy 2): If moreover $0$ is an eigenvalue of $X$ at $O$ with
multiplicity $k\geq 1$, then the differentials of the functions
$F_{1},\ldots,F_{k}$ are linearly independent at $O$:
$dF_{1}(O)\wedge\ldots\wedge dF_{k}(O)\neq 0.$
Then there exists a local smooth coordinate system $(x_{1},\ldots,x_{n})$ in
which $X$ can be written as
(1.1) $X=FX^{(1)},$
where $X^{(1)}$ is a semisimple linear vector field in $(x_{1},\ldots,x_{n})$,
and $F$ is a smooth first integral of $X^{(1)}$, i.e. $X^{(1)}(F)=0,$ with
$F(O)=1.$
The above theorem is in fact more than mere orbital linearization: not only
that $X$ is orbitally equivalent to its linear part $X^{(1)},$ but also the
factor $F$ in the expression $X=FX^{(1)}$ in a normalized coordinate system is
a first integral of $X$ and $X^{(1)}.$ In [9], this kind of linearization is
called _geometrical linearization_.
The formal and analytic case of the above theorem also holds and was shown in
[9] in a more general context of integrable non-Hamiltonian systems of type
$(p,q)$, i.e. with $p$ commuting vector fields and $q$ common first integrals,
where $p+q=n$ is the dimension of the manifold. The vector fields that we
study in this paper are integrable of type $(1,n-1),$ i.e. just one vector
field and $n-1$ first integrals.
The nondegeneracy condition in Theorem 1.1 is a bit stronger than the
nondegeneracy condition in [9]: in [9] the (formal or analytic) vector field
$X$ is called integrable _nondegenerate_ if it satisifes the above conditions
i) and ii), without the need of condition iii). In fact, in the formal and
analytic case, condition iii) is a simple consequence of the first two
conditions and the theorem about the existence of (formal or analytic)
Poincaré-Dulac normalization [8, 9]. However, in the smooth case, we don’t
have a proof of the fact that condition iii) follows from conditions i) and
ii) in general, though we do have a proof of this fact for dimension 2.
The rest of this paper is organized as follows. Section 2 is devoted to some
preliminary results, including the classification of the nondegenerate
singularities of completely integrable vector fields into (strong/weak)
elliptic and hyperbolic cases (Lemma 2.2), and the normalization up to a flat
term (Proposition 2.3). These preliminary results are used in the proof of
Theorem 1.1 which is presented in Section 3. Finally, in Section 4, we show
that, at least in the case $n=2$, condition iii) Theorem 1.1 is a consequence
of the first two conditions, and can be dropped from the formulation of the
theorem (Theorem 4.1). We conjecture that condition iii) is redundant in the
higher-dimensional case as well.
This paper is part of our program of systematic study of the geometry and
topology of integrable non-Hamiltonian systems. In particular, Theorem 4.1,
which is a refinement of Theorem 1.1 in the case of dimension 2, is the
starting point of our joint work with Nguyen Van Minh on the local and global
smooth invariants of integrable dynamical systems on 2-dimensional surfaces
[10].
## 2\. Preliminary results
### 2.1. Adapted first integrals
We have the following simple lemma, which is similar to the well-known
Ziglin’s lemma [7]
###### Lemma 2.1.
Let $G_{1},\ldots,G_{m}$ be $m$ formal series in $n$ variables which are
functionally independent. Then there exists $m$ polynomial functions of $m$
variables $P_{1},\ldots,P_{m}$ such that the homogeneous (i.e. lowest degree)
parts of the formal series of
$P_{1}(G_{1},\ldots,G_{m}),\ldots,P_{1}(G_{1},\ldots,G_{m})$ are functionally
independent.
The proof of the above lemma follows exactly the same lines as the proof of
Ziglin of his lemma in [7], and our situation is simpler than the situation of
meromorphic functions considered by Ziglin.
Let $X$ be a smooth completely integrable vector field with a singularity at
$O$. We will say that the smooth first integrals $F_{1},\ldots,F_{n-1}$ of $X$
are _adapted_ first integrals if
(2.1) $dH_{1}\wedge\ldots\wedge dH_{n-1}\neq 0\ \ a.e.,$
where $H_{i}=F_{i}^{(h_{i})}$ denotes the homogeneous part (consisting of non-
constant terms of lowest degree in the Taylor expansion) of $F_{i}$ at $O$.
Using the above lemma to replace the first integrals $F_{1},\ldots,F_{n-1}$ of
$X$ by appropriate polynomial functions of them if necessary, from now on we
can assume that $F_{1},\ldots,F_{n-1}$ are adapted.
### 2.2. The eigenvalues of $X$
The fact that $X$ admits $n-1$ first integrals implies that the $X$ is very
resonant at $O$. More precisely, we have:
###### Lemma 2.2.
Let $(X,F_{1},\ldots,F_{n-1})$ be smooth nondegenerate at $O$, i.e. they
satisfy the conditions of Theorem 1.1. Then the linear part of $X$ at $O$ is
semisimple, and there is a positive number $\lambda>0$ such that either all
the eigenvalues of $X$ at $O$ belong to $\lambda\mathbb{Z},$ or all of them
belong to $\sqrt{-1}\lambda\mathbb{Z}.$
###### Proof.
We can assume that $H_{1},\ldots,H_{n-1}$ are functionally independent, where
$H_{i}$ denotes the homogeneous part of $F_{i}$. The equality $X(F_{i})=0$
implies that
(2.2) $X^{ss}(H_{i})=X^{(1)}(H_{i})=0\ \forall i=1,\ldots,n-1,$
where $X^{(1)}$ is the linear part of $X$, and $X^{ss}$ is the semisimple part
of $X^{(1)}$ in the Jordan-Dunford decomposition. We can write
(2.3) $X^{ss}=\sum_{i=1}^{n}\lambda_{j}z_{j}\frac{\partial}{\partial z_{i}}$
in a complex coordinate system. Recall that the ring of polynomial first
integrals of $\sum_{i=1}^{n}\lambda_{j}z_{j}\frac{\partial}{\partial z_{i}}$
is generated by the monomial functions $\prod_{i=1}^{n}z_{i}^{a_{i}}$ which
satisfies the resonance relation
(2.4) $\sum_{i=1}^{n}a_{i}\lambda_{i}=0.$
The fact that $H_{1},\ldots,H_{n-1}$ are independent implies that Equation
(2.4) has $n-1$ linearly independent solutions which belong to
$\mathbb{Z}^{n}_{+},$ which in turn implies that there is a complex number
$\lambda$ such that $\lambda_{1},\ldots,\lambda_{n}\in\lambda\mathbb{Z}.$
Remark that if the spectrum of $X^{ss}$ contains a complex eigenvalue
$\lambda_{1}\in\mathbb{C}\setminus(\mathbb{R}\cup\sqrt{-1}\mathbb{R}),$ then
its complex conjugate $\overline{\lambda}_{1}$ is also in the spectrum because
$X$ is real, and $\lambda_{1}$ and $\overline{\lambda}_{1}$ cannot belong to
$\lambda\mathbb{Z}$ at the same time for any $\lambda.$ Thus any eigenvalue of
$X^{ss}$ is either real or pure imaginary. If there is one real non-zero
eigenvalue, then we can choose $\lambda\in\mathbb{R}_{+},$ otherwise we can
choose $\lambda\in\sqrt{-1}\mathbb{R}_{+}.$ Notice that $\lambda\neq 0$
because at least one eigenvalue of $X^{ss}$ is non-zero by our assumptions.
The common level sets of $H_{1},\ldots,H_{n-1}$ are 1-dimensional almost
everywhere, and since both $X^{(1)}$ and $X^{ss}$ are tangent to these common
level sets, we have that $X^{(1)}\wedge X^{ss}=0,$ which implies that
$X^{(1)}$ is semisimple, i.e. $X^{(1)}=X^{ss}.$ ∎
With the above lemma, we can divide the problem into 4 cases (here
$\mathbb{R}^{*}=\mathbb{R}\setminus\\{0\\}$):
I. Strongly hyperbolic (or hyperbolic without eigenvalue 0):
$\lambda_{i}\in\lambda\mathbb{R}^{*}\ \forall i.$
II. Weakly hyerbolic (or hyperbolic with eigenvalue 0):
$\lambda_{i}\in\lambda\mathbb{R}^{*}\ \forall i>k\geq 1$,
$\lambda_{1}=\ldots=\lambda_{k}=0.$
III. Strongly elliptic (or elliptic without eigenvalue 0):
$\lambda_{i}\in\sqrt{-1}\lambda\mathbb{R}^{*}\ \forall i.$
IV. Weakly elliptic (or elliptic with eigenvalue 0):
$\lambda_{i}\in\sqrt{-1}\lambda\mathbb{R}^{*}\ \forall i>k\geq 1$,
$\lambda_{1}=\ldots=\lambda_{k}=0.$
### 2.3. Linearization up to a flat term
Using the geometric linearization theorem of [9] in the formal case, we get
the following proposition:
###### Proposition 2.3 (Linearization up to a flat term).
Assume that $X$ satisfies the hypotheses of Theorem 1.1. Then there is a local
smooth coordinate system $(x_{1},\ldots,x_{n})$ in which $X$ can be written as
(2.5) $X=FX^{(1)}+flat$
where $X^{(1)}$ is the linear part of $X$ in the coordinate system
$(x_{1},\ldots,x_{n})$, $F$ is a smooth first integral of $X^{(1)}$, and
$flat$ means a smooth term which is flat at $O$.
###### Proof.
Denote by $\hat{X}$ (resp. $\hat{F}_{i}$) the $\infty$-jet of $X$ (resp.
$F_{i}$) at $O$: $\hat{X}$ is a formal vector field (resp. function) at $O$.
If $(X,F_{1},\ldots,F_{n-1})$ is smooth nondegenerate at $O$, then
$(\hat{X},\hat{F}_{1},\ldots,\hat{F}_{n-1})$ is a nondegenerate formal
integrable system of type $(1,n-1)$ at $p$. According to the geometric
linearization theorem of [9], this formal integrable system can be linearized
geometrically, i.e. there is a formal coordinate system
$(\hat{x}_{1},\ldots,\hat{x}_{n})$ in which we have
(2.6) $\hat{X}=\hat{F}\hat{X}^{(1)}$
where $\hat{X}^{(1)}$ is the linear part of $\hat{X}$ in the formal coordinate
system $(\hat{x}_{1},\ldots,\hat{x}_{n})$, and $\hat{F}$ is a formal first
integral of $\hat{X}^{(1)}.$ By the classical Hilbert–Weyl theorem (see, e.g.,
[3] ), we can write
(2.7)
$\hat{F}=\hat{f}(Q_{1}(\hat{x}_{1},\ldots,\hat{x}_{n}),\ldots,Q_{m}(\hat{x}_{1},\ldots,\hat{x}_{n}))$
where $\hat{f}$ is a formal series and
$Q_{1}(\hat{x}_{1},\ldots,\hat{x}_{n}),\ldots,Q_{m}(\hat{x}_{1},\ldots,\hat{x}_{n})$
are homogeneous polynoms generate the ring of polynomial first integrals of
$\hat{X}^{(1)}.$ Using Borel theorem, we get a smooth coordinate system
$(x_{1},\ldots,x_{n})$ whose $\infty$-jet is
$(\hat{x}_{1},\ldots,\hat{x}_{n})$, and a smooth function $f$ of $m$ variables
whose $\infty$-jet is $\hat{f}.$ Put
(2.8)
$F(x_{1},\ldots,x_{n})=f(Q_{1}(x_{1},\ldots,x_{n}),\ldots,Q_{m}(x_{1},\ldots,x_{n})).$
Then Equations (2.6), (2.7) and (2.8) imply that $X=FX^{(1)}+flat$ in the
smooth coordinate system $(x_{1},\ldots,x_{n}).$ ∎
### 2.4. Reduction to the case without eigenvalue 0
Assume that $X$ has zero eigenvalue at $O$ with multiplicity $k$, and
$dF_{1}\wedge\ldots\wedge F_{k}=0,$ i.e. we can use $F_{1},\ldots,F_{k}$ as
the first $k$ coordinates in our local coordinate systems. Since the vector
field $X$ preserves $x_{1},\ldots,x_{k},$ we can view it as a $k$-dmiensional
family of vector fields on $(n-k)$-dimensional spaces
(2.9) $U_{c_{1},\ldots,c_{k}}=\\{F_{1}=c_{1},\ldots,F_{k}=c_{k}\\}$
(for $c_{1},\ldots,c_{k}$ small enough). It follows from the usual implicit
function theorem that on each $U_{c_{1},\ldots,c_{k}}$ there is a unique point
$O_{c_{1},\ldots,c_{k}}$ such that $X(O_{c_{1},\ldots,c_{k}})=0,$ and moreover
the point $O_{c_{1},\ldots,c_{k}}$ depends smoothly on $c_{1},\ldots,c_{k},$
the eigenvalues of $X$ at $c_{1},\ldots,c_{k},$ are non-zero. It also follows
from the formal independence of $F_{1},\ldots,F_{n}$ at $O$, that the
functions functions $F_{k+1},\ldots,F_{n}$ are formally independent at every
point $O_{c_{1},\ldots,c_{k}}$ provided that $c_{1},\ldots,c_{k}$ are
sufficiently small. In other words, we have a $k$-dimensional family of
nondegenerate singularities of smooth completely integrable
$(n-k)$-dimensional vector fields $X_{c_{1},\ldots,c_{k}}$. In order to
normalize $X$, it suffices to normalize $X_{c_{1},\ldots,c_{k}}$ in a way
which depends smoothly on the parameter.
## 3\. Proof of Theorem 1.1
We will always assume that the vector field $X$ satisfies the hypotheses of
Theorem 1.1. The fact that the linear part of $X$ is semisimple is established
by Lemma 2.2. In view of Subsection 2.4, it suffices to prove Theorem 1.1 for
the cases without zero eigenvalue, by a proof whose parametrized version also
works the same.
### 3.1. The hyperbolic case
Assume that $X$ is hyperbolic without eigenvalue 0. According to Proposition
2.3, we can write $X=Y+flat,$ where $Y=FX^{(1)}$ is a smooth hyperbolic
integrable vector field in normal form. Since $X$ and $Y$ are hyperbolic and
coincide up to a flat term, Sternberg–Chern theorem [6, 2] says that $X$ is
locally smoothly isomorphic to $Y$, i.e. there is a smooth coordinate system
in which $X$ can be written as $X=FX^{(1)},$ where $F$ is a smooth first
integral of $X^{(1)}$. Theorem 1.1 is proved in the hyperbolic case without
eigenvalue 0.
### 3.2. The elliptic case
In this subsection, we will assume that all the eigenvalues of $X$ at $O$ are
non-zero pure imaginary. Using Proposition 2.3, we can assume that
$X=FX^{(1)}+flat$ in a local smooth coordinate system $(x_{1},\ldots,x_{n}),$
where $F$ is a smooth function such that $F(O)=1.$ Put $Y=X/F.$ Then $Y$ has
the same first integrals as $X$, and
(3.1) $Y=X^{(1)}+flat.$
The fact that $X$ is of strong elliptic type implies immediately that the
dimension $n$ is even, the eigenvalues of $X$ at $O$ are
$\pm\sqrt{-1}a_{1},\ldots,\pm\sqrt{-1}a_{n/2}$ where $a_{1},\ldots,a_{n/2}$
are positive real numbers, and we can choose the coordinates
$(x_{1},\ldots,x_{n})$ such that
(3.2) $X^{(1)}=\sum_{i=1}^{n/2}a_{i}(x_{2i-1}\frac{\partial}{\partial
x_{2i}}-x_{2i}\frac{\partial}{\partial x_{2i-1}}).$
According to Lemma 2.2, we can choose $\lambda>0$ such that
$a_{1}/\lambda,\ldots,a_{n/2}/\lambda$ are natural numbers whose greatest
common divisor is 1.
###### Lemma 3.1.
Locally near $O$ all the orbits of $Y=X/F$ (except the fixed point $O$) are
periodic, with periods which are unifromly bounded above and below.
###### Proof.
The vector field $(dF_{1}\wedge\ldots\wedge
dF_{n-1})\lrcorner(\frac{\partial}{\partial
x_{1}}\wedge\ldots\wedge\frac{\partial}{\partial x_{n}})$ is tangent to $Y$,
and therefore it is divisible by $Y$, i.e. we can write
(3.3) $(dF_{1}\wedge\ldots\wedge dF_{n-1})\lrcorner(\frac{\partial}{\partial
x_{1}}\wedge\ldots\wedge\frac{\partial}{\partial x_{n}})=GY,$
where $G$ is a smoth non-flat function at $O$. Notice that the singular locus
of the map $(F_{1},\ldots,F_{n-1}):U\to\mathbb{R}^{n-1},$ where $U\ni O$ is a
small neighborhood of $O$ in $\mathbb{R}^{n},$ coincides with the zero locus
of $G.$
It is clear that, by continuity, the set of all points $x\in U$ such that the
orbit of $Y$ through $x$ is periodic of period $\leq 3\pi/\lambda$ is a closed
subset of $U$. We want to show that this set is actually equal to $U$
(provided that $U$ is small enough). Consider the singular locus
(3.4) $S=\\{x\in U\ |\ G(x)=0\\}=\\{x\in U\ |\ dF_{1}\wedge\ldots\wedge
dF_{n-1}(x)=0\\}$
Since $G$ is non-flat at $O$, we can choose a coordinate system
$(z_{1},\ldots,z_{n})$ which is a linear transformation of the coordinate
system $(x_{1},\ldots,x_{n}),$ such that the homogeneous part $G^{(h)}$ of $G$
has the form
(3.5) $G^{(h)}=z^{h}+\ldots$
which means that $\frac{\partial^{k}G}{\partial z_{1}^{k}}\neq 0$ in $U$.
Because $\frac{\partial^{k}G}{\partial z_{1}^{k}}$ does not vanish in $U$, by
the classical Rolle’s theorem on each line
$\\{z_{2}=const,\ldots,z_{n}=const\\}$ in $U$ there are at most $k$ zeros of
the function $G$, the singular locus $S$ is of dimension at most $n-1,$ and
the function $G$ is not flat at any point of $S.$
Take a point $q_{\epsilon}=(z_{1}=\epsilon,z_{2}=0,\ldots,z_{n}=0)\in U$ with
$\epsilon>0$ small enough. Take the $(n-1)$-dimensional ball
$B^{n-1}(q_{\epsilon},\epsilon^{K})$ of radius $\epsilon^{K}$ which is
orthogonal to the vector $Y(q_{\epsilon})$ at the point $q_{\epsilon}$ in the
coordinate system $(z_{1},\ldots,z_{n})$, for a certain positive number $K$ to
be chosen below. Denote by $\phi$ the Poincaré map of the flow of $Y$ on $D.$
A-priori this map does not necessarily fixes the point $q_{\epsilon}.$ But due
to the fact that $Y=X^{(1)}+flat$ and the flow of $X^{(1)}$ is periodic, the
distance from $q_{\epsilon}$ to $\phi(q_{\epsilon})$ is smaller than
$\epsilon^{K+1}$ (provided that $\epsilon$ is small enough). We can choose $K$
large enough so that the restriction of the map $(F_{1},\ldots,F_{n-1})$ to
$B^{n-1}(q_{\epsilon},\epsilon^{K})$ is injective. Due to the invariance of
the functions $F_{i}$ with respect to the vector field $Y$, the Poincaré map
$\phi$, we also have the points $q_{\epsilon}$ and $\phi(q_{\epsilon})$ have
the same image under the map $(F_{1},\ldots,F_{n-1})$. But this map is
injective on the ball $B^{n-1}(q_{\epsilon},\epsilon^{K})$ which contains
these two point, so in fact these two points must coincide, i.e. we have
$q_{\epsilon}=\phi(q_{\epsilon})$, and the orbit of the flow of $Y$ through
the point $q_{\epsilon}$ is a periodic orbit, and the period of this orbit is
equal to $2\pi/\lambda$ plus a small error term which tends to 0 faster than
anu power of $\epsilon$ when $\epsilon$ tends to 0.
Denote by $V$ the path-connected component of $U\setminus S$ which contains
the points $q_{\epsilon}.$ Then the orbit of $Y$ through any point $q\in V$ is
also periodic and its period is close to $2\pi/\lambda$ (the difference
between the period and $2\pi/\lambda$ tends to 0 uniformly when the radius of
$U$ tends to 0). This fact can be proved easily by showing that the set of
points of $V$ which satisfies the mentioned property is closed and open in $V$
at the same time: closed due to the continuity, and open because
$(F_{1},\ldots,F_{n-1})$ is regular in $V$ and is preserved by the flow of
$Y$.
Let $q\in S$ be a point in the locus $S$ which also lies on the boundary of
$V$. Then by continuity, there is also a number $T$ near $2\pi/\lambda$ such
that the time-$T$ flow of $Y$ fixes the point $q$. In other words, the orbit
of $Y$ through $q$ is also periodic, and the period is equal to $T$ or a
fraction $T/m$ of $T$ for some natural number $m$. As before, consider a
$(n-1)$-dimensional ball $B^{n-1}(q,\delta)$ which is centered at $q$ and
orthogonal to $Y(q)$, for some $\delta>0$ small enough. Consider the Poincaré
map $\phi$ of $Y$ on $B^{n-1}(q,\delta)$ corresponding to the time $T$ (i.e.
if the period of the orbit through $q$ is $T/m$ then consider the $m$-time
itaration of the usual Poincaré map). Since the intersection of
${B^{n-1}(q,\delta)}$ with $V$ contains an open subset of $B^{n-1}(q,\delta)$
whose closure contains $q$, and the Poincaré map is identity on that open
subset by the above considerations, the Poincaré map on $B^{n-1}(q,\delta)$ is
equal to the identity map plus a flat term at $q$. On the other hand, this
Poincaré map must preserve the map
$(F_{1},\ldots,F_{n-1})|_{B^{n-1}(q,\delta)}$, and the determinant of the
differential of this map is not flat at $q$. It implies that the Poincaré map
must be identity in a small neighborhood of $q$ in ${B^{n-1}(q,\delta)}$.
Thus, we can “engulf” the set of points shown to have periodic orbits from $V$
to a larger open subset of $U$ which contains the boundary of $V$. Continuing
this engulfing process, we get that the set of points in $U$ having periodic
orbits is actually the whole $U$. ∎
We will linearize $Y=X/F$ orbitally, and then deduce the normalization of $X$
from this linearization. In order to do that, let us consider the blow-up of
$\mathbb{R}^{n}$ at $O$, which will be denoted by
(3.6) $p:E\to U,$
where $U\ni O$ is a neigborhood of $O$ in $\mathbb{R}^{n}$ and
$p^{-1}(O)\cong\mathbb{R}\mathbb{P}^{n-1}$ is the exceptional divisor of the
blow-up in $E$. We will need the following simple lemma, whose proof is
steaightforward:
###### Lemma 3.2.
With the above notations, a function $G$ or a vector field $Z$ is flat at $O$
in $U$ if and only if its pull-back to $E$ via the projection map $p$ is flat
along $p^{-1}(O)$ in $E$.
Denote by $\tilde{G}$ (resp. $\tilde{Z}$) the pull-back of a function $G$
(resp. vector field $Z$) via the projection map $\pi:E\to U$ of the blow-up.
Then we have
(3.7) $\tilde{Y}=\tilde{X}^{(1)}+\tilde{Z}$
in $E$, where $\tilde{Z}$ is vector field which is flat along $p^{-1}(O),$ and
$\tilde{X}^{(1)}$ is a smooth periodic vector field in $E$ of period
$2\pi/\lambda.$ By Lemma 3.1, the orbits of $\tilde{Y}$ are closed, with
periods close to the period of $\tilde{X}^{(1)}$. Due to the flatness of $Z$
along $p^{-1}(O)$, the period of $\tilde{Y}$ at the points in $E$ is equal to
$2\pi/\lambda$ plus a smooth function on $E$ which is flat along $p^{-1}(O)$.
Projecting $\tilde{Y}$ back to $U$ and using Lemma 3.2, we get a smooth period
function $P=2\pi/\lambda+flat$ (which is invarant on the orbits) such that
$PY$ is periodic of period 1. In other words, $PY$ generates a smooth
$\mathbb{T}^{1}$-action. Using the classical Cartan-Bochner smooth
linearization theorem for compact group actions, we find a smooth coordinate
system, which we will denote again by $(x_{1},\ldots,x_{n})$, in which
$PX/F=PY$ is a linear vector field, i.e. in which we have
(3.8) $X=GX^{(1)}$
where $G$ is a smooth function and $X^{(1)}$ is a linear vector field which
satisfies Formula (3.2).
A-priori, the function $F$ given by Proposition 2.3 is not a first integral of
$X$ (though it is a first integral of the linear part of $X$ in some
coordinate system), and so the function $G=2\pi F/P\lambda$ in Formula (3.8)
is not a first integral of $X$ in either. But we can normalize further in
order to change $G$ into a first integral. Indeed, by the arguments presented
above, we can assume that $G$ is a smooth first integral of $X$ plus a flat
term, or we can write $G=G_{1}(1+flat),$ where $G_{1}$ is a first integral of
$X$. Normalizing the new vector field $Y=X/G_{1}$ instead of the old $Y=X/F,$
we get a new smooth coordinate system in which $PY=(2\pi/\lambda)X^{(1)},$
where $P$ is the period function of the new vector field $Y$, and it is a
smooth first integral of the type $constant+flat.$ In this new coordinate
system we have that $X$ is equal to its linear part times a first integral,
and Theorem 1.1 is proved in the elliptic case i.e. without eigenvalue 0.
Since our proof for the strong hyperbolic case and the strong elliptic case
also works for smooth families of integrable vector fields, Theorem 1.1 is
proved.
###### Remark 3.3.
According to a theorem of Schwarz [5], the smooth first integral $F$ in the
normal form in the elliptic case can also be written as
(3.9) $F=f(Q_{1}(x_{1},\ldots,x_{n}),\ldots,Q_{m}(x_{1},\ldots,x_{n})),$
where $Q_{1}(x_{1},\ldots,x_{n}),\ldots,Q_{m}(x_{1},\ldots,x_{n})$ are
homogeneous polynoms which generatedthe ring of polynomial first integrals of
the linear vector field $X^{(1)}.$
## 4\. The case of dimension 2
The aim of this section is to show that condition iii) in Theorem 1.1 is
redundant at least in the case of dimension 2. More precisely, we have:
###### Theorem 4.1.
Let $X$ be a smooth vector field in a neighborhood of $O=(0,0)$ in
$\mathbb{R}^{2},$ which vanishes at $O$ and satisfies the following
conditions:
i) (Complete integrability): $X$ admits a smooth first integral $F_{1}.$
ii) (Nondegeneracy): The semisimple part of the linear part of $X$ at $O$ is
non-zero, and the $\infty$-jet of $F_{1}$ at $O$ is non-constant.
Then there exists a local smooth coordinate system $(x,y)$ in which $X$ can be
written as
(4.1) $X=FX^{(1)},$
where $X^{(1)}$ is a semisimple linear vector field in $(x,y)$, and $F$ is a
smooth first integral of $X^{(1)}.$
###### Proof.
Remark that, in the case of dimension 2, there are only 3 possibilities:
elliptic without zero eigenvalue, hyperbolic without zero eigenvalue, and
hyperbolic with zero eigenvalue. The first two possibilities are covered by
Theorem 1.1. It remains to prove Theorem 4.1 for the case when $X$ has one
eigenvalue equal to 0. By Proposition 2.3, we can assume that
(4.2) $X=F(y)x\frac{\partial}{\partial x}+flat_{1}\frac{\partial}{\partial
x}+flat_{2}\frac{\partial}{\partial y}$
in a smooth coordinate system $(x,y),$ where $flat_{1}$ and $flat_{2}$ are two
flat functions, and $F(0)\neq 0.$
Denote by
(4.3) $S=\\{q\in U\ |\ X(q)=0\\}$
the singular locus of $X$ near $O$, where $U$ denotes a small neighborhood of
$O$ in $\mathbb{R}^{2}.$ The main point is to prove that $S$ is a smooth
curve. If $S$ is a smooth curve, then we can write $S=\\{y=0\\},$ the vector
field $X$ is divisible by $x$, i.e. $Y=X/x$ is still a smooth vector field,
which is non-zero at $O$, and therefore locally rectifiable and admits a first
integral $G$ such that $dG(0)\neq 0.$ But $G$ is also a first integral of $X,$
so condition iii) of Theorem 1.1 is also satisfied, and Theorem 4.1 is reduced
to a particular case of Theorem 1.1.
Denote by
(4.4) $S_{1}=\\{(x,y)\in U\ |\ F(y)x+flat_{1}(x,y)=0\\}$
the set of points where the $\frac{\partial}{\partial x}$-component of $X$
vanishes. It is clear that $S\subset S_{1},$ and $S_{1}$ is a smooth curve
tangent to the line $\\{x=0\\}$ at $O$ by the inverse function theorem. We
will show that $S=S_{1}.$
Consider the cone
(4.5) $C=\\{(x,y)\in U\ |\ |x|\leq|y|\\}.$
Clearly, $S_{1}\subset C$ (provided that $U$ is small enough). The non-flat
first integral $F_{1}$ of $X$ in the coordinate system $(x,y)$ has the type
(4.6) $F_{1}=f(y)+flat$
where $f(y)=a_{h}y^{h}+h.o.t.$ is a non-flat smooth function. It implis that
the level sets of $F_{1}$ in the cone $C$ are smooth curves which are nearly
tangent to the lines $\\{y=const\\}.$ In particular, each level set of $F_{1}$
in $C$ intersects with $S_{1}$ at exactly 1 point. Since $X$ is tangent to
these level sets, and the $\frac{\partial}{\partial x}$-component of $X$
vanishes at the intersection points of these level sets with $S_{1}$, it
follows that $X$ itself vanishes at these intersetion points. But every point
of $S_{1}$ is an intersection point of $S_{1}$ with a level set of $F_{1}.$
Thus $X$ vanishes on $S_{1}$, and we have $S=S_{1}.$ ∎
###### Remark 4.2.
Two-dimensional elliptic-like vector fields, i.e. those vector fields whose
orbits near a singular point are closed, are also called _centers_ in the
literature. There is a recent interesting theorem of Maksymenko [4] about the
orbital linearization of the center, without the assumption on the existence
of a first integral, but with an assumption on the periods of the periodic
orbits. Maksymenko’s theorem is similar to and a bit stronger than the
elliptic case of Theorem 4.1 because his assumptions are weaker, and the
conclusions are the same. His proof is also based on the formal normalization
and the blowing-up method.
###### Remark 4.3.
Some of the arguments of the proof of Theorem 4.1 are still valid in the
$n$-dimensional case where 0 is an eigenvalue with multiplicity $k\geq 1.$ In
particular, one can still show that, even without condition iii) of Theorem
1.1, the local singular locus of $X$ is still a smooth $k$-dimensional
manifold. However, it is more difficult to show that there is still a local
regular invariant $(n-k)$-dimensional foliation. If one can show the existence
of this regular regular invariant foliation, then one can drop condition iii)
from the statement of Theorem 1.1 because it is a consequence of the first two
conditions. Maybe it is possible to use the techniques of Belitskii-Kopanskii
[1] together with a kind of desingularization of the first integrals in order
to show the existence of an invariant regular foliation, but we don’t have a
proof so far.
## References
* [1] G.R. Belitskii, A.Y. Kopanskii, Equivariant Sternberg-Chen theorem, Journal of Dynamics and Differential Equations, Volume 14 (2002), Number 2, pp. 349–367.
* [2] K.T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., 85 (1963), 693–722.
* [3] M. Golubitsky, I. Stewart, D. Schaeffer, Singularities and Groups in Bifurcation Theory, Volume II (1988), Springer-Verlag, New York.
* [4] S. I. Maksymenko, Symmetries of center singularities of plane vector fields, Nonlinear Oscil. (N. Y.) 13 (2010), no. 2, 196–227.
* [5] G. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology, Vol. 14 (1975), 63–68.
* [6] S. Sternberg, On the structure of local homeomorphisms of Euclidean $n$-space, II, Amer. J. of Math., 80 (1958), 623-631.
* [7] S.L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics, Funcional Anal. Appl. 16 (1982), 181–189.
* [8] N.T. Zung, _Convergence versus integrability in Poincaré-Dulac normal forms_ , Math. Res. Lett. 9 (2002), no. 2-3, 217-228.
* [9] N.T. Zung, Nondegenerate singularities of integrable dynamical systems, preprint arXiv:1108.3551v2 (2012).
* [10] N.T. Zung, N.V. Minh, _Geometry of integrable dynamical systems on 2-dimensional surfaces_ , preprint arXiv:1204.1639 (2012).
|
arxiv-papers
| 2012-04-25T16:35:09 |
2024-09-04T02:49:30.168807
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Nguyen Tien Zung",
"submitter": "Nguyen Tien Zung",
"url": "https://arxiv.org/abs/1204.5701"
}
|
1204.5919
|
11institutetext: US Naval Observatory, Washington DC 20392 USA
11email: michael.efroimsky @ usno.navy.mil
# Justification of the two-bulge method in the theory of bodily tides
Michael Efroimsky 11
(Received: 25 April 2012; accepted: … )
###### Abstract
Mathematical modeling of bodily tides can be carried out in various ways. Most
straightforward is the method of complex amplitudes, which is often used in
the planetary science. Another method, employed both in planetary science and
in astrophysics, is based on decomposition of each harmonic of the tide into
two bulges oriented orthogonally to one another. We prove that the two methods
are equivalent. Specifically, we demonstrate that the two-bulge method is not
a separate approximation, but ensues directly from the Fourier expansion of a
linear tidal theory equipped with an arbitrary rheological model involving a
departure from elasticity. To this end, we use the most general mathematical
formalism applicable to linear bodily tides. To express the tidal amendment to
the potential of the perturbed primary, we act on the tide-raising potential
of the perturbing secondary with a convolution operator. This enables us to
interconnect a complex Fourier component of the tidally generated potential of
the perturbed primary with the appropriate complex Fourier component of the
tide-raising potential of the secondary. Then we demonstrate how this
interrelation entails the two-bulge description. While less economical
mathematically, the two-bulge approach has a good illustrative power, and may
be employed on a par with a more concise method of complex amplitudes. At the
same time, there exist situations where the two-bulge method becomes more
practical for technical calculations.
###### Key Words.:
Celestial Mechanics – Stars: binaries: close – Stars: planetary systems –
Stars: rotation – Planets and satellites: general – Planets and satellites:
dynamical evolution and stability
## 1 Introduction and aim
On several occasions, it was suggested by different authors to model bodily
tides with superposition of two symmetrical bulges. One bulge is always aimed
at the secondary, and thus implements the instantaneous reaction of the
primary’s shape and potential to the tide-rising gravitational pull exerted by
the secondary. This portion of the tide is called “adiabatic tide” (Zahn
1966a,b) or “elastic tide” (Ferraz Mello 2012; Krasinsky 2006). The second
bulge is assumed to align orthogonally to the direction to the tide-raising
secondary, and thus is set to implement the entire nonelastic portion of the
primary’s deformation. This, second bulge is called “dissipative tide” (Zahn
1966a,b; Krasinsky 2006) or “creep tide” (Ferraz Mello 2012).
In this note, we demonstrate that the two-bulge method is not a separate
approximation, but ensues directly from the Fourier expansion of a linear
tidal theory equipped with an arbitrary rheological model involving a
departure from elasticity.
## 2 The static linear theory of bodily tides
Let a spherical primary of radius $\,R\,$ be subject to the gravitational pull
by a secondary of mass $M_{sec}\,$, residing at the position
$\,{\mbox{{\boldmath$\@vec{r}$}}}^{\;*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$,
where $\,r^{*}\geq R\,$. At a surface point
$\mbox{{\boldmath$\@vec{R}$}}=(R,\phi,\lambda)$ of the primary body, the tidal
potential generated by the secondary can be expanded over the Legendre
polynomials $\,P_{\it l}(\cos\gamma)\;$ as
$\displaystyle
W(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*})$ $\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\ *})\leavevmode\nobreak\
=\leavevmode\nobreak\ -\leavevmode\nobreak\
\frac{G\;M_{sec}}{r^{\,*}}\leavevmode\nobreak\
\sum_{{\it{l}}=2}^{\infty}\,\left(\,\frac{R}{r^{\,*}}\,\right)^{\textstyle{{}^{\it{l}}}}\,P_{\it{l}}(\cos\gamma)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (1)
Here $\,G\,$ denotes Newton’s gravity constant, while $\gamma\,$ is the
angular separation between the vectors ${\mbox{{\boldmath$\@vec{r}$}}}^{\;*}$
and $\@vec{R}$ pointing from the primary’s centre. The longitudes
$\lambda,\,\lambda^{*}$ are measured from a fixed meridian on the primary
body, the latitudes $\phi,\,\phi^{*}$ being reckoned from the equator. The
index $\,l\,$ is conventionally named as the degree. In (1) the $\,l=0\,$ term
is missing, because it corresponds to the principal, Newtonian part of the
secondary’s potential, and is not a part of the perturbation. Omission of the
$\,l=1\,$ term is a more subtle point related to the fact that we are
describing the motion of the secondary relative to the primary body, and not
relative to an inertial frame (see equations 8 - 11 in Efroimsky & Williams
2009).
Within the linear theory, the $\,{\emph{l}}^{\leavevmode\nobreak\ th}$ term
$\,W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\ *})\,$ of the secondary’s
potential generates a linear alteration of the primary’s shape. This
alteration, in its turn, causes a linear amendment
$\,U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\,$ to the gravitational potential
of the primary, where linear means: linear in
$\,W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\ *})\,$. The theory of
potential requires that outside the primary body
$\,U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\,$ should scale with the distance
as $\,r^{-(\it{l}+1)}\,$. Hence the said change in the primary’s potential may
be written down as
$\displaystyle
U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\,=\,k_{l}\,\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (2)
where $R\,$ is the mean equatorial radius of the primary,
$\,\mbox{{\boldmath$\@vec{R}$}}\,=\,(R\,,\,\phi\,,\,\lambda)\,$ is a point on
the primary’s surface, while
$\,\mbox{{\boldmath$\@vec{r}$}}\,=\,(r\,,\,\phi\,,\,\lambda)\,$ is an exterior
point right above the surface point $\@vec{R}$ , at a radius $\,r\,\geq\,R\,$.
The numerical factors $\,k_{l}\,$ are the degree-$l$ Love numbers calculated
from the rheology of the primary body.
The overall tidally caused change of the primary’s potential thus amounts to
$\displaystyle U(\mbox{{\boldmath$\@vec{r}$}})\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\
U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\ k_{\it
l}\;\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (3)
## 3 Dynamical linear theories of bodily tides
In realistic situations, the disturbing potential is evolving, so equation (1)
assumes the form of
$\displaystyle
W(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*}(t)\,)$ $\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*}(t)\,)\leavevmode\nobreak\ =\leavevmode\nobreak\ -\leavevmode\nobreak\
\frac{G\;M_{sec}}{r^{\,*}(t)}\leavevmode\nobreak\
\sum_{{\it{l}}=2}^{\infty}\,\left(\,\frac{R}{r^{\,*}(t)}\,\right)^{\textstyle{{}^{\it{l}}}}\,P_{\it{l}}(\cos\gamma(t)\,)\leavevmode\nobreak\
\leavevmode\nobreak\ .\leavevmode\nobreak\ $ (4)
Then one should expect the distortion of the primary, as well as the
corresponding amendment to its potential at an exterior point $\@vec{r}$ , to
become a function of time: $\,U(\mbox{{\boldmath$\@vec{r}$}},\,t)\,$.
### 3.1 Elastic dynamical tides
Had the tides contained only instantaneous, elastic components, the
expressions for the tidal potential would mimic (2 \- 3). At each instant of
time $\,t\,$, the degree-$l$ term of the tide-raising potential $\,W\,$ of the
orbiting secondary would generate instantaneously an appropriate degree-$l$
term of the tidal potential of the primary:
${}^{\textstyle{{}^{(elastic)}}}U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}},\,t)\,=\,k_{l}\,\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t)\,)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (5)
so the total tidal amendment to the potential of the primary would look:
${}^{\textstyle{{}^{(elastic)}}}U(\mbox{{\boldmath$\@vec{r}$}},\,t)\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\
\leavevmode\nobreak\
{}^{\textstyle{{}^{(elastic)}}}U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}},\,t)\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\
k_{l}\leavevmode\nobreak\ \left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*}(t)\,)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (6)
Needless to say, in realistic materials the internal friction prevents the
tidal deformation from being instantaneous.
### 3.2 Realistic dynamical tides
To describe deviation from elasticity, we spell the two basic assumptions
whereon a linear dynamical theory of bodily tides is based:
* [1]
tidal deformation is linear with respect to the stress generated by the tide-
raising potential;
* [2]
the deformation is not fully elastic: it incorporates both an immediate and
delayed portions (delayed – relative to the tide-raising potential).
Mathematically, assumption [1] means that an infinitesimal increment $\,\Delta
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,$,
whereby the perturbing potential increased at the time $\,t\,^{\prime}\,$ in
the past, results in a proportional present-time increment of the tidally
distorted shape of the primary and, accordingly, in a proportional increment
of the tidal amendment to its potential:
$\displaystyle\Delta U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\leavevmode\nobreak\ =\leavevmode\nobreak\
\left(\frac{R}{r}\right)^{{\it
l}+1}{\it{k}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\leavevmode\nobreak\
\,\Delta
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (7)
${\it{k}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ being a function describing
the delayed reaction of the shape. The mathematical form of this function is
defined by the rheology of the body and by its self-gravitation.
Thus the addition $\,U_{l}\,$ to the primary’s potential gets expressed
through the tide-raising potential $\,W_{l}\,$ by a linear integral operator:
$\displaystyle U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\,=\,\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=-\infty}^{t\,^{\prime}=t}k_{\it
l}(t-t\,^{\prime})\stackrel{{\scriptstyle\bf\centerdot}}{{W}}_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,dt\,^{\prime}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (8)
overdot denoting a time derivative. Integration of (8) by parts renders:
$\displaystyle U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\,=\,\left(\frac{R}{r}\right)^{{\it
l}+1}\left[k_{l}(0)W_{l}(t)\,-\,k_{l}(\infty)W_{l}(-\infty)\right]\,+\,\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{-\infty}^{t}{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,dt\,^{\prime}\leavevmode\nobreak\
\,,\leavevmode\nobreak\ \quad$ (9)
where the relaxed term $\,-\,k_{l}(\infty)W(-\infty)\,$ should be neglected.
Indeed, the current events cannot be influenced by the perturbation
$\,W_{l}(-\infty)\,$ in the infinite past, wherefore $\,k_{l}(\infty)=0\,$. Of
the remaining two terms, the unrelaxed term $\,k_{l}(0)W_{l}(t)\,$ reflects
the elastic part of the deformation, while the integral expresses the delayed
components – viscous and anelastic.
Assumption [2] means that both the unrelaxed term $\,k_{l}(0)W_{l}(t)\,$ and
the delayed term given by the integral should be kept. As explained in
Efroimsky (2012a,b), the relaxed term may be easily incorporated into the
integral, where it should show up multiplied with a Heaviside step function
$\,\Theta(t-t\,^{\prime})\,$. Then our expression for the tidal potential will
acquire the simple form of
$\displaystyle U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\;=\;\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{-\infty}^{t}{\stackrel{{\scriptstyle\bf\centerdot}}{{\it{k}}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,dt\,^{\prime}\leavevmode\nobreak\
,$ (10)
${\bf{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})$ now incorporating both
the delayed-reaction terms and the elastic term
$\,{k}_{l}(0)\,\Theta(t-t\,^{\prime})\,$. The elastic part will enter the
kernel
$\,{\stackrel{{\scriptstyle\bf\centerdot}}{{\it{k}}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$
as $\,{k}_{l}(0)\,\delta(t-t\,^{\prime})\,$, with $\,\delta(t-t\,^{\prime})\,$
being the Dirac delta function. Integration of this term will furnish
$\,k_{l}(0)W_{l}(t)\,$, as in (9).
## 4 Fourier components of tidal stresses and strains
### 4.1 Tidal modes and tidal frequencies
The sidereal angle and the spin rate of a tidally-perturbed primary are
normally denoted with $\,\theta\,$ and
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}\,$, while the node,
pericentre, and mean anomaly of a tide-raising secondary, as seen from the
primary, are denoted with $\,\Omega\,$, $\omega\,$, and ${\cal M}$.
In the Darwin-Kaula theory, the tide-raising potential $W$, the primary’s
deformation, and the tidal amendment $U$ to the primary’s potential are
expanded over the modes
$\displaystyle\omega_{lmpq}\;\equiv\;(l-2p)\;\dot{\omega}\,+\,(l-2p+q)\;\dot{\cal{M}}\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})\,\approx\,(l-2p+q)\;n\,-\,m\;\dot{\theta}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (11)
$l,\,m,\,p,\,q\,$ being integers, and $\,n\,$ being the mean motion. Dependent
upon the values of the mean motion, spin rate, and the indices, the tidal
modes $\,\omega_{{\it l}mpq}\,$ may be positive or negative or zero.
The actual forcing frequencies of the resulting stresses and strains in the
primary’s material are the absolute values of the tidal modes:
$\displaystyle\chi_{lmpq}\equiv\,|\,\omega_{lmpq}\,|\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (12)
so these frequencies are always positive.
### 4.2 Fourier expansions
In practical calculations, it is extremely convenient to employ complex
stresses and strains, under the convention that the actual, physical
quantities are the real parts of their complex counterparts. This way, the
Fourier series for the stress and strain look:
$\displaystyle\sigma_{\gamma\nu}(t)\,\,=\,\sum_{s=0}^{\infty}\,\sigma_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,\cos\left[\,\chi_{\textstyle{{}_{s}}}t+\varphi_{\sigma}(\chi_{\textstyle{{}_{s}}})\,\right]\;=\;\sum_{s=0}^{\infty}\,{\cal{R}}{\it{e}}\left[\,{\bar{\sigma}}_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,\;\exp\left({\textstyle{{\,{\it
i}\chi_{\textstyle{{}_{s}}}t}}}\right)\,\;\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (13)
$\displaystyle
u_{\gamma\nu}(t)\,=\,\sum_{s=0}^{\infty}\,u_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,\cos\left[\,\chi_{\textstyle{{}_{s}}}t+\varphi_{u}(\chi_{\textstyle{{}_{s}}})\,\right]\;=\;\sum_{s=0}^{\infty}\,{\cal{R}}{\it{e}}\left[\,{\bar{u}}_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\;\,\exp\left({\textstyle{{\,{\it
i}\chi_{\textstyle{{}_{s}}}t}}}\right)\,\;\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (14)
$\gamma\nu$ being tensor indices, and $s$ being a concise notation for $lmpq$.
The complex amplitudes are
$\displaystyle{\bar{{\sigma}}_{\gamma\nu}}(\chi)={{{\sigma}}_{\gamma\nu}}(\chi)\,\;\exp\left[{{\it
i}\varphi_{\sigma}(\chi)}\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\bar{{u}}_{\gamma\nu}}(\chi)={{{u}}_{\gamma\nu}}(\chi)\,\;\exp\left[{{\it
i}\varphi_{u}(\chi)}\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (15)
where the initial phases $\,\varphi_{\sigma}(\chi)\,$ and
$\,\varphi_{u}(\chi)\,$ are chosen so that the real amplitudes
$\,\sigma_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,$ and
$\,u_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,$ are non-negative.
For a continuous spectrum, the sums get replaced with integrals over
frequency:
$\displaystyle\sigma_{\gamma\nu}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{\sigma}_{\gamma\nu}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\quad\quad\mbox{and}\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad
u_{\gamma\nu}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{u}_{\gamma\nu}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ .$ (16)
Similarly, the tide-raising potential $W_{l}$ and the potential $U_{l}$ of the
primary get expanded into a sum or integral over the tidal modes:
$\displaystyle W_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{-\infty}^{\infty}\,\bar{W}_{l}(\omega)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\omega t}}}\leavevmode\nobreak\
d\omega\quad\quad\mbox{and}\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad U_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{-\infty}^{\infty}\,\bar{U}_{l}(\omega)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\omega t}}}\leavevmode\nobreak\
d\omega\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (17)
where the complex amplitudes are expressed via the real amplitudes and the
initial phases by
$\displaystyle{\bar{{W}}_{l}}(\omega)={{{W}}_{l}}(\omega)\,\;\exp\left[{{\it
i}\varphi_{\textstyle{{}_{W_{l}}}}(\omega)}\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\bar{{U}}_{l}}(\omega)={{{U}}_{l}}(\omega)\,\;\exp\left[{{\it
i}\varphi_{\textstyle{{}_{U_{l}}}}(\omega)}\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (18)
The phases $\,\varphi_{\textstyle{{}_{W_{l}}}}(\omega)\,$ and
$\,\varphi_{\textstyle{{}_{U_{l}}}}(\omega)\,$ can always be set in such a way
that the real amplitudes $\,W_{l}(\chi)\,$ and $\,U_{l}(\chi)\,$ are non-
negative.
Both in (16) and (17), the actual, physical spectral components are the real
parts of the complex ones. However, there also is an important difference
between (16) and (17). While the stresses and strains are habitually expanded
in (16) over positive frequencies $\,\chi\,$ only, the potentials in (17) are
expanded over the tidal modes $\,\omega\,$, which can be positive or negative
or zero, as demonstrated in the Darwin-Kaula theory of tides.
It is of course a common fact that a real function can be decomposed into a
Fourier series or integral over only positive frequencies. This way,
expansions of the potentials over $\,\chi\equiv|\omega|\,$ may appear to be
sufficient, because a contribution from some negative tidal mode
$\,\omega<0\,$ can be shown to coincide with the contribution from the
appropriate positive mode $\,|\omega|>0\,$. Therefore, (17) may be rewritten
simply as
$\displaystyle W_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{W}_{l}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\quad\quad\mbox{and}\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad
U_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{U}_{l}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (19)
where $\,\bar{W}_{l}(\chi)\,=\,2\,\bar{W}_{l}(\omega)\,$ and
$\,\bar{U}_{l}(\chi)\,=\,2\,\bar{U}_{l}(\omega)\,$.
Surprisingly, the theory of tides is a rare exception from the rule, in that
this theory does distinguish between the contribution from a negative tidal
mode and that from a positive mode of the same absolute value. Fortunately,
the difference shows up only at the stage when one calculates tidal forces or
torques (Efroimsky 2012a,b). As in the current paper we discuss potentials
only, we shall ignore this subtlety and shall employ (19) instead of (17).
### 4.3 Dynamical analogues to the Love number
Insertion of (19) into (10) entails
$\displaystyle\bar{U}_{\textstyle{{}_{l}}}(\chi)\;=\;\left(\frac{R}{r}\right)^{l+1}\bar{k}_{\textstyle{{}_{l}}}(\chi)\;\,\bar{W}_{\textstyle{{}_{l}}}(\chi)$
(20)
or, in a more detailed form:
$\displaystyle|\,\bar{U}_{\textstyle{{}_{l}}}(\chi)\,|\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,+\,i\,\varphi_{\textstyle{{}_{{}_{U_{l}}}}}(\chi)}}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \left(\frac{R}{r}\right)^{l+1}\leavevmode\nobreak\
|\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,|\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,-\,i\,\epsilon_{l}(\chi)}}}\;|\,\bar{W}_{\textstyle{{}_{l}}}(\chi)\,|\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,+\,i\,\varphi_{\textstyle{{}_{{}_{W_{l}}}}}(\chi)}}}\;\;\;,$
(21)
where the complex function
$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\,=\,|\bar{k}_{\textstyle{{}_{l}}}(\chi)|\leavevmode\nobreak\
e^{\textstyle{{}^{\,-\,i\,\epsilon_{l}(\chi)\,}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (22)
is a Fourier component of the kernel
$\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ of the integral
operator (10). The kernels
$\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ are named Love
functions, a term suggested by Churkin (1998). The functions
$\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,$ are called complex Love numbers, in
understanding however that these “numbers” change with frequency. An approach
similar to (22) was taken by Mathis and Le Poncin Lafitte (2009). These
authors introduced a complex impedance as the ratio between the complex Love
number and the static Love number. Their equation (107) is equivalent to our
(22), up to a convention on the sign of the argument.
We see from (21) that at each frequency $\,\chi\,$, the negative argument
$\,\epsilon_{l}(\chi)\leavevmode\nobreak\ $ is a measure of lagging of the
spectral component $\,U_{l}(\chi)\,$, relative to appropriate spectral
component $\,W_{l}(\chi)\leavevmode\nobreak\ $:
$\displaystyle\varphi_{\textstyle{{}_{U_{l}}}}(\chi)\leavevmode\nobreak\
=\leavevmode\nobreak\
\varphi_{\textstyle{{}_{W_{l}}}}(\chi)\leavevmode\nobreak\
-\leavevmode\nobreak\ \epsilon_{l}(\chi)\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (23)
## 5 Decomposition of a tidal mode into an in-phase part and an in-quadrature
(lagging by $\bf{90^{o}}$) part
In the expression (21), we may set the initial phase of
$\,\bar{W}_{l}(\chi)\,$ to be zero, and reckon the phase of
$\,\bar{U}_{l}(\chi)\,$ from $\,\bar{W}_{l}(\chi)\,$. This will enable us to
single out, in $\,\bar{U}_{l}(\chi)\,$, a part which is in phase with
$\,\bar{W}_{l}(\chi)\,$, and also to see what part of $\,\bar{U}_{l}(\chi)\,$
is out of phase with $\,\bar{W}_{l}(\chi)\,$.
According to (23), nullification of the phase of $\,\bar{W}_{l}(\chi)\,$ makes
the phase of $\,\bar{U}_{l}(\chi)\,$ equal to the negative argument of
$\,\bar{k}_{l}(\chi)\,$. So (21) will assume the form of
$\displaystyle U_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,-\,i\,\epsilon_{l}(\chi)}}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \left(\frac{R}{r}\right)^{l+1}\leavevmode\nobreak\
k_{l{\textstyle{{}_{\,0}}}}(\chi)\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{\,-\,i\,\epsilon_{l}(\chi)}}}\;W_{l{\textstyle{{}_{\,0}}}}(\chi)\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t}}}\;\;\;,$ (24)
where we introduced simplified notations
$U_{l{\textstyle{{}_{\,0}}}}(\chi)\,$, $\,W_{l{\textstyle{{}_{\,0}}}}(\chi)\,$
and $\,k_{l{\textstyle{{}_{\,0}}}}\,$ for real amplitudes:
$\displaystyle U_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\equiv\leavevmode\nobreak\
|\,\bar{U}_{l}(\chi)\,|\,\quad,\quad\quad\,W_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\equiv\leavevmode\nobreak\
|\,\bar{W}_{l}(\chi)\,|\,\quad,\quad\quad\,k_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ |\,\bar{k}_{l}(\chi)\,|\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (25)
By means of the Euler formula, (24) can be trivially expanded as
$\displaystyle U_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\left[\,\cos\left(\chi t-\epsilon_{l}(\chi)\,\right)+i\sin\left(\chi
t-\epsilon_{l}(\chi)\,\right)\,\right]\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\left(\frac{R}{r}\right)^{2l+1}k_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\left[\,\cos\epsilon_{l}(\chi)-i\sin\epsilon_{l}(\chi)\,\right]\leavevmode\nobreak\
W_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\ \left[\,\cos(\chi
t)+i\sin(\chi t)\,\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .$ (26)
The actual, physical tide-raising potential $W_{l}(\chi)$ is the real part of
the complex $\bar{W}_{l}(\chi)$. So it is rendered by
$\,{W_{l}}_{0}(\chi)\cos(\chi t)\,$, where we set the initial phase nil, as
agreed above. Similarly, the actual, physical tidal potential $U_{l}(\chi)$ is
the real part of the complex $U_{l}(\chi)$, and it reads as:
$\displaystyle{U_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\ \cos(\chi
t)-{U_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\ \sin(\chi t)\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle=\,\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \cos(\chi
t)\,+\,\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \sin(\chi t)\leavevmode\nobreak\
\,.\leavevmode\nobreak\ \leavevmode\nobreak\ $ (27)
In this expression for the actual real $\,U_{l}\,$, we see not one but two
terms. One is the elastic part of the tide, a part that is in phase with the
real $\,W_{l}\,$. This is the term proportional to $\,\cos(\chi
t)\leavevmode\nobreak\ $:
${}^{\textstyle{{}^{(in\leavevmode\nobreak\
phase)}}}{U_{l}}\leavevmode\nobreak\ =\leavevmode\nobreak\
{U_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\ \cos(\chi
t)=\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \cos(\chi t)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (28)
We see that the “response factor” is equal to
${k_{l}}_{0}(\chi)\cos\epsilon_{l}(\chi)\leavevmode\nobreak\ $, where
${k_{l}}_{0}(\chi)$ is the real amplitude of the complex Love number (call
this amplitude dynamical Love number).
The second component is the in-quadrature term, which is proportional to
$\sin(\chi t)\leavevmode\nobreak\ $:
${}^{\textstyle{{}^{(in\leavevmode\nobreak\
quadrature)}}}{U_{l}}\leavevmode\nobreak\ =\leavevmode\nobreak\
-\leavevmode\nobreak\ {U_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\ \sin(\chi
t)=\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \sin(\chi t)\leavevmode\nobreak\
\,.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (29)
Naturally, the expression of this term via $W_{l}$ contains
${k_{l}}_{0}(\chi)\sin\epsilon_{l}(\chi)\leavevmode\nobreak\ $.
Therefore, as soon as we express $U_{l}$ via $W_{l}$ by an integral operator
permitting delayed action (Eqn. 9), we automatically arrive at the two
components of the tidal $U_{l}$, at each frequency involved, – the adiabatic
component and the dissipative component (using the terms of Zahn 1966) or the
elastic one and the creep one (as Ferraz-Mello 2012 named them).111 Also see
the work by Krasinsky (2006), who employed the terms elastic and dissipative.
Similar decomposition will take place for the harmonic modes of the primary’s
surface elevation, except that the Love number $\,h_{l}\,$ will be involved
instead of $\,k_{l}\,$. For the principal, $\,\\{lmpq\\}=\\{2200\\}\,$, tidal
mode, this situation is illustrated, in a very exaggerated manner, by Figure
1.
Figure 1: Decomposition of the semidiurnal tide into the elastic and
dissipative components. The primary is spinning at the rate
$\,\dot{\theta}\,$, with $\,\theta\,$ being its sidereal angle. The secondary
is orbiting the primary with the mean motion $\,n\,$. The ellipse shaded green
depicts the principal, semidiurnal mode of the tides exerted on the primary by
the secondary. This is the mode numbered with $\,\\{lmpq\\}=\\{2200\\}\,$. The
semimajor axis of the green ellipse deviates from the direction towards the
perturber by a geometric lag angle. For the semidiurnal tide, this angle is
equal to $\,\frac{\textstyle 1}{\textstyle 2}\,\epsilon_{2}(\omega_{2200})\,$,
where $\,\epsilon_{2}\,$ is the phase lag as a function of the principal tidal
mode $\,\omega_{2200}\,$. It is shown here, in an exaggerated manner, how the
principal tidal mode gets decomposed into two components. One is the elastic
bulge depicted by the ellipse aligned with the perturber. Another is the
dissipative bulge depicted with the ellipse aligned orthogonally.
It should finally be mentioned that in the case of stars and giant planets it
is, technically, difficult to solve the emerging hydrodynamical problem
analytically. Therefore, in practical calculations of fluid equilibrium tides
in such bodies, the two-bulge method is the only known elegant option of
getting an acceptable analytical solution. One first solves the hydrodynamical
equations governing the adiabatic component (26). The adiabatic adjustment of
the structure is deduced from the hydrostatic balance. Then, since the
velocity field is preserved along isobars, one obtains the radial component of
the adiabatic velocity field induced by the tide. The subsequent calculation
of the horizontal component is based on the velocity field being divergence-
free. The knowledge of the adiabatic velocity field makes it possible to
determine the viscous force. This force drives the dissipative component of
the velocity field. The process leads to mass redistribution inside the body,
and thus to perturbation of the density and gravitational potential (27). The
described, two-bulge approach is available because in the fluid equilibrium
tide the dissipative component is much weaker than the adiabatic component.
The method is implemented, e.g., in the work by Remus et al. (2012a), which
furthers the original two-bulge approach offered by Zahn (1966).
## 6 Conclusions
In this short note, we pointed out a simple rule linking the two methods (or,
possibly better to say, two languages), in which bodily tides have been
described by different authors. The language of complex amplitudes is more
economical and is conventional to those who studied the theory of vibrations,
in physics or engineering. The language of two bulges turns out to be
equivalent to that of complex amplitudes. Although less economical
mathematically, the two-bulges language has some illustrative power, and may
be employed (like, e.g., in Remus et al. 2012a,b) on a par with the more
concise method of complex amplitudes.
Importantly, the existence of two mutually orthogonal bulges at each tidal
frequency is not a separate approximation (as was presumed by some of the
devotees of the two-bulge method), but is a consequence of the linearity
assumption implemented by the integral operator (10). As soon as we say that
the tidal deformation is linear but not fully elastic – this deformation can
be decomposed, at each tidal frequency, into an in-phase and an in-quadrature
part.
## 7 Acknowledgements
It is my pleasure to thank Françoise Remus for her useful advises and for her
help in preparing the figure. I also acknowledge with gratitude the
stimulating conversations on the topic of this work, which I had on various
occasions with Sylvio Ferraz Mello, Valéry Lainey, Valeri Makarov, Stephane
Mathis, and Jean-Paul Zahn.
##
## References
* (1) Churkin, V. A. 1998. “The Love numbers for the models of inelastic Earth.” Preprint No 121. Institute of Applied Astronomy. St.Petersburg, Russia. /in Russian/
* (2) Efroimsky, M., and Williams, J. G. 2009. “Tidal torques. A critical review of some techniques.” Celestial mechanics and Dynamical Astronomy, Vol. 104, pp. 257 - 289. doi: 10.1007/s10569-009-9204-7 arXiv:0803.3299
* (3) Efroimsky, M. 2012a. “Bodily tides near spin-orbit resonances.” Celestial mechanics and Dynamical Astronomy, Vol. 112, pp. 283 - 330. doi: 10.1007/s10569-011-9397-4 . Extended version available at arXiv:1105.6086
* (4) Efroimsky, M. 2012b. “Tidal dissipation compared to seismic dissipation: in small bodies, earths, and superearths.” the Astrophysical Journal, Vol. 746, No 2, article id 150. doi: 10.1088/0004-637X/746/2/150 arXiv:1105.3936
* (5) Ferraz-Mello, S. 2012. “Tidal synchronisation of close-in satellites and exoplanets. A rheophysical approach.” Submitted to: _Celestial Mechanics and Dynamical Astronomy_ . arXiv:1204.3957
* (6) Mathis, S., and Le Poncin-Lafitte, C. 2009. “Tidal dynamics of extended bodies in planetary systems and multiple stars.” Astronomy & Astrophysics, Vol. 497, pp. 889 – 910
* (7) Krasinsky, G. A. 2006. “Numerical theory of rotation of the deformable Earth with the two-layer fluid core. Part 1: Mathematical model.” _Celestial Mechanics and Dynamical Astronomy_ , Vol. 96, pp. 169 - 217. $\left[\right.$In this paper, see formulae (96 - 97) and the subsequent sections 4.2 – 4.3.$\left.\right]$
* (8) Remus, F.; Mathis, S.; and Zahn, J.-P. 2012a. “The Equilibrium Tide in Stars and Giant Planets. I - The Coplanar Case.” Astronomy & Astrophysics. In press.
* (9) Remus, F.; Mathis, S.; Zahn, J.-P.; and Lainey, V. 2012b. “Anelastic tidal dissipation in multi-layer planets.” Astronomy & Astrophysics. In press.
* (10) Zahn, J.-P. 1966a. “Les marées dans une étoile double serrée.” Annales d’Astrophysique, Vol. 29, pp. 313 - 330
* (11) Zahn, J.-P. 1966b. “Les marées dans une étoile double serrée (suite).” Annales d’Astrophysique, Vol. 29, pp. 489 - 506
|
arxiv-papers
| 2012-04-26T13:43:51 |
2024-09-04T02:49:30.180867
|
{
"license": "Public Domain",
"authors": "Michael Efroimsky",
"submitter": "Michael Efroimsky",
"url": "https://arxiv.org/abs/1204.5919"
}
|
1204.5958
|
# Sparse Signal Processing with Frame Theory
Dustin G. Mixon
###### Abstract
Many emerging applications involve sparse signals, and their processing is a
subject of active research. We desire a large class of sensing matrices which
allow the user to discern important properties of the measured sparse signal.
Of particular interest are matrices with the restricted isometry property
(RIP). RIP matrices are known to enable efficient and stable reconstruction of
sufficiently sparse signals, but the deterministic construction of such
matrices has proven very difficult. In this thesis, we discuss this matrix
design problem in the context of a growing field of study known as frame
theory. In the first two chapters, we build large families of equiangular
tight frames and full spark frames, and we discuss their relationship to RIP
matrices as well as their utility in other aspects of sparse signal
processing. In Chapter 3, we pave the road to deterministic RIP matrices,
evaluating various techniques to demonstrate RIP, and making interesting
connections with graph theory and number theory. We conclude in Chapter 4 with
a coherence-based alternative to RIP, which provides near-optimal
probabilistic guarantees for various aspects of sparse signal processing while
at the same time admitting a whole host of deterministic constructions.
Robert Calderbank June 2012
###### Acknowledgements.
This thesis is based on a series of papers I coauthored with a long list of
friends, colleagues and mentors: Boris Alexeev, Waheed U. Bajwa, Afonso S.
Bandeira, Jameson Cahill, Robert Calderbank, Matthew Fickus, Negar Kiyavash,
Christopher J. Quinn, Janet Tremain, and Percy Wong. Each member of this list
taught me a thing or two throughout the course of my thesis research, and I
very much appreciate it! My time at Princeton has been a lot of fun, thanks in
large part to the good friends I’ve made here. From eating sushi, to playing
board games, to solving fun math riddles, the experience has been a blast, and
I’ll always remember it. My wife has a gift for filling my life with beauty
and love, and last year, she gave me a beautiful new life to love. Thank you,
Tessia and Charlotte, for making my life wonderful. Finally, I thank my
parents for their unfailing love and support, and I thank God for His role in
all of these things. This research was supported in part by the A.B. Krongard
Fellowship. The views expressed in this thesis are those of the author and do
not reflect the official policy or position of the United States Air Force,
Department of Defense, or the U.S. Government. To all those who never
dedicated a dissertation to themselves.
And to my daughter, Charlotte.
### 0.1 Overview
In several applications, data is traditionally collected in massive quantities
before employing a reasonable compression strategy. The result is a storage
bottleneck that can be prevented with a data collection alternative known as
_compressed sensing_. The philosophy behind compressed sensing is that we
might as well target the meaningful data features up front instead of spending
our storage budget on less-telling measurements. As an example, natural images
tend to have a highly compressible wavelet decomposition because many of the
wavelet cofficients are typically quite small. In this case, one might
consider targeting large wavelet coefficients as desired image features; in
fact, removing the contribution of the smallest wavelet coefficients will have
little qualitative effect on the image [57], and so using sparsity in this way
is intuitively reasonable.
Let $x$ be an unknown $N$-dimensional vector with the property that at most
$K$ of its entries are nonzero, that is, $x$ is $K$_-sparse_. The goal of
compressed sensing is to construct relatively few non-adaptive linear
measurements along with a stable and efficient reconstruction algorithm that
exploits this sparsity structure. Expressing each measurement as a row of an
$M\times N$ matrix $\Phi$, we have the following noisy system:
$y=\Phi x+z.$ (1)
In the spirit of _compressed_ sensing, we only want a few measurements: $M\ll
N$. Also, in order for there to exist an inversion process for (1), $\Phi$
must map $K$-sparse vectors injectively, or equivalently, every subcollection
of $2K$ columns of $\Phi$ must be linearly independent. Unfortunately, the
natural reconstruction method in this general case, i.e., finding the sparsest
approximation of $y$ from the dictionary of columns of $\Phi$, is known to be
${\mathsf{NP}}$-hard [108]. Moreover, the independence requirement does not
impose any sort of dissimilarity between the columns of $\Phi$, meaning
distinct identity basis elements could lead to similar measurements, thereby
bringing instability in reconstruction.
To get around the ${\mathsf{NP}}$-hardness of sparse approximation, we need
more structure in the matrix $\Phi$. Indeed, several efficient reconstruction
algorithms have been considered (e.g., Basis Pursuit [61, 62, 77], Orthogonal
Matching Pursuit [62, 134], and the Least Absolute Shrinkage and Selection
Operator [20]), and their original performance guarantees depend on the
additional structure that the columns of $\Phi$ are nearly orthogonal to each
other. Depending on the algorithm, this structure in the sensing matrix
enables successful reconstruction when noise term $z$ in (1) is zero,
adversarial, or stochastic, but for any of the original guarantees to apply,
the sparsity level must be $K=\mathrm{O}(\sqrt{M})$. To reconstruct signals
with larger sparsity levels, Candès and Tao [39] impose a much stronger
requirement on the sensing matrix: that every submatrix of $2K$ columns of
$\Phi$ be well-conditioned. To be explicit, we have the following definition:
###### Definition 1.
The matrix $\Phi$ has the _$(K,\delta)$ -restricted isometry property (RIP)_
if
$(1-\delta)\|x\|^{2}\leq\|\Phi x\|^{2}\leq(1+\delta)\|x\|^{2}$
for every $K$-sparse vector $x$. The smallest $\delta$ for which $\Phi$ is
$(K,\delta)$-RIP is the _restricted isometry constant (RIC)_ $\delta_{K}$.
In words, matrices which satisfy RIP act as a near-isometry on sufficiently
sparse vectors. Among other things, this structure imposes near-orthogonality
between the columns of $\Phi$, and so in light of the previous results, it is
not surprising that RIP sensing matrices enable efficient reconstruction:
###### Theorem 2 (Theorem 1.3 in [34]).
Suppose an $M\times N$ matrix $\Phi$ has the $(2K,\delta)$-restricted isometry
property for some $\delta<\sqrt{2}-1$. Assuming $\|z\|\leq\varepsilon$, then
for every $K$-sparse vector $x\in\mathbb{R}^{N}$, the following reconstruction
from (1):
$\tilde{x}=\arg\min\|\hat{x}\|_{1}\qquad\mbox{s.t.
}\|y-\Phi\hat{x}\|\leq\varepsilon$
satisfies $\|\tilde{x}-x\|\leq C\varepsilon$, where $C$ only depends on
$\delta$.
The exciting part about this guarantee is how the sparsity level $K$ of
recoverable signals scales with the number of measurements $M$. Certainly, we
expect at least $K\sim\sqrt{M}$ since RIP is a stronger matrix requirement
than near-orthogonality between columns. In analyzing the sparsity level,
random matrices have found the most success, specifically matrices with
independent Gaussian or Bernoulli entries [17], or matrices whose rows were
randomly selected from the discrete Fourier transform matrix [118]. With high
probability, these random constructions support sparsity levels $K$ on the
order of $\smash{\frac{M}{\log^{\alpha}N}}$ for some $\alpha\geq 1$.
Intuitively, this level of sparsity is near-optimal because $K$ cannot exceed
$\smash{\frac{M}{2}}$ by the linear independence condition. Thus, Theorem 2 is
a substantial improvement over the previous guarantees, and this has prompted
further investigation of RIP matrices. Unfortunately, it is difficult to check
whether a particular instance of a random matrix is $(K,\delta)$-RIP, as this
involves the calculation of singular values for all $\smash{\binom{N}{K}}$
submatrices of $K$ columns of the matrix. For this reason, and for the sake of
reliable sensing standards, many have pursued deterministic RIP matrix
constructions; Tao discusses the significance of this open problem in [132].
Throughout this thesis, we consider the problem from a variety of directions.
In Chapter 1, we observe a technique which is commonly used to analyze the
restricted isometry of deterministic constructions: the Gershgorin circle
theorem. This technique fails to demonstrate RIP for large sparsity levels; it
is only capable of showing RIP for sparity levels on the order of $\sqrt{M}$,
as opposed to $M$. This limitation has become known as the “square-root
bottleneck.” To illustrate that this bottleneck is not merely an artifact of
the Gershgorin analysis, we consider a construction which is optimal in the
Gershgorin sense, and we establish that this construction is $(K,\delta)$-RIP
for every $K\leq\delta\sqrt{M}$ but is not $(K,1-\varepsilon)$-RIP for any
$K>\sqrt{2M}$. The first inequality is proved by the Gershgorin circle
theorem, while the second uses the _spark_ of the matrix, that is, the number
of nonzero entries in the sparsest vector in its nullspace. While this
disparity between $\sqrt{M}$ and $M$ is significant in many applications, such
constructions are particularly well-suited for the sparse signal processing
application of digital fingerprinting, and so we briefly investigate this
application.
For the applications with larger sparsity levels, we note that spark
deficiency is incompatible with restricted isometry; indeed, any matrix which
is $(K,1-\varepsilon)$-RIP necessarily has spark strictly greater than $K$. As
such, in Chapter 2, we consider $M\times N$ _full spark_ matrices, that is,
matrices whose spark is as large as possible: $M+1$. We start by finding
various full spark constructions using Vandermonde matrices and discrete
Fourier transforms. These deterministic constructions are particularly
attractive as RIP candidates because they satisfy the necessary condition of
large spark, a property which is difficult to verify in general. To solidify
this notion of difficulty, we also show that the problem of testing whether a
matrix is full spark is hard for ${\mathsf{NP}}$ under randomized polynomial-
time reductions; this contrasts with the similar problem of testing for RIP,
which currently has unknown computational complexity [93]. To demonstrate that
full spark matrices are useful in their own right, we use them to solve
another important problem in sparse signal processing: signal recovery without
phase.
To date, the only deterministic RIP construction that manages to go beyond the
square-root bottleneck is given by Bourgain et al. [29]. In Chapter 3, we
discuss the technique they use to demonstrate RIP. It is important to stress
the significance of their contribution: Before [29], it was unclear how
deterministic analysis might break the bottleneck, and as such, their result
is a major theoretical achievement. On the other hand, their improvement over
the square-root bottleneck is notably slight compared to what random matrices
provide. However, we show that their technique can actually be used to
demonstrate RIP for sparsity levels much larger than $\sqrt{M}$, meaning one
could very well demonstrate random-like performance given the proper
construction. Our result applies their technique to random matrices, and it
inadvertently serves as a simple alternative proof that certain random
matrices are RIP. We also introduce another technique, and we show that it can
demonstrate RIP for similarly large sparsity levels. Later, we propose a
specific class of full spark matrices as candidates for being RIP. Using a
correspondence between these matrices and the Paley graphs, we observe certain
combinatorial and number-theoretic implications; this lends some probabilistic
intuition for a new bound on the clique number of Paley graphs of prime order.
After investigating deterministic RIP matrices in Chapters 1–3, we have yet to
find deterministic $M\times N$ sensing matrices which provably allow for the
efficient reconstruction of signals with sparsity level
$K\sim\smash{\frac{M}{\log^{\alpha}N}}$ for some $\alpha\geq 1$. To fill this
gap, in Chapter 4, we consider an alternative model for the sparsity in our
signal, namely, that the locations of the nonzero entries are drawn uniformly
at random. With this model, we show that a particularly simple algorithm
called _one-step thresholding_ can reconstruct the signal with high
probability provided $K=\mathrm{O}(\frac{M}{\log N})$. In fact, this
performance guarantee requires relatively modest structure in the sensing
matrix: that the columns are nearly orthogonal to each other and well-
distributed over the unit sphere. Indeed, this structural requirement is much
less stringent than RIP, and we provide a catalog of random and
_deterministic_ sensing matrices which satisfy these conditions. Later, we
further analyze the two conditions separately, finding new fundamental limits
on near-orthogonality and illustrating how to manipulate a given sensing
matrix to achieve good distribution over the sphere.
Throughout this thesis, we use ideas from _frame theory_ , and so it is
fitting to take some time to review the basics:
### 0.2 A brief introduction to frame theory
A _frame_ is a sequence $\\{\varphi_{i}\\}_{i\in\mathcal{I}}$ in a Hilbert
space $\mathcal{H}$ with _frame bounds_ $0<A\leq B<\infty$ that satisfy
$A\|x\|^{2}\leq\sum_{i\in\mathcal{I}}|\langle x,\varphi_{i}\rangle|^{2}\leq
B\|x\|^{2}\qquad\forall x\in\mathcal{H}.$
Frames were introduced by Duffin and Schaeffer [64] in the context of
nonharmonic Fourier analysis, where $\mathcal{H}=L^{2}(-\pi,\pi)$ and the
frame elements $\varphi_{i}$ are sinusoids of irregularly spaced frequencies.
However, the modern application of frame theory to signal processing came
decades later after the landmark paper of Daubechies et al. [55]. This paper
gave the first nontrivial examples of _tight frames_ , that is, frames with
equal frame bounds $A=B$. The utility of tight frames lies partially in their
painless reconstruction formula:
$x=\frac{1}{A}\sum_{i\in\mathcal{I}}\langle x,\varphi_{i}\rangle\varphi_{i}.$
Note that orthonormal bases are tight frames with $A=B=1$; in this way, frames
form a natural and useful generalization. While this founding research in
frame theory concerned frames over infinite-dimensional Hilbert spaces, many
of today’s applications of frames require a finite-dimensional treatment. In
fact, finite frame theory has found some important progress in the past decade
[18, 33, 42, 43, 47, 129], and the remainder of this section will discuss the
basics of this field.
In finite dimensions, say, $\mathcal{H}=\mathbb{C}^{M}$, a frame is given by
the columns of a full-rank $M\times N$ matrix
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$ with $N\geq M$. Here, the extreme
eigenvalues of $\Phi\Phi^{*}$ are the frame bounds, and a tight frame has
equal frame bounds; equivalently, a frame $\Phi$ is tight if
* (i)
the rows are equal-norm and orthogonal.
As established above, tight frames $\Phi$ are useful because they give a
redundant linear encoding $y=\Phi^{*}x$ of a signal $x$ that permits painless
recovery: $x=\frac{1}{A}\Phi y$, where $A$ is the common squared-norm of the
rows. Constructing tight frames is rather simple: perform Gram-Schmidt on the
rows of any frame to orthogonalize with equal norms. For the sake of democracy
in the entries of the encoding $y$, some applications opt for a _unit norm
tight frame (UNTF)_ [45], which has the additional property that
* (ii)
the columns are unit-norm.
Constructing UNTFs has proven a bit more difficult, and there has been a lot
of research to characterize these [18, 33, 127]. As a special example of a
UNTF, take any rows from a discrete Fourier transform matrix and normalize the
resulting columns. In addition to unit-norm tightness, it is often beneficial
to have the columns of $\Phi$ be incoherent, and this occurs when $\Phi$ is an
_equiangular tight frame (ETF)_ , that is, a UNTF with the final property that
* (iii)
the sizes of the inner products between distinct columns are equal.
ETFs do not exist for all matrix dimensions [19], and there are only three
general constructions to date [70, 141, 146]; these invoke block designs,
strongly regular graphs, and difference sets, respectively.
To mitigate any confusion, the reader should be aware that throughout the
literature, both UNTFs and ETFs are referred to as _Welch-bound equality
sequences_ [120]. As one might expect, each achieves equality in one of two
important inequalities, and it is important to review them. Consider $M\times
N$ matrices $\Phi=[\varphi_{1}\cdots\varphi_{N}]$ which have (ii), but not
necessarily (i) or (iii). As such, $\Phi$ might not be a frame, but we can
still take the Hilbert-Schmidt norm of the Gram matrix of its columns:
$\|\Phi^{*}\Phi\|_{\mathrm{HS}}^{2}=\sum_{n=1}^{N}\sum_{n^{\prime}=1}^{N}|\langle\varphi_{n},\varphi_{n^{\prime}}\rangle|^{2}.$
This is oftentimes called the _frame potential_ of $\Phi$ [18], and its
significance will become apparent shortly. Since the columns of $\Phi$ have
unit norm, and since $\Phi^{*}\Phi$ has at most $M$ nonzero eigenvalues, we
have
$N^{2}=\big{(}\mathrm{Tr}(\Phi^{*}\Phi)\big{)}^{2}=\bigg{(}\sum_{m=1}^{M}\lambda_{m}(\Phi^{*}\Phi)\bigg{)}^{2}\leq
M\sum_{m=1}^{M}\big{(}\lambda_{m}(\Phi^{*}\Phi)\big{)}^{2}=M\|\Phi^{*}\Phi\|_{\mathrm{HS}}^{2},$
where the inequality follows from the Cauchy-Schwarz inequality with the all-
ones vector. As such, equality is achieved if and only if the $M$ largest
eigenvalues of $\Phi^{*}\Phi$ are equal; since these are also the eigenvalues
of $\Phi\Phi^{*}$, this implies that $\Phi\Phi^{*}$ is a multiple identity,
and so $\Phi$ satisfies (ii). Thus, the frame potential of $\Phi$ satisfies
$\|\Phi^{*}\Phi\|_{\mathrm{HS}}^{2}\geq\frac{N^{2}}{M}$, with equality if and
only if $\Phi$ is a UNTF. Some call this the _Welch bound_ , and therefore say
that UNTFs have Welch-bound equality.
Another bound is also (more correctly) referred to as the Welch bound, and its
derivation uses the previous one. It concerns the _worst-case coherence_ of an
$M\times N$ matrix $\Phi=[\varphi_{1}\cdots\varphi_{N}]$ that satisfies (ii):
$\mu:=\max_{\begin{subarray}{c}n,n^{\prime}\in\\{1,\ldots,N\\}\\\ n\neq
n^{\prime}\end{subarray}}|\langle\varphi_{n},\varphi_{n^{\prime}}\rangle|.$
Since the columns of $\Phi$ have unit norm, we have
$\frac{N^{2}}{M}\leq\|\Phi^{*}\Phi\|_{\mathrm{HS}}^{2}=\sum_{n=1}^{N}\sum_{n^{\prime}=1}^{N}|\langle\varphi_{n},\varphi_{n^{\prime}}\rangle|^{2}\leq
N+N(N-1)\mu^{2}.$
Again, equality is achieved in the first inequality if and only if $\Phi$
satisfies (i). Also, equality is achieved in the second inequality if and only
if $\Phi$ satisfies (iii). Rearranging gives the following:
###### Theorem 3 (Welch bound [129, 143]).
Every $M\times N$ matrix $\Phi$ with unit-norm columns has worst-case
coherence
$\mu\geq\sqrt{\frac{N-M}{M(N-1)}},$
with equality if and only if $\Phi$ is an equiangular tight frame.
Equiangular lines have long been a subject of interest [97], and since
equiangular tight frames have minimal coherence, they are particularly useful
in a number of applications. Recent work on ETFs was spurred by results
inspired by communication theory [26, 84, 129] that show that the linear
encoders provided by ETFs are optimally robust against channel erasures. In
the real setting, the existence of an ETF of a given size is equivalent to the
existence of a strongly regular graph with certain corresponding parameters
[84, 122]. Such graphs have a rich history and remain an active topic of
research [31]; the specific ETFs which arise from particular graphs are
detailed in [141]. Some of this theory generalizes to the complex-variable
setting in the guise of complex Seidel matrices [25, 27, 65]. Many approaches
to constructing ETFs have focused on the special case in which every entry of
$\Phi$ is a root of unity [88, 115, 128, 130, 146]. Other approaches are given
in [46, 125, 137]. In the complex setting, much attention has focused on the
maximal case of $M^{2}$ vectors in $\mathbb{C}^{M}$ [9, 68, 91, 116, 121].
In the next chapter, we construct one of three known general families of ETFs,
and we evaluate their performance as RIP matrices. Having reviewed the frame-
theoretic background for this thesis, the interested reader is encouraged to
discover more about frame theory in [49].
## Chapter 1 Steiner equiangular tight frames
In this chapter, we provide a new method for constructing equiangular tight
frames (ETFs), that is, matrices $\Phi$ with orthogonal and equal-norm rows,
and unit-norm columns whose inner products are equal in modulus. As discussed
earlier, such frames have minimal worst-case coherence, and are therefore
quite useful in applications. However, up to this point, they have proven
notoriously difficult to construct. By contrast, the construction of _Steiner
equianglar tight frames_ is particularly simple: a tensor-like combination of
a Steiner system and a regular simplex. This simplicity permits us to resolve
an open question regarding ETFs and the restricted isometry property (RIP): we
show that the RIP performance of some ETFs is unfortunately no better than the
so-called “square-root bottleneck.”
In the next section, we provide some simple tests for demonstrating whether a
given matrix is RIP; not only will this clarify the notion of the square-root
bottleneck, it will show how ETFs are in some sense optimal as deterministic
RIP matrices, thereby motivating the construction of ETFs. Later, we provide
the main result of this chapter, namely Theorem 7, which shows how certain
Steiner systems may be combined with regular simplices to produce ETFs [69,
70]. In the third section, we discuss each of the known infinite families of
such Steiner systems, and compute the corresponding infinite families of ETFs
they generate. We further provide some necessary and asymptotically sufficient
conditions, namely Theorem 8, to aid in the quest for discovering other
examples of such frames that lie outside of the known infinite families.
Finally, after demonstrating that Steiner ETFs fail to break the square-root
bottleneck, we consider their application to the design of digital
fingerprints to combat data piracy [103, 104].
### 1.1 Simple tests for restricted isometry
Before formally defining Steiner equiangular tight frames, we motivate their
construction by reviewing a couple common methods for determining whether a
matrix is RIP:
Positive test for RIP: | Apply the Gershgorin circle theorem to the submatrices $\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$.
---|---
Negative test for RIP: | Find a sparse vector in the nullspace of $\Phi$.
In what follows, we discuss each of these tests in more detail, and later, we
will use these tests to analyze Steiner ETFs as RIP matrices.
#### 1.1.1 Applying Gershgorin’s circle thoerem
Take an $M\times N$ matrix $\Phi$, and recall Definition 1. For a given $K$,
we wish to find some $\delta$ for which $\Phi$ is $(K,\delta)$-RIP. To this
end, it is useful to consider the following expression for the restricted
isometry constant:
###### Lemma 4.
The smallest $\delta$ for which $\Phi$ is $(K,\delta)$-RIP is given by
$\delta_{K}=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\|\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}\|_{2},$
(1.1)
where $\Phi_{\mathcal{K}}$ denotes the submatrix consisting of columns of
$\Phi$ indexed by $\mathcal{K}$.
###### Proof.
We first note that $\Phi$ being $(K,\delta)$-RIP trivially implies that $\Phi$
is $(K,\delta+\varepsilon)$-RIP for every $\varepsilon>0$. It therefore
suffices to show that the expression for $\delta_{K}$ in (1.1) satisfies two
criteria: (i) $\Phi$ is $(K,\delta_{K})$-RIP, and (ii) $\Phi$ is not
$(K,\delta)$-RIP for any $\delta<\delta_{K}$. To this end, pick some
$K$-sparse vector $x$. To prove (i), we need to show that
$(1-\delta_{K})\|x\|^{2}\leq\|\Phi x\|^{2}\leq(1+\delta_{K})\|x\|^{2}.$ (1.2)
Let $\mathcal{K}\subseteq\\{1,\dots,N\\}$ be the size-$K$ support of $x$, and
let $x_{\mathcal{K}}$ be the corresponding subvector. Then rearranging (1.2)
gives
$\delta_{K}\geq\Big{|}\tfrac{\|\Phi
x\|^{2}}{\|x\|^{2}}-1\Big{|}=\Big{|}\tfrac{\langle\Phi_{\mathcal{K}}x_{\mathcal{K}},\Phi_{\mathcal{K}}x_{\mathcal{K}}\rangle-\langle
x_{\mathcal{K}},x_{\mathcal{K}}\rangle}{\|x_{\mathcal{K}}\|^{2}}\Big{|}=\Big{|}\Big{\langle}\tfrac{x_{\mathcal{K}}}{\|x_{\mathcal{K}}\|},(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})\tfrac{x_{\mathcal{K}}}{\|x_{\mathcal{K}}\|}\Big{\rangle}\Big{|}.$
(1.3)
Since the expression for $\delta_{K}$ in (1.1) maximizes (1.3) over all
supports $\mathcal{K}$ and entry values $x_{\mathcal{K}}$, the inequality
necessarily holds; that is, $\Phi$ is necessarily $(K,\delta_{K})$-RIP.
Furthermore, equality is achieved by the support $\mathcal{K}$ which maximizes
(1.1) and the eigenvector $x_{\mathcal{K}}$ corresponding to the largest
eigenvalue of $\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}$; this proves
(ii). ∎
Note that we are not tasked with actually computing $\delta_{K}$; rather, we
recognize that $\Phi$ is $(K,\delta)$-RIP for every $\delta\geq\delta_{K}$,
and so we seek an upper bound on $\delta_{K}$. The following classical result
offers a particularly easy-to-calculate bound on eigenvalues:
###### Theorem 5 (Gershgorin circle theorem [73]).
For each eigenvalue $\lambda$ of a $K\times K$ matrix $A$, there is an index
$i\in\\{1,\ldots,K\\}$ such that
$\Big{|}\lambda-A[i,i]\Big{|}\leq\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{K}\Big{|}A[i,j]\Big{|}.$
To use this theorem, take some $\Phi$ with unit-norm columns. Note that
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is the Gram matrix of the columns
indexed by $\mathcal{K}$, and as such, the diagonal entries are $1$, and the
off-diagonal entries are inner products between distinct columns of $\Phi$.
Let $\mu$ denote the _worst-case coherence_ of
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$:
$\mu:=\max_{\begin{subarray}{c}i,j\in\\{1,\ldots,N\\}\\\ i\neq
j\end{subarray}}|\langle\varphi_{i},\varphi_{j}\rangle|.$
Then the size of each off-diagonal entry of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is $\leq\mu$, regardless of our
choice for $\mathcal{K}$. Therefore, for every eigenvalue $\lambda$ of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}$, the Gershgorin circle
theorem gives
$|\lambda|=|\lambda-0|\leq\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{K}|\langle\varphi_{i},\varphi_{j}\rangle|\leq(K-1)\mu.$
(1.4)
Since (1.4) holds for every eigenvalue $\lambda$ of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}$ and every choice of
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$, we conclude from (1.1) that
$\delta_{K}\leq(K-1)\mu$, i.e., $\Phi$ is $(K,(K-1)\mu)$-RIP. This process of
using the Gershgorin circle theorem to demonstrate RIP for deterministic
constructions has become standard in the community [8, 60, 70].
Recall that random RIP constructions support sparsity levels $K$ on the order
of $\smash{\frac{M}{\log^{\alpha}N}}$ for some $\alpha\geq 1$. To see how well
the Gershgorin circle theorem demonstrates RIP, we need to express $\mu$ in
terms of $M$ and $N$. To this end, we consider the Welch bound (Theorem 3):
$\mu\geq\sqrt{\frac{N-M}{M(N-1)}}.$
Since equiangular tight frames (ETFs) achieve equality in the Welch bound (as
demonstrated in Section 0.2), we can further analyze what it means for an
$M\times N$ ETF $\Phi$ to be $(K,(K-1)\mu)$-RIP. In particular, since Theorem
2 requires that $\Phi$ be $(2K,\delta)$-RIP for $\delta<\sqrt{2}-1$, it
suffices to have $\smash{\frac{2K}{\sqrt{M}}<\sqrt{2}-1}$, since this implies
$\delta=(2K-1)\mu=(2K-1)\sqrt{\frac{N-M}{M(N-1)}}\leq\frac{2K}{\sqrt{M}}<\sqrt{2}-1.$
(1.5)
That is, ETFs form sensing matrices that support sparsity levels $K$ on the
order of $\sqrt{M}$. Most other deterministic constructions have identical
bounds on sparsity levels [8, 60, 70]. In fact, since ETFs minimize coherence,
they are necessarily optimal constructions in terms of the Gershgorin
demonstration of RIP, but the question remains whether they are actually RIP
for larger sparsity levels; the Gershgorin demonstration fails to account for
cancellations in the sub-Gram matrices
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$, and so this technique is too weak
to indicate either possibility.
#### 1.1.2 Spark considerations
Recall that, in order for an inversion process for (1) to exist, $\Phi$ must
map $K$-sparse vectors injectively, or equivalently, every subcollection of
$2K$ columns of $\Phi$ must be linearly independent. This linear independence
condition can be nicely expressed in more general terms, as the following
definition provides:
###### Definition 6.
The _spark_ of a matrix $\Phi$ is the size of the smallest linearly dependent
subset of columns, i.e.,
$\mathrm{Spark}(\Phi)=\min\Big{\\{}\|x\|_{0}:\Phi x=0,~{}x\neq 0\Big{\\}}.$
This definition was introduced by Dohono and Elad [61] to help build a theory
of sparse representation that later gave birth to modern compressed sensing.
The concept of spark is also found in matroid theory, where it goes by the
name _girth_. The condition that every subcollection of $2K$ columns of $\Phi$
is linearly independent is equivalent to $\mathrm{Spark}(\Phi)>2K$. Relating
spark to RIP, suppose $\Phi$ is $(K,\delta)$-RIP with
$\mathrm{Spark}(\Phi)\leq K$. Then there exists a nonzero $K$-sparse vector
$x$ such that $(1-\delta)\|x\|^{2}\leq\|\Phi x\|^{2}=0$, and so $\delta\geq
1$. The reason behind this stems from our necessary linear independence
condition: RIP implies linear independence, and so small spark implies linear
dependence, which in turn implies not RIP.
As an example of using spark to test RIP, consider the $M\times 2M$ matrix
$\Phi=[I~{}~{}F]$ that comes from concatenating the identity matrix $I$ with
the unitary discrete Fourier transform matrix $F$. In this example, columns
from a common orthonormal basis are orthogonal, while columns from different
bases have an inner product of size $\frac{1}{\sqrt{M}}$. As such, the
Gershgorin analysis gives that $\Phi$ is $(K,\delta)$-RIP for all
$\delta\geq\frac{K-1}{\sqrt{M}}$. However, when $M$ is a perfect square, the
Dirac comb $x$ of $\sqrt{M}$ Kronecker deltas is an eigenvector of $F$, and so
concatenating $Fx$ with $-x$ produces a $2\sqrt{M}$-sparse vector in the
nullspace of $\Phi$. In other words, $\mathrm{Spark}(\Phi)\leq 2\sqrt{M}$, and
so $\Phi$ is not $(K,1-\varepsilon)$-RIP for any $K\geq 2\sqrt{M}$. After
building Steiner equiangular tight frames, we will see that they perform
similarly as RIP matrices.
### 1.2 Constructing Steiner equiangular tight frames
Steiner systems and block designs have been studied for over a century; the
background facts presented here on these topics are taken from [1, 52]. In
short, a $(v,b,r,k,\lambda)$-block design is a $v$-element set $V$ along with
a collection $\mathcal{B}$ of $b$ size-$k$ subsets of $V$, dubbed blocks, that
have the property that any element of $V$ lies in exactly $r$ blocks and that
any $2$-element subset of $V$ is contained in exactly $\lambda$ blocks. The
corresponding incidence matrix is a $v\times b$ matrix $A$ that is one in a
given entry if that block contains the corresponding point, and is otherwise
zero; in this chapter, it is more convenient for us to work with the $b\times
v$ transpose $A^{\mathrm{T}}$ of this incidence matrix. Our particular
construction of ETFs involves a special class of block designs known as
$(2,k,v)$-Steiner systems. These have the property that any $2$-element subset
of $V$ is contained in exactly one block, that is, $\lambda=1$. With respect
to our purposes, the crucial facts are the following:
> The transpose $A^{\mathrm{T}}$ of the $\\{{0,1}\\}$-incidence matrix $A$ of
> a $(2,k,v)$-Steiner system:
>
> 1. (i)
>
> is of size $\smash{\frac{v(v-1)}{k(k-1)}}\times v$,
>
> 2. (ii)
>
> has $k$ ones in each row,
>
> 3. (iii)
>
> has $\smash{\frac{v-1}{k-1}}$ ones in each column, and
>
> 4. (iv)
>
> has the property that any two of its columns have a inner product of one.
>
>
The first three facts follow immediately from solving for
$\smash{b=\frac{v(v-1)}{k(k-1)}}$ and $\smash{r=\frac{v-1}{k-1}}$, using the
well-known relations $vr=bk$ and $r(k-1)=\lambda(v-1)$. Meanwhile, (iv) comes
from the fact that $\lambda=1$: each column of $A^{\mathrm{T}}$ corresponds to
an element of the set, and the inner product of any two columns computes the
number of blocks that contains the corresponding pair of points. This in hand,
we present the main result of this chapter; here, the density of a matrix is
the ratio of the number of nonzero entries of that matrix to the total number
of its entries:
###### Theorem 7.
Every $(2,k,v)$-Steiner system generates an equiangular tight frame consisting
of $N=v(1+\frac{v-1}{k-1})$ vectors in $M=\frac{v(v-1)}{k(k-1)}$-dimensional
space with redundancy $\smash{\frac{N}{M}=k(1+\frac{k-1}{v-1})}$ and density
$\smash{\frac{k}{v}=(\frac{N-1}{M(N-M)})^{\frac{1}{2}}}$.
Moreover, if there exists a real Hadamard matrix of size
$\smash{1+\frac{v-1}{k-1}}$, then such frames are real.
Specifically, a $\frac{v(v-1)}{k(k-1)}\times v(1+\frac{v-1}{k-1})$ ETF matrix
$\Phi$ may be constructed as follows:
1. 1.
Let $A^{\mathrm{T}}$ be the $\smash{\frac{v(v-1)}{k(k-1)}}\times v$ transpose
of the adjacency matrix of a $(2,k,v)$-Steiner system.
2. 2.
For each $j=1,\dotsc,v$, let $H_{j}$ be any
$(1+\frac{v-1}{k-1})\times(1+\frac{v-1}{k-1})$ matrix that has orthogonal rows
and unimodular entries, such as a possibly complex Hadamard matrix.
3. 3.
For each $j=1,\dotsc,v$, let $\Phi_{j}$ be the
$\smash{\frac{v(v-1)}{k(k-1)}\times(1+\frac{v-1}{k-1})}$ matrix obtained from
the $j$th column of $A^{\mathrm{T}}$ by replacing each of the one-valued
entries with a distinct row of $H_{j}$, and every zero-valued entry with a row
of zeros.
4. 4.
Concatenate and rescale the $\Phi_{j}$’s to form
$\Phi=(\frac{k-1}{v-1})^{\frac{1}{2}}[\Phi_{1}\cdots\Phi_{v}]$.
It is important to note that a version of this ETF construction was previously
employed by Seidel in Theorem 12.1 of [122] to prove the existence of certain
strongly regular graphs. In the context of that result, our contributions are
as follows: (i) the realization that when Seidel’s block design arises from a
particular type of Steiner system, the resulting strongly regular graph indeed
corresponds to a real ETF; (ii) noting that in this case, the graph theory may
be completely bypassed, as the idea itself directly produces the requisite
frame $\Phi$; and (iii) having bypassed the graph theory, realizing that this
construction immediately generalizes to the complex-variable setting if
Seidel’s requisite Hadamard matrix is permitted to become complex. These
realizations permit us to exploit the vast literature on Steiner systems [52]
to construct several new infinite families of ETFs, in both the real and
complex settings. Moreover, these ETFs are extremely sparse in their native
space; sparse tight frames have recently become a subject of interest in their
own right [44].
We refer to the ETFs produced by Theorem 7 as $(2,k,v)$-Steiner ETFs. In
essence, the idea of the construction is that the nonzero rows of any
particular $\Phi_{j}$ form a regular simplex in
$\smash{\frac{v-1}{k-1}}$-dimensional space; these vectors are automatically
equiangular amongst themselves; by requiring the entries of these simplices to
be unimodular, and requiring that distinct blocks have only one entry of
mutual support, one can further control the inner products of vectors arising
from distinct blocks. This idea is best understood by considering a simple
example, such as the ETF that arises from a $(2,2,4)$-Steiner system whose
transposed incidence matrix is
$A^{\mathrm{T}}=\begin{bmatrix}+&+&&\\\ +&&+&\\\ +&&&+\\\ &+&+&\\\ &+&&+\\\
&&+&+\end{bmatrix}.$
One can immediately verify that $A^{\mathrm{T}}$ corresponds to a block
design: there is a set $V$ of $v=4$ elements, each corresponding to a column
of $A^{\mathrm{T}}$; there is also a collection $\mathcal{B}$ of $b=6$ subsets
of $V$, each corresponding to a row of $A^{\mathrm{T}}$; every row contains
$k=2$ elements; every column contains $r=3$ elements; any given pair of
elements is contained in exactly one row, that is, $\lambda=1$, a fact which
is equivalent to having the inner product of any two distinct columns of
$A^{\mathrm{T}}$ being $1$. To form an ETF, for each of the four columns of
$A^{\mathrm{T}}$ we must choose a $4\times 4$ matrix $H$ with unimodular
entries and orthogonal rows; the size of $H$ is always one more than the
number $r$ of ones in a given column of $A^{\mathrm{T}}$. Though in principle
one may choose a different $H$ for each column, we choose them all to be the
same, namely the Hadamard matrix:
$H=\begin{bmatrix}+&+&+&+\\\ +&-&+&-\\\ +&+&-&-\\\ +&-&-&+\end{bmatrix}.$
To form the ETF, for each column of $A^{\mathrm{T}}$ we replace each of its
$1$-valued entries with a distinct row of $H$. Again, though in principle one
may choose a different sequence of rows of $H$ for each column, we simply
decide to use the second, third and fourth rows, in that order. The result is
a real ETF of $N=16$ elements of dimension $M=6$:
$\Phi=\frac{1}{\sqrt{3}}\left[\begin{array}[]{cccccccccccccccc}+&-&+&-&+&-&+&-\\\
+&+&-&-&&&&&+&-&+&-\\\ +&-&-&+&&&&&&&&&+&-&+&-\\\ &&&&+&+&-&-&+&+&-&-\\\
&&&&+&-&-&+&&&&&+&+&-&-\\\ &&&&&&&&+&-&-&+&+&-&-&+\end{array}\right].$ (1.6)
One can immediately verify that the rows of $\Phi$ are orthogonal and have
constant norm, implying $\Phi$ is indeed a tight frame. One can also easily
see that the inner products of two columns from the same block are
$-\frac{1}{3}$, while the inner products of columns from distinct blocks are
$\pm\frac{1}{3}$. Theorem 7 states that this behavior holds in general for any
appropriate choice of $A^{\mathrm{T}}$ and $H$.
###### Proof of Theorem 7.
To verify $\Phi$ is a tight frame, note that the inner product of any two
distinct rows of $\Phi$ is zero, as they are the sum of the inner products of
the corresponding rows of the $\Phi_{j}$’s over all $j=1,\dotsc,v$; for any
$j$, these shorter inner products are necessarily zero, as they either
correspond to inner products of distinct rows of $H_{j}$ or to inner products
with zero vectors. Moreover, the rows of $\Phi$ have constant norm: as noted
in (ii) above, each row of $A^{\mathrm{T}}$ contains $k$ ones; since each
$H_{j}$ has unimodular entries, the squared-norm of any row of $\Phi$ is the
squared-scaling factor $\frac{k-1}{v-1}$ times a sum of
$\smash{k(1+\frac{v-1}{k-1})}$ ones, which, as is necessary for any unit norm
tight frame, equals the redundancy $\smash{\frac{N}{M}=k(1+\frac{k-1}{v-1})}$.
Having that $\Phi$ is tight, we show $\Phi$ is also equiangular. We first note
that the columns of $\Phi$ have unit norm: the squared-norm of any column of
$\Phi$ is $\smash{\frac{k-1}{v-1}}$ times the squared-norm of a column of one
of the $\Phi_{j}$’s; since the entries of $H_{j}$ are unimodular and (iii)
above gives that each column of $A^{\mathrm{T}}$ contains
$\smash{\frac{v-1}{k-1}}$ ones, the squared-norm of any column of $\Phi$ is
$\smash{(\frac{k-1}{v-1})(\frac{v-1}{k-1})1=1}$, as claimed. Moreover, the
inner products of any two distinct columns of $\Phi$ has constant modulus.
Indeed, the fact (iv) that any two distinct columns of $A^{\mathrm{T}}$ have
but a single entry of mutual support implies the same is true for columns of
$\Phi$ that arise from distinct $\Phi_{j}$ blocks, implying the inner product
of such columns is $\smash{\frac{k-1}{v-1}}$ times the product of two
unimodular numbers. That is, the squared-magnitude of the inner products of
two columns that arise from distinct blocks is
$\smash{\frac{N-M}{M(N-1)}=(\frac{k-1}{v-1})^{2}}$, as needed. Meanwhile, the
same holds true for columns that arise from the same block $\Phi_{j}$. To see
this, note that since $H_{j}$ is a scalar multiple of a unitary matrix, its
columns are orthogonal. Moreover, $\Phi_{j}$ contains all but one of the
$H_{j}$’s rows, namely one for each of the $1$-valued entries of
$A^{\mathrm{T}}$, à la (iii). Thus, the inner products of the portions of
$H_{j}$ that lie in $\Phi_{j}$ are their entire inner product of zero, less
the contribution from the left-over entries. Overall, the inner product of two
columns of $\Phi$ that arise from the same $\Phi_{j}$ block is
$\smash{\frac{k-1}{v-1}}$ times the negated product of one entry of $H_{j}$
and the conjugate of another; since the entries of $H_{j}$ are unimodular, we
have that the squared-magnitude of such inner products is
$\smash{\frac{N-M}{M(N-1)}=(\frac{k-1}{v-1})^{2}}$, as needed.
Thus $\Phi$ is an ETF. Moreover, as noted above, its redundancy is
$\smash{\frac{N}{M}=k(1+\frac{k-1}{v-1})}$. All that remains to verify is its
density: as the entries of each $H_{j}$ are all nonzero, the proportion of
$\Phi$’s nonzero entries is the same as that of the incidence matrix $A$,
which is clearly $\frac{k}{v}$, having $k$ ones in each $v$-dimensional row.
Moreover, substituting $\smash{N=v(1+\frac{v-1}{k-1})}$ and
$\smash{M=\frac{v(v-1)}{k(k-1)}}$ into the quantity
$\smash{\frac{N-1}{M(N-M)}}$ reveals it to be $\frac{k^{2}}{v^{2}}$, and so
the density can be alternatively expressed as
$\smash{(\frac{N-1}{M(N-M)})^{\frac{1}{2}}}$. ∎
In the next section, we apply Theorem 7 to produce several infinite families
of Steiner ETFs. Before doing so, however, we pause to remark on the
redundancy and sparsity of such frames. In particular, note that since the
parameters $k$ and $v$ of the requisite Steiner system always satisfy $2\leq
k\leq v$, the redundancy $k(1+\frac{k-1}{v-1})$ of Steiner ETFs is always
between $k$ and $2k$; the redundancy is therefore on the order of $k$, and is
always strictly greater than $2$. If a low-redundancy ETF is desired, one can
always take the Naimark complement [43] of an ETF of $N$ elements in
$M$-dimensional space to produce a new ETF of $N$ elements in
$(N-M)$-dimensional space; though the complement process does not preserve
sparsity, it nevertheless transforms any Steiner ETF into a new ETF whose
redundancy is strictly less than $2$. However, such a loss of sparsity should
not be taken lightly. Indeed, the low density of Steiner ETFs gives them a
large computational advantage over their non-sparse brethren.
To clarify, the most common operation in frame-theoretic applications is the
evaluation of the analysis operator $\Phi^{*}$ on a given
$x\in\mathbb{C}^{M}$. For a non-sparse $\Phi$, this act of computing
$\Phi^{*}x$ requires $\mathrm{O}(MN)$ operations; for a frame $\Phi$ of
density $D$, this cost is reduced to $\mathrm{O}(DMN)$. Indeed, using the
explicit value of $\smash{D=(\frac{N-1}{M(N-M)})^{\frac{1}{2}}}$ given in
Theorem 7 as well as the aforementioned fact that the redundancy of such
frames necessarily satisfies $\frac{N}{M}>2$, we see that the cost of
evaluating $\Phi^{*}x$ when $\Phi$ is a Steiner ETF is on the order of
$\smash{(\frac{M(N-1)}{N-M})^{\frac{1}{2}}N<(2M)^{\frac{1}{2}}N}$ operations,
a dramatic cost savings when $M$ is large. Further efficiency is gained when
$\Phi$ is real, as its nonzero elements are but a fixed scaling factor times
the entries of a real Hadamard matrix, implying $\Phi^{*}x$ can be evaluated
using only additions and subtractions. The fact that every entry of $\Phi$ is
either $0$ or $\pm 1$ further makes real Steiner ETFs potentially useful for
applications that require binary measurements, such as design of experiments.
### 1.3 Examples of Steiner equiangular tight frames
In this section, we apply Theorem 7 to produce several infinite families of
Steiner ETFs. When designing frames for real-world applications, three
considerations reign supreme: size, redundancy and sparsity. As noted above,
every Steiner ETF is very sparse, a serious computational advantage in high-
dimensional signal processing. Moreover, some of these infinite families, such
as those arising from finite affine and projective geometries, provide great
flexibility in choosing the ETF’s size and redundancy. Indeed, these
constructions provide the first known guarantee that for a given application,
one is always able to find ETFs whose frame elements lie in a space whose
dimension matches, up to an order of magnitude, that of one’s desired class of
signals, while simultaneously permitting one to have an almost arbitrary fixed
level of redundancy, a handy weapon in the fight against noise. To be clear,
recall that the redundancy of a Steiner ETF is always strictly greater than
$2$. Moreover, general bounds on the maximal number of equiangular lines [97]
require that any real $M\times N$ ETF satisfy $\smash{N\leq\frac{M(M+1)}{2}}$
and any complex ETF satisfy $N\leq M^{2}$; thus, the redundancy of an ETF is
never truly arbitrary. Nevertheless, if one prescribes a given level of
redundancy in advance, the Steiner method can produce arbitrarily large ETFs
whose redundancy is approximately the prime power closest to the desired
level.
#### 1.3.1 Infinite families of Steiner equiangular tight frames
We now detail eight infinite families of ETFs, each generated by applying
Theorem 7 to one of the eight completely understood infinite families of
$(2,k,v)$-Steiner systems. Table 1.1 summarizes the most important features of
each family, and Table 1.2 gives the first few examples of each type,
summarizing those that lie in 100 dimensions or less.
##### All two-element blocks: $(2,2,v)$-Steiner ETFs for any $v\geq 2$.
The first infinite family of Steiner systems is so simple that it is usually
not discussed in the design-theory literature. For any $v\geq 2$, let $V$ be a
$v$-element set, and let $\mathcal{B}$ be the collection of all $2$-element
subsets of $V$. Clearly, we have $\smash{b=\frac{v(v-1)}{2}}$ blocks, each of
which contains $k=2$ elements; each point is contained in $r=v-1$ blocks, and
each pair of points is indeed contained in but a single block, that is,
$\lambda=1$.
By Theorem 7, the ETFs arising from these $(2,2,v)$-Steiner systems consist of
$N=v(1+\frac{v-1}{k-1})=v^{2}$ vectors in
$\smash{M=\frac{v(v-1)}{k(k-1)}=\frac{v(v-1)}{2}}$-dimensional space. Though
these frames can become arbitrarily large, they do not provide any freedom
with respect to redundancy: $\smash{\frac{N}{M}=2\frac{v}{v-1}}$ is
essentially $2$. These frames have density $\smash{\frac{k}{v}=\frac{2}{v}}$.
Moreover, these ETFs can be real-valued if there exists a real Hadamard matrix
of size $\smash{1+\frac{v-1}{k-1}}=v$. In particular, it suffices to have $v$
to be a power of $2$; should the Hadamard conjecture prove true, it would
suffice to have $v$ divisible by $4$.
One example of such an ETF with $v=4$ was given in the previous section. For a
complex example, consider $v=3$. The $b\times v$ transposed incidence matrix
$A^{\mathrm{T}}$ is $3\times 3$, with each row corresponding to a given
$2$-element subset of $\\{{0,1,2}\\}$:
$A^{\mathrm{T}}=\begin{bmatrix}+&+&\\\ +&&+\\\ &+&+\end{bmatrix}.$
To form the corresponding $3\times 9$ ETF $\Phi$, we need a $3\times 3$
unimodular matrix with orthogonal rows, such as a DFT; letting
$\smash{\omega=\mathrm{e}^{2\pi\mathrm{i}/3}}$, we can take
$H=\left[\begin{array}[]{lll}1&1&1\\\ 1&\omega^{2}&\omega\\\
1&\omega&\omega^{2}\end{array}\right].$
To form $\Phi$, in each column of $A^{\mathrm{T}}$, we replace each $1$-valued
entry with a distinct row of $H$. Always choosing the second and third rows
yields an ETF of $9$ elements in $\mathbb{C}^{3}$:
$\Phi=\frac{1}{\sqrt{2}}\left[\begin{array}[]{lllllllll}1&\omega^{2}&\omega&1&\omega^{2}&\omega&&&\\\
1&\omega&\omega^{2}&&&&1&\omega^{2}&\omega\\\
&&&1&\omega&\omega^{2}&1&\omega&\omega^{2}\end{array}\right].$
This is the only known instance of when the Steiner-based construction of
Theorem 7 produces a maximal ETF, that is, one that has $N=M^{2}$.
##### Steiner triple systems: $(2,3,v)$-Steiner ETFs for any $v\equiv 1,3\bmod
6$.
Steiner triple systems, namely $(2,3,v)$-Steiner systems, have been a subject
of interest for over a century, and are known to exist precisely when $v\equiv
1,3\bmod 6$ [52]. Each of the $\smash{b=\frac{v(v-1)}{6}}$ blocks contains
$k=3$ points, while each point is contained in $\smash{r=\frac{v-1}{2}}$
blocks. The corresponding ETFs produced by Theorem 7 consist of
$\smash{\frac{v(v+1)}{2}}$ vectors in $\smash{\frac{v(v-1)}{6}}$-dimensional
space. The density of such frames is $\frac{3}{v}$. As with ETFs stemming from
$2$-element blocks, Steiner triple systems offer little freedom in terms of
redundancy: $\smash{\frac{N}{M}=3\frac{v+1}{v-1}}$ is always approximately
$3$. Such ETFs can be real if there exists a real Hadamard matrix of size
$\smash{\frac{v+1}{2}}$.
##### Four element blocks: $(2,4,v)$-Steiner ETFs for any $v\equiv 1,4\bmod
12$.
It is known that $(2,4,v)$-Steiner systems exist precisely when $v\equiv
1,4\bmod 12$ [1]. Continuing the trend of the previous two families, these
ETFs can vary in size but not in redundancy: they consist of
$\smash{\frac{v(v+2)}{3}}$ vectors in $\smash{\frac{v(v-1)}{12}}$-dimensional
space, having redundancy $\smash{4\frac{v+2}{v-1}}$ and density
$\smash{\frac{4}{v}}$. Interestingly, such frames can never be real: with the
exception of the trivial $1\times 1$ and $2\times 2$ cases, the dimensions of
all real Hadamard matrices are divisible by $4$; since $v\equiv 1,4\bmod 12$,
the requisite matrices $H$ here are of size $\smash{\frac{v+2}{3}}\equiv
1,2\bmod 4$.
##### Five element blocks: $(2,5,v)$-Steiner ETFs for any $v\equiv 1,5\bmod
20$.
It is also known that $(2,5,v)$-Steiner systems exist precisely when $v\equiv
1,5\bmod 20$ [1]. The corresponding ETFs consist of $\smash{\frac{v(v+3)}{4}}$
vectors in $\smash{\frac{v(v-1)}{20}}$-dimensional space, having redundancy
$\smash{5\frac{v+3}{v-1}}$ and density $\smash{\frac{5}{v}}$. Such frames can
be real whenever there exists a real Hadamard matrix of size $\frac{v+3}{4}$.
In particular, letting $v=45$, we see that there exists a real Steiner ETF of
$540$ vectors in $99$-dimensional space, a fact not obtained from any other
known infinite family.
##### Affine geometries: $(2,q,q^{n})$-Steiner ETFs for any prime power $q$,
$n\geq 2$.
At this point, the constructions depart from those previously considered,
allowing both $k$ and $v$ to vary. In particular, using techniques from finite
geometry, one can show that for any prime power $q$ and any $n\geq 2$, there
exists a $(2,k,v)$-Steiner system with $k=q$ and $v=q^{n}$ [52]. The
corresponding ETFs consist of $\smash{q^{n}(1+\frac{q^{n}-1}{q-1})}$ vectors
in $\smash{q^{n-1}(\frac{q^{n}-1}{q-1}})$-dimensional space. Like the
preceding four classes of Steiner ETFs, these frames can grow arbitrarily
large: fixing any prime power $q$, one may manipulate $n$ to produce ETFs of
varying orders of magnitude. However, unlike the four preceding classes, these
affine Steiner ETFs also provide great flexibility in choosing redundancy.
That is, they provide the ability to pick $M$ and $N$ somewhat independently.
Indeed, the redundancy of such frames $q(1+\frac{q-1}{q^{n}-1})$ is
essentially $q$, which may be an arbitrary prime power. Moreover, as these
frames grow large, they also become increasingly sparse: their density is
$\smash{\frac{1}{q^{n-1}}}$. Because of their high sparsity and flexibility
with regards to size and redundancy, these frames, along with their projective
geometry-based cousins detailed below, are perhaps the best known candidates
for use in ETF-based applications. Such ETFs can be real if there exists a
real Hadamard matrix of size $1+\frac{q^{n}-1}{q-1}$, such as whenever $q=2$,
or when $q=5$ and $n=3$.
##### Projective geometries: $(2,q+1,\frac{q^{n+1}-1}{q-1})$-Steiner ETFs for
any prime power $q$, $n\geq 2$.
With finite geometry, one can show that for any prime power $q$ and any $n\geq
2$, there exists a $(2,k,v)$-Steiner system with $k=q+1$ and
$\smash{v=\frac{q^{n+1}-1}{q-1}}$ [52]. Qualitatively speaking, the ETFs that
these projective geometries generate share much in common with their affinely
generated cousins, possessing very high sparsity and great flexibility with
respect to size and redundancy. The technical details are as follows: they
consist of $\smash{\frac{q^{n+1}-1}{q-1}(1+\frac{q^{n}-1}{q-1})}$ vectors in
$\smash{\frac{(q^{n}-1)(q^{n+1}-1)}{(q+1)(q-1)^{2}}}$-dimensional space, with
density $\smash{\frac{q^{2}-1}{q^{n+1}-1}}$ and redundancy
$\smash{(q+1)(1+\frac{q-1}{q^{n}-1})}$. These frames can be real if there
exists a real Hadamard matrix of size $1+\frac{q^{n}-1}{q-1}$; note this
restriction is identical to the one for ETFs generated by affine geometries
for the same $q$ and $n$, implying that real Steiner ETFs generated by finite
geometries always come in pairs, such as the $6\times 16$ and $7\times 28$
ETFs generated when $q=2$, $n=2$, and the $28\times 64$ and $35\times 120$
ETFs generated when $q=2$, $n=3$.
##### Unitals: $(2,q+1,q^{3}+1)$-Steiner ETFs for any prime power $q$.
For any prime power $q$, one can show that there exists a $(2,k,v)$-Steiner
system with $k=q+1$ and $v=q^{3}+1$ [52]. Though one may pick a redundancy of
one’s liking, such a choice confines one to ETFs of a given size: they consist
of $(q^{2}+1)(q^{3}+1)$ vectors in
$\smash{\frac{q^{2}(q^{3}+1)}{q+1}}$-dimensional space, having redundancy
$\smash{(q+1)(1+\frac{1}{q^{2}})}$ and density $\smash{\frac{q+1}{q^{3}+1}}$.
These ETFs can never be real: the requisite Hadamard matrices are of size
$q^{2}+1$ which is never divisible by $4$ since $0$ and $1$ are the only
squares in $\mathbb{Z}_{4}$.
##### Denniston designs: $(2,2^{r},2^{r+s}+2^{r}-2^{s})$-Steiner ETFs for any
$2\leq r<s$.
For any $2\leq r<s$, one can show that there exists a $(2,k,v)$-Steiner system
with $k=2^{r}$ and $v=2^{r+s}+2^{r}-2^{s}$ [52]. By manipulating $r$ and $s$,
one can independently determine the order of magnitude of redundancy and size:
the corresponding ETFs consist of $(2^{s}+2)(2^{r+s}+2^{r}-2^{s})$ vectors in
$\smash{\frac{(2^{s}+1)(2^{r+s}+2^{r}-2^{s})}{2^{r}}}$-dimensional space,
having redundancy $\smash{2^{r}\frac{2^{s}+2}{2^{s}+1}}$ and density
$\smash{\frac{2^{r}}{2^{r+s}+2^{r}-2^{s}}}$. As such, this family has some
qualitative similarities to the familes of ETFs produced by affine and
projective geometries. However, unlike those families, the ETFs produced by
Denniston designs can never be real: the requisite Hadamard matrices are of
size $2^{s}+2$, which is never divisible by $4$.
Name | $M$ | $N$ | Redundancy | Real? | Restrictions
---|---|---|---|---|---
$2$-blocks | $\frac{v(v-1)}{2}$ | $v^{2}$ | $2\frac{v}{v-1}$ | $v$ | None
$3$-blocks | $\frac{v(v-1)}{6}$ | $\frac{v(v+1)}{2}$ | $3\frac{v+1}{v-1}$ | $\frac{v+1}{2}$ | $v\equiv 1,3\bmod 6$
$4$-blocks | $\frac{v(v-1)}{12}$ | $\frac{v(v+2)}{3}$ | $4\frac{v+2}{v-1}$ | Never | $v\equiv 1,4\bmod 12$
$5$-blocks | $\frac{v(v-1)}{20}$ | $\frac{v(v+3)}{4}$ | $5\frac{v+3}{v-1}$ | $\frac{v+3}{4}$ | $v\equiv 1,5\bmod 20$
Affine | $q^{n-1}(\frac{q^{n}-1}{q-1})$ | $q^{n}(1+\frac{q^{n}-1}{q-1})$ | $q(1+\frac{q-1}{q^{n}-1})$ | $1+\frac{q^{n}-1}{q-1}$ | prime power $q$, $n\geq 2$
Projective | $\frac{(q^{n}-1)(q^{n+1}-1)}{(q+1)(q-1)^{2}}$ | $\frac{q^{n+1}-1}{q-1}(1+\frac{q^{n}-1}{q-1})$ | $(q+1)(1+\frac{q-1}{q^{n}-1})$ | $1+\frac{q^{n}-1}{q-1}$ | prime power $q$, $n\geq 2$
Unitals | $\frac{q^{2}(q^{3}+1)}{q+1}$ | $(q^{2}\\!+1)(q^{3}\\!+1)$ | $(q+1)(1+\frac{1}{q^{2}})$ | Never | prime power $q$
Denniston | $\frac{(2^{s}+1)(2^{r+s}+2^{r}-2^{s})}{2^{r}}$ | $(2^{s}\\!+2)(2^{r+s}\\!+2^{r}\\!-2^{s})$ | $2^{r}\frac{2^{s}+2}{2^{s}+1}$ | Never | $2\leq r<s$
Table 1.1: Eight infinite families of Steiner ETFs, each arising from a known infinite family of $(2,k,v)$-Steiner designs. Each family permits both $M$ and $N$ to grow very large, but only a few families—affine, projective and Denniston—give one the freedom to simultaneously control the proportion between $M$ and $N$, namely the redundancy $\frac{N}{M}$ of the ETF. The column denoted “Real?” indicates the size for which a real Hadamard matrix must exist in order for the resulting ETF to be real; it suffices to have this size be a power of $2$; if the Hadamard conjecture is true, it would suffice for this number to be divisible by $4$. $M$ | $N$ | $k$ | $v$ | $r$ | $\mathbb{R}/\mathbb{C}$ | Construction of the Steiner system
---|---|---|---|---|---|---
6 | 16 | 2 | 4 | 3 | $\mathbb{R}$ | $2$-blocks of $v=4$; Affine with $q=2$, $n=2$
7 | 28 | 3 | 7 | 3 | $\mathbb{R}$ | $3$-blocks of $v=7$; Projective with $q=2$, $n=2$
28 | 64 | 2 | 8 | 7 | $\mathbb{R}$ | $2$-blocks of $v=8$; Affine with $q=2$, $n=3$
35 | 120 | 3 | 15 | 7 | $\mathbb{R}$ | $3$-blocks of $v=15$; Projective with $q=2$, $n=3$
66 | 144 | 2 | 12 | 11 | $\mathbb{R}$ | $2$-blocks of $v=12$
99 | 540 | 5 | 45 | 11 | $\mathbb{R}$ | $5$-blocks of $v=45$
3 | 9 | 2 | 3 | 2 | $\mathbb{C}$ | $2$-blocks of $v=3$
10 | 25 | 2 | 5 | 4 | $\mathbb{C}$ | $2$-blocks of $v=5$
12 | 45 | 3 | 9 | 4 | $\mathbb{C}$ | $3$-blocks of $v=9$; Affine with $q=3$, $n=2$
13 | 65 | 4 | 13 | 4 | $\mathbb{C}$ | $4$-blocks of $v=13$; Projective with $q=3$, $n=2$
15 | 36 | 2 | 6 | 5 | $\mathbb{C}$ | $2$-blocks of $v=6$
20 | 96 | 4 | 16 | 5 | $\mathbb{C}$ | $4$-blocks of $v=16$; Affine with $q=4$, $n=2$
21 | 49 | 2 | 7 | 6 | $\mathbb{C}$ | $2$-blocks of $v=7$
21 | 126 | 5 | 21 | 5 | $\mathbb{C}$ | $5$-blocks of $v=21$; Projective with $q=4$, $n=2$
26 | 91 | 3 | 13 | 6 | $\mathbb{C}$ | $3$-blocks of $v=13$
30 | 175 | 5 | 25 | 6 | $\mathbb{C}$ | $5$-blocks of $v=25$; Affine with $q=5$, $n=2$
31 | 217 | 6 | 31 | 6 | $\mathbb{C}$ | Projective with $q=5$, $n=2$
36 | 81 | 2 | 9 | 8 | $\mathbb{C}$ | $2$-blocks of $v=9$
45 | 100 | 2 | 10 | 9 | $\mathbb{C}$ | $2$-blocks of $v=10$
50 | 225 | 4 | 25 | 8 | $\mathbb{C}$ | $4$-blocks of $v=25$
55 | 121 | 2 | 11 | 10 | $\mathbb{C}$ | $2$-blocks of $v=11$
56 | 441 | 7 | 49 | 8 | $\mathbb{C}$ | Affine with $q=7$, $n=2$
57 | 190 | 3 | 19 | 9 | $\mathbb{C}$ | $3$-blocks of $v=19$
57 | 513 | 8 | 57 | 8 | $\mathbb{C}$ | Projective with $q=7$, $n=2$
63 | 280 | 4 | 28 | 9 | $\mathbb{C}$ | Unital with $q=3$; Denniston with $r=2$, $s=3$
70 | 231 | 3 | 21 | 10 | $\mathbb{C}$ | $3$-blocks of $v=21$
72 | 640 | 8 | 64 | 9 | $\mathbb{C}$ | Affine with $q=8$, $n=2$
73 | 730 | 9 | 73 | 9 | $\mathbb{C}$ | Projective with $q=8$, $n=2$
78 | 169 | 2 | 13 | 12 | $\mathbb{C}$ | $2$-blocks of $v=13$
82 | 451 | 5 | 41 | 19 | $\mathbb{C}$ | $5$-blocks of $v=41$
90 | 891 | 9 | 81 | 10 | $\mathbb{C}$ | Affine with $q=9$, $n=2$
91 | 196 | 2 | 14 | 13 | $\mathbb{C}$ | $2$-blocks of $v=14$
91 | 1001 | 10 | 91 | 10 | $\mathbb{C}$ | Projective with $q=9$, $n=2$
100 | 325 | 3 | 25 | 12 | $\mathbb{C}$ | $3$-blocks of $v=25$
Table 1.2: The ETFs of dimension 100 or less that can be constructed by
applying Theorem 7 to the eight infinite families of Steiner systems detailed
in Section 1.3. That is, these ETFs represent the first few examples of the
general constructions summarized in Table 1.1. For each ETF, we give the
dimension $M$ of the underlying space, the number of frame vectors $N$, as
well as the number $k$ of elements that lie in any block of a $v$-element set
in the corresponding $(2,k,v)$-Steiner system. We further give the value $r$
of the number of blocks that contain a given point; by Theorem 8,
$|{\langle{f_{n}},{f_{n^{\prime}}}\rangle}|=\frac{1}{r}$ measures the angle
between any two frame elements. We also indicate whether the given frame is
real or complex, and the method(s) of constructing the corresponding Steiner
system.
#### 1.3.2 Conditions for the existence of Steiner equiangular tight frames
$(2,k,v)$-Steiner systems have been actively studied for over a century, with
many celebrated results. Nevertheless, much about these systems is still
unknown. In this subsection, we discuss some known partial characterizations
of the Steiner systems which lie outside of the eight families we have already
discussed, as well as what these results tell us about the existence of
certain ETFs. To begin, recall that, for a given $k$ and $v$, if a
$(2,k,v)$-Steiner system exists, then the number $r$ of blocks that contain a
given point is necessarily $\smash{\frac{v-1}{k-1}}$, while the total number
of blocks $b$ is $\smash{\frac{v(v-1)}{k(k-1)}}$. As such, in order for a
$(2,k,v)$-Steiner system to exist, it is necessary for $(k,v)$ to be
admissible, that is, to have the property that $\smash{\frac{v-1}{k-1}}$ and
$\smash{\frac{v(v-1)}{k(k-1)}}$ are integers.
However, this property is not sufficient for existence: it is known that a
$(2,6,16)$-Steiner system does not exist [1] despite the fact that
$\smash{\frac{v-1}{k-1}}=3$ and $\smash{\frac{v(v-1)}{k(k-1)}}=8$. In fact,
letting $v$ be either $16$, $21$, $36$, or $46$ results in an admissible pair
with $k=6$, despite the fact that none of the corresponding Steiner systems
exist; there are twenty-nine additional values of $v$ which form an admissible
pair with $k=6$ and for which the existence of a corresponding Steiner system
remains an open problem [1]. Similar nastiness arises with $k\geq 7$. The good
news is that admissibility, though not sufficient for existence, is, in fact,
asymptotically sufficient: for any fixed $k$, there exists a corresponding
admissible index $v_{0}(k)$ for which for all $v>v_{0}(k)$ such that
$\smash{\frac{v-1}{k-1}}$ and $\smash{\frac{v(v-1)}{k(k-1)}}$ are integers, a
$(2,k,v)$-Steiner system indeed exists [1]. Moreover, explicit values of
$v_{0}(k)$ are known for small $k$: $v_{0}(6)=801$, $v_{0}(7)=2605$,
$v_{0}(8)=3753$, $v_{0}(9)=16497$. We now detail the ramifications of these
design-theoretic results on frame theory:
###### Theorem 8.
If an $M\times N$ Steiner equiangular tight frame exists, then letting
$\smash{\alpha=(\frac{N-M}{M(N-1)})^{\frac{1}{2}}}$, the corresponding block
design has parameters:
$v=\tfrac{N\alpha}{1+\alpha},\qquad b=M,\qquad r=\tfrac{1}{\alpha},\qquad
k=\tfrac{N}{M(1+\alpha)}.$
In particular, if such a frame exists, then these expressions for $v$, $k$ and
$r$ are necessarily integers.
Conversely, for any fixed $k\geq 2$, there exists an index $v_{0}(k)$ for
which for all $v>v_{0}(k)$ such that $\smash{\frac{v-1}{k-1}}$ and
$\smash{\frac{v(v-1)}{k(k-1)}}$ are integers, there exists a Steiner
equiangular tight frame of $\smash{v(1+\frac{v-1}{k-1})}$ vectors for a space
of dimension $\smash{\frac{v(v-1)}{k(k-1)}}$.
In particular, for any fixed $k\geq 2$, letting $v$ be either $jk(k-1)+1$ or
$jk(k-1)+k$ for increasingly large values of $j$ results in a sequence of
Steiner equiangular tight frames whose redundancy is asymptotically $k$; these
frames can be real if there exist real Hadamard matrices of sizes $jk+1$ or
$jk+2$, respectively.
###### Proof.
To prove the necessary conditions on $M$ and $N$, recall that Steiner ETFs,
namely those ETFs produced by Theorem 7, have $\smash{N=v(1+\frac{v-1}{k-1})}$
and $\smash{M=\frac{v(v-1)}{k(k-1)}}$. Together, these two equations imply
$N=v+kM$. Solving for $k$ and substituting the resulting expression into
$\smash{N=v(1+\frac{v-1}{k-1})}$ yields the quadratic equation
$0=(M-1)v^{2}+2(N-M)v-N(N-M)$. With some algebra, the only positive root of
this equation can be found to be $\smash{v=\frac{N\alpha}{1+\alpha}}$, as
claimed. Substituting this expression for $v$ into $N=v+kM$ yields
$\smash{k=\tfrac{N}{M(1+\alpha)}}$. Having $v$ and $k$, the previously
mentioned relations $bk=vr$ and $v-1=r(k-1)$ imply
$\smash{r=\frac{v-1}{k-1}=\frac{1}{\alpha}}$ and $\smash{b=\frac{v}{k}r=M}$,
as claimed.
The second set of conclusions is the result of applying Theorem 7 to the
aforementioned $(2,k,v)$-Steiner ETFs that are guaranteed to exist for all
sufficiently large $v$, provided $\smash{\frac{v-1}{k-1}}$ and
$\smash{\frac{v(v-1)}{k(k-1)}}$ are integers. The final set of conclusions are
then obtained by applying this fact in the special cases where $v$ is either
$jk(k-1)+1$ or $jk(k-1)+k$. In particular, if $v=jk(k-1)+1$ then
$\smash{\frac{v-1}{k-1}=jk}$ and
$M=\smash{\frac{v(v-1)}{k(k-1)}=j\big{(}jk(k-1)+1\big{)}}$ are integers, and
the resulting ETF of $(jk+1)\big{(}jk(k-1)+1\big{)}$ vectors has a redundancy
of $\smash{k+\frac{1}{j}}$ that tends to $k$ for large $j$; such an ETF can be
real if there exists a real Hadamard matrix of size $jk+1$. Meanwhile, if
$v=jk(k-1)+k$ then $\smash{\frac{v-1}{k-1}=jk+1}$ and
$M=\smash{\frac{v(v-1)}{k(k-1)}=(jk+1)\big{(}j(k-1)+1\big{)}}$ are integers,
and the resulting ETF of $k(jk+2)\big{(}j(k-1)+1\big{)}$ vectors has a
redundancy of $\smash{k\frac{jk+2}{jk+1}}$ that tends to $k$ for large $j$;
such an ETF can be real if there exists a real Hadamard matrix of size $jk+2$.
∎
We conclude this section with a few thoughts on Theorems 7 and 8. First, we
emphasize that the method of Theorem 7 is a method for constructing some ETFs,
and by no means constructs them all. Indeed, as noted above, the redundancy of
Steiner ETFs is always strictly greater than $2$; while some of those ETFs
with $\frac{N}{M}<2$ will be the Naimark complements of Steiner ETFs, one must
admit that the Steiner method contributes little towards the understanding of
those ETFs with $\frac{N}{M}=2$, such as those arising from Paley graphs
[141]. Moreover, Theorem 8 implies that not even every ETF with
$\frac{N}{M}>2$ arises from a Steiner system: though there exists an ETF of
$76$-elements in $\mathbb{R}^{19}$ [141], the corresponding parameters of the
design would be $v=\frac{38}{3}$, $r=5$ and $k=\frac{10}{3}$, not all of which
are integers.
That said, the method of Theorem 7 is truly significant: comparing Table 1.2
with a comprehensive list of all real ETFs of dimension $50$ or less [141], we
see the Steiner method produces $4$ of the $17$ ETFs that have redundancy
greater than $2$, namely $6\times 16$, $7\times 28$, $28\times 64$ and
$35\times 120$ ETFs. Interestingly, an additional $4$ of these $17$ ETFs can
also be produced by the Steiner method, but only in complex form, namely those
of $15\times 36$, $20\times 96$, $21\times 126$ and $45\times 100$ dimensions;
it is unknown whether this is the result of a deficit in our analysis or the
true non-existence of real-valued Steiner-based constructions of these sizes.
The plot further thickens when one realizes that an additional $2$ of these
$17$ real ETFs satisfy the necessary conditions of Theorem 8, but that the
corresponding $(2,k,v)$-Steiner systems are known to not exist: if a $28\times
288$ ETF was to arise as a result of Theorem 7, the corresponding Steiner
system would have $k=6$ and $v=36$, while the $43\times 344$ ETF would have
$k=7$ and $v=43$; in fact, $(2,6,36)$\- and $(2,7,43)$-Steiner systems cannot
exist [1]. With our limited knowledge of the rich literature on Steiner
systems, we were unable to resolve the existence of two remaining candidates:
$23\times 276$ and $46\times 736$ ETFs could potentially arise from
$(2,10,46)$\- and $(2,14,92)$-Steiner systems, respectively, provided they
exist.
### 1.4 Restricted isometry and digital fingerprinting
In the previous section, we used Theorem 7 to construct many examples of
Steiner ETFs. In this section, we investigate the feasibility of using such
frames for applications in sparse signal processing. Regarding restricted
isometry, one of the sad consequences of the Steiner construction method in
Theorem 7 is that we now know there is a large class of ETFs for which the
seemingly coarse estimate from the Gershgorin analysis (1.4) is, in fact,
accurate. In particular, recall that Gershgorin guarantees that every $M\times
N$ ETF is $(K,\delta)$-RIP whenever $K\leq\delta\sqrt{M}$. Furthermore, recall
from Theorem 7 that every Steiner ETF is built by carefully overlapping $v$
regular simplices, each consisting of $r+1$ vectors in an $r$-dimensional
subspace of $b$-dimensional space. Thus, the corresponding subcollection of
$r+1$ vectors that lie in a given block are linearly dependent. Considering
the value of $r$ given in Theorem 8, we see that Steiner ETFs $\Phi$ have
$\mathrm{Spark}(\Phi)\leq
r+1=\sqrt{\frac{M(N-1)}{N-M}}+1\leq\sqrt{\frac{MN}{N-N/2}}+1=\sqrt{2M}+1,$
where the last inequality uses the fact that Steiner ETFs have redundancy
$\frac{N}{M}\geq 2$. Therefore, Steiner ETFs are not $(K,1-\varepsilon)$-RIP
for any $K>\sqrt{2M}$, that is, they fail to break the square-root bottleneck.
This begs the open question: Are there any ETFs which are as RIP as random
matrices, or does being optimal in the Gershgorin sense necessarily come at
the cost of being able to support large sparsity levels? In Chapter 3, we
address this problem directly and make some interesting connections with graph
theory and number theory, but we do not give a conclusive answer.
Despite their provably suboptimal performance as RIP matrices, we will see
that Steiner ETFs are particularly well-suited for the application of digital
fingerprints. Digital media protection has become an important issue in recent
years, as illegal distribution of licensed material has become increasingly
prevalent. A number of methods have been proposed to restrict illegal
distribution of media and ensure only licensed users are able to access it.
One method involves cryptographic techniques, which encrypt the media before
distribution. By doing this, only the users with appropriate licensed hardware
or software have access; satellite TV and DVDs are two such examples.
Unfortunately, cryptographic approaches are limited in that once the content
is decrypted (legally or illegally), it can potentially be copied and
distributed freely.
An alternate approach involves marking each copy of the media with a unique
signature. The signature could be a change in the bit sequence of the digital
file or some noise-like distortion of the media. The unique signatures are
called _fingerprints_ , by analogy to the uniqueness of human fingerprints.
With this approach, a licensed user could illegally distribute the file, only
to be implicated by his fingerprint. The potential for prosecution acts as a
deterrent to unauthorized distribution. However, fingerprinting systems are
vulnerable when multiple users form a _collusion_ by combining their copies to
create a forged copy. This attack can reduce and distort the colluders’
individual fingerprints, making identification of any particular user
difficult. Some examples of potential attacks involve comparing the bit
sequences of different copies, averaging copies in the signal space, as well
as introducing noise, rotations, or cropping.
One of the principal approaches to designing fingerprints with robustness to
collusions uses what is called the _distortion assumption_. In this regime,
fingerprints are noise-like distortions to the media in signal space. In order
to preserve the overall quality of the media, limits are placed on the
magnitude of this distortion. The content owner limits the power of the
fingerprint he adds, and the collusion limits the power of the noise they add
in their attack. When applying the distortion assumption, the literature
typically assumes that the collusion linearly averages their individual copies
to forge the host signal. Also, while results using the distortion assumption
tend to accommodate fewer users than those with other assumptions, this
assumption is distinguished by its natural embedding of fingerprints, namely
in the signal space.
Cox et al. introduced one of the first robust fingerprint designs under the
distortion assumption [54]; the robustness was later analytically proven in
[92]. Different fingerprint designs have since been studied, including
orthogonal fingerprints [142] and simplex fingerprints [94]. We propose ETFs
as a fingerprint design under the distortion assumption, and we analyze their
performance against the worst-case collusion [103, 104]. Using analysis from
Ergun et al. [66], we will show that ETFs perform particularly well as
fingerprints; as a matter of fact, Steiner ETF fingerprints perform comparably
to orthogonal and simplex fingerprints on average, while accommodating several
times as many users [104]. We start by formally presenting the fingerprinting
and collusion processes.
#### 1.4.1 Problem setup
A content owner has a host signal that he wishes to share, but he wants to
mark it with fingerprints before distributing it. We view this host signal as
a vector $s\in\mathbb{R}^{M}$, and the marked versions of this vector will be
given to $N>M$ users. Specifically, the $n$th user is given
$\hat{s}_{n}:=s+\varphi_{n},$
where $\varphi_{n}\in\mathbb{R}^{M}$ denotes the $n$th fingerprint; we assume
the fingerprints have equal norm. We wish to design the fingerprints
$\\{\varphi_{n}\\}_{n=1}^{N}$ to be robust to a linear averaging attack. In
particular, let $\mathcal{K}\subseteq\\{1,\ldots,N\\}$ denote a collection of
users who together make a different copy of the host signal. Then their linear
averaging attack produces a forgery:
$f:=\sum_{k\in\mathcal{K}}x_{k}\hat{s}_{k}+z,\qquad\sum_{k\in\mathcal{K}}x_{k}=1,\qquad
x_{k}\geq 0~{}~{}~{}\forall k,$ (1.7)
where $z$ is a noise vector introduced by the colluders. This attack model is
illustrated in Figure 1.1.
$s$fingerprintassignment$\varphi_{1}$$\varphi_{2}$$\varphi_{3}$$\vdots$$\varphi_{N-2}$$\varphi_{N-1}$$\varphi_{N}$$\hat{s}_{1}$$\vdots$$\hat{s}_{N-2}$$\hat{s}_{N-1}$$\underbrace{}_{\mathcal{K}}$linear-
average-plus-noiseforgery
process$\hat{s}_{2}$$x_{2}$$\hat{s}_{3}$$x_{3}$$\vdots$$\vdots$$\hat{s}_{N}$$x_{N}$$z$$f$
Figure 1.1: The fingerprint and forgery processes. First, the content owner
makes different copies of his host signal $s$ by adding fingerprints
$\varphi_{n}$ which are unknown to the users. Next, a subcollection
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$ of the users collude to create a
forgery $f$ by picking a convex combination of their copies and adding noise
$z$. In this example, the forgery coalition $\mathcal{K}$ includes users $2$,
$3$, and $N$.
Certainly, the ultimate goal of the content owner is to detect every member of
the forgery coalition. This can prove difficult in practice, though,
particularly when some individuals contribute little to the forgery, with
$x_{k}\ll\frac{1}{|\mathcal{K}|}$. However, in the real world, if at least one
colluder is caught, then other members could be identified through the legal
process. As such, we consider _focused_ detection, where a test statistic is
computed for each user, and we perform a binary hypothesis test to decide
whether that particular user is guilty.
Our detection procedure is as follows: With the cooperation of the content
owner, the host signal can be subtracted from a forgery to isolate the
fingerprint combination:
$y:=f-s=\sum_{k\in\mathcal{K}}x_{k}\varphi_{k}+z.$ (1.8)
To help the content owner discern who is guilty, we then use a normalized
correlation function as a test statistic for each user $n$:
$T_{n}(y):=\frac{\langle y,\varphi_{n}\rangle}{\|\varphi_{n}\|^{2}}.$
Having devised a test statistic, let $H_{1}(n)$ denote the guilty hypothesis
($n\in\mathcal{K}$) and $H_{0}(n)$ denote the innocent hypothesis
($n\not\in\mathcal{K}$). Then picking some correlation threshold $\tau$, we
use the following detector:
$D_{\tau}(n):=\left\\{\begin{array}[]{ll}H_{1}(n),&T_{n}(y)\geq\tau,\\\
H_{0}(n),&T_{n}(y)<\tau.\end{array}\right.$ (1.9)
To determine the effectiveness of our fingerprint design and focused detector,
we will investigate the corresponding error probabilities, but first, we build
our intuition for fingerprint design using a certain geometric figure of
merit.
#### 1.4.2 A geometric figure of merit for fingerprint design
For each user $n$, consider the distance between forgeries deriving from two
types of potential collusions: those of which $n$ is a member, and those of
which $n$ is not. Intuitively, if every fingerprint combination involving $n$
is distant from every combination not involving $n$, then even with moderate
noise, there should be little ambiguity as to whether the $n$th user was
involved. To make this precise, for each user $n$, we define the “guilty” and
“not guilty” sets of noiseless fingerprint combinations:
$\displaystyle\mathcal{G}_{K,n}$
$\displaystyle:=\bigg{\\{}\frac{1}{|\mathcal{K}|}\sum_{k\in\mathcal{K}}\varphi_{k}:n\in\mathcal{K}\subseteq\\{1,\ldots,N\\},~{}|\mathcal{K}|\leq
K\bigg{\\}},$ $\displaystyle\neg\mathcal{G}_{K,n}$
$\displaystyle:=\bigg{\\{}\frac{1}{|\mathcal{K}|}\sum_{k\in\mathcal{K}}\varphi_{k}:n\not\in\mathcal{K}\subseteq\\{1,\ldots,N\\},~{}|\mathcal{K}|\leq
K\bigg{\\}}.$
In words, $\mathcal{G}_{K,n}$ is the set of size-$K$ fingerprint combinations
of equal weights which include $n$, while $\neg\mathcal{G}_{K,n}$ is the set
of combinations which do not include $n$. Note that in our setup (1.7), the
weights $x_{k}$ were arbitrary values which sum to $1$. We will show in
Theorem 11 that the best attack from the collusion’s perspective uses equal
weights so that no single colluder is particularly vulnerable. From this
perspective, it makes sense to bound the distance between these two sets:
$\mathrm{dist}(\mathcal{G}_{K,n},\neg\mathcal{G}_{K,n}):=\min\\{\|y-y^{\prime}\|_{2}:y\in\mathcal{G}_{K,n},~{}y^{\prime}\in\neg\mathcal{G}_{K,n}\\}.$
(1.10)
Note that by taking $\Phi$ to be the $M\times N$ matrix whose columns are the
fingerprints $\varphi_{n}$, the fingerprint combination (1.8) can be rewritten
as $y=\Phi x+z$, where the entries of $x$ are $x_{k}$ when $k\in\mathcal{K}$
and zero otherwise. Thus, if the matrix of fingerprints $\Phi$ is
$(K,\delta)$-RIP with $\delta<\sqrt{2}-1$, then we can recover the $K$-sparse
vector $x$ using Theorem 2. However, the error in the estimate $\tilde{x}$ of
$x$ will be on the order of $10$ times the size of the noise $z$ [34]. Due to
the potential legal ramifications of false accusations, this order of error is
not tolerable. Note that the methods of compressed sensing recover the entire
vector $x$, the support of which identifies the entire collusion. By contrast,
we will investigate RIP matrices for fingerprint design, but to minimize false
accusations, we will use focused detection (1.9) to identify colluders.
We now investigate how well RIP matrices perform with respect to our geometric
figure of merit. Without loss of generality, we assume the fingerprints are
unit norm; since they have equal norm, the fingerprint combination can be
scaled by $\frac{1}{\|\varphi_{n}\|}$ before the detection phase. With this in
mind, we have the following a lower bound on the distance (1.10) between the
“guilty” and “not guilty” sets corresponding to any user $n$:
###### Theorem 9.
Suppose fingerprints $\Phi=[\varphi_{1}\cdots\varphi_{N}]$ have restricted
isometry constant $\delta_{2K}$. Then
$\mathrm{dist}(\mathcal{G}_{K,n},\neg\mathcal{G}_{K,n})\geq\sqrt{\frac{1-\delta_{2K}}{K(K-1)}}.$
(1.11)
###### Proof.
Take $\mathcal{K},\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ such that
$|\mathcal{K}|,|\mathcal{K}^{\prime}|\leq K$ and
$n\in\mathcal{K}\setminus\mathcal{K}^{\prime}$. Then the left-hand inequality
of the restricted isometry property gives
$\displaystyle\bigg{\|}\frac{1}{|\mathcal{K}|}\sum_{n\in\mathcal{K}}\varphi_{n}-\frac{1}{|\mathcal{K}^{\prime}|}\sum_{n\in\mathcal{K}^{\prime}}\varphi_{n}\bigg{\|}^{2}$
$\displaystyle=\bigg{\|}\Big{(}\frac{1}{|\mathcal{K}|}-\frac{1}{|\mathcal{K}^{\prime}|}\Big{)}\sum_{n\in\mathcal{K}\cap\mathcal{K}^{\prime}}\varphi_{n}+\frac{1}{|\mathcal{K}|}\sum_{n\in\mathcal{K}\setminus\mathcal{K}^{\prime}}\varphi_{n}-\frac{1}{|\mathcal{K}^{\prime}|}\sum_{n\in\mathcal{K}^{\prime}\setminus\mathcal{K}}\varphi_{n}\bigg{\|}^{2}$
$\displaystyle\geq(1-\delta_{|\mathcal{K}\cup\mathcal{K}^{\prime}|})\bigg{(}|\mathcal{K}\cap\mathcal{K}^{\prime}|\Big{(}\frac{1}{|\mathcal{K}|}-\frac{1}{|\mathcal{K}^{\prime}|}\Big{)}^{2}+\frac{|\mathcal{K}\setminus\mathcal{K}^{\prime}|}{|\mathcal{K}|^{2}}+\frac{|\mathcal{K}^{\prime}\setminus\mathcal{K}|}{|\mathcal{K}^{\prime}|^{2}}\bigg{)}$
$\displaystyle=\frac{1-\delta_{|\mathcal{K}\cup\mathcal{K}^{\prime}|}}{|\mathcal{K}||\mathcal{K}^{\prime}|}\bigg{(}|\mathcal{K}|+|\mathcal{K}^{\prime}|-2|\mathcal{K}\cap\mathcal{K}^{\prime}|\bigg{)}.$
(1.12)
For a fixed $|\mathcal{K}|$, we will find a lower bound for
$\frac{1}{|\mathcal{K}|}\bigg{(}|\mathcal{K}|+|\mathcal{K}^{\prime}|-2|\mathcal{K}\cap\mathcal{K}^{\prime}|\bigg{)}=1+\frac{|\mathcal{K}|-2|\mathcal{K}\cap\mathcal{K}^{\prime}|}{|\mathcal{K}^{\prime}|}.$
(1.13)
Since we can have
$|\mathcal{K}\cap\mathcal{K}^{\prime}|>\frac{|\mathcal{K}|}{2}$, we know
$\frac{|\mathcal{K}|-2|\mathcal{K}\cap\mathcal{K}^{\prime}|}{|\mathcal{K}^{\prime}|}<0$
when (1.13) is minimized. That said, $|\mathcal{K}^{\prime}|$ must be as small
as possible, i.e.,
$|\mathcal{K}^{\prime}|=|\mathcal{K}\cap\mathcal{K}^{\prime}|$. Thus, when
(1.13) is minimized, we have
$\frac{1}{|\mathcal{K}|}\bigg{(}|\mathcal{K}|+|\mathcal{K}^{\prime}|-2|\mathcal{K}\cap\mathcal{K}^{\prime}|\bigg{)}=\frac{|\mathcal{K}|}{|\mathcal{K}\cap\mathcal{K}^{\prime}|}-1,$
i.e., $|\mathcal{K}\cap\mathcal{K}^{\prime}|$ must be as large as possible.
Since $n\in\mathcal{K}\setminus\mathcal{K}^{\prime}$, we have
$|\mathcal{K}\cap\mathcal{K}^{\prime}|\leq|\mathcal{K}|-1$. Therefore,
$\frac{1}{|\mathcal{K}|}\bigg{(}|\mathcal{K}|+|\mathcal{K}^{\prime}|-2|\mathcal{K}\cap\mathcal{K}^{\prime}|\bigg{)}\geq\frac{1}{|\mathcal{K}|-1}.$
(1.14)
Substituting (1.14) into (1.12) gives
$\bigg{\|}\frac{1}{|\mathcal{K}|}\sum_{n\in\mathcal{K}}\varphi_{n}-\frac{1}{|\mathcal{K}^{\prime}|}\sum_{n\in\mathcal{K}^{\prime}}\varphi_{n}\bigg{\|}^{2}\geq\frac{1-\delta_{|\mathcal{K}\cup\mathcal{K}^{\prime}|}}{|\mathcal{K}|(|\mathcal{K}|-1)}\geq\frac{1-\delta_{2K}}{K(K-1)}.$
Since this bound holds for every $n$, $\mathcal{K}$ and $\mathcal{K}^{\prime}$
with $n\in\mathcal{K}\setminus\mathcal{K}^{\prime}$, we have (1.11). ∎
Combining Theorem 9 with the Gershgorin estimate $\delta_{2K}\leq(2K-1)\mu$ in
terms of worst-case coherence $\mu$ yields the following:
###### Corollary 10.
Suppose fingerprints $\Phi=[\varphi_{1}\cdots\varphi_{N}]$ are unit-norm with
worst-case coherence $\mu$. Then
$\mathrm{dist}(\mathcal{G}_{K,n},\neg\mathcal{G}_{K,n})\geq\sqrt{\frac{1-(2K-1)\mu}{K(K-1)}}.$
(1.15)
In words, Corrolary 10 says that less coherent fingerprints provide a greater
distance between the “guilty” and “not guilty” sets. It is therefore fitting
to consider minimizers of worst-case coherence, namely equiangular tight
frames. One type of ETF has already been proposed for fingerprint design: the
simplex [94]. The simplex is an ETF with $N=M+1$ and $\mu=\frac{1}{M}$. In
fact, [94] gives a derivation for the exact value of the distance (1.10) in
this case:
$\mathrm{dist}(\mathcal{G}_{K,n},\neg\mathcal{G}_{K,n})=\sqrt{\frac{1}{K(K-1)}\frac{N}{N-1}}.$
(1.16)
The bound (1.15) is lower than (1.16) by a factor of
$\sqrt{1-\frac{2K}{M+1}}$, and for practical cases in which $K\ll M$, the two
are particularly close. Overall, ETF fingerprint design is a natural
generalization of the provably optimal simplex design of [94].
Having applied the Gershgorin analysis to illustrate how ETF fingerprints
perform with respect to our geometric figure of merit, we have yet to
establish any fingerprint-specific consequences of Steiner ETFs not being as
RIP as random matrices. Certainly, whether $K$ scales as $\sqrt{M}$ or $M$ is
an important distinction in the compressed sensing community, but
interestingly, in the context of fingerprints, this difference offers no
advantage. To be clear, Ergun et al. [66] showed that for any fingerprinting
system, there is a tradeoff between the probabilities of successful detection
and false positives imposed by a linear-average-plus-noise attack from
sufficiently large collusions. Specifically, a collusion of size
$K=\mathrm{\Omega}\big{(}\sqrt{\frac{M}{\log M}}\big{)}$ is sufficient to
overcome the fingerprints, as the detector will not be able to identify any
attacker without incurring a false-alarm probability that is too large to be
admissible in court. This constraint is more restrictive than the coherence-
based reconstruction guarantees which require $K=\mathrm{O}(\sqrt{M})$, and so
from this perspective, random RIP constructions are no better for fingerprint
design than deterministic constructions.
#### 1.4.3 Error analysis
We now investigate the errors associated with using ETF fingerprints and a
focused correlation detector with linear-average-plus-noise attacks. To do
this, we assume that the noise $z$ included in the attack (1.7) has
independent Gaussian entries of mean zero and variance $\sigma^{2}$. One type
of error we can expect is the false-positive error, in which an innocent user
$n\notin\mathcal{K}$ is found guilty ($T_{n}(y)\geq\tau$). This could have
significant ramifications in legal proceedings, so this error probability
$\mathrm{Pr}\big{[}T_{n}(y)\geq\tau\big{|}H_{0}(n)\big{]}$ should be kept
extremely low. To ensure this type of error is improbable, we consider the
_worst-case type I error probability_ , which depends on the fingerprint
design $\Phi$, the correlation threshold $\tau$, and the weights
$\\{x_{k}\\}_{k=1}^{K}$ used by the colluders in their linear average:
$\mathrm{P}_{\mathrm{I}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K}):=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\max_{\begin{subarray}{c}\mathcal{K}\rightarrow\\{x_{k}\\}\\\
\mathrm{bijective}\end{subarray}}\max_{n\not\in\mathcal{K}}\mathrm{Pr}\big{[}T_{n}(y)\geq\tau\big{|}H_{0}(n)\big{]}.$
(1.17)
In words, the probability that an innocent user $n$ is found guilty is no
larger than $\mathrm{P}_{\mathrm{I}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$,
regardless of the coalition $\mathcal{K}$ or how the coalition members assign
weights from $\\{x_{k}\\}_{k=1}^{K}$. The other error type is the false-
negative error, in which a guilty user $n\in\mathcal{K}$ is found innocent
($T_{n}(y)<\tau$). In this case, since the goal of our detection is to catch
at least one of the colluders, we define the _worst-case type II error
probability_ as follows:
$\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K}):=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\max_{\begin{subarray}{c}\mathcal{K}\rightarrow\\{x_{k}\\}\\\
\mathrm{bijective}\end{subarray}}\min_{n\in\mathcal{K}}\mathrm{Pr}\big{[}T_{n}(y)<\tau\big{|}H_{1}(n)\big{]}.$
(1.18)
This way, regardless of who the colluders are or how they assign the weights,
at least one of the colluders will have a false-negative probability less than
$\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$, meaning even in
the worst-case scenario, we can correctly identify one of the colluders with
probability $\geq 1-\mathrm{P}_{\mathrm{II}}$.
###### Theorem 11.
Take fingerprints as the columns of an $M\times N$ matrix
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$, which, when normalized by the
fingerprints’ common norm $\gamma$, forms an equiangular tight frame. If the
noise $z$ included in the attack (1.7) has independent Gaussian entries of
mean zero and variance $\sigma^{2}$, then the worst-case type I and type II
error probabilities, (1.17) and (1.18), satisfy
$\displaystyle\mathrm{P}_{\mathrm{I}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$
$\displaystyle\leq
Q\bigg{(}\frac{\gamma}{\sigma}\big{(}\tau-\mu\big{)}\bigg{)},$
$\displaystyle\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$
$\displaystyle\leq
Q\bigg{(}\frac{\gamma}{\sigma}\Big{(}(1+\mu)\max\\{x_{k}\\}_{k=1}^{K}-\mu-\tau\Big{)}\bigg{)},$
where $Q(x):=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-u^{2}/2}du$ and
$\mu=\sqrt{\frac{N-M}{M(N-1)}}$.
###### Proof.
To bound $\mathrm{P}_{\mathrm{I}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$, assume a
given user $n$ is innocent, i.e., $H_{0}(n)$. Then the test statistic for our
detector (1.9) is given by
$T_{n}(y)=\frac{1}{\gamma^{2}}\bigg{\langle}\sum_{k\in\mathcal{K}}x_{k}\varphi_{k}+z,\varphi_{n}\bigg{\rangle}=\sum_{k\in\mathcal{K}}x_{k}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}+\frac{1}{\gamma}\bigg{\langle}z,\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}.$
By the symmetry of $z$’s Gaussian distribution, we know the projection
$\langle z,\frac{\varphi_{n}}{\|\varphi_{n}\|}\rangle$ also has Gaussian
distribution with mean zero and variance $\sigma^{2}$, meaning our test
statistic $T_{n}(y)$ has Gaussian distribution with mean
$\sum_{k\in\mathcal{K}}x_{k}\langle\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\rangle$
and variance $\frac{\sigma^{2}}{\gamma^{2}}$. Furthermore, since the
normalized fingerprints form an ETF with worst-case coherence $\mu$, we can
use the triangle inequality to bound the mean of $T_{n}(y)$:
$\sum_{k\in\mathcal{K}}x_{k}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}\leq\bigg{|}\sum_{k\in\mathcal{K}}x_{k}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}\bigg{|}\leq\sum_{k\in\mathcal{K}}x_{k}\bigg{|}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}\bigg{|}=\mu.$
We use this to bound the false-positive probability for user $n$:
$\mathrm{Pr}\big{[}T_{n}(y)\geq\tau\big{|}H_{0}(n)\big{]}=Q\bigg{(}\frac{\gamma}{\sigma}\Big{(}\tau-\mathbb{E}\big{[}T_{n}(y)|H_{0}(n)\big{]}\Big{)}\bigg{)}\leq
Q\bigg{(}\frac{\gamma}{\sigma}\big{(}\tau-\mu\big{)}\bigg{)}.$
Since this bound holds for all coalitions, weight assignments and innocent
users, this bound must also hold for
$\mathrm{P}_{\mathrm{I}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$.
Next, to bound $\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$,
assume a given user $n$ is guilty, i.e., $H_{1}(n)$. In this case, the test
statistic for our detector (1.9) is given by
$T_{n}(y)=\frac{1}{\gamma^{2}}\bigg{\langle}\sum_{k\in\mathcal{K}}x_{k}\varphi_{k}+z,\varphi_{n}\bigg{\rangle}=x_{n}+\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k\neq
n\end{subarray}}x_{k}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}+\frac{1}{\gamma}\bigg{\langle}z,\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}.$
As before, $T_{n}(y)$ has Gaussian distribution with variance
$\frac{\sigma^{2}}{\gamma^{2}}$, but this time, the mean is
$x_{n}+\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\ k\neq
n\end{subarray}}x_{k}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}\geq
x_{n}-\bigg{|}\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\ k\neq
n\end{subarray}}x_{k}\bigg{\langle}\frac{\varphi_{k}}{\|\varphi_{k}\|},\frac{\varphi_{n}}{\|\varphi_{n}\|}\bigg{\rangle}\bigg{|}\geq
x_{n}-\mu\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\ k\neq
n\end{subarray}}x_{k}=(1+\mu)x_{n}-\mu.$
As such, the false-negative probability for user $n$ is
$\mathrm{Pr}\big{[}T_{n}(y)<\tau\big{|}H_{1}(n)\big{]}=Q\bigg{(}-\frac{\gamma}{\sigma}\Big{(}\tau-\mathbb{E}\big{[}T_{n}(y)|H_{1}(n)\big{]}\Big{)}\bigg{)}\leq
Q\bigg{(}\frac{\gamma}{\sigma}\Big{(}(1+\mu)x_{n}-\mu-\tau\Big{)}\bigg{)}.$
Applying the definition of
$\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$ therefore gives
$\displaystyle\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$
$\displaystyle=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\max_{\begin{subarray}{c}\mathcal{K}\rightarrow\\{x_{k}\\}\\\
\mathrm{bijective}\end{subarray}}\min_{n\in\mathcal{K}}\mathrm{Pr}\big{[}T_{n}(y)<\tau\big{|}H_{1}(n)\big{]}$
$\displaystyle\leq\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\max_{\begin{subarray}{c}\mathcal{K}\rightarrow\\{x_{k}\\}\\\
\mathrm{bijective}\end{subarray}}\min_{n\in\mathcal{K}}Q\bigg{(}\frac{\gamma}{\sigma}\Big{(}(1+\mu)x_{n}-\mu-\tau\Big{)}\bigg{)}$
$\displaystyle=Q\bigg{(}\frac{\gamma}{\sigma}\Big{(}(1+\mu)\max\\{x_{k}\\}_{k=1}^{K}-\mu-\tau\Big{)}\bigg{)}.\qed$
From Theorem 11, we can glean a few interesting insights about ETF
fingerprints. First, the upper bound on
$\mathrm{P}_{\mathrm{I}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$ is independent of
$\\{x_{k}\\}_{k=1}^{K}$, indicating that the coalition cannot pick weights in
a way that frames an innocent user. Additionally, the upper bound on
$\mathrm{P}_{\mathrm{II}}(\Phi,\tau,\\{x_{k}\\}_{k=1}^{K})$ is maximized when
the weights $x_{k}$ are equal, corresponding to our use of equal weights in
the geometric figure of merit. This confirms our intuition that the coalition
has the best chance of not being caught if no member is particularly
vulnerable.
## Chapter 2 Full spark frames
In the previous chapter, we reviewed how to use the Gershgorin circle theorem
to demonstrate the restricted isometry property (RIP), and how identifying
small spark disproves RIP. We then showed that Steiner equiangular tight
frames (ETFs) are optimal in the Gershgorin sense, but have particularly small
spark. Among other things, this illustrates that the “square-root bottleneck”
with deterministic RIP matrices is not merely an artifact of the Gershgorin
analysis. That said, as an intermediate goal to constructing RIP matrices, we
seek deterministic matrices with large spark, understanding that RIP matrices
necessarily have this property. To this end, one is naturally led to consider
_full spark_ matrices, that is, $M\times N$ matrices $\Phi$ with the largest
spark possible: $\mathrm{Spark}(\Phi)=M+1$. Equivalently, $M\times N$ full
spark matrices have the property that every $M\times M$ submatrix is
invertible; as such, a full spark matrix is necessarily full rank, and
therefore a frame.
Interestingly, in sparse signal processing, the specific application of full
spark frames has already been studied for some time. In 1997, Gorodnitsky and
Rao [74] first considered full spark frames, referring to them as matrices
with the _unique representation property_. Since [74], the unique
representation property has been explicitly used to find a variety of
performance guarantees for sparse signal processing [30, 105, 144]. Tang and
Nehorai [133] also obtain performance guarantees using full spark frames, but
they refer to them as _non-degenerate measurement matrices_.
For another application of full spark frames, we consider the problem of
reconstructing a signal from distorted frame coefficients. Specifically, we
observe a scenario in which frame coefficients
$\\{(\Phi^{*}x)[n]\\}_{n=1}^{N}$ are transmitted over a noisy or lossy channel
before reconstructing the signal:
$y=\mathcal{D}(\Phi^{*}x),\qquad\tilde{x}=(\Phi\Phi^{*})^{-1}\Phi y,$ (2.1)
where $\mathcal{D}(\cdot)$ represents the channel’s random and not-
necessarily-linear deformation process. Using an additive white Gaussian noise
model, Goyal [75] established that, of all unit norm frames, unit norm tight
frames minimize mean squared error in reconstruction. For the case of a lossy
channel, Holmes and Paulsen [84] established that, of all tight frames, unit
norm tight frames minimize worst-case error in reconstruction after one
erasure, and that equiangular tight frames minimize this error after two
erasures. We note that the reconstruction process in (2.1), namely the
application of $(\Phi\Phi^{*})^{-1}\Phi$, is inherently blind to the effect of
the deformation process of the channel. This contrasts with Püschel and
Kovačević’s more recent work [113], which describes an adaptive process for
reconstruction after multitudes of erasures. In this context, they reconstruct
the signal after first identifying which frame coefficients were not erased;
with this information, the signal can be estimated provided the corresponding
frame elements span. In this sense, full spark frames are _maximally robust to
erasures_ , as coined in [113]. In particular, an $M\times N$ full spark frame
is robust to $N-M$ erasures since any $M$ of the frame coefficients will
uniquely determine the original signal.
Yet another application of full spark frames is phaseless reconstruction,
which can be viewed in terms of a channel, as in (2.1); in this case,
$\mathcal{D}(\cdot)$ is the entrywise absolute value function. Phaseless
reconstruction has a number of real-world applications including speech
processing [15], X-ray crystallography [37], and quantum state estimation
[116]. As such, there has been a lot of work to reconstruct an $M$-dimensional
vector (up to an overall phase factor) from the magnitudes of its frame
coefficients, most of which involves frames in operator space, which
inherently require $N=\Omega(M^{2})$ measurements [14, 116]. However, Balan et
al. [15] show that if an $M\times N$ real frame $\Phi$ is full spark with
$N\geq 2M-1$, then $\mathcal{D}\circ\Phi^{*}$ is injective, meaning an
inversion process is possible with only $N=\mathrm{O}(M)$ measurements. This
result prompted an ongoing search for efficient phaseless reconstruction
processes [13, 37], but no reconstruction process can succeed without a good
family of frames, such as full spark frames.
Despite the fact that full spark frames have a multitude of applications, to
date, there has not been much progress in constructing deterministic full
spark frames, let alone full spark frames with additional desirable
properties. A noteworthy exception is Püschel and Kovačević’s work [113], in
which real full spark tight frames are constructed using polynomial
transforms. In the present chapter, we start by investigating Vandermonde
frames, harmonic frames, and modifications thereof [2]. While the use of
certain Vandermonde and harmonic frames as full spark frames is not new [30,
36, 72], the fruits of our investigation are new: For instance, we demonstrate
that certain classes of ETFs are full spark, and we characterize the $M\times
N$ full spark harmonic frames for which $N$ is a prime power. Later, we prove
that verifying whether a matrix is full spark is hard for ${\mathsf{NP}}$
under randomized polynomial-time reductions [2]. In other words, assuming
${\mathsf{NP}}\not\subseteq{\mathsf{BPP}}$ (a computational complexity
assumption slightly stronger than ${\mathsf{P}}\neq{\mathsf{NP}}$ and nearly
as widely believed), then there is no method by which one can efficiently test
whether matrices are full spark. As such, the deterministic constructions we
provide are significant in that they guarantee a property which is otherwise
difficult to check. We conclude the chapter by introducing a new technique for
efficient phaseless recovery, which explicitly makes use of deterministic full
spark frames to design $N=\mathrm{O}(M)$ measurements.
### 2.1 Deterministic constructions of full spark frames
A square matrix is invertible if and only if its determinant is nonzero, and
in our quest for deterministic constructions of full spark frames, this
characterization will reign supreme. One class of matrices has a particularly
simple determinant formula: Vandermonde matrices. Specifically, Vandermonde
matrices have the following form:
$V=\begin{bmatrix}1&1&\cdots&1\\\ \alpha_{1}&\alpha_{2}&\cdots&\alpha_{N}\\\
\vdots&\vdots&\cdots&\vdots\\\
\alpha_{1}^{M-1}&\alpha_{2}^{M-1}&\cdots&\alpha_{N}^{M-1}\end{bmatrix},$ (2.2)
and square Vandermonde matrices, i.e., with $N=M$, have the following
determinant:
$\mathrm{det}(V)=\prod_{1\leq i<j\leq M}(\alpha_{j}-\alpha_{i}).$ (2.3)
Consider (2.2) in the case where $N\geq M$. Since every $M\times M$ submatrix
of $V$ is also Vandermonde, we can modify the indices in (2.3) to calculate
the determinant of the submatrices. These determinants are nonzero precisely
when the bases $\\{\alpha_{n}\\}_{n=1}^{N}$ are distinct, yielding the
following result:
###### Lemma 12.
A Vandermonde matrix is full spark if and only if its bases are distinct.
To be clear, this result is not new. In fact, the full spark of Vandermonde
matrices was first exploited by Fuchs [72] for sparse signal processing.
Later, Bourguignon et al. [30] specifically used the full spark of Vandermonde
matrices whose bases are sampled from the complex unit circle. Interestingly,
when viewed in terms of frame theory, Vandermonde matrices naturally point to
the discrete Fourier transform:
###### Theorem 13.
The only $M\times N$ Vandermonde matrices that are equal norm and tight have
bases in the complex unit circle. Among these, the frames with the smallest
worst-case coherence have bases that are equally spaced in the complex unit
circle, provided $N\geq 2M$.
###### Proof.
Suppose a Vandermonde matrix is equal norm and tight. Note that a zero base
will produce the zeroth identity basis element $\delta_{0}$. Letting
$\mathcal{P}$ denote the indices of the nonzero bases, the fact that the
matrix is full rank implies $|\mathcal{P}|\geq M-1$. Also, equal norm gives
that the frame element length
$\|\varphi_{n}\|^{2}=\sum_{m=0}^{M-1}|\varphi_{n}[m]|^{2}=\sum_{m=0}^{M-1}|\alpha_{n}^{m}|^{2}=\sum_{m=0}^{M-1}|\alpha_{n}|^{2m}$
is constant over $n\in\mathcal{P}$. Since $\sum_{m=0}^{M-1}x^{2m}$ is strictly
increasing over $0<x<\infty$, there exists $c>0$ such that
$|\alpha_{n}|^{2}=c$ for all $n\in\mathcal{P}$. Next, tightness gives that the
rows have equal norm, implying that the first two rows have equal norm, i.e.,
$|\mathcal{P}|c=|\mathcal{P}|c^{2}$. Thus $c=1$, and so the nonzero bases are
in the complex unit circle. Furthermore, since the zeroth and first rows have
equal norm by tightness, we have $|\mathcal{P}|=N$, and so every base is in
the complex unit circle.
Now consider the inner product between Vandermonde frame elements whose bases
$\\{e^{2\pi ix_{n}}\\}_{n=1}^{N}$ come from the complex unit circle:
$\langle\varphi_{n},\varphi_{n^{\prime}}\rangle=\sum_{m=0}^{M-1}(e^{2\pi
ix_{n}})^{m}\overline{(e^{2\pi ix_{n^{\prime}}})^{m}}=\sum_{m=0}^{M-1}e^{2\pi
i(x_{n}-x_{n^{\prime}})m}.$
We will show that the worst-case coherence comes from the two closest bases.
Consider the following function:
$g(x):=\bigg{|}\sum_{m=0}^{M-1}e^{2\pi ixm}\bigg{|}^{2}.$ (2.4)
Figure 2.1 gives a plot of this function in the case where $M=5$. We will
prove two things about this function:
* (i)
$\tfrac{d}{dx}g(x)<0$ for every $x\in(0,\tfrac{1}{2M})$,
* (ii)
$g(x)\leq g(\tfrac{1}{2M})$ for every $x\in(\tfrac{1}{2M},1-\tfrac{1}{2M})$.
Figure 2.1: Plot of $g$ defined by (2.4) in the case where $M=5$. Observe (i)
that $g$ is strictly decreasing on the interval $(0,\frac{1}{10})$, and (ii)
that $g(x)\leq g(\frac{1}{10})$ for every $x\in(\frac{1}{10},\frac{9}{10})$.
As established in the proof of Theorem 13, $g$ behaves in this manner for
general values of $M$.
First, we claim that (i) and (ii) are sufficient to prove our result. To
establish this, we first show that the two closest bases $e^{2\pi
ix_{n^{\prime}}}$ and $e^{2\pi ix_{n^{\prime\prime}}}$ satisfy
$|x_{n^{\prime}}-x_{n^{\prime\prime}}|\leq\frac{1}{2M}$. Without loss of
generality, the $n$’s are ordered in such a way that
$\\{x_{n}\\}_{n=0}^{N-1}\subseteq[0,1)$ are nondecreasing. Define
$d(x_{n},x_{n+1}):=\left\\{\begin{array}[]{ll}x_{n+1}-x_{n},&n=0,\ldots,N-2\\\
x_{0}-(x_{N-1}-1),&n=N-1,\end{array}\right.$
and let $n^{\prime}$ be the $n$ which minimizes $d(x_{n},x_{n+1})$. Since the
minimum is less than the average, we have
$d(x_{n^{\prime}},x_{n^{\prime}+1})\leq\frac{1}{N}\bigg{(}(x_{0}-(x_{N-1}-1))+\sum_{n=0}^{N-1}(x_{n+1}-x_{n})\bigg{)}=\frac{1}{N}\leq\frac{1}{2M},$
(2.5)
provided $N\geq 2M$. Note that if we view $\\{x_{n}\\}_{n\in\mathbb{Z}_{N}}$
as members of $\mathbb{R}/\mathbb{Z}$, then $d(x_{n},x_{n+1})=x_{n+1}-x_{n}$.
Since $g(x)$ is even, then (i) implies that
$|\langle\varphi_{n^{\prime}+1},\varphi_{n^{\prime}}\rangle|^{2}=g(x_{n^{\prime}+1}-x_{n^{\prime}})$
is larger than any other
$g(x_{p}-x_{p^{\prime}})=|\langle\varphi_{p},\varphi_{p^{\prime}}\rangle|^{2}$
in which $x_{p}-x_{p^{\prime}}\in[0,\tfrac{1}{2M}]\cup[1-\tfrac{1}{2M},1)$.
Next, (2.5) and (ii) together imply that
$|\langle\varphi_{n^{\prime}+1},\varphi_{n^{\prime}}\rangle|^{2}=g(x_{n^{\prime}+1}-x_{n^{\prime}})\geq
g(\tfrac{1}{2M})$ is larger than any other
$g(x_{p}-x_{p^{\prime}})=|\langle\varphi_{p},\varphi_{p^{\prime}}\rangle|^{2}$
in which $x_{p}-x_{p^{\prime}}\in(\tfrac{1}{2M},1-\tfrac{1}{2M})$, provided
$N\geq 2M$. Combined, (i) and (ii) give that
$|\langle\varphi_{n^{\prime}+1},\varphi_{n^{\prime}}\rangle|$ achieves the
worst-case coherence of $\\{\varphi_{n}\\}_{n\in\mathbb{Z}_{N}}$.
Additionally, (i) gives that the worst-case coherence
$|\langle\varphi_{n^{\prime}+1},\varphi_{n^{\prime}}\rangle|$ is minimized
when $x_{n^{\prime}+1}-x_{n^{\prime}}$ is maximized, i.e., when the $x_{n}$’s
are equally spaced in the unit interval.
To prove (i), note that the geometric sum formula gives
$g(x)=\bigg{|}\sum_{m=0}^{M-1}e^{2\pi ixm}\bigg{|}^{2}=\bigg{|}\frac{e^{2M\pi
ix}-1}{e^{2\pi ix}-1}\bigg{|}^{2}=\frac{2-2\cos(2M\pi x)}{2-2\cos(2\pi
x)}=\bigg{(}\frac{\sin(M\pi x)}{\sin(\pi x)}\bigg{)}^{2},$ (2.6)
where the final expression uses the identity $1-\cos(2z)=2\sin^{2}z$. To show
that $g$ is decreasing over $(0,\frac{1}{2M})$, note that the base of (2.6) is
positive on this interval, and performing the quotient rule to calculate its
derivative will produce a fraction whose denominator is nonnegative and whose
numerator is given by
$M\pi\sin(\pi x)\cos(M\pi x)-\pi\sin(M\pi x)\cos(\pi x).$ (2.7)
This factor is zero at $x=0$ and has derivative:
$-(M^{2}-1)\pi^{2}\sin(\pi x)\sin(M\pi x),$
which is strictly negative for all $x\in(0,\tfrac{1}{2M})$. Hence, (2.7) is
strictly negative whenever $x\in(0,\tfrac{1}{2M})$, and so $g^{\prime}(x)<0$
for every $x\in(0,\tfrac{1}{2M})$.
For (ii), note that for every $x\in(\frac{1}{2M},1-\frac{1}{2M})$, we can
individually bound the numerator and denominator of what the geometric sum
formula gives:
$g(x)=\bigg{|}\sum_{m=0}^{M-1}e^{2\pi ixm}\bigg{|}^{2}=\frac{|e^{2M\pi
ix}-1|^{2}}{|e^{2\pi ix}-1|^{2}}\leq\frac{|e^{\pi i}-1|^{2}}{|e^{\pi
i/M}-1|^{2}}=\bigg{|}\sum_{m=0}^{M-1}e^{\pi
im/M}\bigg{|}^{2}=g(\tfrac{1}{2M}).\qed$
Consider the $N\times N$ discrete Fourier transform (DFT) matrix, scaled to
have entries of unit modulus:
$\begin{bmatrix}1&1&1&\cdots&1\\\ 1&\omega&\omega^{2}&\cdots&\omega^{N-1}\\\
1&\omega^{2}&\omega^{4}&\cdots&\omega^{2(N-1)}\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
1&\omega^{N-1}&\omega^{2(N-1)}&\cdots&\omega^{(N-1)(N-1)}\end{bmatrix},$
where $\omega=e^{-2\pi i/N}$. The first $M$ rows of the DFT form a Vandermonde
matrix of distinct bases $\\{\omega^{n}\\}_{n=0}^{N-1}$; as such, this matrix
is full spark by Lemma 12. In fact, the previous result says that this is in
some sense an optimal Vandermonde frame, but this might not be the best way to
pick rows from a DFT. Indeed, several choices of DFT rows could produce full
spark frames, some with smaller coherence or other desirable properties, and
so the remainder of this section focuses on full spark DFT submatrices. First,
we note that not every DFT submatrix is full spark. For example, consider the
$4\times 4$ DFT:
$\begin{bmatrix}1&1&1&1\\\ 1&-i&-1&i\\\ 1&-1&1&-1\\\ 1&i&-1&-i\end{bmatrix}.$
Certainly, the zeroth and second rows of this matrix are not full spark, since
the zeroth and second columns of this submatrix form the all-ones matrix,
which is not invertible. So what can be said about the set of permissible row
choices? The following result gives some necessary conditions on this set:
###### Theorem 14.
Take an $N\times N$ discrete Fourier transform matrix, and select the rows
indexed by $\mathcal{M}\subseteq\mathbb{Z}_{N}$ to build the matrix $\Phi$. If
$\Phi$ is full spark, then so is the matrix built from rows indexed by
* (i)
any translation of $\mathcal{M}$,
* (ii)
any $A\mathcal{M}$ with $A$ relatively prime to $N$,
* (iii)
the complement of $\mathcal{M}$ in $\mathbb{Z}_{N}$.
###### Proof.
For (i), we first define $D$ to be the $N\times N$ diagonal matrix whose
diagonal entries are $\\{\omega^{n}\\}_{n=0}^{N-1}$. Note that, since
$\omega^{(m+1)n}=\omega^{n}\omega^{mn}$, translating the row indices
$\mathcal{M}$ by $1$ corresponds to multiplying $\Phi$ on the right by $D$.
For some set $\mathcal{K}\subseteq\mathbb{Z}_{N}$ of size $M:=|\mathcal{M}|$,
let $\Phi_{\mathcal{K}}$ denote the $M\times M$ submatrix of $\Phi$ whose
columns are indexed by $\mathcal{K}$, and let $D_{\mathcal{K}}$ denote the
$M\times M$ diagonal submatrix of $D$ whose diagonal entries are indexed by
$\mathcal{K}$. Then since $D_{\mathcal{K}}$ is unitary, we have
$|\mathrm{det}((\Phi
D)_{\mathcal{K}})|=|\mathrm{det}(\Phi_{\mathcal{K}}D_{\mathcal{K}})|=|\mathrm{det}(\Phi_{\mathcal{K}})||\mathrm{det}(D_{\mathcal{K}})|=|\mathrm{det}(\Phi_{\mathcal{K}})|.$
Thus, if $\Phi$ is full spark, $|\mathrm{det}((\Phi
D)_{\mathcal{K}})|=|\mathrm{det}(\Phi_{\mathcal{K}})|>0$, and so $\Phi D$ is
also full spark. Using this fact inductively proves (i) for all translations
of $\mathcal{M}$.
For (ii), let $\Psi$ denote the submatrix of rows indexed by $A\mathcal{M}$.
Then for any $\mathcal{K}\subseteq\mathbb{Z}_{N}$ of size $M$,
$\mathrm{det}(\Psi_{\mathcal{K}})=\mathrm{det}(\omega^{(Am)k})_{m\in\mathcal{M},k\in\mathcal{K}}=\mathrm{det}(\omega^{m(Ak)})_{m\in\mathcal{M},k\in\mathcal{K}}=\mathrm{det}(\Phi_{A\mathcal{K}}).$
Since $A$ is relatively prime to $N$, multiplication by $A$ permutes the
elements of $\mathbb{Z}_{N}$, and so $A\mathcal{K}$ has exactly $M$ distinct
elements. Thus, if $\Phi$ is full spark, then
$\mathrm{det}(\Psi_{\mathcal{K}})=\mathrm{det}(\Phi_{A\mathcal{K}})\neq 0$,
and so $\Psi$ is also full spark.
For (iii), we let $\Psi$ be the $(N-M)\times N$ submatrix of rows indexed by
$\mathcal{M}^{\mathrm{c}}$, so that
$NI_{N}=\begin{bmatrix}\Phi^{*}&\Psi^{*}\end{bmatrix}\begin{bmatrix}\Phi\\\
\Psi\end{bmatrix}=\Phi^{*}\Phi+\Psi^{*}\Psi.$ (2.8)
We will use contraposition to show that $\Phi$ being full spark implies that
$\Psi$ is also full spark. To this end, suppose $\Psi$ is not full spark. Then
$\Psi$ has a collection of $N-M$ linearly dependent columns
$\\{\psi_{i}\\}_{i\in\mathcal{K}}$, and so there exists a nontrivial sequence
$\\{\alpha_{i}\\}_{i\in\mathcal{K}}$ such that
$\sum_{i\in\mathcal{K}}\alpha_{i}\psi_{i}=0.$
Considering $\psi_{i}=\Psi\delta_{i}$, where $\delta_{i}$ is the $i$th
identity basis element, we can use (2.8) to express this linear dependence in
terms of $\Phi$:
$0=\Psi^{*}0=\Psi^{*}\sum_{i\in\mathcal{K}}\alpha_{i}\psi_{i}=\sum_{i\in\mathcal{K}}\alpha_{i}\Psi^{*}\Psi\delta_{i}=\sum_{i\in\mathcal{K}}\alpha_{i}(NI_{N}-\Phi^{*}\Phi)\delta_{i}.$
Rearranging then gives
$x:=N\sum_{i\in\mathcal{K}}\alpha_{i}\delta_{i}=\sum_{i\in\mathcal{K}}\alpha_{i}\Phi^{*}\Phi\delta_{i}.$
(2.9)
Here, we note that $x$ is nonzero since $\\{\alpha_{i}\\}_{i\in\mathcal{K}}$
is nontrivial, and that $x\in\mathrm{Range}(\Phi^{*}\Phi)$. Furthermore,
whenever $j\not\in\mathcal{K}$, we have from (2.9) that
$\langle x,\Phi^{*}\Phi\delta_{j}\rangle=\langle\Phi^{*}\Phi
x,\delta_{j}\rangle=N\bigg{\langle}\Phi^{*}\Phi\sum_{i\in\mathcal{K}}\alpha_{i}\delta_{i},\delta_{j}\bigg{\rangle}=N^{2}\bigg{\langle}\sum_{i\in\mathcal{K}}\alpha_{i}\delta_{i},\delta_{j}\bigg{\rangle}=0,$
and so
$x\perp\mathrm{Span}\\{\Phi^{*}\Phi\delta_{j}\\}_{j\in\mathcal{K}^{\mathrm{c}}}$.
Thus, the containment
$\mathrm{Span}\\{\Phi^{*}\Phi\delta_{j}\\}_{j\in\mathcal{K}^{\mathrm{c}}}\subseteq\mathrm{Range}(\Phi^{*}\Phi)$
is proper, and so
$M=\mathrm{Rank}(\Phi)=\mathrm{Rank}(\Phi^{*}\Phi)>\mathrm{Rank}(\Phi^{*}\Phi_{\mathcal{K}^{\mathrm{c}}})=\mathrm{Rank}(\Phi_{\mathcal{K}^{\mathrm{c}}}).$
Since the $M\times M$ submatrix $\Phi_{\mathcal{K}^{\mathrm{c}}}$ is rank-
deficient, it is not invertible, and therefore $\Phi$ is not full spark. ∎
We note that our proof of (iii) above uses techniques from Cahill et al. [32],
and can be easily generalized to prove that the Naimark complement of a full
spark tight frame is also full spark. Theorem 14 tells us quite a bit about
the set of permissible choices for DFT rows. For example, not only can we pick
the first $M$ rows of the DFT to produce a full spark Vandermonde frame, but
we can also pick any consecutive $M$ rows, by Theorem 14(i). We would like to
completely characterize the choices that produce full spark harmonic frames.
The following classical result does this in the case where $N$ is prime:
###### Theorem 15 (Chebotarëv, see [126]).
Let $N$ be prime. Then every square submatrix of the $N\times N$ discrete
Fourier transform matrix is invertible.
As an immediate consequence of Chebotarëv’s theorem, every choice of rows from
the DFT produces a full spark harmonic frame, provided $N$ is prime. This
application of Chebotarëv’s theorem was first used by Candès et al. [36] for
sparse signal processing. Note that each of these frames are equal-norm and
tight by construction. Harmonic frames can also be designed to have minimal
coherence; Xia et al. [146] produces harmonic equiangular tight frames by
selecting row indices which form a difference set in $\mathbb{Z}_{N}$.
Interestingly, most known families of difference sets in $\mathbb{Z}_{N}$
require $N$ to be prime [87], and so the corresponding harmonic equiangular
tight frames are guaranteed to be full spark by Chebotarëv’s theorem. In the
following, we use Chebotarëv’s theorem to demonstrate full spark for a class
of frames which contains harmonic frames, namely, frames which arise from
concatenating harmonic frames with any number of identity basis elements:
###### Theorem 16 (cf. [131, Theorem 1.1]).
Let $N$ be prime, and pick any $M\leq N$ rows of the $N\times N$ discrete
Fourier transform matrix to form the harmonic frame $H$. Next, pick any $K\leq
M$, and take $D$ to be the $M\times M$ diagonal matrix whose first $K$
diagonal entries are $\sqrt{\frac{N+K-M}{MN}}$, and whose remaining $M-K$
entries are $\sqrt{\frac{N+K}{MN}}$. Then concatenating $DH$ with the first
$K$ identity basis elements produces an $M\times(N+K)$ full spark unit norm
tight frame.
As an example, when $N=5$ and $K=1$, we can pick $M=3$ rows of the $5\times 5$
DFT which are indexed by $\\{0,1,4\\}$. In this case, $D$ makes the entries of
the first DFT row have size $\sqrt{\frac{1}{5}}$ and the entries of the
remaining rows have size $\sqrt{\frac{2}{5}}$. Concatenating with the first
identity basis element then produces an equiangular tight frame which is full
spark:
$\Phi=\left[\begin{array}[]{llllll}\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&1\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&0\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&0\\\
\end{array}\right].$ (2.10)
###### Proof of Theorem 16.
Let $\Phi$ denote the resulting $M\times(N+K)$ frame. We start by verifying
that $\Phi$ is unit norm. Certainly, the identity basis elements have unit
norm. For the remaining frame elements, the modulus of each entry is
determined by $D$, and so the norm squared of each frame element is
$K(\tfrac{N+K-M}{MN})+(M-K)(\tfrac{N+K}{MN})=1.$
To demonstrate that $\Phi$ is tight, it suffices to show that
$\Phi\Phi^{*}=\frac{N+K}{M}I_{M}$. The rows of $DH$ are orthogonal since they
are scaled rows of the DFT, while the rows of the identity portion are
orthogonal because they have disjoint support. Thus, $\Phi\Phi^{*}$ is
diagonal. Moreover, the norm squared of each of the first $K$ rows is
$N(\frac{N+K-M}{MN})+1=\frac{N+K}{M}$, while the norm squared of each of the
remaining rows is $N(\frac{N+K}{MN})=\frac{N+K}{M}$, and so
$\Phi\Phi^{*}=\frac{N+K}{M}I_{M}$.
To show that $\Phi$ is full spark, note that every $M\times M$ submatrix of
$DH$ is invertible since
$|\mathrm{det}((DH)_{\mathcal{K}})|=|\mathrm{det}(DH_{\mathcal{K}})|=|\mathrm{det}(D)||\mathrm{det}(H_{\mathcal{K}})|>0,$
by Chebotarëv’s theorem. Also, in the case where $K=M$, we note that the
$M\times M$ submatrix of $\Phi$ composed solely of identity basis elements is
trivially invertible. The only remaining case to check is when identity basis
elements and columns of $DH$ appear in the same $M\times M$ submatrix
$\Phi_{\mathcal{K}}$. In this case, we may shuffle the rows of
$\Phi_{\mathcal{K}}$ to have the form
$\begin{bmatrix}A&0\\\ B&I_{K}\end{bmatrix}.$
Since shuffling rows has no impact on the size of the determinant, we may
further use a determinant identity on block matrices to get
$|\mathrm{det}(\Phi_{\mathcal{K}})|=\left|\mathrm{det}\begin{bmatrix}A&0\\\
B&I_{K}\end{bmatrix}\right|=|\mathrm{det}(A)\mathrm{det}(I_{K})|=|\mathrm{det}(A)|.$
Since $A$ is a multiple of a square submatrix of the $N\times N$ DFT, we are
done by Chebotarëv’s theorem. ∎
As an example of Theorem 16, pick $N$ to be a prime congruent to $1\bmod 4$,
and select $\frac{N+1}{2}$ rows of the $N\times N$ DFT according to the index
set $\mathcal{M}:=\\{k^{2}:k\in\mathbb{Z}_{N}\\}$. If we take $K=1$, the
process in Theorem 16 produces an equiangular tight frame of redundancy $2$,
which we will verify in the next chapter using quadratic Gauss sums; in the
case where $N=5$, this construction produces (2.10). Note that this
corresponds to a special case of a construction in Zauner’s thesis [150],
which was later studied by Renes [115] and Strohmer [128]. Theorem 16 says
that this construction is full spark.
Maximally sparse frames have recently become a subject of active research [44,
70]. We note that when $K=M$, Theorem 16 produces a maximally sparse
$M\times(N+K)$ full spark frame, having a total of $M(M-1)$ zero entries. To
see that this sparsity level is maximal, we note that if the frame had any
more zero entries, then at least one of the rows would have $M$ zero entries,
meaning the corresponding $M\times M$ submatrix would have a row of all zeros
and hence a zero determinant. Similar ideas were studied previously by
Nakamura and Masson [107].
Another interesting case is where $K=M=N$, i.e., when the frame constructed in
Theorem 16 is a union of the unitary DFT and identity bases. Unions of
orthonormal bases have received considerable attention in the context of
sparse approximation [61, 136]. In fact, when $N$ is a perfect square,
concatenating the DFT with an identity basis forms the canonical example
$\Phi$ of a dictionary with small spark [61], and we used this example in the
previous chapter. Recall the Dirac comb of $\sqrt{N}$ spikes is an eigenvector
of the DFT, and so concatenating this comb with the negative of its Fourier
transform produces a $2\sqrt{N}$-sparse vector in the nullspace of $\Phi$. In
stark contrast, when $N$ is prime, Theorem 16 shows that $\Phi$ is full spark.
The vast implications of Chebotarëv’s theorem leads one to wonder whether the
result admits any interesting generalization. In this direction, Candès et al.
[36] note that any such generalization must somehow account for the nontrivial
subgroups of $\mathbb{Z}_{N}$ which are not present when $N$ is prime.
Certainly, if one could characterize the full spark submatrices of a general
DFT, this would provide ample freedom to optimize full spark frames for
additional considerations. While we do not have a characterization for the
general case, we do have one for the case where $N$ is a prime power. Before
stating the result, we require a definition:
###### Definition 17.
We say a subset $\mathcal{M}\subseteq\mathbb{Z}_{N}$ is _uniformly distributed
over the divisors of $N$_ if, for every divisor $d$ of $N$, the $d$ cosets of
$\langle d\rangle$ partition $\mathcal{M}$ into subsets, each of size
$\lfloor\frac{|\mathcal{M}|}{d}\rfloor$ or
$\lceil\frac{|\mathcal{M}|}{d}\rceil$.
At first glance, this definition may seem rather unnatural, but we will
discover some important properties of uniformly distributed rows from the DFT.
As an example, we briefly consider uniform distribution in the context of the
restricted isometry property (RIP). Recall that a matrix of random rows from a
DFT and normalized columns is RIP with high probability [118]. We will show
that harmonic frames satisfy RIP only if the selected row indices are nearly
uniformly distributed over sufficiently small divisors of $N$.
To this end, recall that for any divisor $d$ of $N$, the Fourier transform of
the $d$-sparse normalized Dirac comb
$\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}$ is the
$\frac{N}{d}$-sparse normalized Dirac comb $\sqrt{\frac{d}{N}}\chi_{\langle
d\rangle}$. Let $F$ be the $N\times N$ unitary DFT, and let $\Phi$ be the
harmonic frame which arises from selecting rows of $F$ indexed by
$\mathcal{M}$ and then normalizing the columns. In order for $\Phi$ to be
$(K,\delta)$-RIP, $\mathcal{M}$ must contain at least one member of $\langle
d\rangle$ for every divisor $d$ of $N$ which is $\leq K$, since otherwise
$\Phi\tfrac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}=\sqrt{\tfrac{N}{|\mathcal{M}|}}(F\tfrac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle})_{\mathcal{M}}=\sqrt{\tfrac{N}{|\mathcal{M}|}}\Big{(}\sqrt{\tfrac{d}{N}}\chi_{\langle
d\rangle}\Big{)}_{\mathcal{M}}=\sqrt{\tfrac{d}{|\mathcal{M}|}}\chi_{\mathcal{M}\cap\langle
d\rangle}=0,$
which violates the lower RIP bound at
$x=\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}$. In fact, the RIP
bounds indicate that
$\|\Phi
x\|^{2}=\|\Phi\tfrac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}\|^{2}=\Big{\|}\sqrt{\tfrac{d}{|\mathcal{M}|}}\chi_{\mathcal{M}\cap\langle
d\rangle}\Big{\|}^{2}=\tfrac{d}{|\mathcal{M}|}|\mathcal{M}\cap\langle
d\rangle|$
cannot be more than $\delta$ away from $\|x\|^{2}=1$. Similarly, taking $x$ to
be $\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}$ modulated by $a$,
i.e., $x[n]:=\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}[n]e^{2\pi
ian/N}$ for every $n\in\mathbb{Z}_{N}$, gives that $\|\Phi
x\|^{2}=\frac{d}{|\mathcal{M}|}|\mathcal{M}\cap(a+\langle d\rangle)|$ is also
no more than $\delta$ away from $1$. This observation gives the following
result:
###### Theorem 18.
Select rows indexed by $\mathcal{M}\subseteq\mathbb{Z}_{N}$ from the $N\times
N$ discrete Fourier transform matrix and then normalize the columns to produce
the harmonic frame $\Phi$. Then $\Phi$ satisfies the $(K,\delta)$-restricted
isometry property only if
$\Big{|}\big{|}\mathcal{M}\cap(a+\langle
d\rangle)\big{|}-\tfrac{|\mathcal{M}|}{d}\Big{|}\leq\tfrac{|\mathcal{M}|}{d}\delta$
for every divisor $d$ of $N$ with $d\leq K$ and every $a=0,\ldots,d-1$.
Now that we have an intuition for uniform distribution in terms of modulated
Dirac combs and RIP, we take this condition to the extreme by considering
uniform distribution over all divisors. Doing so produces a complete
characterization of full spark harmonic frames when $N$ is a prime power:
###### Theorem 19.
Let $N$ be a prime power, and select rows indexed by
$\mathcal{M}\subseteq\mathbb{Z}_{N}$ from the $N\times N$ discrete Fourier
transform matrix to build the submatrix $\Phi$. Then $\Phi$ is full spark if
and only if $\mathcal{M}$ is uniformly distributed over the divisors of $N$.
Note that, perhaps surprisingly, an index set $\mathcal{M}$ can be uniformly
distributed over $p$ but not over $p^{2}$, and vice versa. For example,
$\mathcal{M}=\\{0,1,4\\}$ is uniformly distributed over $2$ but not $4$, while
$\mathcal{M}=\\{0,2\\}$ is uniformly distributed over $4$ but not $2$.
Since the first $M$ rows of a DFT form a full spark Vandermonde matrix, let’s
check that this index set is uniformly distributed over the divisors of $N$.
For each divisor $d$ of $N$, we partition the first $M$ indices into the $d$
cosets of $\langle d\rangle$. Write $M=qd+r$ with $0\leq r<d$. The first $qd$
of the $M$ indices are distributed equally amongst all $d$ cosets, and then
the remaining $r$ indices are distributed equally amongst the first $r$
cosets. Overall, the first $r$ cosets contain
$q+1=\lfloor\frac{M}{d}\rfloor+1$ indices, while the remaining $d-r$ cosets
have $q=\lfloor\frac{M}{d}\rfloor$ indices; thus, the first $M$ indices are
indeed uniformly distributed over the divisors of $N$. Also, when $N$ is
prime, _every_ subset of $\mathbb{Z}_{N}$ is uniformly distributed over the
divisors of $N$ in a trivial sense. In fact, Chebotarëv’s theorem follows
immediately from Theorem 19. In some ways, portions of our proof of Theorem 19
mirror recurring ideas in the existing proofs of Chebotarëv’s theorem [59, 67,
126, 131]. For the sake of completeness, we provide the full argument and save
the reader from having to parse portions of proofs from multiple references.
We start with the following lemmas, whose proofs are based on the proofs of
Lemmas 1.2 and 1.3 in [131].
###### Lemma 20.
Let $N$ be a power of some prime $p$, and let $P(z_{1},\ldots,z_{M})$ be a
polynomial with integer coefficients. Suppose there exists $N$th roots of
unity $\\{\omega_{m}\\}_{m=1}^{M}$ such that
$P(\omega_{1},\ldots,\omega_{M})=0$. Then $P(1,\ldots,1)$ is a multiple of
$p$.
###### Proof.
Denoting $\omega:=e^{-2\pi i/N}$, then for every $m=1,\ldots,M$, we have
$\omega_{m}=\omega^{k_{m}}$ for some $0\leq k_{m}<N$. Defining the polynomial
$Q(z):=P(z^{k_{1}},\ldots,z^{k_{M}})$, we have $Q(\omega)=0$ by assumption.
Also, $Q(z)$ is a polynomial with integer coefficients, and so it must be
divisible by the minimal polynomial of $\omega$, namely, the cyclotomic
polynomial $\Phi_{N}(z)$. Evaluating both polynomials at $z=1$ then gives that
$p=\Phi_{N}(1)$ divides $Q(1)=P(1,\ldots,1)$. ∎
###### Lemma 21.
Let $N$ be a power of some prime $p$, and pick
$\mathcal{M}=\\{m_{i}\\}_{i=1}^{M}\subseteq\mathbb{Z}_{N}$ such that
$\frac{\displaystyle{\prod_{1\leq i<j\leq
M}(m_{j}-m_{i})}}{\displaystyle{\prod_{m=0}^{M-1}m!}}$ (2.11)
is not a multiple of $p$. Then the rows indexed by $\mathcal{M}$ in the
$N\times N$ discrete Fourier transform form a full spark frame.
###### Proof.
We wish to show that $\mathrm{det}(\omega_{n}^{m})_{m\in\mathcal{M},1\leq
n\leq M}\neq 0$ for all $M$-tuples of distinct $N$th roots of unity
$\\{\omega_{n}\\}_{n=1}^{M}$. Define the polynomial
$D(z_{1},\ldots,z_{M}):=\mathrm{det}(z_{n}^{m})_{m\in\mathcal{M},1\leq n\leq
M}$. Since columns $i$ and $j$ of $(z_{n}^{m})_{m\in\mathcal{M},1\leq n\leq
M}$ are identical whenever $z_{i}=z_{j}$, we know that $D$ vanishes in each of
these instances, and so we can factor:
$D(z_{1},\ldots,z_{M})=P(z_{1},\ldots,z_{M})\prod_{1\leq i<j\leq
M}(z_{j}-z_{i})$
for some polynomial $P(z_{1},\ldots,z_{M})$ with integer coefficients. By
Lemma 20, it suffices to show that $P(1,\ldots,1)$ is not a multiple of $p$,
since this implies $D(\omega_{1},\ldots,\omega_{M})$ is nonzero for all
$M$-tuples of distinct $N$th roots of unity $\\{\omega_{n}\\}_{n=1}^{M}$.
To this end, we proceed by considering
$A:=\bigg{(}z_{1}\frac{\partial}{\partial
z_{1}}\bigg{)}^{0}\bigg{(}z_{2}\frac{\partial}{\partial
z_{2}}\bigg{)}^{1}\cdots\bigg{(}z_{M}\frac{\partial}{\partial
z_{M}}\bigg{)}^{M-1}D(z_{1},\ldots,z_{M})\bigg{|}_{z_{1}=\cdots=z_{M}=1}.$
(2.12)
To compute $A$, we note that each application of
$z_{j}\frac{\partial}{\partial z_{j}}$ produces terms according to the product
rule. For some terms, a linear factor of the form $z_{j}-z_{i}$ or
$z_{i}-z_{j}$ is replaced by $z_{j}$ or $-z_{j}$, respectively. For each the
other terms, these linear factors are untouched, while another factor, such as
$P(z_{1},\ldots,z_{M})$, is differentiated and multiplied by $z_{j}$. Note
that there are a total of $M(M-1)/2$ linear factors, and only $M(M-1)/2$
differentiation operators to apply. Thus, after expanding every product rule,
there will be two types of terms: terms in which every differentiation
operator was applied to a linear factor, and terms which have at least one
linear factor remaining untouched. When we evaluate at $z_{1}=\cdots=z_{M}=1$,
the terms with linear factors vanish, and so the only terms which remain came
from applying every differentiation operator to a linear factor. Furthermore,
each of these terms before the evaluation is of the form
$P(z_{1},\ldots,z_{M})\prod_{1\leq i<j\leq M}z_{j}$, and so evaluation at
$z_{1}=\cdots=z_{M}=1$ produces a sum of terms of the form $P(1,\ldots,1)$; to
determine the value of $A$, it remains to count these terms. The $M-1$ copies
of $z_{M}\frac{\partial}{\partial z_{M}}$ can only be applied to linear
factors of the form $z_{M}-z_{i}$, of which there are $M-1$, and so there are
a total of $(M-1)!$ ways to distribute these operators. Similarly, there are
$(M-2)!$ ways to distribute the $M-2$ copies of
$z_{M-1}\frac{\partial}{\partial z_{M-1}}$ amongst the $M-2$ linear factors of
the form $z_{M-1}-z_{i}$. Continuing in this manner produces an expression for
$A$:
$A=(M-1)!(M-2)!\cdots 1!0!~{}P(1,\ldots,1).$ (2.13)
For an alternate expression of $A$, we substitute the definition of
$D(z_{1},\ldots,z_{M})$ into $\eqref{eq.A defn}$. Here, we exploit the
multilinearity of the determinant and the fact that
$(z_{n}\frac{\partial}{\partial z_{n}})z_{n}^{m}=mz_{n}^{m}$ to get
$A=\mathrm{det}(m^{n-1})_{m\in\mathcal{M},1\leq n\leq M}=\prod_{1\leq i<j\leq
M}(m_{j}-m_{i}),$ (2.14)
where the final equality uses the fact that $(m^{n-1})_{m\in\mathcal{M},1\leq
n\leq M}$ is the transpose of a Vandermonde matrix. Equating (2.13) to (2.14)
reveals that (2.11) is an expression for $P(1,\ldots,1)$. Thus, by assumption,
$P(1,\ldots,1)$ is not a multiple of $p$, and so we are done. ∎
###### Proof of Theorem 19.
($\Leftarrow$) We will use Lemma 21 to demonstrate that $\Phi$ is full spark.
To apply this lemma, we need to establish that (2.11) is not a multiple of
$p$, and to do this, we will show that there are as many $p$-divisors in the
numerator of (2.11) as there are in the denominator. We start by counting the
$p$-divisors of the denominator:
$\prod_{m=0}^{M-1}m!=\prod_{m=1}^{M-1}\prod_{\ell=1}^{m}\ell=\prod_{\ell=1}^{M-1}\prod_{m=1}^{M-l}\ell.$
(2.15)
For each pair of integers $k,a\geq 1$, there are $\max\\{M-ap^{k},~{}0\\}$
factors in (2.15) of the form $\ell=ap^{k}$. By adding these, we count each
factor $\ell$ as many times as it can be expressed as a multiple of a power of
$p$, which equals the number of $p$-divisors in $\ell$. Thus, the number of
$p$-divisors of (2.15) is
$\sum_{k=1}^{\lfloor\log_{p}M\rfloor}\sum_{a=1}^{\lfloor\frac{M}{p^{k}}\rfloor}(M-ap^{k}).$
(2.16)
Next, we count the $p$-divisors of the numerator of (2.11). To do this, we use
the fact that $\mathcal{M}$ is uniformly distributed over the divisors of $N$.
Since $N$ is a power of $p$, the only divisors of $N$ are smaller powers of
$p$. Also, the cosets of $\langle p^{k}\rangle$ partition $\mathcal{M}$ into
subsets $S_{k,b}:=\\{m_{i}\equiv b\mod p^{k}\\}$. We note that $m_{j}-m_{i}$
is a multiple of $p^{k}$ precisely when $m_{i}$ and $m_{j}$ belong to the same
subset $S_{k,b}$ for some $0\leq b<p^{k}$. To count $p$-divisors, we again
count each factor $m_{j}-m_{i}$ as many times as it can be expressed as a
multiple of a prime power:
$\sum_{k=1}^{\lfloor\log_{p}M\rfloor}\sum_{b=0}^{p^{k}-1}\binom{|S_{k,b}|}{2}.$
(2.17)
Write $M=qp^{k}+r$ with $0\leq r<p^{k}$. Then
$q=\lfloor\frac{M}{p^{k}}\rfloor$. Since $\mathcal{M}$ is uniformly
distributed over $p^{k}$, there are $r$ subsets $S_{k,b}$ with $q+1$ elements
and $p^{k}-r$ subsets with $q$ elements. We use this to get
$\sum_{b=0}^{p^{k}-1}\binom{|S_{k,b}|}{2}=\binom{q+1}{2}r+\binom{q}{2}(p^{k}-r)=\frac{q}{2}\Big{(}(q-1)p^{k}+2r+(qp^{k}-qp^{k})\Big{)}.$
Rearranging and substituting $M=qp^{k}+r$ then gives
$\sum_{b=0}^{p^{k}-1}\binom{|S_{k,b}|}{2}=\frac{q}{2}\Big{(}2M-(q+1)p^{k}\Big{)}=Mq-\binom{q+1}{2}p^{k}=\sum_{a=1}^{\lfloor\frac{M}{p^{k}}\rfloor}(M-ap^{k}).$
Thus, there are as many $p$-divisors in the numerator (2.17) as there are in
the denominator (2.16), and so (2.11) is not divisible by $p$. Lemma 21
therefore gives that $\Phi$ is full spark.
($\Rightarrow$) We will prove that this direction holds regardless of whether
$N$ is a prime power. Suppose $\mathcal{M}\subseteq\mathbb{Z}_{N}$ is not
uniformly distributed over the divisors of $N$. Then there exists a divisor
$d$ of $N$ such that one of the cosets of $\langle d\rangle$ intersects
$\mathcal{M}$ with $\leq\lfloor\frac{M}{d}\rfloor-1$ or
$\geq\lceil\frac{M}{d}\rceil+1$ indices. Notice that if a coset of $\langle
d\rangle$ intersects $\mathcal{M}$ with $\leq\lfloor\frac{M}{d}\rfloor-1$
indices, then the complement $\mathcal{M}^{\mathrm{c}}$ intersects the same
coset with
$\geq\lceil\frac{N-M}{d}\rceil+1=\lceil\frac{|\mathcal{M}^{\mathrm{c}}|}{d}\rceil+1$
indices. By Theorem 14(iii), $\mathcal{M}$ produces a full spark harmonic
frame precisely when $\mathcal{M}^{\mathrm{c}}$ produces a full spark harmonic
frame, and so we may assume without loss of generality that there exists a
coset of $\langle d\rangle$ which intersects $\mathcal{M}$ with
$\geq\lceil\frac{M}{d}\rceil+1$ indices.
To prove that the rows with indices in $\mathcal{M}$ are not full spark, we
find column entries which produce a singular submatrix. Writing $M=qd+r$ with
$0\leq r<d$, let $\mathcal{K}$ contain $q=\lfloor\frac{M}{d}\rfloor$ cosets of
$\langle\frac{N}{d}\rangle$ along with $r$ elements from an additional coset.
We claim that the DFT submatrix with row entries $\mathcal{M}$ and column
entries $\mathcal{K}$ is singular. To see this, shuffle the rows and columns
to form a matrix $A$ in which the row entries are grouped into common cosets
of $\langle d\rangle$ and the column entries are grouped into common cosets of
$\langle\frac{N}{d}\rangle$. This breaks $A$ into rank-1 submatrices: each
pair of cosets $a+\langle d\rangle$ and $b+\langle\frac{N}{d}\rangle$ produces
a submatrix
$(\omega^{(a+id)(b+j\frac{N}{d})})_{i\in\mathcal{I},j\in\mathcal{J}}=\omega^{ab}(\omega^{bdi}\omega^{a\frac{N}{d}j})_{i\in\mathcal{I},j\in\mathcal{J}}$
for some index sets $\mathcal{I}$ and $\mathcal{J}$; this is a rank-1 outer
product. Let $\mathcal{L}$ be the largest intersection between $\mathcal{M}$
and a coset of $\langle d\rangle$. Then
$|\mathcal{L}|\geq\lceil\frac{M}{d}\rceil+1$ is the number of rows in the
tallest of these rank-1 submatrices. Define $A_{\mathcal{L}}$ to be the
$M\times M$ matrix with entries $A_{\mathcal{L}}[i,j]=A[i,j]$ whenever
$i\in\mathcal{L}$ and zero otherwise. Then
$\mathrm{Rank}(A)=\mathrm{Rank}(A_{\mathcal{L}}+A-A_{\mathcal{L}})\leq\mathrm{Rank}(A_{\mathcal{L}})+\mathrm{Rank}(A-A_{\mathcal{L}}).$
(2.18)
Since $A-A_{\mathcal{L}}$ has $|\mathcal{L}|$ rows of zero entries, we also
have
$\mathrm{Rank}(A-A_{\mathcal{L}})\leq M-|\mathcal{L}|\leq
M-(\lceil\tfrac{M}{d}\rceil+1).$ (2.19)
Moreover, since we can decompose $A_{\mathcal{L}}$ into a sum of
$\lceil\frac{M}{d}\rceil$ zero-padded rank-1 submatrices, we have
$\mathrm{Rank}(A_{\mathcal{L}})\leq\lceil\frac{M}{d}\rceil$. Combining this
with (2.18) and (2.19) then gives that $\mathrm{Rank}(A)\leq M-1$, and so the
DFT submatrix is not invertible. ∎
Note that our proof of Theorem 19 establishes the necessity of having row
indices uniformly distributed over the divisors of $N$ in the general case.
This leaves some hope for completely characterizing full spark harmonic
frames. Naturally, one might suspect that the uniform distribution condition
is sufficient in general, but this suspicion fails when $N=10$. Indeed, the
following DFT submatrix is singular despite the row indices being uniformly
distributed over the divisors of $10$:
$(e^{-2\pi imn/10})_{m\in\\{0,1,3,4\\},n\in\\{0,1,2,6\\}}.$
Just as we used Chebotarëv’s theorem to analyze the harmonic equiangular tight
frames from Xia et al. [146], we can also use Theorem 19 to determine whether
harmonic equiangular tight frames with a prime power number of frame elements
are full spark. Unfortunately, none of the infinite families in [146] have the
number of frame elements in the form of a prime power (other than primes).
Luckily, there is at least one instance in which the number of frame elements
happens to be a prime power: the harmonic frames that arise from Singer
difference sets have $M=\frac{q^{d}-1}{q-1}$ and $N=\frac{q^{d+1}-1}{q-1}$ for
a prime power $q$ and an integer $d\geq 2$; when $q=3$ and $d=4$, the number
of frame elements $N=11^{2}$ is a prime power. In this case, the row indices
we select are
$\displaystyle\mathcal{M}=$
$\displaystyle\\{1,2,3,6,7,9,11,18,20,21,25,27,33,34,38,41,44,47,53,54,55,56,$
$\displaystyle~{}~{}~{}58,59,60,63,64,68,70,71,75,81,83,89,92,99,100,102,104,114\\},$
but these are not uniformly distributed over 11, and so the corresponding
harmonic frame is not full spark by Theorem 19.
### 2.2 The computational complexity of verifying full spark
In the previous section, we constructed a large collection of deterministic
full spark frames. To see how special these constructions are, we consider the
following question: How much computation is required to check whether any
given frame is full spark? At the heart of the matter is computational
complexity theory, which provides a rigorous playing field for expressing how
hard certain problems are. In this section, we consider the complexity of the
following problem:
###### Problem 22 (Full Spark).
Given a matrix, is it full spark?
For the lay mathematician, Full Spark is “obviously” ${\mathsf{NP}}$-hard
because the easiest way he can think to solve it for a given $M\times N$
matrix is by determining whether each of the $M\times M$ submatrices is
invertible; computing $\binom{N}{M}$ determinants would do, but this would
take a lot of time, and so Full Spark must be ${\mathsf{NP}}$-hard. However,
computing $\binom{N}{M}$ determinants may not necessarily be the fastest way
to test whether a matrix is full spark. For example, perhaps there is an easy-
to-calculate expression for the product of the determinants; after all, this
product is nonzero precisely when the matrix is full spark. Recall that
Theorem 19 gives a very straightforward litmus test for Full Spark in the
special case where the matrix is formed by rows of a DFT of prime-power
order—who’s to say that a version of this test does not exist for the general
case? If such a test exists, then it would suffice to find it, but how might
one disprove the existence of any such test? Indeed, since we are concerned
with the necessary amount of computation, as opposed to a sufficient amount,
the lay mathematician’s intuition is a bit misguided.
To discern how much computation is necessary, the main feature of interest is
a problem’s _complexity_. We use complexity to compare problems and determine
whether one is harder than the other. As an example of complexity,
intuitively, doubling an integer is no harder than adding integers, since one
can use addition to multiply by $2$; put another way, the complexity of
doubling is somehow “encoded” in the complexity of adding, and so it must be
lesser (or equal). To make this more precise, complexity theorists use what is
called a _polynomial-time reduction_ , that is, a polynomial-time algorithm
that solves problem $A$ by exploiting an oracle which solves problem $B$; the
reduction indicates that solving problem $A$ is no harder than solving problem
$B$ (up to polynomial factors in time), and we say “$A$ reduces to $B$,” or
$A\leq B$. Since we can use the polynomial-time routine $x+x$ to produce $2x$,
we conclude that doubling an integer reduces to adding integers, as expected.
In complexity theory, problems are categorized into complexity classes
according to the amount of resources required to solve them. For example, the
complexity class ${\mathsf{P}}$ contains all problems which can be solved in
polynomial time, while problems in ${\mathsf{EXP}}$ may require as much as
exponential time. Problems in ${\mathsf{NP}}$ have the defining quality that
solutions can be verified in polynomial time given a certificate for the
answer. As an example, the graph isomorphism problem is in ${\mathsf{NP}}$
because, given an isomorphism between graphs (a certificate), one can verify
that the isomorphism is legit in polynomial time. Clearly,
${\mathsf{P}}\subseteq{\mathsf{NP}}$, since we can ignore the certificate and
still solve the problem in polynomial time. Finally, a problem $B$ is called
${\mathsf{NP}}$-_hard_ if every problem $A$ in ${\mathsf{NP}}$ reduces to $B$,
and a problem is called ${\mathsf{NP}}$-_complete_ if it is both
${\mathsf{NP}}$-hard and in ${\mathsf{NP}}$. In plain speak,
${\mathsf{NP}}$-hard problems are harder than every problem in
${\mathsf{NP}}$, while ${\mathsf{NP}}$-complete problems are the hardest of
problems in ${\mathsf{NP}}$.
At this point, it should be clear that ${\mathsf{NP}}$-hard problems are not
merely problems that seem to require a lot of computation to solve. Certainly,
${\mathsf{NP}}$-hard problems have this quality, as an ${\mathsf{NP}}$-hard
problem can be solved in polynomial time only if ${\mathsf{P}}={\mathsf{NP}}$;
this is an open problem, but it is widely believed that
${\mathsf{P}}\neq{\mathsf{NP}}$. However, there are other problems which seem
hard but are not known to be ${\mathsf{NP}}$-hard (e.g., the graph isomorphism
problem). Rather, to determine whether a problem is ${\mathsf{NP}}$-hard, one
must find a polynomial-time reduction that compares the problem to all
problems in ${\mathsf{NP}}$. To this end, notice that $A\leq B$ and $B\leq C$
together imply $A\leq C$, and so to demonstrate that a problem $C$ is
${\mathsf{NP}}$-hard, it suffices to show that $B\leq C$ for some
${\mathsf{NP}}$-hard problem $B$.
Unfortunately, it can sometimes be difficult to find a deterministic reduction
from one problem to another. One example is reducing the satisfiability
problem (SAT) to the unique satisfiability problem (Unique SAT). To be clear,
SAT is an ${\mathsf{NP}}$-hard problem [89] that asks whether there exists an
input for which a given Boolean function returns “true,” while Unique SAT asks
the same question with an additional promise: that the given Boolean function
is satisfiable only if there is a _unique_ input for which it returns “true.”
Intuitively, Unique SAT is easier than SAT because we might be able to exploit
the additional structure of uniquely satisfiable Boolean functions; thus, it
could be difficult to find a reduction from SAT to Unique SAT. Despite this
intuition, there is a _randomized_ polynomial-time reduction from SAT to
Unique SAT [138]. Defined over all Boolean functions of $n$ variables, the
reduction maps functions that are not satisfiable to other functions that are
not satisfiable, and with probability $\geq\frac{1}{8n}$, it maps satisfiable
functions to uniquely satisfiable functions. After applying this reduction to
a given Boolean function, if a Unique SAT oracle declares “uniquely
satisfiable,” then we know for certain that the original Boolean function was
satisfiable. But the reduction will only map a satisfiable problem to a
uniquely satisfiable problem with probability $\geq\frac{1}{8n}$, so what good
is this reduction? The answer lies in something called _amplification_ ; since
the success probability is, at worst, polynomially small in $n$ (i.e.,
$\geq\frac{1}{p(n)}$), we can repeat our oracle-based randomized algorithm a
polynomial number of times $np(n)$ and achieve an error probability
$\leq(1-\frac{1}{p(n)})^{np(n)}\sim e^{-n}$ which is exponentially small.
In this section, we give a randomized polynomial-time reduction from a problem
in matroid theory. Before stating the problem, we first briefly review some
definitions. To each bipartite graph with bipartition $(E,E^{\prime})$, we
associate a _transversal matroid_ $(E,\mathcal{I})$, where $\mathcal{I}$ is
the collection of subsets of $E$ whose vertices form the ends of a matching in
the bipartite graph; subsets in $\mathcal{I}$ are called independent. Next,
just as spark is the size of the smallest linearly dependent set, the _girth_
of a matroid is the size of the smallest subset of $E$ that is not in
$\mathcal{I}$. In fact, this analogy goes deeper: A matroid is _representable
over a field_ $\mathbb{F}$ if, for some $M$, there exists a mapping
$\varphi\colon E\rightarrow\mathbb{F}^{M}$ such that $\varphi(A)$ is linearly
independent if and only if $A\in\mathcal{I}$; as such, the girth of
$(E,\mathcal{I})$ is the spark of $\varphi(E)$. In our reduction, we make use
of the fact that every transversal matroid is representable over $\mathbb{R}$
[112]. We are now ready to state the problem from which we will reduce Full
Spark:
###### Problem 23.
Given a bipartite graph, what is the girth of its transversal matroid?
Before giving the reduction, we note that Problem 23 is ${\mathsf{NP}}$-hard.
This is demonstrated in McCormick’s thesis [100], which credits the proof to
Stockmeyer; since [100] is difficult to access, we refer the reader to [2]. We
now turn to the main result of this section; note that our proof is
specifically geared toward the case where the matrix in question has integer
entries—this is stronger than manipulating real (complex) numbers exactly as
well as with truncations and tolerances.
###### Theorem 24.
Full Spark is hard for ${\mathsf{NP}}$ under randomized polynomial-time
reductions.
###### Proof.
We will give a randomized polynomial-time reduction from Problem 23 to Full
Spark. As such, suppose we are given a bipartite graph $G$, in which every
edge is between the disjoint sets $A$ and $B$. Take $M:=|B|$ and $N:=|A|$.
Using this graph, we randomly draw an $M\times N$ matrix $\Phi$ using the
following process: for each $i\in B$ and $j\in A$, pick the entry $\Phi_{ij}$
randomly from $\\{1,\ldots,N2^{N+1}\\}$ if $i\leftrightarrow j$ in $G$;
otherwise set $\Phi_{ij}=0$. In Proposition 3.11 of [99], it is shown that the
columns of $\Phi$ form a representation of the transversal matroid of $G$ with
probability $\geq\frac{1}{2}$. For the moment, we assume that $\Phi$ succeeds
in representing the matroid.
Since the girth of the original matroid equals the spark of its
representation, for each $K=1,\ldots,M$, we test whether
$\mathrm{Spark}(\Phi)>K$. To do this, take $H$ to be some $M\times P$ full
spark frame. We will determine an appropriate value for $P$ later, but for
simplicity, we can take $H$ to be the Vandermonde matrix formed from bases
$\\{1,\ldots,P\\}$; see Lemma 12. We claim we can randomly select $K$ indices
$\mathcal{K}\subseteq\\{1,\ldots,P\\}$ and test whether
$H_{\mathcal{K}}^{*}\Phi$ is full spark to determine whether
$\mathrm{Spark}(\Phi)>K$. Moreover, after performing this test for each
$K=1,\ldots,M$, the probability of incorrectly determining
$\mathrm{Spark}(\Phi)$ is $\leq\frac{1}{2}$, provided $P$ is sufficiently
large.
We want to test whether $H_{\mathcal{K}}^{*}\Phi$ is full spark and use the
result as a proxy for whether $\mathrm{Spark}(\Phi)>K$. For this to work, we
need to have $\mathrm{Rank}(H_{\mathcal{K}}^{*}\Phi_{\mathcal{K}^{\prime}})=K$
precisely when $\mathrm{Rank}(\Phi_{\mathcal{K}^{\prime}})=K$ for every
$\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ of size $K$. To this end, it
suffices to have the nullspace $\mathcal{N}(H_{\mathcal{K}}^{*})$ of
$H_{\mathcal{K}}^{*}$ intersect trivially with the column space of
$\Phi_{\mathcal{K}^{\prime}}$ for every $\mathcal{K}^{\prime}$. To be clear,
it is always the case that
$\mathrm{Rank}(H_{\mathcal{K}}^{*}\Phi_{\mathcal{K}^{\prime}})\leq\mathrm{Rank}(\Phi_{\mathcal{K}^{\prime}})$,
and so $\mathrm{Rank}(\Phi_{\mathcal{K}^{\prime}})<K$ implies
$\mathrm{Rank}(H_{\mathcal{K}}^{*}\Phi_{\mathcal{K}^{\prime}})<K$. If we
further assume that
$\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})=\\{0\\}$,
then the converse also holds. To see this, suppose
$\mathrm{Rank}(H_{\mathcal{K}}^{*}\Phi_{\mathcal{K}^{\prime}})<K$. Then by the
rank-nullity theorem, there is a nontrivial
$x\in\mathcal{N}(H_{\mathcal{K}}^{*}\Phi_{\mathcal{K}^{\prime}})$. Since
$H_{\mathcal{K}}^{*}\Phi_{\mathcal{K}^{\prime}}x=0$, we must have
$\Phi_{\mathcal{K}^{\prime}}x\in\mathcal{N}(H_{\mathcal{K}}^{*})$, which in
turn implies $x\in\mathcal{N}(\Phi_{\mathcal{K}^{\prime}})$ since
$\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})=\\{0\\}$
by assumption. Thus, $\mathrm{Rank}(\Phi_{\mathcal{K}^{\prime}})<K$ by the
rank-nullity theorem.
Now fix $\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ of size $K$ such that
$\mathrm{Rank}(\Phi_{\mathcal{K}^{\prime}})=K$. We will show that the vast
majority of choices $\mathcal{K}\subseteq\\{1,\ldots,P\\}$ of size $K$ satisfy
$\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})=\\{0\\}$.
To do this, we consider the columns $\\{h_{k}\\}_{k\in\mathcal{K}}$ of
$H_{\mathcal{K}}$ one at a time, and we make use of the fact that
$\mathcal{N}(H_{\mathcal{K}}^{*})=\bigcap_{k\in\mathcal{K}}\mathcal{N}(h_{k}^{*})$.
In particular, since $H$ is full spark, there are at most $M-K$ columns of $H$
in the orthogonal complement of $\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})$,
and so there are at least $P-(M-K)$ choices of $h_{k_{1}}$ for which
$\mathcal{N}(h_{k_{1}}^{*})$ does not contain
$\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})$, i.e.,
$\mathrm{dim}\Big{(}\mathcal{N}(h_{k_{1}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})\Big{)}=K-1.$
Similarly, after selecting the first $J$ $h_{k}$’s, we have
$\mathrm{dim}(S)=K-J$, where
$S:=\bigcap_{j=1}^{J}\mathcal{N}(h_{k_{j}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}}).$
Again, since $H$ is full spark, there are at most $M-(K-J)$ columns of $H$ in
the orthogonal complement of $S$, and so the remaining $P-(M-(K-J))$ columns
are candidates for $h_{k_{J+1}}$ that give
$\mathrm{dim}\bigg{(}\bigcap_{j=1}^{J+1}\mathcal{N}(h_{k_{j}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})\bigg{)}=\mathrm{dim}\Big{(}\mathcal{N}(h_{k_{J+1}}^{*})\cap
S\Big{)}=K-(J+1).$
Overall, if we randomly pick $\mathcal{K}\subseteq\\{1,\ldots,P\\}$ of size
$K$, then
$\displaystyle\mathrm{Pr}\Big{(}\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})=\\{0\\}\Big{)}$
$\displaystyle\geq(1-\tfrac{M-K}{P})(1-\tfrac{M-(K-1)}{P})\cdots(1-\tfrac{M-1}{P})$
$\displaystyle\geq(1-\tfrac{M}{P})^{K}$ $\displaystyle\geq 1-\tfrac{MK}{P},$
where the final step is by Bernoulli’s inequality. Taking a union bound over
all choices of $\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ and all values
of $K=1,\ldots,M$ then gives
$\displaystyle\mathrm{Pr}\bigg{(}\mbox{fail to determine
$\mathrm{Spark}(\Phi)$}\bigg{)}$
$\displaystyle\leq\sum_{K=1}^{M}\binom{N}{K}\mathrm{Pr}\Big{(}\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(\Phi_{\mathcal{K}^{\prime}})\neq\\{0\\}\Big{)}$
$\displaystyle\leq\sum_{K=1}^{M}\binom{N}{K}\frac{MK}{P}$
$\displaystyle\leq\frac{M^{3}2^{N}}{P}.$
Thus, to make the probability of failure $\leq\frac{1}{2}$, it suffices to
have $P=M^{3}2^{N+1}$.
In summary, we succeed in representing the original matroid with probability
$\geq\frac{1}{2}$, and then we succeed in determining the spark of its
representation with probability $\geq\frac{1}{2}$. The probability of overall
success is therefore $\geq\frac{1}{4}$. Since our success probability is, at
worst, polynomially small, we can apply amplification to achieve an
exponentially small error probability. ∎
Our use of random linear projections in the above reduction to Full Spark is
similar in spirit to Valiant and Vazirani’s use of random hash functions in
their reduction to Unique SAT [138]. Since their randomized reduction is the
canonical example thereof, we find our reduction to be particularly natural.
To conclude this section, we clarify that Theorem 24 is a statement about the
amount of computation necessary in the _worst case_. Indeed, the hardness of
Full Spark does not rule out the existence of smaller classes of matrices for
which full spark is easily determined. As an example, Theorem 19 determines
Full Spark in the special case where the matrix is formed by rows of a DFT of
prime-power order. This illustrates the utility of applying additional
structure to efficiently solve the Full Spark problem, and indeed, such
classes of matrices are rather special for this reason.
### 2.3 Phaseless recovery with polarization
In the previous sections, we constructed deterministic full spark frames and
showed that checking for full spark in general is computationally hard. In
this section, we provide a new technique for phaseless recovery which makes
use of full spark frames in the measurement design. We are particularly
interested in using the fewest measurements necessary for recovery, namely
$N=\mathrm{O}(M)$, where $M$ is the dimension of the signal [15].
Take a finite set $V$, and suppose we take phaseless measurements of
$x\in\mathbb{C}^{M}$ with a frame $\Phi_{V}:=\\{\varphi_{i}\\}_{i\in
V}\subseteq\mathbb{C}^{M}$ with the task of recovering $x$ up to a global
phase factor. For notational convenience, we take $\sim$ to be the equivalence
relation of being identical up to a global phase factor, and we say $y$ is a
member of the equivalence class $[x]\in\mathbb{C}^{M}/\\!\\!\sim$ if $y\sim
x$. Having $|\langle x,\varphi_{i}\rangle|$ for every $i\in V$, we claim it
suffices to determine the relative phase between all pairs of frame
coefficients. If we had this information, we could arbitrarily assign some
nonzero frame coefficient $c_{i}=|\langle x,\varphi_{i}\rangle|$ to have
positive phase. If $\langle x,\varphi_{j}\rangle$ is also nonzero, then it has
well-defined relative phase
$\omega_{ij}:=\big{(}\tfrac{\langle x,\varphi_{i}\rangle}{|\langle
x,\varphi_{i}\rangle|}\big{)}^{-1}\tfrac{\langle
x,\varphi_{j}\rangle}{|\langle x,\varphi_{j}\rangle|},$
which determines the frame coefficent by multiplication:
$c_{j}=\omega_{ij}|\langle x,\varphi_{j}\rangle|$. Otherwise when $\langle
x,\varphi_{j}\rangle=0$, we naturally take $c_{j}=0$, and for notational
convenience, we arbitrarily take $\omega_{ij}=1$. From here,
$[x]\in\mathbb{C}^{M}/\\!\\!\sim$ can be identified by applying the canonical
dual frame $\\{\tilde{\varphi}_{j}\\}_{j\in V}$ of $\Phi_{V}$:
$\sum_{j\in V}c_{j}\tilde{\varphi}_{j}=\sum_{j\in V}\omega_{ij}|\langle
x,\varphi_{j}\rangle|\tilde{\varphi}_{j}=\big{(}\tfrac{\langle
x,\varphi_{i}\rangle}{|\langle x,\varphi_{i}\rangle|}\big{)}^{-1}\sum_{j\in
V}\langle x,\varphi_{j}\rangle\tilde{\varphi}_{j}=\big{(}\tfrac{\langle
x,\varphi_{i}\rangle}{|\langle x,\varphi_{i}\rangle|}\big{)}^{-1}x\in[x].$
To find the relative phase between frame coefficients, we turn to the
polarization identity:
$\overline{\langle x,\varphi_{i}\rangle}\langle
x,\varphi_{j}\rangle=\frac{1}{4}\sum_{k=0}^{3}\mathrm{i}^{k}\big{|}\langle
x,\varphi_{i}\rangle+\mathrm{i}^{-k}\langle
x,\varphi_{j}\rangle\big{|}^{2}=\frac{1}{4}\sum_{k=0}^{3}\mathrm{i}^{k}\big{|}\langle
x,\varphi_{i}+\mathrm{i}^{k}\varphi_{j}\rangle\big{|}^{2}.$
Thus, if in addition to $\Phi_{V}$, we measure with
$\\{\varphi_{i}+\mathrm{i}^{k}\varphi_{j}\\}_{k=0}^{3}$, we can use the above
calculation to determine $\overline{\langle x,\varphi_{i}\rangle}\langle
x,\varphi_{j}\rangle$ and then normalize to get the relative phase
$\omega_{ij}$, provided both $\langle x,\varphi_{i}\rangle$ and $\langle
x,\varphi_{j}\rangle$ are nonzero. To summarize, if we measure with $\Phi_{V}$
and $\\{\varphi_{i}+\mathrm{i}^{k}\varphi_{j}\\}_{k=0}^{3}$ for every pair
$i,j\in V$, then we can recover $[x]$. However, such a method uses
$|V|+4\binom{|V|}{2}$ measurements, and since $\Phi_{V}$ is a frame, we
necessarily have $|V|\geq M$ and thus a total of $\Omega(M^{2})$ measurements.
In pursuit of $\mathrm{O}(M)$ measurements, take some simple graph $G=(V,E)$,
and only take measurements with $\Phi_{V}$ and $\Phi_{E}:=\bigcup_{(i,j)\in
E}\\{\varphi_{i}+\mathrm{i}^{k}\varphi_{j}\\}_{k=0}^{3}$. To recover $[x]$, we
again arbitrarily assign some nonzero vertex measurement to have positive
phase, and then we propagate relative phase information along the edges by
multiplication to determine the phase of the other vertex measurements
relative to the original vertex measurement. However, if $x$ is orthogonal to
a given vertex vector, then that measurement is zero, and so relative phase
information cannot propagate through the corresponding vertex; indeed, such
orthogonality has the effect of removing the vertex from the graph, and for
some graphs, this will prevent recovery. For example, if $G$ is a star, then
$x$ could be orthogonal to the vector corresponding to the internal vertex,
whose removal would render the remaining graph edgeless. That said, we should
select $\Phi_{V}$ and $G$ so as to minimize the impact of orthogonality with
vertex vectors.
First, we can take $\Phi_{V}$ to be full spark so that every subcollection of
$M$ frame elements spans. This implies that $x$ is orthogonal to at most $M-1$
members of $\Phi_{V}$, thereby limiting the extent of $x$’s damage to our
graph. Additionally, $\Phi_{V}$ being full spark frees us from requiring the
graph to be connected after the removal of vertices; indeed, any remaining
component of size $M$ or more will correspond to a subframe of $\Phi_{V}$ that
necessarily has a dual frame to reconstruct with. It remains to find a graph
of $\mathrm{O}(M)$ vertices and edges that maintains a size-$M$ component
after the removal of any $M-1$ vertices.
To this end, we consider a well-studied family of sparse graphs known as
_expander graphs_. We choose these graphs for their notably strong
connectivity properties. There is a combinatorial definition of expander
graphs, but we will focus on the spectral definition. Given a $d$-regular
graph $G$ of $n$ vertices, consider the eigenvalues of its adjacency matrix:
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$. We say $G$ has
_expansion_ $\lambda(G):=\frac{1}{d}\max\\{|\lambda_{2}|,|\lambda_{n}|\\}$.
Furthermore, a family of $d$-regular graphs $\\{G_{i}\\}_{i=1}^{\infty}$ is a
_spectral expander family_ if there exists $c<1$ such that every $G_{i}$ has
expansion $\lambda(G_{i})\leq c$. Since $d$ is constant over an expander
family, we see that expanders with many vertices are particularly sparse.
There are many results which describe the connectivity of expanders, but the
following is particularly relevant to our application:
###### Lemma 25 ([78]).
Consider a $d$-regular graph $G$ of $n$ vertices with spectral expansion
$\leq\lambda$. For all $\varepsilon\leq\frac{1-\lambda}{6}$, removing any
$\varepsilon dn$ edges from $G$ results in a connected component of size
$\geq(1-\frac{2\varepsilon}{1-\lambda})n$.
For our application, removing $\varepsilon n$ vertices from a $d$-regular
graph necessarily removes $\leq\varepsilon dn$ edges, and so this lemma
directly applies. Also,
$\varepsilon\leq\frac{1-\lambda}{6}<\frac{1}{6}<\frac{2}{3}\leq
1-\frac{2\varepsilon}{1-\lambda},$
where the last inequality is a rearrangement of
$\varepsilon\leq\frac{1-\lambda}{6}$. Since we want to guarantee that the
removal of any $M-1$ vertices maintains a size-$M$ component, we must
therefore take $M\leq\varepsilon n+1$. Overall, we use the following criteria
to pick our expander graph: Given the signal dimension $M$, use a $d$-regular
graph $G=(V,E)$ of $n$ vertices with spectral expansion $\lambda$ such that
$M\leq(\frac{1-\lambda}{6})n+1$. Then by the previous discussion, the total
number of measurements is $N=|V|+4|E|=(2d+1)n$. We wish to find choices of
graphs which yield only $N=\mathrm{O}(M)$ measurements.
To minimize the redundancy $\frac{N}{M}$, we see that for a fixed degree $d$,
we would like minimal spectral expansion $\lambda$. Spectral graph families
known as _Ramanujan graphs_ are asymptotically optimal in this sense; taking
$\mathcal{G}_{n}^{d}$ to be the set of connected $d$-regular graphs with $\geq
n$ vertices, Alon and Boppana (see [4]) showed that for any fixed $d$,
$\lim_{n\rightarrow\infty}\inf_{G\in\mathcal{G}_{n}^{d}}\lambda(G)\geq\frac{2\sqrt{d-1}}{d},$
while Ramanujan graphs are defined to have spectral expansion
$\leq\frac{2\sqrt{d-1}}{d}$. To date, Ramanujan graphs have only been
constructed for certain values of $d$. One important construction was given by
Lubotzky et al. [98], which produces a Ramanujan family whenever $d-1\equiv
1\bmod 4$ is prime. Among these graphs, we get the smallest redundancy
$\frac{N}{M}$ when $d=6$ and $M=\lfloor(\frac{1-\lambda}{6})n+1\rfloor$:
$\frac{N}{M}\leq\frac{(2d+1)n}{(1-\lambda)n/6}\leq\frac{6d(2d+1)}{d-2\sqrt{d-1}}=\frac{234}{3-\sqrt{5}}\approx
306.31.$
Thus, in such cases, we may perform phaseless recovery with only $N\leq 307M$
measurements. However, the number of vertices in each Ramanujan graph from
[98] is of the form $q(q^{2}-1)$ or $\frac{q(q^{2}-1)}{2}$, where $q\equiv
1\bmod 4$ is prime, and so any bound on redundancy $\frac{N}{M}$ using graphs
from [98] will only be valid for particular values of $M$.
In order to get $N=\mathrm{O}(M)$ in general, we use the fact that random
graphs are nearly Ramanujan with high probability. In particular, for every
$\varepsilon>0$ and even $d$, a random $d$-regular graph has spectral
expansion $\lambda\leq\frac{1}{d}(2\sqrt{d-1}+\varepsilon)$ with high
probability as $n\rightarrow\infty$ [71]. Thus, picking $\varepsilon$ and $d$
to satisfy $\frac{1}{d}(2\sqrt{d-1}+\varepsilon)<1$, we may again take
$M=\lfloor(\frac{1-\lambda}{6})n+1\rfloor$ to get
$\frac{N}{M}\leq\frac{6(2d+1)}{1-\lambda}\leq\frac{6d(2d+1)}{d-(2\sqrt{d-1}+\varepsilon)}$
with high probability. Note that in this case, $n$ can be any sufficiently
large integer, and so the above bound is valid for all sufficiently large $M$,
i.e., our procedure can perform phaseless recovery with $N=\mathrm{O}(M)$
measurements in general.
Note that this section has only considered the case in which the phaseless
measurements were not corrupted by noise. For the noisy case, Candès et al.
[37] used semidefinite programming to stably reconstruct from
$N=\mathrm{O}(M\log M)$ measurements. Our technique also appears to be stable,
and we expect positive results in this vein using synchronization-type
analysis [124]; we leave this for future work.
## Chapter 3 Deterministic matrices with the restricted isometry property
In Chapter 1, we observed how to use the Gershgorin circle theorem to
demonstrate that certain $M\times N$ matrices have the restricted isometry
property (RIP) for sparsity levels $K=\mathrm{O}(\sqrt{M})$. In this chapter,
we consider better demonstration techniques which promise to break this
“square-root bottleneck” [16]. To date, the only deterministic construction
that manages to go beyond the bottleneck is given by Bourgain et al. [29]; in
the following section, we discuss what they call _flat RIP_ , which is the
technique they use to demonstrate RIP. We will see that their technique can be
used to demonstrate RIP for sparsity levels much larger than $\sqrt{M}$,
meaning one could very well demonstrate random-like performance given the
proper construction. Later, we introduce an alternate technique, which can
also demonstrate RIP for large sparsity levels.
After considering the efficacy of these techniques to demonstrate RIP, it
remains to find a deterministic construction that is amenable to analysis. To
this end, we discuss various properties of certain equiangular tight frames
(ETFs). Specifically, real ETFs can be characterized in terms of their Gram
matrices using strongly regular graphs [141]. By applying our demonstration
techniques to real ETFs, we derive equivalent combinatorial statements in
graph theory. By focussing on the ETFs which correspond to Paley graphs of
prime order, we are able to make important statements about their clique
numbers and provide some intuition for an open problem in number theory. We
conclude by conjecturing that the Paley ETFs are RIP in a manner similar to
random matrices.
### 3.1 Flat restricted orthogonality
In [29], Bourgain et al. provided a deterministic construction of $M\times N$
RIP matrices that support sparsity levels $K$ on the order of
$M^{1/2+\varepsilon}$ for some small value of $\varepsilon$. To date, this is
the only known deterministic RIP construction that breaks the square-root
bottleneck. In this section, we analyze their technique for demonstrating RIP,
but first, we provide some historical context. We begin with a definition:
###### Definition 26.
The matrix $\Phi$ has _$(K,\theta)$ -restricted orthogonality (RO)_ if
$|\langle\Phi x,\Phi y\rangle|\leq\theta\|x\|\|y\|$
for every pair of $K$-sparse vectors $x,y$ with disjoint support. The smallest
$\theta$ for which $\Phi$ has $(K,\theta)$-RO is the _restricted orthogonality
constant (ROC)_ $\theta_{K}$.
In the past, restricted orthogonality was studied to produce reconstruction
performance guarantees for both $\ell_{1}$-minimization and the Dantzig
selector [38, 40]. Intuitively, restricted orthogonality is important to
compressed sensing because any stable inversion process for (1) would require
$\Phi$ to map vectors of disjoint support to particularly dissimilar
measurements. For the present chapter, we are interested in upper bounds on
RICs; in this spirit, the following result illustrates some sort of
equivalence between RICs and ROCs:
###### Lemma 27 (Lemma 1.2 in [38]).
$\theta_{K}\leq\delta_{2K}\leq\theta_{K}+\delta_{K}$.
To be fair, the above upper bound on $\delta_{2K}$ does not immediately help
in estimating $\delta_{2K}$, as it requires one to estimate $\delta_{K}$.
Certainly, we may iteratively apply this bound to get
$\delta_{2K}\leq\theta_{K}+\theta_{\lceil K/2\rceil}+\theta_{\lceil
K/4\rceil}+\cdots+\theta_{1}+\delta_{1}\leq(1+\lceil\log_{2}K\rceil)\theta_{K}+\delta_{1}.$
(3.1)
Note that $\delta_{1}$ is particularly easy to calculate:
$\delta_{1}=\max_{n\in\\{1,\ldots,N\\}}\Big{|}\|\varphi_{n}\|^{2}-1\Big{|},$
which is zero when the columns of $\Phi$ have unit norm. In pursuit of a
better upper bound on $\delta_{2K}$, we use techniques from [29] to remove the
log factor from (3.1):
###### Lemma 28.
$\delta_{2K}\leq 2\theta_{K}+\delta_{1}$.
###### Proof.
Given a matrix $\Phi=[\varphi_{1}\cdots\varphi_{N}]$, we want to upper-bound
the smallest $\delta$ for which $(1-\delta)\|x\|^{2}\leq\|\Phi
x\|^{2}\leq(1+\delta)\|x\|^{2}$, or equivalently:
$\delta\geq\Big{|}\|\Phi\tfrac{x}{\|x\|}\|^{2}-1\Big{|}$ (3.2)
for every nonzero $2K$-sparse vector $x$. We observe from (3.2) that we may
take $x$ to have unit norm without loss of generality. Letting $\mathcal{K}$
denote a size-$2K$ set that contains the support of $x$, and letting
$\\{x_{k}\\}_{k\in\mathcal{K}}$ denote the corresponding entries of $x$, the
triangle inequality gives
$\displaystyle\Big{|}\|\Phi x\|^{2}-1\Big{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{K}}x_{i}\varphi_{i},\sum_{j\in\mathcal{K}}x_{j}\varphi_{j}\bigg{\rangle}-1\bigg{|}$
$\displaystyle=\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle+\sum_{i\in\mathcal{K}}\|x_{i}\varphi_{i}\|^{2}-1\bigg{|}$
$\displaystyle\leq\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}+\bigg{|}\sum_{i\in\mathcal{K}}\|x_{i}\varphi_{i}\|^{2}-1\bigg{|}.$
(3.3)
Since $\sum_{i\in\mathcal{K}}|x_{i}|^{2}=1$, the second term of (3.3)
satisfies
$\bigg{|}\sum_{i\in\mathcal{K}}\|x_{i}\varphi_{i}\|^{2}-1\bigg{|}\leq\sum_{i\in\mathcal{K}}|x_{i}|^{2}\Big{|}\|\varphi_{i}\|^{2}-1\Big{|}\leq\sum_{i\in\mathcal{K}}|x_{i}|^{2}\delta_{1}=\delta_{1},$
(3.4)
and so it remains to bound the first term of (3.3). To this end, we note that
for each $i,j\in\mathcal{K}$ with $j\neq i$, the term $\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle$ appears in
$\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{K}\setminus\mathcal{I}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle$
as many times as there are size-$K$ subsets of $\mathcal{K}$ which contain $i$
but not $j$, i.e., $\binom{2K-2}{K-1}$ times. Thus, we use the triangle
inequality and the definition of restricted orthogonality to get
$\displaystyle\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}$
$\displaystyle=\bigg{|}\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{K}\setminus\mathcal{I}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}$
$\displaystyle\leq\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{K}\setminus\mathcal{I}}x_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\theta_{K}\bigg{(}\sum_{i\in\mathcal{I}}|x_{i}|^{2}\bigg{)}^{1/2}\bigg{(}\sum_{j\in\mathcal{K}\setminus\mathcal{I}}|x_{j}|^{2}\bigg{)}^{1/2}.$
At this point, $x$ having unit norm implies
$(\sum_{i\in\mathcal{I}}|x_{i}|^{2})^{1/2}(\sum_{j\in\mathcal{K}\setminus\mathcal{I}}|x_{j}|^{2})^{1/2}\leq\frac{1}{2}$,
and so
$\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}\leq\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\frac{\theta_{K}}{2}=\frac{\binom{2K}{K}}{\binom{2K-2}{K-1}}\frac{\theta_{K}}{2}=\bigg{(}4-\frac{2}{K}\bigg{)}\frac{\theta_{K}}{2}.$
Applying both this and (3.4) to (3.3) gives the result. ∎
Having discussed the relationship between restricted isometry and restricted
orthogonality, we are now ready to introduce the property used in [29] to
demonstrate RIP:
###### Definition 29.
The matrix $\Phi=[\varphi_{1}\cdots\varphi_{N}]$ has _$(K,\hat{\theta})$ -flat
restricted orthogonality_ if
$\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
for every disjoint pair of subsets
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$.
Note that $\Phi$ has $(K,\theta_{K})$-flat restricted orthogonality (FRO) by
taking $x$ and $y$ in Definition 26 to be the characteristic functions
$\chi_{\mathcal{I}}$ and $\chi_{\mathcal{J}}$, respectively. Also to be clear,
_flat restricted orthogonality_ is called _flat RIP_ in [29]; we feel the name
change is appropriate considering the preceeding literature. Moreover, the
definition of flat RIP in [29] required $\Phi$ to have unit-norm columns,
whereas we strengthen the corresponding results so as to make no such
requirement. Interestingly, FRO bears some resemblence to the cut-norm of the
Gram matrix $\Phi^{*}\Phi$, defined as the maximum value of
$|\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\langle\varphi_{i},\varphi_{j}\rangle|$
over _all_ subsets $\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$; the
cut-norm has received some attention recently for the hardness of its
approximation [6]. The following theorem illustrates the utility of flat
restricted orthogonality as an estimate of the RIC:
###### Theorem 30.
A matrix with $(K,\hat{\theta})$-flat restricted orthogonality has a
restricted orthogonality constant $\theta_{K}$ which is $\leq
C\hat{\theta}\log K$, and we may take $C=75$.
Indeed, when combined with Lemma 28, this result gives an upper bound on the
RIC: $\delta_{2K}\leq 2C\hat{\theta}\log K+\delta_{1}$. The noteworthy benefit
of this upper bound is that the problem of estimating singular values of
submatrices is reduced to a combinatorial problem of bounding the coherence of
disjoint sums of columns. Furthermore, this reduction comes at the price of a
mere log factor in the estimate. In [29], Bourgain et al. managed to satisfy
this combinatorial coherence property using techniques from additive
combinatorics. While we will not discuss their construction, we find the proof
of Theorem 30 to be instructive; our proof is valid for all values of $K$ (as
opposed to sufficiently large $K$ in the original [29]), and it has near-
optimal constants where appropriate. The proof can be found in the Appendix.
To reiterate, Bourgain et al. [29] used flat restricted orthogonality to build
the only known deterministic construction of $M\times N$ RIP matrices that
support sparsity levels $K$ on the order of $M^{1/2+\varepsilon}$ for some
small value of $\varepsilon$. We are particularly interested in the efficacy
of FRO as a technique to demonstrate RIP in general. Certainly, [29] shows
that FRO can produce at least an $\varepsilon$ improvement over the Gershgorin
technique discussed in the previous section, but it remains to be seen whether
FRO can do better.
In the remainder of this section, we will show that flat restricted
orthogonality is actually capable of demonstrating RIP with much higher
sparsity levels than indicated by [29]. Hopefully, this realization will spur
further research in deterministic constructions which satisfy FRO. To evaluate
FRO, we investigate how well it performs with random matrices; in doing so, we
give an alternative proof that certain random matrices satisfy RIP with high
probability:
###### Theorem 31.
Construct an $M\times N$ matrix $\Phi$ by drawing each of its entries
independently from a Gaussian distribution with mean zero and variance
$\frac{1}{M}$, take $C$ to be the constant from Theorem 30, and set
$\alpha=0.01$. Then $\Phi$ has $(K,\frac{(1-\alpha)\delta}{2C\log K})$-flat
restricted orthogonality and $\delta_{1}\leq\alpha\delta$, and therefore the
$(2K,\delta)$-restricted isometry property, with high probability provided
$M\geq\frac{33C^{2}}{\delta^{2}}K\log^{2}K\log N$.
In proving this result, we will make use of the following Bernstein
inequality:
###### Theorem 32 (see [23, 148]).
Let $\\{Z_{m}\\}_{m=1}^{M}$ be independent random variables of mean zero with
bounded moments, and suppose there exists $L>0$ such that
$\mathbb{E}|Z_{m}|^{k}\leq\frac{\mathbb{E}|Z_{m}|^{2}}{2}L^{k-2}k!$ (3.5)
for every $k\geq 2$. Then
$\mathrm{Pr}\bigg{[}\sum_{m=1}^{M}Z_{m}\geq
2t\bigg{(}\sum_{m=1}^{M}\mathbb{E}|Z_{m}|^{2}\bigg{)}^{1/2}\bigg{]}\leq
e^{-t^{2}}$ (3.6)
provided
$\displaystyle{t\leq\frac{1}{2L}\bigg{(}\sum_{m=1}^{M}\mathbb{E}|Z_{m}|^{2}\bigg{)}^{1/2}}$.
###### Proof of Theorem 31.
Considering Lemma 28, it suffices to show that $\Phi$ has restricted
orthogonality and that $\delta_{1}$ is sufficiently small. First, to
demonstrate restricted orthogonality, it suffices to demonstrate FRO by
Theorem 30, and so we will ensure that the following quantity is small:
$\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}=\sum_{m=1}^{M}\bigg{(}\sum_{i\in\mathcal{I}}\varphi_{i}[m]\bigg{)}\bigg{(}\sum_{j\in\mathcal{J}}\varphi_{j}[m]\bigg{)}.$
(3.7)
Notice that $X_{m}:=\sum_{i\in\mathcal{I}}\varphi_{i}[m]$ and
$Y_{m}:=\sum_{j\in\mathcal{J}}\varphi_{j}[m]$ are mutually independent over
all $m=1,\ldots,M$ since $\mathcal{I}$ and $\mathcal{J}$ are disjoint. Also,
$X_{m}$ is Gaussian with mean zero and variance $\frac{|\mathcal{I}|}{M}$,
while $Y_{m}$ similarly has mean zero and variance $\frac{|\mathcal{J}|}{M}$.
Viewed this way, (3.7) being small corresponds to the sum of independent
random variables $Z_{m}:=X_{m}Y_{m}$ having its probability measure
concentrated at zero. To this end, Theorem 32 is naturally applicable, as the
absolute central moments of a Gaussian random variable $X$ with mean zero and
variance $\sigma^{2}$ are well known:
$\mathbb{E}|X|^{k}=\left\\{\begin{array}[]{rl}\sqrt{\frac{2}{\pi}}\sigma^{k}(k-1)!!&\mbox{
if $k$ odd},\\\ \sigma^{k}(k-1)!!&\mbox{ if $k$ even}.\end{array}\right.$
Since $Z_{m}=X_{m}Y_{m}$ is a product of independent Gaussian random
variables, this gives
$\mathbb{E}|Z_{m}|^{k}=\mathbb{E}|X_{m}|^{k}~{}\mathbb{E}|Y_{m}|^{k}\leq\Big{(}\frac{|\mathcal{I}|}{M}\Big{)}^{k/2}\Big{(}\frac{|\mathcal{J}|}{M}\Big{)}^{k/2}\Big{(}(k-1)!!\Big{)}^{2}\leq\bigg{(}\frac{(|\mathcal{I}||\mathcal{J}|)^{1/2}}{M}\bigg{)}^{k}k!.$
Further since
$\mathbb{E}|Z_{m}|^{2}=\frac{|\mathcal{I}||\mathcal{J}|}{M^{2}}$, we may
define $L:=2\frac{(|\mathcal{I}||\mathcal{J}|)^{1/2}}{M}$ to get (3.5). Later,
we will take $\hat{\theta}<\delta<\sqrt{2}-1<\frac{1}{2}$. Considering
$t:=\frac{\hat{\theta}\sqrt{M}}{2}<\frac{\sqrt{M}}{4}=\frac{1}{2L}\Big{(}M\frac{|\mathcal{I}||\mathcal{J}|}{M^{2}}\Big{)}^{1/2}=\frac{1}{2L}\bigg{(}\sum_{m=1}^{M}\mathbb{E}|Z_{m}|^{2}\bigg{)}^{1/2},$
we therefore have (3.6), which in this case has the form
$\mathrm{Pr}\Bigg{[}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}\geq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}\Bigg{]}\leq
2e^{-M\hat{\theta}^{2}/4},$
where the probability is doubled due to the symmetric distribution of
$\sum_{m=1}^{M}Z_{m}$. Since we need to account for all possible choices of
$\mathcal{I}$ and $\mathcal{J}$, we will perform a union bound. The total
number of choices is given by
$\sum_{|\mathcal{I}|=1}^{K}\sum_{|\mathcal{J}|=1}^{K}\binom{N}{|\mathcal{I}|}\binom{N-|\mathcal{I}|}{|\mathcal{J}|}\leq
K^{2}\binom{N}{K}^{2}\leq N^{2K},$
and so the union bound gives
$\mathrm{Pr}\Big{[}\mbox{$\Phi$ does not have
$(K,\hat{\theta})$-FRO}\Big{]}\leq
2e^{-M\hat{\theta}^{2}/4}~{}N^{2K}=2\exp\Big{(}-\frac{M\hat{\theta}^{2}}{4}+2K\log
N\Big{)}.$ (3.8)
Thus, Gaussian matrices tend to have FRO, and hence restricted orthogonality
by Theorem 30; this is made more precise below.
Again by Lemma 28, it remains to show that $\delta_{1}$ is sufficiently small.
To this end, we note that $M\|\varphi_{n}\|^{2}$ has chi-squared distribution
with $M$ degrees of freedom, and so we can use another (simpler)
concentration-of-measure result; see Lemma 1 of [95]:
$\mathrm{Pr}\bigg{[}\Big{|}\|\varphi_{n}\|^{2}-1\Big{|}\geq
2\Big{(}\sqrt{\frac{t}{M}}+\frac{t}{M}\Big{)}\bigg{]}\leq 2e^{-t}$
for any $t>0$. Specifically, we pick
$\delta^{\prime}:=2\Big{(}\sqrt{\frac{t}{M}}+\frac{t}{M}\Big{)}\leq\frac{4t}{M},$
and we perform a union bound over the $N$ choices for $\varphi_{n}$:
$\mathrm{Pr}\Big{[}\delta_{1}>\delta^{\prime}\Big{]}\leq
2\exp\Big{(}-\frac{M\delta^{\prime}}{4}+\log N\Big{)}.$ (3.9)
To summarize, Lemma 28, the union bound, Theorem 30, and (3.8) and (3.9) give
$\displaystyle\mathrm{Pr}\Big{[}\delta_{2K}>\delta\Big{]}$
$\displaystyle\leq\mathrm{Pr}\Big{[}\theta_{K}>\frac{(1-\alpha)\delta}{2}\mbox{
or }\delta_{1}>\alpha\delta\Big{]}$
$\displaystyle\leq\mathrm{Pr}\Big{[}\theta_{K}>\frac{(1-\alpha)\delta}{2}\Big{]}+\mathrm{Pr}\Big{[}\delta_{1}>\alpha\delta\Big{]}$
$\displaystyle\leq\mathrm{Pr}\Big{[}\mbox{$\Phi$ does not have
$\displaystyle{\Big{(}K,\frac{(1-\alpha)\delta}{2C\log
K}\Big{)}}$-FRO}\Big{]}+\mathrm{Pr}\Big{[}\delta_{1}>\alpha\delta\Big{]}$
$\displaystyle\leq
2\exp\Big{(}-\frac{M}{4}\Big{(}\frac{(1-\alpha)\delta}{2C\log
K}\Big{)}^{2}+2K\log N\Big{)}+2\exp\Big{(}-\frac{M\alpha\delta}{4}+\log
N\Big{)},$
and so $M\geq\frac{33C^{2}}{\delta^{2}}K\log^{2}K\log N$ gives that $\Phi$ has
$(2K,\delta)$-RIP with high probability. ∎
We note that a version of Theorem 31 also holds for matrices whose entries are
independent Bernoulli random variables taking values $\pm\frac{1}{\sqrt{M}}$
with equal probability. In this case, one can again apply Theorem 32 by
comparing moments with those of the Gaussian distribution; also, a union bound
with $\delta_{1}$ will not be necessary since the columns have unit norm,
meaning $\delta_{1}=0$.
### 3.2 Restricted isometry by the power method
In the previous section, we established the efficacy of flat restricted
orthogonality as a technique to demonstrate RIP. While flat restricted
orthogonality has proven useful in the past [29], future deterministic RIP
constructions might not use this technique. Indeed, it would be helpful to
have other techniques available that demonstrate RIP beyond the square-root
bottleneck. In pursuit of such techniques, we recall that the smallest
$\delta$ for which $\Phi$ is $(K,\delta)$-RIP is given in terms of operator
norms in (1.1). In addition, we notice that for any self-adjoint matrix $A$,
$\|A\|_{2}=\|\lambda(A)\|_{\infty}\leq\|\lambda(A)\|_{p},$
where $\lambda(A)$ denotes the spectrum of $A$ with multiplicities. Let
$A=UDU^{*}$ be the eigenvalue decomposition of $A$. When $p$ is even, we can
express $\|\lambda(A)\|_{p}$ in terms of an easy-to-calculate trace:
$\|\lambda(A)\|_{p}^{p}=\mathrm{Tr}[D^{p}]=\mathrm{Tr}[(UDU^{*})^{p}]=\mathrm{Tr}[A^{p}].$
Combining these ideas with the fact that
$\|\cdot\|_{p}\rightarrow\|\cdot\|_{\infty}$ pointwise leads to the following:
###### Theorem 33.
Given an $M\times N$ matrix $\Phi$, define
$\delta_{K;q}:=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]^{\frac{1}{2q}}.$
Then $\Phi$ has the $(K,\delta_{K;q})$-restricted isometry property for every
$q\geq 1$. Moreover, the restricted isometry constant of $\Phi$ is approached
by these estimates: $\lim_{q\rightarrow\infty}\delta_{K;q}=\delta_{K}$.
Similar to flat restricted orthogonality, this _power method_ has a
combinatorial aspect that prompts one to check every sub-Gram matrix of size
$K$; one could argue that the power method is slightly _less_ combinatorial,
as flat restricted orthogonality is a statement about all pairs of disjoint
subsets of size $\leq K$. Regardless, the work of Bourgain et al. [29]
illustrates that combinatorial properties can be useful, and there may exist
constructions to which the power method would be naturally applied. Moreover,
we note that since $\delta_{K;q}$ approaches $\delta_{K}$, a sufficiently
large choice of $q$ should deliver better-than-$\varepsilon$ improvement over
the Gershgorin analysis. How large should $q$ be? If we assume $\Phi$ has
unit-norm columns, taking $q=1$ gives
$\delta_{K;1}^{2}=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2}]=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}|\langle\varphi_{i},\varphi_{j}\rangle|^{2}\leq
K(K-1)\mu^{2},$ (3.10)
where $\mu$ is the worst-case coherence of $\Phi$. Equality is achieved above
whenever $\Phi$ is an ETF, in which case (3.10) along with reasoning similar
to (1.5) demonstrates that $\Phi$ is RIP with sparsity levels on the order of
$\sqrt{M}$, as the Gershgorin analysis established. It remains to be shown how
$\delta_{K;2}$ compares. To make this comparison, we apply the power method to
random matrices:
###### Theorem 34.
Construct an $M\times N$ matrix $\Phi$ by drawing each of its entries
independently from a Gaussian distribution with mean zero and variance
$\frac{1}{M}$, and take $\delta_{K;q}$ to be as defined in Theorem 33. Then
$\delta_{K;q}\leq\delta$, and therefore $\Phi$ has the $(K,\delta)$-restricted
isometry property, with high probability provided
$M\geq\frac{81}{\delta^{2}}K^{1+1/q}\log\frac{eN}{K}$.
While flat restricted orthogonality comes with a negligible penalty of
$\log^{2}K$ in the number of measurements, the power method has a penalty of
$K^{1/q}$. As such, the case $q=1$ uses the order of $K^{2}$ measurements,
which matches our calculation in (3.10). Moreover, the power method with $q=2$
can demonstrate RIP with $K^{3/2}$ measurements, i.e., $K\sim M^{1/2+1/6}$,
which is considerably better than an $\varepsilon$ improvement over the
Gershgorin technique.
###### Proof of Theorem 34.
Take $t:=\frac{\delta}{3K^{1/2q}}-(\frac{K}{M})^{1/2}$ and pick
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$. Then Theorem II.13 of [58] states
$\mathrm{Pr}\bigg{[}1-\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\leq\sigma_{\min}(\Phi_{\mathcal{K}})\leq\sigma_{\max}(\Phi_{\mathcal{K}})\leq
1+\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{]}\geq 1-2e^{-Mt^{2}/2}.$
Continuing, we use the fact that
$\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})=\sigma(\Phi_{\mathcal{K}})^{2}$
to get
$\displaystyle 1-2e^{-Mt^{2}/2}$
$\displaystyle\leq\mathrm{Pr}\bigg{[}\bigg{(}1-\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{)}^{2}\leq\lambda_{\min}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq\lambda_{\max}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq\bigg{(}1+\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{)}^{2}\bigg{]}$
$\displaystyle\leq\mathrm{Pr}\bigg{[}1-3\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\leq\lambda_{\min}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq\lambda_{\max}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq
1+3\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{]},$ (3.11)
where the last inequality follows from the fact that
$(\frac{K}{M})^{1/2}+t<1$. Since $\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$
and $I_{K}$ are simultaneously diagonalizable, the spectrum of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}$ is given by
$\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})=\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})-1$.
Combining this with (3.11) then gives
$\mathrm{Pr}\bigg{[}\Big{\|}\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})\Big{\|}_{\infty}\leq
3\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{]}\geq 1-2e^{-Mt^{2}/2}.$
Considering $\mathrm{Tr}[A^{2q}]^{\frac{1}{2q}}=\|\lambda(A)\|_{2q}\leq
K^{\frac{1}{2q}}\|\lambda(A)\|_{\infty}$, we continue:
$\mathrm{Pr}\bigg{[}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]^{\frac{1}{2q}}\leq\delta\bigg{]}\geq\mathrm{Pr}\bigg{[}K^{\frac{1}{2q}}\Big{\|}\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})\Big{\|}_{\infty}\leq\delta\bigg{]}\geq
1-2e^{-Mt^{2}/2}.$
From here, we perform a union bound over all possible choices of
$\mathcal{K}$:
$\displaystyle\mathrm{Pr}\bigg{[}\exists\mathcal{K}\mbox{ s.t.
}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]^{\frac{1}{2q}}>\delta\bigg{]}$
$\displaystyle\leq\binom{N}{K}\mathrm{Pr}\bigg{[}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]^{\frac{1}{2q}}>\delta\bigg{]}$
$\displaystyle\leq 2\exp\Big{(}-\frac{Mt^{2}}{2}+K\log\frac{eN}{K}\Big{)}.$
(3.12)
Rearranging $M\geq\frac{81}{\delta^{2}}K^{1+1/q}\log\frac{eN}{K}$ gives
$K^{1/2}\leq\frac{\delta M^{1/2}}{9K^{1/2q}\log^{1/2}(eN/K)}\leq\frac{\delta
M^{1/2}}{9K^{1/2q}}$, and so
$\frac{Mt^{2}}{2}=\frac{1}{2}\bigg{(}\frac{\delta
M^{1/2}}{3K^{1/2q}}-K^{1/2}\bigg{)}^{2}\geq\frac{1}{2}\bigg{(}\frac{2\delta
M^{1/2}}{9K^{1/2q}}\bigg{)}^{2}\geq 2K\log\frac{eN}{K}.$ (3.13)
Combining (3.12) and (3.13) gives the result. ∎
### 3.3 Equiangular tight frames as RIP candidates
In Chapter 1, we observed that equiangular tight frames (ETFs) are optimal RIP
matrices under the Gershgorin analysis. In the present section, we reexamine
ETFs as prospective RIP matrices. Specifically, we consider the possibility
that certain classes of $M\times N$ ETFs support sparsity levels $K$ larger
than the order of $\sqrt{M}$. Before analyzing RIP, let’s first observe some
important features of ETFs. Recall that Section 0.2 characterized ETFs in
terms of their rows and columns. Interestingly, _real_ ETFs have a natural
alternative characterization.
Let $\Phi$ be a real $M\times N$ ETF, and consider the corresponding Gram
matrix $\Phi^{*}\Phi$. Observing Section 0.2, we have from (ii) that the
diagonal entries of $\Phi^{*}\Phi$ are 1’s. Also, (iii) indicates that the
off-diagonal entries are equal in absolute value (to the Welch bound); since
$\Phi$ has real entries, the phase of each off-diagonal entry of
$\Phi^{*}\Phi$ is either positive or negative. Letting $\mu$ denote the
absolute value of the off-diagonal entries, we can decompose the Gram matrix
as $\Phi^{*}\Phi=I_{N}+\mu S$, where $S$ is a matrix of zeros on the diagonal
and $\pm 1$’s on the off-diagonal. Here, $S$ is referred to as a _Seidel
adjacency matrix_ , as $S$ encodes the adjacency rule of a simple graph with
$i\leftrightarrow j$ whenever $S[i,j]=-1$; this correspondence originated in
[139].
There is an important equivalence class amongst ETFs: given an ETF $\Phi$, one
can negate any of the columns to form another ETF $\Phi^{\prime}$. Indeed, the
ETF properties in Section 0.2 are easily verified to hold for this new matrix.
For obvious reasons, $\Phi$ and $\Phi^{\prime}$ are called _flipping
equivalent_. This equivalence plays a key role in the following result, which
characterizes real ETFs in terms of a particular class of strongly regular
graphs:
###### Definition 35.
We say a simple graph $G$ is _strongly regular_ of the form
$\mathrm{srg}(v,k,\lambda,\mu)$ if
* (i)
$G$ has $v$ vertices,
* (ii)
every vertex has $k$ neighbors (i.e., $G$ is $k$_-regular_),
* (iii)
every two adjacent vertices have $\lambda$ common neighbors, and
* (iv)
every two non-adjacent vertices have $\mu$ common neighbors.
###### Theorem 36 (Corollary 5.6 in [141]).
Every real $M\times N$ equiangular tight frame with $N>M+1$ is flipping
equivalent to a frame whose Seidel adjacency matrix corresponds to the join of
a vertex with a strongly regular graph of the form
$\mathrm{srg}\bigg{(}N-1,L,\frac{3L-N}{2},\frac{L}{2}\bigg{)},\qquad
L:=\frac{N}{2}-1+\bigg{(}1-\frac{N}{2M}\bigg{)}\sqrt{\frac{M(N-1)}{N-M}}.$
Conversely, every such graph corresponds to flipping equivalence classes of
equiangular tight frames in the same manner.
The first chapter illustrated the main issue with the Gershgorin analysis: it
ignores important cancellations in the sub-Gram matrices. We suspect that such
cancellations would be more easily observed in a real ETF, since Theorem 36
neatly represents the Gram matrix’s off-diagonal oscillations in terms of
adjacencies in a strongly regular graph. The following result gives a taste of
how useful this graph representation can be:
###### Theorem 37.
Take a real equiangular tight frame $\Phi$ with worst-case coherence $\mu$,
and let $G$ denote the corresponding strongly regular graph in Theorem 36.
Then the restricted isometry constant of $\Phi$ is given by
$\delta_{K}=(K-1)\mu$ for every $K\leq\omega(G)+1$, where $\omega(G)$ denotes
the size of the largest clique in $G$.
###### Proof.
The Gershgorin analysis (1.4) gives the bound $\delta_{K}\leq(K-1)\mu$, and so
it suffices to prove $\delta_{K}\geq(K-1)\mu$. Since $K\leq\omega(G)+1$, there
exists a clique of size $K$ in the join of $G$ with a vertex. Let
$\mathcal{K}$ denote the vertices of this clique, and take $S_{\mathcal{K}}$
to be the corresponding Seidel adjacency submatrix. In this case,
$S_{\mathcal{K}}=I_{K}-J_{K}$, where $J_{K}$ is the $K\times K$ matrix of all
1’s. Observing the decomposition
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}=I_{K}+\mu S_{\mathcal{K}}$, it
follows from (1.1) that
$\delta_{K}\geq\|\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}\|_{2}=\|\mu
S_{\mathcal{K}}\|_{2}=\mu\|I_{K}-J_{K}\|_{2}=(K-1)\mu,$
which concludes the proof. ∎
This result indicates that the Gershgoin analysis is tight for all real ETFs,
at least for sufficiently small values of $K$. In particular, in order for a
real ETF to be RIP beyond the square-root bottleneck, its graph must have a
small clique number. As an example, note that the first four columns of the
Steiner ETF in (1.6) have negative inner products with each other, and thus
the corresponding subgraph is a clique. In general, each block of an $M\times
N$ Steiner ETF, whose size is guaranteed to be $\mathrm{O}(\sqrt{M})$, is a
lower-dimensional simplex and therefore has this property; this is an
alternative proof that the Gershgorin analysis of Steiner ETFs is tight for
$K=\mathrm{O}(\sqrt{M})$.
#### 3.3.1 Equiangular tight frames with flat restricted orthogonality
To find ETFs that are RIP beyond the square-root bottleneck, we must apply
better techniques than Gershgorin. We first consider what it means for an ETF
to have $(K,\hat{\theta})$-flat restricted orthogonality. Take a real ETF
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$ with worst-case coherence $\mu$, and
note that the corresponding Seidel adjacency matrix $S$ can be expressed in
terms of the usual $\\{0,1\\}$-adjacency matrix $A$ of the same graph:
$S[i,j]=1-2A[i,j]$ whenever $i\neq j$. Therefore, for every disjoint
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$, we want
$\displaystyle\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
$\displaystyle\geq\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}=\bigg{|}\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\mu
S[i,j]\bigg{|}$
$\displaystyle\qquad=\mu\bigg{|}|\mathcal{I}||\mathcal{J}|-2\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}A[i,j]\bigg{|}=2\mu\bigg{|}E(\mathcal{I},\mathcal{J})-\frac{1}{2}|\mathcal{I}||\mathcal{J}|\bigg{|},$
(3.14)
where $E(\mathcal{I},\mathcal{J})$ denotes the number of edges between
$\mathcal{I}$ and $\mathcal{J}$ in the graph. This condition bears a striking
resemblence to the following well-known result in graph theory:
###### Lemma 38 (Expander mixing lemma [85]).
Given a $d$-regular graph of $n$ vertices, the second largest eigenvalue
$\lambda$ of its adjacency matrix satisfies
$\bigg{|}E(\mathcal{I},\mathcal{J})-\frac{d}{n}|\mathcal{I}||\mathcal{J}|\bigg{|}\leq\lambda(|\mathcal{I}||\mathcal{J}|)^{1/2}$
for every pair of vertex subsets $\mathcal{I},\mathcal{J}$.
In words, the expander mixing lemma says that the number of edges between
vertex subsets of a regular graph is roughly what you would expect in a
_random_ regular graph. For this lemma to be applicable to (3.14), we need the
strongly regular graph of Theorem 36 to satisfy
$\frac{L}{N-1}=\frac{d}{n}\approx\frac{1}{2}$. Using the formula for $L$, it
is not difficult to show that
$|\frac{L}{N-1}-\frac{1}{2}|=\mathrm{O}(M^{-1/2})$ provided $N=\mathrm{O}(M)$
and $N\geq 2M$. Furthermore, the second largest eigenvalue of the strongly
regular graph will be $\lambda\approx\frac{1}{2}N^{1/2}$, and so the expander
mixing lemma says the optimal $\hat{\theta}$ is $\leq
2\mu\lambda\approx(\frac{N-M}{M})^{1/2}$ since
$\mu=(\frac{N-M}{M(N-1)})^{1/2}$. This is a rather weak estimate for
$\hat{\theta}$ because the expander mixing lemma does not account for the
sizes of $\mathcal{I}$ and $\mathcal{J}$ being $\leq K$. Put in this light, a
real ETF that has flat restricted orthogonality corresponds to a strongly
regular graph that satisfies a particularly strong version of the expander
mixing lemma.
#### 3.3.2 Equiangular tight frames and the power method
Next, we try applying the power method to ETFs. Given a real ETF
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$, let $H:=\Phi^{*}\Phi-I_{N}$ denote the
“hollow” Gram matrix. Also, take $E_{\mathcal{K}}$ to be the $N\times K$
matrix built from the columns of $I_{N}$ that are indexed by $\mathcal{K}$.
Then
$\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]=\mathrm{Tr}[(E_{\mathcal{K}}^{*}\Phi^{*}\Phi
E_{\mathcal{K}}-I_{K})^{2q}]=\mathrm{Tr}[(E_{\mathcal{K}}^{*}HE_{\mathcal{K}})^{2q}]=\mathrm{Tr}[(HE_{\mathcal{K}}E_{\mathcal{K}}^{*})^{2q}].$
Since
$E_{\mathcal{K}}E_{\mathcal{K}}^{*}=\sum_{k\in\mathcal{K}}\delta_{k}\delta_{k}^{*}$,
where $\delta_{k}$ is the $k$th identity basis element, we continue:
$\displaystyle\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]$
$\displaystyle=\mathrm{Tr}\bigg{[}\bigg{(}H\sum_{k\in\mathcal{K}}\delta_{k}\delta_{k}^{*}\bigg{)}^{2q}\bigg{]}$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\cdots\sum_{k_{2q-1}\in\mathcal{K}}\mathrm{Tr}[H\delta_{k_{0}}\delta_{k_{0}}^{*}\cdots
H\delta_{k_{2q-1}}\delta_{k_{2q-1}}^{*}]$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\cdots\sum_{k_{2q-1}\in\mathcal{K}}\delta_{k_{0}}^{*}H\delta_{k_{1}}\cdots\delta_{k_{2q-1}}^{*}H\delta_{k_{0}},$
(3.15)
where the last step used the cyclic property of the trace. From here, note
that $H$ has a zero diagonal, meaning several of the terms in (3.15) are zero,
namely, those for which $k_{\ell+1}=k_{\ell}$ for some
$\ell\in\mathbb{Z}_{2q}$. To simplify (3.15), take $\mathcal{K}^{(2q)}$ to be
the set of $2q$-tuples satisfying $k_{\ell+1}\neq k_{\ell}$ for every
$\ell\in\mathbb{Z}_{2q}$:
$\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{2q}]=\sum_{\\{k_{\ell}\\}\in\mathcal{K}^{(2q)}}\prod_{\ell\in\mathbb{Z}_{2q}}\langle\varphi_{k_{\ell}},\varphi_{k_{\ell+1}}\rangle=\mu^{2q}\sum_{\\{k_{\ell}\\}\in\mathcal{K}^{(2q)}}\prod_{\ell\in\mathbb{Z}_{2q}}S[k_{\ell},k_{\ell+1}],$
(3.16)
where $\mu$ is the wost-case coherence of $\Phi$, and $S$ is the corresponding
Seidel adjacency matrix. Note that the left-hand side is necessarily
nonnegative, while it is not immediate why the right-hand side should be. This
indicates that more simplification can be done, but for the sake of clarity,
we will perform this simplification in the special case where $q=2$; the
general case is very similar. When $q=2$, we are concerned with 4-tuples
$\\{k_{0},k_{1},k_{2},k_{3}\\}\in\mathcal{K}^{(4)}$. Let’s partition these
4-tuples according to the value taken by $k_{0}$ and $k_{q}=k_{2}$. Note, for
a fixed $k_{0}$ and $k_{2}$, that $k_{1}$ can be any value other than $k_{0}$
or $k_{2}$, as can $k_{3}$. This leads to the following simplification:
$\displaystyle\sum_{\\{k_{\ell}\\}\in\mathcal{K}^{(4)}}\prod_{\ell\in\mathbb{Z}_{4}}S[k_{\ell},k_{\ell+1}]$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\sum_{k_{2}\in\mathcal{K}}\bigg{(}\sum_{\begin{subarray}{c}k_{1}\in\mathcal{K}\\\
k_{0}\neq k_{1}\neq
k_{2}\end{subarray}}S[k_{0},k_{1}]S[k_{1},k_{2}]\bigg{)}\bigg{(}\sum_{\begin{subarray}{c}k_{3}\in\mathcal{K}\\\
k_{2}\neq k_{3}\neq k_{0}\end{subarray}}S[k_{2},k_{3}]S[k_{3},k_{0}]\bigg{)}$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\sum_{k_{2}\in\mathcal{K}}~{}~{}~{}~{}\bigg{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k_{0}\neq k\neq k_{2}\end{subarray}}S[k_{0},k]S[k,k_{2}]\bigg{|}^{2}$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\bigg{|}\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k\neq
k_{0}\end{subarray}}S[k_{0},k]S[k,k_{0}]\bigg{|}^{2}+\sum_{k_{0}\in\mathcal{K}}\sum_{\begin{subarray}{c}k_{2}\in\mathcal{K}\\\
k_{2}\neq
k_{0}\end{subarray}}~{}~{}~{}~{}\bigg{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k_{0}\neq k\neq k_{2}\end{subarray}}S[k_{0},k]S[k,k_{2}]\bigg{|}^{2}.$
The first term above is $K(K-1)^{2}$, while the other term is not as easy to
analyze, as we expect a certain degree of cancellation. Substituting this
simplification into (3.16) gives
$\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})^{4}]=\mu^{4}\bigg{(}K(K-1)^{2}+\sum_{k_{0}\in\mathcal{K}}\sum_{\begin{subarray}{c}k_{2}\in\mathcal{K}\\\
k_{2}\neq
k_{1}\end{subarray}}~{}~{}~{}~{}\bigg{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k_{0}\neq k\neq k_{2}\end{subarray}}S[k_{0},k]S[k,k_{2}]\bigg{|}^{2}\bigg{)}.$
If there were no cancellations in the second term, then it would equal
$K(K-1)(K-2)^{2}$, thereby dominating the expression. However, if oscillations
occured as a $\pm 1$ Bernoulli random variable, we could expect this term to
be on the order of $K^{3}$, matching the order of the first term. In this
hypothetical case, since $\mu\leq M^{-1/2}$, the parameter $\delta_{K;2}^{4}$
defined in Theorem 33 scales as $\frac{K^{3}}{M^{2}}$, and so $M\sim K^{3/2}$;
this corresponds to the behavior exhibited in Theorem 34. To summarize, much
like flat restricted orthogonality, applying the power method to ETFs leads to
interesting combinatorial questions regarding subgraphs, even when $q=2$.
#### 3.3.3 The Paley equiangular tight frame as an RIP candidate
Pick some prime $p\equiv 1\bmod 4$, and build an $M\times p$ matrix $H$ by
selecting the $M:=\frac{p+1}{2}$ rows of the $p\times p$ discrete Fourier
transform matrix which are indexed by $Q$, the quadratic residues modulo $p$
(including zero). To be clear, the entries of $H$ are scaled to have unit
modulus. Next, take $D$ to be an $M\times M$ diagonal matrix whose zeroth
diagonal entry is $\sqrt{\frac{1}{p}}$, and whose remaining $M-1$ entries are
$\sqrt{\frac{2}{p}}$. Now build the matrix $\Phi$ by concatenating $DH$ with
the zeroth identity basis element; for example, when $p=5$, we have a $3\times
6$ matrix:
$\Phi=\left[\begin{array}[]{llllll}\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&1\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&0\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&0\\\
\end{array}\right].$
We claim that in general, this process produces an $M\times 2M$ equiangular
tight frame, which we call the _Paley ETF_ [115]. Presuming for the moment
that this claim is true, we have the following result which lends hope for the
Paley ETF as an RIP matrix:
###### Lemma 39.
An $M\times 2M$ Paley equiangular tight frame has restricted isometry constant
$\delta_{K}<1$ for all $K\leq M$.
###### Proof.
First, we note that Theorem 16 used Chebotarëv’s theorem [126] to prove that
the spark of the $M\times 2M$ Paley ETF $\Phi$ is $M+1$, that is, every
size-$M$ subcollection of columns of $\Phi$ forms a spanning set. Thus, for
every $\mathcal{K}\subseteq\\{1,\ldots,2M\\}$ of size $\leq M$, the smallest
singular value of $\Phi_{\mathcal{K}}$ is positive. It remains to show that
the square of the largest singular value is strictly less than 2. Let $x$ be a
unit vector for which
$\|\Phi_{\mathcal{K}}^{*}x\|=\|\Phi_{\mathcal{K}}^{*}\|_{2}$. Then since the
spark of $\Phi$ is $M+1$, the columns of $\Phi_{\mathcal{K}^{\mathrm{c}}}$
span, and so
$\|\Phi_{\mathcal{K}}\|_{2}^{2}=\|\Phi_{\mathcal{K}}^{*}\|_{2}^{2}=\|\Phi_{\mathcal{K}}^{*}x\|^{2}<\|\Phi_{\mathcal{K}}^{*}x\|^{2}+\|\Phi_{\mathcal{K}^{\mathrm{c}}}^{*}x\|^{2}=\|\Phi^{*}x\|^{2}\leq\|\Phi^{*}\|_{2}^{2}=\|\Phi\Phi^{*}\|_{2}=2,$
where the final step follows from (i) and (ii) of Section 0.2, which imply
$\Phi\Phi^{*}=2I_{M}$. ∎
Now that we have an interest in the Paley ETF $\Phi$, we wish to verify that
it is, in fact, an ETF. It suffices to show that the columns of $\Phi$ have
unit norm, and that the inner products between distinct columns equal the
Welch bound in absolute value. Certainly, the zeroth identity basis element is
unit-norm, while the squared norm of each of the other columns is given by
$\frac{1}{p}+(M-1)\frac{2}{p}=\frac{2M-1}{p}=1$. Also, the inner product
between the zeroth identity basis element and any other column equals the
zeroth entry of that column: $p^{-1/2}=(\frac{N-M}{M(N-1)})^{1/2}$. It remains
to calculate the inner product between distinct columns which are not identity
basis elements. To this end, note that since $a^{2}=b^{2}$ if and only if
$a=\pm b$, the sequence $\\{k^{2}\\}_{k=1}^{p-1}\subseteq\mathbb{Z}_{p}$
doubly covers $Q\setminus\\{0\\}$, and so
$\langle\varphi_{n},\varphi_{n^{\prime}}\rangle=\frac{1}{p}+\sum_{m\in
Q\setminus\\{0\\}}\bigg{(}\sqrt{\frac{2}{p}}e^{-2\pi\mathrm{i}mn/p}\bigg{)}\bigg{(}\sqrt{\frac{2}{p}}e^{2\pi\mathrm{i}mn^{\prime}/p}\bigg{)}=\frac{1}{p}\sum_{k=0}^{p-1}e^{2\pi\mathrm{i}(n^{\prime}-n)k^{2}/p}.$
This well-known expression is called a quadratic Gauss sum, and since $p\equiv
1\bmod 4$, its value is determined by the Legendre symbol in the following
way:
$\langle\varphi_{n},\varphi_{n^{\prime}}\rangle=\frac{1}{\sqrt{p}}(\frac{n^{\prime}-n}{p})$
for every $n,n^{\prime}\in\mathbb{Z}_{p}$ with $n\neq n^{\prime}$, where
$\bigg{(}\frac{k}{p}\bigg{)}:=\left\\{\begin{array}[]{rl}+1&\mbox{ if $k$ is a
nonzero quadratic residue modulo $p$,}\\\ 0&\mbox{ if $k=0$,}\\\ -1&\mbox{
otherwise.}\end{array}\right.$
Having established that $\Phi$ is an ETF, we notice that the inner products
between distinct columns of $\Phi$ are real. This implies that the columns of
$\Phi$ can be unitarily rotated to form a real ETF $\Psi$; indeed, one may
take $\Psi$ to be the $M\times 2M$ matrix formed by taking the nonzero rows of
$L^{\mathrm{T}}$ in the Cholesky factorization $\Phi^{*}\Phi=LL^{\mathrm{T}}$.
As such, we consider the Paley ETF to be real. From here, Theorem 36 prompts
us to find the corresponding strongly regular graph. First, we can flip the
identity basis element so that its inner products with the other columns of
$\Phi$ are all negative. As such, the corresponding vertex in the graph will
be adjacent to each of the other vertices; naturally, this will be the vertex
to which the strongly regular graph is joined. For the remaining vertices,
$n\leftrightarrow n^{\prime}$ precisely when $(\frac{n^{\prime}-n}{p})=-1$,
that is, when $n^{\prime}-n$ is not a quadratic residue. The corresponding
subgraph is therefore the complement of the Paley graph, namely, the Paley
graph [119]. In general, Paley graphs of order $p$ necessarily have $p\equiv
1\bmod 4$, and so this correspondence is particularly natural.
One interesting thing about the Paley ETF’s restricted isometry is that it
lends insight into important properties of the Paley graph. The following is
the best known upper bound for the clique number of the Paley graph of prime
order (see Theorem 13.14 of [28] and discussion thereafter), and we give a new
proof of this bound using restricted isometry:
###### Theorem 40.
Let $G$ denote the Paley graph of prime order $p$. Then the size of the
largest clique is $\omega(G)<\sqrt{p}$.
###### Proof.
We start by showing $\omega(G)+1\leq M$. Suppose otherwise: that there exists
a clique $\mathcal{K}$ of size $M+1$ in the join of a vertex with $G$. Then
the corresponding sub-Gram matrix of the Paley ETF has the form
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}=(1+\mu)I_{M+1}-\mu J_{M+1}$, where
$\mu=p^{-1/2}$ is the worst-case coherence and $J_{M+1}$ is the
$(M+1)\times(M+1)$ matrix of 1’s. Since the largest eigenvalue of $J_{M+1}$ is
$M+1$, the smallest eigenvalue of $\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$
is $1+p^{-1/2}-(M+1)p^{-1/2}=1-\frac{1}{2}(p+1)p^{-1/2}$, which is negative
when $p\geq 5$, contradicting the fact that
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is positive semidefinite.
Since $\omega(G)+1\leq M$, we can apply Lemma 39 and Theorem 37 to get
$1>\delta_{\omega(G)+1}=\Big{(}\omega(G)+1-1\Big{)}\mu=\frac{\omega(G)}{\sqrt{p}},$
(3.17)
and rearranging gives the result. ∎
It is common to apply probabilistic and heuristic reasoning to gain intuition
in number theory. For example, consecutive entries of the Legendre symbol are
known to mimic certain properties of a $\pm 1$ Bernoulli random variable
[110]. Moreover, Paley graphs enjoy a certain quasi-random property that was
studied in [50]. On the other hand, Graham and Ringrose [76] showed that,
while random graphs of size $p$ have an expected clique number of
$(1+o(1))2\log p/\log 2$, Paley graphs of prime order deviate from this random
behavior, having a clique number $\geq c\log p\log\log\log p$ infinitely
often. The best known universal lower bound, $(1/2+o(1))\log p/\log 2$, is
given in [51], which indicates that the random graph analysis is at least
tight in some sense. Regardless, this has a significant difference from the
upper bound $\sqrt{p}$ in Theorem 40, and it would be nice if probabilistic
arguments could be leveraged to improve this bound, or at least provide some
intuition.
Note that our proof (3.17) hinged on the fact that $\delta_{\omega(G)+1}<1$,
courtesy of Lemma 39. Hence, any improvement to our estimate for
$\delta_{\omega(G)+1}$ would directly lead to the best known upper bound on
the Paley graph’s clique number. To approach such an improvement, note that
for large $p$, the Fourier portion of the Paley ETF $DH$ is not significatly
different from the normalized partial Fourier matrix $(\frac{2}{p+1})^{1/2}H$;
indeed,
$\|H_{\mathcal{K}}^{*}D^{2}H_{\mathcal{K}}-\frac{2}{p+1}H_{\mathcal{K}}^{*}H_{\mathcal{K}}\|_{2}\leq\frac{2}{p}$
for every $\mathcal{K}\subseteq\mathbb{Z}_{p}$ of size $\leq\frac{p+1}{2}$,
and so the difference vanishes. If we view the quadratic residues modulo $p$
(the row indices of $H$) as random, then a random partial Fourier matrix
serves as a proxy for the Fourier portion of the Paley ETF. This in mind, we
appeal to the following:
###### Theorem 41 (Theorem 3.2 in [114]).
Draw rows from the $N\times N$ discrete Fourier transform matrix uniformly at
random with replacement to construct an $M\times N$ matrix, and then normalize
the columns to form $\Phi$. Then $\Phi$ has restricted isometry constant
$\delta_{K}\leq\delta$ with probability $1-\varepsilon$ provided
$\frac{M}{\log M}\geq\frac{C}{\delta^{2}}K\log^{2}K\log
N\log\varepsilon^{-1}$, where $C$ is a universal constant.
In our case, both $M$ and $N$ scale as $p$, and so picking $\delta$ to achieve
equality above gives
$\delta^{2}=\frac{C^{\prime}}{p}K\log^{2}K\log^{2}p\log\varepsilon^{-1}.$
Continuing as in (3.17), denote $\omega=\omega(G)$ and take $K=\omega$ to get
$\frac{C^{\prime}}{p}\omega\log^{2}\omega\log^{2}p\log\varepsilon^{-1}\geq\delta_{\omega}^{2}=\frac{(\omega-1)^{2}}{p}\geq\frac{\omega^{2}}{2p},$
and then rearranging gives $\omega/\log^{2}\omega\leq
C^{\prime\prime}\log^{2}p\log\varepsilon^{-1}$ with probability
$1-\varepsilon$. Interestingly, having
$\omega/\log^{2}\omega=\mathrm{O}(\log^{3}p)$ with high probability (again,
under the model that quadratic residues are random) agrees with the results of
Graham and Ringrose [76]. This gives some intuition for what we can expect the
size of the Paley graph’s clique number to be, while at the same time
demonstrating the power of Paley ETFs as RIP candidates. We conclude with the
following, which can be reformulated in terms of both flat restricted
orthogonality and the power method:
###### Conjecture 42.
The Paley equiangular tight frame has the $(K,\delta)$-restricted isometry
property with some $\delta<\sqrt{2}-1$ whenever
$K\leq\frac{Cp}{\log^{\alpha}p}$, for some universal constants $C$ and
$\alpha$.
### 3.4 Appendix
In this section, we prove Theorem 30, which states that a matrix with
$(K,\hat{\theta})$-flat restricted orthogonality has $\theta_{K}\leq
C\hat{\theta}\log K$, that is, it has restricted orthogonality. The proof
below is adapted from the proof of Lemma 3 in [29]. Our proof has the benefit
of being valid for all values of $K$ (as opposed to sufficiently large $K$ in
the original [29]), and it has near-optimal constants where appropriate.
Moreover in this version, the columns of the matrix are not required to have
unit norm.
###### Proof of Theorem 30.
Given arbitrary disjoint subsets
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$, we will bound the following quantity
three times, each time with different constraints on
$\\{x_{i}\\}_{i\in\mathcal{I}}$ and $\\{y_{j}\\}_{j\in\mathcal{J}}$:
$\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}.$
(3.18)
To be clear, our third bound will have no constraints on
$\\{x_{i}\\}_{i\in\mathcal{I}}$ and $\\{y_{j}\\}_{j\in\mathcal{J}}$, thereby
demonstrating restricted orthogonality. Note that by assumption, (3.18) is
$\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$ whenever the $x_{i}$’s
and $y_{j}$’s are in $\\{0,1\\}$. We first show that this bound is preserved
when we relax the $x_{i}$’s and $y_{j}$’s to lie in the interval $[0,1]$.
Pick a disjoint pair of subsets
$\mathcal{I}^{\prime},\mathcal{J}^{\prime}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}^{\prime}|,|\mathcal{J}^{\prime}|\leq K$. Starting with some
$k\in\mathcal{I}^{\prime}$, note that flat restricted orthogonality gives that
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2},$
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}\setminus\\{k\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\hat{\theta}(|\mathcal{I}\setminus\\{k\\}||\mathcal{J}|)^{1/2}\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
for every disjoint $\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$ and $k\in\mathcal{I}$. Thus, we may take
any $x_{k}\in[0,1]$ to form a convex combination of these two expressions, and
then the triangle inequality gives
$\displaystyle\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
$\displaystyle\geq
x_{k}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}+(1-x_{k})\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}\setminus\\{k\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\geq\bigg{|}x_{k}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}+(1-x_{k})\bigg{\langle}\sum_{i\in\mathcal{I}\setminus\\{k\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\bigg{\\{}\begin{array}[]{cc}x_{k},&i=k\\\
1,&i\neq
k\end{array}\bigg{\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}.$
(3.21)
Since (3.21) holds for every disjoint
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$ and $k\in\mathcal{I}$, we can do the same
thing with an additional index $i\in\mathcal{I}^{\prime}$ or
$j\in\mathcal{J}^{\prime}$, and replace the corresponding unit coefficient
with some $x_{i}$ or $y_{j}$ in $[0,1]$. Continuing in this way proves the
claim that (3.18) is $\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
whenever the $x_{i}$’s and $y_{j}$’s lie in the interval $[0,1]$.
For the second bound, we assume the $x_{i}$’s and $y_{j}$’s are nonnegative
with unit norm:
$\sum_{i\in\mathcal{I}}x_{i}^{2}=\sum_{j\in\mathcal{J}}y_{j}^{2}=1$. To bound
(3.18) in this case, we partition $\mathcal{I}$ and $\mathcal{J}$ according to
the size of the corresponding coefficients:
$\mathcal{I}_{k}:=\\{i\in\mathcal{I}:2^{-(k+1)}<x_{i}\leq
2^{-k}\\},\qquad\mathcal{J}_{k}:=\\{j\in\mathcal{J}:2^{-(k+1)}<y_{j}\leq
2^{-k}\\}.$
Note the unit-norm constraints ensure that
$\mathcal{I}=\bigcup_{k=0}^{\infty}\mathcal{I}_{k}$ and
$\mathcal{J}=\bigcup_{k=0}^{\infty}\mathcal{J}_{k}$. The triangle inequality
thus gives
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{k_{1}=0}^{\infty}\sum_{i\in\mathcal{I}_{k_{1}}}x_{i}\varphi_{i},\sum_{k_{2}=0}^{\infty}\sum_{j\in\mathcal{J}_{k_{2}}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}2^{-(k_{1}+k_{2})}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}_{k_{1}}}\frac{x_{i}}{2^{-k_{1}}}\varphi_{i},\sum_{j\in\mathcal{J}_{k_{2}}}\frac{y_{j}}{2^{-k_{2}}}\varphi_{j}\bigg{\rangle}\bigg{|}.$
(3.22)
By the definitions of $\mathcal{I}_{k_{1}}$ and $\mathcal{J}_{k_{2}}$, the
coefficients of $\varphi_{i}$ and $\varphi_{j}$ in (3.22) all lie in $[0,1]$.
As such, we continue by applying our first bound:
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}2^{-(k_{1}+k_{2})}\hat{\theta}(|\mathcal{I}_{k_{1}}||\mathcal{J}_{k_{2}}|)^{1/2}$
$\displaystyle=\hat{\theta}\bigg{(}\sum_{k=0}^{\infty}2^{-k}|\mathcal{I}_{k}|^{1/2}\bigg{)}\bigg{(}\sum_{k=0}^{\infty}2^{-k}|\mathcal{J}_{k}|^{1/2}\bigg{)}.$
(3.23)
We now observe from the definition of $\mathcal{I}_{k}$ that
$1=\sum_{i\in\mathcal{I}}x_{i}^{2}=\sum_{k=0}^{\infty}\sum_{i\in\mathcal{I}_{k}}x_{i}^{2}>\sum_{k=0}^{\infty}4^{-(k+1)}|\mathcal{I}_{k}|.$
Thus for any positive integer $t$, the Cauchy-Schwarz inequality gives
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|\mathcal{I}_{k}|^{1/2}$
$\displaystyle=\sum_{k=0}^{t-1}2^{-k}|\mathcal{I}_{k}|^{1/2}+\sum_{k=t}^{\infty}2^{-k}|\mathcal{I}_{k}|^{1/2}$
$\displaystyle\leq
t^{1/2}\bigg{(}\sum_{k=0}^{t-1}4^{-k}|\mathcal{I}_{k}|\bigg{)}^{1/2}+\sum_{k=t}^{\infty}2^{-k}K^{1/2}$
$\displaystyle<2(t^{1/2}+K^{1/2}2^{-t}),$ (3.24)
and similarly for the $\mathcal{J}_{k}$’s. For a fixed $K$, we note that
(3.24) is minimized when $K^{1/2}2^{-t}=\frac{t^{-1/2}}{2\log 2}$, and so we
pick $t$ to be the smallest positive integer such that
$K^{1/2}2^{-t}\leq\frac{t^{-1/2}}{2\log 2}$. With this, we continue (3.23):
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle<\hat{\theta}\Big{(}2(t^{1/2}+K^{1/2}2^{-t})\Big{)}^{2}$
$\displaystyle\leq 4\hat{\theta}\bigg{(}t^{1/2}+\frac{t^{-1/2}}{2\log
2}\bigg{)}^{2}=4\hat{\theta}\bigg{(}t+\frac{1}{\log 2}+\frac{1}{(2\log
2)^{2}t}\bigg{)}.$ (3.25)
From here, we claim that $t\leq\lceil\frac{\log K}{\log 2}\rceil$. Considering
the definition of $t$, this is easily verified for $K=2,3,\ldots,7$ by showing
$K^{1/2}2^{-s}\leq\frac{s^{-1/2}}{2\log 2}$ for $s=\lceil\frac{\log K}{\log
2}\rceil$. For $K\geq 8$, one can use calculus to verify the second inequality
of the following:
$K^{1/2}2^{-\lceil\frac{\log K}{\log 2}\rceil}\leq K^{1/2}2^{-\frac{\log
K}{\log 2}}\leq\frac{1}{2\log 2}\bigg{(}\frac{\log K}{\log
2}+1\bigg{)}^{-1/2}\leq\frac{1}{2\log 2}\bigg{\lceil}\frac{\log K}{\log
2}\bigg{\rceil}^{-1/2},$
meaning $t\leq\lceil\frac{\log K}{\log 2}\rceil$. Substituting
$t\leq\frac{\log K}{\log 2}+1$ and $t\geq 1$ into (3.25) then gives
$\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}<4\hat{\theta}\bigg{(}\frac{\log
K}{\log 2}+1+\frac{1}{\log 2}+\frac{1}{(2\log 2)^{2}}\bigg{)}\\\
=\hat{\theta}(C_{0}\log K+C_{1}),$
with $C_{0}\approx 5.77$, $C_{1}\approx 11.85$. As such, (3.18) is $\leq
C^{\prime}\hat{\theta}\log K$ with $C^{\prime}=C_{0}+\frac{C_{1}}{\log 2}$ in
this case.
We are now ready for the final bound on (3.18) in which we apply no
constraints on the $x_{i}$’s and $y_{j}$’s. To do this, we consider the
positive and negative real and imaginary parts of these coefficients:
$x_{i}=\sum_{k=0}^{3}x_{i,k}\mathrm{i}^{k}\quad\mbox{s.t.}\quad x_{i,k}\geq
0\quad\forall k,$
and similarly for the $y_{j}$’s. With this decomposition, we apply the
triangle inequality to get
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\sum_{k_{1}=0}^{3}x_{i,k_{1}}\mathrm{i}^{k_{1}}\varphi_{i},\sum_{j\in\mathcal{J}}\sum_{k_{2}=0}^{3}y_{j,k_{2}}\mathrm{i}^{k_{2}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{3}\sum_{k_{2}=0}^{3}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i,k_{1}}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j,k_{2}}\varphi_{j}\bigg{\rangle}\bigg{|}.$
Finally, we normalize the coefficients by
$(\sum_{i\in\mathcal{I}}x_{i,k_{1}}^{2})^{1/2}$ and
$(\sum_{j\in\mathcal{J}}y_{j,k_{2}}^{2})^{1/2}$ so we can apply our second
bound:
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{3}\sum_{k_{2}=0}^{3}\bigg{(}\sum_{i\in\mathcal{I}}x_{i,k_{1}}^{2}\bigg{)}^{1/2}\bigg{(}\sum_{j\in\mathcal{J}}y_{j,k_{2}}^{2}\bigg{)}^{1/2}C^{\prime}\hat{\theta}\log
K$ $\displaystyle\leq(C\hat{\theta}\log K)\|x\|\|y\|,$
where $C=4C^{\prime}\approx 74.17$ by the Cauchy-Schwarz inequality, and so we
are done. ∎
## Chapter 4 Two fundamental parameters of frame coherence
Chapters 1–3 of this thesis were dedicated to a particularly popular
understanding of compressed sensing: that matrices which satisfy the
restricted isometry property (RIP) are very well-suited as sensing matrices.
However, as these chapters show, it is very difficult to deterministically
construct matrices which are provably RIP. It is therefore desirable to find a
worthy alternative to RIP which admits deterministic sensing matrices. The
present chapter is dedicated to one such alternative, namely the _strong
coherence property_ , but before we define this property, we first motivate it
in the context of a support recovery method known as _one-step thresholding
(OST)_.
The main idea behind OST is that the noiseless measurement vector $y=\Phi x$
will look similar to the active columns of
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$, provided the sparsity level is
sufficiently small and the nonzero members of $x$ are sufficiently large in
some sense. Using this intuition, it makes sense to find the support of $x$ by
finding the large values of
$|\langle\varphi_{i},y\rangle|=\bigg{|}\bigg{\langle}\varphi_{i},\sum_{j=1}^{N}x_{j}\varphi_{j}\bigg{\rangle}\bigg{|}=\bigg{|}\sum_{j=1}^{N}x_{j}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|}=\bigg{|}x_{i}+\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}x_{j}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|},$
assuming the columns of $\Phi$ have unit norm. Indeed, if the nonzero entries
of $x$ are larger than the contribution of the cross-column interactions, then
the above calculation serves as a reasonable test for the support of $x$. The
magnitude of this contribution can be assessed using two measures of
coherence. Indeed, if the columns are incoherent, then each term of this sum
is small, and so it makes sense to consider the worst-case coherence of
$\Phi$:
$\mu:=\max_{\begin{subarray}{c}i,j\in\\{1,\ldots,N\\}\\\ i\neq
j\end{subarray}}|\langle\varphi_{i},\varphi_{j}\rangle|.$ (4.1)
However, this measure of coherence does not account for sign fluxuations in
the inner products, which should bring significant cancellations in the sum.
If we assume the support of $x$ is drawn randomly, then by a concentration-of-
measure argument, this sum will typically be close to its expectation, and so
its size will rarely exceed some multiple of $\|x\|_{1}$ times the following
maximum average:
$\nu:=\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\frac{1}{N-1}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|}.$
(4.2)
For this reason, this notion of coherence, called _average coherence_ , was
recently introduced in [11].
Intuitively, worst-case coherence is a measure of dissimilarity between frame
elements, whereas average coherence measures how well the frame elements are
distributed in the unit hypersphere. As we will see, both worst-case and
average coherence play an important role in various portions of sparse signal
processing, provided we describe the sparse signal’s support with a
probabilistic model. In fact, [11] used worst-case and average coherence to
produce probabilistic reconstruction guarantees for OST, permitting sparsity
levels on the order of $\smash{\frac{M}{\log N}}$ (akin to the RIP-based
guarantees). In accordance with our motivation above, these probabilistic
guarantees require that worst-case and average coherence together satisfy the
following property:
###### Definition 43.
We say an $M\times N$ unit norm frame $\Phi$ satisfies the _strong coherence
property_ if
$\mbox{(SCP-1)}~{}~{}~{}\mu\leq\frac{1}{164\log
N}\qquad\mbox{and}\qquad\mbox{(SCP-2)}~{}~{}~{}\nu\leq\frac{\mu}{\sqrt{M}},$
where $\mu$ and $\nu$ are given by (4.1) and (4.2), respectively.
The reader should know that the constant $164$ is not particularly essential
to the above definition; it is used in [11] to simplify some analysis and make
certain performance guarantees explicit, but the constant is by no means
optimal. In the next section, we will use the strong coherence property to
continue the work of [11]. Where [11] provided guarantees for noiseless
reconstruction, we will produce near-optimal guarantees for signal detection
and reconstruction from _noisy_ measurements of sparse signals. These
guarantees are related to those in [35, 62, 135, 136], and we will also
elaborate on this relationship.
The results given in [11] and the following section, as well as the
applications discussed in [35, 62, 84, 103, 129, 134, 136, 149] demonstrate a
pressing need for nearly tight frames with small worst-case and average
coherence, especially in sparse signal processing. This chapter offers three
additional contributions in this regard [12, 102]. In Section 4.2, we provide
a sizable catalog of frames that exhibit small spectral norm, worst-case
coherence, and average coherence. With all three frame parameters provably
small, these frames are guaranteed to perform well in relevant applications.
Next, performance in many applications is dictated by worst-case coherence. It
is therefore particularly important to understand which worst-case coherence
values are achievable. To this end, the Welch bound (Theorem 3) is commonly
used in the literature. However, the Welch bound is only tight when the number
of frame elements $N$ is less than the square of the spatial dimension $M$
[129]. Another lower bound, given in [106, 146], beats the Welch bound when
there are more frame elements, but it is known to be loose for real frames
[53]. Given this context, Section 4.3 gives a new lower bound on the worst-
case coherence of real frames. Our bound beats both the Welch bound and the
bound in [106, 146] when the number of frame elements far exceeds the spatial
dimension. Finally, since average coherence is so new, there is currently no
intuition as to when (SCP-2) is satisfied. In Section 4.4, we use ideas akin
to the switching equivalence of graphs to transform a frame that satisfies
(SCP-1) into another frame with the same spectral norm and worst-case
coherence that additionally satisfies (SCP-2).
### 4.1 Implications of worst-case and average coherence
Frames with small spectral norm, worst-case coherence, and/or average
coherence have found use in recent years with applications involving sparse
signals. Donoho et al. used the worst-case coherence in [62] to provide
uniform bounds on the signal and support recovery performance of combinatorial
and convex optimization methods and greedy algorithms. Later, Tropp [136] and
Candès and Plan [35] used both the spectral norm and worst-case coherence to
provide tighter bounds on the signal and support recovery performance of
convex optimization methods for most support sets under the additional
assumption that the sparse signals have independent nonzero entries with zero
median. Recently, Bajwa et al. [11] made use of the spectral norm and both
coherence parameters to report tighter bounds on the noisy model selection and
noiseless signal recovery performance of an incredibly fast greedy algorithm
called _one-step thresholding (OST)_ for most support sets and _arbitrary_
nonzero entries. In this section, we discuss further implications of the
spectral norm and worst-case and average coherence of frames in applications
involving sparse signals.
#### 4.1.1 The weak restricted isometry property
A common task in signal processing applications is to test whether a
collection of measurements corresponds to mere noise [90]. For applications
involving sparse signals, one can test measurements $y\in\mathbb{C}^{M}$
against the null hypothsis $H_{0}:y=z$ and alternative hypothesis
$H_{1}:y=\Phi x+z$, where the entries of the noise vector $z\in\mathbb{C}^{M}$
are independent, identical zero-mean complex-Gaussian random variables and the
signal $x\in\mathbb{C}^{N}$ is $K$-sparse. The performance of such signal
detection problems is directly proportional to the energy in $\Phi x$ [56, 80,
90]. In particular, existing literature on the detection of sparse signals
[56, 80] leverages the fact that $\|\Phi x\|^{2}\approx\|x\|^{2}$ when $\Phi$
satisfies the restricted isometry property (RIP) of order $K$. In contrast, we
now show that the strong coherence property also guarantees $\|\Phi
x\|^{2}\approx\|x\|^{2}$ for most $K$-sparse vectors. We start with a
definition:
###### Definition 44.
We say an $M\times N$ frame $\Phi$ satisfies the _$(K,\delta,p)$ -weak
restricted isometry property (weak RIP)_ if for every $K$-sparse vector
$y\in\mathbb{C}^{N}$, a random permutation $x$ of $y$’s entries satisfies
$(1-\delta)\|x\|^{2}\leq\|\Phi x\|^{2}\leq(1+\delta)\|x\|^{2}$ (4.3)
with probability exceeding $1-p$.
At first glance, it may seem odd that we introduce a random permutation when
we might as well define weak RIP in terms of a $K$-sparse vector whose support
is drawn randomly from all $\smash{\binom{N}{K}}$ possible choices. In fact,
both versions would be equivalent in distribution, but we stress that in the
present definition, the values of the nonzero entries of $x$ are _not_ random;
rather, the only randomness we have is in the locations of the nonzero
entries. We wish to distinguish our results from those in [35], which
explicitly require randomness in the values of the nonzero entries. We also
note the distinction between RIP and weak RIP—weak RIP requires that $\Phi$
preserves the energy of _most_ sparse vectors. Moreover, the manner in which
we quantify “most” is important. For each sparse vector, $\Phi$ preserves the
energy of most permutations of that vector, but for different sparse vectors,
$\Phi$ might not preserve the energy of permutations with the same support.
That is, unlike RIP, weak RIP is _not_ a statement about the singular values
of submatrices of $\Phi$. Certainly, matrices for which most submatrices are
well-conditioned, such as those discussed in [135, 136], will satisfy weak
RIP, but weak RIP does not require this. That said, the following theorem
shows, in part, the significance of the strong coherence property.
###### Theorem 45.
Any $M\times N$ unit norm frame $\Phi$ with the strong coherence property
satisfies the $(K,\delta,\frac{4K}{N^{2}})$-weak restricted isometry property
provided $N\geq 128$ and
$\smash{2K\log{N}\leq\min\\{\frac{\delta^{2}}{100\mu^{2}},M\\}}$.
###### Proof.
Let $x$ be as in Definition 44. Note that (4.3) is equivalent to
$\big{|}\|\Phi x\|^{2}-\|x\|^{2}\big{|}\leq\delta\|x\|^{2}$. Defining
$\mathcal{K}:=\\{n:|x_{n}|>0\\}$, then the Cauchy-Schwarz inequality gives
$\displaystyle\big{|}\|\Phi x\|^{2}-\|x\|^{2}\big{|}$
$\displaystyle=|x_{\mathcal{K}}^{*}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})x_{\mathcal{K}}|$
$\displaystyle\leq\|x_{\mathcal{K}}\|\|(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})x_{\mathcal{K}}\|\leq\sqrt{K}\|x_{\mathcal{K}}\|\|(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})x_{\mathcal{K}}\|_{\infty},$
(4.4)
where the last inequality uses the fact that
$\|\cdot\|\leq\sqrt{K}\|\cdot\|_{\infty}$ in $\mathbb{C}^{K}$. We now consider
Lemma 3 of [11], which states that for any $\varepsilon\in[0,1)$ and $a\geq
1$,
$\|(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K})x_{\mathcal{K}}\|_{\infty}\leq\varepsilon\|x_{\mathcal{K}}\|$
with probability exceeding
$1-4Ke^{-(\varepsilon-\sqrt{K}\nu)^{2}/16(2+a^{-1})^{2}\mu^{2}}$ provided
$K\leq\min\\{\varepsilon^{2}\nu^{-2},(1+a)^{-1}N\\}$. We claim that (4.4)
together with Lemma 3 of [11] guarantee $\big{|}\|\Phi
x\|^{2}-\|x\|^{2}\big{|}\leq\delta\|x\|^{2}$ with probability exceeding
$1-\frac{4K}{N^{2}}$. In order to establish this claim, we fix
$\varepsilon=10\mu\sqrt{2\log{N}}$ and $a=2\log{128}-1$. It is then easy to
see that (SCP-1) gives $\varepsilon<1$, and also that (SCP-2) and
$2K\log{N}\leq M$ give $K\leq\varepsilon^{2}\nu^{-2}/9$. Therefore, since the
assumption that $N\geq 128$ together with $2K\log{N}\leq M$ implies
$K\leq(1+a)^{-1}N$, we obtain
$e^{-(\varepsilon-\sqrt{K}\nu)^{2}/16(2+a^{-1})^{2}\mu^{2}}\leq\frac{1}{N^{2}}$.
The result now follows from the observation that
$2K\log{N}\leq\frac{\delta^{2}}{100\mu^{2}}$ implies
$\sqrt{K}\varepsilon\leq\delta$. ∎
This theorem shows that having small worst-case and average coherence is
enough to guarantee weak RIP. This contrasts with related results by Tropp
[135, 136] that require $\Phi$ to be nearly tight. In fact, the proof of
Theorem 45 does not even use the full power of the strong coherence property;
instead of (SCP-1), it suffices to have $\mu\leq 1/(15\sqrt{\log N})$, part of
what [11] calls the coherence property. Also, if $\Phi$ has worst-case
coherence $\mu=\mathrm{O}(1/\sqrt{M})$ and average coherence
$\nu=\mathrm{O}(1/M)$, then even if $\Phi$ has large spectral norm, Theorem 45
states that $\Phi$ preserves the energy of most $K$-sparse vectors with
$K=\mathrm{O}(M/\log N)$, i.e., the sparsity regime which is linear in the
number of measurements.
#### 4.1.2 Reconstruction of sparse signals from noisy measurements
Another common task in signal processing applications is to reconstruct a
$K$-sparse signal $x\in\mathbb{C}^{N}$ from a small collection of linear
measurements $y\in\mathbb{C}^{M}$. Recently, Tropp [136] used both the worst-
case coherence and spectral norm of frames to find bounds on the
reconstruction performance of _basis pursuit (BP)_ [48] for most support sets
under the assumption that the nonzero entries of $x$ are independent with zero
median. In contrast, [11] used the spectral norm and worst-case and average
coherence of frames to find bounds on the reconstruction performance of OST
for most support sets and _arbitrary_ nonzero entries. However, both [11] and
[136] limit themselves to recovering $x$ in the absence of noise,
corresponding to $y=\Phi x$, a rather ideal scenario.
Our goal in this section is to provide guarantees for the reconstruction of
sparse signals from noisy measurements $y=\Phi x+z$, where the entries of the
noise vector $z\in\mathbb{C}^{M}$ are independent, identical complex-Gaussian
random variables with mean zero and variance $\sigma^{2}$. In particular, and
in contrast with [62], our guarantees will hold for arbitrary unit norm frames
$\Phi$ without requiring the signal’s sparsity level to satisfy
$K=\mathrm{O}(\mu^{-1})$. The reconstruction algorithm that we analyze here is
the OST algorithm of [11], which is described in Algorithm 1. The following
theorem extends the analysis of [11] and shows that the OST algorithm leads to
near-optimal reconstruction error for certain important classes of sparse
signals.
Before proceeding further, we first define some notation. We use
$\textsc{snr}:=\|x\|^{2}/\mathbb{E}[\|z\|^{2}]$ to denote the _signal-to-noise
ratio_ associated with the signal reconstruction problem. Also, we use
$\mathcal{T}_{\sigma}(t):=\bigg{\\{}n:|x_{n}|>\frac{2\sqrt{2}}{1-t}\sqrt{2\sigma^{2}\log{N}}\bigg{\\}}$
for any $t\in(0,1)$ to denote the locations of all the entries of $x$ that,
roughly speaking, lie above the _noise floor_ $\sigma$. Finally, we use
$\mathcal{T}_{\mu}(t):=\bigg{\\{}n:|x_{n}|>\frac{20}{t}\mu\|x\|\sqrt{2\log{N}}\bigg{\\}}$
to denote the locations of entries that, roughly speaking, lie above the
_self-interference floor_ $\mu\|x\|$.
Algorithm 1 One-Step Thresholding (OST) for sparse signal reconstruction [11]
Input: An $M\times N$ unit norm frame $\Phi$, a vector $y=\Phi x+z$, and a
threshold $\lambda>0$
Output: An estimate $\hat{x}\in\mathbb{C}^{N}$ of the true sparse signal $x$
$\hat{x}\leftarrow 0$ {Initialize}
$\tilde{x}\leftarrow\Phi^{*}y$ {Form signal proxy}
$\hat{\mathcal{K}}\leftarrow\\{n:|\tilde{x}_{n}|>\lambda\\}$ {Select indices
via OST}
$\hat{x}_{\hat{\mathcal{K}}}\leftarrow(\Phi_{\hat{\mathcal{K}}})^{\dagger}y$
{Reconstruct signal via least-squares}
###### Theorem 46 (Reconstruction of sparse signals).
Take an $M\times N$ unit norm frame $\Phi$ which satisfies the strong
coherence property, pick $t\in(0,1)$, and choose
$\lambda=\sqrt{2\sigma^{2}\log{N}}\max\\{\frac{10}{t}\mu\sqrt{M\textsc{snr}},\frac{\sqrt{2}}{1-t}\\}$.
Further, suppose $x\in\mathbb{C}^{N}$ has support $\mathcal{K}$ drawn
uniformly at random from all possible $K$-subsets of $\\{1,\ldots,N\\}$. Then
provided
$K\leq\frac{N}{c_{1}^{2}\|\Phi\|_{2}^{2}\log{N}},$ (4.5)
Algorithm 1 produces $\hat{\mathcal{K}}$ such that
$\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}\subseteq\mathcal{K}$
and $\hat{x}$ such that
$\|x-\hat{x}\|\leq
c_{2}\sqrt{\sigma^{2}|\hat{\mathcal{K}}|\log{N}}+c_{3}\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|$
(4.6)
with probability exceeding $1-10N^{-1}$. Finally, defining
$T:=|\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)|$, we further have
$\|x-\hat{x}\|\leq c_{2}\sqrt{\sigma^{2}K\log{N}}+c_{3}\|x-x_{T}\|$ (4.7)
in the same probability event. Here, $c_{1}=37e$,
$c_{2}=\frac{2}{1-e^{-1/2}}$, and $c_{3}=1+\frac{e^{-1/2}}{1-e^{-1/2}}$ are
numerical constants.
###### Proof.
To begin, note that since $\|\Phi\|_{2}^{2}\geq\frac{N}{M}$, we have from
(4.5) that $K\leq M/(2\log{N})$. It is then easy to conclude from Theorem 5 of
[11] that $\hat{\mathcal{K}}$ satisfies
$\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}\subseteq\mathcal{K}$
with probability exceeding $1-6N^{-1}$. Therefore, conditioned on the event
$\mathcal{E}_{1}:=\\{\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}\subseteq\mathcal{K}\\}$,
we can make use of the triangle inequality to write
$\|x-\hat{x}\|\leq\|x_{\hat{\mathcal{K}}}-\hat{x}_{\hat{\mathcal{K}}}\|+\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|.$
(4.8)
Next, we may use (4.5) and the fact that $\Phi$ satisfies the strong coherence
property to conclude from [135] (see, e.g., Proposition 3 of [11]) that
$\|\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}\|_{2}<e^{-1/2}$ with
probability exceeding $1-2N^{-1}$. Hence, conditioning on $\mathcal{E}_{1}$
and
$\mathcal{E}_{2}:=\\{\|\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}\|_{2}<e^{-1/2}\\}$,
we have that
$(\Phi_{\hat{\mathcal{K}}})^{\dagger}=(\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\hat{\mathcal{K}}})^{-1}\Phi_{\hat{\mathcal{K}}}^{*}$
since $\Phi_{\hat{\mathcal{K}}}$ is a submatrix of a full column rank matrix
$\Phi_{\mathcal{K}}$. Therefore, given $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$, we may write
$\hat{x}_{\hat{\mathcal{K}}}=(\Phi_{\hat{\mathcal{K}}})^{\dagger}(\Phi
x+z)=x_{\hat{\mathcal{K}}}+(\Phi_{\hat{\mathcal{K}}})^{\dagger}\Phi_{\mathcal{K}\setminus\hat{\mathcal{K}}}x_{\mathcal{K}\setminus\hat{\mathcal{K}}}+(\Phi_{\hat{\mathcal{K}}})^{\dagger}z,$
(4.9)
and so substituting (4.9) into (4.8) and applying the triangle inequality
gives
$\displaystyle\|x-\hat{x}\|$
$\displaystyle\leq\|(\Phi_{\hat{\mathcal{K}}})^{\dagger}\Phi_{\mathcal{K}\setminus\hat{\mathcal{K}}}x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|+\|(\Phi_{\hat{\mathcal{K}}})^{\dagger}z\|+\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|$
$\displaystyle\leq\Big{(}1+\|(\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\hat{\mathcal{K}}})^{-1}\|_{2}\|\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|_{2}\Big{)}\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|+\|(\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\hat{\mathcal{K}}})^{-1}\|_{2}\|\Phi_{\hat{\mathcal{K}}}^{*}z\|.$
(4.10)
Since, given $\mathcal{E}_{1}$, we have that
$\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\hat{\mathcal{K}}}-I_{K}$ and
$\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\mathcal{K}\setminus\hat{\mathcal{K}}}$ are
submatrices of $\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-I_{K}$, and since the
spectral norm of a matrix provides an upper bound for the spectral norms of
its submatrices, we have the following given $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$:
$\|\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|_{2}\leq
e^{-1/2}$ and
$\|(\Phi_{\hat{\mathcal{K}}}^{*}\Phi_{\hat{\mathcal{K}}})^{-1}\|_{2}\leq\frac{1}{1-e^{-1/2}}$.
We can now substitute these bounds into (4.10) and make use of the fact that
$\|\Phi_{\hat{\mathcal{K}}}^{*}z\|\leq|\hat{\mathcal{K}}|^{1/2}\|\Phi_{\hat{\mathcal{K}}}^{*}z\|_{\infty}$
to conclude that
$\|x-\hat{x}\|\leq\frac{|\hat{\mathcal{K}}|^{1/2}}{1-e^{-1/2}}\|\Phi_{\hat{\mathcal{K}}}^{*}z\|_{\infty}+\Big{(}1+\frac{e^{-1/2}}{1-e^{-1/2}}\Big{)}\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|,$
given $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. At this point, define the event
$\mathcal{E}_{3}=\\{\|\Phi_{\hat{\mathcal{K}}}^{*}z\|_{\infty}\leq
2\sqrt{\sigma^{2}\log{N}}\\}$ and note from Lemma 6 of [11] that
$\Pr(\mathcal{E}_{3}^{\mathrm{c}})\leq 2(\sqrt{2\pi\log{N}}~{}N)^{-1}$. A
union bound therefore gives (4.6) with probability exceeding $1-10N^{-1}$. For
(4.7), note that $\hat{\mathcal{K}}\subseteq\mathcal{K}$ implies
$|\hat{\mathcal{K}}|\leq K$, and so
$\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}$
implies that
$\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|\leq\|x_{\mathcal{K}\setminus(\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t))}\|=\|x-x_{T}\|$.
∎
A few remarks are in order now for Theorem 46. First, if $\Phi$ satisfies the
strong coherence property _and_ $\Phi$ is nearly tight, then OST handles
sparsity that is almost linear in $M$: $K=\mathrm{O}(M/\log{N})$ from (4.5).
Second, we do not impose any control over the size of $T$, but rather we state
the result in generality in terms of $T$; its size is determined by the signal
class $x$ belongs to, the worst-case coherence of the frame $\Phi$ we use to
measure $x$, and the magnitude of the noise that perturbs $\Phi x$. Third, the
$\ell_{2}$ error associated with the OST algorithm is the near-optimal (modulo
the $\log$ factor) error of $\sqrt{\sigma^{2}K\log{N}}$ _plus_ the best
$T$-term approximation error caused by the inability of the OST algorithm to
recover signal entries that are smaller than
$\mathrm{O}(\mu\|x\|\sqrt{2\log{N}})$. In particular, if the $K$-sparse signal
$x$, the worst-case coherence $\mu$, and the noise $z$ together satisfy
$\|x-x_{T}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$, then the OST algorithm
succeeds with a near-optimal $\ell_{2}$ error of
$\|x-\hat{x}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$. To see why this error
is near-optimal, note that a $K$-dimension vector of random entries with mean
zero and variance $\sigma^{2}$ has expected squared norm $\sigma^{2}K$; in our
case, we pay an additional log factor to find the locations of the $K$ nonzero
entries among the entire $N$-dimensional signal. It is important to recognize
that the optimality condition
$\|x-x_{T}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$ depends on the signal
class, the noise variance, and the worst-case coherence of the frame; in
particular, the condition is satisfied whenever
$\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$,
since
$\|x-x_{T}\|\leq\|x_{\mathcal{K}\setminus\mathcal{T}_{\sigma}(t)}\|+\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=\mathrm{O}\Big{(}\sqrt{\sigma^{2}K\log{N}}\Big{)}+\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|.$
The following lemma provides classes of sparse signals that satisfy
$\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$
given sufficiently small noise variance and worst-case coherence, and
consequently the OST algorithm is near-optimal for the reconstruction of such
signal classes.
###### Lemma 47.
Take an $M\times N$ unit norm frame $\Phi$ with worst-case coherence
$\mu\leq\frac{c_{0}}{\sqrt{M}}$ for some $c_{0}>0$, and suppose that
$K\leq\frac{N}{c_{1}^{2}\|\Phi\|_{2}^{2}\log N}$ for some $c_{1}>0$. Fix a
constant $\beta\in(0,1]$, and suppose the magnitudes of $\beta K$ nonzero
entries of $x$ are some $\alpha=\Omega(\sqrt{\sigma^{2}\log{N}})$, while the
magnitudes of the remaining $(1-\beta)K$ nonzero entries are not necessarily
same, but are smaller than $\alpha$ and scale as
$\mathrm{O}(\sqrt{\sigma^{2}\log{N}})$. Then
$\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$,
provided $c_{0}\leq\frac{tc_{1}}{20\sqrt{2}}$.
###### Proof.
Let $\mathcal{K}$ be the support of $x$, and define
$\mathcal{I}:=\\{n:|x_{n}|=\alpha\\}$. We wish to show that
$\mathcal{I}\subseteq\mathcal{T}_{\mu}(t)$, since this implies
$\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|\leq\|x_{\mathcal{K}\setminus\mathcal{I}}\|=\mathrm{O}(\sqrt{\sigma^{2}K\log{N}})$.
In order to prove $\mathcal{I}\subseteq\mathcal{T}_{\mu}(t)$, notice that
$\|x\|^{2}=\|x_{\mathcal{I}}\|^{2}+\|x_{\mathcal{K}\setminus\mathcal{I}}\|^{2}<\beta
K\alpha^{2}+(1-\beta)K\alpha^{2}=K\alpha^{2},$
and so combining this with the fact that $\|\Phi\|_{2}^{2}\geq\frac{N}{M}$
gives
$\mu\|x\|\sqrt{\log{N}}<\frac{c_{0}}{\sqrt{M}}\sqrt{K}\alpha\sqrt{\log{N}}\leq\frac{c_{0}}{\sqrt{M}}\sqrt{\frac{N}{c_{1}^{2}\|\Phi\|_{2}^{2}\log
N}}~{}\alpha\sqrt{\log{N}}\leq\frac{c_{0}}{c_{1}}\alpha.$
Therefore, provided $c_{0}\leq\frac{tc_{1}}{20\sqrt{2}}$, we have that
$\mathcal{I}\subseteq\mathcal{T}_{\mu}(t)$. ∎
In words, Lemma 47 implies that OST is near-optimal for those $K$-sparse
signals whose entries above the noise floor have roughly the same magnitude.
This subsumes a very important class of signals that appears in applications
such as multi-label prediction [86], in which all the nonzero entries take
values $\pm\alpha$. Theorem 46 is the first result in the sparse signal
processing literature that does not require RIP and still provides near-
optimal reconstruction guarantees for such signals from noisy measurements,
while using either random or deterministic frames, even when
$K=\mathrm{O}(M/\log{N})$.
Note that our techniques can be extended to reconstruct noisy signals, that
is, we may consider measurements of the form $y=\Phi(x+n)+z$, where
$n\in\mathbb{C}^{N}$ is also a noise vector of independent, identical zero-
mean complex-Gaussian random variables. In particular, if the frame $\Phi$ is
tight, then our measurements will not color the noise, and so noise in the
signal may be viewed as noise in the measurements: $y=\Phi x+(\Phi n+z)$; if
the frame is not tight, then the noise will become correlated in the
measurements, and performance would be depend nontrivially on the frame’s Gram
matrix. Also, Theorem 46 can be generalized to approximately sparse signals;
the analysis follows similiar lines, but is rather cumbersome, and it appears
as though the end result is only strong enough in the case of very nearly
sparse signals. As such, we omit this result.
### 4.2 Frame constructions
In this section, we consider a range of nearly tight frames with small worst-
case and average coherence. We investigate various ways of selecting frames at
random from different libraries, and we show that for each of these frames,
the spectral norm, worst-case coherence, and average coherence are all small
with high probability. Later, we will consider deterministic constructions
that use Gabor and chirp systems, spherical designs, equiangular tight frames,
and error-correcting codes. For the reader’s convenience, all of these
constructions are summarized in Table 4.1. Before we go any further, we
consider the following lemma, which gives three different sufficient
conditions for a frame to satisfy (SCP-2). These conditions will prove quite
useful in this section and throughout the chapter.
###### Lemma 48.
For any $M\times N$ unit norm frame $\Phi$, each of the following conditions
implies $\nu\leq\frac{\mu}{\sqrt{M}}$:
1. (i)
$\langle\varphi_{k},\sum_{n=1}^{N}\varphi_{n}\rangle=\frac{N}{M}$ for every
$k=1,\ldots,N$,
2. (ii)
$N\geq 2M$ and $\sum_{n=1}^{N}\varphi_{n}=0$,
3. (iii)
$N\geq M^{2}+3M+3$ and $\|\sum_{n=1}^{N}\varphi_{n}\|^{2}\leq N$.
###### Proof.
For condition (i), we have
$\nu=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|}=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\bigg{\langle}\varphi_{i},\sum_{j=1}^{N}\varphi_{j}\bigg{\rangle}-1\bigg{|}=\frac{1}{N-1}\bigg{(}\frac{N}{M}-1\bigg{)}.$
The Welch bound (Theorem 3) therefore gives
$\nu=\frac{1}{N-1}\big{(}\frac{N}{M}-1\big{)}=\frac{N-M}{M(N-1)}\leq\mu\sqrt{\frac{N-M}{M(N-1)}}\leq\frac{\mu}{\sqrt{M}}$.
For condition (ii), we have
$\nu=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|}=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\bigg{\langle}\varphi_{i},\sum_{j=1}^{N}\varphi_{j}\bigg{\rangle}-1\bigg{|}=\frac{1}{N-1}.$
Considering the Welch bound, it suffices to show
$\frac{1}{N-1}\leq\frac{1}{\sqrt{M}}\sqrt{\frac{N-M}{M(N-1)}}$. Rearranging
gives
$N^{2}-(M+1)N-M(M-1)\geq 0.$ (4.11)
When $N=2M$, the left-hand side of (4.11) becomes $(M-1)^{2}$, which is
trivially nonnegative. Otherwise, we have
$N\geq 2M+1\geq
M+1+\sqrt{M(M-1)}\geq\frac{M+1}{2}+\sqrt{\Big{(}\frac{M+1}{2}\Big{)}^{2}+M(M-1)}.$
In this case, by the quadratic formula and the fact that the left-hand side of
(4.11) is concave up in $N$, we have that (4.11) is indeed satisfied. For
condition (iii), we use the triangle and Cauchy-Schwarz inequalities to get
$\nu=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\bigg{\langle}\varphi_{i},\sum_{j=1}^{N}\varphi_{j}\bigg{\rangle}-1\bigg{|}\leq\frac{1}{N-1}\bigg{(}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\bigg{\langle}\varphi_{i},\sum_{j=1}^{N}\varphi_{j}\bigg{\rangle}\bigg{|}+1\bigg{)}\leq\frac{\sqrt{N}+1}{N-1}.$
Considering the Welch bound, it suffices to show
$\frac{\sqrt{N}+1}{N-1}\leq\frac{1}{\sqrt{M}}\sqrt{\frac{N-M}{M(N-1)}}$.
Taking $x:=\sqrt{N}$ and rearranging gives a polynomial:
$x^{4}-(M^{2}+M+1)x^{2}-2M^{2}x-M(M-1)\geq 0$. By convexity and monotonicity
of the polynomial in $[M+\frac{3}{2},\infty)$, it can be shown that the
largest real root of this polynomial is always smaller than $M+\frac{3}{2}$.
Also, considering it is concave up in $x$, it suffices that $\sqrt{N}=x\geq
M+\frac{3}{2}$, which we have since $N\geq M^{2}+3M+3\geq(M+\frac{3}{2})^{2}$.
∎
#### 4.2.1 Normalized Gaussian frames
Construct a matrix with independent, Gaussian-distributed entries that have
zero mean and unit variance. By normalizing the columns, we get a matrix
called a _normalized Gaussian frame_. This is perhaps the most widely studied
type of frame in the signal processing and statistics literature. To be clear,
the term “normalized” is intended to distinguish the results presented here
from results reported in earlier works, such as [11, 17, 38, 140], which only
ensure that Gaussian frame elements have unit norm in expectation. In other
words, normalized Gaussian frame elements are independently and uniformly
distributed on the unit hypersphere in $\mathbb{R}^{M}$. The following theorem
characterizes the spectral norm and the worst-case and average coherence of
normalized Gaussian frames.
###### Theorem 49 (Geometry of normalized Gaussian frames).
Build a real $M\times N$ frame $\Psi$ by drawing entries independently at
random from a Gaussian distribution of zero mean and unit variance. Next,
construct a normalized Gaussian frame $\Phi$ by taking
$\varphi_{n}:=\frac{\psi_{n}}{\|\psi_{n}\|}$ for every $n=1,\ldots,N$.
Provided $60\log{N}\leq M\leq\frac{N-1}{4\log{N}}$, then the following
simultaneously hold with probability exceeding $1-11N^{-1}$:
1. (i)
$\mu\leq\frac{\sqrt{15\log{N}}}{\sqrt{M}-\sqrt{12\log{N}}}$,
2. (ii)
$\nu\leq\frac{\sqrt{15\log{N}}}{M-\sqrt{12M\log{N}}}$,
3. (iii)
$\|\Phi\|_{2}\leq\frac{\sqrt{M}+\sqrt{N}+\sqrt{2\log{N}}}{\sqrt{M-\sqrt{8M\log{N}}}}$.
###### Proof.
Theorem 49(i) can be shown to hold with probability exceeding $1-2N^{-1}$ by
using a bound on the norm of a Gaussian random vector in Lemma 1 of [95] and a
bound on the magnitude of the inner product of two independent Gaussian random
vectors in Lemma 6 of [79]. Specifically, pick any two distinct indices
$i,j\in\\{1,\dots,N\\}$, and define probability events
$\mathcal{E}_{1}:=\\{|\langle\psi_{i},\psi_{j}\rangle|\leq\varepsilon_{1}\\}$,
$\mathcal{E}_{2}:=\\{\|\psi_{i}\|^{2}\geq M(1-\varepsilon_{2})\\}$, and
$\mathcal{E}_{3}:=\\{\|\psi_{j}\|^{2}\geq M(1-\varepsilon_{2})\\}$ for
$\varepsilon_{1}=\sqrt{15M\log{N}}$ and
$\varepsilon_{2}=\sqrt{(12\log{N})/M}$. Then it follows from the union bound
that
$\Pr\bigg{(}|\langle\varphi_{i},\varphi_{j}\rangle|>\frac{\varepsilon_{1}}{M(1-\varepsilon_{2})}\bigg{)}=\Pr\bigg{(}\frac{|\langle\psi_{i},\psi_{j}\rangle|}{\|\psi_{i}\|\|\psi_{j}\|}>\frac{\varepsilon_{1}}{M(1-\varepsilon_{2})}\bigg{)}\leq\Pr(\mathcal{E}_{1}^{\mathrm{c}})+\Pr(\mathcal{E}_{2}^{\mathrm{c}})+\Pr(\mathcal{E}_{3}^{\mathrm{c}}).$
One can verify that
$\Pr(\mathcal{E}_{2}^{\mathrm{c}})=\Pr(\mathcal{E}_{3}^{\mathrm{c}})\leq
N^{-3}$ because of Lemma 1 of [95], and we further have
$\Pr(\mathcal{E}_{1}^{\mathrm{c}})\leq 2N^{-3}$ because of Lemma 6 of [79] and
the fact that $M\geq 60\log{N}$. Thus, for any fixed $i$ and $j$,
$|\langle\varphi_{i},\varphi_{j}\rangle|\leq\sqrt{15\log{N}}/(\sqrt{M}-\sqrt{12\log{N}})$
with probability exceeding $1-4N^{-3}$. It therefore follows by taking a union
bound over all $\binom{N}{2}$ choices for $i$ and $j$ that Theorem 49(i) holds
with probability exceeding $1-2N^{-1}$.
Theorem 49(ii) can be shown to hold with probability exceeding $1-6N^{-1}$ by
appealing to the preceding analysis and Hoeffding’s inequality for a sum of
independent, bounded random variables [83]. Specifically, fix any index
$i\in\\{1,\dots,N\\}$, and define random variables
$Z_{ij}:=\frac{1}{N-1}\langle\varphi_{i},\varphi_{j}\rangle$. Next, define the
probability event
$\mathcal{E}_{4}:=\bigcap_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\bigg{\\{}|Z_{ij}|\leq\frac{1}{N-1}\frac{\sqrt{15\log{N}}}{\sqrt{M}-\sqrt{12\log{N}}}\bigg{\\}}.$
Using the analysis for the worst-case coherence of $\Phi$ and taking a union
bound over the $N-1$ possible $j$’s gives
$\Pr(\mathcal{E}_{4}^{\mathrm{c}})\leq 4N^{-2}$. Furthermore, taking
$\varepsilon_{3}:=\sqrt{15\log{N}}/(M-\sqrt{12M\log{N}})$, then elementary
probability analysis gives
$\displaystyle\Pr\Bigg{(}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{N}Z_{ij}\bigg{|}>\varepsilon_{3}\Bigg{)}$
$\displaystyle\leq\Pr\Bigg{(}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{N}Z_{ij}\bigg{|}>\varepsilon_{3}~{}\Bigg{|}~{}\mathcal{E}_{4}\Bigg{)}+\Pr(\mathcal{E}_{4}^{\mathrm{c}})$
$\displaystyle\leq\int_{\mathbb{S}^{M-1}}\\!\\!\\!\Pr\Bigg{(}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{N}Z_{ij}\bigg{|}>\varepsilon_{3}~{}\Bigg{|}~{}\mathcal{E}_{4},\varphi_{i}=x\Bigg{)}~{}p_{\varphi_{i}}(x)~{}\mathrm{dH}^{M-1}(x)+4N^{-2},$
(4.12)
where $\mathbb{S}^{M-1}$ denotes the unit hypersphere in $\mathbb{R}^{M}$,
$\mathrm{H}^{M-1}$ denotes the $(M-1)$-dimensional Hausdorff measure on
$\mathbb{S}^{M-1}$, and $p_{\varphi_{i}}(x)$ denotes the probability density
function for the random vector $\varphi_{i}$. The first thing to note here is
that the random variables $\\{Z_{ij}:j\not=i\\}$ are bounded and jointly
independent when conditioned on $\mathcal{E}_{4}$ and $\varphi_{i}$. This
assertion mainly follows from Bayes’ rule and the fact that
$\\{\varphi_{j}:j\not=i\\}$ are jointly independent when conditioned on
$\varphi_{i}$. The second thing to note is that
$\mathbb{E}[Z_{ij}~{}|~{}\mathcal{E}_{4},\varphi_{i}]=0$ for every $j\neq i$.
This comes from the fact that the random vectors $\\{\varphi_{n}\\}_{n=1}^{N}$
are independent and have a uniform distribution over $\mathbb{S}^{M-1}$, which
in turn guarantees that the random variables $\\{Z_{ij}:j\not=i\\}$ have a
symmetric distribution around zero when conditioned on $\mathcal{E}_{4}$ and
$\varphi_{i}$. We can therefore make use of Hoeffding’s inequality [83] to
bound the probability expression inside the integral in (4.12) as
$\Pr\Bigg{(}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{N}Z_{ij}\bigg{|}>\varepsilon_{3}~{}\Bigg{|}~{}\mathcal{E}_{4},\varphi_{i}=x\Bigg{)}\leq
2e^{-(N-1)/2M},$ (4.13)
which is bounded above by $2N^{-2}$ provided $M\leq\frac{N-1}{4\log{N}}$. We
can now substitute (4.13) into (4.12) and take the union bound over the $N$
possible choices for $i$ to conclude that Theorem 49(ii) holds with
probability exceeding $1-6N^{-1}$.
Lastly, Theorem 49(iii) can be shown to hold with probability exceeding
$1-3N^{-1}$ by using a bound on the spectral norm of standard Gaussian random
matrices reported in [117] along with Lemma 1 of [95]. Specifically, define an
$N\times N$ diagonal matrix
$D:=\mathrm{diag}(\|\psi_{1}\|^{-1},\dots,\|\psi_{N}\|^{-1})$, and note that
the entries of $\Psi:=\Phi D^{-1}$ are independently and normally distributed
with zero mean and unit variance. We therefore have from (2.3) in [117] that
$\Pr\Big{(}\|\Psi\|_{2}>\sqrt{M}+\sqrt{N}+\sqrt{2\log{N}}\Big{)}\leq 2N^{-1}.$
(4.14)
In addition, we can appeal to the preceding analysis for the probability bound
on Theorem 49(i) and conclude using Lemma 1 of [95] and a union bound over the
$N$ possible choices for $i$ that
$\Pr\Big{(}\|D\|_{2}>\Big{(}M-\sqrt{8M\log{N}}\Big{)}^{-1/2}\Big{)}\leq
N^{-1}.$ (4.15)
Finally, since $\|\Phi\|_{2}\leq\|\Psi\|_{2}\|D\|_{2}$, we can take a union
bound over (4.14) and (4.15) to argue that Theorem 49(iii) holds with
probability exceeding $1-3N^{-1}$.
The complete result now follows by taking a union bound over the failure
probabilities for the conditions (i)-(iii) in Theorem 49. ∎
###### Example 50.
To illustrate the bounds in Theorem 49, we ran simulations in MATLAB. Picking
$N=50000$, we observed $30$ realizations of normalized Gaussian frames for
each $M=700,900,1100$. The distributions of $\mu$, $\nu$, and $\|\Phi\|_{2}$
were rather tight, so we only report the ranges of values attained, along with
the bounds given in Theorem 49:
$\begin{array}[]{rrcll}M=700:&\qquad\mu&\in&[0.1849,0.2072]&\qquad\leq
0.8458\\\ &\qquad\nu&\in&[0.5643,0.6613]\times 10^{-3}&\qquad\leq 0.0320\\\
&\qquad\|\Phi\|_{2}&\in&[8.0521,8.0835]&\qquad\leq 11.9565\\\ \\\
M=900:&\qquad\mu&\in&[0.1946,0.2206]&\qquad\leq 0.6848\\\
&\qquad\nu&\in&[0.5800,0.7501]\times 10^{-3}&\qquad\leq 0.0229\\\
&\qquad\|\Phi\|_{2}&\in&[8.4352,8.4617]&\qquad\leq 10.3645\\\ \\\
M=1100:&\qquad\mu&\in&[0.1807,0.1988]&\qquad\leq 0.5852\\\
&\qquad\nu&\in&[0.5260,0.6713]\times 10^{-3}&\qquad\leq 0.0177\\\
&\qquad\|\Phi\|_{2}&\in&[7.7262,7.7492]&\qquad\leq 9.2927\end{array}$
These simulations seem to indicate that our bounds on $\mu$ and $\|\Phi\|_{2}$
reflect real-world behavior, at least within an order of magnitude, whereas
the bound on $\nu$ is rather loose.
#### 4.2.2 Random harmonic frames
Random harmonic frames, constructed by randomly selecting rows of a discrete
Fourier transform (DFT) matrix and normalizing the resulting columns, have
received considerable attention lately in the compressed sensing literature
[36, 39, 118]. However, there is no result in the literature that gives the
worst-case coherence of random harmonic frames. To fill this gap, the
following theorem gives the spectral norm and the worst-case and average
coherence of random harmonic frames.
###### Theorem 51 (Geometry of random harmonic frames).
Let $F$ be an $N\times N$ non-normalized discrete Fourier transform matrix,
explicitly, $F_{k\ell}:=e^{2\pi\mathrm{i}k\ell/N}$ for each
$k,\ell=0,\ldots,N-1$. Next, let $\\{B_{i}\\}_{i=0}^{N-1}$ be a collection of
independent Bernoulli random variables with mean $\frac{M}{N}$, and take
$\mathcal{M}:=\\{i:B_{i}=1\\}$. Finally, construct an $|\mathcal{M}|\times N$
harmonic frame $\Phi$ by collecting rows of $F$ which correspond to indices in
$\mathcal{M}$ and normalizing the columns. Then $\Phi$ is a unit norm tight
frame: $\|\Phi\|_{2}^{2}=\frac{N}{|\mathcal{M}|}$. Also, provided
$16\log{N}\leq M\leq\frac{N}{3}$, the following simultaneously hold with
probability exceeding $1-4N^{-1}-N^{-2}$:
1. (i)
$\frac{1}{2}M\leq|\mathcal{M}|\leq\frac{3}{2}M$,
2. (ii)
$\nu\leq\frac{\mu}{\sqrt{|\mathcal{M}|}}$,
3. (iii)
$\mu\leq\sqrt{\frac{118(N-M)\log{N}}{MN}}$.
###### Proof.
The claim that $\Phi$ is tight follows trivially from the fact that the rows
of $F$ are orthogonal and that the rows of $\Phi$ correspond to a subset of
the rows of $F$. Next, we define the probability events
$\mathcal{E}_{1}:=\\{|\mathcal{M}|\leq\frac{3}{2}M\\}$ and
$\mathcal{E}_{2}:=\\{|\mathcal{M}|\geq\frac{1}{2}M\\}$, and claim that
$\Pr(\mathcal{E}_{1}^{\mathrm{c}}\cup\mathcal{E}_{2}^{\mathrm{c}})\leq
N^{-1}+N^{-2}$. The proof of this claim follows from a Bernstein-like large
deviation inequality. Specifically, note that
$|\mathcal{M}|=\sum_{i=0}^{N-1}B_{i}$ with $\mathbb{E}[|\mathcal{M}|]=M$, and
so we have from Theorems A.1.12 and A.1.13 of [7] and page 4 of [118] that for
any $\varepsilon_{1}\in[0,1)$,
$\Pr\Big{(}|\mathcal{M}|>(1+\varepsilon_{1})M\Big{)}\leq
e^{-M\varepsilon_{1}^{2}(1-\varepsilon_{1})/2}\qquad\mbox{and}\qquad\Pr\Big{(}|\mathcal{M}|<(1-\varepsilon_{1})M\Big{)}\leq
e^{-M\varepsilon_{1}^{2}/2}.$ (4.16)
Taking $\varepsilon_{1}:=\frac{1}{2}$, then a union bound gives
$\Pr(\mathcal{E}_{1}^{\mathrm{c}}\cup\mathcal{E}_{2}^{\mathrm{c}})\leq
N^{-1}+N^{-2}$ provided $M\geq 16\log{N}$. Conditioning on
$\mathcal{E}_{1}\cap\mathcal{E}_{2}$, we have that Theorem 51(i) holds
trivially, while Theorem 51(ii) follows from Lemma 48. Specifically, we have
that $\frac{N}{3}\geq M$ guarantees $N\geq 2|\mathcal{M}|$ because of the
conditioning on $\mathcal{E}_{1}\cap\mathcal{E}_{2}$, which in turn implies
that $\Phi$ satisfies either condition (i) or (ii) of Lemma 48, depending on
whether $0\in\mathcal{M}$. This therefore establishes that Theorem 51(i)-(ii)
simultaneously hold with probability exceeding $1-N^{-1}-N^{-2}$.
The only remaining claim is that
$\mu\leq\varepsilon_{2}:=\sqrt{(118(N-M)\log{N})/MN}$ with high probability.
To this end, define $p:=\frac{M}{N}$, and pick any two distinct indices
$i,j\in\\{0,\dots,N-1\\}$. Note that
$\langle\varphi_{i},\varphi_{j}\rangle=\frac{1}{|\mathcal{M}|}\sum_{k=0}^{N-1}B_{k}F_{ki}\overline{F_{kj}}=\frac{1}{|\mathcal{M}|}\sum_{k=0}^{N-1}(B_{k}-p)F_{ki}\overline{F_{kj}},$
(4.17)
where the last equality follows from the fact that $F$ has orthogonal columns.
Next, we write
$F_{ki}\overline{F_{kj}}=\cos(\theta_{k})+\mathrm{i}\sin(\theta_{k})$ for some
$\theta_{k}\in[0,2\pi)$. Then applying the union bound to (4.17) and to the
real and imaginary parts of $F_{ki}\overline{F_{kj}}$ gives
$\displaystyle\Pr\Big{(}|\langle\varphi_{i},\varphi_{j}\rangle|>\varepsilon_{2}\Big{)}$
$\displaystyle\leq\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)F_{ki}\overline{F_{kj}}\Big{|}>\frac{M\varepsilon_{2}}{2\sqrt{2}}\bigg{)}+\Pr\Big{(}|\mathcal{M}|<\frac{M}{2\sqrt{2}}\Big{)}$
$\displaystyle\leq\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)\cos(\theta_{k})\Big{|}>\frac{M\varepsilon_{2}}{4}\bigg{)}+\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)\sin(\theta_{k})\Big{|}>\frac{M\varepsilon_{2}}{4}\bigg{)}+N^{-3},$
(4.18)
where the last term follows from (4.16) and the fact that $M\geq 16\log{N}$.
Define random variables $Z_{k}:=(B_{k}-p)\cos(\theta_{k})$. Note that the
$Z_{k}$’s have zero mean and are jointly independent. Also, the $Z_{k}$’s are
bounded by $1-p$ almost surely since
$|(B_{k}-p)\cos(\theta_{k})|\leq\max\\{p,1-p\\}$ and $N\geq 2M$. Moreover, the
variance of each $Z_{k}$ is bounded: $\mathrm{Var}(Z_{\ell})\leq p(1-p)$.
Therefore, we may use the Bernstein inequality for a sum of independent,
bounded random variables [21] to bound the probability that
$|\sum_{k=0}^{N-1}Z_{k}|$ deviates from
$\varepsilon_{3}:=\frac{M\varepsilon_{2}}{4}$:
$\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)\cos(\theta_{k})\Big{|}>\varepsilon_{3}\bigg{)}\leq
2e^{-\varepsilon_{3}^{2}/(2Np(1-p)+2(1-p)\varepsilon_{3}/3)}\leq 2N^{-3}.$
Similarly, the probability that
$|\sum_{k=0}^{N-1}(B_{k}-p)\sin(\theta_{k})|>\varepsilon_{3}$ is also bounded
above by $2N^{-3}$. Substituting these probability bounds into (4.18) gives
$|\langle\varphi_{i},\varphi_{j}\rangle|>\varepsilon_{2}$ with probability at
most $5N^{-3}$ provided $M\geq 16\log{N}$. Finally, we take a union bound over
the $\binom{N}{2}$ possible choices for $i$ and $j$ to get that Theorem
51(iii) holds with probability exceeding $1-3N^{-1}$.
The result now follows by taking a final union bound over
$\mathcal{E}_{1}^{\mathrm{c}}\cup\mathcal{E}_{2}^{\mathrm{c}}$ and
$\\{\mu>\varepsilon_{2}\\}$. ∎
As stated earlier, random harmonic frames are not new to sparse signal
processing. Interestingly, for the application of compressed sensing, [38,
118] provides performance guarantees for both random harmonic and Gaussian
frames, but requires more rows in a random harmonic frame to accommodate the
same level of sparsity. This suggests that random harmonic frames may be
inferior to Gaussian frames as compressed sensing matrices, but practice
suggests otherwise [63]. In a sense, Theorem 51 helps to resolve this gap in
understanding; there exist compressed sensing algorithms whose performance is
dictated by worst-case coherence [11, 62, 134, 136], and Theorem 51 states
that random harmonic frames have near-optimal worst-case coherence, being on
the order of the Welch bound with an additional $\sqrt{\log N}$ factor.
###### Example 52.
To illustrate the bounds in Theorem 51, we ran simulations in MATLAB. Picking
$N=5000$, we observed $30$ realizations of random harmonic frames for each
$M=1000,1250,1500$. The distributions of $|\mathcal{M}|$, $\nu$, and $\mu$
were rather tight, so we only report the ranges of values attained, along with
the bounds given in Theorem 51. Notice that Theorem 51 gives a bound on $\nu$
in terms of both $|\mathcal{M}|$ and $\mu$. To simplify matters, we show that
$\nu\leq\frac{\min\mu}{\sqrt{\max|\mathcal{M}|}}\leq\frac{\mu}{\sqrt{|\mathcal{M}|}}$,
where the minimum and maximum are taken over all realizations in the sample:
$\begin{array}[]{rrcll}M=1000:&\qquad|\mathcal{M}|&\in&[961,1052]&\qquad\subseteq[500,1500]\\\
&\qquad\nu&\in&[0.2000,0.8082]\times 10^{-3}&\qquad\leq
0.0023\approx\tfrac{0.0746}{\sqrt{1052}}\\\
&\qquad\mu&\in&[0.0746,0.0890]&\qquad\leq 0.8967\\\ \\\
M=1250:&\qquad|\mathcal{M}|&\in&[1207,1305]&\qquad\subseteq[625,1875]\\\
&\qquad\nu&\in&[0.2000,0.6273]\times 10^{-3}&\qquad\leq
0.0018\approx\tfrac{0.0623}{\sqrt{1305}}\\\
&\qquad\mu&\in&[0.0623,0.0774]&\qquad\leq 0.7766\\\ \\\
M=1500:&\qquad|\mathcal{M}|&\in&[1454,1590]&\qquad\subseteq[750,2250]\\\
&\qquad\nu&\in&[0.2000,0.4841]\times 10^{-3}&\qquad\leq
0.0015\approx\tfrac{0.0571}{\sqrt{1590}}\\\
&\qquad\mu&\in&[0.0571,0.0743]&\qquad\leq 0.6849\end{array}$
The reader may have noticed how consistently the average coherence value of
$\nu\approx 0.2000\times 10^{-3}$ was realized. This occurs precisely when the
zeroth row of the DFT is not selected, as the frame elements sum to zero in
this case:
$\nu:=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|}=\frac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\bigg{\langle}\varphi_{i},\sum_{j=1}^{N}\varphi_{j}\bigg{\rangle}-\|\varphi_{i}\|^{2}\bigg{|}=\frac{1}{N-1}.$
These simulations seem to indicate that our bounds on $|\mathcal{M}|$, $\nu$,
and $\mu$ leave room for improvement. The only bound that lies within an order
of magnitude of real-world behavior is our bound on $|\mathcal{M}|$.
#### 4.2.3 Gabor and chirp frames
Gabor frames constitute an important class of frames, as they appear in a
variety of applications such as radar [82], speech processing [145], and
quantum information theory [121]. Given a nonzero seed function
$f:\mathbb{Z}_{M}\rightarrow\mathbb{C}$, we produce all time- and frequency-
shifted versions: $f_{xy}(t):=f(t-x)e^{2\pi\mathrm{i}yt/M}$,
$t\in\mathbb{Z}_{M}$. Viewing these shifted functions as vectors in
$\mathbb{C}^{M}$ gives an $M\times M^{2}$ Gabor frame. The following theorem
characterizes the spectral norm and the worst-case and average coherence of
Gabor frames generated from either a deterministic Alltop vector [3] or a
random Steinhaus vector.
###### Theorem 53 (Geometry of Gabor frames).
Take an Alltop function defined by
$f(t):=\frac{1}{\sqrt{M}}e^{2\pi\mathrm{i}t^{3}/M}$, $t\in\mathbb{Z}_{M}$.
Also, take a random Steinhaus function defined by
$g(t):=\frac{1}{\sqrt{M}}e^{2\pi\mathrm{i}\theta_{t}}$, $t\in\mathbb{Z}_{M}$,
where the $\theta_{t}$’s are independent random variables distributed
uniformly on the unit interval. Then the $M\times M^{2}$ Gabor frames $\Phi$
and $\Psi$ generated by $f$ and $g$, respectively, are unit norm and tight,
i.e., $\|\Phi\|_{2}=\|\Psi\|_{2}=\sqrt{M}$. Also, both frames have average
coherence $\leq\frac{1}{M+1}$. Furthermore, if $M\geq 5$ is prime, then
$\mu_{\Phi}=\frac{1}{\sqrt{M}}$, while if $M\geq 13$, then
$\mu_{\Psi}\leq\sqrt{(13\log{M})/M}$ with probability exceeding $1-4M^{-1}$.
###### Proof.
The tightness claim follows from [96], in which it was shown that Gabor frames
generated by nonzero seed vectors are tight. The bound on average coherence is
a consequence of Theorem 7 of [11] concerning arbitrary Gabor frames. The
claim concerning $\mu_{\Phi}$ follows directly from [129], while the claim
concerning $\mu_{\Psi}$ is a simple consequence of Theorem 5.1 of [111]. ∎
Instead of taking all translates and modulates of a seed function, [41]
constructs _chirp frames_ by taking all powers and modulates of a chirp
function. Picking $M$ to be prime, we start with a chirp function
$h_{M}:\mathbb{Z}_{M}\rightarrow\mathbb{C}$ defined by
$h_{M}(t):=e^{\pi\mathrm{i}t(t-M)/M}$, $t\in\mathbb{Z}_{M}$. The $M^{2}$ frame
elements are then defined entrywise by
$h_{ab}(t):=\frac{1}{\sqrt{M}}h_{M}(t)^{a}e^{2\pi\mathrm{i}bt/M}$,
$t\in\mathbb{Z}_{M}$. Certainly, chirp frames are, at the very least, similar
in spirit to Gabor frames. As a matter of fact, the chirp frame is in some
sense equivalent to the Gabor frame generated by the Alltop function: it is
easy to verify that
$h_{(-6x,y-3x^{2})}(t)=e^{2\pi\mathrm{i}(t^{3}+x^{3})/M}f_{xy}(t)$, and when
$M\geq 5$, the map $(x,y)\mapsto(-6x,y-3x^{2})$ is a permutation over
$\mathbb{Z}_{M}^{2}$. Using terminology from Definition 67, we say the chirp
frame is _wiggling equivalent_ to a unitary rotation of permuted Alltop Gabor
frame elements. As such, by Lemma 68, the chirp frame has the same spectral
norm and worst-case coherence as the Alltop Gabor frame, but the average
coherence may be different. In this case, the average coherence still
satisfies (SCP-2). Indeed, adding the frame elements gives
$\displaystyle\sum_{a=0}^{M-1}\sum_{b=0}^{M-1}h_{ab}(t)$
$\displaystyle=\frac{1}{\sqrt{M}}\sum_{a=0}^{M-1}h_{M}(t)^{a}\sum_{b=0}^{M-1}e^{2\pi\mathrm{i}bt/M}$
$\displaystyle=\frac{1}{\sqrt{M}}\sum_{a=0}^{M-1}h_{M}(t)^{a}M\delta_{0}(t)=\sqrt{M}\bigg{(}\sum_{a=0}^{M-1}h_{M}(0)^{a}\bigg{)}~{}\delta_{0}(t)=M^{3/2}\delta_{0}(t),$
and so $\langle
h_{a^{\prime}b^{\prime}},\sum_{a=0}^{M-1}\sum_{b=0}^{M-1}h_{ab}\rangle=\langle
h_{a^{\prime}b^{\prime}},M^{3/2}\delta_{0}\rangle=M^{3/2}h_{a^{\prime}b^{\prime}}(0)=M=\frac{M^{2}}{M}$.
Therefore, applying Lemma 48(i) gives the result:
###### Theorem 54 (Geometry of chirp frames).
Pick $M$ prime, and let $\Phi$ be the $M\times M^{2}$ frame of all powers and
modulates of the chirp function $h_{M}$. Then $\Phi$ is a unit norm tight
frame with $\|\Phi\|_{2}=\sqrt{M}$, and has worst case coherence
$\mu=\frac{1}{\sqrt{M}}$ and average coherence $\nu\leq\frac{\mu}{\sqrt{M}}$.
###### Example 55.
To illustrate the bounds in Theorems 53 and 54, we consider the examples of an
Alltop Gabor frame and a chirp frame, each with $M=5$. In this case, the Gabor
frame has $\nu\approx 0.1348\leq 0.1667\approx\frac{1}{M+1}$, while the chirp
frame has $\nu=\frac{1}{6}\leq\frac{1}{5}=\frac{\mu}{\sqrt{M}}$. Note the
Gabor and chirp frames have different average coherences despite being
equivalent in some sense. For the random Steinhaus Gabor frame, we ran
simulations in MATLAB and observed $30$ realizations for each $M=60,70,80$.
The distributions of $\nu$ and $\mu$ were rather tight, so we only report the
ranges of values attained, along with the bounds given in Theorem 53:
$\begin{array}[]{rrcll}M=60:&\qquad\nu&\in&[0.3916,0.5958]\times
10^{-2}&\qquad\leq 0.0164\\\ &\qquad\mu&\in&[0.3242,0.4216]&\qquad\leq
0.9419\\\ \\\ M=70:&\qquad\nu&\in&[0.3151,0.4532]\times 10^{-2}&\qquad\leq
0.0141\\\ &\qquad\mu&\in&[0.2989,0.3814]&\qquad\leq 0.8883\\\ \\\
M=80:&\qquad\nu&\in&[0.2413,0.3758]\times 10^{-2}&\qquad\leq 0.0124\\\
&\qquad\mu&\in&[0.2711,0.3796]&\qquad\leq 0.8439\end{array}$
These simulations seem to indicate that bound on $\nu$ is conservative by an
order of magnitude.
#### 4.2.4 Spherical 2-designs
Lemma 48(ii) leads one to consider frames of vectors that sum to zero. In
[84], it is proved that real unit norm tight frames with this property make up
another well-studied class of vector packings: spherical 2-designs. To be
clear, a collection of unit-norm vectors $\Phi\subseteq\mathbb{R}^{M}$ is
called a spherical $t$-design if, for every polynomial $g(x_{1},\ldots,x_{M})$
of degree at most $t$, we have
$\frac{1}{\mathrm{H}^{M-1}(\mathbb{S}^{M-1})}\int_{\mathbb{S}^{M-1}}g(x)~{}\mathrm{d}\mathrm{H}^{M-1}(x)=\frac{1}{|\Phi|}\sum_{\varphi\in\Phi}g(\varphi),$
where $\mathbb{S}^{M-1}$ is the unit hypersphere in $\mathbb{R}^{M}$ and
$\mathrm{H}^{M-1}$ denotes the $(M-1)$-dimensional Hausdorff measure on
$\mathbb{S}^{M-1}$. In words, vectors that form a spherical $t$-design serve
as good representatives when calculating the average value of a degree-$t$
polynomial over the unit hypersphere. Today, such designs find application in
quantum state estimation [81].
Since real unit norm tight frames always exist for $N\geq M+1$, one might
suspect that spherical 2-designs are equally common, but this intuition is
faulty—the sum-to-zero condition introduces certain issues. For example, there
is no spherical 2-design when $M$ is odd and $N=M+2$. In [101], spherical
2-designs are explicitly characterized by construction. The following theorem
gives a construction based on harmonic frames:
###### Theorem 56 (Geometry of spherical 2-designs).
Pick $M$ even and $N\geq 2M$. Take an $\frac{M}{2}\times N$ harmonic frame
$\Psi$ by collecting rows from a discrete Fourier transform matrix according
to a set of nonzero indices $\mathcal{M}$ and normalizing the columns. Let
$m(n)$ denote $n$th largest index in $\mathcal{M}$, and define a real $M\times
N$ frame $\Phi$ by
$\Phi_{k\ell}:=\left\\{\begin{array}[]{ll}\sqrt{\frac{2}{M}}\cos(\frac{2\pi
m((k+1)/2)\ell}{N}),&k\mbox{ odd}\\\ \sqrt{\frac{2}{M}}\sin(\frac{2\pi
m(k/2)\ell}{N}),&k\mbox{ even}\end{array}\right.,\qquad
k=1,\ldots,M,~{}\ell=0,\ldots,N-1.$
Then $\Phi$ is unit norm and tight, i.e., $\|\Phi\|_{2}^{2}=\frac{N}{M}$, with
worst-case coherence $\mu_{\Phi}\leq\mu_{\Psi}$ and average coherence
$\nu\leq\frac{\mu}{\sqrt{M}}$.
###### Proof.
It is easy to verify that $\Phi$ is a unit norm tight frame using the
geometric sum formula. Also, since the frame elements sum to zero and $N\geq
2M$, the claim regarding average coherence follows from Lemma 48(ii). It
remains to prove $\mu_{\Phi}\leq\mu_{\Psi}$. For each pair of indices
$i,j\in\\{1,\ldots,N\\}$, we have
$\displaystyle\langle\varphi_{i},\varphi_{j}\rangle$
$\displaystyle=\frac{2}{M}\sum_{m\in\mathcal{M}}\bigg{(}\cos\Big{(}\frac{2\pi
mi}{N}\Big{)}\cos\Big{(}\frac{2\pi mj}{N}\Big{)}+\sin\Big{(}\frac{2\pi
mi}{N}\Big{)}\sin\Big{(}\frac{2\pi mj}{N}\Big{)}\bigg{)}$
$\displaystyle=\frac{2}{M}\sum_{m\in\mathcal{M}}\cos\Big{(}\frac{2\pi
m(i-j)}{N}\Big{)}$ $\displaystyle=\mathrm{Re}\langle\psi_{i},\psi_{j}\rangle,$
and so
$|\langle\varphi_{i},\varphi_{j}\rangle|=|\mathrm{Re}\langle\psi_{i},\psi_{j}\rangle|\leq|\langle\psi_{i},\psi_{j}\rangle|$.
This gives the result. ∎
###### Example 57.
To illustrate the bounds in Theorem 56, we consider the spherical 2-design
constructed from a $9\times 37$ harmonic equiangular tight frame [146].
Specifically, we take a $37\times 37$ DFT matrix, choose nonzero row indices
$\mathcal{M}=\\{1,7,9,10,12,16,26,33,34\\},$
and normalize the columns to get a harmonic frame $\Psi$ whose worst-case
coherence achieves the Welch bound:
$\smash{\mu_{\Psi}=\sqrt{\frac{37-9}{9(37-1)}}\approx 0.2940}$. Following
Theorem 56, we produce a spherical 2-design $\Phi$ with $\mu_{\Phi}\approx
0.1967\leq\mu_{\Psi}$ and $\nu\approx 0.0278\leq
0.0464\approx\frac{\mu}{\sqrt{M}}$.
#### 4.2.5 Steiner equiangular tight frames
We now consider the construction of Chapter 1: Steiner equiangular tight
frames (ETFs). Recall that these fail to break the square-root bottleneck as
deterministic RIP matrices. By contrast, Steiner ETFs are particularly well-
suited as sensing matrices for one-step thresholding. To be clear, every
Steiner ETF satisfies $N\geq 2M$. Moreover, if in step (iii) of Theorem 7, we
choose the distinct rows to be the $\frac{v-1}{k-1}$ rows of the (complex)
Hadamard matrix $H$ that are not all-ones, then the sum of columns of each
$F_{j}$ is zero, meaning the sum of columns of $F$ is also zero. This was done
in (1.6), and the columns sum to zero, accordingly. Therefore, by Lemma
48(ii), Steiner ETFs satisfy (SCP-2). This gives the following theorem:
###### Theorem 58 (Geometry of Steiner equiangular tight frames).
Build an $M\times N$ matrix $\Phi$ according to Theorem 7, and in step (iii),
choose rows from the (complex) Hadamard matrix $H$ that are not all-ones. Then
$\Phi$ is an equiangular tight frame, meaning $\|\Phi\|_{2}^{2}=\frac{N}{M}$
and $\mu^{2}=\frac{N-M}{M(N-1)}$, and has average coherence
$\nu\leq\frac{\mu}{\sqrt{M}}$.
###### Example 59.
To illustrate the bound in Theorem 58, we note that the example given in (1.6)
has $\nu=\frac{1}{11}\leq\frac{1}{3\sqrt{2}}=\frac{\mu}{\sqrt{M}}$.
#### 4.2.6 Code-based frames
Many structures in coding theory are also useful in frame theory. In this
section, we build frames from a code that originally emerged with Berlekamp in
[22], and found recent reincarnation with [147]. We build a $2^{m}\times
2^{(t+1)m}$ frame, indexing rows by elements of $\mathbb{F}_{2^{m}}$ and
indexing columns by $(t+1)$-tuples of elements from $\mathbb{F}_{2^{m}}$. For
$x\in\mathbb{F}_{2^{m}}$ and $\alpha\in\mathbb{F}_{2^{m}}^{t+1}$, the
corresponding entry of the matrix $\Phi$ is given by
$\Phi_{x\alpha}=\frac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}},$
(4.19)
where $\mathrm{Tr}:\mathbb{F}_{2^{m}}\rightarrow\mathbb{F}_{2}$ denotes the
trace map, defined by $\mathrm{Tr}(z)=\sum_{i=0}^{m-1}z^{2^{i}}$. The
following theorem gives the spectral norm and the worst-case and average
coherence of this frame.
Name | $\mathbb{R}/\mathbb{C}$ | Size | $\mu_{F}$ | $\nu_{F}$
---|---|---|---|---
Normalized Gaussian | $\mathbb{R}$ | $M\times N$ | $\leq\frac{\sqrt{15\log{N}}}{\sqrt{M}-\sqrt{12\log{N}}}$ | $\leq\frac{\sqrt{15\log{N}}}{M-\sqrt{12M\log{N}}}$
Random harmonic | $\mathbb{C}$ | $|\mathcal{M}|\times N$, $\frac{1}{2}M\leq|\mathcal{M}|\leq\frac{3}{2}M$ | $\leq\sqrt{\frac{118(N-M)\log{N}}{MN}}$ | $\leq\frac{\mu_{F}}{\sqrt{|\mathcal{M}|}}$
Alltop Gabor | $\mathbb{C}$ | $M\times M^{2}$ | $=\frac{1}{\sqrt{M}}$ | $\leq\frac{1}{M+1}$
Steinhaus Gabor | $\mathbb{C}$ | $M\times M^{2}$ | $\leq\sqrt{\frac{13\log M}{M}}$ | $\leq\frac{1}{M+1}$
Chirp | $\mathbb{C}$ | $M\times M^{2}$ | $=\frac{1}{\sqrt{M}}$ | $\leq\frac{\mu_{F}}{\sqrt{M}}$
$\overset{\mbox{Spherical 2-design}}{\mbox{from harmonic }G}$ | $\mathbb{R}$ | $M\times N$ | $\leq\mu_{G}$ | $\leq\frac{\mu_{F}}{\sqrt{M}}$
Steiner | $\mathbb{C}$ | $M\times N$, $M=\frac{v(v-1)}{k(k-1)}$, $N=v(1+\frac{v-1}{k-1})$ | $=\sqrt{\frac{N-M}{M(N-1)}}$ | $\leq\frac{\mu_{F}}{\sqrt{M}}$
Code-based | $\mathbb{R}$ | $2^{m}\times 2^{(t+1)m}$ | $\leq\frac{1}{\sqrt{2^{m-2t-1}}}$ | $\leq\frac{\mu_{F}}{\sqrt{2^{m}}}$
Table 4.1: Eight constructions detailed in this chapter. The bounds given for
the normalized Gaussian, random harmonic and Steinhaus Gabor frames are
satisfied with high probability. All of the frames above are unit norm tight
frames except for the normalized Gaussian frame, which has squared spectral
norm
$\|\Phi\|_{2}^{2}\leq(\\!\sqrt{M}+\\!\sqrt{N}+\\!\sqrt{2\log{N}})^{2}/(M-\\!\sqrt{8M\log{N}})$
in the same probability event.
###### Theorem 60 (Geometry of code-based frames).
The $2^{m}\times 2^{(t+1)m}$ frame defined by (4.19) is unit norm and tight,
i.e., $\|\Phi\|_{2}^{2}=2^{tm}$, with worst-case coherence
$\mu\leq\frac{1}{\sqrt{2^{m-2t-1}}}$ and average coherence
$\nu\leq\frac{\mu}{\sqrt{2^{m}}}$.
###### Proof.
For the tightness claim, we use the linearity of the trace map to write the
inner product of rows $x$ and $y$:
$\displaystyle\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}\\!\\!\frac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}\frac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}y+\sum_{i=1}^{t}\alpha_{i}y^{2^{i}+1}\big{]}}$
$\displaystyle\qquad=\frac{1}{2^{m}}\bigg{(}\\!\sum_{\alpha_{0}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}[\alpha_{0}(x+y)]}\bigg{)}\\!\\!\sum_{\alpha_{1}\in\mathbb{F}_{2^{m}}}\\!\\!\cdots\\!\\!\sum_{\alpha_{t}\in\mathbb{F}_{2^{m}}}\\!\\!(-1)^{\mathrm{Tr}\big{[}\sum_{i=1}^{t}\alpha_{i}(x^{2^{i}+1}+y^{2^{i}+1})\big{]}}.$
This expression is $2^{tm}$ when $x=y$. Otherwise, note that
$\alpha_{0}\mapsto(-1)^{\mathrm{Tr}[\alpha_{0}(x+y)]}\in\\{\pm 1\\}$ defines a
homomorphism on $\mathbb{F}_{2^{m}}$. Since $(x+y)^{-1}\mapsto-1$, the inverse
images of $\pm 1$ under this homomorphism must form two cosets of equal size,
and so
$\sum_{\alpha_{0}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}[\alpha_{0}(x+y)]}=0$,
meaning distinct rows in $\Phi$ are orthogonal. Thus, $\Phi$ is a unit norm
tight frame.
For the worst-case coherence claim, we first note that the linearity of the
trace map gives
$(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}(-1)^{\mathrm{Tr}\big{[}\alpha^{\prime}_{0}x+\sum_{i=1}^{t}\alpha^{\prime}_{i}x^{2^{i}+1}\big{]}}=(-1)^{\mathrm{Tr}\big{[}(\alpha_{0}+\alpha^{\prime}_{0})x+\sum_{i=1}^{t}(\alpha_{i}+\alpha^{\prime}_{i})x^{2^{i}+1}\big{]}},$
i.e., every inner product between columns of $\Phi$ is a sum over another
column. Thus, there exists $\alpha\in\mathbb{F}_{2^{m}}^{t+1}$ such that
$\displaystyle 2^{2m}\mu^{2}$
$\displaystyle=\bigg{(}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}\bigg{)}^{2}$
$\displaystyle=2^{m}+\sum_{x\in\mathbb{F}_{2^{m}}}\sum_{\begin{subarray}{c}y\in\mathbb{F}_{2^{m}}\\\
y\neq
x\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}(x+y)+\sum_{i=1}^{t}\alpha_{i}\big{(}(x+y)^{2^{i}+1}+\sum_{j=0}^{i-1}(xy)^{2^{j}}(x+y)^{2^{i}-2^{j+1}+1}\big{)}\big{]}},$
where the last equality is by the identity
$(x+y)^{2^{i}+1}=x^{2^{i}+1}+y^{2^{i}+1}+\sum_{j=0}^{i-1}(xy)^{2^{j}}(x+y)^{2^{i}-2^{j+1}+1}$,
whose proof is a simple exercise of induction. From here, we perform a change
of variables: $u:=x+y$ and $v:=xy$. Notice that $(u,v)$ corresponds to $(x,y)$
for some $x\neq y$ whenever $(z+x)(z+y)=z^{2}+uz+v$ has two solutions, that
is, whenever $\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$. Since $(u,v)$
corresponds to both $(x,y)$ and $(y,x)$, we must correct for under-counting:
$\displaystyle 2^{2m}\mu^{2}$
$\displaystyle=2^{m}+2\sum_{\begin{subarray}{c}u\in\mathbb{F}_{2^{m}}\\\ u\neq
0\end{subarray}}\sum_{\begin{subarray}{c}v\in\mathbb{F}_{2^{m}}\\\
\mathrm{Tr}(v/u^{2})=0\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}u+\sum_{i=1}^{t}\alpha_{i}\big{(}u^{2^{i}+1}+\sum_{j=0}^{i-1}v^{2^{j}}u^{2^{i}-2^{j+1}+1}\big{)}\big{]}}$
$\displaystyle=2^{m}+2\sum_{\begin{subarray}{c}u\in\mathbb{F}_{2^{m}}\\\ u\neq
0\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}u+\sum_{i=1}^{t}\alpha_{i}u^{2^{i}+1}\big{]}}\sum_{\begin{subarray}{c}v\in\mathbb{F}_{2^{m}}\\\
\mathrm{Tr}(v/u^{2})=0\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\big{(}\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{-j}}u^{2^{i-j}-2+2^{-j}}\big{)}v\big{]}}$
$\displaystyle\leq 2^{m}+2\sum_{\begin{subarray}{c}u\in\mathbb{F}_{2^{m}}\\\
u\neq
0\end{subarray}}~{}\bigg{|}\\!\\!\\!\sum_{\begin{subarray}{c}v\in\mathbb{F}_{2^{m}}\\\
\mathrm{Tr}(v/u^{2})=0\end{subarray}}\\!\\!\\!(-1)^{\mathrm{Tr}[p(u)v]}~{}\bigg{|},$
(4.20)
where the second equality is by repeated application of
$\mathrm{Tr}(z)=\mathrm{Tr}(z^{2})$, and
$p(u):=\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{-j}}u^{2^{i-j}-2+2^{-j}}.$
To bound $\mu$, we will count the $u$’s that produce nonzero summands in
(4.20).
For each $u\neq 0,$ we have a homomorphism
$\chi_{u}\colon\\{v\in\mathbb{F}_{2^{m}}:\mathrm{Tr}(\frac{v}{u^{2}})=0\\}\rightarrow\\{\pm
1\\}$ defined by $\chi_{u}(v):=(-1)^{\mathrm{Tr}[p(u)v]}$. Pick $u\neq 0$ for
which there exists a $v$ such that both
$\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$ and $\mathrm{Tr}[p(u)v]=1$. Then
$\chi_{u}(v)=-1$, and so the kernel of $\chi_{u}$ is the same size as the
coset
$\smash{\\{v\in\mathbb{F}_{2^{m}}:\mathrm{Tr}(\frac{v}{u^{2}})=0,\chi_{u}(v)=-1\\}}$,
meaning the summand associated with $u$ in (4.20) is zero. Hence, the nonzero
summands in (4.20) require $\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$ and
$\mathrm{Tr}[p(u)v]=0$. This is certainly possible whenever $p(u)=0$.
Exponentiation gives
$p(u)^{2^{t-1}}=\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{t-j-1}}u^{2^{t+i-j-1}-2^{t}+2^{t-j-1}},$
which has degree $2^{2t-1}-2^{t-1}$. Thus, $p(u)=0$ has at most
$2^{2t-1}-2^{t-1}$ solutions, and each such $u$ produces a summand in (4.20)
of size $2^{m-1}$. Next, we consider the $u$’s for which
$\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$, $\mathrm{Tr}[p(u)v]=0$, and
$p(u)\neq 0$. In this case, the hyperplanes defined by
$\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$ and $\mathrm{Tr}[p(u)v]=0$ are
parallel, and so $\smash{p(u)=\frac{1}{u^{2}}}$. Here,
$1=(u^{2}p(u))^{2^{t-1}}=\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{t-j-1}}u^{2^{t+i-j-1}+2^{t-j-1}},$
which has degree $2^{2t-1}+2^{t-1}$. Thus, $\smash{p(u)=\frac{1}{u^{2}}}$ has
at most $2^{2t-1}+2^{t-1}$ solutions, and each such $u$ produces a summand in
(4.20) of size $2^{m-1}$. We can now continue the bound from (4.20):
$2^{2m}\mu^{2}\leq 2^{m}+2(2^{2t-1}-2^{t-1}+2^{2t-1}+2^{t-1})2^{m-1}\leq
2^{m+2t+1}$. From here, isolating $\mu$ gives the claim.
Lastly, for average coherence, pick some $x\in\mathbb{F}_{2^{m}}$. Then
summing the entries in the $x$th row gives
$\displaystyle\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}\frac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}$
$\displaystyle\qquad=\frac{1}{\sqrt{2^{m}}}\bigg{(}\sum_{\alpha_{0}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}(\alpha_{0}x)}\bigg{)}\sum_{\alpha_{1}\in\mathbb{F}_{2^{m}}}\cdots\sum_{\alpha_{t}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}$
$\displaystyle\qquad=\left\\{\begin{array}[]{lc}2^{(t+1/2)m},&x=0\\\ 0,&x\neq
0\end{array}\right..$
That is, the frame elements sum to a multiple of an identity basis element:
$\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}\varphi_{\alpha}=2^{(t+1/2)m}\delta_{0}$.
Since every entry in row $x=0$ is $\smash{\frac{1}{\sqrt{2^{m}}}}$, we have
$\langle\varphi_{\alpha^{\prime}},\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}\varphi_{\alpha}\rangle=\frac{2^{(t+1)m}}{2^{m}}$
for every $\alpha^{\prime}\in\mathbb{F}_{2^{m}}^{t+1}$, and so by Lemma 48(i),
we are done. ∎
###### Example 61.
To illustrate the bounds in Theorem 60, we consider the example where $m=4$
and $t=1$. This is a $16\times 256$ code-based frame $\Phi$ with
$\smash{\mu=\frac{1}{2}\leq\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2^{m-2t-1}}}}$
and $\smash{\nu=\frac{1}{17}\leq\frac{1}{8}=\frac{\mu}{\sqrt{2^{m}}}}$.
### 4.3 Fundamental limits on worst-case coherence
In many applications of frames, performance is dictated by worst-case
coherence [11, 35, 62, 84, 103, 129, 134, 136, 149]. It is therefore
particularly important to understand which worst-case coherence values are
achievable. To this end, the Welch bound is commonly used in the literature.
When worst-case coherence achieves the Welch bound, the frame is equiangular
and tight [129]. However, equiangular tight frames cannot have more vectors
than the square of the spatial dimension [129], meaning the Welch bound is not
tight whenever $N>M^{2}$. When the number of vectors $N$ is exceedingly large,
the following theorem gives a better bound:
###### Theorem 62 ([5, 109]).
Every sufficiently large $M\times N$ unit norm frame with $N\geq 2M$ and
worst-case coherence $\mu<\frac{1}{2}$ satisfies
$\mu^{2}\log\frac{1}{\mu}\geq\frac{C\log N}{M}$ (4.21)
for some constant $C>0$.
For a fixed worst-case coherence $\mu<\frac{1}{2}$, this bound indicates that
the number of vectors $N$ cannot exceed some exponential in the spatial
dimension $M$, that is, $N\leq a^{M}$ for some $a>0$. However, since the
constant $C$ is not established in this theorem, it is unclear which base $a$
is appropriate for each $\mu$. The following theorem is a little more explicit
in this regard:
###### Theorem 63 ([106, 146]).
Every $M\times N$ unit norm frame has worst-case coherence $\mu\geq
1-2N^{-1/(M-1)}$. Furthermore, taking $N=\Theta(a^{M})$, this lower bound goes
to $1-\frac{2}{a}$ as $M\rightarrow\infty$.
For many applications, it does not make sense to use a complex frame, but the
bound in Theorem 63 is known to be loose for real frames [53]. We therefore
improve Theorems 62 and 63 for the case of real unit norm frames:
###### Theorem 64.
Every real $M\times N$ unit norm frame has worst-case coherence
$\mu\geq\cos\bigg{[}\pi\bigg{(}\frac{M-1}{N\pi^{1/2}}\cdot\frac{\Gamma(\frac{M-1}{2})}{\Gamma(\frac{M}{2})}\bigg{)}^{\frac{1}{M-1}}\bigg{]}.$
(4.22)
Furthermore, taking $N=\Theta(a^{M})$, this lower bound goes to
$\cos(\frac{\pi}{a})$ as $M\rightarrow\infty$.
Before proving this theorem, we first consider the special case where the
dimension is $M=3$:
###### Lemma 65.
Given $N$ points on the unit sphere $\mathbb{S}^{2}\subseteq\mathbb{R}^{3}$,
the smallest angle between points is $\leq
2\cos^{-1}\big{(}1-\frac{2}{N}\big{)}$.
###### Proof.
We first claim there exists a closed spherical cap in $\mathbb{S}^{2}$ with
area $\smash{\frac{4\pi}{N}}$ that contains two of the $N$ points. Suppose
otherwise, and take $\gamma$ to be the angular radius of a spherical cap with
area $\smash{\frac{4\pi}{N}}$. That is, $\gamma$ is the angle between the
center of the cap and every point on the boundary. Since the cap is closed, we
must have that the smallest angle $\alpha$ between any two of our $N$ points
satisfies $\alpha>2\gamma$. Let $C(p,\theta)$ denote the closed spherical cap
centered at $p\in\mathbb{S}^{2}$ of angular radius $\theta$, and let $P$
denote our set of $N$ points. Then we know for $p\in P$, the $C(p,\gamma)$’s
are disjoint, $\frac{\alpha}{2}>\gamma$, and $\bigcup_{p\in
P}C(p,\tfrac{\alpha}{2})\subseteq\mathbb{S}^{2}$, and so taking 2-dimensional
Hausdorff measures on the sphere gives
$\mathrm{H}^{2}(\mathbb{S}^{2})=4\pi=\mathrm{H}^{2}\bigg{(}\bigcup_{p\in
P}C(p,\gamma)\bigg{)}<\mathrm{H}^{2}\bigg{(}\bigcup_{p\in
P}C(p,\tfrac{\alpha}{2})\bigg{)}\leq\mathrm{H}^{2}(\mathbb{S}^{2}),$
a contradiction.
Since two of the points reside in a spherical cap of area
$\smash{\frac{4\pi}{N}}$, we know $\alpha$ is no more than twice the radius of
this cap. We use spherical coordinates to relate the cap’s area to the radius:
$\smash{\mathrm{H}^{2}(C(\cdot,\gamma))=2\pi\int_{0}^{\gamma}\sin\phi~{}\mathrm{d}\phi=2\pi(1-\cos\gamma)}$.
Therefore, when $\smash{\mathrm{H}^{2}(C(\cdot,\gamma))=\frac{4\pi}{N}}$, we
have $\gamma=\cos^{-1}(1-\frac{2}{N})$, and so $\alpha\leq 2\gamma$ gives the
result. ∎
###### Theorem 66.
Every real $3\times N$ unit norm frame has worst-case coherence $\mu\geq
1-\frac{4}{N}+\frac{2}{N^{2}}$.
###### Proof.
Packing $N$ unit vectors in $\mathbb{R}^{3}$ corresponds to packing $2N$
antipodal points in $\mathbb{S}^{2}$, and so Lemma 65 gives $\alpha\leq
2\cos^{-1}(1-\frac{1}{N})$. Applying the double angle formula to
$\mu=\cos\alpha\geq\cos[2\cos^{-1}(1-\tfrac{1}{N})]$
gives the result. ∎
$N$$\mu_{F}$Numerically optimalWelch boundTheorem 63Theorem 64Theorem 66
Figure 4.1: Different bounds on worst-case coherence for $M=3$,
$N=3,\ldots,55$. Stars give numerically determined optimal worst-case
coherence of $N$ real unit vectors, found in [53]. Dotted curve gives Welch
bound, dash-dotted curve gives bound from Theorem 63, dashed curve gives bound
from Theorem 64, and solid curve gives bound from Theorem 66.
Now that we understand the special case where $M=3$, we tackle the general
case:
###### Proof of Theorem 64.
As in the proof of Theorem 66, we relate packing $N$ unit vectors to packing
$2N$ points in the hypersphere $\mathbb{S}^{M-1}\subseteq\mathbb{R}^{M}$. The
argument in the proof of Lemma 65 generalizes so that two of the $2N$ points
must reside in some closed hyperspherical cap of hypersurface area
$\frac{1}{2N}\mathrm{H}^{M-1}(\mathbb{S}^{M-1})$. Therefore, the smallest
angle $\alpha$ between these points is no more than twice the radius of this
cap. Let $C(\gamma)$ denote a hyperspherical cap of angular radius $\gamma$.
Then we use hyperspherical coordinates to get
$\displaystyle\mathrm{H}^{M-1}(C(\gamma))$
$\displaystyle=\int_{\phi_{1}=0}^{\gamma}\int_{\phi_{2}=0}^{\pi}\cdots\int_{\phi_{M-2}=0}^{\pi}\int_{\phi_{M-1}=0}^{2\pi}\sin^{M-2}(\phi_{1})\cdots\sin^{1}(\phi_{M-2})~{}\mathrm{d}\phi_{M-1}\cdots\mathrm{d}\phi_{1}$
$\displaystyle=2\pi\bigg{(}\prod_{j=1}^{M-3}\pi^{1/2}\frac{\Gamma(\frac{j+1}{2})}{\Gamma(\frac{j}{2}+1)}\bigg{)}\int_{0}^{\gamma}\sin^{M-2}\phi~{}\mathrm{d}\phi$
$\displaystyle=\frac{2\pi^{(M-1)/2}}{\Gamma(\frac{M-1}{2})}\int_{0}^{\gamma}\sin^{M-2}\phi~{}\mathrm{d}\phi.$
(4.23)
We wish to solve for $\gamma$, but analytically inverting
$\int_{0}^{\gamma}\sin^{M-2}\phi~{}\mathrm{d}\phi$ is difficult. Instead, we
use $\sin\phi\geq\frac{2\phi}{\pi}$ for $\phi\in[0,\frac{\pi}{2}]$. Note that
we do not lose generality by forcing $\gamma\leq\frac{\pi}{2}$, since this is
guaranteed with $N\geq 2$. Continuing (4.23) gives
$\mathrm{H}^{M-1}(C(\gamma))\geq\frac{2\pi^{(M-1)/2}}{\Gamma(\frac{M-1}{2})}\int_{0}^{\gamma}\Big{(}\frac{2\phi}{\pi}\Big{)}^{M-2}\mathrm{d}\phi=\frac{(2\gamma)^{M-1}}{(M-1)\pi^{(M-3)/2}\Gamma(\frac{M-1}{2})}.$
(4.24)
Using the formula for a hypersphere’s hypersurface area, we can express the
left-hand side of (4.24):
$\frac{(2\gamma)^{M-1}}{(M-1)\pi^{(M-3)/2}\Gamma(\frac{M-1}{2})}\leq\mathrm{H}^{M-1}(C(\gamma))=\frac{1}{2N}\mathrm{H}^{M-1}(\mathbb{S}^{M-1})=\frac{\pi^{M/2}}{N\Gamma(\frac{d}{2})}.$
Isolating $2\gamma$ above and using $\alpha\leq 2\gamma$ and $\mu=\cos\alpha$
gives (4.22). The second part of the result comes from a simple application of
Stirling’s approximation. ∎
In [53], numerical results are given for $M=3$, and we compare these results
to Theorems 63 and 64 in Figure 4.1. Considering this figure, we note that the
bound in Theorem 63 is inferior to the maximum of the Welch bound and the
bound in Theorem 64, at least when $M=3$. This illustrates the degree to which
Theorem 64 improves the bound in Theorem 63 for real frames. In fact, since
$\cos(\frac{\pi}{a})\geq 1-\frac{2}{a}$ for all $a\geq 2$, the bound for real
frames in Theorem 64 is asymptotically better than the bound for complex
frames in Theorem 63. Moreover, for $M=2$, Theorem 64 says
$\mu\geq\cos(\frac{\pi}{N})$, and [19] proved this bound to be tight for every
$N\geq 2$. Lastly, Figure 4.1 illustrates that Theorem 66 improves the bound
in Theorem 64 for the case $M=3$.
In many applications, large dictionaries are built to obtain sparse
reconstruction, but the known guarantees on sparse reconstruction place
certain requirements on worst-case coherence. Asymptotically, the bounds in
Theorems 63 and 64 indicate that certain exponentially large dictionaries will
not satisfy these requirements. For example, if $N=\Theta(3^{M})$, then
$\mu_{F}=\Omega(\frac{1}{3})$ by Theorem 63, and if the frame is real, we have
$\mu=\Omega(\frac{1}{2})$ by Theorem 64. Such a dictionary will only work for
sparse reconstruction if the sparsity level $K$ is sufficiently small;
deterministic guarantees require $K<\mu^{-1}$ [62, 134], while probabilistic
guarantees require $K<\mu^{-2}$ [11, 135], and so in this example, the
dictionary can, at best, only accommodate sparsity levels that are smaller
than 10. Unfortunately, in real-world applications, we can expect the sparsity
level to scale with the signal dimension. This in mind, Theorems 63 and 64
tell us that dictionaries can only be used for sparse reconstruction if
$N=O((2+\varepsilon)^{M})$ for some sufficiently small $\varepsilon>0$. To
summarize, the Welch bound is known to be tight only if $N\leq M^{2}$, and
Theorems 63 and 64 give bounds which are asympotically better than the Welch
bound whenever $N=\Omega(2^{M})$. When $N$ is between $M^{2}$ and $2^{M}$, the
best bound to date is the (loose) Welch bound, and so more work needs to be
done to bound worst-case coherence in this parameter region.
### 4.4 Reducing average coherence
In [11], average coherence is used to derive a number of guarantees on sparse
signal processing. Since average coherence is so new to the frame theory
literature, this section will investigate how average coherence relates to
worst-case coherence and the spectral norm. We start with a definition:
###### Definition 67 (Wiggling and flipping equivalent frames).
We say the frames $\Phi$ and $\Psi$ are _wiggling equivalent_ if there exists
a diagonal matrix $D$ of unimodular entries such that $\Psi=\Phi D$.
Furthermore, they are _flipping equivalent_ if $D$ is real, having only $\pm
1$’s on the diagonal.
The terms “wiggling” and “flipping” are inspired by the fact that individual
frame elements of such equivalent frames are related by simple unitary
operations. Note that every frame with $N$ nonzero frame elements belongs to a
flipping equivalence class of size $2^{N}$, while being wiggling equivalent to
uncountably many frames. The importance of this type of frame equivalence is,
in part, due to the following lemma, which characterizes the shared geometry
of wiggling equivalent frames:
###### Lemma 68 (Geometry of wiggling equivalent frames).
Wiggling equivalence preserves the norms of frame elements, the worst-case
coherence, and the spectral norm.
###### Proof.
Take two frames $\Phi$ and $\Psi$ such that $\Psi=\Phi D$. The first claim is
immediate. Next, the Gram matrices are related by
$\Psi^{*}\Psi=D^{*}\Phi^{*}\Phi D$. Since corresponding off-diagonal entries
are equal in modulus, we know the worst-case coherences are equal. Finally,
$\|\Psi\|_{2}^{2}=\|\Psi\Psi^{*}\|_{2}^{2}=\|\Phi
DD^{*}\Phi^{*}\|_{2}=\|\Phi\Phi^{*}\|_{2}=\|\Phi\|_{2}^{2}$, and so we are
done. ∎
Wiggling and flipping equivalence are not entirely new to frame theory. For a
real equiangular tight frame $\Phi$, the Gram matrix $\Phi^{*}\Phi$ is
completely determined by the sign pattern of the off-diagonal entries, which
can in turn be interpreted as the Seidel adjacency matrix of a graph
$G_{\Phi}$. As such, flipping a frame element $\varphi\in\Phi$ has the effect
of negating the corresponding row and column in the Gram matrix, which further
corresponds to _switching_ the adjacency rule for that vertex $v_{\varphi}\in
V(G_{\Phi})$ in the graph—vertices are adjacent to $v_{\varphi}$ after
switching precisely when they were not adjacent before switching. Graphs are
called _switching equivalent_ if there is a sequence of switching operations
that produces one graph from the other; this equivalence was introduced in
[139] and was later extensively studied by Seidel in [122, 123]. Since
flipping equivalent real equiangular tight frames correspond to switching
equivalent graphs, the terms have become interchangeable. For example, [24]
uses switching (i.e., wiggling and flipping) equivalence to make progress on
an important problem in frame theory called the _Paulsen problem_ , which asks
how close a nearly unit norm, nearly tight frame must be to a unit norm tight
frame.
Now that we understand wiggling and flipping equivalence, we are ready for the
main idea behind this section. Suppose we are given a unit norm frame with
acceptable spectral norm and worst-case coherence, but we also want the
average coherence to satisfy (SCP-2). Then by Lemma 68, all of the wiggling
equivalent frames will also have acceptable spectral norm and worst-case
coherence, and so it is reasonable to check these frames for good average
coherence. In fact, the following theorem guarantees that at least one of the
flipping equivalent frames will have good average coherence, with only modest
requirements on the original frame’s redundancy.
###### Theorem 69 (Constructing frames with low average coherence).
Let $\Phi$ be an $M\times N$ unit norm frame with $\smash{M<\frac{N-1}{4\log
4N}}$. Then there exists a frame $\Psi$ that is flipping equivalent to $\Phi$
and satisfies $\smash{\nu\leq\frac{\mu}{\sqrt{M}}}$.
###### Proof.
Take $\\{R_{n}\\}_{n=1}^{N}$ to be a Rademacher sequence that independently
takes values $\pm 1$, each with probability $\frac{1}{2}$. We use this
sequence to randomly flip $\Phi$; define
$Z:=\Phi~{}\mathrm{diag}\\{R_{n}\\}_{n=1}^{N}$. Note that if
$\smash{\Pr(\nu_{Z}\leq\frac{\mu_{\Phi}}{\sqrt{M}})>0}$, we are done. Fix some
$i\in\\{1,\ldots,N\\}$. Then
$\Pr\Bigg{(}\frac{1}{N-1}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\langle
z_{i},z_{j}\rangle\bigg{|}>\frac{\mu_{\Phi}}{\sqrt{M}}\Bigg{)}=\Pr\Bigg{(}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}R_{j}\langle\varphi_{i},\varphi_{j}\rangle\bigg{|}>\frac{(N-1)\mu_{\Phi}}{\sqrt{M}}\Bigg{)}.$
(4.25)
We can view $\sum_{j\neq i}R_{j}\langle\varphi_{i},\varphi_{j}\rangle$ as a
sum of $N-1$ independent zero-mean complex random variables that are bounded
by $\mu_{\Phi}$. We can therefore use a complex version of Hoeffding’s
inequality [83] (see, e.g., Lemma 3.8 of [10]) to bound the probability
expression in (4.25) as $\leq 4e^{-(N-1)/4M}$. From here, a union bound over
all $N$ choices for $i$ gives $\Pr(\nu_{Z}\leq\frac{\mu_{\Phi}}{\sqrt{M}})\geq
1-4Ne^{-(N-1)/4M}$, and so $M<\frac{N-1}{4\log 4N}$ implies
$\Pr(\nu_{Z}\leq\frac{\mu_{\Phi}}{\sqrt{M}})>0$, as desired. ∎
While Theorem 69 guarantees the existence of a flipping equivalent frame with
good average coherence, the result does not describe how to find it.
Certainly, one could check all $2^{N}$ frames in the flipping equivalence
class, but such a procedure is computationally slow. As an alternative, we
propose a linear-time flipping algorithm (Algorithm 2). The following theorem
guarantees that linear-time flipping will produce a frame with good average
coherence, but it requires the original frame’s redundancy to be higher than
what suffices in Theorem 69.
Algorithm 2 Linear-time flipping
Input: An $M\times N$ unit norm frame $\Phi$
Output: An $M\times N$ unit norm frame $\Psi$ that is flipping equivalent to
$\Phi$
$\psi_{1}\leftarrow\varphi_{1}$ {Keep first frame element}
for $n=2$ to $N$ do
if
$\|\sum_{i=1}^{n-1}\psi_{i}+\varphi_{n}\|\leq\|\sum_{i=1}^{n-1}\psi_{i}-\varphi_{n}\|$
then
$\psi_{n}\leftarrow\varphi_{n}$ {Keep frame element to make sum length
shorter}
else
$\psi_{n}\leftarrow-\varphi_{n}$ {Flip frame element to make sum length
shorter}
end if
end for
###### Theorem 70.
Suppose $N\geq M^{2}+3M+3$. Then Algorithm 2 outputs an $M\times N$ frame
$\Psi$ that is flipping equivalent to $\Phi$ and satisfies
$\nu\leq\frac{\mu}{\sqrt{M}}$.
###### Proof.
Considering Lemma 48(iii), it suffices to have
$\|\sum_{n=1}^{N}\psi_{n}\|^{2}\leq N$. We will use induction to show
$\|\sum_{n=1}^{k}\psi_{n}\|^{2}\leq k$ for $k=1,\ldots,N$. Clearly,
$\|\sum_{n=1}^{1}\psi_{n}\|^{2}=\|\varphi_{n}\|^{2}=1\leq 1$. Now assume
$\|\sum_{n=1}^{k}\psi_{n}\|^{2}\leq k$. Then by our choice for $\psi_{k+1}$ in
Algorithm 2, we know that
$\|\sum_{n=1}^{k}\psi_{n}+\psi_{k+1}\|^{2}\leq\|\sum_{n=1}^{k}\psi_{n}-\psi_{k+1}\|^{2}$.
Expanding both sides of this inequality gives
$\bigg{\|}\sum_{n=1}^{k}\psi_{n}\bigg{\|}^{2}+2\mathrm{Re}\bigg{\langle}\sum_{n=1}^{k}\psi_{n},\psi_{k+1}\bigg{\rangle}+\|\psi_{k+1}\|^{2}\leq\bigg{\|}\sum_{n=1}^{k}\psi_{n}\bigg{\|}^{2}-2\mathrm{Re}\bigg{\langle}\sum_{n=1}^{k}\psi_{n},\psi_{k+1}\bigg{\rangle}+\|\psi_{k+1}\|^{2},$
and so $\mathrm{Re}\langle\sum_{n=1}^{k}\psi_{n},\psi_{k+1}\rangle\leq 0$.
Therefore,
$\bigg{\|}\sum_{n=1}^{k+1}\psi_{n}\bigg{\|}^{2}=\bigg{\|}\sum_{n=1}^{k}\psi_{n}\bigg{\|}^{2}+2\mathrm{Re}\bigg{\langle}\sum_{n=1}^{k}\psi_{n},\psi_{k+1}\bigg{\rangle}+\|\psi_{k+1}\|^{2}\leq\bigg{\|}\sum_{n=1}^{k}\psi_{n}\bigg{\|}^{2}+\|\psi_{k+1}\|^{2}\leq
k+1,$
where the last inequality uses the inductive hypothesis. ∎
###### Example 71.
Apply linear-time flipping to reduce average coherence in the following
matrix:
$\Phi:=\frac{1}{\sqrt{5}}\left[\begin{array}[]{cccccccccc}+&+&+&+&-&+&+&+&+&-\\\
+&-&+&+&+&-&-&-&+&-\\\ +&+&+&+&+&+&+&+&-&+\\\ -&-&-&+&-&+&+&-&-&-\\\
-&+&+&-&-&+&-&-&-&-\end{array}\right].$
Here, $\smash{\nu_{\Phi}\approx
0.3778>0.2683\approx\frac{\mu_{\Phi}}{\sqrt{M}}}$, and linear-time flipping
produces the flipping pattern $D:=\mathrm{diag}(+-+--++-++)$. Then $\Phi D$
has average coherence $\smash{\nu_{\Phi D}\approx
0.1556<\frac{\mu_{\Phi}}{\sqrt{M}}=\frac{\mu_{\Phi D}}{\sqrt{M}}}$. This
illustrates that the condition $N\geq M^{2}+3M+3$ in Theorem 70 is sufficient
but not necessary.
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|
arxiv-papers
| 2012-04-26T15:18:43 |
2024-09-04T02:49:30.199341
|
{
"license": "Public Domain",
"authors": "Dustin G. Mixon",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1204.5958"
}
|
1204.5991
|
# 2D Radiative MHD Simulations of the Importance of Partial Ionization in the
Chromosphere
Juan Martínez-Sykora 1,2 j.m.sykora@astro.uio.no Bart De Pontieu 1 Viggo
Hansteen 1,2 1 Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto,
CA 94304 2 Institute of Theoretical Astrophysics, University of Oslo, P.O.
Box 1029 Blindern, N-0315 Oslo, Norway
###### Abstract
The bulk of the solar chromosphere is weakly ionized and interactions between
ionized particles and neutral particles likely have significant consequences
for the thermodynamics of the chromospheric plasma. We investigate the
importance of introducing neutral particles into the MHD equations using
numerical 2.5D radiative MHD simulations obtained with the Bifrost code. The
models span the solar atmosphere from the upper layers of the convection zone
to the low corona, and solve the full MHD equations with non-grey and non-LTE
radiative transfer, and thermal conduction along the magnetic field. The
effects of partial ionization are implemented using the generalized Ohm’s law,
i.e., we consider the effects of the Hall term and ambipolar diffusion in the
induction equation. The approximations required in going from three fluids to
the generalized Ohm’s law are tested in our simulations. The Ohmic diffusion,
the Hall term, and ambipolar diffusion show strong variations in the
chromosphere. These strong variations of the various magnetic diffusivities
are absent or significantly underestimated when, as has been common for these
types of studies, using the semi-empirical VAL-C model as a basis for
estimates. In addition, we find that differences in estimating the magnitude
of ambipolar diffusion arise depending on which method is used to calculate
the ion-neutral collision frequency. These differences cause uncertainties in
the different magnetic diffusivity terms. In the chromosphere, we find that
the ambipolar diffusion is of the same order of magnitude or even larger than
the numerical diffusion used to stabilize our code. As a consequence,
ambipolar diffusion produces a strong impact on the modeled atmosphere.
Perhaps more importantly, it suggests that at least in the chromospheric
domain, self-consistent simulations of the solar atmosphere driven by magneto-
convection can accurately describe the impact of the dominant form of
resistivity, i.e., ambipolar diffusion. This suggests that such simulations
may be more realistic in their approach to the lower solar atmosphere (which
directly drives the coronal volume) than previously assumed.
Magnetohydrodynamics MHD —Methods: numerical — Radiative transfer — Sun:
atmosphere — Sun: magnetic field
## 1 Introduction
Most of the models and simulations of the solar atmosphere solve the
magnetohydrodynamics (MHD) equations, implicitly assuming that the plasma is
magnetized, i.e., fully ionized or with the ion-neutral collision frequency
lower than the ion gyrofrequency (Schaffenberger et al., 2005; Vögler et al.,
2005; Stein & Nordlund, 2006; Gudiksen et al., 2011, among others). However,
since the photosphere and parts of the chromosphere are unmagnetized, i.e.,
the ions are not necessarily tied to the field lines, we expect that neutral
particles can have a significant impact on the dynamics of this region
(Vernazza et al., 1981; Fontenla et al., 1990, 1993). Therefore, it is likely
that, under some conditions, the photosphere and chromosphere should be
treated as a three component fluid, where the dynamics of the neutrals, ions,
and electrons are treated separately. Under the assumption of a weakly ionized
plasma, one can return to a one-component fluid. However, new terms known as
the Hall term and ambipolar diffusion appear in the induction equation
(Parker, 1963, 2007; Pandey & Wardle, 2008). The latter term is a consequence
of the ion-neutral dissipation which can be derived from the Cowling
resistivity (Khodachenko et al., 2004, 2006; Leake & Arber, 2006). This form
of the induction equation is known as the generalized Ohm’s law (Cowling,
1957).
A large number of papers in recent years have investigated the effects of the
ion-neutral interactions on single fluid MHD. Leake & Arber (2006) simulated
2.5D simulations of flux emergence and observed that the ambipolar diffusion
leads to an increase of the rates of magnetic field emergence and a resultant
magnetic field that is much more diffuse than the case with only Ohmic
diffusivity. In addition, the magnetic field that emerges into the corona is
found to be more force-free, since currents are aligned to the field. This is
because ambipolar diffusion acts on the currents perpendicular to the magnetic
field. Arber et al. (2007) extended this simulation to 3D where the previous
results were confirmed, and in addition found that, as a result of including
neutrals, flux emergence lifts less chromospheric material to great heights.
This effect suppresses the Rayleigh-Taylor instability between the emerging
flux and the corona.
The interaction between ions and neutrals can also dissipate Alfvén waves as
result of the small but finite coupling time between ions and neutrals. This
type of damping can heat and accelerate the plasma in the upper chromosphere
and in spicules (De Pontieu & Haerendel, 1998; De Pontieu, 1999; James &
Erdélyi, 2002; James et al., 2003; Erdélyi & James, 2004) and incur wave
energy leakage at the footpoints of coronal loops (De Pontieu et al., 2001)
and in the network (Goodman, 2000). Khodachenko et al. (2004), using the
temperature and density structure from the 1D VAL-C model, concluded that the
collisional friction damping of MHD waves is often more important than the
viscous damping for waves propagating in the partially ionized plasmas of the
solar photosphere, chromosphere and prominences. Estimates of the efficiency
of the damping of waves were made by Leake et al. (2005) as well as by the
previous authors.
Pandey & Wardle (2008) determine that waves can be affected by the Hall term
at both low and high fractional ionization, because the Hall regime wave
damping is inversely proportional to the fractional ionization. Thus Hall term
may also be important at high fractional ionization in contrast to ambipolar
diffusion which is important only at low fractional ionization.
Khomenko & Collados (2012) performed various simplified scenarios where they
studied the impact of the ambipolar diffusion in the chromosphere. They
conclude that current dissipation enhanced by the action of ambipolar
diffusion is an important process that is able to provide a significant energy
input into the chromosphere. Heating from ambipolar diffusion leads to
thermodynamic evolution in the chromosphere on timescales of about 10–100
seconds.
All the models above, even the 2D and 3D models, are based on a 1D semi-
empirical atmosphere (e.g., VAL-C), and/or a simplified approach to the energy
balance in the chromosphere (adiabatic or Newtonian cooling). In addition,
none of the partial ionization effects have been considered in full magneto-
convection simulations. Cheung & Cameron (2012) have made progress in this
direction and performed full magneto-convection simulations of an umbra taking
into account partial-ionization effects. However, their simulations only
extend up to the upper photosphere.
In this paper, we use the Bifrost code (Gudiksen et al., 2011) to create a
self-consistent and fully dynamic model atmosphere of the sun, from the
convection zone to the corona, to consider the importance of the Hall term and
ambipolar diffusion relative to the Ohmic and artificial diffusion. Unlike
other models, Bifrost includes an advanced treatment of radiative losses in
the chromosphere based on recipes derived from dynamic non-LTE radiative 1D
hydrodynamic simulations. Such a treatment is crucial for a consideration of
the effects of partial ionization, as shown in what follows. The code and the
implementation of the generalized Ohm’s law are described in Section 2. The
tests performed for the code validation are discussed in Section 2.2. We
describe the different forms of diffusion in 2D MHD simulations in Section
3.1. Finally, the various simplifications made in order to obtain the
generalized Ohm’s law following Pandey & Wardle (2008) have been investigated
and tested for the 2D MHD simulations in Section 3.2. The paper finishes by
addressing the conclusions and discussion.
## 2 Equations and numerical method
The magnetic upper-photosphere and chromosphere is weakly ionized and the
interaction between ionized particles and neutral particles potentially has
important consequences for the thermodynamics (Fontenla et al., 1993) of this
region. We investigate these consequences in the solar atmosphere. In order to
model the solar atmosphere we solve the MHD equations in $2.5$D. The model
spans from the upper layers of the convection zone to the low corona. We have
implemented the effects of partial ionization into the induction equation
through the Hall and ambipolar diffusion terms as described below.
The Bifrost (Gudiksen et al., 2011) code is a staggered mesh, explicit code
that solves the MHD partial differential equations, including non-LTE and non-
grey radiative transfer with scattering, and conduction along the magnetic
field lines. A lookup table, based on LTE, is used to compute the temperature,
pressure, opacities and other radiation quantities, and ionization state,
given the pressure and the internal energy of the plasma. Spatial derivatives
and the interpolation of variables are done using high order polynomials. The
equations are stepped forward in time using the explicit third order
predictor-corrector procedure described by Hyman et al. (1979). In order to
suppress numerical noise, high-order artificial diffusion is added both in the
forms of a viscosity and in the form of a magnetic diffusivity (see Gudiksen
et al., 2011, for details).
The Bifrost code includes an advanced treatment of the effects of radiation on
the local energy balance, which is crucial if one wants to accurately
determine the ionization degree. The radiative flux divergence from the
photosphere and lower chromosphere is obtained by angle and wavelength
integration of the transport equation assuming isotropic opacities and
emissivities. The transport equation assumes that opacities are in LTE using
four group mean opacities to cover the entire spectrum (Nordlund, 1982). This
is done by formulating the transfer equation for each of the four bins,
calculating a mean source function in each bin. These source functions contain
an approximate coherent scattering term and an exact contribution from thermal
emissivity. The resulting 3D scattering problems are solved by iteration,
based on one-ray approximation in the angle integral for the mean intensity, a
method developed by Skartlien (2000).
In the mid and upper chromosphere, the Bifrost code includes non-LTE radiative
losses from tabulated hydrogen continua, hydrogen lines, and lines from singly
ionized calcium as functions of temperature and column mass (Carlsson &
Leenaarts, 2012). These radiative losses depend on the computed non-LTE escape
probability as a function of column mass and are based on a 1D dynamical
chromospheric model in which the radiative losses are computed in detail
(Carlsson & Stein, 1992, 1994, 1997, 2002).
The energy dissipated by Joule heating is given by $Q_{Joule}={\bf E\cdot J}$
where the electric field ${\bf E}$ is calculated from the current ${\bf J}$,
taking into account high-order artificial resistivity. The resistivity is
computed using a hyper-diffusion operator (Gudiksen et al., 2011). This
entails that the Joule heating due artificial diffusion is set proportional to
the current squared times a factor that becomes large (of order 10) when
magnetic field gradients are large, and is unity otherwise.
### 2.1 Generalized Ohm’s Law theory
#### 2.1.1 Multi-fluid
Most codes treat the solar atmosphere as a single fluid where collisional
frequencies are considered sufficient to ensure that all species are well
coupled and that the momentum and energy equations can be added without the
introduction of frictional terms or similar. However, as chromospheric
temperatures are likely to drop to a few $10^{3}$ K or even lower (Leenaarts
et al., 2011), there is a high probability that plasma is only partially
ionized and that “slippage” effects could become important. In this case the
MHD equations should be treated by considering the plasma to consist of three
fluids: ions, electrons and neutral particles. The mass density for each type
of particle is governed by the continuity equation applied to each species
separately:
$\displaystyle\frac{\partial\rho_{j}}{\partial t}+\nabla\cdot\rho_{j}{\bf
u_{j}}=0$ (1)
where $\rho_{j}=m_{j}n_{j}$, ${\bf u_{j}}$, $n_{j}$ and $m_{j}$ are the mass
density, velocity, number density and particle mass of the ion, electron, and
neutral species, i.e., $j=\mathrm{i,e,n}$ respectively. The mass transfer term
as result of ionization and recombination has been neglected. This
approximation is valid for a one-fluid approach if the system is in ionization
balance, and there is no decoupling of ions and neutrals.
The momentum equation, written in SI units, for each species, is as follows:
$\displaystyle\rho_{i}\left(\frac{\partial}{\partial t}+{\bf
u_{i}}\cdot\nabla\right){\bf u_{i}}=-\nabla P_{i}+n_{i}Zq_{e}\left({\bf
E}+{\bf u_{i}}\times{\bf B}\right)-\rho_{i}\sum_{j=e,n}\nu_{ij}({\bf
u_{i}-u_{j}})$ (2) $\displaystyle\rho_{e}\left(\frac{\partial}{\partial
t}+{\bf u_{e}}\cdot\nabla\right){\bf u_{e}}=-\nabla P_{e}-n_{i}q_{e}\left({\bf
E}+{\bf u_{e}}\times{\bf B}\right)-\rho_{e}\sum_{j=i,n}\nu_{ej}({\bf
u_{e}-u_{j}})$ (3) $\displaystyle\rho_{n}\left(\frac{\partial}{\partial
t}+{\bf u_{n}}\cdot\nabla\right){\bf u_{n}}=-\nabla
P_{n}-\rho_{n}\sum_{j=e,i}\nu_{nj}({\bf u_{n}-u_{j}})$ (4)
where $q_{e}$ and $Z$ are respectively the electron charge and ion charge.
${\bf E}$ and ${\bf B}$ are the electric and magnetic field and
$P_{j}=n_{i}kT_{i}$ is the partial pressure of the $j$th species, $k$ is
Boltzmann’s constant, and $\nu_{ij}$ is the collision frequency for species
$i$ with species $j$. We assume that collisions are sufficiently numerous that
the ion and electron temperature can be considered the same ($T_{i}=T_{e}$).
All three equations are linked through the last term, i.e., the exchange of
momentum between the particles, where we have ignored the thermal force. In a
similar manner as for the continuity equation, the momentum transfer term as
result of ionization and recombination has been neglected.
The number of equations thus increases considerably compared with single fluid
MHD, but by considering some simplifications, as described by Cowling (1957);
Parker (2007); Pandey & Wardle (2008), one can easily generalize the MHD
equations for each species to a single fluid (see below too). Therefore, the
mass density is governed by the continuity equation for the bulk fluid as
follows:
$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot\rho{\bf u}=0$ (5)
where the density for the bulk fluid is the sum of the different particle
densities ($\rho=\rho_{i}+\rho_{e}+\rho_{n}$), and considering
$\rho_{i}/\rho>>\rho_{e}/\rho$, then $\rho\approx\rho_{i}+\rho_{n}$. In a
similar manner, the velocity of the bulk fluid is ${\bf u}\approx(\rho_{i}{\bf
u_{i}}+\rho_{n}{\bf u_{n}})/\rho$, where the electron inertia is implicitly
neglected in the definition of the bulk velocity. If we define the neutral
density fraction ($D=\rho_{n}/\rho$), then ${\bf u}\approx(1-D){\bf
u_{i}}+D{\bf u_{n}}$. Finally, the current density is given by ${\bf
J}=n_{e}q_{e}({\bf u_{i}-u_{e}})$ (assuming singly charged ions). Since
Equation 5 is the same as for the single fluid formulation, the continuity
equation does not need any modification in the Bifrost code.
Following Pandey & Wardle (2008), the single fluid momentum equation can be
recovered if we neglect the effects of the electron inertia. Because it is
implicitly neglected in the definition of the bulk velocity, it can also be
neglected in the continuity and momentum equations. For simplicity, the ions
are assumed to be singly charged, and we adopt charge neutrality
($n_{i}=n_{e}$). In addition, the drift momentum is assumed to be considerably
smaller than the fast momentum ($\rho\sqrt{v_{a}^{2}+c_{s}^{2}}$) so that:
$\displaystyle\rho_{i}\rho_{n}u_{D}^{2}<<\rho^{2}(v_{a}^{2}+c_{s}^{2}),$ (6)
where ${\bf u_{D}}={\bf u_{i}}-{\bf u_{n}}$, $v_{a}=B/\sqrt{\mu_{o}\rho}$, and
$c_{s}=\sqrt{\gamma P/\rho}$ are respectively the drift, Alfvén and sound
velocities in the bulk fluid; $\mu_{o}$ is the vacuum permeability, and
$\gamma$ is the ratio of specific heats. When the plasma does not fulfill
Equation 6, the fluids are strongly decoupled. This happens when the ion-
neutral collision frequency is low. When the drift momentum is low, the drift
momentum can be neglected for small dynamical frequencies (i.e., changes of
the plasma properties on timescales commensurate with such frequencies):
$\displaystyle\omega\leq\frac{\rho}{\sqrt{\rho_{i}\rho_{n}}}\left(\frac{D\beta_{e}}{1+D\beta_{e}}\right)\nu_{ni}$
(7)
where $\beta_{\rm e}=\frac{\omega_{c\mathrm{e}}}{\nu_{\rm e}}$, the ratio of
the cyclotron frequency and the collisional frequency. With these assumptions
we recover the single fluid momentum equation as it is implemented in the
Bifrost code (see Pandey & Wardle, 2008, for details):
$\displaystyle\rho\left(\frac{\partial}{\partial t}+{\bf
u}\cdot\nabla\right){\bf u}=-\nabla P+{\bf J}\times{\bf B}$ (8)
#### 2.1.2 Induction equation
The Ohmic diffusion, Hall term, and ambipolar diffusion are given by
$\displaystyle\eta_{ohm}=\frac{1}{\sigma}$ (9)
$\displaystyle\eta_{hall}=\frac{|B|}{q_{e}n_{e}}$ (10)
$\displaystyle\eta_{amb}=\frac{(|B|\rho_{n}/\rho)^{2}}{\rho_{i}\nu_{in}}=\frac{(|B|\rho_{n}/\rho)^{2}}{\rho_{n}\nu_{ni}}$
(11)
The electrical conductivity ($\sigma$) in the absence of a magnetic field is
$\displaystyle\sigma=\frac{q_{e}^{2}n_{e}}{m_{e}\nu_{e}}$ (12)
where the sums of the collision frequencies are written
$\displaystyle\nu_{e}=\nu_{en}+\nu_{ei}$ (13)
$\displaystyle\nu_{i}=\nu_{in}+\nu_{ie}$ (14)
In order to obtain the induction equation the following assumptions are made:
* •
First, the electric field
$\displaystyle{\bf E}+{\bf u_{i}}\times{\bf B}=-\frac{\nabla
P_{e}}{n_{e}q_{e}}+\frac{{\bf J}}{\sigma}+\frac{\bf J\times
B}{q_{e}n_{e}}-\frac{m_{e}\nu_{en}}{q_{e}}{\bf u_{D}}$ (15)
is deduced from the electron momentum equation assuming zero electron inertia
and is expressed in the ion’s rest frame.
* •
The plasma obeys
$\displaystyle\rho_{e}\nu_{en}<<\rho_{i}\nu_{in}.$ (16)
* •
The term
$\displaystyle\frac{\rho_{i}\rho_{n}}{\rho}\left[\frac{d{\bf u_{D}}}{dt}-({\bf
u_{D}}\cdot\nabla){\bf u_{i}}-({\bf u_{i}}\cdot\nabla){\bf u_{D}}\right]$ (17)
can be neglected when the dynamical frequency of the plasma is small
$\displaystyle\omega\leq\nu_{ni}\rho/\rho_{i}$ (18)
* •
Biermann’s battery contribution, from the $\nabla P_{e}/q_{e}n_{e}$ term in
Equation 15, is neglected.
* •
The term $D\beta_{\mathrm{i}}/\beta_{\mathrm{e}}$ is small and of order $\leq
10^{-3}$, so it is also neglected, where $\beta_{j}=\omega_{cj}/\nu_{j}$ is
the ratio of the cyclotron frequency to the sum of the $j$th particle
collision frequency.
* •
Finally, terms due to the pressure gradient $\nabla P\times{\bf B}$ are
negligible compared to the induction term ${\bf u\times B}$ when the dynamical
frequency is small:
$\displaystyle\omega\leq\left(\frac{v_{a}^{2}}{c_{s}^{2}}\right)\frac{\rho^{2}}{\rho_{i}\rho_{n}}\nu_{ni}$
(19)
Under these assumptions the electric field is defined as
$\displaystyle{\bf E}=\frac{{\bf J}}{\sigma}+\frac{\bf J\times
B}{q_{e}n_{e}}-D^{2}\frac{{\bf J\times B\times B}}{\rho_{i}\nu_{in}},$ (20)
and the magnetic field evolution is governed by the induction equation,
derived from the Maxwell equations, and under the considerations listed above
(see Parker, 2007; Pandey & Wardle, 2008, for details).
$\displaystyle\frac{\partial{\bf B}}{\partial t}=\nabla\times\left[{\bf
u\times B}-\eta{\bf J}-\frac{\eta_{hall}}{|B|}{\bf J\times
B}+\frac{\eta_{amb}}{B^{2}}({\bf J\times B})\times{\bf B}\right]$ (21)
The right hand side of the induction equation has the convective, Ohmic, Hall,
and ambipolar terms, from left to right respectively. Note that for simplicity
we are referring to the Ohmic and ambipolar terms as diffusion terms, but
strictly speaking none of them can be cast in the form of a diffusion equation
(Parker, 1963, already used ambipolar diffusion terminology). The two new
terms (Hall and ambipolar) are implemented in the Bifrost code in the
induction equation and in the electric field.
Note that from Equation 21, the Hall and ambipolar terms can be considered as
advection terms:
$\displaystyle\frac{\partial{\bf B}}{\partial t}=\nabla\times\left[{\bf
u\times B}-\eta{\bf J}-{\bf u_{H}\times B}+{\bf u_{A}\times B}\right]$ (22)
where the Hall velocity is ${\bf u_{H}}=(\eta_{hall}{\bf J})/|B|$ and the
ambipolar velocity is ${\bf u_{A}}=(\eta_{amb}{\bf J\times B})/B^{2}$.
The generalized Ohm’s law is implemented in the code using the same scheme as
used for the MHD equations, i.e., a 6th order explicit method (Gudiksen et
al., 2011). From the expression 22 it is clear that the Hall term and
ambipolar diffusivity give rise to two new constraints on the CFL condition
which restrict the timestep interval (Courant et al., 1928) ($\Delta
t_{H}=\Delta x/{\bf u_{H}}$ and $\Delta t_{A}=\Delta x/{\bf u_{A}}$). Both
velocities are a function of the current ($\nabla\times\mathbf{B}$), i.e.,
both CFL conditions are quadratic functions in $\Delta x$, and the timestep
will decrease quadratically with increasing spatial resolution. We note that
for the simulation with mean magnetic field strength in the photosphere of the
order of 100 G, the ambipolar and Hall velocities are maximal in the cold
regions in the chromosphere with, respectively, $u_{A}\approx 100$ km s-1 and
$u_{H}\approx 1$ km s-1. As result of this, the CFL criteria are approximately
$\Delta t_{A}\approx 0.3$ s and $\Delta t_{H}\approx 20$ s with $\Delta
x\approx 32$ km, compared with the strictest CFL condition in the simulation
of $\Delta t\approx 3\,10^{-3}$ s. Therefore, as long as we do not increase
the magnetic field and/or the spatial resolution too much, we do not need to
change to an implicit implementation of our equations.
#### 2.1.3 The energy equation
As mentioned above, the energy dissipated by Joule heating is given by
$Q_{Joule}={\bf E}\cdot{\bf J}$. In the previous section, the Hall term and
ambipolar diffusivity were shown to lead to changes in the electric field.
These changes need to be taken into account when computing the energy due to
the dissipation of the magnetic field. Note however, that because the Hall
term in the electric field is a function of ${\bf J}\times{\bf B}$, then
$({\bf J}\times{\bf B})\cdot{\bf J}$ is zero, i.e., the Hall term does not
produce any energy dissipation at all. The only terms which directly dissipate
electromagnetic energy by dissipation are by Ohmic and by ambipolar diffusion.
In the Bifrost code the former is negligible compared to the artificial
diffusion needed to stabilize the code at numerically resolvable scales and is
therefore set to zero.
In contrast to the artificial resistivity present in the code, the Hall term
and ambipolar diffusion are calculated as a function of the electron density
and, for the latter, of the collision frequency between the different species
in the solar atmosphere. In order to avoid instabilities from rapid heating
processes due to the new terms, it is sometimes necessary to further limit the
time steps (beyond the CFL condition) because the timescales of the energy
dissipation of the ambipolar diffusion are short. As a result, the energy
distribution in the chromosphere changes rapidly and the source and sink terms
in the energy equation, such as radiative processes, need to be updated more
often than is the case without ambipolar diffusion.
#### 2.1.4 Collision frequencies
The collision frequency between electrons and ions can be found in e.g. Priest
(1982) and is given by
$\displaystyle\nu_{ei}=3.759\,10^{-6}n_{e}T^{-{3/2}}\ln\Lambda$ (23)
and
$\displaystyle\frac{\nu_{en}}{\nu_{ei}}=5.2\,10^{-11}\frac{n_{n}}{n_{e}}\frac{T^{2}}{\ln\Lambda}$
(24)
where $\ln\Lambda$ is the Coulomb logarithm (all in SI units).
As in De Pontieu et al. (2001), we follow three different approximations in
computing the collision frequency between ions and neutral particles: as
described by Osterbrock (1961); De Pontieu & Haerendel (1998) (hereafter case
A), as described by von Steiger & Geiss (1989) (hereafter case B) and as
described by Fontenla et al. (1993) (hereafter case C), (see Appendix A).
Table 1 lists the 2D simulations for which we investigate the effects of these
different methods to calculate $\nu_{in}$. We note that the appendix of (De
Pontieu et al., 2001) contains two typos: their formula A6 should be divided
by 2 to provide the correct expression for the collision frequency between
neutral hydrogen and protons, and formula A12 should be replaced by our
formula . Our formula provides the correct equation for the collision
frequency between neutral hydrogen and protons, according to the recipe
derived by De Pontieu & Haerendel (1998) and Osterbrock (1961).
Throughout the paper we will focus on the results of case B since it is more
recent, and the most extensive.
In order to calculate the various collision frequencies, the ion and neutral
fractions are calculated from the Saha-Boltzman equation. The electron density
is also computed on the basis of LTE, in practice this is done via a table
lookup in the Bifrost code. In the pre-computed table, the 16 most important
atomic species in the solar atmosphere are taken into account. Table 2 lists
the atomic species, abundances and ionization fraction ($X_{i}$).
### 2.2 Tests
One of the main objectives of this work is to study the importance and
validity of the generalized Ohm’s law in a “realistic” 2.5D simulation of the
solar atmosphere. We describe three different tests done for the
implementation of the generalized Ohm’s law, which also illustrate the role
and importance of each form of diffusivity. For two of the tests, we imposed a
velocity equal to zero at all times in the full domain. We also run separate
tests using only the Hall term or ambipolar diffusion.
#### 2.2.1 1D Hall test
First, we test that our code correctly includes the Hall term. In this test
case, we set the velocities and ambipolar diffusion to zero and consider the
induction equation in 1D
$\displaystyle\frac{\partial B_{y}}{\partial
t}=-\eta_{hall}B_{x}\frac{\partial^{2}B_{z}}{\partial x^{2}}$ (25)
$\displaystyle\frac{\partial B_{z}}{\partial
t}=\eta_{hall}B_{x}\frac{\partial^{2}B_{y}}{\partial x^{2}}$ (26)
For this test, we set $B_{x}$ constant ($B_{x}=0,1121,2242$ G are shown with
orange diamond symbols, and blue and green lines in Figure 1). With higher
$B_{x}$, the rate at which $B_{y}$ and $B_{z}$ change with time increases.
However, the total magnetic flux should remain the same at all times, since
the Hall term cannot convert the magnetic flux into thermal or kinetic energy.
Note also that the Hall term will give rise to a non-zero $B_{z}$ (and
therefore a non-zero $u_{z}$ in a dynamic simulation) even if the field
originally has no component in the $z$-direction. Figure 1 shows $B_{y}$ in
the top panel and $B_{z}$ at the bottom panel for four different runs. All
cases have the same jump in $B_{y}$ (black triangle symbols in the top panel)
and a constant Hall term. In the test shown in red line in Figure 1 does not
have the Hall term and $B_{x}=2242$ G. All of these tests are shown at the
same instant ($t=20$ s).
The rate of change of $B_{y}$ and $B_{z}$ is as expected, i.e., the case with
$B_{x}=2242$ G leads to an increase of unsigned total flux of $B_{z}$
(integrated along the x-axis) that is twice as large as the case where
$B_{x}=1121$ G. Moreover, the case $B_{x}=0$ G behaves similarly to the case
with no Hall term. The magnetic flux is in all cases conserved. This gives us
confidence that our implementation of the Hall term in the code is
satisfactory.
#### 2.2.2 1D Ambipolar test
In 1D, the induction equation for $B_{y}$ is:
$\displaystyle\frac{\partial B_{y}}{\partial
t}=\eta_{amb}\frac{\partial}{\partial x}\left(B_{y}^{2}\frac{\partial
B_{y}}{\partial x}\right)$ (27)
Apart from the trivial solution, $B_{y}=constant$, it is clear that Equation
27 also permits a steady solution of the form of $B_{y}\propto x^{1/3}$ (see
Brandenburg & Zweibel, 1994, for details). In this expression we should keep
in mind that the code includes numerical diffusivity in addition to ambipolar
diffusion. We consider the evolution of an initially sinusoidal profile of
$B_{y}$. This profile evolves, and strong gradients become stronger
approaching the form $B_{y}\propto x^{1/3}$ as time progresses. Figure 2 shows
the initial condition of $B_{y}$ (solid line) and at $t=50$ s (dashed line)
which is close to the steady solution. Observe that where the gradient of
$B_{y}$ is large $B_{y}$ closely follows the expression $B_{y}\propto x^{1/3}$
(dash-dotted line).
Ambipolar diffusion converts magnetic energy into thermal energy as discussed
above. In this 1D test, we turn off the heating from artificial diffusion and
only allow heating from the ambipolar diffusivity. Such heating in this simple
simulation must follow the expression
$\displaystyle\frac{\partial e}{\partial t}=\eta_{amb}J_{z}^{2}B_{y}^{2}.$
(28)
Figure 3 shows the energy profile with $x$ of this test (black diamonds) at
$t=2.1$ s. Calculating the right hand side of this expression using the
sinusoidal shape of $B_{y}$ from the initial condition, then deriving $J_{z}$,
we calculate the energy to be
$\displaystyle e=e_{init}+\eta_{amb}J_{z}^{2}B_{y}^{2}\Delta t,$ (29)
where $\Delta t$ is the time increment. This relationship is shown with the
red line in Figure 3. The black diamonds overlaps the red line as would be
expected. This indicates we have correctly implemented ambipolar diffusion in
the code.
#### 2.2.3 Collision frequencies test: VAL-C model
In order to test whether the absolute values of the diffusion terms are
calculated correctly, we use three different sources for the neutral-ion
collision frequency ($\nu_{ni}$, see Section 2.1.4). This also allows us to
study the uncertainties involved in the various formulas for the collision
frequency, as already studied for 1D-static models by De Pontieu et al.
(2001). We test our implementation in the Bifrost code by using the densities
and temperatures from the VAL-C atmospheric model (Vernazza et al., 1981)
which allows us to compare our results with those found in the published
literature. Indeed, we correctly obtain the neutral-ion collision frequencies
as a function of height as can be seen by comparing our Figure 4 with Figure 2
in De Pontieu et al. (2001). The large dip in the collision frequency at $0.5$
Mm is due to low number of ions (mostly non-hydrogen species) in this region..
### 2.3 Initial and boundary conditions
Let us now consider the importance and validity of the generalized Ohm’s law
in a “realistic” 2.5D simulation of the solar atmosphere. The 2.5D
computational domain stretches from the upper convection zone to the lower
corona and is evaluated on a non-uniform grid of $512\times 325$ points
spanning $16\times 16$ Mm2. The frame of reference for the model is chosen so
that $x$ is the horizontal direction and $z$ is the vertical direction (Figure
5). The grid is non-uniform in the vertical $z$-direction to ensure that the
vertical resolution is good enough to resolve the photosphere and the
transition region with a grid spacing of $28$ km, while becoming larger at
coronal heights where gradients are smaller.
We run two different initial conditions with different values for the unsigned
magnetic field strength but with similar field configurations (Figure 5).
$B_{y}$ is originally set to zero. The initial model starts with a magnetic
field that is inclined some 5 degrees with respect to the vertical axis and
the two different setups for the unsigned field strengths in the photosphere
are $0.25$ G, the other $90$ G. These two initial conditions are run for the
three different formulas that were mentioned above to calculate the collision
frequency $\nu_{in}$. The simulations with the initially weak magnetic field
using cases A, B or C for the neutral-ion collision frequency are labeled WA,
WB or WC, while the strong field simulations using cases A, B, or C for the
neutral-ion collision frequency are labeled SA, SB, or SC, respectively. Table
1 list the different simulations.
In the following we will refer in our analysis to simulations WB and SB,
unless otherwise noted.
## 3 Results
The simulations presented include the dynamic processes (including radiative
losses) of the photosphere and chromosphere and a self-maintained chromosphere
and corona. This is a very different type of model as compared to semi-
empirical models such as the VAL-C model, or previous simulations
investigating the effects of partial ionization which had a simplified
treatment of the energy balance (and ionization degree). It is thus of
significant interest to determine how dynamic atmospheres such as from our
simulations impact the importance of the ambipolar diffusion and the Hall
terms (also assuming different approximations to the collision frequency), and
to compare the results with those from the models based on a VAL-C type
atmosphere.
The basic structure of our modeled chromospheres are shown in Figure 5. A full
description of their properties fall outside the scope of this paper but we
will mention the most important as concerns ambipolar diffusion and the Hall
term: the basic thermodynamic state of the chromosphere is maintained by the
continual injection of acoustic shocks from the photosphere. These
perturbations are due to the chaotic generation of waves in the convection
zone, of which waves with periods of order $3$ minutes will propagate and
steepen in the chromosphere, as is well known and as extensively studied by
Carlsson & Stein (1992, 1994). The propagation of waves will be modified in
the presence of a magnetic field (Bogdan et al., 2003; De Pontieu et al.,
2004; Heggland et al., 2007; McIntosh et al., 2011; Heggland et al., 2011,
among others)
but will nevertheless steepen and form strong shocks, with high temperatures
in the shock fronts and very low temperatures in the regions behind (Leenaarts
et al., 2011). These “cold chromospheric bubbles” can be seen in both panels
of Figure 5 which show temperatures as low as $2\,000$ K or lower. In the
strong field case the Lorentz force is clearly important, pushing the corona
upwards and allowing cool material to exist at great heights, much higher, up
to 5 Mm above the photosphere, than that found in semi-empirical models where
the maximum chromospheric height is found to be of order 2-2.5 Mm. The
distribution of density and temperature with height in dynamical “realistic”
simulations is discussed in much greater detail in e.g. Leenaarts et al.
(2011).
### 3.1 Collision frequencies and diffusivities
As mentioned, most studies of the effects of ion-neutral collisions in the
chromosphere have been based, in some form, on semi-empirical models (VAL-C or
FAL-C models as shown in Figure 4). However, the chromosphere and transition
region are clearly highly dynamic, and it is of great importance to know the
effects of the neutral-ion interactions in such dynamic atmospheres. First of
all, we are interested in studying the relative importance of the different
diffusivities in the chromosphere and transition region. Figures 6 and 7 show
the Ohmic diffusion, artificial diffusion, Hall term, and ambipolar diffusion
from top to bottom and left to right for the simulations WB and SB,
respectively.
On comparing the different diffusivities, we find that in the entire
chromosphere, the Hall term is on average two orders of magnitude larger than
Ohmic diffusion. This is true for both simulations WB and SB and is perhaps
more easily seen by considering the ratios of the diffusivities plotted in
Figure 8. Ambipolar diffusion is roughly four orders of magnitude larger in
the weak field (WB) case and fully six orders of magnitude larger than Ohmic
diffusion for the strong field SB case. Ambipolar diffusion is considerably
larger for SB than for WB because ambipolar diffusion depends quadratically on
the magnetic field strength. Note that while Ohmic diffusion has a significant
magnitude throughout the atmosphere, ambipolar diffusion is important only in
the chromosphere.
Numerical simulations must include some form of artificial diffusion in order
to compensate for the fact that they do not have infinite spatial resolution.
In the Bifrost code this is done through a so called hyper-diffusivity which
in practice means that the diffusion coefficient is increased in regions that
require high diffusivity, i.e., where gradients are large. The magnitude of
this artificial diffusion is set by the spatial resolution. To some degree
this behavior is similar to Ohmic diffusion, but there are also significant
differences. Simulations run at the highest possible spatial resolution cannot
even come close to the diffusion values found in Ohmic diffusion. This is a
well known problem for numerical simulations of the solar atmosphere.
For the simulations reported here, we find an artificial diffusivity in the
chromosphere that is three orders of magnitude larger than the Ohmic
diffusivity and that is up to five orders of magnitude larger than the Ohmic
diffusivity in the corona. By design, the artificial diffusion is largest
where the shocks and other high gradient phenomena are located. Moreover,
since the grid is non-uniform and the grid resolution decreases with height,
artificial diffusion will on average be larger in the corona than in the lower
layers. In contrast, the Ohmic diffusion is largest in the upper photosphere
and chromosphere. As result of these differences between artificial diffusion
and Ohmic diffusion, the magnetic Reynolds number on the Sun and in the
simulations is completely different: the magnetic Reynolds number is several
orders of magnitude higher in the solar atmosphere than even the highest
resolution simulations. Since Ohmic diffusion is negligible compared to
artificial diffusion we do not include its effects either in the induction
equation nor in the energy equation. In a similar manner, as result of the low
resolution of these simulations, the artificial diffusion will mask the Hall
term, but here we are interested in describing the stratification of the Hall
term and ambipolar diffusion in self-consistent magneto-convection
simulations.
One of the most interesting results of our calculations is that, on the other
hand, ambipolar diffusion is of the same order or, in some regions, even
larger than the artificial diffusion in the chromosphere. This is a perhaps
surprising, but crucial property of the chromosphere. It allows us to use our
numerical simulations to study the effects of ambipolar diffusion while using
the correct physical magnitudes of the coefficients. As a result, the
chromosphere may be the only region where simulations are close to reality
once all the physics are included in the code, despite the necessarily limited
resources of todays computing technology. This has an impact beyond the
chromosphere, since it directly affects discussions on whether these self-
consistent magneto-convective simulations provide a realistic driver and
boundary to the corona. For example, recent simulations by Hansteen et al.
(2010) suggest a preponderance of heating in the lower atmosphere (first few
Mm above the photosphere), implying that much of the coronal heating occurs
towards the footpoints (Martínez-Sykora et al., 2011). The large ambipolar
dissipation we find here suggests that such simulations (which only include
artificial resistivity) are actually much more realistic than previously
thought, including the predictions of heating low down.
The Ohmic diffusivity, Hall term, and ambipolar diffusivity depend on the
electron density, while the Ohmic and ambipolar diffusivites also depend on
collision frequencies which are shown in Figures 9 and 10 for the weak field
WB and strong field SB simulations (see Appendix A). (Note that the Ohmic
diffusivity is proportional to the collision frequency, while ambipolar
diffusivity is inversely proportional to the collision frequency.) On average,
the collision frequency between electrons and ions is larger than both the
ion-neutral, electron-neutral, and neutral-ion collision frequencies in the
chromosphere, in the WB simulation one order of magnitude larger and in the SB
simulation two orders of magnitude. This difference in collision frequencies
between the WB and SB simulations is mainly because the chromosphere is hotter
in the SB simulation as a result of ambipolar heating.
The electron-ion collision frequency also shows strong variation throughout
the chromosphere, by almost 5 orders of magnitude in both simulations. This
variation is due to the electron density variation in the chromosphere (see
second row in Figure 11 and Equation 23). As a result, the electron-ion
collision frequency is lower inside the cold chromospheric bubbles than in the
shock fronts. In the chromosphere, the electron-neutral and ion-neutral
collision frequencies are similar in magnitude. However, the ion-neutral
collision frequency shows a stronger variation in space in the middle
chromosphere than the electron-neutral collision frequency. This is especially
true in the cold chromospheric bubbles, where the ion-neutral collision
frequency is significantly lower. What causes these differences? First, we
note that $\rho_{n}$ shows less variation in horizontal cuts in the lower
chromosphere than $\rho_{i}$ because the region is mostly dominated by
neutrals. As result of this, the neutral density is almost similar to the
total density. In the cold bubbles, hydrogen is mostly neutral, and the only
ions are provided by the heavier metals. While both the electron-neutral and
ion-neutral collision frequency are dependent on the neutral density (which
does not vary much in the lower chromosphere), the dominance of metals in
providing ions implies that the average mass per ion increases significantly
in the cold bubbles (compared to the rest of the chromosphere). The associated
drop of average thermal speed (for the heavy ions compared to protons) is the
reason for the sharp drop in ion-neutral collision frequency in the bubbles
(compared to the rest of the chromosphere). The neutral-ion collision
frequency is even lower than the ion-neutral collision frequency in the
bubbles, because there are so few ions available to collide with (bottom
panels in Figure 11).
We now consider which parameters are responsible for the changes in
diffusivities throughout the solar atmosphere. In both simulations (WB and
SB), the strongest Ohmic diffusivity is concentrated in the lower-middle
chromosphere while it is weaker in the corona and convection zone. In the
chromosphere, the Ohmic diffusivity varies over a range of almost four orders
of magnitude. This variation in the chromosphere is due to the strong
variation of the electron density and collision frequency of electrons with
neutrals and ions (Figures 9-11). Ohmic diffusion is large in the expanding
cool bubbles and low where temperatures are higher. This is because the Ohmic
diffusion variations are dominated by the variations in electron density,
which is very low in the cool bubbles, and large in shock fronts. The
collision frequency of electrons with ions and neutrals does not drop as
precipitously in the cold bubbles since there are plenty of neutrals to
collide with in these bubbles.
The Hall term is largest in the lower-middle chromosphere and in the corona
(Figure 8). This is because it is inversely proportional to the electron
density which is small in both regions. We see that for both simulations, the
Hall term is larger than the Ohmic diffusivity in the chromosphere and corona,
but not in the photosphere nor in the convection zone. In the cooler regions
of the chromosphere, the Hall term is relatively even higher than in the shock
fronts, and up to three orders of magnitude greater than the Ohmic
diffusivity. Such differences are a bit larger in the WB simulation, since
electron density is smaller in the cold chromospheric bubbles in the weak
field model. This difference in the electron density between WB and SB is
because the cold bubbles have cooler temperatures in WB simulation than in the
simulation SB. In the intergranular lanes in the photosphere, the Hall term is
the most important diffusion term after the artificial diffusion. Therefore,
since the Hall term is proportional to the strength of the magnetic field,
this term may be important to consider in magnetoconvective simulations that
include strong magnetic fields.
Ambipolar diffusion is important in the region from the upper-photosphere to
the upper-chromosphere. In the photosphere, ambipolar diffusion shows some
importance in intergranular lanes which have strong concentrations of magnetic
field (Figure 11). Therefore, the strong field in the SB simulation shows
considerably more diffusivity in intergranules with high magnetic flux
concentrations than in the weak field WB simulation. In the chromosphere,
ambipolar diffusion dominates almost everywhere except for in the lower
chromosphere in shock fronts. The largest difference between ambipolar and
Ohmic diffusion is located in the cold chromospheric bubbles and near the
upper-chromosphere/lower transition region. Note that for the SB simulation
the ratio between ambipolar and Ohmic diffusion is almost four orders of
magnitude larger than in the WB simulation due to the quadratic dependence of
the ambipolar diffusivity on the magnetic field strength. The ambipolar
diffusivity is large in the cold bubbles since the ion density and the ion-
neutral collision frequencies are low, but mainly because the ion density is
extremely low (5 orders of magnitude lower than in the chromospheric shock
fronts). In the upper chromosphere, the ambipolar diffusivity becomes
relatively strong due to low densities — which lead to low ion-neutral
collision frequencies — but only in those regions where the magnetic field
strength is high. In the cold bubbles the ion density is low because of the
adiabatic expansion and cooling, whereas in the upper chromosphere, it is
because the density drops by 2–3 orders of magnitude compared to the lower
chromosphere. As shown in Figure 11, the ambipolar diffusivity depends
strongly on the ion and neutral density and thus on the ionization state of
the chromospheric plasma. However, it is well known that in the middle and
upper chromosphere the ionization and recombination rates are fairly slow for
hydrogen which will not be in ionizational equilibrium (Carlsson & Stein,
2002). This suggests that, in order to treat ambipolar diffusion
realistically, it is necessary to solve the full time dependent rate equations
for hydrogen ionization (Leenaarts et al., 2007).
#### 3.1.1 Comparison with VAL-C model
The VAL-C model does not provide a good description of the strong temporal and
spatial variations found in the physical variables of the chromosphere. In the
section above, we have seen that the different collision frequencies and
diffusivities show spatial variations of several orders of magnitude at the
same height in the chromosphere. The lower-chromosphere changes rapidly due to
the shock fronts; these lead to changes in the thermodynamic structure of the
lower chromosphere on times-scales shorter than a minute. As a result of this,
cold chromospheric bubbles appear and disappear in minutes. Due to the
ambipolar diffusion, plasma is heated in the cold bubbles on timescales
shorter than those characterizing the shock front. As a result of the spatial
and temporal variations, the neutral-ion collision frequency varies by almost
eight orders of magnitude in the chromosphere in the 2D simulation, whereas
the VAL-C model has a unique value for the collision frequency at every height
(Figure 12). In the cold chromospheric bubbles, the collision frequency drops
to considerably lower values than those found in the VAL-C model. This is a
result of the low ion number density in these areas, which are overestimated
in the VAL-C model.
We use the maximum, minimum and median magnetic field of the 2D models (SB and
WB) as a function of height in order to calculate the range of ambipolar
diffusivities in the semi-empirical VAL-C model. The ambipolar diffusion has a
very wide range of values, 8 orders of magnitude in simulation SB and 11
orders of magnitude for simulation WB. These variations are almost 6 or 8
orders of magnitude larger than in the VAL-C model. The ambipolar diffusivity
in the 2D simulations is much higher in most of the chromosphere compared to
what is found in the VAL-C model. The reason for the large difference of the
neutral-ion collision frequency and the ambipolar diffusivity in the VAL-C
model (compared to the 2D model) is because the VAL-C model does not capture
the thermodynamics of the cold chromospheric bubbles where the neutral-ion
collision frequency drops precipitously.
These large differences in both the neutral-ion collision frequency and
ambipolar diffusivities, found between the VAL-C model and our simulation
should lead to a re-examination of previous results related to the generalized
Ohm’s law (see references) using semi-empirical models to define the density
and temperature structure. We also reiterate the importance of taking into
account the likely dynamic state of hydrogen ionization (Leenaarts et al.,
2007).
#### 3.1.2 Other methods to calculate collision frequencies
We considered three different methods to calculate $\nu_{in}$ (Section 2.1.4),
and thus, the ambipolar diffusivity. Do the different methods give similar
values of the collision frequency and/or diffusion for the different models?
We note that the evolution of the simulations using the different methods to
calculate the collision frequency (WA, WB and WC, and SA, SB, and SC) diverge
within a few minutes. We therefore integrate the properties of the models in
time in order to study the different values of the collision frequency and
ambipolar diffusion, and proceed as follows. In Figure 13 we show the joint
probability distribution function (JPDF) of the temperature versus the ion-
neutral collision frequency (top) and the ambipolar diffusivity (bottom) for
the simulations labeled WB (left) and WC (right). In Figure 14 we show the
cases SB and SC in a similar manner. We have integrated over 4 minutes. Note
that the variation of the $y$-axes are logarithmic and cover more than 10
orders of magnitude. We mostly focus on cases B and C since those are the most
recent, and include more advanced calculations of the collision frequencies.
The values for $\nu_{in}$ and $\eta_{amb}$ differ in range and mean values for
the different cases in each simulation. These differences are significant in
certain temperature ranges. The differences between the different methods to
calculate the ion-neutral collision frequency are similar for both atmospheres
(weak and strong magnetic field strength). For instance, at $\log(T)\approx
3.7$ ($5000$ K), case B shows ion-neutral collision frequencies that are a
factor two larger than for case C. At temperatures larger than $\log(T)\approx
3.7$, the collision frequency for case B is almost two orders of magnitude
smaller than in case C. As a result, in certain temperature ranges, the
largest values of the ambipolar diffusion for case B are almost 2 orders of
magnitude larger than for case C. For temperatures lower than $\log(T)\approx
3.6$ ($4000$ K), the median collision frequency as a function of temperature
is roughly similar between cases B and C, but not the distribution, as can be
seen: case B reaches collision frequencies smaller than case C.
At temperatures larger than $\log(T)\approx 3.8$ ($6300$ K), the collision
frequencies for case C are one order of magnitude larger than for case B. As a
result, the median of the ambipolar diffusion for cases WC and SC is one order
of magnitude smaller than for WB and SB.
In order to have a better impression where in the atmosphere the ion-neutral
collision frequency and ambipolar diffusion differ between the different
cases, we take the same atmospheric model (simulation WB or SB at $t=2500$ s)
and calculate from these two models the collision frequency and ambipolar
diffusivity using the different methods (Figures 15-16). It is interesting and
important to see that at the precise location where the ambipolar diffusion is
really high (in cold chromospheric bubbles and in the upper chromosphere), the
different methods differ most. In the cold chromospheric bubbles for both
atmospheres (weak and strong magnetic field), the collision frequency using
case B is almost 4 times smaller than case A, but similar to case C. As
result, the ambipolar diffusivity is more than 2 times smaller using the
method of case B than for case A. In the upper chromosphere or shock fronts,
the collision frequency using case B is almost two times larger than case A
and slightly larger than with case C. As a result, the ambipolar diffusivity
using case B is more than 3 times smaller than case A, and almost 3 times
smaller than case C. However, in the proximity of the transition region, the
collision frequency for case B is a bit smaller than case A, and more than 10
times smaller than case C. As result of this, the ambipolar diffusivity using
case B is a bit larger than case A, and more than 10 times larger than case C.
These large differences between each method are due to the different
temperature dependences (see Appendix A). As mentioned, these differences lead
to rapidly diverging thermodynamic evolution in the various models. Thus, it
is important to take into account this uncertainty in calculating the
collision frequencies when the generalized Ohm’s law is modeled.
### 3.2 Approximations to the generalized Ohm’s law
The generalized Ohm’s law is based on several approximations and
considerations. In this section we describe where these approximations fail
and the implications of this failure. We employ the atmospheres of the WB and
SB simulations in this discussion.
#### 3.2.1 Approximations in the momentum equation
Let us establish and validate the different assumptions underlying the
generalized Ohm’s law as implemented in the code, and see if they are
fulfilled in the fully dynamic self-consistent simulations. One of the first
consideration is that the ion density dominates over the electron density
($\rho_{i}/\rho>>\rho_{e}/\rho$). Everywhere in the atmosphere, the values of
ion and electron densities remain within the range that fulfill
$\rho_{i}/\rho>>\rho_{e}/\rho$ so that electron inertia can be neglected.
In order to neglect the effects of drift momentum in the momentum equation,
the drift momentum has to be smaller than the fast momentum
($\rho\sqrt{v_{a}^{2}+c_{s}^{2}}$, see Equation 6 and Pandey & Wardle (2008)).
This approximation is fulfilled in most of the atmosphere under both strong
and weak field conditions. The only exception is in the weak field atmosphere,
where some low density areas just below the transition region show a ratio of
order 0.1-1 (see Figure 17). This is because the ion-neutral collision
frequency drops significantly there, so that the drift between ions and
neutrals becomes rather large. As a result, in these small regions the plasma
becomes decoupled from the neutrals, and it may be necessary to add the drift
momentum to the momentum equation, and/or solve the MHD equations using
multiple fluids. In the weak field atmosphere, a few of the cold, expanding
bubbles show ratios of order 0.1, so that the fast momentum does stay in
excess of the drift momentum. This suggests that the region below the
transition region is the only one of concern for this particular condition.
#### 3.2.2 Approximations in the induction equation
To allow the removal of the time dependence of the drift momentum equation
(Pandey & Wardle, 2008), the electron density times the collision frequency of
electrons with neutrals has to be smaller than the ion density times the
collision frequency of ions with neutrals (
$\rho_{e}\nu_{en}<<\rho_{i}\nu_{in}$). This approximation is fulfilled in most
of the atmosphere with the exception of some areas in the upper photosphere
and in the cold chromospheric bubbles (Figure 18). In the cold bubbles, the
electron-neutral collision frequency is almost similar to the ion-neutral
collision frequency. In the upper photosphere, and the collision frequency of
electrons with neutrals is relatively large so
$\rho_{e}\nu_{en}<<\rho_{i}\nu_{in}$ is not fulfilled. Therefore, in these
regions, the proper way to solve the ambipolar term in the induction equation
is by calculating the drift velocity using the fully time dependent equation
of ${\bf u_{D}}$ (Pandey & Wardle, 2008).
Some of the approximations used in deriving the equations require that the
dynamical frequency remains smaller than the frequencies shown in Figure 19.
The typical timescales on which the simulated atmosphere evolves is of order
10s or longer, i.e., a dynamic frequency of $\approx 0.5$ Hz or lower: if the
frequencies shown in Figure 19 are higher than $\approx 0.5$ Hz, the
assumptions underlying the generalized Ohm’s law are fulfilled.
The first assumption is that the time derivative of the drift velocity can be
neglected. Following Equation 18, this can be done only if the dynamical
frequency is smaller than the frequency $(\rho/\rho_{i})\nu_{ni}$ shown in the
top panels of Figure 19. The latter frequency is very high in the upper
photosphere and the chromosphere, and stay well above the dynamical frequency
of our simulations ($\approx 0.5$ Hz). Only in the vicinity of the transition
region does $(\rho/\rho_{i})\nu_{ni}$ become small enough that it is of the
same order as the dynamical frequency of the simulations. As a result, we may
need to take into account the derivative terms shown in Equation 17 only in
this small region in the vicinity of the transition region.
A second assumption is that the dyadic product of the drift velocity in the
momentum equation can be neglected (Pandey & Wardle, 2008). This term can only
be neglected if the dynamic frequency stays well below the frequency defined
in Equation 7 and shown in the middle panels in Figure 19. We find that in the
cold chromospheric bubbles (in the weak field case) and in the upper
chromosphere (in both weak and strong field cases), this assumption sometimes
fails. In these regions, we may thus need to take into account the momentum
drift term in the momentum equation.
A final assumption is that the terms of the form $\nabla P\times{\bf B}$ in
the induction equation can be neglected. This can only be done when the
dynamic frequency stays below the frequency defined in Equation 19 and shown
in the bottom panels in Figure 19. This bound for the dynamical frequency
strongly depends on magnetic field strength of the model. We find that for the
weakly magnetic atmosphere case (WB) this limit is low, and the assumption
fails in the upper chromosphere and cold bubbles.In the strongly magnetic
atmosphere (SB) the assumptions only fails in the upper part of the
chromosphere.
In summary, for the weak field atmosphere we cannot neglect the time
derivative and dyadic product of the drift velocity, and the $\nabla
P\times{\bf B}$ terms (in the momentum and induction equation respectively) in
the cold bubbles and just below the transition region. In all other regions in
the weak field atmosphere, the assumptions underlying the generalized Ohm’s
law are fulfilled.
For the strong field atmosphere, the generalized Ohm’s law works well in most
of the chromosphere, except in the region just below the transition region
where the time derivative and dyadic product of the drift velocity and the
$\nabla P\times{\bf B}$ terms cannot be neglected.
## 4 Discussion and Conclusions
We have implemented the partial ionization effects in the Bifrost code in the
form of the Hall term and ambipolar diffusion. The code has been tested and
verified with different tests that are presented in this paper. The code
allows the simulation of the solar atmosphere, from the upper convection zone
to the lower corona, with a magnetoconvective photosphere, and a fully-dynamic
and self-maintained chromosphere and corona. We studied the different
diffusivities in two different models, one is weakly magnetic, and the other
is rather strongly magnetic. The magnetic field strength of the latter model
is similar to that found in the quiet sun, including the network.
In short, the Ohmic diffusion is roughly three orders of magnitude smaller
than the Hall term in the chromosphere, and the latter is three orders of
magnitude smaller than the artificial diffusion. Unlike Ohmic diffusion, the
Hall term depends on the magnetic field, as does ambipolar diffusion which is
strongly dependent on the magnetic field strength. As a result of this, the
ambipolar diffusivity is clearly different for the two models; in regions with
large ambipolar diffusivity we find it is of the same order as the artificial
diffusion in the chromosphere for the weakly magnetic model (WB), and more
than one order of magnitude larger than the artificial diffusivity for the
strongly magnetic model (SB). The fact that the artificial diffusivity is
actually smaller than the ambipolar diffusivity under many chromospheric
conditions has some very important consequences. It means that these
simulations are capable of providing a surprisingly realistic view of the
consequences of the ambipolar diffusion in the chromosphere and corona. This
has an impact beyond the chromosphere, since it directly affects discussions
on whether these self-consistent magneto-convective simulations provide a
realistic driver and boundary to the corona. These results will be described
in detail in a follow up paper.
Another important result is that both, the Hall term and ambipolar
diffusivity, vary by several orders of magnitude in the chromosphere as result
of the time varying dynamics and the strong variations in temperature,
electron, ion and neutral density, and magnetic field strength in this region.
This strong variation is not taken into account in any of the previous studies
which use either 1D semi-empirical VAL-C type models, or lack more
sophisticated approaches to the radiation, ionization and energy balance. The
largest values of the ambipolar diffusivity are located in the cold
chromospheric bubbles that have low temperatures due to strong adiabatic
expansion, and in the upper chromosphere because the neutral-ion collision
frequency is small. However, the ambipolar diffusion is strongly dependent on
the ionization degree, and as shown by Leenaarts et al. (2007), time dependent
hydrogen will change the ratio between neutrals and ions compared to LTE
conditions. The Bifrost code can treat the time-dependent ionization of
hydrogen and we plan to run new simulations taking into account both the
generalized Ohm’s law and time-dependent hydrogen ionization.
We have compared different methods to calculate the collision frequency
between neutrals and ions. Both the ion-neutral collision frequency and
ambipolar diffusivity differ considerably as a function of the method used to
calculate this collision frequency. Since ambipolar diffusion has a
significant impact on the thermodynamic evolution of these models, the
simulations rapidly diverge. When comparing each method we find the largest
differences are located in regions where the ambipolar diffusivity is large:
in the cold chromospheric bubbles and in the upper chromosphere in the
vicinity of the transition region. These differences bring a new uncertainty
to the results (Section 3.1.2), and highlight the need for a detailed
consideration of the relevant collisional processes in the chromosphere.
Finally, we investigated the different approximations underlying the
generalized Ohm’s law as described in detail by Pandey & Wardle (2008). In
both models, most of the simplifications are applicable with some exceptions.
In the upper-chromosphere the collision frequency is too low, as a
consequence, the velocity drift can be large. Therefore, we may need to define
the velocity drift and add an extra term in the momentum equation related to
the momentum drift between ions and neutrals. In the upper photosphere, and in
cold chromospheric bubbles the ambipolar term in the induction equation may
need to be calculated using the drift velocity. Moreover, the drift velocity
should be calculated using the time dependent form (as shown in Pandey &
Wardle, 2008). This is necessary because the ion density and the ion-neutrals
collision frequency drop in these cold areas as opposed to the electron
density and the electron-neutral collision frequency.
## 5 Acknowledgments
The 2D simulations have been run with the Njord and Stallo cluster from the
Notur project, and the Pleiades cluster through computing grants SMD-07-0434,
SMD-08-0743, SMD-09-1128, SMD-09-1336, SMD-10-1622, SMD-10-1869, SMD- 11-2312,
and SMD-11-2752 from the High End Computing (HEC) division of NASA. We
thankfully acknowledge the computer and supercomputer resources of the
Research Council of Norway through grant 170935/V30 and through grants of
computing time from the Programme for Supercomputing. This work has benefited
from discussions at the International Space Science Institute (ISSI) meeting
on “Heating of the magnetized chromosphere” from 21-24 February, 2012, where
many aspects of this paper were discussed with other colleagues. To analyze
the data we have used IDL. B.D.P. was supported through NASA grants
NNX08BA99G, NNX08AH45G and NNX11AN98G.
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## Appendix A Collision frequencies
In order to calculate the collision frequency between ions and neutral
particles we use three different approximations (following the approach by De
Pontieu et al. 2001): one described by Osterbrock (1961) (hereafter case A),
one described by von Steiger & Geiss (1989) (hereafter case B) and one by
Fontenla et al. (1993) (hereafter case C).
As a first approach (case A), we take the formulas from Osterbrock (1961) and
De Pontieu & Haerendel (1998), where the collision frequency between neutral
hydrogen and protons ($\nu_{Hp}$) are given by:
$\displaystyle\nu_{Hp}=5\,10^{-19}\sqrt{\frac{1}{2}}\sqrt{\frac{8kT}{\pi
m_{H}}}n_{p}$ (A1)
$m_{H}$ is the hydrogen atom mass, and $n_{p}$ is the proton number density.
Note that De Pontieu et al. (2001) had a typo with a factor of 2. The
collision frequency ($\nu_{Hm}$) of neutral hydrogen with an ionized metal is
defined as:
$\displaystyle\nu_{Hm}=8\,10^{-20}\sqrt{\frac{m_{m}}{m_{m}+1}}\sqrt{\frac{8kT}{\pi
m_{H}}}n_{m}$ (A2)
where $m_{m}$ and $n_{m}$ are the atomic mass number of metals ions and the
number density of metals ions of type $m$, respectively. The collisions
between neutral helium and ions is given by:
$\displaystyle\nu_{Hep}=4\,10^{-20}\sqrt{\frac{1}{5}}\sqrt{\frac{8kT}{\pi
m_{H}}}n_{p}$ (A3)
$\displaystyle\nu_{Hep}=4\,10^{-20}\sqrt{\frac{m_{m}}{m_{m}+1}}\sqrt{\frac{8kT}{\pi
m_{H}}}n_{m}$ (A4)
For the second approach (Case B), following De Pontieu et al. (2001), von
Steiger & Geiss (1989) describe the collision rate as follows:
$\displaystyle\nu_{Hp}=118\sqrt{\frac{T}{10^{4}}}\left(1-0.125\log{\frac{T}{10^{4}}}\right)^{2}\frac{n_{p}}{10^{16}}$
(A5)
$\displaystyle\nu_{Hm}=21.05\sqrt{\frac{A_{m}}{A_{m}+1}}Z_{m}\frac{n_{m}}{10^{16}}$
(A6)
For the helium-proton and helium-metal collision frequency we follow Geiss &
Buergi (1986):
$\displaystyle\nu_{Hep}=2.2\frac{n_{p}/10^{6}}{\sqrt{T/10^{4}}}Z_{m}$ (A7)
$\displaystyle\nu_{Hem}=5.84\sqrt{\frac{A_{m}}{A_{m}+1}}Z_{m}\frac{n_{m}}{10^{16}}$
(A8)
where $Z_{m}$ is the ionization weight and we considered that the ions have
only one ionization state, i.e., $Z_{m}=1$. Note that De Pontieu et al. (2001)
have a typo where the expression for $\nu_{Hep}$ is missing the square root
symbol for the temperature and the constant $2.2$ is also different.
Finally, we find the collision frequencies for the Case C in the appendix of
Fontenla et al. (1993).
Using these collision frequencies (Eq. A1-A8), the collision frequency of
neutral Hydrogen with all ions is given by:
$\displaystyle\nu_{Hi}$ $\displaystyle=$
$\displaystyle\nu_{Hp}+\nu_{HC}+\nu_{HN}+\nu_{HO}+\nu_{HNe}+\nu_{HNa}+\nu_{HMg}+\nu_{HAl}$
(A10)
$\displaystyle+\nu_{HSi}+\nu_{HS}+\nu_{HK}+\nu_{HCa}+\nu_{HCr}+\nu_{HFe}+\nu_{HNi}$
and similarly for the collision frequency of neutral Helium with all ions.
Finally, the average neutral-ion collision frequency is given by
$\displaystyle\nu_{ni}=\frac{\rho_{H}}{\rho_{n}}\nu_{Hi}+\frac{\rho_{He}}{\rho_{n}}\nu_{Hei}$
(A11)
Note that in the main text we often use $\nu_{in}$, which can be derived from
$\nu_{ni}$ using momentum conservation ($\rho_{j}\nu_{jk}=\rho_{k}\nu_{kj}$).
Figure 1: $B_{y}$ (top panel) and $B_{z}$ (bottom panel) as a function of $x$
are shown for the different 1D simulations with constant Hall term at time
$t=20$ s. The initial condition is the same for all simulations (shown with
black triangles). The runs have different constant $B_{x}$ values: $B_{x}=0$ G
(orange diamonds), $B_{x}=1121$ G (blue line), $B_{x}=2242$ G with the Hall
term (green line), and $B_{x}=2242$ G without the Hall term (red line). Note
that the orange diamonds, red line, and black triangles overlap. Figure 2:
From the ambipolar test, $B_{y}$ is shown as a function of $x$ at t=50 s. The
initial condition is shown in solid line. The dashed line shows $B_{y}$ at
$t=50$ s. The dash dotted line shows a function proportional to $x^{1/3}$
which is what would be expected from the analytical considerations. Figure 3:
Test of ambipolar diffusion on the energy balance. Energy is shown as a
function of $x$ at t=2.1 s. The energy from the model is shown with the black
diamonds and the energy extracted from Equation 29 is shown with the red line.
Note that the red line is overlapping with the black diamonds. Figure 4:
Neutral-ion collision frequency as a function of height for the quiet sun
model of Vernazza et al. (1981), using different formulas for $\nu_{ni}$: the
dotted line is case A, solid line is case B and dashed line is case C. Figure
5: 2D snapshots of the two initial 2.5D MHD models. The initial conditions
with weak (simulations labeled WA, WB, and WC) and strong magnetic field (SA,
SB, and SC) are shown respectively in the left and right panel. The color
scale shows the temperature in logarithmic scale and the magnetic field is
shown with white lines.
Figure 6: Comparison of the different diffusivity terms for the simulation WB
at $t=500~{}s$. $\eta_{ohm}$, $\eta_{art}$, $\eta_{hall}$, and $\eta_{amb}$
are shown from top to bottom and left to right respectively in logarithmic
scale. Note that more than 9 orders of magnitude are shown. Figure 7:
Comparison of the different diffusivities for the simulation SB at $t=500$ s.
The layout is the same as in Figure 6. Figure 8: Ratio between the Hall term
and Ohmic diffusion (left panels) and ambipolar and Ohmic diffusion (right
panels) for the SB simulation (top panels) and WB (bottom panels) at $t=500$
s. Figure 9: Comparison of the different collision frequencies for the WB
simulation at $t=500~{}s$. $\nu_{ei}$, $\nu_{en}$, $\nu_{in}$, and $\nu_{ni}$
are shown from top to bottom and left to right respectively in logarithmic
scale. Figure 10: Comparison of the different collision frequencies for the
SB simulation at $t=500$ s. The layout is the same as in Figure 9. Figure 11:
The ambipolar diffusion, electron density, absolute value of the magnetic
field, ratio between neutral and total density and ion density are shown from
top to bottom for the WB simulation (left panels) and SB (right panels) at
$t=500$ s. Figure 12: Minimum (dashed line), median (solid line) and maximum
(dashed line) of $\nu_{in}$ (top panels) and $\eta_{amb}$ (bottom panels) as
function of height are shown for the simulation labeled WB (black in left
panels), SB (black in right panels) and for the VAL-C atmosphere (red). The
VAL-C ambipolar diffusion is calculated taking into account the maximum,
minimum and median magnetic field of the 2D models as a function of height.
The minimum, median and maximum are calculated in horizontal planes for the
instant $t=500$ s. Figure 13: Joint probability distribution function (JPDF)
of the temperature against of the ion-neutral collision frequency (top) and
ambipolar diffusivity (bottom) for the simulations labeled WB (left) and WC
(right). JPDF is calculated integrated over 220 s above the photosphere. The
colorbar is in logarithmic scale. The median as a function of temperature for
the WB and WC cases are shown in solid, and dashed lines respectively.
Figure 14: The layout is the same as Figure 13. However, the simulations are
SB (left) and SC (right). Figure 15: Ratio of the ion-neutral collision
frequency (left panels) and ambipolar diffusivity (right panels) between case
A to case B (top panels) and case C to case B (bottom panels), for the
atmosphere with weak magnetic field strength. The white contours show where
these ratios are equal to one. Note that the color scheme is in a logarithmic
scale. We used the same atmospheric model for all three cases, i.e., before
the simulations diverge with time, but then calculated the collision frequency
and ambipolar diffusion using the different formulas of each case. Figure 16:
Same as Figure 15 for the atmosphere with strong magnetic field. Figure 17:
The drift momentum has to be smaller than the fast momentum (Equation 6). The
ratio between both terms, i.e., $\rho_{i}\rho_{n}{\bf u_{D}^{2}}$ and
$\rho^{2}(v_{a}^{2}+c_{s}^{2})$ is shown for the simulations labeled WB (top
panel) and SB (bottom panel) at $t=500$ s. The colorbar is in logarithmic
scale and it is the same for both panels. Figure 18: Following the Equation
16, the ratio between $\rho_{e}\nu_{en}$ and $\rho_{i}\nu_{in}$ is shown for
the simulations labeled WB (top panel) and SB (bottom panel) at $t=500$ s. The
colorbar is the same for both panels and it is in logarithmic scale. Figure
19: A study of the validity of the assumptions underlying the generalized
Ohm’s law. The dynamic frequency of the simulations ($\approx 0.5$ Hz) should
remain lower than the frequency limits shown in the different panels for
simulations WB (left panels) and SB (right panels) at $t=500~{}s$. The
frequencies are following the expressions Equation 18 (top panels), Equation 7
(middle panels) and Equation 19 (bottom panels). The colorbar for each
frequency is located at the top side and is in logarithmic scales. The white
color is where the temperature is above $3\,10^{4}$K. Table 1: Simulation
description
Name | Collision frequency | Min/Mean/Max $|B|$ [G]
---|---|---
WA | Case A | 0.003/0.25/3
WB | Case B | 0.003/0.25/3
WC | Case C | 0.003/0.25/3
SA | Case A | 0.1/90/920
SB | Case B | 0.1/90/920
SC | Case C | 0.1/90/920
Note. — The left column lists the names of the different 2D simulations,
middle column lists the method used to calculate the collision frequency
between ions and neutrals. The last column shows the minimum, mean and maximum
value of the unsigned magnetic field strength in the photosphere.
Table 2: Atomic info
name | H | He | C | N | O | Ne | Na | Mg
---|---|---|---|---|---|---|---|---
abund | 12. | 11. | 8.55 | 7.93 | 8.77 | 8.51 | 6.18 | 7.48
mass ion | 1.008 | 4.003 | 12.01 | 14.01 | 16. | 20.18 | 23. | 24.32
$X_{i}$ | 13.595 | 24.58 | 11.256 | 14.529 | 13.614 | 21.559 | 5.138 | 7.644
name | Al | Si | S | K | Ca | Cr | Fe | Ni
abund | 6.4 | 7.55 | 7.21 | 5.05 | 6.33 | 5.47 | 7.5 | 5.08
mass ion | 26.97 | 28.06 | 32.06 | 39.1 | 40.08 | 52.01 | 55.85 | 58.69
$X_{i}$ | 5.984 | 8.149 | 10.357 | 4.339 | 6.111 | 6.763 | 7.896 | 7.633
Note. — The atomic species, abundances ($\log$ of number of atoms per
$10^{12}$ Hydrogen atoms), mass ion (uma), and ionization fraction (eV) of the
16 most important atomic species in the solar atmosphere are listed from the
top to the bottom row. The various collision frequencies and the electron
density are calculated taking into account the atomic species in this table.
|
arxiv-papers
| 2012-04-26T17:42:20 |
2024-09-04T02:49:30.235864
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan Martinez-Sykora (1,2), Bart De Pontieu (1) and Viggo Hansteen\n (2,1)",
"submitter": "Juan Mart\\'inez-Sykora",
"url": "https://arxiv.org/abs/1204.5991"
}
|
1204.6064
|
# Eigenfunction expansions for a fundamental solution of Laplace’s equation on
${\mathbf{R}}^{3}$ in parabolic and elliptic cylinder coordinates
H S Cohl1 and H Volkmer2 1Information Technology Laboratory, National
Institute of Standards and Technology, Gaithersburg, MD, USA 2Department of
Mathematical Sciences, University of Wisconsin–Milwaukee, P. O. Box 413,
Milwaukee, WI 53201, U.S.A. hcohl@nist.gov
###### Abstract
A fundamental solution of Laplace’s equation in three dimensions is expanded
in harmonic functions that are separated in parabolic or elliptic cylinder
coordinates. There are two expansions in each case which reduce to expansions
of the Bessel functions $J_{0}(kr)$ or $K_{0}(kr)$,
$r^{2}=(x-x_{0})^{2}+(y-y_{0})^{2}$, in parabolic and elliptic cylinder
harmonics. Advantage is taken of the fact that $K_{0}(kr)$ is a fundamental
solution and $J_{0}(kr)$ is the Riemann function of partial differential
equations on the Euclidean plane.
###### pacs:
02.30.Em, 02.30.Gp, 02.30.Hq, 02.30.Jr, 02.30.Mv, 02.40.Dr
###### ams:
35A08, 35J05, 42C15, 33E10, 33C15
## 1 Introduction
A fundamental solution of Laplace’s equation
$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}+\frac{\partial^{2}U}{\partial z^{2}}=0$ (1)
is given by (apart from a multiplicative factor of $4\pi$)
$U({\bf x},{\mathbf{x}}_{0})=\frac{1}{\|{\bf
x}-{\mathbf{x}}_{0}\|},\quad\textup{where ${\bf
x}=(x,y,z)\neq{\mathbf{x}}_{0}=(x_{0},y_{0},z_{0})$},$ (2)
and $\|{\bf x}-{\bf x}_{0}\|$ denotes the Euclidean distance between ${\bf x}$
and ${\bf x}_{0}$. In many applications it is required to expand a fundamental
solution in the form of a series or an integral, in terms of solutions of (1)
that are separated in suitable curvilinear coordinates. Examples of such
applications include electrostatics, magnetostatics, quantum direct and
exchange Coulomb interactions, Newtonian gravity, potential flow, and steady
state heat transfer. Morse & Feshbach (1953) [16] (see also Hobson (1955)
[10]; MacRobert (1947) [12]; Heine (1881) [9]) provide a list of such
expansions for various coordinate systems but the formulas for several
coordinate systems are missing. It is the goal of this paper to provide these
expansions for parabolic and elliptic cylinder coordinates. Although these
expansions are partially known from Buchholz (1953) [2] and Lebedev (1972)
[11] for parabolic cylinder coordinates, and from Meixner & Schäfke (1954)
[14] for elliptic cylinder coordinates, we found it desirable to investigate
these expansions in a systematic fashion and provide direct proofs for them
based on eigenfunction expansions.
There will be two expansions for both of these coordinate systems. The first
expansion for a cylindrical coordinate system on ${\mathbf{R}}^{3}$ starts
from the known formula in terms of the integral of Lipschitz (see Watson
(1944) [22, section 13.2]; Cohl et al. (2000) [5, (8)])
$\frac{1}{\|\mathbf{x}-\mathbf{x_{0}}\|}=\int_{0}^{\infty}J_{0}(k\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}})e^{-k|z-z_{0}|}\,dk.$
(3)
The Bessel function $J_{\nu}$ can be defined by (see for instance (10.2.2) in
Olver et al. (2010) [17])
$J_{\nu}(z):=\biggl{(}\frac{z}{2}\biggr{)}^{\nu}\sum_{n=0}^{\infty}\frac{(-z^{2}/4)^{n}}{n!\Gamma(\nu+n+1)},$
(4)
Note that for $W_{k}:{\mathbf{R}}^{2}\times{\mathbf{R}}^{2}\to{\mathbf{R}}$
$W_{k}(x,y,x_{0},y_{0}):=J_{0}(kr),$ (5)
where $r^{2}:=(x-x_{0})^{2}+(y-y_{0})^{2},$ solves the partial differential
equation
$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}+k^{2}U=0.$ (6)
Therefore, in a cylindrical coordinate system on ${\mathbf{R}}^{3}$ involving
the Cartesian coordinate $z$, the first expansion (3) reduces to expanding
$W_{k}$ from (5) in terms of solutions of (6) that are separated in a
curvilinear coordinate system on the plane.
The second expansion for a cylindrical coordinate system on ${\mathbf{R}}^{3}$
is based on the known formula given in terms of the Lipschitz-Hankel integral
(see Watson (1944) [22, section 13.21]; Cohl et al. (2000) [5, (9)])
$\frac{1}{\|\mathbf{x}-\mathbf{x_{0}}\|}=\frac{2}{\pi}\int_{0}^{\infty}K_{0}(k\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}})\cos
k(z-z_{0})\,dk,$ (7)
where $K_{\nu}:(0,\infty)\to{\mathbf{R}}$ (cf. (10.32.9) in Olver et al.
(2010) [17]), the modified Bessel function of the second kind (Macdonald’s
function), of order $\nu\in{\mathbf{R}}$, is defined by
$K_{\nu}(z):=\int_{0}^{\infty}e^{-z\cosh t}\cosh(\nu t)\,dt.$ (8)
Now $V_{k}:{\mathbf{R}}^{2}\times{\mathbf{R}}^{2}\setminus\\{({\bf x},{\bf
x}):{\bf x}\in{\mathbf{R}}^{2}\\}\to(0,\infty),$ defined by
$V_{k}(x,y,x_{0},y_{0}):=K_{0}(kr)$ (9)
solves the partial differential equation
$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}-k^{2}U=0.$ (10)
In a cylindrical coordinate system on ${\mathbf{R}}^{3}$ involving the
Cartesian coordinate $z$, the second expansion (7) reduces to expanding
$V_{k}$ in terms of solutions of (10) that are separated in curvilinear
coordinates on the plane.
The paper is organized as follows. In section 2 we derive the desired
expansion of $K_{0}(kr)$ in parabolic cylinder coordinates. This expansion is
given in terms of series over Hermite functions. In section 3 we obtain an
integral representation for $J_{0}(kr)$ in terms of separated solutions of
(10) in terms of (modified) parabolic cylinder functions. We show how these
results are based on a general expansion theorem in terms of solutions of the
differential equation
$-u^{\prime\prime}-\frac{1}{4}\xi^{2}u=\lambda u.$ (11)
In sections 4 and 5 we derive the fundamental solution expansions in elliptic
cylinder coordinates for $J_{0}(kr)$ and $K_{0}(kr)$, respectively.
Throughout this paper we rely on the following definitions. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,3,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, and the set
${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\}.$ The set ${\mathbf{R}}$ represents
the real numbers and the set ${\mathbf{C}}$ represents the complex numbers.
## 2 Expansion of $K_{0}(kr)$ for parabolic cylinder coordinates
Parabolic coordinates on the plane $(\xi,\eta)$ (see for instance Chapter 10
in Lebedev (1972) [11]) are connected to Cartesian coordinates $(x,y)$ by
$x=\frac{1}{2}(\xi^{2}-\eta^{2}),\quad y=\xi\eta,$ (12)
where $\xi\in{\mathbf{R}}$ and $\eta\in[0,\infty)$. To simplify notation we
will first set $k=1$ in (9). Then $V_{1}$ satisfies
$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}-U=0\quad\textup{if $(x,y)\neq(x_{0},y_{0})$.}$ (13)
Let $(\xi,\eta)$, $(\xi_{0},\eta_{0})$ be parabolic coordinates on
${\mathbf{R}}^{2}$ for $(x,y)$ and $(x_{0},y_{0})$, respectively. Then $V_{1}$
transforms to
$v(\xi,\eta,\xi_{0},\eta_{0}):=K_{0}(r(\xi,\eta,\xi_{0},\eta_{0})),\quad(\xi,\eta)\neq\pm(\xi_{0},\eta_{0})$
and $r(\xi,\eta,\xi_{0},\eta_{0})$ is defined by
$r^{2}=\frac{1}{4}\left[(\xi+\xi_{0})^{2}+(\eta+\eta_{0})^{2}\right]\left[(\xi-\xi_{0})^{2}+(\eta-\eta_{0})^{2}\right],\quad
r>0.$ (14)
Here and in the following we allow all
$\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$ such that
$(\xi,\eta)\neq\pm(\xi_{0},\eta_{0})$. From (13) or by direct computation we
obtain that $v$ solves the equation
$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}-(\xi^{2}+\eta^{2})u=0.$
(15)
Separating variables $u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta)$ in (15), we obtain
the ordinary differential equations
$\displaystyle u_{1}^{\prime\prime}+(2n+1-\xi^{2})u_{1}$ $\displaystyle=$
$\displaystyle 0,$ (16) $\displaystyle
u_{2}^{\prime\prime}-(2n+1+\eta^{2})u_{2}$ $\displaystyle=$ $\displaystyle 0,$
(17)
where we will use only $n\in{\mathbf{N}}_{0}$. Equations (16), (17) have the
general solutions
$\displaystyle u_{1}(\xi)$ $\displaystyle=$ $\displaystyle
c_{1}e^{-\xi^{2}/2}H_{n}(\xi)+c_{2}e^{\xi^{2}/2}H_{-n-1}(i\xi),$
$\displaystyle u_{2}(\eta)$ $\displaystyle=$ $\displaystyle
c_{3}e^{\eta^{2}/2}H_{n}(i\eta)+c_{4}e^{-\eta^{2}/2}H_{-n-1}(\eta),$
where $H_{\nu}:{\mathbf{C}}\to{\mathbf{C}}$ is the Hermite function which can
be defined in terms of Kummer’s function of the first kind $M$ as (cf.
(10.2.8) in Lebedev (1972) [11])
$H_{\nu}(z):=\frac{2^{\nu}\sqrt{\pi}}{\Gamma\left(\frac{1-\nu}{2}\right)}M\left(-\frac{\nu}{2},\frac{1}{2},z^{2}\right)-\frac{2^{\nu+1}\sqrt{\pi}}{\Gamma\left(-\frac{\nu}{2}\right)}zM\left(\frac{1-\nu}{2},\frac{3}{2},z^{2}\right),$
and
$M(a,b,z):=\sum_{n=0}^{\infty}\frac{(a)_{n}z^{n}}{(b)_{n}n!}$ (18)
(see for instance (13.2.2) in Olver et al. (2010) [17]). Note that the Kummer
function of the first kind is entire in $z$ and $a,$ and is a meromorphic
function of $b.$ The Hermite function is an entire function of both $z$ and
$\nu$. If $\nu=n\in{\mathbf{N}}_{0}$ then $H_{\nu}(z)$ reduces to the Hermite
polynomial of degree $n$.
We will expand $v$ as a function of $\xi$ in an orthogonal series of functions
$e^{-\xi^{2}/2}H_{n}(\xi)$, $n\in{\mathbf{N}}_{0}$, so that the coefficients
$f_{n}(\eta,\xi_{0},\eta_{0}):=\int_{-\infty}^{\infty}v(\xi,\eta,\xi_{0},\eta_{0})e^{-\xi^{2}/2}H_{n}(\xi)\,d\xi,$
are to be evaluated. We do this based on the observation that the function
$(\xi,\eta)\mapsto v(\xi,\eta,\xi_{0},\eta_{0})$ is a fundamental solution of
equation (15). It has logarithmic singularities at the points
$\pm(\xi_{0},\eta_{0})$. Arguing as in Volkmer (1984) [21, Theorem 1.11], we
obtain the following integral representation for solutions of (15).
###### Theorem 2.1.
Let $u\in C^{2}({\mathbf{R}}^{2})$ be a solution of (15). Let
$(\xi_{0},\eta_{0})\in{\mathbf{R}}^{2}$, and let $C$ be a a closed rectifiable
curve on ${\mathbf{R}}^{2}$ which does not pass through
$\pm(\xi_{0},\eta_{0})$, and let $n^{\pm}$ be the winding number of $C$ with
respect to $\pm(\xi_{0},\eta_{0})$. Then we have
$2\pi\left[n^{+}u(\xi_{0},\eta_{0})+n^{-}u(-\xi_{0},-\eta_{0})\right]=\int_{C}(u\partial_{2}v-v\partial_{2}u)\,d\xi+(v\partial_{1}u-u\partial_{1}v)\,d\eta,$
where $\partial_{1},\partial_{2}$ denote partial derivatives with respect to
$\xi$, $\eta$, respectively.
We apply Theorem 2.1 to the solution
$u(\xi,\eta)=e^{-\xi^{2}/2}H_{n}(\xi)e^{\eta^{2}/2}H_{n}(i\eta),$
of (15), and for $C$ we take the positively oriented boundary of the rectangle
$|\xi|\leq\xi_{1}$, $|\eta|\leq\eta_{1}$, where $|\xi_{0}|<\xi_{1}$,
$|\eta_{0}|<\eta_{1}$, so $n^{+}=n^{-}=1$. Then let $\xi_{1}\to\infty$ and
note that the integrals over the vertical sides converge to $0$. The integrals
over the horizontal sides of the rectangle give the same contribution because
the integrand changes sign when $(\xi,\eta)$ is replaced by $(-\xi,-\eta)$.
Therefore, we obtain, for $\eta=\eta_{1}>|\eta_{0}|$,
$2\pi
u(\xi_{0},\eta_{0})=\int_{-\infty}^{\infty}(v(\xi,\eta,\xi_{0},\eta_{0})\partial_{2}u(\xi,\eta)-u(\xi,\eta)\partial_{2}v(\xi,\eta,\xi_{0},\eta_{0}))d\xi.$
(19)
Set $f(\eta)=f_{n}(\eta,\xi_{0},\eta_{0})$ and
$g(\eta)=e^{\eta^{2}/2}H_{n}(i\eta)$. Then (19) can be written as
$2\pi u(\xi_{0},\eta_{0})=g^{\prime}(\eta)f(\eta)-g(\eta)f^{\prime}(\eta).$
(20)
By differentiating both sides of (20) with respect to $\eta$ we see that $f$
satisfies (17). Since $f(\eta)$ goes to $0$ as $\eta\to\infty$, it follows
that
$f(\eta)=ce^{-\eta^{2}/2}H_{-n-1}(\eta),$
where $c$ is a constant. Going back to (20), we find that
$2\pi u(\xi_{0},\eta_{0})=cW,$
where $W=i^{n}$ is the (constant) Wronskian of $e^{-\eta^{2}/2}H_{-n-1}(\eta)$
and $g(\eta)$. Therefore, if $\eta>|\eta_{0}|$, we obtain
$f_{n}(\eta,\xi_{0},\eta_{0})=2\pi(-i)^{n}e^{-\xi_{0}^{2}/2}H_{n}(\xi_{0})e^{\eta_{0}^{2}/2}H_{n}(i\eta_{0})e^{-\eta^{2}/2}H_{-n-1}(\eta).$
Using the well-known series expansion in terms of Hermite functions (Lebedev
(1972) [11, Theorem 2, page 71]), we obtain the following result.
###### Theorem 2.2.
For $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$ with $\eta>|\eta_{0}|$,
$K_{0}(r(\xi,\eta,\xi_{0},\eta_{0}))=\displaystyle{\sqrt{\pi}}e^{(\eta_{0}^{2}-\xi_{0}^{2}-\eta^{2}-\xi^{2})/2}\sum_{n=0}^{\infty}\frac{(-i)^{n}}{2^{n-1}n!}H_{n}(\xi)H_{-n-1}(\eta)H_{n}(\xi_{0})H_{n}(i\eta_{0}),$
where $r$ is defined by (14).
The special case $\xi_{0}=\eta_{0}=0$ of Theorem 2.2 can be found in [11,
Problem 7, page 298]. If we multiply each $\xi,\eta,\xi_{0},\eta_{0}$ by
$\sqrt{k}$ then we obtain an expansion for $K_{0}(kr)$. Inserting this
expansion into (7) yields our final result.
###### Theorem 2.3.
Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with parabolic
cylinder coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$,
respectively. If $\eta_{\lessgtr}:={\min\atop\max}\\{\eta,\eta_{0}\\}$ then
$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-k/2(\xi^{2}+\eta^{2}+\xi_{0}^{2}-\eta_{0}^{2})}\cos
k(z-z_{0})$ $\displaystyle\hskip
56.9055pt\times\sum_{n=0}^{\infty}\frac{(-i)^{n}}{2^{n-1}n!}H_{n}(\sqrt{k}\xi)H_{-n-1}(\sqrt{k}\eta_{>})H_{n}(\sqrt{k}\xi_{0})H_{n}(i\sqrt{k}\eta_{<})\,dk,$
where ${\bf x}\neq{\bf x}_{0}$.
If we reverse the order of the infinite series and the definite integral in
the above expression we obtain
$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-i)^{n}}{2^{n-1}n!}\int_{0}^{\infty}e^{-k/2(\xi^{2}+\eta^{2}+\xi_{0}^{2}-\eta_{0}^{2})}$
$\displaystyle\hskip 56.9055pt\times
H_{n}(\sqrt{k}\xi)H_{-n-1}(\sqrt{k}\eta)H_{n}(\sqrt{k}\xi_{0})H_{n}(i\sqrt{k}\eta_{0})\cos
k(z-z_{0})\,dk.$
It would be interesting to know the value of the above definite integral.
## 3 Expansion of $J_{0}(kr)$ for parabolic cylinder coordinates
Transforming equation (6) to parabolic coordinates (12) we obtain
$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}+k^{2}(\xi^{2}+\eta^{2})u=0.$
In order to simplify notation we will (temporarily) set $k=\frac{1}{2}$ and
$\zeta=i\eta$ with $\zeta$ real. Thus we consider
$\frac{\partial^{2}u}{\partial\xi^{2}}-\frac{\partial^{2}u}{\partial\zeta^{2}}+\frac{1}{4}(\xi^{2}-\zeta^{2})u=0,\quad\xi,\zeta\in{\mathbf{R}}.$
(21)
If $u_{1}(\xi)$ and $u_{2}(\zeta)$ are solutions of the ordinary differential
equation (11) for some $\lambda$ then $u(\xi,\zeta)=u_{1}(\xi)u_{2}(\zeta)$
solves (21).
The function $W_{1/2}$ from (5) transformed to $(\xi,\zeta)$ becomes
$w(\xi,\zeta,\xi_{0},\zeta_{0})=J_{0}\left(\frac{1}{2}\tilde{r}(\xi,\zeta,\xi_{0},\zeta_{0})\right),$
(22)
where $\tilde{r}^{2}$ is a symmetric polynomial defined by
$\displaystyle
4\tilde{r}^{2}:=\left[(\xi-\xi_{0})^{2}-(\zeta-\zeta_{0})^{2}\right]\left[(\xi+\xi_{0})^{2}-(\zeta+\zeta_{0})^{2}\right]$
$\displaystyle=8\xi\xi_{0}\zeta\zeta_{0}+\xi^{4}+\xi_{0}^{4}+\zeta^{4}+\zeta_{0}^{4}-2\xi_{0}^{2}\zeta^{2}-2\xi_{0}^{2}\zeta_{0}^{2}-2\zeta^{2}\zeta_{0}^{2}-2\xi^{2}\xi_{0}^{2}-2\xi^{2}\zeta^{2}-2\xi^{2}\zeta_{0}^{2},$
and $J_{0}$ is the order zero Bessel function of the first kind (see (4)). For
fixed $\zeta,\xi_{0},\zeta_{0}\in{\mathbf{R}}$ consider the function
$f:{\mathbf{R}}\to{\mathbf{R}}$ defined by
$f(\xi):=w(\xi,\zeta,\xi_{0},\zeta_{0}).$ (23)
We wish to expand this function in terms of (modified) parabolic cylinder
harmonics according to a general expansion theorem that is derived in the
following subsection.
### 3.1 Spectral theory of (modified) parabolic cylinder harmonics – A
singular Sturm-Liouville problem
We discuss the Sturm-Liouville problem
$-u^{\prime\prime}-{\textstyle\frac{1}{4}}x^{2}u=\lambda
u,\quad-\infty<x<\infty,$ (24)
involving the spectral parameter $\lambda$ subject to
$u\in L^{2}(-\infty,\infty).$
By replacing $\frac{1}{4}x^{2}$ by $-\frac{1}{4}x^{2}$, we obtain the equation
describing the harmonic oscillator whose eigenfunctions (Hermite functions)
and the corresponding spectral theory is well-known. The spectral problem
associated with (24) is far less known.
A discussion of differential equation (24) and its solutions can be found in
[15] and in section 8.2 of (24) Erdélyi et al. (1982) [7] (see also Magnus
(1941) [13]; Wells & Spence (1945) [23]; Cherry (1948) [3]; Darwin (1949)
[6]).
We will follow Chapter 9 in Coddington & Levinson (1955) [4]. First note that,
by [4, Corollary 2, page 231], equation (24) is in the limit-point case at
$x=\pm\infty$. Therefore, one can apply section 5 of Chapter 9 in [4].
For $\lambda,x\in{\mathbf{C}}$ we define the functions $u_{1}(\lambda,x)$,
$u_{2}(\lambda,x)$ as the solutions of (24) uniquely determined by the initial
conditions
$u_{1}(\lambda,0)=u_{2}^{\prime}(\lambda,0)=1,\quad
u_{1}^{\prime}(\lambda,0)=u_{2}(\lambda,0)=0.$
These functions may be expressed in terms of Kummer’s function of the first
kind (18)
$\displaystyle u_{1}(\lambda,x)$ $\displaystyle=$ $\displaystyle
e^{-\frac{i}{4}x^{2}}M\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda,{\textstyle\frac{1}{2}},{\textstyle\frac{i}{2}}x^{2}\right),$
(25) $\displaystyle u_{2}(\lambda,x)$ $\displaystyle=$ $\displaystyle
e^{-{\textstyle\frac{i}{4}}x^{2}}xM\left({\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda,{\textstyle\frac{3}{2}},{\textstyle\frac{i}{2}}x^{2}\right).$
(26)
For $x>0$, the function
$\displaystyle u_{3}(\lambda,x)$ $\displaystyle=$ $\displaystyle
e^{-{\textstyle\frac{i}{4}}x^{2}}U\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda,{\textstyle\frac{1}{2}},{\textstyle\frac{i}{2}}x^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}}{\Gamma\bigl{(}{\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda\bigr{)}}u_{1}(\lambda,x)-(1+i)\frac{\sqrt{\pi}}{\Gamma\bigl{(}{\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda\bigr{)}}u_{2}(\lambda,x)$
is another solution of (24). Here the Kummer function of the second kind
$U:{\mathbf{C}}\times{\mathbf{C}}\times({\mathbf{C}}\setminus(-\infty,0])\to{\mathbf{C}}$
can be defined as (see Olver et al. (2010) [17, (13.2.42)])
$U(a,b,z):=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z).$
Except when $z=0$, each branch of $U$ is entire in $a$ and $b.$ We assume that
$U(a,b,z)$ has its principal value. The asymptotic behavior of the Kummer
function of the second kind [17, (13.2.6)] shows that $u_{3}(\lambda,\cdot)\in
L^{2}(0,\infty)$ provided that $\Im\lambda<0$. Since (24) is in the limit-
point case at $+\infty$, $u_{3}$ is the only solution with this property
except for a constant factor. The Titchmarsh-Weyl functions
$m_{\pm\infty}(\lambda)$, $\Im\lambda\neq 0$, are defined by the property that
$u_{1}(\lambda,x)+m_{\pm\infty}(\lambda)u_{2}(\lambda,x)$ is square-integrable
at $x=\pm\infty$. Therefore,
$m_{\infty}(\lambda)=\left\\{\begin{array}[]{ll}\displaystyle\displaystyle{-(1+i)\frac{\Gamma(\frac{3}{4}+\frac{i}{2}\lambda)}{\Gamma(\frac{1}{4}+\frac{i}{2}\lambda)}}&\quad\mathrm{if}\
\Im\lambda<0,\\\\[5.0pt]
\displaystyle\displaystyle{-(1-i)\frac{\Gamma(\frac{3}{4}-\frac{i}{2}\lambda)}{\Gamma(\frac{1}{4}-\frac{i}{2}\lambda)}}&\quad\mathrm{if}\
\Im\lambda>0.\end{array}\right.$
By symmetry, we have
$m_{-\infty}(\lambda)=-m_{\infty}(\lambda).$
Using the notation of [4, Theorem 5.1, page 251], on obtains
$M_{11}(\lambda)=\frac{-1}{2m_{\infty}(\lambda)},\quad M_{12}=M_{21}=0,\quad
M_{22}(\lambda)=\frac{m_{\infty}(\lambda)}{2}.$
Now using [4, page 250, last line], we find that, for
$\lambda\in{\mathbf{R}}$,
$\rho^{\prime}_{1}(\lambda):=\rho^{\prime}_{11}(\lambda)=\frac{1}{\pi}\lim_{\epsilon\to
0+}\Im\left(M_{11}(\lambda+i\epsilon)\right)=\frac{e^{\frac{\pi}{2}\lambda}}{4\sqrt{2}\pi^{2}}\left|\Gamma({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda)\right|^{2},$
since [17, (5.4.5)]
$\Gamma({\textstyle\frac{1}{4}}+iy)\Gamma({\textstyle\frac{3}{4}}-iy)=\frac{\pi\sqrt{2}}{\cosh(\pi
y)+i\sinh(\pi y)}.$
Moreover, $\rho_{12}(\lambda)=\rho_{21}(\lambda)=0$ and
$\rho^{\prime}_{2}(\lambda):=\rho^{\prime}_{22}(\lambda)=\frac{1}{\pi}\lim_{\epsilon\to
0+}\Im\left(M_{22}(\lambda+i\epsilon)\right)=\frac{e^{\frac{\pi}{2}\lambda}}{2\sqrt{2}\pi^{2}}\left|\Gamma({\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda)\right|^{2}.$
Since the $\rho$-functions are real-analytic functions (with no jumps), we see
that the spectrum of (24) is the whole real line ${\mathbf{R}}$ and there are
no eigenvalues. The latter also follows from the known asymptotic behavior of
the solutions of (24) (see [17, Chapter 12]).
Using a variant of Stirling’s formula (see (5.11.9) in Olver et al. (2010)
[17])
$|\Gamma(x+iy)|\sim\sqrt{2\pi}|y|^{x-1/2}e^{-\pi|y|/2}\quad\textup{as
$x,y\in{\mathbf{R}}$, $|y|\to\infty$},$ (28)
we can determine the asymptotic behavior of $\rho^{\prime}_{j}$, namely
$\displaystyle\rho_{1}^{\prime}(\lambda)$ $\displaystyle\sim$
$\displaystyle\frac{1}{2\pi}|\lambda|^{-1/2}e^{\pi(\lambda-|\lambda|)/2}\quad\textup{as
$|\lambda|\to\infty$,}$ (29) $\displaystyle\rho_{2}^{\prime}(\lambda)$
$\displaystyle\sim$
$\displaystyle\frac{1}{2\pi}|\lambda|^{1/2}e^{\pi(\lambda-|\lambda|)/2}\quad\textup{as
$|\lambda|\to\infty$.}$ (30)
Applying [4, Theorem 5.2, page 251] to the analysis above, we obtain the
following result on the spectral resolution associated with equation (24).
###### Theorem 3.1.
For a given function $f\in L^{2}({\mathbf{R}}),$ form the functions
$g_{j}(\lambda)=\int_{-\infty}^{\infty}u_{j}(\lambda,x)f(x)\,dx,\quad j=1,2,\
\lambda\in{\mathbf{R}}.$ (31)
Then
$g_{j}\in L^{2}({\mathbf{R}},\rho_{j}),\quad j=1,2,$
or, equivalently,
$\int_{-\infty}^{\infty}|g_{j}(\lambda)|^{2}\rho^{\prime}_{j}(\lambda)\,d\lambda<\infty,\quad
j=1,2.$
The function $f$ can be represented in the form
$f(x)=\sum_{j=1}^{2}\int_{-\infty}^{\infty}u_{j}(\lambda,x)g_{j}(\lambda)\rho_{j}^{\prime}(\lambda)\,d\lambda.$
(32)
Moreover, we have Parseval’s equation
$\int_{-\infty}^{\infty}|f(x)|^{2}\,dx=\sum_{j=1}^{2}\int_{-\infty}^{\infty}|g_{j}(\lambda)|^{2}\rho^{\prime}_{j}(\lambda)\,d\lambda.$
Equations (31), (32) establish a one-to-one correspondence between $f$ and
$(g_{1},g_{2})$. The integrals appearing in (31), (32) have to be interpreted
in the $L^{2}$-sense. For instance, (31) means that
$\int_{-n}^{n}u_{j}(\lambda,x)f(x)dx$
converges to $g_{j}(\lambda)$ in $L^{2}({\mathbf{R}},\rho_{j})$ as
$n\to\infty$. Of course, if
$\int_{-\infty}^{\infty}\left|u_{j}(\lambda,x)f(x)\right|dx<\infty\quad\textup{for
every $\lambda\in{\mathbf{R}}$,}$
then (31) is also true pointwise.
### 3.2 Applying the spectral theory to the expansion of $J_{0}(kr)$ in
parabolic cylinder coordinates
Since $f(\xi)=O(|\xi|^{-1})$ in (23) as $|\xi|\to\infty$, we have that $f\in
L^{2}({\mathbf{R}})$. Therefore, we can expand $f$ using Theorem 3.1,
according to (32). For $\lambda\in{\mathbf{R}}$, we form the integrals
$g_{j}(\lambda,\zeta,\xi_{0},\zeta_{0})=\int_{-\infty}^{\infty}w(\xi,\zeta,\xi_{0},\zeta_{0})u_{j}(\lambda,\xi)\,d\xi,\quad
j=1,2.$ (33)
These are absolutely convergent integrals because $f(\xi)=O(|\xi|^{-1})$ and
$u_{j}(\lambda,\xi)=O(|\xi|^{-1/2})$ as $|\xi|\to\infty$.
Since $w(\xi,\zeta,\xi_{0},\zeta_{0})$ from (22) solves (21) for fixed
$(\xi_{0},\zeta_{0}),$ and $u_{j}(\lambda,\xi)$ from (25), (26) solves (24),
it follows from differentiation under the integral sign followed by
integration by parts (see for instance Schäfke (1963) [18, Satz 8, page 26])
that $g_{j}$ solves (24) as a function of $\zeta$. Since the function $w$ is
symmetric in its four variables, it is also true that $g_{j}$ solves (24) as
function of $\xi_{0}$ for fixed $\zeta_{0},\zeta$ and as function of
$\zeta_{0}$ for fixed $\xi_{0},\zeta$. From these properties of $g_{j}$ it
follows easily that there are functions $c_{jk\ell
m}:{\mathbf{R}}\to{\mathbf{R}}$ with $j,k,\ell,m=1,2$, depending on $\lambda$
but not on $\zeta,\xi_{0},\zeta_{0}$ such that
$g_{j}(\lambda,\zeta,\xi_{0},\zeta_{0})=\sum_{k,\ell,m=1}^{2}c_{jk\ell
m}(\lambda)u_{k}(\lambda,\zeta)u_{\ell}(\lambda,\xi_{0})u_{m}(\lambda,\zeta_{0}).$
(34)
This formula holds for all $\lambda,\zeta,\xi_{0},\zeta_{0}\in{\mathbf{R}}$.
Substituting $\zeta=\xi_{0}=\zeta_{0}=0$ in (33), (34), we obtain
$c_{j111}(\lambda)=\int_{-\infty}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{j}(\lambda,\xi)\,d\xi.$
If $j=2,$ we integrate over an odd function, so $c_{2111}(\lambda)=0$. By
differentiating (34) with respect to $\zeta$ and/or $\xi_{0}$ and/or
$\zeta_{0}$ and then substituting $\zeta=\xi_{0}=\zeta_{0}=0$ we find (after
some calculations)
$c_{jk\ell m}(\lambda)=\left\\{\begin{array}[]{ll}\displaystyle
c_{1}(\lambda)&\quad\mathrm{if}\ j=k=\ell=m=1,\\\\[2.0pt] \displaystyle
c_{2}(\lambda)&\quad\mathrm{if}\ j=k=\ell=m=2,\\\\[2.0pt] \displaystyle
0&\quad\textup{otherwise}.\end{array}\right.$
Here $c_{j}:{\mathbf{R}}\to{\mathbf{R}}$ for $j=1,2$ is given by (see A)
$\displaystyle
c_{1}(\lambda):=\int_{-\infty}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{1}(\lambda,\xi)\,d\xi=\frac{2\sqrt{2}\pi
e^{-\pi\lambda/2}}{\cosh(\pi\lambda)|\Gamma\left(\frac{3}{4}+\frac{i\lambda}{2}\right)|^{2}},$
(35) $\displaystyle
c_{2}(\lambda):=-\int_{-\infty}^{\infty}\xi^{-1}J_{1}({\textstyle\frac{1}{4}}\xi^{2})u_{2}(\lambda,\xi)\,d\xi=\frac{-4\sqrt{2}\pi
e^{-\pi\lambda/2}}{\cosh(\pi\lambda)|\Gamma\left(\frac{1}{4}+\frac{i\lambda}{2}\right)|^{2}}.$
(36)
According to (32),
$\displaystyle w(\xi,\zeta,\xi_{0},\zeta_{0})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}\int_{-\infty}^{\infty}\\!\\!\\!c_{j}(\lambda)\rho^{\prime}_{j}(\lambda)u_{j}(\lambda,\xi)u_{j}(\lambda,\zeta)u_{j}(\lambda,\xi_{0})u_{j}(\lambda,\zeta_{0})d\lambda.$
(37)
Note that
$\displaystyle c_{1}(\lambda)\rho_{1}^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\cosh(\pi\lambda)}\left|\frac{\Gamma\left(\frac{1}{4}+i\frac{\lambda}{2}\right)}{\Gamma\left(\frac{3}{4}+i\frac{\lambda}{2}\right)}\right|^{2}=\frac{1}{4\pi^{3}}\left|\Gamma\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i\lambda}{2}}\right)\right|^{4},$
(38) $\displaystyle c_{2}(\lambda)\rho_{2}^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi\cosh(\pi\lambda)}\left|\frac{\Gamma\left(\frac{3}{4}+i\frac{\lambda}{2}\right)}{\Gamma\left(\frac{1}{4}+i\frac{\lambda}{2}\right)}\right|^{2}=\frac{-1}{\pi^{3}}\left|\Gamma\left({\textstyle\frac{3}{4}}+{\textstyle\frac{i\lambda}{2}}\right)\right|^{4},$
(39)
where we used [17, (5.4.4), (5.5.5)]. It follows from (28) that
$\displaystyle c_{1}(\lambda)\rho_{1}^{\prime}(\lambda)$ $\displaystyle\sim$
$\displaystyle\frac{1}{\pi|\lambda|\cosh(\pi\lambda)}\quad\textup{as
$|\lambda|\to\infty$,}$ (40) $\displaystyle
c_{2}(\lambda)\rho_{2}^{\prime}(\lambda)$ $\displaystyle\sim$
$\displaystyle-\frac{|\lambda|}{\pi\cosh(\pi\lambda)}\quad\textup{as
$|\lambda|\to\infty$}.$ (41)
It is known (see Atkinson (1964), [1, (8.2.5)]) that, for fixed
$x\in{\mathbf{C}}$, there is a constant $C$ such that
$u_{j}(\lambda,x)=O(e^{C|\lambda|^{1/2}})$. It follows from (40), (41), that
the integrands in (37) decay exponentially and therefore the corresponding
integrals are absolutely convergent. By the identity theorem for analytic
functions we see that equation (37) is true for all
$\xi,\zeta,\xi_{0},\zeta_{0}\in{\mathbf{C}}$.
After setting $\zeta=i\eta$ and $\zeta_{0}=i\eta_{0}$ in (37), one obtains the
following result.
###### Theorem 3.2.
Let $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$. Then
$\displaystyle J_{0}\left(\frac{1}{2}r(\xi,\eta,\xi_{0},\eta_{0})\right)$
$\displaystyle\hskip
19.91684pt=\sum_{j=1}^{2}\int_{-\infty}^{\infty}c_{j}(\lambda)\rho^{\prime}_{j}(\lambda)u_{j}(\lambda,\xi)u_{j}(\lambda,i\eta)u_{j}(\lambda,\xi_{0})u_{j}(\lambda,i\eta_{0})\,d\lambda,$
where $r$ is given by (14) and $c_{j}(\lambda)\rho_{j}^{\prime}(\lambda)$ is
given by (38), (39).
In the special case $\xi_{0}=\eta_{0}=0$ (or correspondingly $\xi=\eta=0$),
Theorem 3.2 can be found in Buchholz (1953) [2, (16), page 175]. Of course, if
we multiply each $\xi,\eta,\xi_{0},\eta_{0}$ by $\sqrt{2k}$ we get the
expansion of $J_{0}(k\,r(\xi,\eta,\xi_{0},\eta_{0}))$. This leads to the
$J_{0}(kr)$ expansion of a fundamental solution for the three-dimensional
Laplace equation in parabolic cylindrical coordinates.
###### Theorem 3.3.
Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with parabolic
coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$, respectively, Then
$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\sum_{j=1}^{2}\int_{0}^{\infty}\int_{-\infty}^{\infty}c_{j}(\lambda)\rho^{\prime}_{j}(\lambda)$
$\displaystyle\hskip 14.22636pt\times
u_{j}(\lambda,2\sqrt{k}\xi)u_{j}(\lambda,2i\sqrt{k}\eta)u_{j}(\lambda,2\sqrt{k}\xi_{0})u_{j}(\lambda,2i\sqrt{k}\eta_{0})e^{-k|z-z_{0}|}\,d\lambda\,dk.$
### 3.3 The Riemann method of integration
The function $w(\xi,\zeta,\xi_{0},\zeta_{0})$ (as a function of $(\xi,\zeta)$)
is a solution of the partial differential equation (21) and it satisfies the
condition $w(\xi,\zeta,\xi_{0},\zeta_{0})=1$ if
$\xi-\xi_{0}=\pm(\zeta-\zeta_{0})$. This shows that $w$ is the Riemann
function of (21); for instance, see Garabedian (1986) [8].
The Riemann method of integration applied to the partial differential equation
(21) as in Volkmer (1980) [19] gives, for all
$\zeta,\xi_{0},\zeta_{0}\in{\mathbf{R}}$,
$\displaystyle
2u(\xi_{0},\zeta_{0})=u(\zeta-\zeta_{0}+\xi_{0},\zeta)+u(-\zeta+\zeta_{0}+\xi_{0},\zeta)$
(42)
$\displaystyle\int_{-\zeta+\zeta_{0}+\xi_{0}}^{\zeta-\zeta_{0}+\xi_{0}}\left[w(\xi,\zeta,\xi_{0},\zeta_{0})\partial_{2}u(\xi,\zeta)-\partial_{2}w(\xi,\zeta,\xi_{0},\zeta_{0})u(\xi,\zeta)\right]\,d\xi,$
where $u\in C^{2}({\mathbf{R}}^{2})$ is a solution of (21). This formula for
$\xi_{0}=\zeta=0$ and $u(\xi,\zeta)=u_{1}(\lambda,\xi)u_{2}(\lambda,\zeta)$
with $u_{1},u_{2}$ from (25), (26) (after replacing $\zeta_{0}$ by $\zeta$),
implies that
$\int_{-\zeta}^{\zeta}J_{0}({\textstyle\frac{1}{4}}(\xi^{2}-\zeta^{2}))u_{1}(\lambda,\xi)\,d\xi=2u_{2}(\lambda,\zeta).$
(43)
This equation allows us to transform the even solution $u_{1}$ into the odd
solution $u_{2}$ of equation (24). One can prove (43) directly by denoting the
left-hand side of (43) by $f(\zeta)$ and then showing that $f$ is an odd
solution of (24) with $f^{\prime}(0)=2$.
By differentiating (42) first with respect to $\xi_{0},\zeta_{0}$, and using
$w_{1}=u_{2}$, $w_{2}=u_{1}$, we obtain (after a lengthy calculation) that
$\int_{-\zeta}^{\zeta}\frac{\xi\zeta}{\xi^{2}-\zeta^{2}}J_{1}({\textstyle\frac{1}{4}}(\xi^{2}-\zeta^{2}))u_{2}(\lambda,\xi)\,d\xi=2u_{1}(\lambda,\zeta)-2u_{2}^{\prime}(\lambda,\zeta).$
This formula can also be proved directly.
Let $h:{\mathbf{R}}^{2}\to{\mathbf{R}}$ be the function defined by
$h(\xi,\zeta):=\left\\{\begin{array}[]{ll}\displaystyle
J_{0}\left({\textstyle\frac{1}{4}}(\xi^{2}-\zeta^{2})\right)&\quad\mathrm{if}\
|\xi|<|\zeta|,\\\\[5.0pt] \displaystyle
0&\quad\textup{otherwise.}\end{array}\right.$
For fixed $\zeta$ this is an even function in $L^{2}({\mathbf{R}})$ which can
be expanded according to Theorem 3.1 (without knowing $c_{j}(\lambda)$), so
that
${\rm
sign}(\zeta)h(\xi,\zeta)=2\int_{-\infty}^{\infty}u_{1}(\lambda,\xi)u_{2}(\lambda,\zeta)\rho^{\prime}_{1}(\lambda)\,d\lambda.$
If $\xi=0$, we obtain
${\rm
sign}(\zeta)J_{0}({\textstyle\frac{1}{4}}\zeta^{2})=2\int_{-\infty}^{\infty}u_{2}(\lambda,\zeta)\rho_{1}^{\prime}(\lambda)\,d\lambda.$
By Theorem 3.1, this formulas allows us to conclude
$\int_{0}^{\infty}J_{0}({\textstyle\frac{1}{4}}\zeta^{2})u_{2}(\lambda,\zeta)\,d\zeta=\frac{\rho_{1}^{\prime}(\lambda)}{\rho_{2}^{\prime}(\lambda)},$
and this is in agreement with (62).
## 4 Expansion of $J_{0}(kr)$ for elliptic cylinder coordinates
Consider equation (6) for $k>0$ and elliptic coordinates on the plane
$x=c\cosh\xi\cos\eta,\quad y=c\sinh\xi\sin\eta,$ (44)
where $\xi\in[0,\infty)$, $\eta\in{\mathbf{R}},$ and $c>0$. Transforming
$u(\xi,\eta)=U(x,y)$ we obtain
$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}+k^{2}c^{2}(\cosh^{2}\xi-\cos^{2}\eta)u=0.$
(45)
Separating variables $u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta),$ leads to
$\displaystyle-u_{1}^{\prime\prime}(\xi)+(\lambda-2q\cosh 2\xi)u_{1}(\xi)$
$\displaystyle=$ $\displaystyle 0,$ (46) $\displaystyle\hskip
14.22636ptu_{2}^{\prime\prime}(\eta)+(\lambda-2q\cos 2\eta)u_{2}(\eta)$
$\displaystyle=$ $\displaystyle 0,$ (47)
where $q=\frac{1}{4}c^{2}k^{2}>0$.
For $q\in{\mathbf{R}}$, Mathieu’s equation (47) is a Hill’s differential
equation with period $\pi$; see [17] or [14]. Here $q$ is positive but in the
next section $q$ will be negative. As a Hill’s equation, Mathieu’s equation
(47) admits nontrivial $2\pi$-periodic solutions if and only if $\lambda$ is
equal to one of its eigenvalues $a_{n}(q)$, $n\in{\mathbf{N}}_{0}$ or
$b_{n}(q)$, $n\in{\mathbf{N}}$. If $\lambda=a_{n}(q),$ then Mathieu’s equation
has an even $2\pi$-periodic solution ${\rm ce}_{n}(\eta,q)$, and if
$\lambda=b_{n}(q)$ then Mathieu’s equation has an odd $2\pi$-periodic solution
${\rm se}_{n}(\eta,q)$. These functions are normalized according to
$\int_{0}^{2\pi}{\rm ce}_{n}^{2}(\eta,q)\,d\eta=\int_{0}^{2\pi}{\rm
se}_{n}^{2}(\eta,q)\,d\eta=\pi.$
Moreover,
${\rm ce}_{n}(\eta+\pi,q)=(-1)^{n}{\rm ce}(\eta,q),\quad{\rm
se}_{n}(\eta+\pi,q)=(-1)^{n}{\rm se}(\eta,q).$
Note that all solutions of Mathieu’s equation are entire functions of $\eta$.
For these $2\pi$-periodic solutions of the Mathieu equation, we have the
following expansion theorem.
###### Theorem 4.1.
Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-
infinite strip $S$ that contains the real axis. Then
$f(z)=\alpha_{0}{\rm ce}_{0}(z,q)+\sum_{n=1}^{\infty}(\alpha_{n}{\rm
ce}_{n}(z,q)+\beta_{n}{\rm se}_{n}(z,q)),$ (48)
where
$\alpha_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x){\rm
ce}_{n}(x,q)\,dx,\quad\beta_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x){\rm
se}_{n}(x,q)\,dx.$
The series (48) converges absolutely and uniformly on any compact subset of
the strip $S$.
Let $(x_{0},y_{0})$ and $(x,y)$ be points on ${\mathbf{R}}^{2}$ with distance
$r$. Let $(x_{0},y_{0})$, $(x,y)$ have elliptic coordinates
$(\xi_{0},\eta_{0})$ and $(\xi,\eta)$, respectively. Then
$\displaystyle
r^{2}=c^{2}\left[(\cosh\xi\cos\eta-\cosh\xi_{0}\cos\eta_{0})^{2}+(\sinh\xi\sin\eta-\sinh\xi_{0}\sin\eta_{0})^{2}\right].$
(49)
Clearly, $J_{0}(kr)$ as a function of $(\xi,\eta)$ solves (45). We substitute
$\zeta=i\xi$ and $\zeta_{0}=i\xi_{0}$. Then (45) changes to
$\frac{\partial^{2}u}{\partial\zeta^{2}}-\frac{\partial^{2}u}{\partial\eta^{2}}+k^{2}c^{2}(\cos^{2}\eta-\cos^{2}\zeta)u=0,$
(50)
and $J_{0}(kr)$ is transformed to
$w(\zeta,\eta,\zeta_{0},\eta_{0})=J_{0}(k\tilde{r}),$
where
$\displaystyle\tilde{r}^{2}$ $\displaystyle=$ $\displaystyle
c^{2}[(\cos\zeta\cos\eta-\cos\zeta_{0}\cos\eta_{0})^{2}-(\sin\zeta\sin\eta-\sin\zeta_{0}\sin\eta_{0})^{2}]$
$\displaystyle=$ $\displaystyle
c^{2}(\cos(\zeta-\eta)-\cos(\zeta_{0}-\eta_{0}))(\cos(\zeta+\eta)-\cos(\zeta_{0}+\eta_{0})).$
The function $w(\zeta,\eta,\zeta_{0},\eta_{0})$ is an analytic function on
${\mathbf{C}}^{4}$ and it solves equation (50) as a function of
$(\zeta,\eta)$. It is the Riemann function of this differential equation
because $w(\zeta,\eta,\zeta_{0},\eta_{0})=1$ if
$\zeta-\zeta_{0}=\pm(\eta-\eta_{0})$. For fixed $\eta,\zeta_{0},\eta_{0}$ we
wish to expand the function $\zeta\mapsto w(\zeta,\eta,\zeta_{0},\eta_{0})$ in
a series of Mathieu functions according to Theorem 4.1. To this end we have to
evaluate the integral
$\int_{-\pi}^{\pi}w(\zeta,\eta,\zeta_{0},\eta_{0}){\rm
ce}_{n}(\zeta,q)\,d\zeta,$
and a similar integral with ${\rm ce}_{n}$ replaced by ${\rm se}_{n}$. Using
Riemann’s method of integration applied to a pentagonal curve, it has been
shown in [20] that
$\displaystyle\int_{-\pi}^{\pi}w(\zeta,\eta,\zeta_{0},\eta_{0}){\rm
ce}_{n}(\zeta,q)\,d\zeta$ $\displaystyle=$ $\displaystyle\mu_{n}(q){\rm
ce}_{n}(\eta,q){\rm ce}_{n}(\zeta_{0},q){\rm ce}_{n}(\eta_{0},q),$ (51)
$\displaystyle\int_{-\pi}^{\pi}w(\zeta,\eta,\zeta_{0},\eta_{0}){\rm
se}_{n}(\zeta,q)\,d\zeta$ $\displaystyle=$ $\displaystyle\nu_{n}(q){\rm
se}_{n}(\eta,q){\rm se}_{n}(\zeta_{0},q){\rm se}_{n}(\eta_{0},q).$ (52)
There do not exist explicit formulas for the quantities $\mu_{n}(q)$ and
$\nu_{n}(q)$ but they can be determined as follows. Mathieu’s equation (47)
with $\lambda=a_{n}(q)$, $n\in{\mathbf{N}}_{0}$, has the solution
$u_{1}(\eta)={\rm ce}_{n}(\eta,q)$. We choose a second linear independent
solution $u_{2}(\eta)$. Then there is $\sigma$ such that
$u_{2}(\eta+\pi)=\sigma u_{1}(\eta)+(-1)^{n}u_{2}(\eta)$
and
$\mu_{n}(q)=\frac{2(-1)^{n}\sigma}{W[u_{1},u_{2}]},$ (53)
where $W[u_{1},u_{2}]$ denotes the Wronskian of $u_{1}$ and $u_{2}$.
Similarly, Mathieu’s equation (47) with $\lambda=b_{n}(q)$,
$n\in{\mathbf{N}}$, has the solution $u_{3}(\eta)={\rm se}_{n}(\eta,q)$. We
choose a second linear independent solution $u_{4}(\eta)$. Then there is
$\tau$ such that
$u_{4}(\eta+\pi)=\tau u_{3}(\eta)+(-1)^{n}u_{4}(\eta)$
and
$\nu_{n}(q)=\frac{2(-1)^{n}\tau}{W[u_{3},u_{4}]}.$ (54)
Now applying Theorem 4.1 and substituting $\zeta=i\xi$, $\zeta_{0}=i\xi_{0},$
we obtain the following result.
###### Theorem 4.2.
Let $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{C}}$, and let $k>0$, $c>0$,
$q=\frac{1}{4}c^{2}k^{2}$. Then
$\displaystyle J_{0}(kr)=\frac{1}{\pi}\sum_{n=0}^{\infty}\mu_{n}(q){\rm
ce}_{n}(i\xi,q){\rm ce}_{n}(\eta,q){\rm ce}_{n}(i\xi_{0},q){\rm
ce}_{n}(\eta_{0},q)$ $\displaystyle\hskip
34.14322pt+\frac{1}{\pi}\sum_{n=1}^{\infty}\nu_{n}(q){\rm se}_{n}(i\xi,q){\rm
se}_{n}(\eta,q){\rm se}_{n}(i\xi_{0},q){\rm se}_{n}(\eta_{0},q),$
where $r$ is given by (49).
Theorem 4.2 agrees with expansion (23) (for $j=1$ and $\nu=0$), section 2.66
in Meixner & Schäfke [14], who have a slightly different notation. They use
${\rm me}_{n}(z,q)$, $n\in{\mathbf{Z}}$, where
$\displaystyle{\rm me}_{n}(z,q)$ $\displaystyle:=$ $\displaystyle\sqrt{2}{\rm
ce}_{n}(z,q)\quad\textup{if $n\in{\mathbf{N}}_{0},$}$ $\displaystyle{\rm
me}_{-n}(z,q)$ $\displaystyle:=$ $\displaystyle-\sqrt{2}i{\rm
se}_{n}(z,q)\quad\textup{if $n\in{\mathbf{N}}.$}$
Moreover, the coefficients $\mu_{n}(q)$ and $\nu_{n}(q)$ are represented in a
different form. The proof of Theorem 4.2 based on Riemann’s method of
integration appears to be new.
We now use (3) to obtain our final result in this section.
###### Theorem 4.3.
Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with elliptic
cylinder coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$,
respectively. Then
$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{1}{\pi}\sum_{n=0}^{\infty}\int_{0}^{\infty}\mu_{n}(q){\rm
ce}_{n}(i\xi,q){\rm ce}_{n}(\eta,q){\rm ce}_{n}(i\xi_{0},q){\rm
ce}_{n}(\eta_{0},q)e^{-k|z-z_{0}|}\,dk$
$\displaystyle+\frac{1}{\pi}\sum_{n=1}^{\infty}\int_{0}^{\infty}\nu_{n}(q){\rm
se}_{n}(i\xi,q){\rm se}_{n}(\eta,q){\rm se}_{n}(i\xi_{0},q){\rm
se}_{n}(\eta_{0},q)e^{-k|z-z_{0}|}\,dk,$
where $q=\frac{1}{4}c^{2}k^{2}$.
## 5 Expansion of $K_{0}(kr)$ for elliptic cylinder coordinates
Consider equation (10) for $k>0$. Transforming to elliptic coordinates (44),
we obtain
$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}-k^{2}c^{2}(\cosh^{2}\xi-\cos^{2}\eta)u=0.$
(55)
Separating variables $u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta),$ leads again to (46),
(47) but now $q=-\frac{1}{4}c^{2}k^{2}$ is negative.
We will need the following solutions of the modified Mathieu equation (46)
when $q<0$; see [17, §28.20]. Set $q=-h^{2}$ with $h>0$. For
$n\in{\mathbf{N}}_{0}$, ${\rm Ie}_{n}(\xi,h)$ is the even solution of (46)
with $\lambda=a_{n}(q)$ with asymptotic behavior
${\rm Ie}_{n}(\xi,h)\sim I_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$},$
while ${\rm Ke}_{n}(\xi,h)$ is the recessive solution determined by
${\rm Ke}_{n}(\xi,h)\sim K_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$},$
where $I_{n}(z):=i^{-n}J_{n}(iz)$ and $K_{n}(z)$ are the modified Bessel
functions of the first [17, (10.27.6)] and second kinds respectively, with
integer order $n$ (see (4), (8)). Similarly, for $n\in{\mathbf{N}}$, ${\rm
Io}_{n}(\xi,h)$ is the odd solution of (46) with $\lambda=b_{n}(q)$ with
asymptotic behavior
${\rm Io}_{n}(\xi,h)\sim I_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$},$
while ${\rm Ko}_{n}(\xi,h)$ is the recessive solution determined by
${\rm Ko}_{n}(\xi,h)\sim K_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$}.$
For fixed $\xi,\xi_{0},\eta_{0}\in{\mathbf{R}}$ we wish to expand the function
$v(\xi,\eta,\xi_{0},\eta_{0}):=K_{0}(kr(\xi,\eta,\xi_{0},\eta_{0}))$
with $r$ given by (49) into a series of periodic Mathieu functions according
to Theorem 4.1. The corresponding integrals appearing in the expansion will be
computed based on the observation that $(\xi,\eta)\mapsto
v(\xi,\eta,\xi_{0},\eta_{0})$ is a fundamental solution of (55). In fact, it
is a solution of (55) and it has logarithmic singularities at the points
$\pm(\xi_{0},\eta_{0}+2m\pi)$, where $m$ is any integer. Arguing as in Volkmer
(1984) [21, Theorem 1.11], we have the following representation theorem for a
solution of (55).
###### Theorem 5.1.
Let $u\in C^{2}({\mathbf{R}}^{2})$ be a solution of (55). Let
$(\xi_{0},\eta_{0})\in{\mathbf{R}}^{2}$, and let $C$ be a a closed rectifiable
curve on ${\mathbf{R}}^{2}$ which does not pass through any of the points
$\pm(\xi_{0},\eta_{0}+2m\pi)$, $m\in{\mathbf{Z}}$. Let $n^{\pm}_{m}$ be the
winding number of $C$ with respect to $\pm(\xi_{0},\eta_{0}+2m\pi)$. Then we
have
$\displaystyle
2\pi\sum_{m}\left[n^{+}_{m}u(\xi_{0},\eta_{0}+2m\pi)+n^{-}_{m}u(-\xi_{0},-\eta_{0}-2m\pi))\right]$
$\displaystyle\hskip
22.76219pt=\int_{C}(u\partial_{2}v-v\partial_{2}u)\,d\xi+(v\partial_{1}u-u\partial_{1}v)\,d\eta,$
where $\partial_{1},\partial_{2}$ denote partial derivatives with respect to
$\xi$, $\eta$, respectively.
In Theorem 5.1 we choose
$u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta),$
where
$u_{1}(\xi)={\rm Ke}_{n}(\xi,h),\quad u_{2}(\eta)={\rm ce}_{n}(\eta,q).$
Let $\xi_{0}>0$ and $\eta_{0}\in{\mathbf{R}}$. We take the curve $C$ to be the
positively oriented boundary of the rectangle $\xi_{1}\leq\xi\leq\xi_{2}$,
$\eta_{0}-\pi\leq\eta\leq\eta_{0}+\pi$, where $|\xi_{1}|<\xi_{0}<\xi_{2}$.
Consider the line integral $\int_{C}$ in Theorem 5.1. Since $u_{2}$ has period
$2\pi,$ the line integrals along the horizontal segments of $C$ cancel each
other. When $\xi_{2}\to+\infty$ the asymptotic behavior of $u_{1}(\xi)$ shows
that the integral along the right-hand vertical segment of $C$ tends to $0$ as
$\xi_{2}\to+\infty$. Therefore, setting
$f(\xi)=\int_{-\pi}^{\pi}v(\xi,\eta,\xi_{0},\eta_{0})u_{2}(\eta)\,d\eta=\int_{\eta_{0}-\pi}^{\eta_{0}+\pi}v(\xi,\eta,\xi_{0},\eta_{0})u_{2}(\eta)\,d\eta$
for $|\xi|<\xi_{0}$, one obtains
$2\pi
u_{1}(\xi_{0})u_{2}(\eta_{0})=u_{1}(\xi)f^{\prime}(\xi)-u_{1}^{\prime}(\xi)f(\xi).$
(56)
We now argue as in section 2. By differentiating (56) with respect to $\xi$,
we find that $f(\xi)$ satisfies the modified Mathieu equation (46). It is easy
to see that $f(\xi)$ is an even function, so $f(\xi)=cu_{3}(\xi)$, where
$u_{3}(\xi)={\rm Ie}_{n}(\xi,h)$ and $c$ is a constant. Then (56) implies that
$2\pi u_{1}(\xi_{0})u_{2}(\eta_{0})=cW[u_{1},u_{3}].$
Since $W[u_{1},u_{3}]=1$,
$\frac{1}{\pi}\int_{-\pi}^{\pi}v(\xi,\eta,\xi_{0},\eta_{0})u_{2}(\eta)\,d\eta=2u_{1}(\xi_{0})u_{2}(\eta_{0})u_{3}(\xi)\quad\textup{if
$|\xi|<\xi_{0}$}.$ (57)
By the same reasoning, we see that (57) is also true when $u_{1}(\xi)={\rm
Ko}_{n}(\xi,h)$, $u_{2}(\eta)={\rm se}_{n}(\eta,q)$, $u_{3}(\xi)={\rm
Io}_{n}(\xi,h)$.
Expanding the function $\eta\mapsto v(\xi,\eta,\xi_{0},\eta_{0})$ according to
Theorem 4.1, the following result is obtained.
###### Theorem 5.2.
Let $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$ such that $|\xi|<\xi_{0}$, and
let $k>0$, $c>0,q=-\frac{1}{4}c^{2}k^{2}$, $h=\frac{1}{2}ck$. Then
$\displaystyle K_{0}(kr)$ $\displaystyle=$ $\displaystyle
2\sum_{n=0}^{\infty}{\rm Ie}_{n}(\xi,h){\rm ce}_{n}(\eta,q){\rm
Ke}_{n}(\xi_{0},h){\rm ce}_{n}(\eta_{0},q)$ $\displaystyle+$ $\displaystyle
2\sum_{n=1}^{\infty}{\rm Io}_{n}(\xi,h){\rm se}_{n}(\eta,q){\rm
Ko}_{n}(\xi_{0},h){\rm se}_{n}(\eta_{0},q),$
where $r$ is given by (49).
Theorem 5.2 agrees with Meixner & Schäfke (1954) [14, section 2.66] although
our notation and proof are different.
Inserting this result in (7), we obtain our final result.
###### Theorem 5.3.
Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with elliptic
cylinder coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$,
respectively. If $\xi_{\lessgtr}:={\min\atop\max}\\{\xi,\xi_{0}\\}$ then
$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{4}{\pi}\sum_{n=0}^{\infty}\int_{0}^{\infty}{\rm
Ie}_{n}(\xi_{<},h){\rm ce}_{n}(\eta,q){\rm Ke}_{n}(\xi_{>},h){\rm
ce}_{n}(\eta_{0},q)\cos k(z-z_{0})\,dk$
$\displaystyle+\frac{4}{\pi}\sum_{n=1}^{\infty}\int_{0}^{\infty}{\rm
Io}_{n}(\xi_{<},h){\rm se}_{n}(\eta,q){\rm Ko}_{n}(\xi_{>},h){\rm
se}_{n}(\eta_{0},q)\cos k(z-z_{0})\,dk,$
where $q=-\frac{1}{4}c^{2}k^{2}$, $h=\frac{1}{2}ck$, and ${\bf x}\neq{\bf
x}_{0}$.
## Appendix A Integrals for the (modified) parabolic cylinder harmonics
expansion of $J_{0}(kr)$
The following formulas are valid for $\Im\lambda<0$ :
$\displaystyle
I_{1}:=\int_{0}^{\infty}J_{0}\left({\textstyle\frac{1}{4}}\xi^{2}\right)u_{3}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\pi}(1-i)\frac{G_{1}}{G_{2}^{2}},$ (58)
$\displaystyle
I_{2}:=\int_{0}^{\infty}\xi^{-1}J_{1}\left({\textstyle\frac{1}{4}}\xi^{2}\right)u_{3}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle-\sqrt{\pi}\left(\lambda\frac{1}{G_{2}}+2i\frac{G_{2}}{G_{1}^{2}}\right),$
(59)
where
$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle
G_{1}(\lambda)=\Gamma\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda\right),$
$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
G_{2}(\lambda)=\Gamma\left({\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda\right).$
We believe that these integrals may be known but do not have a reference. We
will derive (59). In (3.1) we use the integral representation [17, (13.4.4)]
$\Gamma(a)U(a,b,z)=\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt,\quad\Re
z,\Re a>0.$
Substituting $4s=\xi^{2}$ and changing the order of integration, one obtains
$I_{2}=\frac{1}{2G_{1}}\int_{0}^{\infty}t^{-\frac{3}{4}+\frac{i}{2}\lambda}(1+t)^{-\frac{3}{4}-\frac{i}{2}\lambda}\int_{0}^{\infty}s^{-1}J_{1}(s)e^{-is(2t+1)}\,ds\,dt.$
(60)
From [22, page 405], we have, for $t>0$,
$\displaystyle\int_{0}^{\infty}s^{-1}J_{1}(s)\cos(s(2t+1))\,ds$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\int_{0}^{\infty}s^{-1}J_{1}(s)\sin(s(2t+1))\,ds$
$\displaystyle=$ $\displaystyle t+(t+1)-2\sqrt{t}\sqrt{t+1}.$
Substituting these formulas in (60), we can evaluate $I_{2}$ using three times
the formula for the beta function [17, (5.12.3)]
$B(z,w)=\frac{\Gamma(z)\Gamma(w)}{\Gamma(z+w)}=\int_{0}^{\infty}t^{z-1}(1+t)^{-z-w}\,dt,\quad\Re
z,\Re w>0.$
This gives (59).
The proof of (58) is similar, but in (3.1) one should first use [17,
(13.2.40)]
$U(a,b,z)=z^{1-b}U(1+a-b,2-b,z).$
The formulas (58), (59) remain valid for real $\lambda$. By separating real
and imaginary parts, we obtain for $\lambda\in{\mathbf{R}}$,
$\displaystyle\int_{0}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{1}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle\frac{\Re(G_{1}\overline{G}_{2})+\Im(G_{1}\overline{G_{2}})}{|G_{2}|^{2}},$
(61)
$\displaystyle\int_{0}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{2}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\left|\frac{G_{1}}{G_{2}}\right|^{2},$
(62)
$\displaystyle\int_{0}^{\infty}\xi^{-1}J_{1}({\textstyle\frac{1}{4}}\xi^{2})u_{1}(\lambda,\xi)\,d\xi$
$\displaystyle=$ $\displaystyle
2\left|\frac{G_{2}}{G_{1}}\right|^{2}-\lambda,$ (63)
$\displaystyle\int_{0}^{\infty}\xi^{-1}J_{1}({\textstyle\frac{1}{4}}\xi^{2})u_{2}(\lambda,\xi)\,d\xi$
$\displaystyle=$ $\displaystyle
2\frac{\Re(G_{1}\overline{G}_{2})+\Im(G_{1}\overline{G_{2}})}{|G_{1}|^{2}},.$
(64)
We may use
$\Re(G_{1}\overline{G}_{2})+\Im(G_{1}\overline{G_{2}})=\frac{\pi\sqrt{2}e^{-\frac{1}{2}\pi\lambda}}{\cosh(\pi\lambda)}.$
Formulas (61), (64) give us the integrals (35), (36) noting that we integrate
even functions in (35), (36).
## Acknowledgements
Part of this work was conducted while H. S. Cohl was a National Research
Council Research Postdoctoral Associate in the Information Technology
Laboratory at the National Institute of Standards and Technology,
Gaithersburg, Maryland, U.S.A.
## References
## References
* [1] F. V. Atkinson. Discrete and continuous boundary problems. Mathematics in Science and Engineering, Vol. 8. Academic Press, New York, 1964.
* [2] H. Buchholz. Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen. Ergebnisse der angewandten Mathematik. Bd. 2. Springer-Verlag, Berlin, 1953.
* [3] T. M. Cherry. Expansions in terms of parabolic cylinder functions. Proceedings of the Edinburgh Mathematical Society. Series II, 8:50–65, 1948.
* [4] E. A. Coddington and N. Levinson. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
* [5] H. S. Cohl, J. E. Tohline, A. R. P. Rau, and H. M. Srivastava. Developments in determining the gravitational potential using toroidal functions. Astronomische Nachrichten, 321(5/6):363–372, 2000.
* [6] C. G. Darwin. On Weber’s function. The Quarterly Journal of Mechanics and Applied Mathematics, 2:311–320, 1949.
* [7] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher transcendental functions. Vol. II. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
* [8] P. R. Garabedian. Partial differential equations. Chelsea Publishing Co., New York, second edition, 1986.
* [9] E. Heine. Handbuch der Kugelfunctionen, Theorie und Anwendungen (volume 2). Druck und Verlag von G. Reimer, Berlin, 1881.
* [10] E. W. Hobson. The theory of spherical and ellipsoidal harmonics. Chelsea Publishing Company, New York, 1955.
* [11] N. N. Lebedev. Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.
* [12] T. M. MacRobert. Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. Methuen & Co. Ltd., London, second edition, 1947.
* [13] W. Magnus. Zur Theorie des zylindrisch-parabolischen Spiegels. Zeitschrift für Physik. C. Particles and Fields, 118:343–356, 1941.
* [14] J. Meixner and F. W. Schäfke. Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXI. Springer-Verlag, Berlin, 1954.
* [15] J. C. P. Miller, editor. Tables of Weber parabolic cylinder functions, giving solutions of the differential equation $d^{2}y/dx^{2}+(\frac{1}{4}{x^{2}}-a)y=0$. Her Majesty’s Stationery Office, London (British Information Services, 30 Rockefeller Plaza, New York 20, N. Y.), 1955. Computed by Scientific Computing Service Limited, Mathematical introduction by J. C. P. Miller.
* [16] P. M. Morse and H. Feshbach. Methods of theoretical physics. 2 volumes. McGraw-Hill Book Co., Inc., New York, 1953.
* [17] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical functions. Cambridge University Press, Cambridge, 2010.
* [18] F. W. Schäfke. Einführung in die Theorie der speziellen Funktionen der mathematischen Physik. Die Grundlehren der mathematischen Wissenschaften, Bd. 118. Springer-Verlag, Berlin, 1963.
* [19] H. Volkmer. Integralrelationen mit variablen Grenzen für spezielle Funktionen der mathematischen Physik. Journal für die Reine und Angewandte Mathematik, 319:118–132, 1980.
* [20] H. Volkmer. Integralgleichungen für periodische Lösungen Hillscher differentialgleichungen. Analysis. International Journal of Analysis and its Application, 3(1-4):189–203, 1983.
* [21] H. Volkmer. Integral representations for products of Lamé functions by use of fundamental solutions. SIAM Journal on Mathematical Analysis, 15(3):559–569, 1984.
* [22] G. N. Watson. A treatise on the theory of Bessel functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 1944.
* [23] C. P. Wells and R. D. Spence. The parabolic cylinder functions. Journal of Mathematics and Physics, 24:51–64, 1945.
|
arxiv-papers
| 2012-04-26T21:24:21 |
2024-09-04T02:49:30.250548
|
{
"license": "Public Domain",
"authors": "Howard S. Cohl and Hans Volkmer",
"submitter": "Howard Cohl",
"url": "https://arxiv.org/abs/1204.6064"
}
|
1204.6092
|
# On SDE associated with continuous-state branching processes conditioned to
never be extinct
M.C. Fittipaldi DIM–CMM, UMI 2807 UChile-CNRS, Universidad de Chile, Casilla
170-3, Correo 3, Santiago, Chile ; Supported by Basal-CONICTY;
mfittipaldi@dim.uchile.cl J. Fontbona DIM–CMM, UMI 2807 UChile-CNRS,
Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile ; Partially
supported by Basal-CONICTY; fontbona@dim.uchile.cl.
###### Abstract
We study the pathwise description of a (sub-)critical continuous-state
branching process (CSBP) conditioned to be never extinct, as the solution to a
stochastic differential equation driven by Brownian motion and Poisson point
measures. The interest of our approach, which relies on applying Girsanov
theorem on the SDE that describes the unconditioned CSBP, is that it points
out an explicit mechanism to build the immigration term appearing in the
conditioned process, by randomly selecting jumps of the original one. These
techniques should also be useful to represent more general $h$-transforms of
diffusion-jump processes.
Key words Stochastic Differential Equations; Continuous-state branching
processes;
Non-extinction; Immigration.
AMS 2010 Subject Classification 60J80, 60H20,60H10.
## 1 Introduction and preliminaries
Stochastic differential equations (SDE) representing continuous-state
branching processes (CSBP) or CSBP with immigration (CBI) have attracted
increasing attention in the last years, as powerful tools for studying
pathwise and distributional properties of these processes as well as some
scaling limits, see e.g. Dawson and Li [5], [6] , Lambert [17], Fu and Li [10]
and Caballero et al. [4].
In this note, we are interested in SDE representations for (sub)-critical CSBP
conditioned to never be extinct. It is well known that such conditioned CSBP
correspond to CBIs with particular immigration mechanisms (see [25]). Thus, it
is possible to obtain SDE representations for them by using general results
and techniques developed in some of the aforementioned works, see [5] and
[10]. However, our goal is to directly obtain such representation by rather
using the fact that the law of the conditioned CSBP is obtained from the one
of the non conditioned process, by means of an explicit $h-$transform. In
accordance with that relation between the laws and to the “spine” or immortal
particle picture of the conditioned process ([25], [9]), one should expect to
identify, after measure change, copies of the original driving random
processes and an independent subordinator accounting for immigration. Our
proof will show how to obtain these processes by using Girsanov theorem and an
enlargement of the probability space in order to select by a suitable marking
procedure those jumps of the original (non conditioned) process that will
constitute (or will not) the immigrants. The enlargement of the probability
space and the marking procedure are both inspired in a construction of Lambert
[17] on stable Lévy processes. They are also reminiscent of the sized biased
tree representation of measure changes for Galton-Watson trees (Lyons et. al
[22]) or for branching Brownian motions (see e.g. Kyprianou [15] and Englänger
and Kyprianou [8]), but we do not aim at fully developing those ideas in the
present framework. In a related direction, in a recently posted article [12]
Hénard obtains the same SDE description of the conditioned CSBP, using the
look-down particle representation of CSBP of Donnelly and Kurtz [7].
We start by recalling some definitions and classic results about CSBPs and
Lévy processes along the lines of [16, Chap. 1,2 and 10], in particular the
relationship between them through the Lamperti transform. (We also refer the
reader to Le Gall [20] and Li [21] for further background on CSBP).
### 1.1 Continuous-state branching processes
Continuous-state branching processes (CSBP) were introduced by Jirina [13] in
1958. Later, Lamperti [19] showed that they can be obtained as scaling limits
of a sequence of Galton-Watson processes. A CSBP with probability laws given
the initial state $\\{\mathbb{P}_{x}:x\geq 0\\}$ is a càdlàg
$[0,\infty)$-valued strong Markov processes $Z=\\{Z_{t}:t\geq 0\\}$ satisfying
the branching property. That is, for any $t\geq 0$ and
$z_{1},z_{2}\in[0,\infty)$, $Z_{t}$ under $\mathbb{P}_{z_{1}+z_{2}}$ has the
same law as the independent sum $Z_{t}^{(1)}+Z_{t}^{(2)}$, where the
distribution of $Z_{t}^{(i)}$ is equal to that of $Z_{t}$ under
$\mathbb{P}_{z_{i}}$ for $i=1,2$. Usually, $Z_{t}$ represents the population
at time $t$ descending from an initial population $x$. The law of $Z$ is
completely characterized by its Laplace transform
$\mathbb{E}_{x}(e^{-\theta
Z_{t}})\,=\,e^{-xu_{t}(\theta)},\,\,\forall\,x>0,\,t\geq 0,$
where $u$ is a differentiable function in $t$ satisfying
$\left\\{\begin{array}[]{ll}\displaystyle{\frac{\partial u_{t}}{\partial
t}(\theta)+\psi(u_{t}(\theta))}=0\\\ \\\
u_{0}(\theta)=\theta,\end{array}\right.$ (1)
and $\psi$ is called the branching mechanism of $Z$, which has the form
$\psi(\lambda)=-q-a\lambda+\frac{1}{2}\sigma^{2}\lambda^{2}+\int_{(0,\infty)}(e^{-\lambda
x}-1+\lambda x\textbf{1}_{(x<1)})\Pi(dx)\quad\lambda\geq 0,$ (2)
for some $q\geq 0,\,a\in\mathbb{R},\,\sigma\geq 0$ and $\Pi$ a measure
supported in $(0,\infty)$ such that
$\int_{(0,\infty)}(1\wedge x^{2})\Pi(dx)\,<\,\infty.$ In particular, $\psi$ is
the characteristic exponent of a spectrally positive Lévy process, i.e. one
with no negative jumps. Since clearly,
$\mathbb{E}_{x}(Z_{t})=xe^{-\psi^{\prime}(0+)t}$, defining
$\rho:=\psi^{\prime}(0+)$ one has the following classification of CSBPs :
* (i)
subcritical, if $\rho>0$,
* (ii)
critical, if $\rho=0$ and
* (iii)
supercritical, if $\rho<0$,
according to whether the process will, on average, decrease, remain constant
or increase.
In the following, we will assume that $Z$ is conservative, i.e. $\forall$
$t>0$, $\mathbb{P}_{x}(Z_{t}<\infty)=1$. By Grey (1974), this is true if and
only if $\int_{0^{+}}\frac{d\xi}{|\psi(\xi)|}=\infty$, so it is sufficient to
asume $\psi(0)=0$ and $|\psi^{\prime}(0+)|<\infty$.
### 1.2 Lévy Processes and their connection with CSBP
Let $X=\\{X_{t}:t\geq 0\\}$ be a spectrally positive Lévy process with
characteristic exponent $\psi$ given by (2) with $q=0$, and initial state
$x\geq 0$ . By the Lévy-Ito decomposition it is well known that it can be
written as the following sum of independent processes
$X_{t}=x+at+\sigma
B_{t}^{X}+\int_{0}^{t}\int_{1}^{\infty}rN^{X}(ds,dr)+\int_{0}^{t}\int_{0}^{1}r\tilde{N}^{X}(ds,dr),$
where $a$ is a real number, $\sigma\geq 0$, $B^{X}$ is a Brownian motion,
$N^{X}$ is an independent Poisson measure on $[0,\infty)\times(0,\infty)$ with
intensity measure $dt\times\Pi(dr)$ and
$\tilde{N}^{X}(dt,dr):=N^{X}(dt,dr)-dt\Pi(dr)$ denotes the compensated measure
associated to $N^{X}$ (the last integral thus being a square integrable
martingale of compensated jumps of magnitude less than unity).
Lamperti [18] established a one-to-one correspondence between CSBPs and
spectrally positive Lévy processes via a random time change. More precisely,
for a Lévy process $X$ as above the process
$Z:=\\{Z_{t}=X_{\theta_{t}\wedge T_{0}}:t\geq 0\\},$
where $T_{0}=\inf\\{t>0:X_{t}=0\\}$ and
$\theta_{t}=\inf\left\\{s>0:\int_{0}^{s}\frac{du}{X_{u}}>t\right\\}$, is a
continuous-state branching process with branching mechanism $\psi$ and initial
value $Z_{0}=x$. Conversely, given $Z=\\{Z_{t}:t\geq 0\\}$ a CSBP with
branching mechanism $\psi$, such that $Z_{0}=x>0$, we have that
$X:=\\{X_{t}=Z_{\varphi_{t}\wedge T}:t\geq 0\\},$
where $T=\inf\\{t>0:Z_{t}=0\\}$ and
$\varphi_{t}=\inf\left\\{s>0:\int_{0}^{s}Z_{u}du>t\right\\}$, is a Lévy
process with no negative jumps, stopped at $T_{0}$ and satisfying
$\psi(\lambda)=\log\mathrm{I\\!E}(e^{-\lambda X_{1}})$, with initial position
$X_{0}=x$.
Relying on this relationship, Caballero et al. [4, Prop 4] provide a pathwise
description of the dynamics of a CSBP: for $(Z_{t},t\geq 0)$ there exist a
standard Brownian motion $B^{Z}$, and an independent Poisson measure $N^{Z}$
on $[0,\infty)\times(0,\infty)\times(0,\infty)$ with intensity measure
$dt\times d\nu\times\Pi(dr)$ in an enlarged probability space such that
$\begin{split}Z_{t}=&x+a\int_{0}^{t}Z_{s}ds+\sigma\int_{0}^{t}\sqrt{Z_{s}}dB_{s}^{Z}+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{Z}(ds,d\nu,dr)\\\
&+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{Z}(ds,d\nu,dr),\end{split}$
(3)
where $\tilde{N}^{Z}$ is the compensated Poisson measure associated with
$N^{Z}$. Pathwise properties of stochastic differential equations driven by
Brownian motion and Poisson point processes have been studied in more general
settings in [5], [10] and [6]. In particular, strong existence and pathwise
uniqueness for (3) is established [10]. Related SDE have also been considered
in Bertoin and Le Gall [2], [3].
## 2 CSBPs conditioned to be never extinct as solutions of SDEs
### 2.1 CSBP conditioned to be never extinct
We assume from now on that $Z$ is a (sub-)critical CSBP such that
$\psi(\infty)=\infty$ and $\int^{\infty}\frac{d\xi}{\psi(\xi)}<\infty$. Under
these and the previous conditions, the process does not explode and there is
almost surely extinction in finite time. Branching processes conditioned to
stay positive were first studied in the continuous-state framework by Roelly
and Rouault [25], who proved that for $Z$ as before,
$\mathbb{P}_{x}^{\uparrow}(A):=\lim_{s\uparrow\infty}\mathbb{P}_{x}(A|T>t+s),\quad
A\in\sigma(Z_{s}:s\leq t)$ (4)
is a well defined probability measure which satisfies
$\mathbb{P}_{x}^{\uparrow}(A)=\mathbb{E}(\textbf{1}_{A}e^{\rho
t}\frac{Z_{t}}{x}).$
In particular, $\mathbb{P}_{x}^{\uparrow}(T<\infty)=0$, and $\\{e^{\rho
t}Z_{t}:t\geq 0\\}$ is a martingale under $\mathbb{P}_{x}$. Note that
$\mathbb{P}_{x}^{\uparrow}$ is the law of the so-called $Q$-process (for in-
depth looks at this type of processes, we refer the reader to [17], [23] and
references therein). They also proved that $(Z,\mathbb{P}^{\uparrow})$ has the
same law as a CBI with branching mechanism $\psi$ and immigration mechanism
$\phi(\theta)=\psi^{\prime}(\theta)-\rho,$ $\theta\geq 0.$ This means that
$(Z,\mathbb{P}^{\uparrow})$ is a càdlàg $[0,\infty)$-valued process, and for
all $x,t>0$ and $\theta\geq 0$
$\mathbb{E}_{x}^{\uparrow}(e^{-\theta
Z_{t}})=\exp\\{-xu_{t}(\theta)-\int_{0}^{t}\phi(u_{t-s}(\theta))ds\\},$
where $u_{t}(\theta)$ is the unique solution to (1). Note also that $\phi$ is
the Laplace exponent of a subordinator.
### 2.2 Main Result
The above result is the key for the study of CSBP conditioned on non-
extinction, but we seek a more explicit description for the paths of $Z$ under
$\mathbb{P}^{\uparrow}$. To this end, we shall prove that
$(Z,\mathbb{P}^{\uparrow})$ has a SDE representation, which agrees with the
interpretation of a CSBP conditioned on non-extinction as a CBI, but also
gives us a pathwise description for the conditioned process. In particular,
this result extends Lambert’s results for the stable case [17, Theorem 5.2]
(see below for details) as well as equation (3).
###### Theorem 2.1
Under $\mathbb{P}^{\uparrow}$, the process $Z$ is the unique strong solution
of the following stochastic differential equation:
$\begin{array}[]{lcl}Z_{t}&=&x+a\int_{0}^{t}Z_{s}ds+\sigma\int_{0}^{t}\sqrt{Z_{s}}dB^{\uparrow}_{s}+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)\\\
\\\ &&+\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr)+\sigma^{2}t\end{array}$
(5)
where $\\{B^{\uparrow}_{t}:t\geq 0\\}$ is a Brownian motion, $N^{\uparrow}$
and $N^{\star}$ are Poisson measures on $[0,\infty)\times(0,\infty)^{2}$ and
$[0,\infty)\times(0,\infty)$ with intensities measures $ds\times
d\nu\times\Pi(dr)$ and $ds\times r\Pi(dr)$, respectively, and these objects
are mutually independent (as usual, $\tilde{N}^{\uparrow}$ stands for the
compensated measure associated with $N^{\uparrow}$). Moreover, the point
processes $N^{\uparrow}$ and $N^{\star}$ can be constructed by change of
measure and a marking procedure on an enlargement of the probability space
where $B^{Z}$ and $N^{Z}$ in (3) are defined, which supports and independent
i.i.d. sequence of uniform random variables in the unit interval.
This result implies that we can recover $Z$ conditioned on non-extinction as
the solution of a SDE driven by a copy of $B^{Z}$, a copy of $N^{Z}$, and a
Poisson random measure with intensity $ds\times r\Pi(dr)$, plus a drift.
(Notice that taking out the last line, corresponding to a subordinator with
drift, one again obtains equation (3).)
## 3 Relations to previous results
### 3.1 Stable processes
We will show that, as pointed out before, Lambert’s SDE representation of
stable branching processes given in [17, Theorem 5.2] can be seen as a special
case of Theorem 2.1.
Let $X$ be a spectrally positive $\alpha$-stable process with characteristic
exponent $\psi$ and characteristic measure $\Pi(dr)=kr^{-(\alpha+1)}dr$, where
$k$ is some positive constant and $1<\alpha\leq 2$. Let $Z$ be the branching
process with branching mechanism $\psi$. Thanks to Theorem 2.1 we know that,
under $\mathbb{P}^{\uparrow}$, $Z$ satisfies the following stochastic
differential equation:
$Z_{t}=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr),$
(6)
where $N^{\uparrow}$ is a Poisson random measure with intensity $ds\times
d\nu\times\Pi(dr)$ and $N^{\star}$ is an independent Poisson random measure
with intensity $ds\times r\Pi(dr)$. Now, we define
$\theta_{n}=\displaystyle{\frac{r^{\uparrow}_{n}\textbf{1}_{(\nu^{\uparrow}_{n}\leq
Z_{t_{n}-})}}{Z_{t_{n}-}^{1/\alpha}}},$
where $\\{(t_{n},r_{n}^{\uparrow},\nu_{n}^{\uparrow}):n\in\mathbb{N}\\}$ are
the atoms of $N^{\uparrow}$. We claim that, under $\mathbb{P}^{\uparrow}$,
$\\{(t_{n},\theta_{n}):n\in\mathbb{N}\\}$ are atoms of a Poisson random
measure $N^{\prime}$ with intensity $ds\times\Pi(du)$. Indeed, for any bounded
non-negative predictable process $H$, and any positive bounded function $f$
vanishing at zero,
$M_{t}:=\sum_{t_{n}\leq
t}H_{t_{n}}f(\theta_{n})-\int_{0}^{t}H_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f\left(\frac{r}{Z_{s}^{1/\alpha}}\right)\textbf{1}_{(\nu\leq
Z_{s})}d\nu\Pi(dr)$
is a martingale. If we change variables, the particular form of $\Pi$ implies
that
$M_{t}=\sum_{t_{n}\leq
t}H_{s}f(\theta_{n})-\int_{0}^{t}H_{s}ds\int_{0}^{\infty}f(u)\Pi(du).$
Taking expectations, our claim follows thanks to Lemma 4.2 below. Since
$\sum_{t_{n}\leq t}r^{\uparrow}_{n}\textbf{1}_{(\nu^{\uparrow}_{n}\leq
Z_{t_{n}-})}=\sum_{t_{n}\leq t}Z^{1/\alpha}_{t_{n}-}\theta_{n},$ we can
rewrite (6) as
$Z_{t}=\int_{0}^{t}\int_{1}^{\infty}Z^{1/\alpha}_{s-}uN^{\prime}(ds,du)+\int_{0}^{t}\int_{0}^{1}Z^{1/\alpha}_{s-}u\tilde{N}^{\prime}(ds,du)+\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr).$
Defining
$X_{t}:=\int_{0}^{t}\int_{1}^{\infty}uN^{\prime}(ds,du)+\int_{0}^{t}\int_{0}^{1}u\tilde{N}^{\prime}(ds,du),$
by the Lévy-Ito decomposition it is easy to see that $X$ is an $\alpha$-stable
Lévy process with characteristic exponent $\psi$. Similarly,
$S_{t}:=\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr)$
is seen to be an $(\alpha-1)$-stable subordinator. Independence of $X$ and $S$
is granted by construction, because the two processes do not have simultaneous
jumps. Thus, we have
$dZ_{t}=Z^{1/\alpha}_{t}dX_{t}+dS_{t},$
which corresponds to Lambert’s result.
### 3.2 CSBP flows as SDE solutions
A family of CSBP processes $Z=\\{Z_{t}(a):t\geq 0,a\geq 0\\}$ allowing the
initial population size $Z_{0}(a)=a$ to vary, can be constructed
simultaneously as a two parameter process or stochastic flow satisfying the
branching property. This was done by Bertoin and Le-Gall [1] by using families
of subordinators. In [2], [3] they later used Poisson measure driven SDE to
formulate such type of flows in related contexts, including equations close to
(3). In the same line, Dawson and Li [6] proved the existence of strong
solutions for stochastic flows of continuous-state branching processes with
immigration, as SDE families driven by white noise processes and Poisson
random measures with joint regularity properties. The stochastic equations
they study (in particular equation (1.5) ) are close to equation (5), the main
difference being the immigration behavior which in their case only covers
linear drifts. For simplicity reasons Theorem 2.1 is presented in the case of
a Brownian motion and Poisson measure driven SDE, but our arguments can be
extended to the white-noise and Poisson measure driven stochastic flow
considered in [6] (in absence of immigration).
## 4 Proof of the main theorem
In [17], a suitable marking of Poisson point processes was used to firstly
construct a stable Lévy process, conditioned to stay positive, out of the
realization of the unconditioned one. After time-changing the author takes
advantage of the scaling property of $\alpha$-stable processes to derive an
SDE for the branching process. Our proof is inspired in his marking argument
but in turn it is carried out directly in the time scale of the CSBP. We will
need the following version of Girsanov’s theorem (c.f. Theorem 37 in Chapter
III.8 of [24]):
###### Theorem 4.1
Let $(\varOmega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ be a filtered
probability space, and let $M$ be a $\mathbb{P}$-local martingale with
$M_{0}=0$. Let $\mathbb{P}^{\star}$ be another probability measure absolutely
continuous with respect to $\mathbb{P}$, and let
$D_{t}=\mathbb{E}(\frac{d\mathbb{P}^{\star}}{d\mathbb{P}}|\mathcal{F}_{t})$.
Assume that $\langle M,D\rangle$ exists for $\mathbb{P}$. Then
$A_{t}=\int_{0}^{t}\frac{1}{Z_{s^{-}}}d\langle M,Z\rangle_{s}$ exists a.s. for
the probability $\mathbb{P}^{\star}$, and $M_{t}-A_{t}$ is a
$\mathbb{P}^{\star}$-local martingale.
The following well-known characterization of Poisson point processes will also
be useful:
###### Lemma 4.2
Let $(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ be a filtered
probability space, $(S,\mathcal{S},\eta)$ an arbitrary $\sigma$-finite measure
space, and $\\{(t_{n},\delta_{n})\in\mathbb{R_{+}}\times S\\}$ a countable
family of random variables such that $\\{t_{n}\leq t,\delta_{n}\in
A\\}\in{\cal F}_{t}$ for all $n\in\mathbb{N}$, $t\geq 0$ and $A\in{\cal S}$,
and moreover
$\mathbb{E}\sum\limits_{n:t_{n}\leq
t}F_{t_{n}}g(\delta_{n})=\mathbb{E}\int\limits_{0}^{t}F_{s}ds\int\limits_{S}g(x)m(dx)$
(7)
for any nonnegative predictable process $F_{s}$ and any nonnegative function
$g:S\rightarrow\mathrm{I\\!R}$. Then, $(t_{n},\delta_{n})_{n\in\mathbb{N}}$
are the atoms of a Poisson random measure $N$ on $\mathbb{R_{+}\times S}$ with
intensity $dt\times m(dx)$.
###### Proof.
Writing
$\displaystyle{e^{\left\\{\sum\limits_{t_{n}\leq
t}f(\delta_{n}))\right\\}}}=\sum\limits_{n:t_{n}\leq
t}\left[\prod\limits_{k:t_{k}<t_{n}}e^{f(\delta_{k})}\right](e^{f(\delta_{n})}-1)=\sum\limits_{n:t_{n}\leq
t}\left[e^{\sum\limits_{k:t_{k}\leq
s}f(\delta_{k})}\right](e^{f(\delta_{n})}-1)$
we get from (7) that
$\mathbb{E}\left[e^{\sum\limits_{n:t_{n}\leq
t}f(\delta_{n})}\right]=\int\limits_{0}^{t}\mathbb{E}\left[e^{\sum\limits_{k:t_{k}\leq
s}f(\delta_{k})}\right]ds\int_{S}(e^{f(x)}-1)m(dx)$
since $F_{s}:=\prod\limits_{t_{k}<s}e^{f(\delta_{k})}$ is a predictable
process. Solving this differential equation yields
$\mathbb{E}\left[e^{\sum\limits_{t_{n}\leq
t}f(\delta_{n})}\right]=e^{-t\int\limits_{S}(1-e^{f(x)})m(dx)},$
and the statement follows by Campbell’s formula (see e.g. [14]) ∎
###### Proof of Theorem 3.1.
We will prove that under the laws $\mathbb{P}_{x}^{\uparrow}$ the process $Z$
in equation (3) is a weak solution of (5). Pathwise uniqueness, which then
classically implies also strong existence, can be shown as in [10].
We write $B=B^{Z}$ and $N=N^{Z}$, and we denote by $\\{\mathcal{F}_{t}\\}$ the
filtration
$\mathcal{F}_{t}:=\sigma(B_{s},(r_{n},\nu_{n})\textbf{1}_{t_{n}\leq
s};n\in\mathbb{N},s\leq t),$
where
$\\{(t_{n},r_{n},\nu_{n})\in[0,\infty)\times(0,\infty)\times(0,\infty)\\}_{n\in\mathbb{N}}$
are the atoms of the Poisson point process $N$. We will use the absolute
continuity of $\mathbb{P}^{\uparrow}$ w.r.t. $\mathbb{P}$ and the Radon-
Nikodym density $D_{t}=\frac{e^{\rho t}Z_{t}}{x}$ applying the previous
theorem to the process $\\{B_{t}:t\geq 0\\}$ and, indirectly, to the Poisson
random measure $N$ and its compensated measure.
Dealing with the diffusion part is standard since $d\langle
D,B\rangle_{t}=\frac{e^{\rho t}}{x}\sigma\sqrt{Z_{t}}dt,$ so that
$B_{t}^{\uparrow}:=B_{t}-\int_{0}^{t}\frac{d\langle
D,B\rangle_{s}}{D_{s}}=B_{t}-\sigma\int_{0}^{t}Z_{s}^{-\frac{1}{2}}ds$
is a Brownian motion under $\mathbb{P}^{\uparrow}$ by Girsanov theorem.
We next study the way the Poisson random measure $N$ is affected by the change
of probability, which is the main part of the proof. Enlarging the probability
space and filtration if needed, we may and shall assume that there is a
sequence $(u_{n})_{n\geq 1}$ of independent random variables uniformly
distributed on $[0,1]$, independent of $B$ and $N$ and such that
$u_{n}\textbf{1}_{t_{n}\leq t}$ is $\mathcal{F}_{t}$-measurable. Define random
variables $(\Delta_{n},\delta_{n})\in[0,\infty)^{2}\times[0,\infty)$ by
$(\Delta_{n},\delta_{n}):=\left\\{\begin{array}[]{lcl}((0,0),r_{n}\textbf{1}_{(\nu_{n}\leq
Z_{t_{n}^{-}})})&\mbox{ if
}&\displaystyle{u_{n}>\frac{D_{t_{n}-}}{D_{t_{n}}}\,=\frac{Z_{t_{n}-}}{Z_{t_{n}}}}\mbox{
and }Z_{t_{n}}>0,\\\ ((r_{n},\nu_{n}),0)&\mbox{ if
}&\displaystyle{u_{n}\leq\frac{D_{t_{n}-}}{D_{t_{n}}}}\mbox{ and
}Z_{t_{n}}>0,\\\ ((0,0),0)&\mbox{ if }&Z_{t_{n}}=0.\\\ \end{array}\right.$
Let $f_{R,\epsilon}$ be a nonnegative function such that for all $(r,\nu,s)$
* -
$f_{R,\epsilon}((r,\nu),s)=0$ when $\nu\geq R$, for some fixed $R\geq 0$,
* -
$f_{R,\epsilon}((r,\nu),s)=0$ when $r<\epsilon$, for some fixed
$0<\epsilon\leq 1$, and
* -
$f_{R,\epsilon}((0,0),0)=0$.
For any non-negative predictable process $F$, we have
$\begin{array}[]{lcl}\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((0,0),r_{n}\textbf{1}_{\\{\nu\leq
Z_{t_{n}-}\\}})\textbf{1}_{\\{u_{n}>\frac{Z_{t_{n}-}}{Z_{t_{n}}}\\}}\\\ \\\
&&+\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((r_{n},\nu_{n}),0)\textbf{1}_{\\{u_{n}\leq\frac{Z_{t_{n}-}}{Z_{t_{n}}}\\}}.\end{array}$
Therefore, since
$1-\frac{Z_{t_{n}-}}{Z_{t_{n}}}=\frac{r_{n}\textbf{1}_{\\{\nu_{n}\leq
Z_{t_{n}-}\\}}}{Z_{t_{n}}}$, the process
$\begin{array}[]{lcl}S_{t}&:=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})-\int_{0}^{t}dsF_{s}\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((0,0),r\textbf{1}_{\\{\nu\leq
Z_{s}\\}})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}+r\textbf{1}_{(\nu\leq Z_{s})}}}\Pi(dr)d\nu\\\ \\\
&&-\int_{0}^{t}dsF_{s}\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((r,\nu),0)\displaystyle{\frac{Z_{s}}{Z_{s}+r\textbf{1}_{(\nu\leq
Z_{s})}}}\Pi(dr)d\nu\end{array}$
is a martingale under $\mathbb{P}$. The quadratic covariation of $S$ and $D$
is given by
$\begin{array}[]{lcl}[S,D]_{t}&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})\frac{e^{\rho
t_{n}}}{x}r_{n}\textbf{1}_{(\nu_{n}\leq Z_{t_{n}-})}\\\ \\\
&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((0,0),r_{n}\textbf{1}_{\\{\nu\leq
Z_{t_{n}-}\\}})\frac{e^{\rho t_{n}}}{x}r_{n}\textbf{1}_{\\{\nu_{n}\leq
Z_{t_{n}-}\\}}\textbf{1}_{\left\\{u_{n}>\frac{Z_{t_{n}-}}{Z_{t_{n}}}\right\\}}\\\
\\\ &&+\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((r_{n},\nu_{n}),0)\frac{e^{\rho
t_{n}}}{x}r_{n}\textbf{1}_{\\{\nu_{n}\leq
Z_{t_{n}-}\\}}\textbf{1}_{\left\\{u_{n}\leq\frac{Z_{t_{n}-}}{Z_{t_{n}}}\right\\}}\,,\par\end{array}$
since $S$ is a jump process. Thus, the conditional quadratic covariation is
$\begin{array}[]{lcl}\langle D,S\rangle_{t}&=&\int_{0}^{t}\frac{e^{\rho
s}}{x}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((0,0),r\textbf{1}_{\\{\nu\leq
Z_{s}\\}})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}+r\textbf{1}_{(\nu\leq Z_{s})}}r}d\Pi(dr)d\nu\\\ \\\
&&+\int_{0}^{t}\frac{e^{\rho
s}}{x}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((r,\nu),0)\displaystyle{\frac{Z_{s}}{Z_{s}+r\textbf{1}_{(\nu\leq
Z_{s})}}r\textbf{1}_{(\nu\leq Z_{s})}}d\Pi(dr)d\nu.\end{array}$
Then, using Girsanov’s theorem, we see that the process
$\begin{array}[]{lcl}S_{t}^{\uparrow}:&=&S_{t}-\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\infty}F_{s}f_{R,\epsilon}((0,0),r\textbf{1}_{(\nu\leq
Z_{s})})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}+r\textbf{1}_{(\nu\leq Z_{s})}}\frac{r}{Z_{s}}}\Pi(dr)d\nu ds\\\
\\\
&&-\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\infty}F_{s}f_{R,\epsilon}((r,\nu),0)\displaystyle{\frac{Z_{s}}{Z_{s}+r\textbf{1}_{(\nu\leq
Z_{s})}}\frac{r\textbf{1}_{(\nu\leq Z_{s})}}{Z_{s}}}\Pi(dr)d\nu ds\end{array}$
is a $(\mathcal{F}_{t})$-martingale under $\mathbb{P}^{\uparrow}$. By the
definition of $S$,
$\begin{array}[]{lcl}S_{t}^{\uparrow}&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})-\int_{0}^{t}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}\left[f_{R,\epsilon}((0,0),r\textbf{1}_{(\nu\leq
Z_{s})})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}}}+f_{R,\epsilon}((r,\nu),0)\right]\Pi(dr)d\nu\\\ \\\
&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})-\int_{0}^{t}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}\left[f_{R,\epsilon}((0,0),r)\displaystyle{\frac{r}{Z_{s}}\textbf{1}_{(\nu\leq
Z_{s})}}+f_{R,\epsilon}((r,\nu),0)\right]\Pi(dr)d\nu\\\ \\\ \end{array}$
since $f_{R,\epsilon}((0,0),0)=0$. Recalling that $S^{\uparrow}$ is a
$(\mathcal{F}_{t})$-martingale on $\mathbb{P}^{\uparrow}$ starting from $0$,
we deduce that
$\begin{array}[]{lcl}\mathbb{E}^{\uparrow}\left[\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})\right]&=&\mathbb{E}^{\uparrow}\left[\int_{0}^{t}F_{s}ds\int_{0}^{\infty}f_{R,\epsilon}((0,0),r)r\Pi(dr)\right]\\\
\\\
&&+\mathbb{E}^{\uparrow}\left[\int_{0}^{t}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((r,\nu),0)\Pi(dr)d\nu\right].\end{array}$
By standard arguments, this formula is also true for any nonnegative function
$f$ such that $f((0,0),0)=0$. By Lemma 4.2 we see that, under
$\mathbb{P}^{\uparrow}$, $(t_{n},\Delta_{n})_{n\geq 0}$ and
$(t_{n},\delta_{t})_{n\geq 0}$ are atoms of two Poisson point processes
$N^{\uparrow}$ and $N^{\star}$ with intensity measures $dt\times
d\nu\times\Pi(dr)$ and $dt\times r\Pi(dr)$ on
$[0,\infty)\times(0,\infty)\times(0,\infty)$ and $[0,\infty)\times(0,\infty)$
respectively. By construction, $N^{\uparrow}$ and $N^{\star}$ are independent
because they never jump simultaneously. Now set
$J_{t}:=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN(ds,d\nu,dr)=\sum\limits_{t_{n}\leq
t}r_{n}\textbf{1}_{(\nu_{n}\leq Z_{t_{n}^{-}})}\textbf{1}_{(r_{n}\geq 1)}.$
From above, we have
$J_{t}=\sum_{t_{n}\leq t}\Delta_{n}^{(1)}\textbf{1}_{(\Delta^{(2)}_{n}\leq
Z_{t_{n}^{-}})}\textbf{1}_{(\Delta_{n}^{(1)}\geq 1)}+\sum_{t_{n}\leq
t}\delta_{n}\textbf{1}_{(\delta_{n}\geq 1)},$
where $\Delta_{n}^{(i)}$ is the $i-$th coordinate of $\Delta_{n}$, $i=1,2$.
Therefore
$J(t)=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{1}^{\infty}rN^{\star}(ds,dr).$
Finally, given $0<\varepsilon<1$, let $\\{\tilde{M}_{t}^{(\varepsilon)},t\geq
0\\}$ be the $\mathbb{P}$-martingale
$\begin{array}[]{lcl}\tilde{M}_{t}^{(\varepsilon)}&:=&\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{\varepsilon}^{1}rN^{Z}(ds,d\nu,dr)-\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{\varepsilon}^{1}r\,dsd\nu\Pi(dr)\\\
\\\ &=&\sum\limits_{t_{n}\leq t}r_{n}\textbf{1}_{(\nu_{n}\leq
Z_{t_{n}^{-}})}\textbf{1}_{(\varepsilon<r_{n}<1)}-\int_{0}^{t}\int_{0}^{Z_{s}}\int_{\varepsilon}^{1}r\,dsd\nu\Pi(dr),\end{array}$
which converges in the $L^{2}(\mathbb{P})$ sense when $\varepsilon\rightarrow
0$ to
$\tilde{M}_{t}:=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{Z}(ds,d\nu,dr).$
In terms of $(\Delta_{n})$ and $(\delta_{n})$, we can write
$\begin{array}[]{lcl}\tilde{M}^{(\varepsilon)}&=&\left(\sum\limits_{t_{n}\leq
t}\Delta^{(1)}_{n}\textbf{1}_{(\Delta^{(2)}_{n}\leq
Z_{t_{n}^{-}})}\textbf{1}_{(\varepsilon<\Delta^{(1)}_{n}<1)}-\int_{0}^{t}\int_{0}^{Z_{s}}\int_{\varepsilon}^{1}rdsd\nu\Pi(dr)\right)\\\
\\\ &&+\sum\limits_{t_{n}\leq
t}\delta_{n}\textbf{1}_{(\varepsilon<\delta_{n}<1)}\\\ \\\
&=&\left(\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{\varepsilon}^{1}rN^{\uparrow}(ds,d\nu,dr)-\int_{0}^{t}\int_{0}^{Z_{s}}\int_{\varepsilon}^{1}rdsd\nu\Pi(dr)\right)\\\
\\\ &&+\int_{0}^{t}\int_{\varepsilon}^{1}rN^{\star}(ds,dr).\end{array}$
Thanks to [16, Theorem 2.10], the limit as $\varepsilon\to 0$ in the
$L^{2}(\mathbb{P}^{\uparrow})$ sense of the $\mathbb{P}^{\uparrow}$-martingale
given by the first term on the right hand side exists, and it is equal to the
martingale
$\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)$,
where $\tilde{N}^{\uparrow}$ is the compensated measure associated with
$N^{\uparrow}$. Also, as $\int_{0}^{\infty}(1\wedge x^{2})\Pi(dx)<\infty$, by
[16, Theorem 2.9] the second term on the right hand side converges
$\mathbb{P}^{\uparrow}$-a.s., so we have
$\tilde{M}_{t}=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{1}rN^{\star}(ds,dr).$
Bringing all parts together, we have shown that $Z$ satisfies under
$\mathbb{P}^{\uparrow}$ the desired SDE, except for the independence of the
processes $B^{\uparrow}$ and $(N^{\uparrow},N^{\star}$), which we shall
establish in what follows.
Since $N^{\uparrow}$ and $N^{\star}$ have $\sigma$-finite intensities and
thanks to the Markov property of the three processes with respect to the
filtration $(\mathcal{F}_{t})$, it is enough to show that for every $t>s\geq
0$, $\zeta\in\mathbb{R}$, $\lambda_{k},\gamma_{k}\in\mathbb{R}_{+}$,
$k\in\\{1,...,m\\}$, and $m\in\mathbb{N}$
$\begin{array}[]{lcl}\mathbb{E}^{\uparrow}\left[\displaystyle{e^{-\zeta(B^{\uparrow}_{t}-B^{\uparrow}_{s})}e^{-\sum\limits_{k=1}^{m}\lambda_{k}N^{\uparrow}((s,t]\times
W_{k})}e^{-\sum\limits_{k=1}^{m}\gamma_{k}N^{\star}((s,t]\times
V_{k})}}\bigg{|}\mathcal{F}_{s}\right]\\\ \\\
=\displaystyle{e^{-\frac{\zeta^{2}}{2}(t-s)}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{W_{k}}(e^{-\lambda_{k}}-1)\Pi(dr)d\nu
du}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{V_{k}}(e^{-\gamma_{k}}-1)r\Pi(dr)du}},\end{array}$
where $\\{W_{k}\\}_{k=1}^{m}$ and $\\{V_{k}\\}_{k=1}^{m}$ are disjoint sets of
$(0,\infty)\times(0,\infty)$ and $(0,\infty)$ such that
$\int_{0}^{t}\int_{W_{k}}\Pi(dr)d\nu du$ and
$\int_{0}^{t}\int_{V_{K}}r\Pi(dr)du$ are finite. To that end, set
$F(x,y_{1},..,y_{m},z_{1},..,z_{m}):=e^{-\zeta
x}e^{-\sum_{k=1}^{m}\lambda_{k}y_{k}}e^{-\sum_{k=1}^{m}\gamma_{k}z_{k}}.$
Applying Itô’s formula to the semimartingale
$X(t)=\left(B^{\uparrow}(t),N^{\uparrow}((0,t]\times
W_{1}),..,N^{\uparrow}((0,t]\times W_{m}),N^{\star}((0,t]\times
V_{1}),..,N^{\star}((0,t]\times V_{m})\right),$
we obtain:
$\begin{array}[]{lcl}F(X(t))&=&F(X(s))-\int_{s}^{t}\zeta
F(X(u))dB^{\uparrow}_{u}-\sum\limits_{j=1}^{m}\int_{s}^{t}\int_{W_{j}}\lambda_{j}F(X(u))N^{\uparrow}(du,d\nu,dr)\\\
\\\
&&-\sum\limits_{j=1}^{m}\int_{s}^{t}\int_{V_{j}}\gamma_{j}F(X(u))N^{\star}(du,dr)+\frac{\zeta^{2}}{2}\int_{s}^{t}F(X(u))du+\sum\limits_{s<u\leq
t}F(X(u))-F(X(u^{-}))\\\ \\\ &&+\sum\limits_{s<t_{n}\leq
t}\sum\limits_{j=1}^{m}\left[\lambda_{j}F(X(t_{n}))\textbf{1}_{\\{\Delta_{n}\in
W_{j}\\}}+\gamma_{j}F(X(t_{n}))\textbf{1}_{\\{\delta_{n}\in
V_{j}\\}}\right].\end{array}$
From above, we deduce that
$\begin{array}[]{lcl}F(X(t))-F(X(s))&=&\bar{M}_{t}-\bar{M}_{s}+\frac{\zeta^{2}}{2}\int_{s}^{t}F(X(u))du\\\
\\\ &&+\sum\limits_{s<t_{n}\leq
t}\left[F\left(X(t_{n}-)+(0,\textbf{1}_{\\{\Delta_{n}\in
W_{1}\\}},..,\textbf{1}_{\\{\delta_{n}\in
V_{m}\\}})\right)-F(X(t_{n}-))\right],\end{array}$
where $(\bar{M}_{t})$ is a $(\mathcal{F}_{t})$-martingale. Defining
$f(\Delta_{n},\delta_{n}):=e^{-\sum\limits_{k=1}^{m}\lambda_{k}\textbf{1}_{\\{\Delta_{n}\in
W_{k}\\}}-\sum\limits_{k=1}^{m}\lambda_{k}\textbf{1}_{\\{\delta_{n}\in
V_{k}\\}}}-1,$ we have
$F(X(t))-F(X(s))=\bar{M}_{t}-\bar{M}_{s}+\frac{\zeta^{2}}{2}\int_{s}^{t}F(X(u))du+\sum\limits_{s<t_{n}\leq
t}F(X(t_{n}-))[f(\Delta_{n},\delta_{n})].$
Let now $A\in\mathcal{F}_{s}$. Multiplying both sides by
$F(-(X(s)))\textbf{1}_{A}$, yields:
$\begin{array}[]{lcl}\mathbb{E}^{\uparrow}[F(X(t-s))\textbf{1}_{A}]-\mathbb{P}^{\uparrow}(A)&=&\frac{\zeta^{2}}{2}\int_{s}^{t}\mathbb{E}^{\uparrow}\left[F(X(u-s))\textbf{1}_{A}\right]du\\\
\\\
&&+\int_{s}^{t}\mathbb{E}^{\uparrow}\left[F(X(u-s))\textbf{1}_{A}\right]du\sum\limits_{k=1}^{m}\int_{W_{k}}(e^{-\lambda_{k}}-1)\Pi(dr)d\nu\\\
\\\
&&+\int_{s}^{t}\mathbb{E}^{\uparrow}\left[F(X(u-s))\textbf{1}_{A}\right]du\sum\limits_{k=1}^{m}\int_{V_{k}}(e^{-\gamma_{k}}-1)r\Pi(dr).\par\end{array}$
Thus,
$\mathbb{E}^{\uparrow}\left[F(X(t-s))\textbf{1}_{A}\right]=\mathbb{P}^{\uparrow}(A)e^{-\frac{\zeta^{2}}{2}(s-t)}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{W_{k}}(e^{-\lambda_{k}}-1)\Pi(dr)d\nu
du}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{V_{k}}(e^{-\gamma_{k}}-1)r\Pi(dr)du}$
which means that the three processes are mutually independent, which ends the
proof of weak existence.
As concerns pathwise uniqueness, we just remark that the proof of Theorem 3.2
in [10] covers the case of equation (5). Indeed, if $B^{\uparrow}$,
$N^{\uparrow}$ and $N^{\star}$ are independent processes as before driving two
solutions $\\{Z_{t}^{(1)}\\}$ and $\\{Z_{t}^{(2)}\\}$ of (5), setting
$\zeta_{t}:=Z_{t}^{(1)}-Z_{t}^{(2)}$ one gets that
$\begin{array}[]{lcl}\zeta_{t}&=&\zeta_{0}+\int_{0}^{t}a\left(Z_{s}^{(1)}-Z_{s}^{(2)}\right)ds+\int_{0}^{t}\sigma\left(\sqrt{Z_{s}^{(1)}}-\sqrt{Z_{s}^{(2)}}\right)dB^{\uparrow}_{s}\\\
\\\
&&+\int_{0}^{t}\int_{U_{0}}r\left(\textbf{1}_{(\nu<Z_{s}^{(1)})}-\textbf{1}_{(\nu<Z_{s}^{(2)})}\right)N^{\uparrow}(ds,d\nu,dr)\\\
\\\
&&+\int_{0}^{t}\int_{U_{1}}r\left(\textbf{1}_{(\nu<Z_{s}^{(1)})}-\textbf{1}_{(\nu<Z_{s}^{(2)})}\right)\tilde{N}^{\uparrow}(ds,d\nu,dr),\end{array}$
(8)
where $U_{0}=[0,\infty)\times[1,\infty)$ and $U_{1}=[0,\infty)\times(0,1)$.
From this point on, the proof of Theorem 3.2 in [10] applies, since conditions
(2.a,b) and (3.a,b) therein are satisfied. Indeed, in their notations, we have
the intensity measure $\mu(du)=\Pi(dr)d\nu$ for $N^{0}=N^{\uparrow}|_{U_{0}}$
and $N^{1}=N^{\uparrow}|_{U_{1}}$ (where $u=(r,\nu)$), continuous functions on
$\mathbb{R}$ given by $b(x):=ax\textbf{1}_{0\leq x}$ and
$\sigma(x):=\sigma\sqrt{x}\textbf{1}_{0\leq x}$, and Borel functions on
$\mathbb{R}\times U_{i}$, $i=\\{0,1\\}$ given by
$g(x,u)=g_{0}(x,u)=g_{1}(x,u)=r\textbf{1}_{\nu<x}$ such that $g(x,u)+x\geq 0$
for $x>0$ and $g(x,u)=0$ for $x\leq 0$. Moreover,
1. 1.
there is a constant $K:=|a|+M\geq 0$ , where
$\int_{1}^{\infty}r\Pi(dr)=M<\infty$, such that
$|ax|+\int_{0}^{\infty}\int_{1}^{\infty}r\textbf{1}_{\nu<x}\Pi(dr)d\nu\leq
K(x+1)\,;$
2. 2.
there is a non-negative and non-decreasing function $L(x)=(\sigma^{2}+I)(x)$
on $\mathbb{R}_{+}$, with $I=\int_{0}^{1}r^{2}\Pi(dr)$, so that
$\sigma^{2}x+\int_{0}^{\infty}\int_{0}^{1}r^{2}\textbf{1}_{\nu<x}\Pi(dr)d\nu\leq
L(x);$
3. 3.
there is a continuous non-decreasing function $x\rightarrow b_{2}(x):=x$ on
$\mathbb{R}_{+}$ such that for $b_{1}(x)=b(x)+b_{2}(x)$, on has
$|(a+1)(b_{1}(x)-b_{1}(y))|+\int_{0}^{\infty}\int_{1}^{\infty}r\textbf{1}_{y<\nu<x}\Pi(dr)d\nu\leq
r(|x-y|)\,;$
where $r$ is the non-decreasing and concave function $r(z)=:(|a+1|+M)z$ on
$\mathbb{R}_{+}$ satisfying $\int_{0_{+}}r(z)^{-1}dz=\infty$; and
4. 4.
for every fixed $u\in U_{0}$ the function $x\rightarrow g(x,u)$ is non-
decreasing, and there is a non-negative and non-decreasing function
$\rho(z):=[\sigma^{2}+I]\sqrt{z}$ on $\mathbb{R}_{+}$ so that
$\int_{0_{+}}\rho(z)^{-2}dz=\infty$ and
$(\sigma\sqrt{x}-\sigma\sqrt{y})^{2}+\int_{0}^{\infty}\int_{0}^{1}r^{2}\textbf{1}_{y<\nu<x}\Pi(dr)d\nu\leq\rho(|x−y|)^{2}.$
Conditions 1,2,3 and 4 respectively ensure that hypotheses (2.a,b) and (3.a,b)
in [10] hold, and pathwise uniqueness follows. ∎
Acknowledgements We would like to thank Julien Berestycki for pointing out to
us relevant references and for several remarks that helped us to improve
earlier versions of this work.
## References
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* [3] J. BERTOIN, J.-F. LE GALL (2006), Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math.50, no. 1-4, 147-181 (electronic).
* [4] M. E. CABALLERO, A. LAMBERT, and G. URIBE BRAVO (2009), Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surveys, Volume 6, 62-89.
* [5] D. A. DAWSON, Z. LI (2006), Skew convolution semigroups and affine Markov processes. Ann. Probab., 34, no. 3, 1103-1142.
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* [7] P. DONNELL, T. G. KURTZ (1999), Particle representations for measure-valued population models. Ann. Probab., 27 (1):166-205.
* [8] J. ENGLÄNDER, A. E. KYPRIANOU (2004), Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32, no. 1A, 78-99.
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|
arxiv-papers
| 2012-04-27T00:37:10 |
2024-09-04T02:49:30.260789
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.C. Fittipaldi (1), J. Fontbona (2) ((1) DIM-CMM, UMI 2807\n UChile-CNRS, Universidad de Chile, Santiago, Chile, (2) DIM-CMM, UMI 2807\n UChile-CNRS, Universidad de Chile, Santiago, Chile)",
"submitter": "Mar\\'ia Clara Fittipaldi",
"url": "https://arxiv.org/abs/1204.6092"
}
|
1204.6152
|
Further analysis on the total number of subtrees of trees***Financially
supported by the National Natural Science Foundation of China (Grant No.
11071096) and the Special Fund for Basic Scientific Research of Central
Colleges (CCNU11A02015).
Shuchao Li†††E-mail: lscmath@mail.ccnu.edu.cn (S.C. Li), wang06021@126.com
(S.J. Wang), Shujing Wang
Faculty of Mathematics and Statistics, Central China Normal University, Wuhan
430079, P.R. China
Abstract: We study that over some types of trees with a given number of
vertices, which trees minimize or maximize the total number of subtrees. Trees
minimizing (resp. maximizing) the total number of subtrees usually maximize
(resp. minimize) the Wiener index, and vice versa. Here are some of our
results: (1) Let $\mathscr{T}_{n}^{k}$ be the set of all $n$-vertex trees with
$k$ leaves, we determine the maximum (resp. minimum) value of the total number
of subtrees of trees among $\mathscr{T}_{n}^{k}$ and characterize the extremal
graphs. (2) Let $\mathscr{P}_{n}^{p,q}$ be the set of all $n$-vertex trees,
each of which has a $(p,q)$-bipartition, we determine the maximum (resp.
minimum) value of the total number of subtrees of trees among
$\mathscr{P}_{n}^{p,q}$ and characterize the extremal graphs. (3) Let
$\mathscr{A}_{n}^{q}$ be the set of all $q$-ary trees with $n$ non-leaf
vertices, we determine the minimum value of the total number of subtrees of
trees among $\mathscr{A}_{n}^{q}$ and identify the extremal graph.
Keywords: Subtrees; Leaves; Diameter; Bipartition; Wiener index; $q$-ary tree
AMS subject classification: 05C05, 05C10
## 1 Introduction
We consider only simple connected graphs (i.e. finite, undirected graphs
without loops or multiple edges). Let $G=(V_{G},E_{G})$ be a graph with
$u,v\in V_{G}$, $d_{G}(u)$ (or $d(u)$ for short) denotes the degree of $u$;
the distance $d_{G}(u,v)$ is defined as the length of the shortest path
between $u$ and $v$ in $G$; $D_{G}(v)$ denotes the sum of all distances from
$v$. The eccentricity $\varepsilon(v)$ of a vertex $v$ is the maximum distance
from $v$ to any other vertex.
Throughout the text we denote by $P_{n},\,K_{1,n-1}$ the path and star on $n$
vertices, respectively. $G-v,\,G-uv$ denote the graph obtained from $G$ by
deleting vertex $v\in V_{G}$, or edge $uv\in E_{G}$, respectively (this
notation is naturally extended if more than one vertex or edge is deleted).
Similarly, $G+uv$ is obtained from $G$ by adding vertex edge $uv\not\in
E_{G}$. For $v\in V_{G},$ let $N_{G}(v)$ (or $N(v)$ for short) denote the set
of all the adjacent vertices of $v$ in $G.$ The diameter diam$(G)$ of a graph
is the maximum eccentricity of any vertex in the graph. We refer to vertices
of degree 1 of a tree $T$ as leaves (or pendant vertices), and the edges
incident to leaves are called pendant edges. The unique path connecting two
vertices $v,u$ in $T$ will be denoted by $P_{T}(v,u)$. For a tree $T$ and two
vertices $v,u$ of $T$, the distance $d_{T}(v,u)$ between them counts the
number of edges on the path $P_{T}(v,u)$.
Let
$W(T)=\frac{1}{2}\sum_{v\in V_{T}}D_{T}(v)$
denote the Wiener index of $T,$ which is the sum of distances of all unordered
pairs of vertices. This topological index was introduced by Wiener [15], which
has been one of the most widely used descriptors in quantitative structure-
activity relationships. Since the majority of the chemical applications of the
Wiener index deal with chemical compounds with acyclic molecular graphs, the
Wiener index of trees has been extensively studied over the past years; see
[1, 2, 3, 4, 6] and the references there for details.
Given a tree $T$, a subtree of $T$ is just a connected induced subgraph of
$T$. The number of subtrees as well as related subjects has been studied. Let
$T$ denote a tree with $n$ nodes each of whose non-pendant vertices has degree
at least three, Andrew and Wang [12] showed that the average number of nodes
in the subtrees of $T$ is at least $\frac{n}{2}$ and strictly less than
$\frac{3n}{4}$. Székely and Wang [8] characterized the binary trees with $n$
leaves that have the greatest number of subtrees. Kirk and Wang [5] identified
the tree, for a given size and such that the vertex degree is bounded, having
the greatest number of subtrees. Székely and Wang [11] gave a formula for the
maximal number of subtrees a binary tree can possess over a given number of
vertices. They also show that caterpillar trees (trees containing a path such
that each vertex not belonging to the path is adjacent to a vertex on the
path) have the smallest number of subtrees among binary trees. Yan and Ye [16]
characterized the tree with the diameter at least $d$, which has the maximum
number of subtrees, and they characterized the tree with the maximum degree at
least $\Delta$, which has the minimum number of subtrees. For some related
results on the enumeration of subtrees of trees, one may also see Székely and
Wang [9, 10] and Wang [14]. Consider the collection of rooted labeled trees
with $n$ vertices, Song [7] derived a closed formula for the number of these
trees in which the child of the root with the smallest label has a total of
$p$ descendants. He also derived a recurrence relation for the number of these
trees with the property that for each non-terminal vertex $v$, the child of
$v$ with the smallest label has no descendants.
It is well known that the Wiener index is maximized by the path and minimized
by the star among general trees with the same number of vertices. It is also
known that the counterparts of these simple results for the number of subtrees
do exist. In fact, it is interesting that the Wiener index and the total
number of subtrees of a tree share exactly the same extremal structure (i.e.
the tree that maximizes/minimizes the corresponding index) among trees with a
given number of vertices and maximum degree, although the values of the
indices are in no general functional correspondence. On the other hand, an
acyclic molecule can be expressed by a tree in quantum chemistry (see [2]).
Obviously, the number of subtrees of a tree can be regarded as a topological
index. Hence, Yan and Ye [16] pointed out that to explore the role of the
total number of subtrees in quantum chemistry is an interesting topic.
Motivated by the work of [5, 8, 11, 12, 16], in this paper we continue to
study some types of trees which minimize or maximize the total number of
subtrees.
Let $\mathscr{T}_{n}^{k}$ be the set of all $n$-vertex trees with $k$ leaves
($2\leq k\leq n-1$). A spider is a tree with at most one vertex of degree more
than 2, called the center of the spider (if no vertex of degree more than two,
then any vertex can be the center). A leg of a spider is a path from the
center to a vertex of degree 1. Let $T_{n}^{k}$ be an $n$-vertex tree with $k$
legs satisfying all the lengths of $k$ legs, say $l_{1},l_{2},\ldots,l_{k}$,
are almost equal lengths, i.e., $|l_{i}-l_{j}|\leq 1$ for $1\leq i,j\leq k.$
It is easy to see that $T_{n}^{k}\in\mathscr{T}_{n}^{k}$ and
$l_{i}+l_{j}\in\\{2\lfloor\frac{n-1}{k}\rfloor,\lfloor\frac{n-1}{k}\rfloor+\lceil\frac{n-1}{k}\rceil,2\lceil\frac{n-1}{k}\rceil\\}$,
where $1\leq i,j\leq k.$ Let $P_{k}(a,b)$ be a tree obtained by attaching $a$
and $b$ pendant vertices to the two pendant vertices of $P_{k}$, respectively.
In particular, if $k=1$, then $P_{k}(a,b)=K_{1,a+b}$. It is straightforward to
check that
$P_{n-k}(\lfloor\frac{k}{2}\rfloor,\lceil\frac{k}{2}\rceil)\in\mathscr{T}_{n}^{k}$.
It is known that the Wiener index among $n$-vertex trees with $k$ pendant
vertices is minimized by $T_{n}^{k}$ and is maximized by
$P_{n-k}(\lfloor\frac{k}{2}\rfloor,\lceil\frac{k}{2}\rceil)$; see Dobrynin,
Entringer, Gutman [1]. We are going to show the counterparts of these results
for the number of subtrees, which also gives a confirm answer for a conjecture
proposed by Székely and Wang in [11].
###### Theorem 1.1.
Among ${\mathscr{T}}_{n}^{k}$ with $n\geq 2$.
* (i)
Precisely the graph $T_{n}^{k}$, which has
$(\lfloor\frac{n-1}{k}\rfloor+1)^{i}(\lceil\frac{n-1}{k}\rceil+1)^{j}+i{\lfloor\frac{n-1}{k}\rfloor+1\choose{2}}+j{\lceil\frac{n-1}{k}\rceil+1\choose{2}}$
subtrees, maximizes the total number of subtrees among $\mathscr{T}_{n}^{k}$,
where $i+j=k$ and $n-1\equiv j\pmod{k}$.
* (ii)
Precisely the graph
$P_{n-k}(\lfloor\frac{k}{2}\rfloor,\lceil\frac{k}{2}\rceil)$, which has
$(2^{\lfloor\frac{k}{2}\rfloor}+2^{\lceil\frac{k}{2}\rceil})(n-k-1)+2^{k}+k+{n-k-1\choose{2}}$
subtrees, minimizes the total number of subtrees among $\mathscr{T}_{n}^{k}$.
Let $\mathscr{T}_{n,d}$ denote the set of all $n$-vertex trees of diameter
$d$. Let $\hat{T}_{n,k}^{d}$ be the $n$-vertex tree obtained from
$P_{d+1}=v_{1}v_{2}\ldots v_{d}v_{d+1}$ by attaching $n-d-1$ pendant edges to
$v_{k}$; see Fig. 1.
Figure 1: Tree $\hat{T}_{n,k}^{d}.$
###### Theorem 1.2.
For any $n\geq 2$, precisely the graph $\hat{T}_{n,i}^{d}$, which has
$2^{n-d-1}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right)+{\lfloor\frac{d}{2}\rfloor+1\choose{2}}{\lceil\frac{d}{2}\rceil+1\choose{2}}+n-d-1$
subtrees, maximizes the total number of subtrees among $\mathscr{T}_{n,d}$,
where $i=\lfloor\frac{d}{2}\rfloor+1$ or $i=\lceil\frac{d}{2}\rceil+1$.
Let $G$ be a connected bipartite graph with $n$ vertices. Hence its vertex set
can be partitioned into two subsets $V_{1}$ and $V_{2}$, such that each edge
joins a vertex in $V_{1}$ with a vertex in $V_{2}$. Suppose that $V_{1}$ has
$p$ vertices and $V_{2}$ has $q$ vertices, where $p+q=n$. Then we say that $G$
has a $(p,q)$-bipartition $(p\leq q)$. Denote by $\mathscr{P}_{n}^{p,q}$ the
class of trees with $n$ vertices, each of which has a $(p,q)$-bipartition
$(p+q=n)$. Consider a star $K_{1,p}$ with $p+1$ vertices and attach $q-1$
pendant edges to a non-central vertex of the star $K_{1,p}$. The resulting
tree with $p+q$ vertices has a $(p,q)$-bipartition. Denote the resulting tree
by $D(p,q)$; see Fig. 2. Obviously, $D(p,q)\in\mathscr{P}_{n}^{p,q}$. We call
$D(p,q)$ a double star. If $q\geq p\geq 3$, suppose that $B(p,q)$ is the tree
obtained from $D(p-1,q)$ by attaching a pendant edge to one of the vertices of
degree one which join the vertex of degree $q$ in $D(p-1,q)$ (see Fig. 2). If
$q\geq p=2$, we assume that $B(2,q)$ is the tree obtained from the path
$P_{4}$ by attaching $q-2$ pendant edges to an end vertex of $P_{4}$ (see Fig.
2).
Figure 2: Trees $D(p,q),B(p,q)\,(p\geq q\geq 3)$ and $B(2,q).$
###### Theorem 1.3.
For any $n\geq 2$.
* (i)
Precisely the graph $D(p,q)\ (q\geq p\geq 1)$, which has
$2^{n-2}+2^{p-1}+2^{q-1}+n-2$ subtrees, maximizes the total number of subtrees
among $\mathscr{P}_{n}^{p,q}$.
* (ii)
Precisely the graph $B(p,q)\ (q\geq p\geq 2)$, which has
$3\cdot 2^{n-4}+3\cdot 2^{q-2}+2^{p-2}+n-1,$ if $p>2$ and $2^{n-2}+n+2$,
otherwise
subtrees, maximizes the total number of subtrees among
$\mathscr{P}_{n}^{p,q}\setminus\\{D(p,q)\\}.$
* (iii)
Precisely the graph
$P_{2p-1}(\lfloor\frac{n-2p+1}{2}\rfloor,\lceil\frac{n-2p+1}{2}\rceil)$, which
has
$(2p-1)\left(2^{\lfloor\frac{n-2p+1}{2}\rfloor}+2^{\lceil\frac{n-2p+1}{2}\rceil}\right)+{n-2p+1\choose{2}}+2^{n-2p+1}+n-2p+1$
subtrees, minimizes the total number of subtrees among
$\mathscr{P}_{n}^{p,q}$.
Given positive integers $n,q$ with $q\geq 2$, we call $T$ a complete $q$-ary
tree (or $q$-ary tree for short) if any non-pendant vertex $v$ in $T$ has
exactly $q$ neighbours. Denote by $\mathscr{A}_{n}^{q}$ the class of $q$-ary
trees with $n$ non-leaf vertices ($(q-2)n+2$ leaves). Consider the path
$P_{n+2}$ and attach $q-2$ pendant edges to each of the non-leaf vertices of
$P_{n+2}$. Denote the resulting tree by $\hat{T}_{n}^{q}$ (see Fig. 3). It is
easy to see that $\hat{T}_{n}^{q}\in\mathscr{A}_{n}^{q}$. In view of Theorem
2.3 in [5], it is easy to determine the tree in $\mathscr{A}_{n}^{q}$ which
maximizes the total number of subtrees. It is natural and interesting to
determine the sharp lower bound on the total number of subtrees of trees among
$\mathscr{A}_{n}^{q}.$
Figure 3: Tree $\hat{T}_{n}^{q}.$
###### Theorem 1.4.
For $n\geq 1$, precisely the graph $\hat{T}_{n}^{q}$ (see Fig. 3), which has
$\frac{2^{q-2}(2^{q-1}-1)^{2}(2^{(n-1)(q-2)}-1)}{(2^{q-2}-1)^{2}}-\frac{n-1}{2^{q-2}-1}+2^{q}+nq-3n+3$
subtrees, minimizes the total number of subtrees among $\mathscr{A}_{n}^{q}.$
## 2 Some Lemmas
In this section, we give some necessary results which will be used to prove
our main results. For a set $S,$ let $|S|$ denote its cardinality. For two
graphs $G_{1},G_{2}$, if $G_{1}$ is a connected subgraph of $G_{2}$, then we
denote it by $G_{1}\subseteq G_{2}$. Given a tree $T$ with $u,\,v\in V_{T}$,
let
$\begin{array}[]{ll}\ \ \ f_{T}(u)=|\\{T^{\prime}:\ \ T^{\prime}\subseteq
T,\,u\in V_{T^{\prime}}\\}|,&f_{T}(u*v)=|\\{T^{\prime}:\ \ T^{\prime}\subseteq
T,\,u,\,v\in V_{T^{\prime}}\\}|,\\\ f_{T}(u/v)=|\\{T^{\prime}:\ \
T^{\prime}\subseteq T,\,u\in V_{T^{\prime}},v\not\in V_{T^{\prime}}\\}|,&\ \ \
\ \ F(T)=|\\{T^{\prime}:\,T^{\prime}\subseteq T,\,|V_{T^{\prime}}|\geq
1\\}|.\end{array}$
###### Lemma 2.1 ([11]).
The $n$-vertex path $P_{n}$ has ${{n+1}\choose{2}}$ subtrees, fewer than any
other tree on n vertices.
Figure 4: Path $P_{W}(x,y)$ connecting vertices $x$ and $y$.
Consider the tree $W$ in Fig. 4 with vertices $x$ and $y$, and
$P_{W}(x,y)=x_{0}(x)x_{1}\ldots x_{n}zy_{n}\ldots
y_{1}y_{0}(y)(x_{0}(x)x_{1}\ldots x_{n}y_{n}\ldots y_{1}y_{0}(y))$
if $d_{W}(x,y)$ is even (odd) for any $n\geq 0$. After the deletion of all the
edges of $P_{W}(x,y)$ from $W$, some connected components will remain. Let
$X_{i}(X_{0})$ denote the component that contains $x_{i}(x_{0}=x)$, let
$Y_{i}(Y_{0})$ denote the component that contains $y_{i}(y_{0}=y)$, for
$i=1,2,\ldots,n$, and let $Z$ denote the component that contains $z$.
###### Lemma 2.2 ([8]).
In the above situation, if $f_{X_{i}}(x_{i})\geq f_{Y_{i}}(y_{i})$ for
$i=0,1,\ldots,n$, then $f_{W}(x)\geq f_{W}(y)$. Furthermore,
$f_{W}(x)=f_{W}(y)$ if and only if $f_{X_{i}}(x_{i})=f_{Y_{i}}(y_{i})$ for all
$i$.
If we have a tree $T$ with vertices $x$ and $y$, and two rooted trees $X$ and
$Y$, then we can build two new trees, first $T^{\prime}$, by identifying the
root of $X$ with $x$ and the root of $Y$ with $y$, second $T^{\prime\prime}$,
by identifying the root of $X$ with $y$ and the root of $Y$ with $x$ (as shown
in Fig. 5).
Figure 5: Switching subtrees rooted at $x$ and $y$.
###### Lemma 2.3 ([8]).
In the above situation, if $f_{T}(x)>f_{T}(y),f_{X}(x)<f_{Y}(y)$, then we have
$F(T^{\prime\prime})>F(T^{\prime})$.
###### Corollary 2.4.
In the above situation, if $f_{T}(x)>f_{T}(y)$ and $X$ is a rooted tree that
is not a single vertex, then we have $F(T^{\prime\prime})>F(T^{\prime})$,
where $T^{\prime}$ (resp. $T^{\prime\prime}$) is obtained by identifying the
root of $X$ with $y$ (resp. $x$) of $T$.
###### Lemma 2.5.
Given an $n$-vertex path $P_{n}=v_{1}v_{2}\ldots v_{n}$, one has
$f_{P_{n}}(v_{k})=k(n-k+1)$ for $k\in\\{1,2,\ldots,n\\}$. Furthermore, one has
$\begin{split}f_{P_{n}}(v_{k})&=f_{P_{n}}(v_{n-k+1}),\\\
f_{P_{n}}(v_{1})<f_{P_{n}}(v_{2})<\cdots<f_{P_{n}}(v_{k})&<f_{P_{n}}(v_{k+1})<\cdots<f_{P_{n}}(v_{\lfloor\frac{n+1}{2}\rfloor})=f_{P_{n}}(v_{\lceil\frac{n+1}{2}\rceil}).\end{split}$
(2.1)
###### Proof.
For any $P\subseteq P_{n}$ such that $P$ contains $v_{k}$, it can be denoted
by $P=v_{i}v_{i+1}\ldots v_{k}\ldots v_{j}$, where $i\leq k\leq j$. It is easy
to see that we have $i\in\\{1,2,\ldots,k\\}$ and $j\in\\{k,k+1,\ldots,n\\}.$
Hence, we have $f_{P_{n}}(v_{k})=k(n-k+1)$.
Consider the function $f(x)=x(n-x+1)$ for $x\geq 0$. By the monotonicity of
$f(x)$, we have
$f(k)=f(n-k+1),\ \ \ \
f(1)<f(2)<\cdots<f(k)<f(k+1)<\cdots<f(\left\lfloor\frac{n+1}{2}\right\rfloor)=f(\left\lceil\frac{n+1}{2}\right\rceil),$
which is equivalent to (2.1), as desired. ∎
By Corollary 2.4 and Lemma 2.5, the following lemma follows immediately.
###### Lemma 2.6.
Given a tree $T$ with at least two vertices and a path $P_{k}=v_{1}v_{2}\ldots
v_{k}$, let $T_{i}$ be a tree obtained from $T$ and $P_{k}$ by identifying a
vertex $v$ of $T$ with $v_{i}$ of $P_{k}$,
$i\in\\{2,3,\ldots,\lfloor\frac{k+1}{2}\rfloor-2\\}$. Then we have
$F(T_{i})=F(T_{k-i+1})$
and
$F(T_{1})<F(T_{2})<\cdots<F(T_{i})<\cdots<F(T_{\lfloor\frac{k+1}{2}\rfloor}).$
###### Lemma 2.7.
Given a tree $T$ with $uv\in E_{T}$ and $u$ is a leaf, one has $f_{T}(u)\leq
f_{T}(v)$, with equality if and only if $T\cong K_{2}$.
###### Proof.
For any edge $uv$ in $E_{T}$, we have
$f_{T}(u)=f_{T}(u*v)+f_{T}(u/v),\ \ \ \ f_{T}(v)=f_{T}(u*v)+f_{T}(v/u).$ (2.2)
In particular, if $u$ is a leaf and $uv\in E_{T}$, then we have
$f_{T}(u/v)=1,\,f_{T}(v/u)=f_{T-u}(v)\geq 1,$ with equality if and only if
$T-u$ is a single vertex, i.e., $T\cong K_{2}$. Our result holds immediately.
∎
###### Lemma 2.8.
Given a tree $T$ with $u,v\in V_{T}$ satisfying $f_{T}(u)\leq f_{T}(v),$ let
$T^{\prime}$ be a tree obtained from $T$ by adding a new vertex $v_{s}$ to
some vertex of $T$ such that in $T^{\prime}$ the unique path between $u$ and
$v_{s}$ contains $v,$ then $f_{T^{\prime}}(u)<f_{T^{\prime}}(v)$.
###### Proof.
Note that $f_{T}(u)\leq f_{T}(v)$, hence in view of (2.2), we have
$f_{T}(u/v)\leq f_{T}(v/u)$. By the structure of $T^{\prime}$, it is
straightforward to check that $f_{T^{\prime}}(u/v)=f_{T}(u/v)$, and
$f_{T^{\prime}}(v/u)>f_{T}(v/u)$, hence we have
$f_{T^{\prime}}(u)=f_{T^{\prime}}(u*v)+f_{T^{\prime}}(u/v)<f_{T^{\prime}}(v*u)+f_{T^{\prime}}(v/u)=f_{T^{\prime}}(v),$
as desired. ∎
###### Lemma 2.9.
Let $P=uu_{1}\ldots v_{1}v$ be a path of a tree $T$ with
$N_{T}(u)=\\{u_{1},w,w_{1},\ldots,w_{s}\\}$,
$N_{T}(v)=\\{v_{1},z,z_{1},\ldots,z_{t}\\}$, here $s\geq 1,t\geq 1$. Then
$F(T)<F(T^{\prime})$ or $F(T)<F(T^{\prime\prime})$, where
$\displaystyle T^{\prime}$ $\displaystyle=$ $\displaystyle
T-uw_{1}-uw_{2}-\cdots-uw_{s}+vw_{1}+vw_{2}+\cdots+uw_{s},$ $\displaystyle
T^{\prime\prime}$ $\displaystyle=$ $\displaystyle T-vz_{1}-vz_{2}-\cdots-
vz_{t}+uz_{1}+uz_{2}+\cdots+uz_{t}.$
###### Proof.
Consider the component in $T-uw_{1}-uw_{2}-\cdots-uw_{s}-vz_{1}-vz_{2}-\cdots-
vz_{t}$, say $\hat{T}$, which contains both $u$ and $v$. If
$f_{\hat{T}}(u)\leq f_{\hat{T}}(v)$, then by Lemma 2.8, we have
$f_{\tilde{T}}(u)<f_{\tilde{T}}(v),$
where $\tilde{T}$ is just the component containing both $u$ and $v$ in the
graph $T-uw_{1}-uw_{2}-\cdots-uw_{s}.$ By Corollary 2.4, we have
$F(T)<F(T^{\prime})$.
If $f_{\hat{T}}(v)<f_{\hat{T}}(u)$, similarly we can also show that
$F(T)<F(T^{\prime\prime})$. We omit the procedure here.
This completes the proof. ∎
###### Lemma 2.10.
Given a tree $T$ containing a path $P_{r}=v_{1}v_{2}\ldots v_{r},$ there
exists a vertex $v_{i}\in V_{P_{r}}\setminus\\{v_{1},v_{r}\\}$ such that
$f_{T}(v_{1})<\cdots<f_{T}(v_{i-1})<f_{T}(v_{i})\geq
f_{T}(v_{i+1})>\cdots>f_{T}(v_{r}).$ (2.3)
###### Proof.
Consider three vertices $x,y,z$ such that $xy,yz\in E_{T}$. Let $X,Y,Z$,
respectively, denote the components containing $x,y,z$ after the removal of
the edges $xy$ and $yz$ from $T$. Observe the identities
$\displaystyle f_{T}(x)$ $\displaystyle=$ $\displaystyle
f_{X}(x)+f_{X}(x)f_{Y}(y)+f_{X}(x)f_{Y}(y)f_{Z}(z),$ $\displaystyle f_{T}(z)$
$\displaystyle=$ $\displaystyle
f_{Z}(z)+f_{Z}(z)f_{Y}(y)+f_{Z}(z)f_{Y}(y)f_{X}(x),$ $\displaystyle f_{T}(y)$
$\displaystyle=$ $\displaystyle
f_{Y}(y)+f_{X}(x)f_{Y}(y)+f_{Z}(z)f_{Y}(y)+f_{X}(x)f_{Y}(y)f_{Z}(z).$
This gives
$2f_{T}(y)-f_{T}(x)-f_{T}(z)=2f_{Y}(y)+(f_{X}(x)+f_{Z}(z))(f_{Y}(y)-1)>0.$
(2.4)
Let
$i=\min\\{j:\ 1\leq j\leq r,\,f_{T}(v_{j})\geq f_{T}(u),\,u\in V_{P_{r}}\\}.$
by Lemma 2.7, $i\not=1,r.$ Hence, we have
$f_{T}(v_{i})\geq f_{T}(v_{i+1}),\ \ \ f_{T}(v_{i})>f_{T}(v_{i-1}).$
Next consider three consecutive vertices $v_{i},v_{i+1},v_{i+2}$ on $P_{r},$
in view of (2.4) we have
$2f_{T}(v_{i+1})-f_{T}(v_{i})-f_{T}(v_{i+2})>0.$
Combining with $f_{T}(v_{i})\geq f_{T}(v_{i+1})$ yields
$f_{T}(v_{i})\geq f_{T}(v_{i+1})>f_{T}(v_{i+2}).$
Repeated as above we obtain
$f_{T}(v_{i})\geq f_{T}(v_{i+1})>\cdots>f_{T}(v_{r}).$ (2.5)
Similarly, we obtain
$f_{T}(v_{1})<\cdots<f_{T}(v_{i-1})<f_{T}(v_{i}).$ (2.6)
Hence, (2.5) and (2.6) imply (2.3) immediately. ∎
## 3 Proof of Theorem 1.1
In this section, we shall determine sharp upper and lower bounds on the total
number of subtrees of $n$-vertex tree with $k$ pendants.
Proof of Theorem 1.1. (i) First we enumerate the total number of subtrees of
$T_{n}^{k}$. Consider the unique vertex, say $v_{0}$, the center of
$T_{n}^{k}$ whose degree is $k$, we have
$\displaystyle
F(T_{n}^{k})=f_{T_{n}^{k}}(v_{0})+F(T-v_{0})=f_{T_{n}^{k}}(v_{0})+F(iP_{\lfloor\frac{n-1}{k}\rfloor}\cup
jP_{\lceil\frac{n-1}{k}\rceil}),$ (3.1)
where $i+j=k$ and $n-1\equiv j\pmod{k}$. On the one hand,
$f_{T_{n}^{k}}(v_{0})=\left(\left\lfloor\frac{n-1}{k}\right\rfloor+1\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil+1\right)^{j}.$
On the other hand, by Lemma 2.1 we have
$F(iP_{\lfloor\frac{n-1}{k}\rfloor}\cup
jP_{\lceil\frac{n-1}{k}\rceil})=i{\lfloor\frac{n-1}{k}\rfloor+1\choose{2}}+j{\lceil\frac{n-1}{k}\rceil+1\choose{2}},$
where $i+j=k$ and $j\equiv n-1\pmod{k}$. Together with (3.1), we have
$F(T_{n}^{k})=\left(\left\lfloor\frac{n-1}{k}\right\rfloor+1\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil+1\right)^{j}+i{\lfloor\frac{n-1}{k}\rfloor+1\choose{2}}+j{\lceil\frac{n-1}{k}\rceil+1\choose{2}},$
as desired.
Now we show that $T_{n}^{k}$ is the unique graph which maximizes the total
number of subtrees among $\mathscr{T}_{n}^{k}.$ Choose
$T\in\mathscr{T}_{n}^{k}$ such that the total number of its subtrees is as
large as possible.
If $k=2$ or, $k=n-1$, it is easy to see that
$\mathscr{T}_{n}^{k}=\\{T_{n}^{k}\\}$, our result follows immediately. Hence,
in what follows we consider $2<k<n-1$.
We are to show that $T$ is a spider, i.e., $T$ contains a unique vertex of
degree larger than 2. Assume to the contrary that $T$ contains at least $2$
vertices of degree greater than 2. By Lemma 2.9, there exists an $n$-vertex
tree $T^{\prime}\in\mathscr{T}_{n}^{k}$ such that $F(T)<F(T^{\prime})$, a
contradiction to the choice of $T$. Hence, we assume that $v_{0}$ is the
unique vertex of degree greater than 2.
In order to complete the proof, it suffices to show that all the legs attached
to $v_{0}$ of $T$ are almost equal lengths. Let
$PV(T)=\\{u_{1},u_{2},\ldots,u_{k}\\}$ be the set of all pendant vertices of
$T$. We are to show that for any $u_{i},u_{j}\in PV(T)$, we have
$|d_{T}(v_{0},u_{i})-d_{T}(v_{0},u_{j})|\leq 1.$ Assume to the contrary that
there exist two pendant vertices, say $u_{t},u_{l}$, in $PV(T)$ such that
$|d_{T}(u_{0},u_{t})-d_{T}(u_{0},u_{l})|\geq 2.$ (3.2)
Denote the unique path connecting $u_{t}$ and $u_{l}$ by
$P_{s}=w_{1}w_{2}\ldots w_{i-1}w_{i}w_{i+1}\ldots w_{s},$ where
$w_{1}=u_{t},w_{s}=u_{l}$ and $w_{i}=u_{0},1\leq i\leq s$. In view of (3.2),
we have
$\text{$u_{0}=w_{i}\neq w_{\lfloor\frac{s+1}{2}\rfloor}$\ \ \ and\ \ \
$u_{0}=w_{i}\neq w_{\lceil\frac{s+1}{2}\rceil}$}.$
Hence, by Lemma 2.6 there exists an $n$-vertex tree
$T^{\prime\prime}\in\mathscr{T}_{n}^{k}$ such that $F(T)<F(T^{\prime\prime})$,
a contradiction to the choice of $T$.
This completes the proof of Theorem 1.1(i).
(ii) If $k=2$ or, $k=n-1$, it is easy to see that
$\mathscr{T}_{n}^{k}=\\{P_{n-k-1}(\lfloor\frac{k}{2}\rfloor,\lceil\frac{k}{2}\rceil)\\}$,
our result follows immediately. Hence, in what follows we consider $2<k<n-1$.
In order to complete the proof, it suffices to show the following claims.
###### Claim 1.
If $T$ minimizes the total number of subtrees in $\mathscr{T}_{n}^{k},T\cong
P_{n-k}(a,b)$, where $a\geq b\geq 1$ and $a+b=k$.
Proof of Claim 1 If diam$(T)=3$, the claim follows immediately. Hence we
consider the trees whose diameter is larger than 3. Suppose that
$P_{r}=v_{1}\ldots v_{r}(r\geq 5)$ is one of the longest path in $T$, we are
to show that $d_{T}(v_{3})=d_{T}(v_{4})=\cdots=d_{T}(v_{r-2})=2$. Assume to
the contrary that there exists $v\in\\{v_{3},v_{4},\ldots,v_{r-2}\\}$ such
that $d_{T}(v)\geq 3.$ Let
$i=\min\\{j:\ d_{T}(v_{j})\geq 3,\ \ \ 3\leq j\leq r-2\\},\ \ \ \
N_{T}(v_{i})=\\{v_{i-1},v_{i+1},z_{1},z_{2},\ldots,z_{s}\\},\ \ s\geq 1.$
After the deletion of all the vertices $z_{1},z_{2},\ldots,z_{s}$ from $T$,
let $T_{0}$ denote the component containing $v_{i}$. By Lemma 2.10, there
exists $v_{t}\in V_{P_{r}}$ such that
$f_{T_{0}}(v_{1})<\cdots<f_{T_{0}}(v_{t-1})<f_{T_{0}}(v_{t})\geq
f_{T_{0}}(v_{t+1})>\cdots>f_{T_{0}}(v_{r}).$
If $t<i,$ then we have $f_{T_{0}}(v_{i})>f_{T_{0}}(v_{r-1})$. By Corollary
2.4, we have
$F(T)>F(T^{\prime}),$ (3.3)
where
$T^{\prime}=T-v_{i}z_{1}-\cdots-v_{i}z_{s}+v_{r-1}z_{1}+\cdots+v_{r-1}z_{s}.$
If $t\geq i,$ then we have $f_{T_{0}}(v_{i})>f_{T_{0}}(v_{2})$. By Corollary
2.4, we have
$F(T)>F(T^{\prime\prime}),$ (3.4)
where
$T^{\prime\prime}=T-v_{i}z_{1}-\cdots-
v_{i}z_{s}+v_{2}z_{1}+\cdots+v_{2}z_{s}.$
It is easy to see that $T^{\prime},\,T^{\prime\prime}\in\mathscr{T}_{n}^{k}.$
Hence, (3.3) (resp. (3.4)) is a contradiction to the choice of $T$. This
completes the proof of Claim 1. ∎
###### Claim 2.
For positive integers $a,b$ with $a\geq b$ and $a+b=k$ one has
$F(P_{n-k}(a,b))=(2^{a}+2^{b})(n-k-1)+2^{k}+k+{n-k-1\choose{2}}$ (3.5)
and if $a-b\geq 2$, then
$F(P_{n-k}(a,b))>F(P_{n-k}(a-1,b+1)).$ (3.6)
Proof of Claim 2 For convenience, assume that $d_{P_{n-k}(a,b)}(v_{1})=a$ and
$d_{P_{n-k}(a,b)}(v_{n-k})=b$. Then we have
$\begin{split}F(P_{n-k}(a,b))=&f_{P_{n-k}(a,b)}(v_{1}/v_{n-k})+f_{P_{n-k}(a,b)}(v_{1}*v_{n-k})+f_{P_{n-k}(a,b)}(v_{n-k}/v_{1})\\\
&+F(P_{n-k}(a,b)-v_{1}-v_{n-k}).\end{split}$ (3.7)
By direct calculation, we have
$f_{P_{n-k}(a,b)}(v_{1}/v_{n-k})=2^{a}(n-k-1)$ and
$f_{P_{n-k}(a,b)}(v_{n-k}/v_{1})=2^{b}(n-k-1)$. (3.8)
It is straightforward to check that the total number of subtrees of
$P_{n-k}(a,b)$ containing both $v_{1}$ and $v_{n-k}$ is equal to the total
number of subtrees of $K_{1,a+b}$ each contains the center of $K_{1,a+b}$.
Hence, we have
$f_{P_{n-k}(a,b)}(v_{1}*v_{n-k})=2^{a+b}=2^{k}.$ (3.9)
On the other hand,
$F(P_{n-k}(a,b)-v_{1}-v_{n-k})=F((a+b)P_{1}\cup
P_{n-k-2})=k+{n-k-1\choose{2}}.$ (3.10)
In view of (3.7)-(3.10), (3.5) holds.
By (3.5), we have
$F(P_{n-k}(a,b))-F(P_{n-k}(a-1,b+1))=(2^{a}+2^{b}-2^{a-1}-2^{b+1})(n-k-1)=(2^{a-1}-2^{b})(n-k-1).$
Note that $a-b\geq 2$, hence $(2^{a-1}-2^{b})(n-k-1)>0$, i.e.,
$F(P_{n-k}(a,b))>F(P_{n-k}(a-1,b+1))$, as desired. ∎
By Claims 1 and 2, Theorem 1.1(ii) follows immediately.∎
## 4 Proof of Theorem 1.2
Proof of Theorem 1.2 In view of Theorem 3.7 in [16], we know that
$\hat{T}_{n,i}^{d}$,
$i=\lfloor\frac{d+2}{2}\rfloor=\lfloor\frac{d}{2}\rfloor+1$ or
$i=\lceil\frac{d+2}{2}\rceil=\lceil\frac{d}{2}\rceil+1$, is the unique graph
maximizing the total number of subtrees among $\mathscr{T}_{n,d}$. In order to
complete the proof, it suffices to show that
$F(\hat{T}_{n,i}^{d})=2^{n-d-1}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right)+{\lfloor\frac{d}{2}\rfloor+1\choose{2}}{\lceil\frac{d}{2}\rceil+1\choose{2}}+n-d-1,$
(4.1)
where $i=\lfloor\frac{d}{2}\rfloor+1$ or $i=\lceil\frac{d}{2}\rceil+1$.
In fact, it is easy to see that
$F(\hat{T}_{n,i}^{d})=f_{\hat{T}_{n,i}^{d}}(x)+F(\hat{T}_{n,i}^{d}-x)$ (4.2)
for any $x\in V_{\hat{T}_{n,i}^{d}}$.
Without loss of generality, consider $x:=v_{\lfloor\frac{d}{2}\rfloor+1}$
whose neighbor contains just $n-d-1$ leaves. By Lemma 2.5, we have
$f_{P_{d+1}}(x)=\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(d+1+1-(\left\lfloor\frac{d}{2}\right\rfloor+1)\right)=\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right)$
which leads to
$f_{\hat{T}_{n,i}^{d}}(x)=2^{n-d-1}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right).$
(4.3)
On the other hand,
$\hat{T}_{n,i}^{d}-x=(n-d-1)P_{1}\cup P_{\lfloor\frac{d}{2}\rfloor}\cup
P_{\lceil\frac{d}{2}\rceil}.$
By Lemma 2.1, we have
$F(\hat{T}_{n,i}^{d}-x)=n-d-1+{{\lfloor\frac{d}{2}\rfloor}+1\choose{2}}{{\lceil\frac{d}{2}\rceil}+1\choose{2}}.$
(4.4)
In view of Eqs. (4.2)-(4.4), Eq. (4.1) follows immediately. ∎
## 5 Proof of Theorem 1.3
In this section, we prove Theorem 1.3. For convenience, denote by $\iota(T)$
the number of non-pendant vertices in $T$.
Proof Theorem 1.3 (i) First we show that $D(p,q)$ is the tree in
$\mathscr{P}_{n}^{p,q}$ which has the largest number of subtrees. For any
$T\in\mathscr{P}_{n}^{p,q}$. If $p=1$,
$\mathscr{P}_{n}^{p,q}=\\{K_{1,n-1}\\}=\\{D(1,n-1)\\}$. Our result holds in
this case. Hence, in what follows, we consider $p\geq 2.$ In order to
determine the structure of the extremal graph, say $T$, in this case, it
suffice to show that $\iota(T)=2.$
Hence, we assume to the contrary that $\iota(T)\geq 3.$
Choose three vertices, say $u,v,w$, such that each of them is of degree at
least 3. Let $V_{T}=V_{1}\cup V_{2}$. It is straightforward to check that in
$\\{u,v,w\\}$, there exist two elements are in $V_{1}$ or $V_{2}$. We assume,
without loss of generality, that $u,v\in V_{1}$ with
$N_{T}(u)=\\{u_{1},z_{1},\ldots,z_{t}\\},N_{T}(v)=\\{u_{2k-1},r_{1},\ldots,r_{s}\\}$,
$t\geq 1,s\geq 1$ and the unique path joining $u$ and $v$ is $P=uu_{1}\ldots
u_{2k-1}v$. Let $X_{u}$ be the component that contains $u$ in $T-E_{P}$ and
$Y_{v}$ the component that contains $v$ in $T-E_{P}$. Let $T^{\prime}$ be the
component that contains $u$ in $T-uz_{1}-\cdots-uz_{t}-vr_{1}-\cdots-vr_{s}$.
If $f_{T^{\prime}}(u)\geq f_{T^{\prime}}(v)$, by Lemma 2.8 we have
$f_{T^{\prime\prime}}(u)>f_{T^{\prime\prime}}(v),$ (5.1)
where $T^{\prime\prime}$ is obtained by identifying $u$ of $T^{\prime}$ with
$u$ of $X_{u}$. Let $T^{*}$ be the tree obtained by identifying $u$ of
$T^{\prime\prime}$ with $v$ of $Y_{v}$. Note that $u$ and $v$ are in $V_{1}$,
hence we have $T^{*}\in\mathscr{P}_{n}^{p,q}$. On the other hand, notice that
$Y_{v}$ is not a single vertex, together with (5.1) and Corollary 2.4, we have
$F(T)<F(T^{*})$, a contradiction to the choice of $T$.
Similarly, if $f_{T^{\prime}}(u)<f_{T^{\prime}}(v)$, we can also show there
exists a tree $\hat{T}\in\mathscr{P}_{n}^{p,q}$ such that $F(T)<F(\hat{T})$, a
contradiction. We omit the procedure here. Hence, we get that $\iota(T)=2$,
i.e., $T\cong D(p,q),$ as desired.
Now we show that
$\displaystyle F(D(p,q))=2^{n-2}+2^{p-1}+2^{q-1}+n-2.$ (5.2)
In fact, let $d_{D(p,q)}(u)=p$ and $d_{D(p,q)}(v)=q$, we have
$F(D(p,q)=f_{D(p,q)}(u/v)+f_{D(p,q)}(u*v)+f_{D(p,q)}(v/u)+F(D(p,q)-v-u).$
It is easy to see that $f_{D(p,q)}(u/v)=2^{p-1},f_{D(p,q)}(v/u)=2^{q-1}$ and
any subtree that contains both $u$ and $v$ can be considered be the subtree
that contains the center of $K_{1,n-2}$, so we have $f_{D(p,q)}(u*v)=2^{n-2}$.
On the other hand, as
$F(D(p,q)-v-u)=F((p-1)P_{1}\cup(q-1)P_{1})=n-2$
Therefore, Eq.(5.2) holds. This completes the proof of Theorem 1.3 (i).
(ii) Suppose $T^{\prime}$ is the tree in
$\mathscr{P}_{n}^{p,q}\setminus\\{D(p,q)\\}$ with $q\geq p\geq 2$ which has
the largest number of subtrees. First we show that $\iota(T^{\prime})=3$. Note
that $T^{\prime}\not\cong D(p,q)$, we know that $\iota(T)\neq 1,2$. If
$\iota(T^{\prime})>3$, then by a similar discussion as in the proof of Theorem
1.3 (i), there exists a tree $T^{\prime\prime}\in\mathscr{P}_{n}^{p,q}$ such
that $\iota(T^{\prime\prime})=\iota(T^{\prime})-1\geq 3$ and
$F(T^{\prime})<F(T^{\prime\prime})$. Note that $\iota(T^{\prime\prime})\geq
3$, hence $T^{\prime\prime}\in\mathscr{P}_{n}^{p,q}\setminus\\{D(p,q)\\}$.
Therefore, we find a tree $T^{\prime\prime}$ in
$\mathscr{P}_{n}^{p,q}\setminus\\{D(p,q)\\}$ such that
$F(T^{\prime})<F(T^{\prime\prime})$, a contradiction to the choice of
$T^{\prime}$. Hence, $\iota(T^{\prime})=3.$
Let $T(x,y,z)$ be the graph obtained by identifying one leaf of $K_{1,x+1}$
(resp. $K_{1,z+1}$) with the center of $K_{1,y}$, which is depicted in Fig. 6.
As $T^{\prime}\in\mathscr{P}_{n}^{p,q}\setminus\\{D(p,q)\\}$ with
$\iota(T^{\prime})=3$, we have
$T^{\prime}\cong T(a,p-2,b),\ \ \ a\geq b\geq 1,a+b+1=q,$
or
$T^{\prime}\cong T(a^{\prime},q-2,b^{\prime}),\ \ \ \ a^{\prime}\geq
b^{\prime}\geq 1,a^{\prime}+b^{\prime}+1=p.$
Figure 6: Trees $T(x,y,z)$ and $T(a,0,b)(a\geq b\geq 1).$
###### Claim 1.
If $T^{\prime}\cong T(a,p-2,b)\ ($or $T^{\prime}\cong
T(a^{\prime},q-2,b^{\prime}))$, then $b=1$ $($or $b^{\prime}=1).$
Proof of Claim 1 We only show that if $T^{\prime}\cong T(a,p-2,b)$, we have
$b=1.$ Similarly, we can also show if $T^{\prime}\cong
T(a^{\prime},q-2,b^{\prime})$, then we have $b^{\prime}=1$. We omit the
procedure for the latter here.
Assume $b>1,$ let $u,v$ be two non-adjacent vertices of degree at least 2 in
$T(a,p-2,b)$ with $N_{T(a,p-2,q)}(v)=\\{w,w_{1},\ldots,w_{a}\\}$ and
$N_{T(a,p-2,b)}(u)=\\{w,z_{1},z_{2},\ldots,z_{b}\\}$. By Lemma 2.2, as
$f_{K_{1,a}}(v)\geq f_{K_{1,b}}(u)$, we have
$f_{T(a,p-2,b)}(v)\geq f_{T(a,p-2,b)}(u).$ (5.3)
Let $T^{*}=T(a,p-2,b)-\\{z_{2},z_{3},\ldots,z_{b}\\}$. Then we have
$f_{T^{*}}(v)>f_{T^{*}}(u),$ (5.4)
otherwise $f_{T^{*}}(v)\leq f_{T^{*}}(u)$. By Lemma 2.8, we have
$f_{T(a,p-2,b)}(v)<f_{T(a,p-2,b)}(u),$
which contradicts (5.3). Hence, the inequality in (5.4) holds.
On the other hand, $T(a+b-1,p-2,1)$ can be obtained by identifying the vertex
$v$ in $T^{*}$ with the center vertex of $K_{1,b-1}$; while $T(a,p-2,b)$ can
be obtained by identifying the vertex $u$ in $T^{*}$ with the center vertex of
$K_{1,b-1}$. By Corollary 2.4, we have
$F(T(a+b-1,p-2,1))>F(T(a,p-2,b)),$
which contradicts the choice of $T^{\prime}\,(=T(a,p-2,b)).$ Hence, $b=1$. ∎
If $p=2$, by Claim 1, we have $T^{\prime}\cong D(2,q),$ or $T^{\prime}\cong
B(2,q)$. Note that $T^{\prime}\in\mathscr{P}_{n}^{2,q}\setminus\\{D(2,q\\}$,
hence $T^{\prime}\cong B(2,q)$, as desired in this case.
If $p>2$, by Claim 1, we have $T^{\prime}\cong T(q-2,p-2,1)$ or
$T^{\prime}\cong T(p-2,q-2,1)=B(p,q)$. If $q=p,$ it is easy to see that
$T(q-2,p-2,1)\cong T(p-2,q-2,1)=B(p,q),$ our result follows immediately in
this subcase. Hence, it suffices to consider $q>p.$ In order to determine the
structure of $T^{\prime}$, it suffices to show that
$F(T(p-2,q-2,1))>F((q-2,p-2,1)).$
Note that $T(q-2,p-2,1)$ is obtained by identifying $u$ of $D(p-1,q-1)$ with a
leaf of $P_{3}$, and $T(q-2,p-2,1)$ is obtained by identifying $v$ of
$D(q-1,p-1)$ with a leaf of $P_{3}$, where $u$ is a vertex of degree $p-1$ in
$D(p-1,q-1)$, $v$ is a vertex of degree $q-1$ in $D(p-1,q-1)$. Notice that
$f_{D(p-1,q-1)}(u/v)=2^{p-2}<2^{q-2}=f_{D(p-1,q-1)}(v/u).$
So
$f_{D(p-1,q-1)}(u)<f_{D(p-1,q-1)}(v).$
Hence, by Corollary 2.4 we have $F((q-2,p-2,1))<F(T(p-2,q-2,1))$. Therefore,
for any $T\in\mathscr{P}_{n}^{p,q}\setminus\\{D(p,q)\\}$, $F(T)\leq
F(B(p,q)),q\geq p\geq 2$, with equality if and only if $T\cong B(p,q)$.
In order to complete the proof of Theorem 1.3 (ii), it suffices to show the
following claim.
###### Claim 2.
In the above situation, if $p\geq q\geq 2$ with $p+q=n$, then
$F(B(p,q))=\left\\{\begin{aligned} &3\cdot 2^{n-4}+3\cdot
2^{q-2}+2^{p-2}+n-1,&q\geq p>2,\\\ &2^{n-2}+n+2,&p=2.\end{aligned}\right.$
Proof of Claim 2 First consider $p=2$. Let $v$ be the vertex of degree $n-3$
in $B(2,n-2)$, hence $F(B(2,n-2))=f_{B(2,n-2)}(v)+F(B(2,n-2)-v).$
Note that
$f_{B(2,n-2)}(v)=4\cdot 2^{n-2-2}=2^{n-2}$
and
$F(B(2,n-2)-v)=F(P_{3}\cup(n-4)P_{1})={4\choose{2}}+n-4=n+2.$
Hence, by simple computation our result holds for $p=2$.
Now consider $p>2$. Let $v$ be the vertex of degree $q$ and $u$ the vertex of
degree $p-1$ in $B(p,q)$. Note that
$F(B(p,q))=f_{B(p,q)}(v*u)+f_{B(p,q)}(v/u)+f_{B(p,q)}(u/v)+F(B(p,q)-v-u).$
On the other hand,
$f_{B(p,q)}(v*u)=3\cdot 2^{p-2+q-2}=3\cdot 2^{n-4},\ \ \
f_{B(p,q)}(v/u)=3\cdot 2^{q-2},\ \ \ f_{B(p,q)}(u/v)=2^{p-2}$
and
$F(B(p,q)-v-u)=F((p-2+q-2)P_{1}\cup P_{2})=n-4+3=n-1.$
By simple calculation, our result also holds for $p>2.$ ∎
This completes the proof of Theorem 1.3 (ii).
(iii) If $p=1$, it is easy to see that
$\mathscr{T}_{n}^{k}=\\{P_{1}(\lfloor\frac{n-1}{2})\rfloor,\lceil\frac{n-1}{2}\rceil\\}$,
our result follows immediately. On the other hand, if $p=q$ or $p=q-1$, it is
easy to see that $P_{n}\in\mathscr{P}_{n}^{p,q}$, by Lemma 2.1, it is easy to
see that $P_{n}=P_{2p-1}(1,1)$ or $P_{n}=P_{2p-1}(0,1),$ minimizes the total
number of subtrees among $\mathscr{P}_{n}^{p,q}$. Hence, in what follows we
consider $1<p<\lfloor\frac{n}{2}\rfloor$. In order to complete the proof, it
suffices to show the following claim.
###### Claim 3.
If $T$ minimizes the total number of subtrees in $\mathscr{P}_{n}^{p,q},T\cong
P_{2p-1}(a,b)$, where $a\geq b\geq 1$ and $a+b=n-2p+1$.
Proof of Claim 3 If $1<p<\lfloor\frac{n}{2}\rfloor$, by (i) we know that
$T\not\cong D(p,q)$, so diam$(T)\geq 3$. If diam$(T)=3$, the claim follows
immediately. Hence in what follows we consider the trees whose diameter is
larger than 3. Suppose that $P_{r}=v_{1}\ldots v_{r}\,(r\geq 5)$ is one of the
longest path in $T$, we are to show that
$d_{T}(v_{3})=d_{T}(v_{4})=\cdots=d_{T}(v_{r-2})=2$ and $r=2p+1$. First assume
to the contrary that there exists $v\in\\{v_{3},v_{4},\ldots,v_{r-2}\\}$ such
that $d_{T}(v)\geq 3.$ Let
$i=\min\\{j:\ d_{T}(v_{j})\geq 3,\ \ \ 3\leq j\leq r-2\\},\ \ \ \
N_{T}(v_{i})=\\{v_{i-1},v_{i+1},z_{1},z_{2},\ldots,z_{s}\\},\ \ s\geq 1.$
Let $T_{0}$ be the component that contains $v_{i}$ in
$T-\\{z_{1},z_{2},\ldots,z_{s}\\}$. By Lemma 2.10, there exists $v_{t}\in
V_{P_{r}}$ such that
$f_{T_{0}}(v_{1})<\cdots<f_{T_{0}}(v_{t-1})<f_{T_{0}}(v_{t})\geq
f_{T_{0}}(v_{t+1})>\cdots>f_{T_{0}}(v_{r}).$
If $t<i,$ then we have $f_{T_{0}}(v_{i})>f_{T_{0}}(v_{r-1})>f_{T_{0}}(v_{r})$.
If $v_{i}$ and $v_{r-1}$ are in the same part, By Corollary 2.4, we have
$F(T)>F(T^{\prime}),$ (5.5)
where
$T^{\prime}=T-v_{i}z_{1}-\cdots-v_{i}z_{s}+v_{r-1}z_{1}+\cdots+v_{r-1}z_{s},\
\ \ \ T^{\prime}\in\mathscr{P}_{n}^{p,q}$
otherwise, $v_{i}$ and $v_{r}$ are in the same part, we have
$F(T)>F(T^{\prime\prime}),$ (5.6)
where
$T^{\prime\prime}=T-v_{i}z_{1}-\cdots-
v_{i}z_{s}+v_{r}z_{1}+\cdots+v_{r}z_{s},\ \ \ \
T^{\prime\prime}\in\mathscr{P}_{n}^{p,q}.$
If $t\geq i$, repeat as above, we have a
$T^{\prime\prime\prime}\in\mathscr{P}_{n}^{p,q}$ such that
$F(T)>F(T^{\prime\prime\prime}),\ \ \ \
T^{\prime\prime\prime}\in\mathscr{P}_{n}^{p,q}.$ (5.7)
Hence, (5.5)-(5.7) are contradictions to the choice of $T$. So we have $T\cong
P_{r}(a,b)$.
On the other hand, since $T\in\mathscr{P}_{n}^{p,q}$ with
$1<p<\lfloor\frac{n}{2}\rfloor$, it is easy to see that $r\leq 2p+1$. If
$r<2p+1$, it means that $v_{1}$ and $v_{r}$ are in different parts (otherwise
we have $p<\lceil\frac{r-2}{2}\rceil$ or $q<\lceil\frac{r-2}{2}\rceil$). As
$a\geq b$, we have $v_{1}\in V_{2}$ and $v_{r}\in V_{1}$, where $V_{1}$ and
$V_{2}$ are two parts of $V_{T}$ with $|V_{1}|=p,|V_{2}|=q$. Assume that
$N_{T}(v_{2})=\\{v_{3},v_{1},w_{2},\ldots,w_{a}\\},N_{T}(v_{r-1})=\\{v_{r-2},v_{r},u_{2},\ldots,u_{b}\\}$.
Let $\hat{T}=T-\\{v_{r},u_{2},\ldots,u_{b}\\}$. By Lemma 2.2, we have
$f_{\hat{T}}(v_{r-1})<f_{\hat{T}}(v_{1})$, by Corollary 2.4 we have
$F(T)>F(\tilde{T}),$ (5.8)
where
$\tilde{T}=T-v_{r-1}v_{r}-v_{r-1}u_{2}-\cdots-
v_{r}u_{b}+v_{1}v_{r}+v_{1}u_{2}+\cdots+v_{1}u_{b}.$
As $v_{r-1},v_{1}\in V_{2}$, $\tilde{T}\in\mathscr{P}_{n}^{p,q}$. Hence, (5.8)
is a contradiction to the choice of $T$. So we have $T\cong P_{2p-1}(a,b)$. ∎
By Claims 2 in the proof of Theorem 1.1(ii), together with Claim 3, $T\cong
P_{2p-1}(\lfloor\frac{n-2p+1}{2}\rfloor,\lceil\frac{n-2p+1}{2}\rceil)$, as
desired. ∎
###### Remark 1.
In view of Eq.(5.2), we have
$F(D(p,q))-F(D(p-1,q+1))=2^{p-2}-2^{q-1}<0$
for $q\geq p>1$. Hence, we have
$F(D(p,q))<F(D(p-1,q+1))<\cdots<F(D(1,n-1))=F(K_{1,n-1}),$ (5.9)
for $q\geq p>1$. Note that $D(p,q)$ maximizes the total number of subtrees
among $\mathscr{P}_{n}^{p,q}$, hence in view of (5.9) and Theorem 1.3(i), the
following corollary holds immediately.
###### Corollary 5.1 ([11]).
The star $K_{1,n-1}$ has $2^{n-1}+n-1$ subtrees, more than any other tree on
$n$ vertices.
## 6 Proof of Theorem 1.4
In this section we shall determine the sharp lower bound on the total number
of subtrees contained in a tree among $\mathscr{A}_{n}^{q}$.
Proof of Theorem 1.4 First we characterize the structure of the tree, say $T$,
minimizing the total number of subtrees in $\mathscr{A}_{n}^{q}.$ In order to
do so, it suffices to show that the diameter of $T$ is $n+1$. Without loss of
generality, we assume one of the longest paths in $T$ is
$P_{r+1}=v_{0}v_{1}\ldots v_{r}$. If $r=n+1$, our result holds obviously. So
in what follows, we assume that $r\leq n.$
For convenience, let
$N_{T}(v_{i})\setminus\\{v_{i-1},v_{i+1}\\}=\\{v_{i_{1}},v_{i_{2}},\ldots,v_{i_{q-2}}\\},\
\ \ \ \ \ i=1,2,\ldots,r-1.$
Hence,
$\bigcup_{i=1}^{r-1}N_{T}(v_{i})=\\{v_{1_{1}},v_{1_{2}},\ldots,v_{1_{q-2}},\ldots,v_{i_{1}},v_{i_{2}},\ldots,v_{i_{q-2}},v_{r_{1}},v_{r_{2}},\ldots,v_{r_{q-2}}\\}$
(6.1)
In fact, $(\bigcup_{i=1}^{r-1}N_{T}(v_{i}),\prec)$ is a total ordering set,
where $\prec$ is defined as following: for any
$v_{i_{j}},\,v_{t_{s}}\in\bigcup_{i=1}^{r-1}N_{T}(v_{i})$, we call
$v_{i_{j}}\prec v_{t_{s}}$ if $i<t$ or $i=t,j<s.$ Hence, we can order the
elements in (6.1) as following:
$v_{1_{1}}\prec v_{1_{2}}\prec\cdots\prec v_{1_{q-2}}\prec\cdots\prec
v_{i_{1}}\prec v_{i_{2}}\prec\cdots\prec v_{i_{q-2}}\prec v_{r_{1}}\prec
v_{r_{2}}\prec\cdots\prec v_{r_{q-2}}.$
Note that $r<n+1$, hence there must exists non-pendant vertex in
$\bigcup_{i=1}^{r-1}N_{T}(v_{i}).$ Choose the minimal element, say
$v_{l_{j}}$, under the order $\prec$ such that it is a non-pendant vertex.
Note that $P_{r+1}$ is the longest path in $T$, hence
$1<l<r-1.$ (6.2)
Thus, we can partition $T$ into two subtrees, say $S$ and $T_{0}$, such that
$E_{T}=E_{S}\cup E_{T_{0}},V_{T}=V_{S}\cup V_{T_{0}}$ and $V_{S}\cap
V_{T_{0}}=\\{v_{l_{j}}\\};$ see Fig. 7. For convenience, let
$N_{T_{0}}(v_{l_{j}})=\\{v_{0},w_{1},w_{2},\ldots,w_{q-1}\\}.$
Figure 7: An $q$-arc tree $T$.
Now we are in the position to apply Lemma 2.2 in the following setting:
$x\leftarrow v_{0},\ x_{i}\leftarrow v_{i},\
x_{\lfloor\frac{l}{2}\rfloor}\leftarrow v_{\lfloor\frac{l}{2}\rfloor}$
($z\leftarrow v_{\lceil\frac{l}{2}\rceil}$ if $l$ is odd)
$y\leftarrow v_{l_{j}},\ y_{i}\leftarrow v_{l+1-i},\
y_{\lfloor\frac{l}{2}\rfloor}\leftarrow v_{l+1-\lfloor\frac{l}{2}\rfloor}$
for $i=1,2,\ldots,\lfloor\frac{l}{2}\rfloor$.
Then
$\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle K_{1,q-2},\ \ \ \
i=1,2,\ldots,\lfloor\frac{l}{2}\rfloor,$ $\displaystyle Y_{i}$
$\displaystyle=$ $\displaystyle K_{1,q-2},\ \ \ \
i=2,3,\ldots,\lfloor\frac{l}{2}\rfloor$
and $Y_{1}$ is the component in $T-v_{l-1}v_{l}-v_{l}v_{l_{j}}$ which contains
$v_{l}$. By direct calculation, it is easy to see that
$f_{X_{i}}(x_{i})=f_{Y_{i}}(y_{i})=2^{q-2},\ \ \ \
i=2,\ldots,\lfloor\frac{l}{2}\rfloor.$
On the other hand, in veiw of (6.2) we know that $Y_{1}\not\cong K_{1,q-2}$
and $f_{Y_{1}}(y_{1})>2^{q-2},$ while $f_{X_{1}}(x_{1})=2^{q-2}.$ Therefore,
we have
$f_{X_{1}}(x_{1})<f_{Y_{1}}(y_{1}).$
By Lemma 2.2, we have
$f_{S}(x)<f_{S}(y),$
where $S$ is defined as above. Therefore, by Corollary 2.4, we have
$F(T^{\prime})<F(T),$ (6.3)
where
$T^{\prime}=T-\\{v_{l_{j}}w_{1},v_{l_{j}}w_{2},\ldots,v_{l_{j}}w_{q-1}\\}+\\{v_{0}w_{1},v_{0}w_{2},\ldots,v_{0}w_{q-1}\\}.$
Inequality (6.3) is a contradiction to the choice of $T$. Hence, we obtain
$r=n+1$, i.e., $T\cong\hat{T}_{n}^{q},$ as desired.
In order to complete the proof of Theorem 1.4, it suffices to determine
$F(\hat{T}_{n}^{q})$. Choose one of the longest paths in $\hat{T}_{n}^{q}$ and
denote it by $P$. Let $v_{0}$ be one of its end-vertices. Then denote the
unique neighbour of $v_{0}$ by $v_{1}$; see Fig. 3. Hence, we have
$\displaystyle f_{\hat{T}_{n}^{q}}(v_{1})$ $\displaystyle=$ $\displaystyle
2^{q-1}\left(1+2^{q-2}+\cdots+2^{(n-2)(q-2)}+2^{(n-1)(q-2)}+2^{(n-1)(q-2)}\right),$
$\displaystyle f_{\hat{T}_{n-1}^{q}}(v_{1})$ $\displaystyle=$ $\displaystyle
1\left(1+2^{q-2}+\cdots+2^{(n-2)(q-2)}+2^{(n-1)(q-2)}+2^{(n-1)(q-2)}\right).$
Note that
$\displaystyle F(\hat{T}_{n}^{q})$ $\displaystyle=$ $\displaystyle
f_{\hat{T}_{n}^{q}}(v_{1})+F(\hat{T}_{n}^{q}-v_{1})$ $\displaystyle=$
$\displaystyle
f_{\hat{T}_{n}^{q}}(v_{1})+F((q-1)P_{1}\cup(\hat{T}_{n-1}^{q}-v_{1}))$
$\displaystyle=$ $\displaystyle
f_{\hat{T}_{n}^{q}}(v_{1})+F(\hat{T}_{n-1}^{q}-v_{1})+q-1$ $\displaystyle=$
$\displaystyle
f_{\hat{T}_{n}^{q}}(v_{1})+F(\hat{T}_{n-1}^{q})-f_{\hat{T}_{n-1}^{q}}(v_{1})+q-1.$
This gives
$\displaystyle F(\hat{T}_{n}^{q})-F(\hat{T}_{n-1}^{q})$ $\displaystyle=$
$\displaystyle f_{\hat{T}_{n}^{q}}(v_{1})-f_{\hat{T}_{n-1}^{q}}(v_{1})+q-1$
$\displaystyle=$ $\displaystyle
q-1+(2^{q-1}-1)\left(1+2^{q-2}+\cdots+2^{(n-2)(q-2)}+2^{(n-1)(q-2)}+2^{(n-1)(q-2)}\right)$
$\displaystyle=$ $\displaystyle
q-1+(2^{q-1}-1)\left(\frac{2^{n(q-2)}-1}{2^{q-2}-1}+2^{(n-1)(q-2)}\right)$
$\displaystyle=$ $\displaystyle
q-1-\frac{2^{q-1}-1}{2^{q-2}-1}+\frac{2^{(n-1)(q-2)}(2^{q-1}-1)^{2}}{2^{q-2}-1}.$
As $F(\hat{T}_{1}^{q})=F(K_{1,q})=2^{q}+q$, we have
$\displaystyle F(\hat{T}_{n}^{q})$ $\displaystyle=$ $\displaystyle
F(B_{1})+\sum_{i=2}^{n}\left[q-3-\frac{1}{2^{q-2}-1}+\frac{2^{(i-1)(q-2)}(2^{q-1}-1)^{2}}{2^{q-2}-1}\right]$
$\displaystyle=$
$\displaystyle(2^{q}+q)+(n-1)(q-3)-\frac{n-1}{2^{q-2}-1}+\frac{(2^{q-1}-1)^{2}}{2^{q-2}-1}\sum_{i=2}^{n}2^{(i-1)(q-2)}$
$\displaystyle=$
$\displaystyle\frac{2^{q-2}(2^{q-1}-1)^{2}(2^{(n-1)(q-2)}-1)}{(2^{q-2}-1)^{2}}-\frac{n-1}{2^{q-2}-1}+2^{q}+nq-3n+3.$
This completes the proof. ∎
###### Remark 2.
In particular, let $q=3$ in Theorem 1.4, we can obtain that just the $n$-leaf
binary caterpillar tree minimizes the total number of subtrees among $n$-leaf
binary trees, which is obtained by Székely and Wang in [11].
###### Corollary 6.1 ([11]).
For any $n\geq 2$, precisely the $n$-leaf binary caterpillar tree
$\hat{T}_{n-2}^{3},$ which has $2^{n+1}+2^{n-2}-n-4$ subtrees, minimizes the
number of subtrees among $n$-leaf binary trees.
## 7 Concluding remarks
In view of Theorem 1.3, we conjecture that one may show the counterparts of
these results for the Wiener index among the $n$-vertex trees with a given
$(p,q)$-bipartition. On the other hand, for the Wiener index, sharp upper and
lower bounds of trees with given degree sequence are determined; see [13, 17,
18]. It is natural for us to determine sharp upper and lower bounds on the
total number of subtrees of a tree with given degree sequence. It is difficult
but interesting and it is still open.
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* [9] L.A. Székely, H. Wang, On subtrees of trees, 2004 Industrial Mathematics Institute Research Reports 04:04, University of South Carolina, http://www.math.sc.edu/ imip/04.html, 2004.
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|
arxiv-papers
| 2012-04-27T09:15:17 |
2024-09-04T02:49:30.270470
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuchao Li and Shujing Wang",
"submitter": "Shuchao Li",
"url": "https://arxiv.org/abs/1204.6152"
}
|
1204.6178
|
# Distributed Output-Feedback LQG Control with Delayed Information Sharing
Hamid Reza Feyzmahdavian, Ather Gattami, and Mikael Johansson H. R.
Feyzmahdavian, A. Gattami, and M. Johansson are with ACCESS Linnaeus Center,
School of Electrical Engineering, KTH-Royal Institute of Technology, SE-100 44
Stockholm, Sweden. E-mails: {hamidrez, gattami, mikaelj}@kth.se
###### Abstract
This paper develops a controller synthesis method for distributed LQG control
problems under output-feedback. We consider a system consisting of three
interconnected linear subsystems with a delayed information sharing structure.
While the state-feedback case has previously been solved, the extension to
output-feedback is nontrivial as the classical separation principle fails. To
find the optimal solution, the controller is decomposed into two independent
components: a centralized LQG-optimal controller under delayed state
observations, and a sum of correction terms based on additional local
information available to decision makers. Explicit discrete-time equations are
derived whose solutions are the gains of the optimal controller.111A
preliminary version of this work was presented in [1]
## I Introduction
Control with information constraints imposed on decision makers, sometimes
called team theory or distributed control, has been very challenging for
decision theory researchers. In general, several classes of these problems are
currently computationally intractable [2]. Early work [3] showed that even in
a simple static linear quadratic decision problem, complex nonlinear decisions
could outperform any given linear decision. As a result, much research has
focused on identifying classes of decentralized control problems that are
tractable [4, 5, 6, 7].
Distributed Linear Quadratic Gaussian (LQG) control with communication delays
has a rich literature dating back to the 1970s. Even though the LQG problem
under one-step delay information sharing pattern has been solved in [8, 9, 10,
11], generalizing their approaches to other delay structures is non-trivial.
In [12] and [13], a computationally efficient solution for the LQG output-
feedback problem with communication delays was presented using a state space
formulation and covariance constraints, but the controller structure is not
apparent from the corresponding semi-definite programming solution. In [14],
the authors consider LQG control with communication delays for the three
interconnected systems. While they provide an explicit solution, their
approach is restricted to state-feedback and assumes independence of
disturbances acting on each subsystem.
In this paper, we generalize the results in [14] to output-feedback and
correlated disturbances. We consider three interconnected systems over a
strongly connected graph, which implies information from neighbors is
available with one step delay and the global information is available to all
decision makers with two step delay. We derive an output-feedback law that
minimizes a finite-horizon quadratic cost. The problem considered here
provides the fundamental understanding for general delay structures.
The main contribution of this paper is the explicit state-space realization of
the LQG output-feedback problems with communication delays. The problem is
solved by decomposing the controller into two components. One is the same as
centralized LQG problem under two-step information delay and the other is the
sum of correction terms based on local information available to decision
makers. Specifically, the optimal control has the form
$\displaystyle
u(k)=F(k)\bigl{(}y(k)-C\widehat{x}^{[1]}(k)\bigr{)}+F^{[1]}(k)\bigl{(}y(k-1)-C\hat{x}(k-1|k-2)\bigr{)}+L(k)\hat{x}(k|k-2),$
where $\hat{x}(k-1|k-2)$ and $\hat{x}(k|k-2)$ is the one- and two-step
estimation of the state based on the common two-step delayed information, and
$\widehat{x}^{[1]}(k)$ is an improved state estimate based on local
information up to time $k-1$ available to decision makers at time $k$. While
the gain matrix $L$ might be full (in fact, it is the standard LQR gain
computed via discrete-time Riccati recursion), the gain matrices $F$ and
$F^{[1]}$ have a sparsity structure that complies with the information
constraints. We further show that $F$ and $F^{[1]}$ can be computed via convex
programming.
The paper is organized as follows. Section II defines the general problem
studied in this paper. In Section III, we review the standard discrete time
Kalman filter and derive an optimal estimation algorithm for the three-player
problem. In Section IV, it is shown that the three-player control problem can
be separated into two optimization problems. The main result of this paper is
stated in Section V. Numerical results are given in Section VI and finally
conclusions and future work are outlined in Section VII.
### I-A Notation
Throughout the paper, we use the following notation: matrices are written in
uppercase letters and vectors in lowercase letters. The sequence $x(0)$,
$x(1)$, $\ldots$ , $x(k)$ is denoted by $x(0:k)$. The symbol $I$ denotes the
identity matrix whose size can be determined from its context. For a matrix
$X$ partitioned into blocks, $[X]_{S_{1}S_{2}}$ denotes the sub-matrix of $X$
containing exactly those rows and columns corresponding to the sets $S_{1}$
and $S_{2}$, respectively. For instance
$[X]_{\\{1\\}\\{2,3\\}}=\begin{bmatrix}X_{12}&X_{13}\end{bmatrix}$. The trace
of a square matrix $X$ is denoted by $\textbf{Tr}\\{X\\}$. Given
$A\in\mathbb{R}^{m\times n}$, we can write $A$ in terms of its columns as
$A=\begin{bmatrix}a_{1}&\cdots&a_{n}\end{bmatrix}$. Then operation
$\textup{vec}(A)$ results in an $mn\times 1$ column vector
$\displaystyle\textup{vec}(A)=\begin{bmatrix}a_{1}\\\ \vdots\\\
a_{n}\end{bmatrix}.$
For $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{r\times s}$, the
operation $A\otimes B\in\mathbb{R}^{mr\times ns}$ denotes the Kronecker
product of $A$ and $B$. We denote the expectation of a random variable $x$ by
$\textbf{E}\\{x\\}$. The conditional expectation of $x$ given $y$ is denoted
by $\textbf{E}\\{x|y\\}$. The covariance of zero-mean random vectors $x$ and
$y$, defined by $\textbf{E}\\{xy^{T}\\}$, is denoted by
$\textbf{{Cov}}\\{x,y\\}$.
## II Problem Formulation
Consider the following linear discrete time system composed of $m$
interconnected subsystems
$\displaystyle\begin{split}x_{i}(k+1)&=\sum_{j=1}^{m}A_{ij}x_{j}(k)+B_{i}u_{i}(k)+w_{i}(k)\\\
y_{i}(k)&=C_{i}x_{i}(k)+v_{i}(k),\end{split}$ (1)
for $i=1,\ldots,m$. Here, $x_{i}\in\mathbb{R}^{n_{i}}$ is the state ,
$u_{i}\in\mathbb{R}^{q_{i}}$ is the control signal,
$y_{i}\in\mathbb{R}^{p_{i}}$ is the measurement output, $w_{i}$ is the
disturbance, and $v_{i}$ is the measurement noise of subsystem $i$. Here,
$A_{ij}\in\mathbb{R}^{n_{i}\times n_{j}}$, $B_{i}\in\mathbb{R}^{n_{i}\times
q_{i}}$ and $C_{i}\in\mathbb{R}^{p_{i}\times n_{i}}$ are constant matrices.
Let us define
$\displaystyle x=\begin{bmatrix}x_{1}\\\ \vdots\\\ x_{m}\\\
\end{bmatrix},\;u=\begin{bmatrix}u_{1}\\\ \vdots\\\ u_{m}\\\
\end{bmatrix},\;y=\begin{bmatrix}y_{1}\\\ \vdots\\\ y_{m}\\\
\end{bmatrix},\;w=\begin{bmatrix}w_{1}\\\ \vdots\\\ w_{m}\\\
\end{bmatrix},\;v=\begin{bmatrix}v_{1}\\\ \vdots\\\ v_{m}\\\ \end{bmatrix}.$
Then the system dynamics (1) can be written as
$\displaystyle\begin{split}x(k+1)&=Ax(k)+Bu(k)+w(k)\\\
y(k)&=Cx(k)+v(k),\end{split}$ (2)
where $A=[A_{ij}]\in\mathbb{R}^{n\times n}$,
$B=\textbf{diag}(B_{1},\ldots,B_{m})\in\mathbb{R}^{n\times q}$ and
$C=\textbf{diag}(C_{1},\ldots,C_{m})\in\mathbb{R}^{p\times n}$. Both $w$ and
$v$ are assumed to be Gaussian white noises with covariance matrix
$\displaystyle\textbf{E}\left\\{\begin{bmatrix}{w(k)}\\\
{v(k)}\end{bmatrix}{\begin{bmatrix}{w(l)}\\\
{v(l)}\end{bmatrix}}^{T}\right\\}=\delta(k-l)\begin{bmatrix}W&0\\\
0&V\end{bmatrix},$
where $\delta(k-l)=1$ if $k=l$ and $\delta(k-l)=0$ if $k\neq l$.
###### Assumption 1
$V$ is positive definite.
The interconnection structure of system (2) can be represented by a graph
$\mathcal{G}$ whose nodes correspond to subsystems. The graph $\mathcal{G}$
has an arrow from node $j$ to node $i$ if and only if $A_{ij}\neq 0$ (_i.e._
if $x_{j}(k)$ influences $x_{i}(k+1)$). Assume that $\mathcal{G}$ is strongly
connected and passing information from one node to another along the graph
takes one time step. Let $d_{ij}$ be the length of the shortest path from node
$i$ to node $j$ with $d_{ii}=0$. Then node $i$ receives the information
available to node $j$ after $d_{ji}$ time steps, and hence the available
information set of subsystem $i$ at time $k$ is given by
$\displaystyle\mathcal{I}_{i}(k)=\bigl{\\{}y_{1}(0:k-d_{1i}),\;\ldots\;,y_{i}(0:k),\;\ldots\;,y_{m}(0:k-d_{mi})\bigr{\\}}.$
(3)
The control problem is to minimize finite-horizon cost
$\displaystyle J=\textbf{E}\left\\{\sum_{k=0}^{N-1}{\begin{bmatrix}{x(k)}\\\
{u(k)}\end{bmatrix}}^{T}Q{\begin{bmatrix}{x(k)}\\\
{u(k)}\end{bmatrix}}+x(N)^{T}Q_{0}x(N)\right\\},$ (4)
subject to inputs of the form
$\displaystyle
u_{i}(k)=\mu_{i}\bigl{(}\mathcal{I}_{i}(k)\bigr{)},\;i=1,\ldots,m,$
where $\mu_{i}$ is the Borel-measurable function. Matrix $Q$ is partitioned
according to the dimensions of $x$ and $u$ as
$Q=\begin{bmatrix}Q_{xx}&Q_{xu}\\\ {Q}^{T}_{xu}&Q_{uu}\end{bmatrix}.$
###### Assumption 2
The matrices $Q_{0}$ and $Q$ are positive semi-definite, and $Q_{uu}$ is
positive definite.
The information structure (3) can be viewed as the consequence of delays in
the communication channels between the controllers. The assumptions about the
information structure and the sparsity of dynamics guarantee that information
propagates at least as fast as the dynamics on the graph. This information
pattern is a simple case of partially nested information structure that has
been studied in [4]. The optimal controller with this information pattern
exists and it is unique and linear.
While the approach proposed in this paper applies for linear systems over
strongly connected graphs, we will concentrate on a simple delayed information
control problem referred to as the three-player problem shown in Figure 1. For
this problem, the system matrices have the structure
$\displaystyle A=\begin{bmatrix}A_{11}&0&A_{13}\\\ A_{21}&A_{22}&0\\\
0&A_{32}&A_{33}\end{bmatrix},\;B=\begin{bmatrix}B_{1}&0&0\\\ 0&B_{2}&0\\\
0&0&B_{3}\end{bmatrix},\;C=\begin{bmatrix}C_{1}&0&0\\\ 0&C_{2}&0\\\
0&0&C_{3}\end{bmatrix},$
and the information available to each player at time $k$ is
$\displaystyle\mathcal{I}_{1}(k)$
$\displaystyle=\\{y_{1}(k),y_{1}(k-1),y_{3}(k-1),y(0:k-2)\\},$
$\displaystyle\mathcal{I}_{2}(k)$
$\displaystyle=\\{y_{2}(k),y_{1}(k-1),y_{2}(k-1),y(0:k-2)\\},$
$\displaystyle\mathcal{I}_{3}(k)$
$\displaystyle=\\{y_{3}(k),y_{2}(k-1),y_{3}(k-1),y(0:k-2)\\}.$
Since the information structure is partially nested, the optimal controller of
each player is a linear function of the elements of its information set.
Hence,
$\displaystyle u_{1}(k)$
$\displaystyle=f_{11}\bigl{(}y_{1}(k)\bigr{)}+f_{12}\bigl{(}y_{1}(k-1),y_{3}(k-1)\bigr{)}+f_{13}\bigl{(}y(0:k-2)\bigr{)},$
$\displaystyle u_{2}(k)$
$\displaystyle=f_{21}\bigl{(}y_{2}(k)\bigr{)}+f_{22}\bigl{(}y_{1}(k-1),y_{2}(k-1)\bigr{)}+f_{23}\bigl{(}y(0:k-2)\bigr{)},$
$\displaystyle u_{3}(k)$
$\displaystyle=f_{31}\bigl{(}y_{3}(k)\bigr{)}+f_{32}\bigl{(}y_{2}(k-1),y_{3}(k-1)\bigr{)}+f_{33}\bigl{(}y(0:k-2)\bigr{)},$
where $f_{ij}$ is a linear function for all $i$, $j$. Therefore, $u(k)$ can be
expressed as
$\displaystyle u(k)=F(k)y(k)+G(k){y(k-1)}+f\bigl{(}{y}(0:k-2)\bigr{)},$ (5)
where
$\displaystyle f={\begin{bmatrix}f_{13}\\\ f_{23}\\\ f_{33}\end{bmatrix}},\
F(k)=\begin{bmatrix}F_{11}(k)&0&0\\\ 0&F_{22}(k)&0\\\ 0&0&F_{33}(k)\\\
\end{bmatrix},\;G(k)=\begin{bmatrix}G_{11}(k)&0&G_{13}(k)\\\
G_{21}(k)&G_{22}(k)&0\\\ 0&G_{32}(k)&G_{33}(k)\\\ \end{bmatrix}.$
Note that the sparsity structures of $F$ and $G$ comply with the information
constraints at time $k$ and $k-1$, respectively. The control problem is now to
find matrices $F$ and $G$, as well as a linear function $f$, that minimize
$J$.
Figure 1: The graph illustrates the interconnection structure of three
players. The state of Player $1$ at time $k+1$ depends directly on the state
of Player $3$ at time $k$ since $A_{13}\neq 0$, hence there is an arc from
node $3$ to node $1$ in the interconnection graph. On the other hand, since
$A_{12}=0$, Player $1$ is not affected directly by the state of Player $2$,
and there is no arc from node $2$ to node $1$ in the interconnection graph.
## III Estimation Structure
This section presents an optimal estimation algorithm for the three-player
problem. First, we provide a short summary of standard Kalman filtering in
Subsection III-A. Next, Subsection III-B sketches a derivation of the
estimation algorithm. Finally, some properties of the algorithm are given in
Subsection III-C.
### III-A Preliminaries on Standard Kalman Filtering
Consider a linear system on the form (2), whose initial state $x(0)$ is
Gaussian with zero mean and covariance matrix $P_{0}$. Let us define the
following variables
$\displaystyle\widehat{x}(k|k-1)$ $\displaystyle\mathrel{\mathop{:}}=$
$\displaystyle\textbf{E}\\{x(k)|y(0:k-1)\\}$ $\displaystyle e(k)$
$\displaystyle\mathrel{\mathop{:}}=$ $\displaystyle x(k)-\widehat{x}(k|k-1)$
$\displaystyle P(k)$ $\displaystyle\mathrel{\mathop{:}}=$
$\displaystyle\textbf{E}\\{e(k)e^{T}(k)\\}.$
Here, $\widehat{x}(k|k-1)$ is the one-step prediction of the state, $e(k)$ is
the prediction error, and $P(k)$ is the covariance matrix of the prediction
error at time $k$. Assume that $u(k)$ is a deterministic function of $y(0:k)$.
The Kalman filter equations can be written as follows ([15])
$\displaystyle\begin{split}\widehat{x}(k+1|k)&=A\widehat{x}(k|k-1)+Bu(k)+K(k)\bigl{(}y(k)-C\widehat{x}(k|k-1)\bigr{)}\\\
P(k+1)&=AP(k)A^{T}+W-AP(k)C^{T}\bigl{(}CP(k)C^{T}+V\bigr{)}^{-1}CP(k)A^{T},\end{split}$
(6)
with $\widehat{x}(0|-1)=0$ and $P(0)=P_{0}$. Here, $K(k)$ is the optimal
Kalman gain given by
$\displaystyle K(k)$ $\displaystyle=AP(k)C^{T}\left(CP(k)C^{T}+V\right)^{-1}.$
The innovations are defined by
$\widetilde{y}(k)=y(k)-C\widehat{x}(k|k-1).$ (7)
The following proposition will be useful when deriving the optimal estimation
algorithm for the three-player problem.
###### Proposition 1
([15]) The following facts hold:
1. (a)
$\mathbb{{E}}\\{x(k)\widetilde{y}(k)^{T}\\}=P(k)C^{T}$.
2. (b)
$\widetilde{y}(k)$ is an uncorrelated Gaussian process with covariance matrix
$\widetilde{Y}(k)=CP(k)C^{T}+V$. Moreover, under Assumption $1$,
$\widetilde{Y}(k)$ is positive definite.
3. (c)
$\widetilde{y}(k)$ is independent of past measurements
$\displaystyle\mathbb{E}\\{\widetilde{y}(k)y^{T}(j)\\}=0\;\;\;\mbox{for}\;j<k.$
### III-B Kalman Filtering for Three-player Problem
Let $\mathcal{I}^{[1]}_{i}(k)$ be the set of all measurements up to time step
$k-1$ that are available to Player $i$ at time $k$. For example,
$\displaystyle\mathcal{I}^{[1]}_{1}(k)=\left\\{y_{1}(k-1),y_{3}(k-1),y(0:k-2)\right\\}.$
It is easy to verify that $\mathcal{I}^{[1]}_{i}(k)\subset y(0:k-1)$, _i.e._
it does not have access to all measurements taken at time $k-1$. Hence,
players cannot execute the one-step prediction of the standard Kalman filter
$\widehat{x}_{i}(k|k-1)$ at time $k$. Define
$\displaystyle\widehat{x}^{[1]}_{i}(k)\mathrel{\mathop{:}}=\textbf{E}\left\\{x_{i}(k)|\mathcal{I}^{[1]}_{i}(k)\right\\},\;\;i=1,2,3.$
We will now derive explicit expressions for these quantities.
Note that $y(0:k-2)$ is the piece of information available to all players.
Thus, $\widehat{x}(k-1|k-2)$ can be computed by each player at time $k$. To
see how the optimal estimation algorithm for the three-player problem is
derived, consider Player $1$. Let $[A]_{i}$ denote the $i$th block row of $A$.
Then,
$\displaystyle\widehat{x}^{[1]}_{1}(k)$
$\displaystyle=\textbf{E}\left\\{x_{1}(k)|\mathcal{I}^{[1]}_{1}(k)\right\\}$
$\displaystyle=[A]_{1}\textbf{E}\left\\{x(k-1)|\mathcal{I}^{[1]}_{1}(k)\right\\}+B_{1}\textbf{E}\left\\{u_{1}(k-1)|\mathcal{I}^{[1]}_{1}(k)\right\\}$
$\displaystyle=[A]_{1}\textbf{E}\left\\{x(k-1)|y_{1}(k-1),y_{3}(k-1),y(0:k-2)\right\\}+B_{1}u_{1}(k-1),$
where we used the independence of $w_{1}(k-1)$ and $\mathcal{I}^{[1]}_{1}(k)$,
and the fact that $u_{1}(k-1)$ is a deterministic function of the information
set $\mathcal{I}^{[1]}_{1}(k)$. To evaluate the expected value of $x(k-1)$
given $\mathcal{I}^{[1]}_{1}(k)$, we will first change the variables so that
we get independent variables. According to Proposition $1(\mbox{c})$, the
innovations $\widetilde{y}_{1}(k-1)$ and $\widetilde{y}_{3}(k-1)$ are
independent of $y(0:k-2)$. Thus,
$\displaystyle\widehat{x}^{[1]}_{1}(k)=$
$\displaystyle[A]_{1}\textbf{E}\\{x(k-1)|y(0:k-2)\\}+[A]_{1}\textbf{E}\\{x(k-1)|\widetilde{y}_{1}(k-1),\widetilde{y}_{3}(k-1)\\}+B_{1}u_{1}(k-1)$
$\displaystyle=$
$\displaystyle[A]_{1}\widehat{x}(k-1|k-2)+B_{1}u_{1}(k-1)+[A]_{1}\textbf{E}\\{x(k-1)|\widetilde{y}_{1}(k-1),\widetilde{y}_{3}(k-1)\\},$
(8)
where we used Proposition $\mbox{4}(\mbox{a})$ to get the first equality. We
will now calculate the last term of Equation (8). Let $S_{t}=\\{1,2,3\\}$ and
$S_{1}=\\{1,3\\}$. Then
$\displaystyle\textbf{E}\\{x(k-1)$
$\displaystyle|\widetilde{y}_{1}(k-1),\widetilde{y}_{3}(k-1)\\}$
$\displaystyle=\textbf{{Cov}}\left\\{x(k-1),\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix}\right\\}\textbf{{Cov}}^{-1}\left\\{\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix},\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix}\right\\}\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}[P(k-1)]_{11}C_{1}^{T}&[P(k-1)]_{13}C_{3}^{T}\\\
[P(k-1)]_{21}C_{1}^{T}&[P(k-1)]_{23}C_{3}^{T}\\\
[P(k-1)]_{31}C_{1}^{T}&[P(k-1)]_{33}C_{3}^{T}\end{bmatrix}\begin{bmatrix}C_{1}[P(k-1)]_{11}C_{1}^{T}+[V]_{11}&C_{1}[P(k-1)]_{13}C_{3}^{T}+[V]_{13}\\\
C_{3}[P(k-1)]_{31}C_{1}^{T}+[V]_{31}&C_{3}[P(k-1)]_{33}C_{3}^{T}+[V]_{33}\end{bmatrix}^{-1}\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix}$
$\displaystyle=[P(k-1)]_{S_{t}S_{1}}[C]_{S_{1}S_{1}}^{T}\bigl{(}[C]_{S_{1}S_{1}}[P(k-1)]_{S_{1}S_{1}}[C]_{S_{1}S_{1}}^{T}+[V]_{S_{1}S_{1}}\bigr{)}^{-1}\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix},$ (9)
where we used Proposition $\mbox{4}(\mbox{b})$ to get the first equality and
Proposition $\mbox{1}(\mbox{a})$-$\mbox{(}\mbox{b})$ to obtain the second
equality. Substituting Equation (9) into Equation (8) shows that
$\widehat{x}^{[1]}_{1}(k)$ is computed as
$\displaystyle\widehat{x}^{[1]}_{1}(k)=$
$\displaystyle[A]_{1}\widehat{x}(k-1|k-2)+B_{1}u_{1}(k-1)+\begin{bmatrix}{K}^{[1]}_{11}(k-1)&{K}^{[1]}_{13}(k-1)\end{bmatrix}\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix},$ (10)
where
$\displaystyle\begin{bmatrix}{K}^{[1]}_{11}(k-1)&{K}^{[1]}_{13}(k-1)\end{bmatrix}=[A]_{1}[P(k-1)]_{S_{t}S_{1}}[C]_{S_{1}S_{1}}^{T}\bigl{(}[C]_{S_{1}S_{1}}[P(k-1)]_{S_{1}S_{1}}[C]_{S_{1}S_{1}}^{T}+[V]_{S_{1}S_{1}}\bigr{)}^{-1}.$
Similar results can be obtained for Player $2$ and Player $3$. Let
$S_{2}=\\{1,2\\}$ and $S_{3}=\\{2,3\\}$. Then
$\displaystyle\widehat{x}^{[1]}_{2}(k)=$
$\displaystyle[A]_{2}\widehat{x}(k-1|k-2)+B_{2}u_{2}(k-1)+\begin{bmatrix}{K}^{[1]}_{21}(k-1)&{K}^{[1]}_{22}(k-1)\end{bmatrix}\begin{bmatrix}\widetilde{y}_{1}(k-1)\\\
\widetilde{y}_{2}(k-1)\end{bmatrix},$ (11)
$\displaystyle\widehat{x}^{[1]}_{3}(k)=$
$\displaystyle[A]_{3}\widehat{x}(k-1|k-2)+B_{3}u_{3}(k-1)+\begin{bmatrix}{K}^{[1]}_{32}(k-1)&{K}^{[1]}_{33}(k-1)\end{bmatrix}\begin{bmatrix}\widetilde{y}_{2}(k-1)\\\
\widetilde{y}_{3}(k-1)\end{bmatrix},$ (12)
where
$\displaystyle\begin{bmatrix}{K}^{[1]}_{21}(k-1)&{K}^{[1]}_{22}(k-1)\end{bmatrix}$
$\displaystyle=[A]_{2}[P(k-1)]_{S_{t}S_{2}}[C]_{S_{2}S_{2}}^{T}\left([C]_{S_{2}S_{2}}[P(k-1)]_{S_{2}S_{2}}[C]_{S_{2}S_{2}}^{T}+[V]_{S_{2}S_{2}}\right)^{-1},$
$\displaystyle\begin{bmatrix}{K}^{[1]}_{32}(k-1)&{K}^{[1]}_{33}(k-1)\end{bmatrix}$
$\displaystyle=[A]_{3}[P(k-1)]_{S_{t}S_{3}}[C]_{S_{3}S_{3}}^{T}\left([C]_{S_{3}S_{3}}[P(k-1)]_{S_{3}S_{3}}[C]_{S_{3}S_{3}}^{T}+[V]_{S_{3}S_{3}}\right)^{-1}.$
Define the matrix ${K}^{[1]}$ by
$\displaystyle{K}^{[1]}(k)=\begin{bmatrix}{K}^{[1]}_{11}(k)&0&{K}^{[1]}_{13}(k)\vspace{1mm}\\\
{K}^{[1]}_{21}(k)&{K}^{[1]}_{22}(k)&0\vspace{1mm}\\\
0&{K}^{[1]}_{32}(k)&{K}^{[1]}_{33}(k)\end{bmatrix}.$
Then equations (10)-(12) can be combined and written in the compact form as
$\displaystyle\widehat{x}^{[1]}(k)$
$\displaystyle=A\widehat{x}(k-1|k-2)+Bu(k-1)+{K}^{[1]}(k-1)\bigl{(}{y}(k-1)-C\widehat{x}(k-1|k-2)\bigr{)}.$
(13)
The Kalman filter iterations for the three-player problem at time $k$ is
summarized as follows
$\displaystyle\widehat{x}(k-1|k-2)$
$\displaystyle=A\widehat{x}(k-2|k-3)+Bu(k-2)+{K}(k-2)\bigl{(}{y}(k-2)-C\widehat{x}(k-2|k-3)\bigr{)}$
$\displaystyle\widehat{x}^{[1]}(k)$
$\displaystyle=A\widehat{x}(k-1|k-2)+Bu(k-1)+{K}^{[1]}(k-1)\bigl{(}{y}(k-1)-C\widehat{x}(k-1|k-2)\bigr{)}.$
(14)
Note that ${K}^{[1]}$ is not the usual Kalman filter gain and that it has a
the same sparsity pattern as $G$. Figure 2 shows the overall estimation scheme
of Player 1 at time $k$.
Figure 2: Optimal estimation scheme of Player 1 at time $k$.
###### Remark 1
Both ${K}^{[1]}$ and $K$ can be calculated off-line without knowing the
control input history $u(0:N-1)$.
### III-C Estimator properties
Here we compute some quantities that will help us in the following section.
Define
$\displaystyle{e}^{[1]}(k)$
$\displaystyle\mathrel{\mathop{:}}=x(k)-\widehat{x}^{[1]}(k)$
$\displaystyle\widetilde{y}^{\;[1]}(k)$
$\displaystyle\mathrel{\mathop{:}}=y(k)-C\widehat{x}^{[1]}(k).$
We denote the covariance matrices of ${e}^{[1]}(k)$ and
$\widetilde{y}^{\;[1]}(k)$ by $P^{[1]}(k)$ and $\widetilde{Y}^{[1]}(k)$,
respectively.
###### Lemma 1
Let $\bigtriangleup K(k)=K(k)-K^{[1]}(k)$. Then the following facts hold:
1. (a)
$P^{[1]}(k)=P(k)+\bigtriangleup K(k-1)\widetilde{Y}(k-1)\bigtriangleup
K^{T}(k-1).$
2. (b)
$\widetilde{Y}^{[1]}(k)=CP^{[1]}(k)C^{T}+V$. Also, under Assumption 1,
$\widetilde{Y}^{[1]}(k)$ is positive definite.
3. (c)
$\widetilde{P}(k)\mathrel{\mathop{:}}=\textbf{E}\left\\{{e}^{[1]}(k)\widetilde{y}^{\;T}(k-1)\right\\}=\bigtriangleup
K(k-1)\widetilde{Y}(k-1)$.
###### Proof:
See Appendix. ∎
## IV Optimal Controller Derivation
This section shows that finding optimal controller for the three-player
problem is equivalent to solving two separate optimization problems. Before
proceeding, we state the following proposition.
###### Proposition 2
([15]) Define the matrices
$\displaystyle S(k)$
$\displaystyle=A^{T}S(k+1)A+Q_{xx}-\bigl{(}A^{T}S(k+1)B+Q_{xu}\bigr{)}\bigl{(}B^{T}S(k+1)B+Q_{uu}\bigr{)}^{-1}\bigl{(}B^{T}S(k+1)A+Q^{T}_{xu}\bigr{)}$
$\displaystyle H(k)$ $\displaystyle=B^{T}S(k+1)B+Q_{uu}$ (15) $\displaystyle
L(k)$ $\displaystyle={H^{-1}(k)}\bigl{(}B^{T}S(k+1)A+Q_{xu}^{T}\bigr{)},$
for $k=0,\cdots,N-1$ and where $S(N)=Q_{0}$. Then the cost function (4) can be
written as
$\displaystyle J=$
$\displaystyle\underbrace{\sum_{k=0}^{N-1}\mathbb{E}\left\\{\bigl{(}u(k)-L(k)x(k)\bigr{)}^{T}H(k)\bigl{(}u(k)-L(k)x(k)\bigr{)}\right\\}}_{J_{u}}$
$\displaystyle+\underbrace{\mathbb{Tr}\\{S(0)P_{0}\\}+\sum_{k=0}^{N-1}\mathbb{Tr}\\{S(k+1)W\\}}_{J_{w}}.$
Moreover, $J_{w}$ is independent of the control.
From Proposition 2, it can be seen that minimizing $J$ is equivalent to
minimizing $J_{u}$. Also, under Assumption 2, $H(k)$ is positive definite for
all $k$.
The first step towards finding the structure of the optimal controller is to
decompose the state vector into independent terms using the following lemma:
###### Lemma 2
The state vector can be decomposed as
$\displaystyle x(k)$ $\displaystyle=\widetilde{x}(k)+\widehat{x}(k),$
where $\widehat{x}(k)$ and $\widetilde{x}(k)$ are independent and given by
$\displaystyle\widehat{x}(k)$ $\displaystyle=\mathbb{E}\\{x(k)|y(0:k-2)\\}$
$\displaystyle\widetilde{x}(k)$
$\displaystyle={e}^{[1]}(k)+\bigl{(}BF(k-1)+{K}^{[1]}(k-1)\bigr{)}\widetilde{y}(k-1).$
###### Proof:
See appendix. ∎
Note that the term $\widehat{x}(k)$ is the conditional estimate of the state
$x(k)$ given the information shared by all players, and $\widetilde{x}(k)$ is
the estimation error. Now that the state vector has been decomposed into
independent terms, the control input $u(k)$ can be decomposed in an analogue
manner.
###### Lemma 3
The control input $u(k)$ can be decomposed into two independent terms
$\displaystyle u(k)=\widetilde{u}(k)+\widehat{u}(k),$
where
$\displaystyle\widehat{u}(k)$ $\displaystyle=\mathbb{E}\\{u(k)|y(0:k-2)\\}$
$\displaystyle\widetilde{u}(k)$
$\displaystyle=F(k)\widetilde{y}^{[1]}(k)+F^{[1]}(k)\widetilde{y}(k-1),$
and $F^{[1]}$ is given by
$F^{[1]}(k)=G(k)+F(k)C\bigl{(}{K}^{[1]}(k-1)+BF(k-1)\bigr{)}.$ (16)
###### Proof:
See appendix. ∎
###### Remark 2
Since $B$, $C$ and $F$ are diagonal matrices, $G(k)$ and $F(k)C{K}^{[1]}(k-1)$
have the same sparsity pattern. Similarly, $F^{[1]}(k)$ and $G(k)$ have the
same sparsity pattern.
From lemmas 2 and 3, both $\widehat{x}(k)$ and $\widehat{u}(k)$ are functions
of $y(0:k-2)$ which is independent of $\widetilde{x}(k)$ and
$\widetilde{u}(k)$. As a result the cost function $J_{u}$ can be decomposed as
$\displaystyle J_{u}$
$\displaystyle=\underbrace{\sum_{k=0}^{N-1}\textbf{E}\left\\{\bigl{(}\widetilde{u}(k)-L(k)\widetilde{x}(k)\bigr{)}^{T}H(k)\bigl{(}\widetilde{u}(k)-L(k)\widetilde{x}(k)\bigr{)}\right\\}}_{\widetilde{J}}$
$\displaystyle+\underbrace{\sum_{k=0}^{N-1}\textbf{E}\left\\{\bigl{(}\widehat{u}(k)-L(k)\widehat{x}(k)\bigr{)}^{T}H(k)\bigl{(}\widehat{u}(k)-L(k)\widehat{x}(k)\bigr{)}\right\\}}_{\widehat{J}},$
and the optimal control problem reduces to solving
Problem 1.minimize $\displaystyle\;\;\;\widehat{J}(\widehat{x},\widehat{u})$
subject to $\displaystyle\;\;\;\widehat{u}(k)\;\mbox{is a function
of}\;y(0:k-2).$ Problem 2.minimize
$\displaystyle\;\;\;\widetilde{J}(\widetilde{x},\widetilde{u})$ subject to
$\displaystyle\;\;\;\widetilde{u}(k)=F(k)\widetilde{y}^{[1]}(k)+F^{[1]}(k)\widetilde{y}(k-1),$
$\displaystyle\;\;\;F(k)\;\mbox{and}\;F^{[1]}(k)\;\mbox{have specified
sparsity structures.}$
The following lemma shows that the optimal solution $\widehat{u}(k)$ for
Problem 1 is exactly the optimal controller for centralized information
structure with two-step delay, where the information set of each player is
$y(0:k-2)$.
###### Lemma 4
Suppose assumptions $1$ and $2$ hold. An optimal solution for Problem 1 is
given by
$\displaystyle\widehat{u}(k)$ $\displaystyle=$ $\displaystyle
L(k)\widehat{x}(k)$ (17) $\displaystyle=$ $\displaystyle
L(k)\mathbb{E}\\{x(k)|y(0:k-2)\\}.$
Moreover, the optimal value of the cost function $\widehat{J}$ is zero.
###### Proof:
See appendix. ∎
We now focus on Problem 2, namely the computation of
$\left\\{F(k)\right\\}_{k=0,\ldots,N-1}$ and
$\left\\{F^{[1]}(k)\right\\}_{k=1,\ldots,N-1}$. Recalling the expansions of
$\widetilde{x}(k)$ and $\widetilde{u}(k)$ in terms of
$\widetilde{y}^{[1]}(k)$, ${e}^{[1]}(k)$, and $\widetilde{y}(k-1)$,
$\widetilde{J}$ can be expanded as follows
$\displaystyle\widetilde{J}=$
$\displaystyle\sum_{k=0}^{N-1}\textbf{E}\left\\{\bigl{(}\widetilde{u}(k)-L(k)\widetilde{x}(k)\bigr{)}^{T}H(k)\bigl{(}\widetilde{u}(k)-L(k)\widetilde{x}(k)\bigr{)}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\textbf{Tr}\left\\{H(k)F(k)VF^{T}(k)\right\\}+\textbf{Tr}\left\\{H(k)\bigl{(}F(k)C-L(k)\bigr{)}P^{[1]}(k)\bigl{(}F(k)C-L(k)\bigr{)}^{T}\right\\}$
$\displaystyle+\textbf{Tr}\left\\{H(k)\bigl{(}F^{[1]}(k)-L(k)(BF(k-1)+{K}^{[1]}(k-1)\bigr{)}\widetilde{Y}(k-1)\bigl{(}F^{[1]}(k)-L(k)(BF(k-1)+{K}^{[1]}(k-1)\bigr{)}^{T}\right\\}$
$\displaystyle+2\textbf{Tr}\left\\{H(k)\bigl{(}F(k)C-L(k)\bigr{)}\widetilde{P}(k)\bigl{(}F^{[1]}(k)-L(k)(BF(k-1)+{K}^{[1]}(k-1)\bigr{)}^{T}\right\\},$
(18)
where we used Proposition 4(c). A point worth noticing is that according to
Proposition 1 and Lemma 1, $P^{[1]}$, $\widetilde{P}$, and $\widetilde{Y}$ are
independent of $F(k)$ and $F^{[1]}(k)$. To minimize $\widetilde{J}$ with
respect to $F(k)$ and $F^{[1]}(k)$, we face two difficulties: the first is
that $F(k)$ and $F^{[1]}(k)$ must satisfy given sparsity constraints; the
second difficulty is the existence of coupling terms between $F(k-1)$ and
$F(k)$. To overcome these difficulties, we will use the vec operator and the
following lemma:
###### Lemma 5
Assume that $A\in\mathbb{R}^{n\times m}$ is split into sub-blocks as follows:
$\displaystyle A=\begin{bmatrix}A_{11}&\cdots&A_{1q}\\\ \vdots&&\vdots\\\
A_{p1}&\cdots&A_{pq}\end{bmatrix},$
where $A\in\mathbb{R}^{n_{i}\times m_{j}}$ for $i=1,\ldots,p$ and
$j=1,\ldots,q$. Let $S$ be the set of non-zero sub-blocks of $A$,
$S=\\{A_{ij}\mid A_{ij}\neq 0\\},\;\mid S\mid=s.$
Then there always exists a full column rank matrix $E$ of an appropriate
dimension such that
$\textup{vec}(A)=E\begin{bmatrix}\textup{vec}(A_{i_{1}j_{1}})\\\ \vdots\\\
\textup{vec}(A_{i_{s}j_{s}})\end{bmatrix},$
where $A_{i_{k}j_{k}}\in S$ for all $k=1,\ldots,s$.
###### Proof:
See appendix. ∎
The way to construct matrix $E$ is described in Appendix. Lemma 5 ensures the
existence of $E_{1}$ and $E_{2}$ such that
$\displaystyle\textup{vec}\bigl{(}F(k)\bigr{)}$
$\displaystyle=E_{1}\xi_{1}(k),$
$\displaystyle\textup{vec}\left(F^{[1]}(k)\right)$
$\displaystyle=E_{2}\xi_{2}(k),$
where $\xi_{1}$ and $\xi_{2}$ are vectors formed by stacking all nonzero sub-
blocks of $F$ and $F^{[1]}$, respectively. That is,
$\displaystyle\textrm{vec}\bigl{(}F(k)\bigr{)}$
$\displaystyle=E_{1}\underbrace{\begin{bmatrix}\textrm{vec}^{T}\left(F_{11}\right)&\textrm{vec}^{T}\left(F_{22}\right)&\textrm{vec}^{T}\left(F_{33}\right)\end{bmatrix}^{T}}_{\xi_{1}(k)},$
$\displaystyle\textrm{vec}\left(F^{[1]}(k)\right)$
$\displaystyle=E_{2}\underbrace{\begin{bmatrix}\textrm{vec}^{T}\left(F^{[1]}_{11}\right)&\textrm{vec}^{T}\left(F^{[1]}_{21}\right)&\textrm{vec}^{T}\left(F^{[1]}_{22}\right)&\textrm{vec}^{T}\left(F^{[1]}_{32}\right)&\textrm{vec}^{T}\left(F^{[1]}_{13}\right)&\textrm{vec}^{T}\left(F^{[1]}_{33}\right)\end{bmatrix}^{T}}_{\xi_{2}(k)}.$
We now show how vectorization allows to convert Problem 2 into an
unconstrained convex optimization problem.
###### Lemma 6
Let $E=\mathbb{diag}(E_{1},E_{2})$, $\zeta(k)=\begin{bmatrix}\xi_{1}(k-1)\\\
\xi_{2}(k)\end{bmatrix}$ for $k=1,\ldots,N-1$, and $\zeta(N)=\xi_{1}(N-1)$.
Define
$\displaystyle Z_{1}(k)$ $\displaystyle=E^{T}\begin{bmatrix}I&0\\\ -I\otimes
L(k)B&I\end{bmatrix}^{T}\begin{bmatrix}\widetilde{Y}^{[1]}(k-1)\otimes
H(k-1)&0\\\ 0&\widetilde{Y}(k-1)\otimes H(k)\end{bmatrix}\begin{bmatrix}I&0\\\
-I\otimes L(k)B&I\end{bmatrix}E,$ $\displaystyle Z_{2}(k)$
$\displaystyle=E\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}^{T}\left(\widetilde{P}^{T}(k)C^{T}\otimes
H(k)\right)\begin{bmatrix}I&0\end{bmatrix}E,$ $\displaystyle b(k)$
$\displaystyle=E^{T}\begin{bmatrix}I&0\end{bmatrix}^{T}\left(CP^{[1]}(k-1)\otimes
H(k-1)\right)\textup{vec}\bigl{(}L(k-1)\bigr{)}$
$\displaystyle+E^{T}\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}^{T}\left(\widetilde{Y}(k-1)\otimes
H(k)\right)\textup{vec}\bigl{(}L(k)K^{[1]}(k-1)\bigr{)},$
with
$\displaystyle Z_{1}(N)=$ $\displaystyle
E_{1}^{T}\bigl{(}\widetilde{Y}^{[1]}(N-1)\otimes H(N-1)\bigr{)}E,$
$\displaystyle b(N)=$ $\displaystyle E_{1}^{T}\left(CP^{[1]}(N-1)\otimes
H(N-1)\right)\textup{vec}\bigl{(}L(N-1)\bigr{)}.$
Then Problem 2 is equivalent to
$\displaystyle\min_{\zeta(1),\ldots,\zeta(N)}=$
$\displaystyle\sum_{k=1}^{N-1}{1\over
2}\zeta^{T}(k){Z}_{1}(k)\zeta(k)+\zeta^{T}(k){Z}_{2}(k)\zeta(k+1)-\zeta^{T}(k){b}(k)$
$\displaystyle+{1\over 2}\zeta^{T}(N){Z}_{1}(N)\zeta(N)-\zeta^{T}(N){b}(N)$
(19)
Moreover, ${Z}_{1}(k)$ is positive definite for all $k$.
###### Proof:
See Appendix. ∎
Consider the two time-step case of (19)
$\displaystyle\min_{\zeta(1),\zeta(2)}\underbrace{{1\over
2}\zeta^{T}(1){Z}_{1}(1)\zeta(1)-\zeta^{T}(1){b}(1)}_{g_{1}(\zeta(1))}+\underbrace{\zeta^{T}(1){Z}_{2}(1)\zeta(2)+{1\over
2}\zeta^{T}(2){Z}_{1}(2)\zeta(2)-\zeta^{T}(2){b}(2)}_{g_{2}(\zeta(1),\zeta(2))}.$
(20)
The optimal $\zeta(2)$ is the one which minimizes $g_{2}$, _i.e._
$\displaystyle\zeta^{\star}(2)$
$\displaystyle=\operatorname*{arg\,min}_{\zeta(2)}g_{2}\bigl{(}\zeta(1),\zeta(2)\bigr{)}$
$\displaystyle=-Z_{1}^{-1}(2)\bigl{(}{Z}^{T}_{2}(1)\zeta(1)-b(2)\bigr{)}.$
If we substitute the optimal $\zeta^{\star}(2)$ into (20), then we can
minimize
$g_{1}\bigl{(}\zeta(1)\bigr{)}+g_{2}\bigl{(}\zeta(1),\zeta^{\star}(2)\bigr{)}$
with respect to $\zeta(1)$. Therefore,
$\displaystyle\zeta^{\star}(1)$
$\displaystyle=\operatorname*{arg\,min}_{\zeta(1)}g_{1}\bigl{(}\zeta(1)\bigr{)}+g_{2}\bigl{(}\zeta(1),\zeta^{\star}(2)\bigr{)}$
$\displaystyle=R_{1}^{-1}(1)c(1),$
where
$\displaystyle R(1)$ $\displaystyle=$
$\displaystyle{Z}_{1}(1)-{Z}_{2}(1)Z_{1}^{-1}(2){Z}^{T}_{2}(1),$
$\displaystyle c(1)$ $\displaystyle=$
$\displaystyle{b}(1)-{Z}_{2}(1)Z_{1}^{-1}(2)b(2).$
The extension to more time steps is straightforward. The result is stated in
the following lemma.
###### Lemma 7
Suppose assumptions $1$ and $2$ hold. Define
$\displaystyle R(k)$ $\displaystyle=$
$\displaystyle{Z}_{1}(k)-{Z}_{2}(k)R^{-1}(k+1){Z}^{T}_{2}(k)$ $\displaystyle
c(k)$ $\displaystyle=$ $\displaystyle{b}(k)-{Z}_{2}(k)R^{-1}(k+1)c(k+1),$
with the end condition $R(N)={Z}_{1}(N)$ and $c(N)=b(N)$. Then optimization
problem (19) has the unique solution
$\displaystyle\zeta(k+1)=-R^{-1}(k+1)\bigl{(}{Z}^{T}_{2}(k)\zeta(k)-c(k+1)\bigr{)},$
(21)
with initial condition $\zeta(1)=R^{-1}(1)c(1)$. Moreover, ${R}(k)$ is
positive definite for all $k$.
## V Main Results
We can now state our main result, Theorem 1, which gives the optimal
controller for the three-player problem.
###### Theorem 1
Suppose assumptions $1$ and $2$ hold. Let
$\hat{x}(k)=\mathbb{E}\\{x(k)|y(0:k-2)\\}$. Then optimal controller for the
three-player problem is given by
$\displaystyle u(k)$
$\displaystyle=F(k)\bigl{(}y(k)-C\widehat{x}^{[1]}(k)\bigr{)}+F^{[1]}(k)\bigl{(}y(k-1)-C\hat{x}(k-1|k-2)\bigr{)}+L(k)\hat{x}(k),$
(22)
where $\widehat{x}^{[1]}(k)$ and $\hat{x}(k-1|k-2)$ are the optimal state
estimates obtained using the Kalman filter iterations (14), $L$ is given by
Equation (15), and $F$ and $F^{[1]}$ are given by Equation (21). Moreover,
$\displaystyle\hat{x}(k)=$
$\displaystyle\widehat{x}^{[1]}(k)-\bigl{(}BF(k-1)+{K}^{[1]}(k-1)\bigr{)}\bigl{(}y(k-1)-C\hat{x}(k-1|k-2)\bigr{)}.$
Having derived the optimal controller, a number of remarks are in order.
###### Remark 3
A physical interpretation of the optimal control policy is given as follows:
The third term of optimal controller, $L(k)\hat{x}(k)$, is exactly the optimal
policy for centralized information structure with two-step delay, where the
information set of each player is $y(0:k-2)$. The first and second terms are
correction terms based on local measurements from time $k$ and $k-1$,
respectively, which are available to each player.
###### Remark 4
The recursive equation (21) reveals a new feature present neither in LQG
control with one-step delay sharing information pattern nor in the state-
feedback case: the optimal control gain at time $k$, $\zeta(k)$, is an affine
function of $\zeta(k-1)$. For example, in the state-feedback case where
$y_{i}(k)=x_{i}(k)$ for $i=1,2,3$, we have
$\displaystyle\widetilde{P}(k)=\mathbf{E}\\{w(k-1)w^{T}(k-2)\\}=0.$
According to Lemma 6, ${Z}_{2}(k)=0$, and hence Equation (21) reduces to
$\zeta(k)=Z_{1}^{-1}(k)b(k).$
###### Remark 5
Equating the right hand side of equations (5) and (22) shows that the linear
function $f$ is given by
$\displaystyle f=\bigl{(}L(k)-F(k)C\bigr{)}\hat{x}(k)-G(k)C\hat{x}(k-1|k-2),$
where $G$ is given by Equation (16). Note that both $\hat{x}(k)$ and
$\hat{x}(k-1|k-2)$ are linear functions of $y(0:k-2)$.
###### Remark 6
If $A\in\mathbb{R}^{n\times n}$, then the optimal controller for the three-
player problem has at most $2n$ states.
## VI Numerical Example
We conclude our discussion of the three-player problem with an example.
Consider a simple system specified by
$\displaystyle A=\begin{bmatrix}2&0&1\\\ 1&2&0\\\
0&1&2\end{bmatrix},\;B=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix},\;C=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix}.$
$w$ and $v$ are Gaussian with zero mean and identity covariance matrix. The
time horizon ${N}$ is chosen to be $1000$ and the cost weight matrices are
given by
$\displaystyle Q_{xx}=\begin{bmatrix}3&1&1\\\ 1&3&1\\\
1&1&3\end{bmatrix},Q_{xu}=\begin{bmatrix}1&0&-1\\\ -1&1&0\\\
0&-1&1\end{bmatrix},Q_{uu}=\begin{bmatrix}2&0&0\\\ 0&2&0\\\
0&0&2\end{bmatrix},$
and $Q_{0}=Q_{xx}$.
We will compare the optimal controller for the three-player problem to
controllers for the following information structures
1. 1.
Centralized with two-step delay: $u_{i}(k)=\mu_{i}\bigl{(}y(0:k-2)\bigr{)}$,
2. 2.
One-step delay sharing information pattern:
$u_{i}(k)=\mu_{i}\bigl{(}y_{i}(k),y(0:k-1)\bigr{)}$,
3. 3.
Centralized without delay: $u_{i}(k)=\mu_{i}\bigl{(}y(0:k)\bigr{)}$.
The one-step delay sharing information pattern studied in [8, 9, 10, 11] is
specified by the graph in Figure 3.
Figure 3: The graph illustrates the communication structure of one-step delay
information pattern. Each controller passes information to both neighbors
after one-step delay.
By minimizing cost function (4), we obtain Table 1. Centralized controller
without delay has the lowest cost as expected. The three-player controller
outperforms the centralized controller with two-step delay by a substantial
margin, and only around $\mbox{1.74}\%$ higher than one-step delay sharing
information pattern control. In other words, for three-player problem, there
is a slight benefit of having two-way communication between controllers.
TABLE I: Simulation Results for Total Cost Control law | Cost mean
---|---
Centralized with delay | 14757
Three-player | 339.9
One-step delay information pattern | 334.1
Centralized without delay | 188.8
Comparison of the costs shows the benefits of using all available information.
## VII Conclusion
In this paper, we presented an explicit solution for a distributed LQG problem
in which three players communicate their information with delays. This was
accomplished via decomposition of the state and input vectors into two
independent terms and using this decomposition to separate the optimal control
problem to two subproblems. Computing the gains of the optimal controller
requires solving one standard discrete-time Riccati equation and one recursive
equation. Future work will continue to extend our approach to the infinite-
horizon and more general networks.
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## VIII Appendix
### VIII-A Preliminaries
###### Proposition 3
([16]) If $A$, $B$, $C$, $D$, $X$ and $Y$ are suitably dimensioned matrices,
then
1. (a)
$\textup{vec}(AXB)=(B^{T}\otimes A)\textup{vec}(X)$,
2. (b)
If $A$ and $B$ are positive definite, then so is $A\otimes B$,
3. (c)
$\mathbb{Tr}\\{AXBY^{T}\\}=\textup{vec}^{T}(Y)(B^{T}\otimes
A)\textup{vec}(X)$,
4. (d)
$(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$.
5. (e)
Let $X\in\mathbb{R}^{m\times n}$, then there exists a unique permutation
matrix $P_{m,n}\in\mathbb{R}^{mn\times mn}$ such that
$\textup{vec}(X^{T})=P_{m,n}\textup{vec}(X)$. The matrix $P_{m,n}$ is given by
$P_{m,n}=\sum_{i=1}^{m}\sum_{j=1}^{n}E_{ij}\otimes E_{ij}^{T},$
where $E_{ij}\in\mathbb{R}^{m\times n}$ has a one in the $(i,j)$ entry and
every other entry is zero.
###### Proposition 4
[15]) Let $x$, $y$ and $z$ be zero-mean random vectors with a jointly Gaussian
distribution, and let $y$ and $z$ be independent. Also, let $S$ be a symmetric
matrix. Then the following facts hold:
1. (a)
$\mathbb{E}\\{x|y,z\\}=\mathbb{E}\\{x|y\\}+\mathbb{E}\\{x|z\\}$.
2. (b)
$\mathbb{E}\\{x|y\\}=\mathbb{{Cov}}\\{x,y\\}\mathbb{{Cov}}^{-1}\\{y,y\\}y$.
3. (c)
$\mathbb{E}\\{x^{T}Sx\\}=\mathbb{Tr}\left\\{S\mathbb{{Cov}}\\{x,x\\}\right\\}$.
4. (d)
$\mathbb{E}\\{x|y\\}$ and $x-\mathbb{E}\\{x|y\\}$ are independent.
### VIII-B Proof Lemma 1
To express the conditional estimate $\widehat{x}(k|k-1)$ in terms of
$\widehat{x}^{[1]}(k)$, we substitute Equation (13) into Equation (6) to
eliminate $A\widehat{x}(k|k-1)+Bu(k)$. We have
$\displaystyle\widehat{x}(k|k-1)$
$\displaystyle=\widehat{x}^{[1]}(k)+\bigl{(}K(k-1)-{K}^{[1]}(k-1)\bigr{)}\bigl{(}{y}(k-1)-C\widehat{x}(k-1|k-2)\bigr{)}$
$\displaystyle=\widehat{x}^{[1]}(k)+\bigtriangleup K(k-1)\widetilde{y}(k-1).$
(23)
Plugging $\widehat{x}(k|k-1)=x(k)-e(k)$ and
$\widehat{x}^{[1]}(k)=x(k)-e^{[1]}(k)$ into Equation (23) leads to
$\displaystyle e^{[1]}(k)=e(k)+\bigtriangleup K(k-1)\widetilde{y}(k-1).$ (24)
Since $e(k)$ is independent of $y(0:k-1)$, the two terms on the right hand
side of Equation (24) are independent. Thus,
$\displaystyle P^{[1]}(k)$
$\displaystyle=\textbf{E}\left\\{{e}^{[1]}(k){{e}^{[1]}(k)}^{T}\right\\}$
$\displaystyle=P(k)+\bigtriangleup K(k-1)\widetilde{Y}(k-1)\bigtriangleup
K^{T}(k-1),$ $\displaystyle\widetilde{Y}^{[1]}(k)$
$\displaystyle=\textbf{E}\left\\{\widetilde{y}^{[1]}(k){\widetilde{y}^{[1]}(k)}^{T}\right\\}$
$\displaystyle=CP^{[1]}(k)C^{T}+V,$ $\displaystyle\widetilde{P}(k)$
$\displaystyle=\textbf{E}\left\\{{e}^{[1]}(k)\widetilde{y}^{\;T}(k-1)\right\\}$
$\displaystyle=\bigtriangleup K(k-1)\widetilde{Y}(k-1).$
### VIII-C Proof Lemma 2
The independence between $x(k)-\widehat{x}(k)$ and $\widehat{x}(k)$ can be
established by Proposition $\mbox{4}(\mbox{d})$. To calculate
$\widetilde{x}(k)$, we proceed in three steps. First consider
$\displaystyle
u(k-1)=F(k-1)y(k-1)+G(k-1){y(k-2)}+f\bigl{(}{y}(0:k-3)\bigr{)},$
where we used Equation (5). Since $G(k-1){y(k-2)}+f\bigl{(}{y}(0:k-3)\bigr{)}$
is a deterministic function of $y(0:k-2)$, we have
$\displaystyle u(k-1)-\textbf{E}\left\\{u(k-1)|y(0:k-2)\right\\}$
$\displaystyle=F(k-1)\bigl{(}y(k-1)-\textbf{E}\\{y(k-1)|y(0:k-2)\\}\bigr{)}$
$\displaystyle=F(k-1)\widetilde{y}(k-1),$ (25)
where we used the definition of $\widetilde{y}$ (Equation (7)) to get the
second equality. Second, consider
$\displaystyle\widehat{x}^{[1]}(k)$
$\displaystyle=A\widehat{x}(k-1|k-2)+Bu(k-1)+{K}^{[1]}(k-1)\widetilde{y}(k-1),$
where we used Equation (13). Since $\widehat{x}(k-1|k-2)$ is a linear function
of $y(0:k-2)$, we have
$\displaystyle\widehat{x}^{[1]}(k)-\textbf{E}\\{\widehat{x}^{[1]}(k)|y(0:k-2)\\}$
$\displaystyle={K}^{[1]}(k-1)\widetilde{y}(k-1)+B\bigl{(}u(k-1)-\textbf{E}\\{u(k-1)|y(0:k-2)\\}\bigr{)}$
$\displaystyle=({K}^{[1]}(k-1)+BF(k-1))\widetilde{y}(k-1),$ (26)
where we used the independence of $\widetilde{y}(k-1)$ and $y(0:k-2)$ to get
the first equality, and Equation (25) to obtain the second equality. Finally,
note that $x(k)={e}^{[1]}(k)+\widehat{x}^{[1]}(k)$. Thus,
$\displaystyle\widetilde{x}(k)$
$\displaystyle=x(k)-\textbf{E}\\{x(k)|y(0:k-2)\\}$
$\displaystyle={e}^{[1]}(k)+\bigl{(}\widehat{x}^{[1]}(k)-\textbf{E}\\{\widehat{x}^{[1]}(k)|y(0:k-2)\\}\bigr{)}$
$\displaystyle={e}^{[1]}(k)+\left({K}^{[1]}(k-1)+BF(k-1)\right)\widetilde{y}(k-1),$
(27)
where we used the independence of ${e}^{[1]}(k)$ and $y(0:k-2)$ to get the
second equality and Equation (26) to obtain the last equality.
### VIII-D Proof Lemma 3
According to Proposition $\mbox{4}(\mbox{d})$, $\widehat{u}(k)$ is independent
of $u(k)-\widehat{u}(k)$. Note that $v(k)$ is independent of the previous
outputs, so
$\displaystyle y(k)-\textbf{E}\\{y(k)|y(0:k-2)\\}$
$\displaystyle=v(k)+C\bigl{(}x(k)-\textbf{E}\\{x(k)|y(0:k-2)\\}\bigr{)}$
$\displaystyle=v(k)+C\bigl{(}{e}^{[1]}(k)+\left(BF(k-1)+{K}^{[1]}(k-1)\right)\widetilde{y}(k-1)\bigr{)}$
$\displaystyle=\widetilde{y}^{[1]}(k)+C\bigl{(}BF(k-1)+{K}^{[1]}(k-1)\bigr{)}\widetilde{y}(k-1),$
(28)
where we used Equation (27) to get the second equality and the definition of
$\widetilde{y}^{[1]}$ (Equation(7)) to obtain the last equality. Since
$f(y(0:k-2))$ is a linear function of $y(0:k-2)$, we have
$\displaystyle\widetilde{u}(k)=$ $\displaystyle
u(k)-\textbf{E}\\{u(k)|y(0:k-2)\\}$ $\displaystyle=$ $\displaystyle
F(k)\bigl{(}y(k)-\textbf{E}\\{y(k)|y(0:k-2)\\}\bigr{)}+G(k)\bigl{(}y(k-1)-\textbf{E}\\{y(k-1)|y(0:k-2)\\}\bigr{)}$
$\displaystyle=$ $\displaystyle
F(k)\left(\widetilde{y}^{[1]}(k)+C\bigl{(}BF(k-1)+{K}^{[1]}(k-1)\bigr{)}\widetilde{y}(k-1)\right)+G(k)\widetilde{y}(k-1)$
$\displaystyle=$ $\displaystyle
F(k)\widetilde{y}^{[1]}(k)+\bigl{(}F(k)C(BF(k-1)+{K}^{[1]}(k-1))+G(k)\bigr{)}\widetilde{y}(k-1),$
where we used Equation (28) and the definition of $\widetilde{y}$ (Equation
(7)) to get the third equality. The proof is completed by defining
$\displaystyle F^{[1]}(k)=G(k)+F(k)C\bigl{(}{K}^{[1]}(k-1)+BF(k-1)\bigr{)}.$
### VIII-E Proof Lemma 4
Due to the assumptions, $H(k)$ is positive definite and hence all terms in the
$\widehat{J}$ are positive. Since $\widehat{u}(k)$ and $\widehat{x}(k)$ are
functions of $y(0:k-2)$, the optimal controller is given by (17).
### VIII-F Proof Lemma 5
Let $A_{j}\in\mathbb{R}^{n\times m_{j}}$ denote the $j^{th}$ block column of
matrix $A$. According to Proposition 3(e), we have
$\displaystyle\textup{vec}(A_{j})$
$\displaystyle=\textup{vec}\left(\begin{bmatrix}A_{1j}\\\ \vdots\\\
A_{pj}\end{bmatrix}\right)=P_{m_{j},n}\textup{vec}\left(\begin{bmatrix}A^{T}_{1j}&\ldots&A^{T}_{pj}\end{bmatrix}\right)$
$\displaystyle=P_{m_{j},n}\begin{bmatrix}\textup{vec}(A^{T}_{1j})\\\ \vdots\\\
\textup{vec}(A^{T}_{pj})\end{bmatrix}=P_{m_{j},n}\begin{bmatrix}P_{n_{1},m_{j}}\textup{vec}(A_{1j})\\\
\vdots\\\ P_{n_{p},m_{j}}\textup{vec}(A_{pj})\end{bmatrix}$
$\displaystyle=P_{m_{j},n}\textbf{diag}(P_{n_{1},m_{j}},\ldots,P_{n_{p},m_{j}})\begin{bmatrix}\textup{vec}(A_{1j})\\\
\vdots\\\ \textup{vec}(A_{pj})\end{bmatrix}.$
Let $P_{j}=P_{m_{j},n}\textbf{diag}(P_{n_{1},m_{j}},\ldots,P_{n_{p},m_{j}})$.
Then
$\displaystyle\textup{vec}(A)=\begin{bmatrix}\textup{vec}(A_{1})\\\ \vdots\\\
\textup{vec}(A_{q})\end{bmatrix}=\underbrace{\textbf{diag}(P_{1},\ldots,P_{q})}_{P}\underbrace{\begin{bmatrix}\textup{vec}(A_{11})\\\
\vdots\\\ \textup{vec}(A_{p1})\\\ \vdots\\\ \textup{vec}(A_{1q})\\\ \vdots\\\
\textup{vec}(A_{pq})\end{bmatrix}}_{a_{A}}.$ (29)
Note that vector $a_{A}$ consists of all $pq$ sub-vectors
$\textup{vec}(A_{11}),\ldots,\textup{vec}(A_{pq})$. Let $a^{\star}_{A}$ denote
the vector containing only nonzero sub-vectors of $a_{A}$. We define
$\mathcal{A}=\left\\{i|[a_{A}]_{i}\neq 0\right\\}$. Let
$T_{i}=\begin{bmatrix}0&\ldots&I&\ldots&0\end{bmatrix}^{T}$ be the block
matrix with an identity in the $i^{th}$ block row. It is easy to see that
there exists full column rank matrix $T$ whose columns are $T_{j}$ for
$j\in\mathcal{A}$ such that $a_{A}=Ta^{\star}_{A}$. This implies that Equation
(29) can be written as
$\textup{vec}(A)=PTa^{\star}_{A}.$
The proof is completed by defining $E=PT$.
### VIII-G Proof Lemma 6
The equivalence of optimization problems follows simply by using the vec
operator. First note that
$\text{vec}\bigl{(}F(k)\bigr{)}=E_{1}\xi_{1}(k)=\begin{bmatrix}I&0\end{bmatrix}E\zeta(k+1)$.
Using Proposition 3(c), the first term on the right-hand side of (18) can be
written as
$\displaystyle\textbf{Tr}\left\\{H(k)F(k)VF(k)^{T}\right\\}$
$\displaystyle=\text{vec}^{T}\bigl{(}F(k)\bigr{)}\bigl{(}V\otimes
H(k)\bigr{)}\text{vec}\bigl{(}F(k)\bigr{)}$
$\displaystyle=\zeta^{T}(k+1)E^{T}\begin{bmatrix}I&0\end{bmatrix}^{T}\bigl{(}V\otimes
H(k)\bigr{)}\begin{bmatrix}I&0\end{bmatrix}E\zeta(k+1).$ (30)
The second term on the right-hand side of (18) can be written as
$\displaystyle\textbf{Tr}\left\\{H(k)\bigl{(}F(k)C-L(k)\bigr{)}P^{[1]}(k)\bigl{(}F(k)C-L(k)\bigr{)}^{T}\right\\}=\text{vec}^{T}\bigl{(}F(k)\bigr{)}\bigl{(}CP^{[1]}(k)C^{T}\otimes
H(k)\bigr{)}\text{vec}\bigl{(}F(k)\bigr{)}$ $\displaystyle\hskip
99.58464pt-2\text{vec}^{T}\bigl{(}F(k)\bigr{)}\bigl{(}CP^{[1]}(k)\otimes
H(k)\bigr{)}\text{vec}\bigl{(}L(k)\bigr{)}+\text{vec}^{T}\bigl{(}L(k)\bigr{)}\bigl{(}P^{[1]}(k)\otimes
H(k)\bigr{)}\text{vec}\bigl{(}L(k)\bigr{)}$ $\displaystyle\hskip
56.9055pt=\zeta^{T}(k+1)E^{T}\begin{bmatrix}I&0\end{bmatrix}^{T}\bigl{(}CP^{[1]}(k)C^{T}\otimes
H(k)\bigr{)}\begin{bmatrix}I&0\end{bmatrix}E\zeta(k+1)$ $\displaystyle\hskip
59.75095pt-2\zeta^{T}(k+1)E^{T}\begin{bmatrix}I&0\end{bmatrix}^{T}\bigl{(}CP^{[1]}(k)\otimes
H(k)\bigr{)}\text{vec}\bigl{(}L(k)\bigr{)}+\textup{vec}^{T}\bigl{(}L(k)\bigr{)}\bigl{(}P^{[1]}(k)\otimes
H(k)\bigr{)}\text{vec}\bigl{(}L(k)\bigr{)}.$ (31)
Likewise,
$\textup{vec}\bigl{(}F^{[1]}(k)\bigr{)}=E_{2}\xi_{2}(k)=\begin{bmatrix}0&I\end{bmatrix}E\zeta(k)$
and
$\displaystyle\text{vec}\left(F^{[1]}(k)-L(k)\bigl{(}{K}^{[1]}(k-1)+BF(k-1)\bigr{)}\right)$
$\displaystyle=\text{vec}\bigl{(}F^{[1]}(k)\bigr{)}-\bigl{(}I\otimes
L(k)B\bigr{)}\text{vec}\bigl{(}F(k-1)\bigr{)}-\text{vec}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)}$
$\displaystyle=\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}E\zeta(k)-\text{vec}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)},$
where we used Proposition 3(a) to obtain the second equality. The third term
on the right-hand side of (18) can be written as
$\displaystyle\textbf{Tr}\left\\{H(k)\left(F^{[1]}(k)-L(k)\bigl{(}{K}^{[1]}(k-1)+BF(k-1)\bigr{)}\right)\vspace{-2mm}\widetilde{Y}(k-1)\left(F^{[1]}(k)-L(k)\bigl{(}{K}^{[1]}(k-1)+BF(k-1)\bigr{)}\right)^{T}\right\\}\vspace{-2mm}$
$\displaystyle\hskip 79.6678pt=\zeta^{T}(k)E^{T}\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}^{T}\bigl{(}\widetilde{Y}(k-1)\otimes
H(k)\bigr{)}\begin{bmatrix}-I\otimes L(k)B&I\end{bmatrix}E\zeta(k)$
$\displaystyle\hskip 85.35826pt-2\zeta^{T}(k)E^{T}\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}^{T}\bigl{(}\widetilde{Y}(k-1)\otimes
H(k)\bigr{)}\textup{vec}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)}$
$\displaystyle\hskip
85.35826pt+\textup{vec}^{T}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)}\bigl{(}\widetilde{Y}(k-1)\otimes
H(k)\bigr{)}\text{vec}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)}.$ (32)
The last term on the right-hand side of (18) can be written as
$\displaystyle\textbf{Tr}\left\\{H(k)\left(F(k)C-L(k)\right)\widetilde{P}(k)\left(F^{[1]}(k)-L(k)\bigl{(}{K}^{[1]}(k-1)+BF(k-1)\bigr{)}\right)^{T}\right\\}$
$\displaystyle=$ $\displaystyle\zeta^{T}(k)E^{T}\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}^{T}\bigl{(}\widetilde{P}^{T}(k)C^{T}\otimes
H(k)\bigr{)}\begin{bmatrix}I&0\end{bmatrix}E\zeta(k+1)$
$\displaystyle-\zeta^{T}(k)E^{T}\begin{bmatrix}-I\otimes
L(k)B&I\end{bmatrix}^{T}\bigl{(}\widetilde{P}^{T}(k)\otimes
H(k)\bigr{)}\textup{vec}\bigl{(}L(k)\bigr{)}$
$\displaystyle-\zeta^{T}(k+1)E^{T}\begin{bmatrix}I&0\end{bmatrix}^{T}\bigl{(}C\widetilde{P}(k)\otimes
H(k)\bigr{)}\textup{vec}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)}$
$\displaystyle+\textup{vec}^{T}\bigl{(}L(k){K}^{[1]}(k-1)\bigr{)}\bigl{(}\widetilde{P}^{T}(k)\otimes
H(k)\bigr{)}\textup{vec}\bigl{(}L(k)\bigr{)}.$ (33)
Substituting (30)-(33) back into (18), noting that
$\widetilde{Y}^{[1]}(k)~{}=CP^{[1]}(k)C^{T}+V$, and omitting constant terms we
arrive at (19).
The proof can be completed by showing that $Z_{1}(k)$ is positive definite.
Since $\widetilde{Y}^{[1]}(k)$, $\widetilde{Y}(k)$, and $H(k)$ are positive
definite according to assumptions 1 and 2, $\widetilde{Y}^{[1]}(k-1)\otimes
H(k-1)$ and $\widetilde{Y}(k-1)\otimes H(k)$ are positive definite according
to Proposition 3(b). Therefore, Since $E$ has full column rank, $Z_{1}(k)$ is
positive definite.
### VIII-H Proof Lemma 7
To prove the theorem, we start from the endpoint and iterate backwards in
time. Define
$\displaystyle\Pi(N)=\min_{\zeta(N)}\biggl{\\{}{1\over 2}\zeta^{T}(N)$
$\displaystyle{Z}_{1}(N)\zeta(N)+\zeta^{T}(N-1){Z}_{2}(N-1)\zeta(N)-\zeta^{T}(N){b}(N)\biggr{\\}}.$
(34)
Since $Z_{1}(N)$ is positive definite, by taking derivative with respect to
$\zeta(N)$, the optimal value of $\zeta(N)$ is given by
$\displaystyle\zeta^{\star}(N)=-R^{-1}(N)\bigl{(}{Z}^{T}_{2}(N-1)\zeta(N-1)-c(N)\bigr{)},$
where $R(N)={Z}_{1}(N)$ and $c(N)={b}(N)$.
By substituting the optimal value of $\zeta(N)$ into Equation (34), we have
$\displaystyle\Pi(N)=$ $\displaystyle-{1\over
2}\zeta^{T}(N-1){Z}_{2}(N-1)R^{-1}(N){Z}^{T}_{2}(N-1)\zeta(N-1)$
$\displaystyle+\zeta^{T}(N-1){Z}_{2}(N-1)R^{-1}(N)c(N)-{1\over
2}c^{T}(N)R^{-1}(N){c}(N).$
Note that the last term is constant and independent of $\zeta(N-1)$.
We proceed similarly and define
$\displaystyle\Pi(N-1)=\min_{\zeta(N-1)}\biggl{\\{}{1\over
2}\zeta^{T}(N-1){Z}_{1}(N-1)\zeta(N-1)$
$\displaystyle+\zeta^{T}(N-2){Z}_{2}(N-2)\zeta(N-1)$ $\displaystyle\hskip
62.59596pt-\zeta^{T}(N-1){b}(N-1)+\Pi(N)\biggr{\\}}$
$\displaystyle=\min_{\zeta(N-1)}\biggl{\\{}{1\over
2}\zeta^{T}(N-1)R(N-1)\zeta(N-1)+\zeta^{T}(N-2){Z}_{2}(N-2)\zeta(N-1)-\zeta^{T}(N-1)c(N-1)\biggr{\\}},$
(35)
where
$\displaystyle R(N-1)$
$\displaystyle={Z}_{1}(N-1)-{Z}_{2}(N-1)R^{-1}(N){Z}^{T}_{2}(N-1),$
$\displaystyle c(N-1)$ $\displaystyle={b}(N-1)-{Z}_{2}(N-1)R^{-1}(N)c(N-1).$
Equation (35) is the same as Equation (34), but with the time arguments
shifted one step. Thus,
$\zeta^{\star}(N-1)=-R^{-1}(N-1)\bigl{(}{Z}^{T}_{2}(N-2)\zeta(N-2)-c(N-1)\bigr{)}.$
The procedure can now be repeated, and
$\displaystyle\Pi(1)=$ $\displaystyle\min_{\zeta(1)}\left\\{{1\over
2}\zeta^{T}(1)R(1)\zeta(1)-\zeta^{T}(1)c(1)\right\\}.$
Therefore,
$\zeta^{\star}(1)=R^{-1}(1)c(1).$
|
arxiv-papers
| 2012-04-27T11:55:26 |
2024-09-04T02:49:30.280356
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hamid Reza Feyzmahdavian, Ather Gattami, Mikael Johansson",
"submitter": "Hamid Reza Feyzmahdavian",
"url": "https://arxiv.org/abs/1204.6178"
}
|
1204.6197
|
# SDSS quasars in the WISE preliminary data release and quasar candidate
selection with optical/infrared colors
Xue-Bing Wu11affiliation: Department of Astronomy, School of Physics, Peking
University, Beijing 100871, China; wuxb@pku.edu.cn , Guoqiang
Hao11affiliation: Department of Astronomy, School of Physics, Peking
University, Beijing 100871, China; wuxb@pku.edu.cn , Zhendong
Jia11affiliation: Department of Astronomy, School of Physics, Peking
University, Beijing 100871, China; wuxb@pku.edu.cn , Yanxia
Zhang22affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Datun Road A20, Beijing 100012, China , Nanbo Peng22affiliation:
National Astronomical Observatories, Chinese Academy of Sciences, Datun Road
A20, Beijing 100012, China
###### Abstract
We present a catalog of 37842 quasars in the Sloan Digital Sky Survey (SDSS)
Data Release 7, which have counterparts within 6′′ in the Wide-field Infrared
Survey Explorer (WISE) Preliminary Data Release. The overall WISE detection
rate of the SDSS quasars is 86.7%, and it decreases to less than 50.0% when
the quasar magnitude is fainter than $i=20.5$. We derive the median color-
redshift relations based on this SDSS-WISE quasar sample and apply them to
estimate the photometric redshifts of the SDSS-WISE quasars. We find that by
adding the WISE W1- and W2-band data to the SDSS photometry we can increase
the photometric redshift reliability, defined as the percentage of sources
with the photometric and spectroscopic redshift difference less than 0.2, from
70.3% to 77.2%. We also obtain the samples of WISE-detected normal and late-
type stars with SDSS spectroscopy, and present a criterion in the $z-W1$
versus $g-z$ color-color diagram, $z-W1>0.66(g-z)+2.01$, to separate quasars
from stars. With this criterion we can recover 98.6% of 3089 radio-detected
SDSS-WISE quasars with redshifts less than four and overcome the difficulty in
selecting quasars with redshifts between 2.2 and 3 from SDSS photometric data
alone. We also suggest another criterion involving the WISE color only,
$W1-W2>0.57$, to efficiently separate quasars with redshifts less than 3.2
from stars. In addition, we compile a catalog of 5614 SDSS quasars detected by
both WISE and UKIDSS surveys and present their color-redshift relations in the
optical and infrared bands. By using the SDSS $ugriz$, UKIDSS YJHK and WISE
W1- and W2-band photometric data, we can efficiently select quasar candidates
and increase the photometric redshift reliability up to 87.0%. We discuss the
implications of our results on the future quasar surveys. An updated SDSS–WISE
quasar catalog consisting of 101,853 quasars with the recently released WISE
all-sky data is also provided.
catalogs — galaxies: active — galaxies: photometry — quasars: general —
surveys
## 1 Introduction
Since the discovery of quasars in 1960s (Schmidt, 1963), more and more quasars
have been found in the last four decades. More than 120,000 quasars have been
discovered from the recent large optical spectroscopic surveys, such as the
Two-Degree Fields(2dF) survey (Boyle et al., 2000) and the Sloan Digital Sky
Survey (SDSS)(York et al., 2000; Schneider et al., 2010). Quasar candidates in
these surveys were mainly selected by optical colors. Because of the strong UV
and optical emission, quasars can be usually distinguished from the stellar
objects in the color-color and color-magnitude diagrams based on optical
photometry (Smith et al., 2005; Richards et al., 2002; Fan et al., 2000).
However, quasar selections based on the SDSS optical photometry alone become
very inefficient for identifying $\rm 2.2<z<3.0$ quasars since they have
optical colors similar to those of normal stars (Fan, 1999; Richards et al.,
2002, 2006; Schneider et al., 2007). In order to get a more complete sample of
quasars, we have to think about other ways to find these SDSS missing quasars
in the ’redshift desert’ ($\rm 2.2<z<3.0$).
An important way to identify the missing quasars with $\rm 2.2<z<3.0$ has been
suggested by using the infrared K-band excess based on the UKIRT (UK Infrared
Telescope) Infrared Deep Sky Survey (UKIDSS) (Warren et al., 2000; Hewett et
al., 2006; Maddox et al., 2008).111The UKIDSS project is defined in Lawrence
et al. (2007). UKIDSS uses the UKIRT Wide Field Camera (WFCAM; Casali et al.
(2007) and a photometric system described in Hewett et al. (2006). The
pipeline processing and science archive are described in Hambly et al. (2008).
Although the $z\sim$2.7 quasars have optical colors similar to stars, they are
usually more luminous in the infrared K-band. In addition, combining the
optical colors in SDSS with the infrared colors in UKIDSS, it should be more
efficient to separate stars from both lower redshift quasars ($z<$3)(Chiu et
al., 2007) and higher redshift ones ($z>$6) (Hewett et al., 2006) in the
color-color diagrams. Recent studies by Wu & Jia (2010) and Wu et al. (2011)
have demonstrated that with the SDSS-UKIDSS photometric data we can
efficiently select quasar candidates at $z<4$ with the selection criterion in
the Y-K versus $g-z$ color-color diagrams. The spectroscopic observations
carried out by Wu et al. (2010a, b) and Wu et al. (2011) have confirmed the
effectiveness of using SDSS-UKIDSS colors to discover the missing SDSS quasars
with $\rm 2.2<z<3.0$.
However, the sky coverage of UKIDSS is limited and the extragalactic survey of
UKIDSS/LAS will finally cover about 4000 $deg^{2}$ of the sky. Therefore, for
a large part of the sky we can not use the UKIDSS data in identifying quasars.
The same problem exists for using the Spitzer Infrared Array Camera (IRAC)
mid-infrared photometric data (Fazio et al., 2004), which also cover only very
limited sky area, though the IRAC colors have been suggested to efficiently
select quasars and other AGNs, including optically obscured quasars (Lacy et
al., 2004; Stern et al., 2005). In the near-IR bands, the Two-Micron All-Sky
Survey (2MASS)(Skrutskie et al., 2006)) made an all-sky survey at J,H,K bands,
reaching a depth of $K_{s}$=16 for sources with high Galactic latitudes. The
2MASS data have been used to select quasars with B-J/J-K colors (Barkhouse &
Hall, 2001). However, 2MASS is too shallow for most quasars with magnitude
$i>17$. Only 13930 of the 105783 quasars in SDSS DR7 have secure 2MASS
photometry, though 53564 of them have 2$\sigma$ detections in at least one
2MASS band (Schneider et al., 2010).
Recently, the preliminary data release of NASA’s Wide-field Infrared Survey
Explorer (WISE) became publicly available (Wright et al., 2010). WISE has
mapped all the sky at 3.4, 4.6, 12, and 22 $\micron$ in 2010 with an angular
resolution of 6.1, 6.4, 6.5 and 12.0 arcsec and 5$\sigma$ photometric
sensitivity better than 0.08, 0.11, 1 and 6 mJy (corresponding to 16.5, 15.5,
11.2, and 7.9 Vega magnitudes) in these four bands. The WISE preliminary data
release includes the positional and photometric data for over 257 million
objects with signal-to-noise ratio (S/N) greater than 7 in at least one band,
covering 23600 $deg^{2}$ of the sky with the ecliptic longitude at
$27.8^{o}<\lambda<133.4^{o}$ and $201.9^{o}<\lambda<309.6^{o}$. The final data
release is scheduled in the spring of 2012. Therefore, the WISE all-sky
photometric data will be very useful in helping us to select quasar candidates
for the future quasar surveys if some selection criteria can be obtained with
the known quasars with WISE detections in the preliminary data release. In
addition, since the WISE bands are similar to those of Spitzer IRAC, we expect
that the WISE colors can be also adopted to select the reddened and optical
obscured quasars, as already demonstrated by using the IRAC colors (Lacy et
al., 2004; Stern et al., 2005). Taking the advantage of the full sky coverage
of WISE data, a more complete quasar sample should be obtained. Although SDSS,
2dF and the ongoing SDSS III/BOSS surveys (Eisenstein et al., 2011) have
discovered almost 200,000 quasars, the currently available quasar samples are
still incomplete due to the differences in quasar selection criteria and
magnitude limits adopted for different surveys. Future efforts are still
needed to find more currently missing quasars at different redshifts,
including the reddened and obscured quasars, and construct a larger, deeper
and more complete sample of quasars.
The paper is organized as follows. In Section 2 we present the SDSS-WISE
quasar catalog. In Section 3 we obtain the color-redshift relations for this
SDSS-WISE quasar sample and investigate how they can improve the photometric
redshift estimations of quasars. In Section 4 we propose the quasar candidate
selection criteria based on our SDSS-WISE samples of spectroscopically
confirmed quasars and stars. After analyzing a SDSS-UKIDSS-WISE quasar sample
in Section 5, we give our summary and discussion in Section 6.
## 2 The SDSS-WISE quasar catalog
We cross-correlate the sources in the quasar catalog of SDSS DR7 (Schneider et
al., 2010), which consists of 105,783 SDSS quasars, with the sources in the
WISE preliminary data release (Wright et al., 2010), which covers the sky area
with ecliptic longitude of $27.8<\lambda<133.4$ or $201.9<\lambda<309.6$ and
presents photometric information for over 257 million objects. Because the
angular resolution of WISE is 6.1, 6.4, 6.5 and 12.0 arcsec in the four bands
respectively, we use 6 arcsec as the position offset for finding the WISE
counterparts of SDSS quasars. Using a larger offset would lead to significant
increase of duplicate WISE sources around SDSS quasars and higher rate of
false positives in matching the SDSS-WISE catalogs. Because five SDSS quasars
have more than one WISE counterparts within a 6 arcsec offset, we carefully
exclude these duplicated WISE sources. The small number of duplicated WISE
sources within a 6 arcseconds offset to the positions of SDSS quasars also
indicates that the rate of false positives in our catalog matching should be
very low. By also excluding also another five quasars without the full
detections in SDSS $ugriz$ bands, we create a catalog of 37842 SDSS quasars
with WISE detection in at least one of four WISE bands.
In Figure 1 we show the histograms of the SDSS and WISE magnitudes, magnitude
uncertainties, redshifts and position offsets of these SDSS-WISE quasars. The
median value of each quantity is also indicated in these histograms. The
redshift range of these 37842 SDSS-WISE quasars is from 0.064 to 5.414, with a
median value of 1.442, and the $i$-band magnitude range is from 14.793 to
21.855, with a median value of 18.883. All the SDSS $ugriz$ magnitudes have
been corrected from the Galactic extinction using a map from (Schlegel et al.,
1998). Throughout the paper the SDSS magnitudes are given in AB magnitudes,
while the WISE magnitudes are given in Vega magnitudes. The significant
decreases of quasar numbers in the histograms of W3- and W4-band magnitude
uncertainties are due to the relatively lower sensitivities of the WISE W3-
and W4-band detectors. The median values of uncertainties of WISE W3 and W4
magnitudes are 0.127 and 0.283 respectively, which are significantly larger
than the median values 0.050 and 0.053 in the WISE W1 and W2 bands and the
median values ($<0.043$) of the magnitude uncertainties in the SDSS $ugriz$
bands. Although we use 6 arcsec for the cross-correlation radius between the
SDSS and WISE sources, from the position offset distributions we see that
majority of sources actually have offsets smaller than 2 arcsec. Therefore, we
are confident about the reliability of such an SDSS-WISE quasar catalog.
We also check the detection rate of SDSS quasars by WISE. There are 43662
sources in the SDSS DR7 quasar catalog within the sky coverage of the WISE
preliminary data release, so the overall detection rate by WISE is about
86.7%. In Figure 2 we show the redshift and $i$-band magnitude histograms of
both SDSS quasars in the sky area of the WISE preliminary data release and the
WISE detected SDSS quasars, as well as the dependences of the WISE detection
rate on the redshift and magnitude. From Figure 2 we can see that the WISE
detection rate of SDSS quasars is higher than 66% at all redshift and is
higher than 80% at $z<2.2$, while it is higher than 85% at $i<19.5$ and
decreases to lower than 50% at $i>20.5$. The lower detection rate at $i>20.5$
is understandable because of the limited sensitivity of WISE detectors in the
mid-infrared bands.
In Table 1 we give the catalog of 37842 SDSS quasars detected in the WISE
preliminary data release. The properties of these quasars, including the
coordinates, offsets between the SDSS-WISE positions, redshifts, SDSS and WISE
magnitudes and their uncertainties, and radio and X-ray properties (adopted
from the SDSS DR7 quasar catalog of Schneider et al. 2010), are listed.
Using the recently released WISE all-sky data we have compiled a new SDSS-WISE
quasar catalog consisting of 101853 quasars, which is available in the online
version of Table 5.
## 3 Color-redshift relations and photometric redshift estimations
With the SDSS-WISE quasar catalog, we can investigate their color-redshift
relations, which are helpful to understand the quasar properties and can be
used to estimate the photometric redshifts of quasars.
From SDSS $ugriz$ and WISE W1,W2,W3,W4 magnitudes we can obtain eight colors
for quasars. In Figure 3 we plot the color versus redshift diagrams for all
the SDSS-WISE quasars. To obtain the reliable color-redshift relations, we
derive the median color-redshift relations based on the quasars with magnitude
uncertainties smaller than 0.2mag in $ugriz$ and the W1,W2 bands and smaller
than 0.4mag in the W3,W4 bands (corresponding to the black dots in Figure 3).
The SDSS quasars without detections in the WISE W3 and W4 bands are not
included when calculating the colors related to these two bands. Clearly we
fail to obtain reliable $u-g$ color at $z>3.4$ and $g-r$ color at $z>4.5$
because of the larger uncertainties of $u$ and $g$ magnitudes at larger
redshifts as the quasar Ly$\alpha$ emission line moves out of the $u$ and $g$
filter bands. The larger magnitude uncertainties in the WISE W3 and W4 bands
also lead to substantial scatters in the color-redshift relations related to
these magnitudes. To obtain the reliable color-redshift relations, we only
focus on quasars with $z<5$ because there are no enough quasars at $z>5$ in
our SDSS-WISE catalog. We adopt the bin size of 0.05 for $z<3$ and 0.1 for
$z>3$ in order to have enough sources in each redshift bin to derive the
median color. In Table 2, we give the median SDSS and WISE colors for quasars
at redshifts from 0.075 to 5.
We need to keep in mind that using both SDSS and WISE data introduce selection
bias to the SDSS-WISE quasar sample and also bias to the color-redshift
relationships, since it is biased toward quasars that are intrinsically bluer
and were selected by SDSS as quasars. These biases are difficult to avoid when
the currently largest quasar sample based on SDSS is adopted. In addition, our
requirement for quasars detected in all SDSS $ugriz$ bands and using only
quasars with smaller magnitude uncertainties in constructing the median color-
redshift relations can introduce bias too, which also make the median color
bluer. Although we can treat the low S/N measurements with some statistical
tools, we believe that using the relatively more accurate observational data
is still the most direct and efficient way to derive reliable color-redshift
relationships. Therefore, our derived color-redshift relations based on the
SDSS-WISE quasars may not be applicable to the reddened quasars and optically
obscured Type 2 quasars. Reliable color-redshift relations of these special
quasars will be obtained and compared with the current results only when large
samples of them are available in the future. Currently, it is still unclear
what the fractions of reddened quasars and Type 2 quasars are in the total
quasar population (Richards et al., 2003; Glikman et al., 2007; Polleta et
al., 2008; Reyes et al., 2008). Some techniques, involving mid-infrared and
near-infrared data, have been proposed to find the obscured quasars and
reddened quasars (Lacy et al., 2004; Maddox et al., 2008). Similarly, we
expect that the WISE data can also provide such helps in constructing a more
complete quasar sample.
With the derived color-redshift relations, we can use our previously
established $\chi^{2}$-minimization method to estimate the most probable
photometric redshifts of quasars (Wu et al., 2004; Wu & Jia, 2010). Here the
$\chi^{2}$ is defined as (see Wu et al. (2004)):
$\chi^{2}=\sum_{ij}{\frac{[(m_{i,cz}-m_{j,cz})-(m_{i,observed}-m_{j,observed})]^{2}}{\sigma_{m_{i,observed}}^{2}+\sigma_{m_{j,observed}}^{2}}},$
(1)
where the sum is obtained for all four SDSS colors and z-W1 and W1-W2 colors,
$m_{i,cz}-m_{j,cz}$ is the color in the color-redshift relations,
$m_{i,observed}-m_{j,observed}$ is the observed color of a quasar, and
$\sigma_{m_{i,observed}}$ and $\sigma_{m_{j,observed}}$ are the uncertainties
of observed magnitudes in two SDSS-WISE bands. We do not use the colors
related to WISE W3 and W4 magnitudes because their uncertainties are
substantially larger and only two third of sources in our SDSS-WISE catalog
have available values for the uncertainties of W4 magnitudes.
We note that in using the simple form (Equation (1)) to calculate the
$\chi^{2}$ values we need to assume that the measurements of colors and
magnitude uncertainties are roughly in Gaussian distribution and uncorrelated,
which may not be true in the real case. Although the measurement of individual
flux does follow a Gaussian distribution, the measurements of magnitude and
color generally do not follow it. Note that the SDSS magnitude is asinh
magnitude (Lupton et al., 1999), which introduces an extra complexity. Future
efforts will be needed to improve the $\chi^{2}$ calculation by considering
the exact distributions of colors and magnitude uncertainties, though the
result may not change significantly. On the other hand, the colors of quasars
appear to be less correlated when comparing with stars, which can be easily
observed from the color-color diagrams in the optical and near-infrared bands
(Richards et al., 2002; Chiu et al., 2007; Maddox et al., 2008). The
uncertainties of magnitudes have also been shown to be minimally correlated
(Weinstein et al., 2004). Therefore, the assumptions of non-correlations of
colors and magnitude uncertainties are believed to be reasonable. Richards et
al. (2001) also adopted a similar formulae as in Equation (1) to calculate the
$\chi^{2}$ values (see their Equation (1)) by considering the constant
scatters of the median color-redshift relations.
As in Wu & Jia (2010), in order to compare the $\chi^{2}$ values for the cases
where different number of colors at different redshifts were used, we actually
use the $\chi^{2}/N$ (where N is the number of colors and is 6,5 and 4
respectively for the input redshift of $z<3.4$,$3.4<z<4.5$ and $z>4.5$)
instead of $\chi^{2}$ to determine the photometric redshift by obtaining the
minimum of $\chi^{2}/N$ at a certain redshift. An IDL program is made to
search the photometric redshifts of SDSS-WISE quasars by taking the above
factors into account.
In order to see more clearly whether using the WISE colors can improve the
photometric redshift estimation, we also estimate the photometric redshifts of
these quasars using the SDSS colors only. In Figure 4 we compare the results
obtained by SDSS and by SDSS+WISE colors. We can see that by adding WISE W1
and W2 magnitudes we can improve the photometric redshift estimations
substantially. If we use the SDSS colors alone, the photometric redshift
reliability, defined as the percentage of sources with the photometric and
spectroscopic redshift difference ($|z_{photo}-z_{spec}|$) less than 0.2, is
70.3%. If we add z-W1 and W1-W2 colors to SDSS colors, such reliability
increases to 77.2%. Especially for SDSS-WISE quasars with $i<19.1$ and
$i<20.5$, which correspond roughly to the depth of the SDSS and WISE quasar
samples respectively, using the SDSS colors alone leads to the photometric
redshifts reliability of 72.82% and 70.46%, while adding the WISE colors can
increase the reliability to 79.97% and 77.48%, respectively. For quasars with
redshifts between 2.2 and 3, which have optical colors similar to stars and
are difficult to select by optical colors, the photometric redshift
reliability can increase from 67.89% (62.64%) using when SDSS colors alone to
75.25% (69.41%) using when SDSS+WISE colors if they are brighter than
$i=19.1$(20.5). This clearly demonstrates the effectiveness of adding the WISE
infrared colors in the photometric redshift estimations for quasar samples
with different magnitude limits and redshift ranges.
## 4 Quasar candidate selections with SDSS and WISE photometric data
One of the most important things for optical quasar surveys is to efficiently
select the quasar candidates. In SDSS, quasar candidates are mostly selected
based on the multi-band optical photometric data (Richards et al., 2002).
However, SDSS quasar selection is very inefficient at redshifts between 2.2
and 3 due to the similar optical colors of quasars with redshifts in this
range to those of stars (Warren et al., 2000). One possible way to improve
this situation is to use the near-IR colors (Warren et al., 2000; Hewett et
al., 2006; Maddox et al., 2008). Because quasars usually have a much flatter
spectral energy distribution over a wide range of wavelengths, their spectral
shapes in the near-IR bands are different from those of normal stars even if
their optical spectra are similar to stars (e.g. for quasars with redshifts
between 2.2 and 3). Wu & Jia (2010) have demonstrated that by combining the
UKIDSS near-IR colors with SDSS optical colors we can separate well quasars
from stars, and efficiently select quasars with redshift less than five.
However, because UKIDSS/LAS will only cover the sky area of 4000 $deg^{2}$, we
have to think about other ways to improve the quasar selection method. Here we
investigate the cases of using the data in the WISE W1 and W2 bands, which are
close to the near-IR bands.
Wu & Jia (2010) have suggested using $Y-K>0.46(g-z)+0.82$ (here g and z are AB
magnitudes and Y and K are Vega magnitudes, see also Wu et al. (2011)) to
efficiently separate quasars with redshifts $z<4$ from stars in the Y-K versus
$g-z$ color-color diagram. We think that this may still be the case in the
$z$-W1 versus $g-z$ diagram because $z$ and W1 bands are close to Y and K
bands respectively. In order to check this idea, we obtain a sample of normal
stars and a sample of late-type stars with both SDSS DR7 spectroscopy and WISE
data. Similar to what we did for SDSS-WISE quasars, we adopted 6′′ as the
offset between the SDSS and WISE positions for the star and late-type star
samples, and deleted a few duplicated WISE sources with relatively larger
offsets. In order to get a reliable quasar selection criterion, we also
include only the quasars and stars with magnitude uncertainties in the $g$,
$z$ and W1 bands less than 0.2 mag. Finally we adopt the SDSS-WISE samples of
37,535 quasars, 19,765 normal stars and 15,359 late-type stars to investigate
the criterion of separating quasars from stars. We note that the SDSS
spectroscopically identified star sample is very biased as many of them have
similar optical colors to quasars, but we believe that including more stars
with very different optical colors from quasars will not affect our quasar
selection criterion because these stars should be well separated from quasars
in our color-color diagram than the stars with optical colors similar to
quasars.
In Figure 5 we plot the distributions of these SDSS-WISE quasars and stars in
the z-W1 versus $g-z$ diagram. Obviously, most quasars with redshifts less
than four can be separated from both normal and late-type stars on this color-
color diagram. This is also confirmed by the median z-W1 and $g-z$ colors at
different redshift, shown as yellow solid line in Figure 5, which is obtained
from the median color-redshift relation of SDSS-WISE quasars. However, there
are significant overlaps, especially between quasars with $z>4$ and stars.
Similar to in Wu & Jia (2010), we perform an automatic search for the best
criterion to efficiently separate quasars and stars. We obtain this criterion
as: $z-W1=0.66(g-z)+2.01$. With this criterion, we can select 36895 of 37,535
quasars (with a percentage of 98.30%) and select 33,442 of 35,124 stars (with
a percentage of 95.21%). The false positive rate, defined as the ratio between
the number of stars (1682) incorrectly selected as quasars and the number of
all sources selected as quasars (38,577) by our criterion, is 4.36%. However,
we must keep in mind that the actual number of stars could be significantly
larger than what we used here because only a tiny fraction of stars have been
spectroscopically observed by SDSS. Therefore, the real false positive rate of
our quasar selection may be higher. For 37,272 SDSS-WISE quasars with
redshifts less than four, with the proposed criterion we can select 36,827 of
them, with completeness of 98.81%. This demonstrates the very high efficiency
in selecting $z<4$ quasars with the $z-W1/g-z$ criterion, similar to using the
$Y-K/g-z$ criterion (Wu & Jia, 2010).
We also explore the case where we use WISE colors only to select quasars. For
this purpose, we selected 37,816 quasars, 19,369 normal stars and 18,127 late-
type stars with both SDSS DR7 spectroscopy and WISE W1,W2 and W3 band data. In
the upper panel of Figure 6 we show the distributions of SDSS-WISE quasars and
stars in the W1-W2 versus W2-W3 diagram. In the lower panel of Figure 6 we
show the histograms of W1-W2 colors of stars and quasars with different
redshifts. Clearly, quasars with redshifts smaller than 3.5 are separated from
stars by the W1-W2 color, but quasars with higher redshifts largely overlap
with stars. Such distributions are similar as that found by the Spitzer IRAC
colors (Lacy et al., 2004; Stern et al., 2005). We also did a search for the
best criterion to separate the quasars and stars and obtained this criterion
as $W1-W2>0.57$ (shown as the dashed line in Figure 6). Using this criterion,
we can select 36,565 from 37,816 quasars, 18,837 from 19,369 normal stars, and
17,922 from 18,127 late-type stars. 96.69% of the SDSS-WISE quasars and 98.04%
of SDSS-WISE stars can be separated with this criterion. The false positive
rate, due to the incorrect classification of 737 stars as quasars by this
W1-W2 criterion, is 1.98%. Comparing with the case of using the previous
criterion in the z-W1/g-z diagram, using the W1-W2 criterion can lead to lower
false positive rate and also lower completeness of selecting quasars.
In order to check whether using our proposed two quasar selection criteria can
avoid the color bias in the SDSS quasar selection algorithm, we obtain a
sample of 3089 FIRST radio selected SDSS-WISE quasars. Because these radio-
detected quasars are spectroscopically identified in SDSS without involving
the optical/infrared color selections, they are often adopted to check the
quasar selection efficiency in using the proposed selection criteria (Richards
et al., 2006). In Figure 7 we demonstrate the results of using two quasar
selection criteria. From Figure 7 we can clearly see that with our $z-W1/g-z$
selection criterion, we can recover 3035 of the 3089 radio detected quasars at
a completeness of 98.25%. This completeness raises to 98.63% for radio quasars
with $z<4$, which confirms the robustness of the $z-W1/g-z$ selection
criterion. With the W1-W2 selection criterion, we can recover 2989 of the 3089
radio detected quasars at a completeness of 96.76% and such completeness
raises to 97.97% for radio quasars with $z<3.2$ (note that the ’noise’ at
$z>3.5$ in Figure 7 is due to the small number statistics). Our investigations
demonstrate that the W1-W2 criterion and the $z-W1/g-z$ criterion can be
adopted to efficiently select $z<3.2$ and $z<4$ quasars respectively with very
high completeness. However, to selecte high redshift quasars with $z>4$,
obviously we still need to use other selection criteria (see Fan et al.
(2001); Hewett et al. (2006); Wu & Jia (2010)).
To better understand whether our proposed quasar selection criteria are
efficient in selecting quasars down to the magnitude limits of $i=19.1$ and
$i=20.5$, which correspond roughly to the depth of the SDSS and WISE quasar
samples respectively, we made the following checks with our SDSS-WISE quasar
sample. For 37842 quasars in this sample, there are 26397 quasars with
magnitudes brighter than $i=19.1$. Using our proposed $z-W1/g-z$ and W1-W2
selection criteria we can recover 26091 and 25977 of them at a completeness of
98.84% and 98.41%, respectively. For 1927 quasars with magnitudes brighter
than $i=19.1$ and redshifts between 2.2 and 3, we can recover 97.15% and
97.46% of them with these two criteria. For 37609 quasars with magnitudes
brighter than $i=20.5$ in our SDSS-WISE sample, using the $z-W1/g-z$ and W1-W2
criteria we can recover 36863 and 36373 of them at a completeness of 98.02%
and 96.71%, respectively. For 3043 quasars with magnitudes brighter than
$i=20.5$ and redshifts between 2.2 and 3, we can recover 96.94% and 95.40% of
them with these two criteria. Obviously these checks demonstrate that with our
proposed quasar selection criteria we can efficiently select WISE detected
quasars even at the magnitude limit down to $i=20.5$. Specifically, our
criteria can be used to recover the quasars with redshifts between 2.2 and 3
very efficiently, which can be also seen from Figure 7. This may have
important implications on the quasar candidate selections for future
spectroscopic quasar surveys.
## 5 The SDSS-UKIDSS-WISE quasars
With the SDSS-WISE quasar sample, we can also find the UKIDSS counterparts for
some of these quasars. In this case, we are able to construct a quasar sample
with the photometric data from SDSS, UKIDSS to WISE bands. Using the DR6
public data of the UKIDSS/LAS, we obtain this SDSS-UKIDSS-WISE quasar sample,
which consists of 5614 quasars with offsets between the SDSS and UKIDSS
positions within 3 arcsec and with all detections at SDSS $ugriz$, UKIDSS YJHK
and WISE W1 and W2 bands. We do not include the WISE W3 and W4 data because of
the relatively lower sensitivities in these two bands. Requiring the
detections in W3 and W4 bands will substantially reduce the quasar number of
our sample. The data of these SDSS-UKIDSS-WISE quasars, including the
coordinates, redshifts and 11-band magnitudes ($ugriz$ in AB magnitudes, YJHK
and W1,W2 in Vega magnitudes), are given in Table 3. From the photometric data
in 11 bands, we can construct the color-redshift relations of these SDSS-
UKIDSS-WISE quasars, which are shown in Figure 8. The median relations are
also obtained from the data with magnitude uncertainties at all bands less
than 0.2mag, and are summarized in Table 4.
With the color-redshift relations of SDSS-UKIDSS-WISE quasars, we can obtain
the photometric redshifts with different sets of photometric data using the
techniques described in Section 3, and compare the photometric redshift
reliability obtained with the different photometric data. In Figure 9, we show
the comparisons of photometric redshifts with spectroscopic redshifts and the
distributions of the differences between them for the cases using the SDSS,
SDSS+UKIDSS, UKIDSS+WISE W1,W2, and SDSS+UKIDSS+WISE W1,W2 photometric data,
respectively. The photometric redshift reliability, defined as the fraction of
the sources with the difference between the photometric and spectroscopic
redshifts smaller than 0.2, is 70.4%, 84.8%, 67.4% and 87.0% respectively for
the above four cases. Therefore, by using the SDSS $ugriz,$ UKIDSS YJHK and
WISE W1 and W2 band photometric data, we can efficiently improve the
photometric redshift reliability up to 87.0%, which is significantly higher
than using the SDSS photometric data alone (70.4%). Moreover, as mentioned in
Section 4 and in Wu & Jia (2010), using the SDSS, UKIDSS and WISE photometric
data can also help us select quasar candidates more efficiently. However, we
noticed that this can be done only in the UKIDSS surveyed area, which is much
smaller than the sky coverage of both SDSS and WISE surveys.
## 6 Summary and Discussion
In this paper, we present a catalog of 37842 SDSS quasars having counterparts
in the WISE Preliminary Data Release within 6′′. The overall WISE detection
rate of the SDSS quasars in the sky area of the WISE preliminary data release
is 86.7%, which demonstrates that the WISE data can be very helpful in
identifying quasars, especially those with magnitudes brighter than $i=20.5$.
By deriving the median color-redshift relations of this SDSS-WISE quasar
sample, we develop a method to estimate the photometric redshifts of quasars
and find that the photometric redshift reliability can increase from 70.3% to
77.2% if the WISE W1- and W2-band data are added to the SDSS photometry. We
also obtain a criterion in the z-W1 versus g-z color-color diagram,
$z-W1>0.66(g-z)+2.01$, to separate quasars from stars. With this criterion we
can recover 98.6% of 3089 radio-detected SDSS-WISE quasars with redshifts less
than four and overcome the difficulty in selecting quasars with redshifts
between 2.2 and 3 from the SDSS photometric data alone. We also suggest
another criterion involving the WISE color only, $W1-W2>0.57$, to separate
quasars with redshifts less than 3.2 from stars. In addition, we compile a
catalog of 5614 SDSS quasars detected by both WISE and UKIDSS surveys and
present their color-redshift relations. By using the SDSS $ugriz$, UKIDSS YJHK
and WISE W1- and W2-band photometric data, we can efficiently select quasar
candidates and increase the photometric redshift reliability up to 87.0%.
Considering the advantages of all-sky coverage of the WISE mid-infrared
photometry, the WISE data will be very helpful in finding new quasars in
future quasar survey and constructing a more complete quasar sample than that
currently available. The ongoing BOSS project in SDSS III has identified
29,000 quasars with $z>2.2$ and expects to obtain the spectra of 150,000
quasars at $2.2<z<4$ (Eisenstein et al., 2011), with the updated quasar target
selection techniques including K-band excess (Ross et al., 2012). The WISE
data may be also helpful to the BOSS quasar target selection, especially in
selecting quasars with $i<20.5$. The Chinese GuoShouJing telescope (LAMOST)(Su
et al., 1998), which is a 4-meter size spectroscopic telescope with 4000
fibers and 5-degree field of view and is currently in the commissioning phase,
is also aiming to discover 0.4 million quasars with magnitudes bright than
$i=20.5$ in the next four years (Wu et al., 2010b). The WISE data will be
adopted to select quasar candidates in the LAMOST quasar survey. Obviously,
the results obtained from this paper, especially the proposed quasar selection
criteria and the photometric redshift estimation methods, will provide
significant help in selecting LAMOST quasar candidates in a large sky areas.
We have demonstrated the advantages of combining the SDSS optical and UKIDSS,
WISE infrared photometric data in finding quasars. However, in order to fully
use this advantage to discover more quasars we still need much wider and
deeper photometry in the optical and infrared bands. Fortunately, several
ongoing and upcoming photometric sky surveys will provide such helps to us.
SDSS III (Eisenstein et al., 2011) has taken 2500 deg2 further imaging in
$ugriz$ bands in the south galactic cap aside from the SDSS I and II
photometry. The SkyMapper (Keller et al., 2007) and Dark Energy Survey (The
Dark Energy Survey Collaboration,, 2005) will also present the multi-band
optical photometry in 20000/5000 deg2 of the southern sky, reaching the
magnitude limit of 22/24 mag in the $i$-band, respectively. The Visible and
Infrared Survey Telescope for Astronomy (VISTA)(Arnaboldi et al., 2007) will
carry out its VISTA Hemisphere Survey in the near-IR YJHK bands for 20000 deg2
of the southern sky with a magnitude limit at K=20.0, which is about 5 mag and
2 mag deeper than the 2MASS and UKIDSS limits, respectively. Therefore, the
WISE data, when combined with the optical and infrared photometric data
obtained with these ongoing and upcoming surveys, will provide us with a large
database for quasar candidate selections using the proposed optical/infrared
selection criteria. We expect that a much larger and more complete quasar
sample covering a wider range of redshift will be constructed in the near
future, which will play an important role in the near future in studying
extragalactic astrophysics, including AGN physics, galaxy evolution, large
scale structure and cosmology.
We thank the anonymous referee for very helpful comments and suggestions. The
work is supported by the National Natural Science Foundation of China
(11033001) and the National Key Basic Research Science Foundation of China
(2007CB815405). This publication makes use of data products from the Wide-
field Infrared Survey Explorer, which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration. Funding for the SDSS and SDSS-II has been provided by the
Alfred P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, the US Department of Energy, the National Aeronautics and
Space Administration, the Japanese Monbukagakusho, the Max Planck Society and
the Higher Education Funding Council for England. The SDSS web site is
http://www.sdss.org/. The SDSS is managed by the Astrophysical Research
Consortium for the Participating Institutions. The Participating Institutions
are the American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve University,
University of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the Joint
Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute
for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New
Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval
Observatory and the University of Washington. Facilities: Sloan (SDSS), UKIRT,
WISE
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Table 1: A catalog of 37,842 SDSS-WISE quasars
RA | Dec | offset | redshift | u | g | r | i | z | W1 | W2 | W3 | W4
---|---|---|---|---|---|---|---|---|---|---|---|---
deg | deg | ′′ | | | | | | | | | |
18.984833 | 31.799980 | 0.853 | 1.265 | 20.158 | 19.792 | 19.192 | 19.027 | 18.944 | 15.163 | 13.754 | 10.814 | 8.511
19.005892 | 32.376339 | 0.141 | 0.662 | 21.666 | 20.586 | 20.334 | 20.145 | 20.031 | 15.177 | 14.193 | 11.805 | 8.927
19.012709 | 31.423994 | 1.979 | 2.645 | 20.677 | 19.831 | 19.630 | 19.550 | 19.345 | 15.690 | 15.013 | 11.940 | 8.370
19.063026 | 32.578110 | 0.859 | 2.389 | 21.790 | 20.524 | 20.390 | 20.116 | 19.824 | 16.458 | 15.487 | 11.813 | 9.065
19.193468 | 31.414030 | 2.501 | 2.005 | 21.009 | 20.409 | 20.071 | 19.769 | 19.424 | 15.665 | 14.573 | 11.804 | 8.836
Notes: This table is available in its entirety in machine-readable and Virtual
Observatory (VO) forms in the online journal. A portion is shown here for
guidance regarding its form and content.
Table 2: The color-redshift relations of SDSS-WISE quasars
Redshift | u-g | g-r | r-i | i-z | z-W1 | W1-W2 | W2-W3 | W3-W4
---|---|---|---|---|---|---|---|---
0.075 | 0.259 | 0.097 | 0.297 | -0.141 | 4.387 | 0.963 | 2.860 | 2.406
0.125 | 0.037 | 0.112 | 0.475 | -0.140 | 4.343 | 0.962 | 2.763 | 2.281
0.175 | 0.049 | 0.205 | 0.387 | -0.043 | 4.349 | 0.943 | 2.743 | 2.356
0.225 | 0.093 | 0.278 | 0.323 | 0.014 | 4.242 | 0.957 | 2.753 | 2.321
0.275 | 0.089 | 0.257 | 0.085 | 0.410 | 4.023 | 0.962 | 2.766 | 2.367
Notes: This table is available in its entirety in machine-readable and Virtual
Observatory (VO) forms in the online journal. A portion is shown here for
guidance regarding its form and content.
Table 3: A catalog of 5614 SDSS-UKIDSS-WISE quasars
RA | Dec | redshift | u | g | r | i | z | Y | J | H | K | W1 | W2
---|---|---|---|---|---|---|---|---|---|---|---|---|---
deg | deg | | | | | | | | | | | |
25.994659 | 14.706349 | 1.634 | 18.984 | 18.821 | 18.749 | 18.634 | 18.608 | 18.389 | 17.868 | 17.413 | 17.148 | 15.907 | 14.114
26.124020 | 14.556283 | 1.615 | 19.385 | 19.237 | 19.133 | 18.863 | 18.906 | 18.568 | 18.168 | 17.429 | 16.943 | 16.043 | 14.848
26.393171 | 14.526940 | 0.635 | 20.106 | 19.670 | 19.444 | 19.131 | 19.037 | 18.361 | 18.046 | 17.198 | 16.063 | 14.442 | 13.328
26.403246 | 14.928178 | 1.147 | 17.873 | 17.702 | 17.539 | 17.535 | 17.584 | 16.944 | 16.932 | 16.488 | 15.802 | 14.386 | 12.932
26.595579 | 14.804535 | 0.971 | 18.787 | 18.757 | 18.547 | 18.729 | 18.706 | 18.677 | 17.815 | 17.631 | 16.605 | 14.966 | 13.755
Notes: This table is available in its entirety in machine-readable and Virtual
Observatory (VO) forms in the online journal. A portion is shown here for
guidance regarding its form and content.
Table 4: The color-redshift relations of SDSS-UKIDSS-WISE quasars
Redshift | u-g | g-r | r-i | i-z | z-Y | Y-J | J-H | H-K | K-W1 | W1-W2
---|---|---|---|---|---|---|---|---|---|---
0.075 | 0.114 | 0.086 | 0.297 | -0.157 | 0.396 | 0.395 | 0.304 | 0.716 | 1.085 | 0.963
0.125 | 0.087 | 0.115 | 0.442 | -0.209 | 0.498 | 0.602 | 0.736 | 1.117 | 1.334 | 0.996
0.175 | 0.024 | 0.178 | 0.391 | -0.070 | 0.759 | 0.607 | 0.670 | 0.963 | 1.347 | 0.963
0.225 | 0.073 | 0.230 | 0.346 | -0.016 | 0.714 | 0.522 | 0.778 | 0.961 | 1.275 | 0.964
0.275 | 0.111 | 0.276 | 0.086 | 0.414 | 0.574 | 0.516 | 0.800 | 0.961 | 1.234 | 0.959
Notes: This table is available in its entirety in machine-readable and Virtual
Observatory (VO) forms in the online journal. A portion is shown here for
guidance regarding its form and content.
Table 5: A catalog of 101,853 SDSS-WISE quasars by atching the WISE all-sky
data
RA | Dec | offset | redshift | u | g | r | i | z | W1 | W2 | W3 | W4
---|---|---|---|---|---|---|---|---|---|---|---|---
deg | deg | ′′ | | | | | | | | | |
0.033900 | 0.276301 | 0.387 | 1.837 | 20.242 | 20.206 | 19.941 | 19.485 | 19.178 | 16.170 | 14.711 | 11.424 | 8.139
0.038604 | 15.298476 | 0.209 | 1.199 | 19.916 | 19.807 | 19.374 | 19.148 | 19.312 | 15.656 | 14.084 | 10.935 | 8.388
0.039089 | 13.938449 | 0.128 | 2.234 | 19.233 | 18.886 | 18.427 | 18.301 | 18.084 | 15.513 | 14.672 | 10.825 | 8.750
0.039271 | -10.464425 | 0.465 | 1.845 | 19.242 | 19.019 | 18.966 | 18.775 | 18.705 | 16.158 | 15.099 | 12.022 | 8.865
0.047549 | 14.929355 | 0.180 | 0.460 | 19.647 | 19.465 | 19.368 | 19.193 | 19.015 | 14.263 | 13.200 | 10.769 | 8.158
Notes: This table is available in its entirety in machine-readable and Virtual
Observatory (VO) forms in the online journal. A portion is shown here for
guidance regarding its form and content.
Figure 1: Histograms of magnitudes, magnitude uncertainties, redshifts and
offsets between the SDSS and WISE positions of the SDSS-WISE quasars. The
dashed line marks the median value of each quantity. Figure 2: Upper panel:
The solid line denotes the redshift distribution of SDSS quasars in the sky
area of WISE preliminary data release, while the dotted line denotes the
redshift histogram of the WISE detected SDSS quasars. The ratio between them
is also plotted as a function of redshift. Lower panel: The solid line denotes
the $i$-band magnitude distribution of SDSS quasars in the sky area of WISE
preliminary data release, while the dotted line denotes the $i$-band magnitude
histogram of the WISE detected SDSS quasars. The ratio between them is also
plotted as a function of magnitude. Figure 3: Color-redshift relations of
SDSS-WISE quasars. The black dots denote the quasars with magnitude
uncertainties smaller than 0.2mag in $ugriz$ and W1, W2 bands and smaller than
0.4 mag in W3, W4 bands. Other quasars with larger magnitude uncertainties are
plotted as cyan dots. The dashed lines represent the median color-redshift
relations obtained from the quasars denoted as black dots. Figure 4:
Comparisons of photometric redshifts with spectroscopic redshifts and the
distributions of the differences between them. The upper panels show the cases
for using the SDSS $ugriz$ magnitudes and the lower panels show the cases for
using SDSS $ugriz$ and WISE W1,W2 magnitudes. The improvements in the later
cases can be clearly seen. Figure 5: Distributions of SDSS-WISE quasars and
stars in the $z-W1$ vs. $g-z$ color-color diagram. The blue, green and red
crosses denote the quasars with redshifts of $z<2.2$, $2.2<z<4$ and $z>4$,
respectively. The black and cyan dots denote the SDSS identified normal stars
and late-type stars. The yellow line represents the median color-color
relation derived from the median color-redshift relation of SDSS-WISE quasars,
and the yellow triangles, from left to right, mark the quasars with redshifts
of $z$=0.1, 4, and 4.5 in the median color-color relation. The dashed line
indicates our proposed quasar selection criterion, $z-W1>0.66(g-z)+2.01$.
Figure 6: Upper panel: The distributions of SDSS-WISE quasars and stars in the
$W1-W2$ vs. $W2-W3$ color-color diagram. The blue, green and red crosses
denote the quasars with redshifts of $z<2.2$, $2.2<z<3.5$ and $z>3.5$,
respectively. The black and cyan dots denote the SDSS identified normal stars
and late-type stars. Lower panel: The histograms of $W1-W2$ colors of SDSS-
WISE quasars and stars. The black and cyan dashed lines denote the normal
stars and late-type stars, while the blue, green and red dotted lines denote
quasars with redshifts of $z<2.2$, $2.2<z<3.5$ and $z>3.5$, respectively. The
black solid line marks the distribution of all quasars. The dashed lines in
these two panel indicate our proposed quasar selection criterion,
$W1-W2>0.57$. Figure 7: Upper panel: Redshift distributions of FIRST radio-
detected SDSS-WISE quasars (black solid line) and those selected by the
criterion $z-W1>0.66(g-z)+2.01$ (blue dashed line) and by the criterion
$W1-W2>0.57$ (red dotted line). Lower panel: The completeness ratio of radio
quasars selected by $z-W1>0.66(g-z)+2.01$ (blue dashed line) and by criterion
$W1-W2>0.57$ (red dotted line) At different redshifts. The ’noise’ behavior of
the red dotted line at $z>3.5$ is due to the small number statistics. Figure
8: Color-redshift relations of SDSS-UKIDSS-WISE quasars. The black dots denote
the quasars with magnitude uncertainties smaller than 0.2mag, and the cyan
dots denote other quasars with magnitude uncertainties larger than 0.2mag. The
dashed lines represent the median color-redshift relations obtained from the
quasars denoted as black dots. Figure 9: Comparisons of photometric redshifts
with spectroscopic redshifts (left panels) and the distributions of the
differences between them (right panels). The panels from top to bottom
correspond to the cases using the SDSS, SDSS plus UKIDSS, UKIDSS plus WISE
W1,W2, and SDSS plus UKIDSS plus WISE W1,W2 photometric data, respectively.
|
arxiv-papers
| 2012-04-27T13:03:46 |
2024-09-04T02:49:30.290915
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-Bing Wu, Guoqiang Hao, Zhendong Jia, Yanxia Zhang, Nanbo Peng",
"submitter": "Xue-Bing Wu",
"url": "https://arxiv.org/abs/1204.6197"
}
|
1204.6231
|
# Quantifying Limits to Detection of Early Warning for Critical Transitions
Carl Boettiger cboettig@ucdavis.edu Alan Hastings Center for Population
Biology, 1 Shields Avenue, University of California, Davis, CA, 95616 United
States. Department of Environmental Science and Policy, University of
California, Davis, CA, 95616 United States
###### Abstract
Catastrophic regime shifts in complex natural systems may be averted through
advanced detection. Recent work has provided a proof-of-principle that many
systems approaching a catastrophic transition may be identified through the
lens of early warning indicators such as rising variance or increased return
times. Despite widespread appreciation of the difficulties and uncertainty
involved in such forecasts, proposed methods hardly ever characterize their
expected error rates. Without the benefits of replicates, controls, or
hindsight, applications of these approaches must quantify how reliable
different indicators are in avoiding false alarms, and how sensitive they are
to missing subtle warning signs. We propose a model based approach in order to
quantify this trade-off between reliability and sensitivity and allow
comparisons between different indicators. We show these error rates can be
quite severe for common indicators even under favorable assumptions, and also
illustrate how a model-based indicator can improve this performance. We
demonstrate how the performance of an early warning indicator varies in
different data sets, and suggest that uncertainty quantification become a more
central part of early warning predictions.
###### keywords:
early warning signals , tipping point , alternative stable states , likelihood
methods
††journal: Journal of the Royal Society Interface
## 1 Introduction
There is an increasing recognition of the importance of regime shifts or
critical transitions at a variety of scales in ecological systems (Holling,
1973; Wissel, 1984; Scheffer et al., 2001, 2009; Drake and Griffen, 2010;
Carpenter, 2011). Many important ecosystems may currently be threatened with
collapse, including corals (Bellwood et al., 2004), fisheries (Berkes et al.,
2006), lakes (Carpenter, 2011), and semi-arid ecosystems (Kéfi et al.,
2007). Given the potential impact of these shifts on the sustainable delivery
of ecosystem services (Folke et al., 2004) and the need for management to
either avoid an undesirable shift or else to adapt to novel conditions, it is
important to develop the ability to predict impending regime shifts based on
early warning signs.
A number of particular systems have demonstrated the kinds of relationships
that would produce regime shifts, including dynamics of coral reefs (Mumby et
al., 2007), and simple models of metapopulations with differing local
population sizes (Hastings, 1991). In cases like these one sensible approach
to understanding whether a regime shift would be likely would be to fit the
model using either a time series or else independent estimates of parameters.
More generally, with a good model of the system, detail-oriented approaches
could be useful (Lade and Gross, 2012). In this treatment we focus on the
situation where these more detailed models are not available.
Indeed, for many ecological systems specific models are not available and
general approaches are needed (Scheffer et al., 2009; Lade and Gross, 2012)
that do not depend on estimating the parameters of a known model of a specific
system. This has led to a variety of approaches based on summary statistics
(_e.g._ Carpenter and Brock, 2006; Held, 2004; Dakos et al., 2008; Guttal and
Jayaprakash, 2008b; Biggs et al., 2009; Carpenter, 2011; Seekell et al., 2011)
that look for generic signs of impending regime shifts. Here we extend earlier
work by providing estimates of the ability of different potential indicators
to accurately signal impending regime shifts, and develop new approaches that
both are more efficient and also lay bare some of the important assumptions
underlying attempts to find general warning signs of regime shifts. We
distinguish this question from the extensive literature involving change-point
analysis for the post-hoc identification of if and when a regime shift has
occurred (Easterling and Peterson, 1995; Rodionov, 2004; Lenton et al., 2009),
which is of little use if the goal is the advanced detection of the shift.
We begin by discussing the limitations of current approaches that rely on
summary statistics and provide a description of assumptions through the
introduction of a model based approach to detect early warning signals. We
then illustrate how stochastic differential equation (SDE) models can be used
to reflect the uncertainty inherent in the detection of early warning signals.
We caution against paradigms that are not useful for capturing uncertainty in
a model-selection based approach, such as information criteria. Finally we use
receiver-operating characteristics (Green and Swets, 1989; Keller et al.,
2009) as a way to illustrate the sensitivity different data sets and different
indicators have in detecting early warning signals and use this to explore a
number of examples. This approach provides a visualization of the types of
errors that arise and how one can trade off between them, and is important for
framing the problem as one focused on prediction.
## 2 The summary statistics approach
Foundational work on early warning signals has operated under the often-
implicit assumption that the system dynamics contain a saddle-node bifurcation
by looking for patterns that are associated with this kind of transition. A
saddle-node bifurcation occurs when a parameter changes and a stable
equilibrium (node) and an unstable equilibrium (saddle) coalesce and
dissapear. The system then moves to a more distant equilbrium. Guckenheimer
and Holmes (1983) or any other textbook on dynamical systems will provide
precise definitions and further explanation.
Typical patterns used as warning signals include an increasing trend in a
summary statistic such as variance (Carpenter and Brock, 2006),
autocorrelation (Held, 2004; Dakos et al., 2008), skew (Guttal and
Jayaprakash, 2008b), spectral ratio (Biggs et al., 2009). While attractive for
their simplicity, such approaches must confront numerous challenges. In this
paper we argue for a model-based approach to warning signals, and describe how
this can be done in a way that best addresses these difficulties. We begin by
enumerating several of the difficulties encountered in approaches lacking an
explicit model.
#### Hidden assumptions
The underlying assumption that the system contains a saddle-node bifurcation
can be easily overlooked in common summary-statistics based approaches. For
instance, variance may increase for reasons that do not signal an approaching
transition (Schreiber, 2003; Schreiber and Rudolf, 2008). Alternatively,
variance may not increase as a bifurcation is approached (Livina et al., 2012;
Dakos et al., 2011b). Some classes of sudden transitions may exhibit no
warning signals Hastings and Wysham (2010). Like saddle-node bifurcations,
transcritical bifurcations involve an eigenvalue passing through zero, and
exhibit the patterns of critical slowing down and increased variance (Drake
and Griffen, 2010). However, transcritical bifurcations involve a change in
stability of a fixed point, rather than the sudden dissapearance of a fixed
point that has made critical transitions so worrisome.While no approach will
be applicable to all classes of sudden transitions, it is certainly still
useful to have an approach that detects transitions driven by saddle-node
bifurcations, which have been found in many contexts (_e.g._ , see Scheffer et
al., 2001).
Even when we can exclude or ignore other dynamics and restrict ourselves to
systems that can produce a saddle-node bifurcation, approaches based on
critical slowing down or rising variance (_e.g._ Held, 2004; Scheffer et al.,
2009; Carpenter, 2011) must further assume that a changing parameter has
brought the system closer to the bifurcation. This assumption excludes at
least three alternative explanations for the transition in system behavior.
The first possibility is that a large perturbation of the system state has
moved the system into the alternative basin of attraction (Scheffer et al.,
2001). This is an exogenous forcing that does not arise from the system
dynamics, so it is not the kind of event we can expect to forecast. (An
example might be a sudden dramatic increase in fishing effort that pushes a
harvested population past a threshold.) The second scenario is a purely noise-
induced transition, a chance fluctuation that happens to carry the system
across the boundary (Ditlevsen and Johnsen, 2010). Livina et al. (2012)
indicate that such noise induced transitions cannot be predicted through early
warning signals – at least they are not expected to exhibit the same early
warning patterns of increased variance and increased autocorrelation
anticipated in the case of a saddle-node bifurcation. The third scenario is
that the system does pass through a saddle-node bifurcation, but rather than
gradually and monotonically approaching the critical point, the bifurcation
parameter moves in a rapid or highly non-linear way, making the detection of
any gradual trend impossible.
#### Arbitrary windows
In addition to the assumption of a saddle-node bifurcation, the calculation of
statistics that would be used to detect an impending transition is subject to
several arbitrary choices. A basic difficulty arises from the need to assume a
time-series is _ergodic_ : that averaging over time is equivalent to averaging
over replicate realizations, while trying to test if it is not. Theoretically,
the increasing trend in variance, autocorrelation, or other statistics is
something that would be measured across an ensemble – across replicates. As
true replicates are seldom available in systems for which developing warning
signals would be most desirable, typical methods average across a single
replicate using a moving window in time. The selection of the size of this
window and whether and by how much to overlap consecutive windows varies
across the literature. Lenton et al. (2012) demonstrates that these
differences can influence the results, and that the different choices each
carry advantages and disadvantages.
In addition to introducing the challenge of selecting a window size, this
ergodic assumption raises further difficulties. While appropriate for a system
that is stationary, or changing slowly enough in the window that it may appear
stationary, the assumption is at odds with the original hypothesis that the
system is approaching a saddle-node bifurcation.
Further, certain statistics such as the critical slowing down measured by
autocorrelation require data that is evenly sampled in time. Interpolating
from existing data to create evenly spaced points is particularly problematic,
as this introduces an artificial autocorrelation into the data.
#### No quantitative measures
Summary statistics typically invoke qualitative patterns such as an increase
in statistic $x$, rather than a quantitative measure of the early warning
pattern. This makes it difficult to compare between signals or to attribute a
statistical significance to the detection. Some authors have suggested
Kendall’s correlation coefficient, $\tau$, could be used to quantify an
increase (Dakos et al., 2008, 2011a) in autocorrelation or variance. Other
measures of increase, such as Pearson’s correlation coefficient have also been
proposed (Drake and Griffen, 2010), while most of the literature simply
forgoes quantifying the increase or estimating significance. While adequate in
experimental systems that can compare patterns between controls and replicates
(_e.g._ Drake and Griffen, 2010; Carpenter, 2011), any real-world application
of these approaches must be useful on a single time-series of observations. In
these cases a quantitative definition of a statistically significant detection
is essential. Without this, we have no assurance that a purported detection is
not, in fact, a false positive. By focusing primarily on examples known to be
approaching a transition when testing warning signals, the probability of
false positives has largely been overlooked.
#### Problematic null models
Specifying an appropriate null model is also difficult. Non-parametric null
hypotheses seem to require the fewest assumptions but in fact can be the most
problematic. For instance, the standard non-parametric hypothesis test with
Kendall’s tau rank correlation coefficient assumes only that the two variables
are independent, but this is an assumption that is violated by the very
experimental design: temporal correlations will exist in any finely-enough
sampled time series, and moving windows introduce temporal correlations in the
statistics. Under such a test any adequately large data set will find a
significant result, regardless of whether a warning signal exists. A similar
problem arises when the points in the time series are reordered to create a
null hypothesis – this destroys the natural autocorrelation in the time
series. More promising parametric null models have been proposed, such as
autoregressive models in Dakos et al. (2008), bringing us closer to a model-
based approach with explicit assumptions. Others have looked for alternative
summary statistics where reasonable null models are more readily available,
such as Seekell et al. (2011)’s proposal to test for conditional
heteroscedasticity.
#### Summary-statistic approaches have less statistical power.
Methods for the detection of early warning signals are continually challenged
by inadequate data (Inman, 2011; Scheffer, 2010; Held, 2004; Dakos et al.,
2008; Scheffer et al., 2009; Guttal and Jayaprakash, 2008b; Carpenter, 2011;
Bestelmeyer et al., 2011). Despite the widespread recognition of the this need
for large data sets, there has been very few studies quantitative studies of
power to determine at how much data is required (Contamin and Ellison, 2009),
how often a particular method would produce a false alarm or fail to detect a
signal, and which tests will be the most powerful or sensitive. The Neyman-
Pearson Lemma demonstrates that the most powerful test between hypotheses
compares the likelihood that the data was produced under each (Neyman and
Pearson, 1933). Such likelihood calculations require a model-based approach.
## 3 A model based approach
Model-based approaches are beginning to play a larger role in early warning
signal detection, though we have not as yet seen the direct fitting and
simulation of models to compare hypotheses. While choosing appropriate models
without system-specific knowledge is challenging, much can be accomplished by
framing the implicit assumptions into equations. Lade and Gross (2012)
introduce the idea of generalized models for early warning signals, and Kuehn
(2011) presents normal forms for bifurcation processes that can give rise to
critical transitions. Carpenter and Brock (2011) and Dakos et al. (2011b)
start by assuming the dynamics obey a generic stochastic differential equation
(SDE), but use this only to derive or define the summary statistics of
interest.
In this section we outline how the detection of early warning signals may be
thought of as a problem of model choice. We next show generic models can be
constructed under the assumptions discussed above and estimated from the data
in a maximum likelihood framework. We highlight the disadvantages of comparing
these estimates by information criteria, and instead introduce a simulation or
bootstrapping approach rooted in Cox (1961) and McLachlan (1987) that
characterizes the rate of missed detections and false alarms expected in the
estimate.
### Early warning signals as model choice
It may be useful to think of the detection of early warning signals as a
problem of model choice rather than one of pattern recognition. The model
choice approach attempts to frame each of the possible scenarios as
structurally different equations, each with unknown parameters that must be
estimated from the data. In any model choice problem, it is important to
identify the goal of the exercise – such as the ability to generalize, to
imitate reality, or to predict (Levins, 1966). In this case generality is more
important than realism or predictive capability: we will write down a general
model that is capable of approximating a wide class of models in which regime
shifts are characterized by a saddle-node bifurcation, and a second generic
model that is capable of representing the behavior of such systems when they
are not approaching a bifurcation. These may be thought of as the hypothesis
and null hypothesis, though they are in fact compound hypotheses, as we must
first estimate the model parameters from the data. In this approach it is not
assumed that “reality” is included in the models being tested, but that one of
the models is a better approximation of the true dynamics than the other.
System whose dynamics violate the assumptions common to both models, such as
in the examples of Hastings and Wysham (2010) where systems exhibit sudden
transitions without warning, fall outside the set of cases where this approach
would be valid; though the inability of either model to match the system
dynamics could be an indication of such a violation.
### Models
In the neighborhood of a bifurcation a system can be transformed into its
_normal form_ by a change of variables to facilitate analysis (Guckenheimer
and Holmes, 1983). The normal form (Guckenheimer and Holmes, 1983; Kuehn,
2011) for the saddle-node bifurcation is
$\frac{\mathrm{d}x}{\mathrm{d}t}=r_{t}-x^{2}.$ (1)
where $x$ is the state variable and $r_{t}$ our bifurcation parameter. We have
added a subscript $t$ to the bifurcation parameter as a reminder that it is
the value which may be slowly varying in time and consequently moving the
system closer to a critical transition or regime shift (Scheffer et al.,
2009). Transforming this canonical form to allow for an arbitrary mean in the
state variable $\theta$, the system near the bifurcation looks like
$dx/dt=r_{t}-(\theta-x)^{2}$, with fixed point
$\hat{x}=\sqrt{r_{t}}+\theta=:\phi(r_{t})$. We expand around the fixed point
and express as a stochastic differential equation (_e.g._ Gardiner, 2009):
$\mathrm{d}X=\sqrt{r_{t}}(\phi(r_{t})-X_{t})\mathrm{d}t+\sigma\sqrt{\phi(r_{t})}\mathrm{d}B_{t}$
(2)
where $B_{t}$ is the standard Brownian motion. This expression captures the
behavior of the system near the stable point as it approaches the bifurcation.
Allowing the stochastic term to scale with the square root of $\phi$ follows
from the assumption that of an internal-noise process, such as demographic
stochasticity, that arises in deriving the SDE from a Markov process, see
Kampen (2007) or Black and McKane (2012). The square root could be removed for
an external noise process, such as environmental noise. In practice it will be
difficult to descriminate between the square root and linear scaling in these
applications, since the average value of the state changes little before the
bifurcation.
As we discuss above, in this paradigm we must include an assumption on how the
bifurcation parameter, $r_{t}$, is changing. We assume a gradual, monotonic
change which we approximate to first order:
$r_{t}=r_{0}-mt.$ (3)
Detecting accelerating or otherwise nonlinear approaches to the bifurcation
will generally require more power. When the underlying system is not changing,
$r_{t}$ is constant ($m=0$) and Equation (2) will reduce to a simple Ornstein-
Uhlenbeck process,
$\mathrm{d}X_{t}=r(\theta-X_{t})\mathrm{d}t+\sigma\mathrm{d}B_{t}$ (4)
This is the continuous time analog of the first-order autoregressive model
considered as a null model elsewhere (_e.g._ Dakos et al., 2008; Guttal and
Jayaprakash, 2008a).
### Likelihood calculations
The probability $P(X|M)$ of the data $X$ given the model $M$ is the product of
the probability of observing each point in the time series given the previous
point and the length of the interval,
$\log P(X|M)=\sum_{i}\log P(x_{i}|x_{i-1},t_{i})$ (5)
For (2) or (4) it is sufficient (Gardiner, 2009) to solve the moment equations
for mean and variance respectively:
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}E(x|M)$ $\displaystyle=f(x)$ (6)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}V(x|M)$
$\displaystyle=-\partial_{x}f(x)V(x|M)+g(x)^{2}$ (7)
For the OU process, we can solve this in closed form over an interval of time
$t_{i}$ between subsequent observations:
$\displaystyle E(x_{i}|M=\text{OU})$
$\displaystyle=X_{i-1}e^{-rt_{i}}+\theta\left(1-e^{-rt_{i}}\right)$ (8)
$\displaystyle V(x_{i}|M=\text{OU})$
$\displaystyle=\frac{\sigma^{2}}{2r}\left(1-e^{-2rt_{i}}\right)$ (9)
For the time dependent model, we have analytic forms only for the dynamical
equations of these moments from equation (7), which we must integrate
numerically over each time interval. The moments of Equation (2) are given by
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}E(x_{i}|M=\text{LSN})$
$\displaystyle=2\sqrt{r(t)}(\sqrt{r(t)}+\theta-x_{i})$ (10)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}V(x_{i}|M=\text{LSN})$
$\displaystyle=-2\sqrt{r(t)}V(x_{i})+\sigma^{2}(\sqrt{r(t)}+\theta)$ (11)
These are numerically integrated using lsoda routine available in R for the
likelihood calculation.
### Comparing Models
Likelihood methods form the basis of much of modern statistics in both
Frequentist and Bayesian paradigms. The ability to evaluate likelihoods
directly by computation has made it possible to treat cases that do not
conform to traditional assumptions more directly. The basis of likelihood
comparisons has its roots in the Neyman-Pearson Lemma, which essentially
asserts that comparing likelihoods is the most powerful test of a choice
between two hypotheses (Neyman and Pearson, 1933), and motivates tests from
the simple likelihood ratio test up through modern model adequacy methods.
The hypotheses considered here are more challenging then the original lemma
provides for, as they are composite in nature: they specify two model forms
(stable and changing stability) but with model parameters that must be first
estimated from the data. Comparing models whose parameters have been estimated
by maximum likelihood is first treated by Cox (1961, 1962), and has since been
developed in this simulation estimation of the null distribution (McLachlan,
1987), by parametric bootstrap estimate (Efron, 1987). Cox’s $\delta$
statistic (often called the deviance between models) is simply the difference
between the log likelihoods of these maximum likelihood estimates, defined as
follows.
Let $L_{0}$ be the likelihood function for model 0, let
$\theta_{0}=\arg\max\theta_{0}\in\Omega_{0}$, ($L_{0}(\theta_{0}|X)$) be the
maximum likelihood estimator for $\theta_{0}$ given $X$, and let
$L_{0}=L_{0}(\theta_{0}|X)$; and define $L_{1}$, $\theta_{1}$, $L_{1}$
similarly for model 1. The statistic we will use is $\delta$, defined to be
twice the difference in log likelihood of observing the data under the two MLE
models,
$\delta=-2(\log L_{0}-\log L_{1}).$ (12)
This approach has been applied to the problem of model adequacy (Goldman,
1993) and model choice (Huelsenbeck and Bull, 1996) in other contexts. We have
extended the approach by generating the test distribution as well as a null
distribution of the statistic $\delta$.
### 3.1 Simulation-based comparisons
We perform the identical analysis procedure described above on each of these
three data sets. First, we estimate parameters for the null and test model to
each data set by maximum likelihood. Comparing the likelihood of these fits
directly gives us only a minimal indication of which model fits better. To
identify if these differences are significant, and by what probability they
could arise as a false alarm or a missed event, we simulate 500 replicate time
series from each estimated model.
The model parameters of both models are re-estimated on both families of
replicates (the null and test, _i.e._ $2\times 2\times 500$ fits). The
differences in the likelihood values between the model estimates produced from
the first set of simulations determines the null distribution for the deviance
statistic $\delta$. As the constant OU process model is nested within the
time-heterogeneous model, these values are always positive, but tend to be not
as large as those produced when the models are fit to the second family of
data.
The extent to which these distributions overlap indicates our inability to
distinguish between these scenarios. The tendency of the observed deviance to
fall more clearly in the domain of one distribution or the other indicates the
probability our observed data corresponds best with that model – either
approaching a critical transition or remaining stable. While it trivial to
assign a statistical significance to this observation based on how far into
the tail of the null distribution it falls, for the reasons we discussed we
prefer the more symmetric comparison of the probability that this value was
observed in either distribution. We visualize the trade-off between false
alarms and failed detection using the ROC curves introduced above.
### Information criteria will not serve.
One will commonly observe models representing alternative processes being
compared through the use of various information criteria such as the Akaike
information criterion. While tempting to apply in this situation, such
approaches are not suited to this problem for several reasons. The first is
that information criteria are not concerned with the model choice objective we
have in mind, as they are typically applied to find an adequate model
description without too many parameters that the system may be over-fit. More
pointedly, information criteria have no inherent notion of uncertainty.
Information criteria tests alone will not tell us our chances of a false
alarm, of missing a real signal, or how much data we need to be confident in
our ability to detect transitions.
### Beyond hypothesis testing
It is possible to frame the question of sensitivity, reliability, and adequate
data in the language of hypothesis testing. This introduces the need for
selecting a statistical significance criterion. In the hypothesis testing
framework, a false positive is a Type I error, which is defined relative to
this arbitrary statistical significance criterion, most commonly 0.05. By
changing the criterion, one can increase or decrease the probability of the
Type I error at the cost of decreasing or increasing false negative or Type II
error, which must also be defined relative to this criterion.
The language of hypothesis testing is built around a bias that false positives
are worse than false negatives, and consequently an emphasis on $p$-values
rather than power. In the context of early warning signals this is perilous –
it suggests that we would rather fail to predict a catastrophe than to sound a
false alarm. To avoid this linguistic bias and the introduction of an nuisance
parameter on which to define statistical significance, we propose the use of
receiver operating characteristic (ROC) curves.
### ROC Curves
We illustrate the trade-off between false alarms and failed detection using
receiver-operating characteristic curves first developed in signal-processing
literature (Green and Swets, 1989; Keller et al., 2009). The curves represent
the corresponding false alarm rate at any detection sensitivity (true positive
rate), Fig 1. The closer these distributions are to one-another, the more
severe the trade-off. If the distributions overlap exactly, the ROC curve has
a constant slope of unity. The ROC curve demonstrates this trade-off between
accuracy and sensitivity. Different early-warning indicators will vary in
their sensitivity to detect differences between stable systems and those
approaching a critical transition, making the ROC curves a natural way to
compare their performance. Since the shape of the curve will also depend on
the duration and frequency of the time-series observations, we can use these
curves to illustrate by how much a given increase in sampling effort can
decrease the rate of false alarms or failed detections.
Figure 1: Top row: The distributions of a hypothetical warning indicator are
shown under the case of a stable system (blue) and a system approaching a
critical transition (red). Bottom row: Points along the ROC curve are
calculated for each possible threshold indicated in the top row. The false
positive rate is the integral of the distribution of the test statistic under
the stable system right of the threshold (blue shaded area, corresponding to
blue vertical line). The true positive rate is the integral of the system
approaching a transition left of the threshold (red shaded area, corresponds
to the red line). Successive columns show the threshold increasing, tracing
out the ROC curve.
## 4 Example Results
We illustrate this approach on simulated data as well as several natural time-
series that have been previously analyzed for early warning signals. All data
and code for simulations and analysis are found in the accompanying R package,
`earlywarning`.
### Data
The simulation implements an individual, continuous-time stochastic birth-
death process with rates given by the master equation (Gardiner, 2009),
$\displaystyle\frac{dP(n,t)}{dt}$
$\displaystyle=b_{n-1}P(n-1,t)+d_{n+1}P(n+1,t)-(b_{n}+d_{n})P(n,t)$ (13)
$\displaystyle b_{n}$ $\displaystyle=\frac{eKn^{2}}{n^{2}+h^{2}}$ (14)
$\displaystyle d_{n}$ $\displaystyle=en+a_{t}$ (15)
where $P(n,t)$ is the probability of having $n$ individuals at time $t$,
$b_{n}$ is the probability of a birth event occurring in a population of $n$
individuals, $d_{n}$ the probability of a death. $e,K,h$ and $a_{t}$ are
parameters. This corresponds to the well-studied ecosystem model of over-
exploitation (Noy-Meir, 1975; May, 1977), with stochasticity introduced
directly through the demographic process. We select this model since it is has
discrete numbers of individuals, nonlinear processes, and the noise is driven
by Poisson process of births and deaths instead of a Gaussian, and thus
provides an illustration that our approach is robust to the violations of
those assumptions in model (2).
This model is forced through a bifurcation by gradually increasing the $a$
parameter, which increases can be thought of as an increasing toxicity of the
environment (from $a_{0}=100$ increasing at constant rate of 0.09 units/unit
time). Other parameters are: $Xo=730$, $e=0.5$, $K=1000$, $h=200$. We run this
model over a time interval from 0 to 500 and sample at 40 evenly spaced time
points, which were used for subsequent analysis. This sampling frequency was
chosen to be representative of reasonable sampling in biological time-series,
and provides enough points to detect a signal while not too many that errors
can be avoided entirely. For the convenience of the inquisitive reader, we
have also provided a simple function in the associated R package where the
user can vary the sampling scheme and parameter values and rerun this
analysis. This time series is shown in the top panel of Figure 2.
Figure 2: A model-based calculation of warning signals for the simulated data
example. Top panel: The original time series data on which model parameters
for Equations (4) and (2) are estimated. Middle panel: replicate simulations
under the maximum likelihood estimated (MLE) parameters of the null model,
Equation (4) and test model, Equation (2). Bottom panel: The distribution of
deviances (differences in log likelihood, Equation (12)), when both null and
test models are fit to each of the replicates from the null model, “null,” in
red, and these differences when estimating for each of the replicates from the
test model, in blue. The overlap of distributions indicate replicates that
will be difficult to tell apart. The observed differences in the original data
are indicated by the vertical line.
The first empirical data set comes from the population dynamics of _Daphnia_
living in the chemostat “H6” in the experiments of Drake & Griffen (Drake and
Griffen, 2010). This individual replicate was chosen as an example that showed
a pattern of increasing variance over the 16 data points where the system was
being manipulated towards a crash. This time series is shown in the top panel
of Figure 3.
Figure 3: A model-based calculation of warning signals for the Daphnia data
analyzed in Drake and Griffen (2010) (Chemostat H6). Panels as in Figure 2.
Our second empirical data set comes from the glaciation record seen in
deuterium levels in Antarctic ice cores (Petit et al., 1999), as analyzed by
Dakos et al. (2008). The data are preprocessed by linear interpolation and de-
trending by Gaussian kernel smoothing to be as consistent as possible with the
original analysis. We focus on the third glaciation event, consisting of 121
sample points. The match is not exact since Dakos et al. (2008) estimates the
de-trending window size manually, but the estimated correlations in the first-
order auto-regression coefficients are in close agreement with that analysis.
De-trending is intended to make the data consistent with the assumptions of
the warning signal detection (Dakos et al., 2008), which did not apply to the
other data sets (Drake and Griffen, 2010). This time series is shown in the
top panel of Figure 4.
Figure 4: A model-based calculation of warning signals for the Glaciation data
analyzed in Dakos et al. (2008) (Glaciation III). Panels as in Figure 2.
### Analysis
The deviances $\delta$ observed are 5.1, 6.0, 83.9 for the simulation, the
chemostat data and the glaciation data, respectively. Based on AIC score each
is large enough to reject the null hypothesis of a stable model with its one
extra parameter, but this does not give the full picture of the anticipated
error rates. The size of these differences reflects not only the magnitude of
the difference in fit between the models but also the arbitrary units of the
raw likelihoods, which are smaller for larger data-sets. Consequently the
glaciation score reflects as much the greater length of its time series as it
does anything else.
Our simulation approach can provide a better sense of the relative trade-off
in error rates associated with these estimates. As described above (Section
3.1), we simulate 500 replicates under each model, shown in the middle panels
of Figures 2, 3 and 4, and determine the distributions in likelihood ratio
under each, shown in the lower panels. The observed deviance from the original
data is also indicated (vertical line).
The ROC curves for each of these data sets are plotted in Figure 5. While
differences in the rate at which the system approaches a transition will also
improve the ratio of true positives to false positives, here we see the best-
sampled data set, Glaciation, with 121 points, also has the clearest signal
with no observed errors in the 500 replicates of each type. Comparing the
chemostat and simulation curves illustrate how the trade-off between false
positives and true positives can vary between data. The chemostat signal,
which estimates a relatively rapid rate of change but has less data, captures
a higher rate of true positives for a given rate of false positives than the
simulation data set with a weaker rate of change but more data, for false
positive rates above 20%. However, the simulated set with more data performs
better if lower false-positive rates are desired.
Figure 5: ROC curves for the Simulation, Chemostat, and Glaciation data,
computed from the distributions shown in Figures 2, 3 and 4, bottom panel.
## 5 Comparing the performance of summary statistics and model-based
approaches
Due to the variety of ways in which early warning signals based on summary
statistics are implemented and evaluated it is difficult to give a straight-
forward comparison between them and the performance of this model-based
approach. However, by adopting one of one of the quantitative measures of a
warning signal pattern, such as Kendall’s $\tau$ (Dakos et al., 2008, 2011a,
2009), we are able to make a side-by-side comparison of the different summary
statistics and the model based approach in the context of false alarms and
failed detections shown by the ROC curve. Values of $\tau$ near unity indicate
a strongly increasing trend in the warning indicator, which is supposed to be
indicative of an approaching transition. Values near zero suggest a lack of a
trend, as expected in stable systems.
Figure 6: Early warning signals in simulated and empirical data sets. The
first two columns are simulated data from (a) a stable system (Stable), and
(b) the same system approaching a saddle-node bifurcation (Deteriorating).
Empirical examples are from (c) _Daphnia magna_ concentrations manipulated
towards a critical transition (Daphnia), and (d) deuterium concentrations
previously cited as an early warning signal of a glaciation period
(Glaciation). Increases in summary statistics, computed over a moving window,
have often been used to indicate if a system is moving towards a critical
transition. The increase is measured by the correlation coefficient $\tau$.
Note that positive correlation does not guarantee the system is moving towards
a transition, as seen in the stable system, first column.
Figure 6 shows the time series for each data set in columns and the early
warning indicators of variance and autocorrelation computed over a sliding
window for each. Kendall’s correlation coefficient $\tau$ is calculated for
each warning indicator and displayed on the graphs, inset. For comparison, the
left-most column includes data simulated under a stable system, which
nevertheless shows a chance increasing autocorrelation with a $\tau=0.7$ We
can adapt the approach we have described above to determine how often such a
strong increase would appear by chance in a stable system as follows.
Figure 7: Box-plots of the distributions of Kendall’s $\tau$ observed for the
summary statistic methods variance and autocorrelation, applied to three
different data sets (from Figures 2, 3, 4). The distributions show extensive
overlap, suggesting that it will be difficult to distinguish early warning
signals by the correlation coefficient in these summary statistics.
By estimating the stable and critical transition models from the data, and
simulating 500 replicate data sets under each as in the analysis above, we can
then calculate the warning signals statistic over a sliding window of size
equal to one-half the length of the time series, and compute the correlation
coefficient $\tau$ measuring the degree to which the statistic shows an
increasing trend. This results in a distribution of $\tau$ values coming from
a model of a stable system, and a corresponding distribution of $\tau$ values
coming from the model with an impending transition. These distributions are
shown in Figure 7. Contrary to the expectation that replicates of the null
model (stable system, Equation (4)) would cluster around zero, while the test
model, Equation (2), would cluster around larger positive $\tau$ values, the
observed $\tau$ values on the replicates extend evenly across the range. This
results in dramatic overlap and offers little ability to distinguish between
the stable replicates and the replicates approaching a transition.
The use of box plots in Figure 7 provide a convenient and familiar way to
visualize the overlap between more than two distributions, though they lack
the resolution of the overlapping density distributions in Figures 2, 3, 4.
The overlapping distributions are the natural representation from which to
introduce the ROC curve, as in Figure 1.
Figure 8: ROC curves compare the performance of the summary statistics
variance and autocorrelation against the likelihood-based approach from Figure
5 on each of three example data sets (Figures 2, 3, 4).
The ROC curves for these data (Fig. 8) show that the summary-statistic based
indicators frequently lack the sensitivity to distinguish reliably between
observed patterns from a stable or unstable system. The large correlations
observed in the empirical examples (Fig. 6) are not uncommon in stable
systems. It is notable that in both empirical examples the summary statistics
approach does little better than chance in distinguishing replicates that have
been simulated from models (2) and (4), despite the fact that these models
correspond to the assumptions of the summary statistics approaches. On the
simulated data, the variance based method approaches the true-positive rate of
our likelihood method at higher levels of false positives, but performs worse
when the desired level of false positives is low. The ROC curve helps us
compare the performance of the different approaches at different tolerances.
For instance, Table 1 shows the fraction of true crashes caught at a 5% false
positive rate. We can instead set a desired True positive rate and read off
the resulting number of false alarms, Table 2.
| Variance | Likelihood
---|---|---
Simulation | 25 % | 61%
Chemostat | 5.0% | 34%
Glaciation | 5.4% | 100%
Table 1: Fraction of _crashes detected_ when the desired false alarm rate is fixed to 5% | Variance | Likelihood
---|---|---
Simulation | 49 % | 55%
Chemostat | 81 % | 35%
Glaciation | 93 % | 0%
Table 2: Fraction of _false alarms_ when the desired detection rate is fixed
to 90%
## 6 Discussion
The challenge of determining early warning signs for impending possible regime
shifts requires real attention to the underlying statistical issues and other
assumptions. Doing this, does, however, open up new possibilities for asking
what the goal of detection should be, and for clearly identifying underlying
assumptions. We consider alternative approaches based either on summary
statistics or a likelihood based model choice. By assuming the underlying
model corresponds to a saddle-node bifurcation, our analysis presents a “best-
case scenario” for both summary statistic and likelihood-based approaches.
Other literature has already begun to address the additional challenges posed
when the underlying dynamics do not correspond to these models (Hastings and
Wysham, 2010). Our results illustrate that even in this best-case scenario,
reliable identification of warning signals from summary statistics can be
difficult.
We have used three examples to illustrate the performance of this approach in
data from simulation, a chemostat experiment, and paleo-atmospheric record;
examples differing in sampling intensity and strength of signal of an
approaching collapse. While the well-sampled geological data shows an
unmistakable signal in this model-based approach, the uncertainty in the
smaller simulated and experimental data forces a trade-off between errors.
As a way to clearly illustrate the choices involved in looking for warning
signals while avoiding false alarms, we introduce an approach based on
receiver operator curves. These curves illustrate the extent to which an
potential warning signal mitigates the trade-off between missed events and
false alarms. The extent of the difficulty in finding reliable indicators of
impending regime shifts based on summary statistics becomes clear from the ROC
curves of these statistics, where a 5% false positive rate often corresponds
to only a 5% true positive rate, performing no better than the flip of a coin.
By estimating the ROC curve for a given set of data, we can better avoid
applying warning signals in cases of inadequate power. By taking advantage of
the assumptions being made to write down a specific likelihood function, we
can develop approaches that get the most information from the data available.
In any application of early warning signals, it is essential to address the
question of model adequacy. Our approach formalizes the assumptions about the
underlying process to match the assumptions of the other warning signals. As
the bifurcation results from the principle eigenvalue passing through zero,
the warning signal is expected in linear-order dynamics; estimation of the
nonlinear model is less powerful and less accurate. The performance of this
approach in the simulated data – which is nonlinear in its dynamics and driven
with non-Gaussian noise introduced by the Poisson demographic events –
demonstrates the accuracy under violation of these assumptions.
The conclusion is not simply that likelihood approaches are more reliable, but
rather more broadly that warning signals should consider the inherent trade-
off between sensitivity and accuracy, and must quantify how this trade-off
depends on both the indicators used and the data available. The approach
developed here estimates the risk of both failed detection and false alarms;
concepts which are critical to prediction-based management. Using the methods
we have outlined when designing early warning strategies for natural systems
can ensure that data collection has adequate power to offer a reasonable
chance of detection.
## 7 Acknowledgments
We would like to thank S. Schreiber, M. Holyoak, M. Baskett, A. Perkins, N.
Ross, M. Holden, and two anonymous reviewers for comments on the manuscript.
This research was supported by funding from NSF Grant EF 0742674 and a
Computational Sciences Graduate Fellowship from the Department of Energy grant
DE-FG02-97ER25308. Data and code are available at
https://github.com/cboettig/earlywarning.
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|
arxiv-papers
| 2012-04-26T19:22:31 |
2024-09-04T02:49:30.301720
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carl Boettiger and Alan Hastings",
"submitter": "Carl Boettiger",
"url": "https://arxiv.org/abs/1204.6231"
}
|
1204.6373
|
Hom-Novikov Algebras and Hom-Novikov-Poisson Algebras 111Corresponding
author: lmyuan@mail.ustc.edu.cn
Lamei Yuan${}^{\,{\ddagger}}$, Hong You${}^{\,{\ddagger}\,{\dagger}}$
${\ddagger}$Academy of Fundamental and Interdisciplinary Sciences,
Harbin Institute of Technology, Harbin 150080, China
†Department of Mathematics, Suzhou University, Suzhou 200092, China
E-mail: lmyuan@mail.ustc.edu.cn, youhong@suda.edu.cn
Abstract. The purpose of this paper is to study Hom-Novikov algebras and Hom-
Novikov-Poisson algebras, both of which were defined by Yau. In the paper, we
give several constructions leading us to some interesting examples of Hom-
Novikov algebras and Hom-Novikov-Poisson algebras. Also, we introduce the
notion of quadratic Hom-Novikov algebras and provide some properties.
Key words: Novikov algebras, Hom-Novikov algebras, quadratic Hom-Novikov
algebras, Novikov-Poisson algebras, Hom-Novikov-Poisson algebras.
2000 Mathematics Subject Classification: 17A30, 17B45, 17B81
§1. Introduction
Novikov algebras were introduced in connection with the Poisson brackets of
hydrodynamic type [3, 8, 9] and Hamiltonian operators in the formal
variational calculus [12, 21]. The theoretical study of Novikov algebras was
started by Zel’manov [31] and Filipov [11]. But the term “Novikov algebra” was
first used by Osborn [19]. The left multiplication operators of a Novikov
algebra form a Lie algebra. Thus, it is effective to relate the study of
Novikov algebras to the theory of Lie algebras [4]. Novikov algebras are a
special class of left-symmetric algebras (or under other names such as pre-Lie
algebras, quasi-associative algebras and Vinberg algebras), arising from the
study of affine manifolds, affine structures and convex homogeneous cones [2,
15, 20]. Left-symmetric algebras have close relations with many important
fields in mathematics and mathematical physics, such as infinite-dimensional
Lie algebras [3, 9], classical and quantum Yang-Baxter equation [10, 13],
quantum field theory [7] and so on.
Novikov-Poisson algebras were originally introduced by Xu [21, 22] motivated
by studying simple Novikov algebras and irreducible modules. Every Novikov-
Poisson algebra with associative commutative unity can be constructed from an
associative commutative derivation algebra. That there exists a relationship
between associative commutative derivation algebras and Novikov algebras was
also pointed out in [12]. Infinite-dimensional Novikov algebras over an
algebraically closed field of characteristic $0$ were classified in [23] under
some restrictions on the structure of right and left multiplication operators.
It turned out that such Novikov algebras are Novikov-Poisson algebras and are
obtained from associative commutative differentially simple algebras with
unity. While it was proved in [32] that a Novikov algebra is simple over a
field of characteristic not $2$ if and only if its associative commutative
derivation algebra is differentially simple.
A quadratic Novikov algebra, introduced in [6, 33], is a Novikov algebra with
a symmetric nondegenerate invariant bilinear form. The motivation for studying
quadratic Novikov algebras came from the fact that Lie algebras or associative
algebras with symmetric nondegenerate invariant bilinear forms have important
applications in several areas of mathematics and physics, such as the
structure theory of finite-dimensional semisimple Lie algebras, the theory of
complete integrable Hamiltonian systems and the classification of statistical
models over two-dimensional graphs.
Hom-Lie algebras were initially introduced by Hartwig, Larsson and Silvestrov
in [14] motivated by examples of deformed Lie algebras coming from twisted
discretizations of vector fields. This kind of algebras includes Lie algebras
as a subclass in which the twisting map is the identity map. As
generalizations of Hom-Lie algebras, quasi-hom-Lie algebras [16], Hom-Lie
superalgebras [1] and Hom-Lie color algebras [30] were introduced,
respectively. Both the Hom-Lie superalgebras and Hom-Lie color algebras are
special cases of the quasi-hom-Lie algebras. There has been a lot of progress
in the study of Hom-Lie algebras (see e.g., [24, 25, 26, 27]).
Hom-associative algebras generalizing associative algebras to a situation
where associativity law is twisted by a linear map were introduced in [18]. It
is well known that there is always a Lie algebra associated to an associative
algebra via the commutator bracket. An analogous result holds for Hom-
associative algebras, i.e., the commutator bracket multiplication defined by
the multiplication in a Hom-associative algebra leads naturally to Hom-Lie
algebras. This result was also extended to the Hom-Lie superalgebra [1] and
the Hom-Lie color algebra [30] cases.
Following the patterns of Hom-Lie algebras and Hom-associative algebras, Yau
in [28] introduced Hom-Novikov algebras, in which the two defining identities
are twisted by a linear map. It turned out that Hom-Novikov algebras can be
constructed from Novikov algebras, commutative Hom-associative algebras and
Hom-Lie algebras along with some suitable linear maps. Later, Yau in [29]
defined a Hom-Novikov-Poisson algebra as a twisted generalization of Novikov-
Poisson algebras.
In the present paper, we consider both Hom-Novikov algebras and Hom-Novikov-
Poisson algebras. In Section 2, we summarize some definitions including
Novikov algebras, Novikov-Poisson algebras, Hom-associative algebras, Hom-Lie
algebras, Hom-Novikov algebras and Hom-Novikov-Poisson algebras and related
results.
In Section 3, we focus on Hom-Novikov algebras. We establish the relationships
between Hom-Novikov algebras and Hom-Lie algebras as well as the relations
between Hom-Novikov algebras and Novikov algebras. Also, we provide a
construction of Hom-Novikov algebras from a commutative Hom-associative
algebra together with a suitable linear map but not a derivation.
In Section 4, we extend the notion of quadratic Novikov algebras to quadratic
Hom-Novikov algebras, which are Hom-Novikov algebras with a symmetric
nondegenerate bilinear form satisfying equation (4.3). We reduce the case
where the twisting map is an automorphism or an involution to the study of the
relationships between quadratic Hom-Novikov algebras and quadratic Novikov
algebras as well as the relations between quadratic Hom-Novikov algebras and
quadratic Hom-Lie algebras.
In Section 5, we study Hom-Novikov-Poisson algebras. We prove that the tensor
of two Hom-Novikov-Poisson algebras is still a Hom-Novikov-Poisson algebra. In
addition, we give a way to construct Hom-Novikov-Poisson algebras by using a
commutative Hom-associative algebra along with a derivation, generalizing a
construction in Novikov algebra case due to Dorfman and Gel’fand, in Novikov-
Poisson algebra case due to Xu and in Hom-Novikov algebra case due to Yau.
Finally, we provide some interesting examples.
Throughout this paper, $\mathbb{F}$ denotes a field of characteristic zero.
All vector spaces and tensor products are assumed to be over $\mathbb{F}$,
except where otherwise indicated.
§2. Preliminaries
In this section, we summarize some definitions concerning Novikov algebras and
Hom-algebras, and related results. For detailed discussions and examples we
refer the reader to the literatures, for instance, [14, 18, 21, 22, 28, 29]
and references therein.
###### Definition 2.1
A Novikov algebra $(\mathcal{A},\mu)$ is a vector space with a bilinear
product $\mu:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$
satisfying
$\displaystyle(xy)z$ $\displaystyle=$ $\displaystyle(xz)y,$ (2.1)
$\displaystyle(xy)z-x(yz)$ $\displaystyle=$ $\displaystyle(yx)z-y(xz),$ (2.2)
for all $x,y,z\in\mathcal{A}$, and where $\mu(x,y)=xy$.
In the sequel, for simplicity, we often write $xy$ instead of $\mu(x,y)$ since
confusion rarely occurs. Note that Novikov algebras are a special class of
left-symmetric algebras with only equation (2.2) satisfied. Define an
associator on $\mathcal{A}$ by
$\displaystyle a(x,y,z)=(xy)z-x(yz),\hskip 8.5359pt\mbox{for all}\
x,y,z\in\mathcal{A}.$
Then equation (2.2) is equivalent to
$\displaystyle a(x,y,z)=a(y,x,z),\hskip 8.5359pt\mbox{for all}\
x,y,z\in\mathcal{A}.$
In other words, the associator is symmetric in $x,y$, from which the left-
symmetric algebra takes its name.
The commutator of a Novikov algebra (or a left-symmetric algebra)
$\mathcal{A}$
$[x,y]=xy-yx,\hskip 8.5359pt\mbox{for all}\ x,y\in\mathcal{A},$
defines a Lie algebra $Lie(\mathcal{A})$, which is called the sub-adjacent Lie
algebra of $\mathcal{A}$. Hence both Novikov algebras and left-symmetric
algebras are Lie-admissible algebras.
In order to establish tensor theory of Novikov algebras, Xu in [22] introduced
Novikov-Poisson algebras, which play important roles in the study of simple
Novikov algebras.
###### Definition 2.2
A Novikov-Poisson algebra is a vector space $\mathcal{A}$ with two operations
$\cdot$ and $\ast$ such that $(\mathcal{A},\cdot)$ forms a commutative
associative algebra and $(\mathcal{A},\ast)$ forms a Novikov algebra for which
$\displaystyle(x\cdot y)\ast z$ $\displaystyle=$ $\displaystyle x\cdot(y\ast
z),$ (2.3) $\displaystyle(x\ast y)\cdot z-x\ast(y\cdot z)$ $\displaystyle=$
$\displaystyle(y\ast x)\cdot z-y\ast(x\cdot z),$ (2.4)
for all $x,y,z\in\mathcal{A}$.
For an endomorphism of a Novikov-Poisson algebra $(\mathcal{A},\cdot,\ast)$,
we mean a linear map $\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ preserving
the two operations, i.e.,
$\displaystyle\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y),\hskip
8.5359pt\alpha(x\ast y)=\alpha(x)\ast\alpha(y),\hskip 8.5359pt\mbox{for all}\
x,y\in\mathcal{A}.$
The notion of Hom-Lie algebras was initially introduced by Hartwig, Larsson
and Silvestrov in[14] motivated by examples of deformed Lie algebras coming
from twisted discretizations of vector fields. A slightly more general
definition of Hom-Lie algebras was given by Makhlouf and Silvestrov in [18],
where the Hom-associative algebras were also introduced.
###### Definition 2.3
A Hom-Lie algebra is a triple $(L,[\cdot,\cdot],\alpha)$ consisting of a
vector space $L$, a bilinear map $[\cdot,\cdot]:L\times L\longrightarrow L$
and a linear map $\alpha:L\longrightarrow L$ satisfying the following two
conditions
$\displaystyle[x,y]=-[y,x],\hskip 14.22636pt\mbox{(skew-symmetry)}$
$\displaystyle[[x,y],\alpha(z)]+[[z,x],\alpha(y)]+[[y,z],\alpha(x)]=0,\hskip
14.22636pt\mbox{(Hom-Jacobi \ identity)}$
for all $x,y,z\in L.$
Clearly, Lie algebras are spacial classes of Hom-Lie algebras when $\alpha$ is
the identity map. If the linear map $\alpha$ is an algebra homomorphism with
respect to the bracket $[\cdot,\cdot]$, then the Hom-Lie algebras are a
spacial case of quasi-hom-Lie algebras [16]. Moreover, all these classes are
special cases of the more general quasi-Lie algebras [17], in which Lie color
algebras and in particular Lie superalgebras are also included.
###### Definition 2.4
A Hom-associative algebra is a triple $(V,\mu,\alpha)$ consisting of a linear
space $V$, a bilinear map $\mu:V\times V\longrightarrow V$ and a linear map
$\alpha:V\longrightarrow V$ satisfying
$\alpha(x)(yz)=(xy)\alpha(z),\hskip 8.5359pt\mbox{for all}\
x,y,z\in\mathcal{A}.$
We say that a Hom-associative algebra $(\mathcal{A},\mu,\alpha)$ is
commutative if $xy=yx$ holds for all $x,y\in\mathcal{A}$. The following result
provides a construction of Hom-Lie algebras from Hom-associative algebras,
extending the fundamental construction of Lie algebras from associative
algebras via commutator bracket multiplication.
###### Proposition 2.5
(see [References, Proposition 1.6]) To any Hom-associative algebra defined by
a multiplication $\mu$ and a linear map $\alpha$ over an $\mathbb{F}$-linear
space $V$, one may associate a Hom-Lie algebra defined by the bracket
$\displaystyle[x,y]=xy-yx,\hskip 8.5359pt\mbox{for all}\ x,y\in V.$
Following the patterns of Hom-Lie algebras and Hom-associative algebras, Yau
in[28] introduced a twisted generalization of Novikov algebras, called Hom-
Novikov algebras.
###### Definition 2.6
A Hom-Novikov algebra is a triple $(\mathcal{A},\mu,\alpha)$ consisting of a
vector space $\mathcal{A}$, a bilinear map
$\mu:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ and an algebra
homomorphism $\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ satisfying
$\displaystyle(xy)\alpha(z)$ $\displaystyle=$ $\displaystyle(xz)\alpha(y),$
(2.5) $\displaystyle(xy)\alpha(z)-\alpha(x)(yz)$ $\displaystyle=$
$\displaystyle(yx)\alpha(z)-\alpha(y)(xz),$ (2.6)
for all $x,y,z\in\mathcal{A}.$
Clearly, Novikov algebras are examples of Hom-Novikov algebras by setting
$\alpha=\rm{id}$, where $\rm{id}$ is the identity map. It was shown in [28]
that any Novikov algebra can be deformed into a Hom-Novikov algebra along with
an algebra endomorphism.
###### Proposition 2.7
(see [References, Theorem 1.1] ) Let $(\mathcal{A},\mu)$ be a Novikov algebra
and $\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ be an algebra homomorphism.
Then $(\mathcal{A},\alpha\circ\mu,\alpha)$ is a Hom-Novikov algebra, where
$\circ$ denotes composition of functions.
Similar to Hom-Novikov algebras, Yau in [29] introduced Hom-Novikov-Poisson
algebras.
###### Definition 2.8
A Hom-Novikov-Poisson algebra is a quadruple
$(\mathcal{A},\cdot,\ast,\alpha)$, where $\mathcal{A}$ is a vector space,
$\cdot$ and $\ast$ are bilinear maps and $\alpha$ is an algebra homomorphism
of $\mathcal{A}$, such that $(\mathcal{A},\ast,\alpha)$ forms a Hom-Novikov
algebra, $(\mathcal{A},\cdot,\alpha)$ forms a commutative Hom-associative
algebra and the following two compatible conditions hold:
$\displaystyle(x\cdot y)\ast\alpha(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\cdot(y\ast z),$ (2.7) $\displaystyle(x\ast
y)\cdot\alpha(z)-\alpha(x)\ast(y\cdot z)$ $\displaystyle=$
$\displaystyle(y\ast x)\cdot\alpha(z)-\alpha(y)\ast(x\cdot z),$ (2.8)
for all $x,y,z\in\mathcal{A}$.
We can recover Novikov-Poisson algebras from Hom-Novikov-Poisson algebras when
$\alpha={\rm id}$. Some more interesting examples of Hom-Novikov-Poisson
algebras will be constructed in Section 5.
§3. Hom-Novikov algebras
In this section, we establish the relationships between Hom-Novikov algebras
and Novikov algebras as well as the relations between Hom-Novikov algebras and
Hom-Lie algebras. Also, we provide a construction of Hom-Novikov algebras from
a commutative Hom-associative algebra along with a suitable linear map but not
a derivation. Thus, it is different from that given in [28].
Just as a Novikov algebra can be deformed into a Lie algebra via the
commutator bracket, we have the similar result for Hom-Novikov algebras and
Hom-Lie algebras, because in Hom-Novikov algebras hold Jacobi-like identities.
###### Proposition 3.1
Let $(\mathcal{A},\mu,\alpha)$ be a Hom-Novikov algebra. For all
$x,y,z\in\mathcal{A}$, one has
$\displaystyle[x,y]\alpha(z)+[y,z]\alpha(x)+[z,x]\alpha(y)$ $\displaystyle=$
$\displaystyle 0,$ (3.1)
$\displaystyle\alpha(x)[y,z]+\alpha(y)[z,x]+\alpha(z)[x,y]$ $\displaystyle=$
$\displaystyle 0,$ (3.2)
where $\mu(x,y)=xy$ and $[x,y]=xy-yx$.
Proof. For all $x,y,z\in\mathcal{A}$, we have
$\displaystyle[x,y]\alpha(z)$ $\displaystyle=$
$\displaystyle(xy)\alpha(z)-(yx)\alpha(z),$ $\displaystyle{[y,z]}\alpha(x)$
$\displaystyle=$ $\displaystyle(yz)\alpha(x)-(zy)\alpha(x),$
$\displaystyle{[z,x]}\alpha(y)$ $\displaystyle=$
$\displaystyle(zx)\alpha(y)-(xz)\alpha(y).$
Since equation (2.5) holds for all $x,y,z\in\mathcal{A}$, we have
$\displaystyle[x,y]\alpha(z)+[y,z]\alpha(x)+[z,x]\alpha(y)$ $\displaystyle=$
$\displaystyle\big{(}(xy)\alpha(z)-(xz)\alpha(y)\big{)}+\big{(}(yz)\alpha(x)-(yx)\alpha(z)\big{)}+\big{(}(zx)\alpha(y)-(zy)\alpha(x)\big{)}$
$\displaystyle=$ $\displaystyle 0,$
which proves equation (3.1). Now using equations (2.6) and (3.1), we have
$\displaystyle\alpha(x)[y,z]+\alpha(y)[z,x]+\alpha(z)[x,y]$ $\displaystyle=$
$\displaystyle\alpha(x)(yz)-\alpha(x)(zy)+\alpha(y)(zx)-\alpha(y)(xz)+\alpha(z)(xy)-\alpha(z)(yx)$
$\displaystyle=$
$\displaystyle\big{(}\alpha(x)(yz)-\alpha(y)(xz)\big{)}+\big{(}\alpha(y)(zx)-\alpha(z)(yx)\big{)}+\big{(}\alpha(z)(xy)-\alpha(x)(zy)\big{)}$
$\displaystyle=$
$\displaystyle\big{(}(xy)\alpha(z)-(yx)\alpha(z)\big{)}+\big{(}(yz)\alpha(x)-(zy)\alpha(x)\big{)}+\big{(}(zx)\alpha(y)-(xz)\alpha(y)\big{)}$
$\displaystyle=$ $\displaystyle[x,y]\alpha(z)+[y,z]\alpha(x)+[z,x]\alpha(y)$
$\displaystyle=$ $\displaystyle 0,$
which proves equation (3.2) and the proposition. $\Box$
###### Corollary 3.2
Let $(\mathcal{A},\mu,\alpha)$ be a Hom-Novikov algebra. Define a bilinear map
$[\cdot,\cdot]:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ by
$[x,y]=\mu(x,y)-\mu(y,x),\hskip 8.5359pt\mbox{for all}\ x,y\in\mathcal{A}.$
Then $HLie(\mathcal{A})=(\mathcal{A},[\cdot,\cdot],\alpha)$ is a Hom-Lie
algebra with the same twisting map $\alpha$.
The Lie admissible algebras were introduced by A.A. Albert in 1948. Makhlouf
and Silvestrov in [18] extended the notions and results about Lie admissible
algebras to Hom-algebras, called Hom-Lie admissible algebras. According to
Corollary 3.2, any Hom-Novikov algeba is Hom-Lie admissible with the same
twisting map. We call such a Hom-Lie algebra $HLie(\mathcal{A})$ the sub-
adjacent Hom-Lie algebra of the Hom-Novikov algebra
$(\mathcal{A},\mu,\alpha)$.
###### Definition 3.3
Let $(\mathcal{A},\mu,\alpha)$ be a Hom-Novikov algebra, which is called
* (i)
regular if $\alpha$ is an algebra automorphism;
* (ii)
involutive if $\alpha$ is an involution, i.e., $\alpha^{2}=\rm{id}.$
Yau in [28] gave a way to construct Hom-Novikov algebras, starting from a
Novikov algebra and an algebra endomorphism. In the following, we provide a
construction of Novikov algebras from Hom-Novikov algebras along with an
algebra automorphism.
###### Proposition 3.4
If $(\mathcal{A},\mu,\alpha)$ is an involutive Hom-Novikov algebra, then
$(\mathcal{A},\alpha\circ\mu)$ is a Novikov algebra.
Proof. For convenience, we write $\mu(x,y)=xy$ and $x\star y=\alpha(xy),$ for
all $x,y\in\mathcal{A}$. Hence, it needs to show
$\displaystyle(x\star y)\star z$ $\displaystyle=$ $\displaystyle(x\star
z)\star y,$ (3.3) $\displaystyle(x\star y)\star z-x\star(y\star z)$
$\displaystyle=$ $\displaystyle(y\star x)\star z-y\star(x\star z),$ (3.4)
for all $x,y,z\in\mathcal{A}$. Since $\alpha$ is an involution, we have
$\displaystyle(x\star y)\star z$ $\displaystyle=$
$\displaystyle\alpha\big{(}\alpha(xy)z\big{)}=\big{(}\alpha^{2}(xy)\big{)}\alpha(z)=(xy)\alpha(z).$
Similarly, we have $(x\star z)\star y=(xz)\alpha(y)$, from which equation
(3.3) follows since $(\mathcal{A},\mu,\alpha)$ is a Hom-Novikov algebra.
Furthermore, using equation (2.6), we have
$\displaystyle(x\star y)\star z-x\star(y\star z)$ $\displaystyle=$
$\displaystyle\alpha\big{(}\alpha(xy)z\big{)}-\alpha\big{(}x\alpha(yz)\big{)}$
$\displaystyle=$ $\displaystyle(xy)\alpha(z)-\alpha(x)(yz)$ $\displaystyle=$
$\displaystyle(yx)\alpha(z)-\alpha(y)(xz)$ $\displaystyle=$
$\displaystyle\alpha\big{(}\alpha(yx)z\big{)}-\alpha\big{(}y\alpha(xz)\big{)}$
$\displaystyle=$ $\displaystyle(y\star x)\star z-y\star(x\star z),$
which proves equation (3.4) and the proposition. $\Box$
###### Proposition 3.5
If $(\mathcal{A},\mu,\alpha)$ is a regular Hom-Novikov algebra, then
$(\mathcal{A},[\cdot,\cdot]_{\alpha^{-1}}=\alpha^{-1}\circ[\cdot,\cdot])$ is a
Lie algebra, where $[x,y]=\mu(x,y)-\mu(y,x)=xy-yx,$ for all
$x,y\in\mathcal{A}.$ In particular, if $\alpha$ is an involution, then
$(\mathcal{A},[\cdot,\cdot]_{\alpha}=\alpha\circ[\cdot,\cdot])$ is a Lie
algebra.
Proof. For any $x,y,z\in\mathcal{A}$, we have
$\displaystyle[[x,y]_{\alpha^{-1}},z]_{\alpha^{-1}}$ $\displaystyle=$
$\displaystyle[\alpha^{-1}(xy-yx),z]_{\alpha^{-1}}$ $\displaystyle=$
$\displaystyle\alpha^{-1}\big{(}\alpha^{-1}(xy-yx)z-z\alpha^{-1}(xy-
yx)\big{)}$ $\displaystyle=$ $\displaystyle\alpha^{-2}\big{(}(xy-
yx)\alpha(z)-\alpha(z)(xy-yx)\big{)}$ $\displaystyle=$
$\displaystyle\alpha^{-2}\big{(}[x,y]\alpha(z)-\alpha(z)[x,y]\big{)}.$
Similarly, we have
$\displaystyle[[y,z]_{\alpha^{-1}},x]_{\alpha^{-1}}$ $\displaystyle=$
$\displaystyle\alpha^{-2}\big{(}[y,z]\alpha(x)-\alpha(x)[y,z]\big{)},$
$\displaystyle{[[z,x]_{\alpha^{-1}},y]_{\alpha^{-1}}}$ $\displaystyle=$
$\displaystyle\alpha^{-2}\big{(}[z,x]\alpha(y)-\alpha(y)[z,x]\big{)}.$
Then it follows from equations (3.1) and (3.2) that
$[[x,y]_{\alpha^{-1}},z]_{\alpha^{-1}}+[[y,z]_{\alpha^{-1}},x]_{\alpha^{-1}}+{[[z,x]_{\alpha^{-1}},y]_{\alpha^{-1}}}=0.$
Clearly, $[x,y]=-[y,x].$ Since $\alpha^{-1}$ is an automorphism, we have
$[x,y]_{\alpha^{-1}}=-[y,x]_{\alpha^{-1}},$
which proves that $(\mathcal{A},[\cdot,\cdot]_{\alpha^{-1}})$ is a Lie
algebra. It follows immediately that $(\mathcal{A},[\cdot,\cdot]_{\alpha})$ is
also a Lie algebra since $\alpha=\alpha^{-1}$ when $\alpha$ is an involution.
$\Box$
Let $(\mathcal{A},\mu)$ be a commutative associative algebra and $D$ be a
derivation of $(\mathcal{A},\mu)$. Then the new product
$\displaystyle x\star_{\lambda}y=xD(y)+\lambda xy,\hskip 8.5359pt\mbox{for
all}\ x,y\in\mathcal{A},$ (3.5)
makes $(\mathcal{A},\star_{\lambda})$ become a Novikov algebra for $\lambda=0$
by Gel’fand and Dorfman [12], for $\lambda\in\mathbb{F}$ by Filipov [11], and
for a fixed element $\lambda\in\mathcal{A}$ by Xu [21]. Yau [28] generalized
this construction to Hom-Novikov algebras $(\mathcal{A},\mu,\alpha)$ in
$\lambda=0$ setting together with an additional condition that $D$ commutes
with $\alpha$ . It is natural to consider other type of linear maps to replace
the derivation $D$ in (3.5), such that $(\mathcal{A},\star_{\lambda})$ forms a
Novikov algebra or a Hom-Novikov algebra.
Let $(\mathcal{A},\mu,\alpha)$ be a commutative Hom-associative algebra with a
linear selfmap $\partial$ commuting with $\alpha$. Consider the following
operation on $\mathcal{A}$:
$\displaystyle x\star y=x\partial(y),\hskip 8.5359pt\mbox{for all}\
x,y\in\mathcal{A}.$ (3.6)
For all $x,y,z\in\mathcal{A}$, by straightforward calculation, we have
$\displaystyle(x\star y)\star\alpha(z)$ $\displaystyle=$
$\displaystyle(x\partial(y))\star\alpha(z)$ $\displaystyle=$
$\displaystyle(x\partial(y))\partial(\alpha(z))$ $\displaystyle=$
$\displaystyle(x\partial(y))\alpha(\partial(z))$ $\displaystyle=$
$\displaystyle\alpha(x)(\partial(y)\partial(z)),$
and
$\displaystyle(x\star z)\star\alpha(y)$ $\displaystyle=$
$\displaystyle(x\partial(z))\star\alpha(y)$ $\displaystyle=$
$\displaystyle(x\partial(z))\partial(\alpha(y))$ $\displaystyle=$
$\displaystyle(x\partial(z))\alpha(\partial(y))$ $\displaystyle=$
$\displaystyle\alpha(x)(\partial(z)\partial(y))$ $\displaystyle=$
$\displaystyle\alpha(x)(\partial(y)\partial(z)),$
where both commutativity and Hom-associativity of $(\mathcal{A},\mu,\alpha)$
are used. Hence, we have
$\displaystyle(x\star y)\star\alpha(z)=(x\star z)\star\alpha(y).$
Similarly, we have
$\displaystyle(x\star y)\star\alpha(z)-\alpha(x)\star(y\star
z)=\alpha(x)(\partial(y)\partial(z))-\alpha(x)\partial(y\partial(z)),$ (3.7)
and
$\displaystyle(y\star x)\star\alpha(z)-\alpha(y)\star(x\star
z)=\alpha(y)(\partial(x)\partial(z))-\alpha(y)\partial(x\partial(z)).$ (3.8)
If, in addition, $\partial$ satisfies the following condition
$\displaystyle\partial(x\partial(y))=\partial(x)\partial(y),\hskip
8.5359pt\mbox{for all}\ x,y\in\mathcal{A}.$ (3.9)
Then it follows from equations (3.7) and (3.8) that
$\displaystyle(x\star y)\star\alpha(z)-\alpha(x)\star(y\star z)=(y\star
x)\star\alpha(z)-\alpha(y)\star(x\star z)=0.$
Now from the discussions above, we obtain
###### Theorem 3.6
Let $(\mathcal{A},\mu,\alpha)$ be a commutative Hom-associative algebra with a
linear selfmap $\partial$ satisfying equation (3.9) and commuting with
$\alpha$. Then $(\mathcal{A},\star,\alpha)$ is a Hom-Novikov algebra, where
$\star$ is defined by (3.6).
We have the following corollary by setting $\alpha={\rm id}$ in the theorem.
###### Corollary 3.7
Let $(\mathcal{A},\mu)$ be a commutative associative algebra with a linear
selfmap $\partial$ satisfying condition (3.9). Then $(\mathcal{A},\star)$ is a
Novikov algebra in which $\star$ is defined by (3.6).
###### Example 3.8
Let $(\mathcal{A}=\mathcal{A}_{0}\oplus\mathcal{A}_{1},\mu)$ be a complex
superalgebra, where $\mathcal{A}_{0}=\mathbb{C}[t,t^{-1}]$ is the Laurent
polynomials in one variable and $\mathcal{A}_{1}=\theta\mathbb{C}[t,t^{-1}]$
in which $\theta$ is the Grassman variable satisfying $\theta^{2}=0$. We
assume that $t$ commutes with $\theta$. The generators of $\mathcal{A}$ are of
the form $t^{n}$ and $\theta t^{n}$ for all $n\in\mathbb{Z}$. Then
$\mathcal{A}$ is a commutative associative superalgebra under the usual
multiplication.
Define a linear map $\partial:\mathcal{A}\longrightarrow\mathcal{A}$ by
$\displaystyle\partial(t^{n}+\theta t^{m})=t^{n},\hskip 8.5359pt\mbox{for
all}\ m,n\in\mathbb{Z}.$ (3.10)
It is easy to check that $\partial$ satisfies condition (3.9). According to
Corollary 3.7, $(\mathcal{A},\star_{1})$ is a Novikov algebra, where the
operation $\star_{1}$ is defined by
$\displaystyle(t^{n_{1}}+\theta t^{m_{1}})\star_{1}(t^{n_{2}}+\theta
t^{m_{2}})=(t^{n_{1}}+\theta t^{m_{1}})\partial(t^{n_{2}}+\theta
t^{m_{2}})=t^{n_{1}+n_{2}}+\theta t^{m_{1}+n_{2}},$
for all $n_{1},n_{2},m_{1},m_{2}\in\mathbb{Z}$.
Furthermore, define an algebra endomorphism $\alpha$ on $\mathcal{A}$ by
$\alpha(t^{n}+\theta t^{m})=(t+c)^{n},\hskip 8.5359pt\mbox{for all}\
m,n\in\mathbb{Z},$
where $c$ is a fixed element in $\mathbb{C}$. Then
$(\mathcal{A},\alpha\circ\mu,\alpha)$ is a commutative Hom-associative
algebra. For convenience, we write $\cdot$ instead of $\alpha\circ\mu$, then
we have
$\displaystyle(t^{n_{1}}+\theta t^{m_{1}})\cdot(t^{n_{2}}+\theta
t^{m_{2}})=(t+c)^{n_{1}+n_{2}},$
for all $n_{1},n_{2},m_{1},m_{2}\in\mathbb{Z}$. Let $\partial$ be as defined
by (3.10). Clearly, $\partial$ commutes with $\alpha$. Moreover, we have
$\displaystyle\partial\big{(}(t^{n_{1}}+\theta
t^{m_{1}})\cdot\partial(t^{n_{2}}+\theta
t^{m_{2}})\big{)}=(t+c)^{n_{1}+n_{2}},$
and
$\displaystyle\partial(t^{n_{1}}+\theta
t^{m_{1}})\cdot\partial(t^{n_{2}}+\theta t^{m_{2}})=(t+c)^{n_{1}+n_{2}}.$
Hence, $\partial$ satisfies condition (3.9). Define a new bilinear operation
on $\mathcal{A}$ by
$\displaystyle(t^{n_{1}}+\theta t^{m_{1}})\star_{2}(t^{n_{2}}+\theta
t^{m_{2}})=(t^{n_{1}}+\theta t^{m_{1}})\cdot\partial(t^{n_{2}}+\theta
t^{m_{2}})=(t+c)^{n_{1}+n_{2}},$
for all $n_{1},n_{2},m_{1},m_{2}\in\mathbb{Z}$. Then
$(\mathcal{A},\star_{2},\alpha)$ forms a Hom-Novikov algebra by Theorem 3.6.
§4. Quadratic Hom-Novikov Algebras
In this section, we extend the notion of quadratic Novikov algebra to
quadratic Hom-Novikov algebras and provide some properties.
Recall that a quadratic Lie algebra is a triple
$(\mathcal{G},[\cdot,\cdot],B)$, where $(\mathcal{G},[\cdot,\cdot])$ is a Lie
algebra and $B:\mathcal{G}\times\mathcal{G}\longrightarrow\mathbb{F}$ is a
symmetric nondegenerate bilinear form satisfying
$\displaystyle B([x,y],z)=B(x,[y,z]),\hskip 8.5359pt\mbox{for all}\
x,y,z\in\mathcal{G},$ (4.1)
which is called the invariance of $B$. It is easy to see that identity (4.1)
is equivalent to
$\displaystyle B([x,y],z)=-B(y,[x,z]),\hskip 8.5359pt\mbox{for all}\
x,y,z\in\mathcal{G}.$
Assume that $(\mathcal{A},\mu)$ is a Novikov algebra. A bilinear form
$B:\mathcal{A}\times\mathcal{A}\longrightarrow\mathbb{F}$ is said to be
invariant or associative if and only if
$B(xy,z)=B(x,yz),\hskip 8.5359pt\mbox{for all}\ x,y,z\in\mathcal{A}.$
A Novikov algebra $(\mathcal{A},\mu)$ with a symmetric nondegenerate invariant
bilinear form is called a quadratic Novikov algebra and denoted by
$(\mathcal{A},\mu,B)$.
Benayadi and Makhlouf in [5] extended the notion of quadratic Lie algebra to
Hom-Lie algebras and obtained quadratic Hom-Lie algebras.
###### Definition 4.1
A quadratic Hom-Lie algebra is a quadruple
$(\mathcal{A},[\cdot,\cdot],\alpha,B)$ such that
$(\mathcal{A},[\cdot,\cdot],\alpha)$ is a Hom-Lie algebra with a symmetric
invariant nondegenerate bilinear form $B$ satisfying
$\displaystyle B(\alpha(x),y)=B(x,\alpha(y)),\hskip 8.5359pt\mbox{for all}\
x,y\in\mathcal{A}.$ (4.2)
We can define quadratic Hom-Novikov algebras as follows.
###### Definition 4.2
A quadratic Hom-Novikov algebra $(\mathcal{A},\mu,\alpha,B)$ is a Hom-Novikov
algebra $(\mathcal{A},\mu,\alpha)$ with a symmetric nondegenerate bilinear
form satisfying
$\displaystyle B(xy,\alpha(z))=B(\alpha(x),yz),\hskip 8.5359pt\mbox{for all}\
x,y,z\in\mathcal{A}.$ (4.3)
We recover quadratic Novikov algebras when $\alpha={\rm id}$. The following
result says the sub-adjacent Hom-Lie algebra of a quadratic Hom-Novikov
algebra is also quadratic.
###### Proposition 4.3
Let $(\mathcal{A},\mu,\alpha,B)$ be a quadratic Hom-Novikov algebra and
$HLie(\mathcal{A})=(\mathcal{A},[\cdot,\cdot],\alpha)$ be the sub-adjacent
Hom-Lie algebra of $\mathcal{A}$. If $\alpha$ is an automorphism satisfying
$\displaystyle B(\alpha(x),y)=B(x,\alpha(y)),\hskip 8.5359pt\mbox{for all}\
x,y\in\mathcal{A}.$ (4.4)
Then $(\mathcal{A},[\cdot,\cdot],\alpha,B_{\alpha})$ is a quadratic Hom-Lie
algebra, where $B_{\alpha}(x,y)=B(\alpha(x),y)$, for all $x,y\in\mathcal{A}$.
Proof. Since $B$ is a nondegenerate bilinear form and $\alpha$ is an
automorphism, $B_{\alpha}$ is a nondegenerate bilinear form on $\mathcal{A}$.
For all $x,y,z\in\mathcal{A}$, using the properties of $B$, we have
$\displaystyle B_{\alpha}([x,y],z)$ $\displaystyle=$ $\displaystyle
B\big{(}\alpha([x,y]),z\big{)}$ $\displaystyle=$ $\displaystyle
B([x,y],\alpha(z))$ $\displaystyle=$ $\displaystyle
B(xy,\alpha(z))-B(yx,\alpha(z))$ $\displaystyle=$ $\displaystyle
B(\alpha(x),yz)-B(\alpha(x),zy)$ $\displaystyle=$ $\displaystyle
B(\alpha(x),[y,z])$ $\displaystyle=$ $\displaystyle B_{\alpha}(x,[y,z]).$
Hence $B_{\alpha}$ is invariant. Using symmetry of $B$ and equation (4.4), we
have
$\displaystyle
B_{\alpha}(x,y)=B(\alpha(x),y)=B(y,\alpha(x))=B(\alpha(y),x)=B_{\alpha}(y,x),$
which proves $B_{\alpha}$ is symmetric. Using equation (4.4) again, we have
$B_{\alpha}(\alpha(x),y)=B(\alpha(\alpha(x)),y)=B(\alpha(x),\alpha(y))=B_{\alpha}(x,\alpha(y)),$
which completes the proof. $\Box$
###### Corollary 4.4
Let $(\mathcal{A},\mu,B)$ be a quadratic Novikov algebra with an automorphism
$\alpha$ satisfying equation (4.4) and $(\mathcal{A},[\cdot,\cdot])$ be the
sub-adjacent Lie algebra. Then
$(\mathcal{A},[\cdot,\cdot]_{\alpha}=\alpha\circ[\cdot,\cdot],\alpha,B_{\alpha})$
forms a quadratic Hom-Lie algebra, where $B_{\alpha}(x,y)=B(\alpha(x),y)$, for
all $x,y\in\mathcal{A}$.
Proof. It follows from [References, Corollary 2.6] that
$(\mathcal{A},[\cdot,\cdot]_{\alpha},\alpha)$ is a Hom-Lie algebra. Using the
similar arguments as those in the proof of Proposition 4.3, we get
$B_{\alpha}$ is a symmetric nondegenerate bilinear form with equation (4.2)
satisfied. It remains to show that $B_{\alpha}$ is invariant. For all
$x,y,z\in\mathcal{A}$, using invariance and symmetry of $B$, we have
$\displaystyle B_{\alpha}([x,y]_{\alpha},z)$ $\displaystyle=$ $\displaystyle
B\big{(}\alpha([x,y]_{\alpha}),z\big{)}$ $\displaystyle=$ $\displaystyle
B\big{(}[x,y]_{\alpha},\alpha(z)\big{)}$ $\displaystyle=$ $\displaystyle
B(\alpha(x)\alpha(y),\alpha(z))-B(\alpha(y)\alpha(x),\alpha(z))$
$\displaystyle=$ $\displaystyle
B(\alpha(x),\alpha(y)\alpha(z))-B(\alpha(z)\alpha(y),\alpha(x))$
$\displaystyle=$ $\displaystyle B(\alpha(x),[y,z]_{\alpha})$ $\displaystyle=$
$\displaystyle B_{\alpha}(x,[y,z]_{\alpha}),$
which proves the invariance of $B_{\alpha}$ and the result. $\Box$
###### Proposition 4.5
Let $(\mathcal{A},\mu,\alpha,B)$ be a quadratic Hom-Novikov algebra, where
$\alpha$ is an involution satisfying equation (4.4). Then
$(\mathcal{A},\star=\alpha\circ\mu,B)$ is a quadratic Novikov algebra.
Proof. $({A},\star=\alpha\circ\mu,\alpha)$ is a Hom-Novikov algebra by
Proposition 3.4. It suffices to show that $B$ is invariant under the operation
$\star$. For all $x,y,z\in\mathcal{A}$, we have
$\displaystyle B(x,y\star
z)=B(x,\alpha(y)\alpha(z))=B(\alpha(x),yz)=B(xy,\alpha(z))=B(\alpha(x)\alpha(y),z)=B(x\star
y,z),$
which completes the proof. $\Box$
###### Proposition 4.6
Let $(\mathcal{A},\mu,\alpha,B)$ be a quadratic Hom-Novikov algebra, where
$\alpha$ is an automorphism satisfying equation (4.4). Then,
$(\mathcal{A},\star=\alpha\circ\mu,\alpha^{2},B_{\alpha^{2}})$ is a quadratic
Hom-Novikov algebra, where $B_{\alpha^{2}}(x,y)=B(\alpha^{2}(x),y)$, for all
$x,y\in\mathcal{A}$.
Proof. It follows from [References, Corollary 2.12]) that
$(\mathcal{A},\alpha\circ\mu,\alpha^{2})$ forms a Hom-Novikov algebra. Since
$B$ is a nondegenerate bilinear form on $\mathcal{A}$ and $\alpha$ is an
automorphism, $B_{\alpha^{2}}$ is a nondegenerate bilinear form. For all
$x,y,z\in\mathcal{A}$, using the hypothesis, we have
$B_{\alpha^{2}}(x,y)=B(\alpha^{2}(x),y)=B(\alpha(x),\alpha(y))=B(x,\alpha^{2}(y))=B(\alpha^{2}(y),x)=B_{\alpha^{2}}(y,x).$
Thus, $B_{\alpha^{2}}$ is symmetric. Moreover, we have
$\displaystyle B_{\alpha^{2}}\big{(}\alpha^{2}(x),y\star z\big{)}$
$\displaystyle=$ $\displaystyle
B\big{(}\alpha^{4}(x),\alpha(y)\alpha(z)\big{)}$ $\displaystyle=$
$\displaystyle B\big{(}\alpha^{3}(x),\alpha^{2}(y)\alpha^{2}(z)\big{)}$
$\displaystyle=$ $\displaystyle
B\big{(}\alpha^{2}(x)\alpha^{2}(y),\alpha^{3}(z)\big{)}$ $\displaystyle=$
$\displaystyle B\big{(}\alpha(x)\alpha(y),\alpha^{4}(z)\big{)}$
$\displaystyle=$ $\displaystyle B_{\alpha^{2}}\big{(}x\star
y,\alpha^{2}(z)\big{)},$
which proves the invariance of $B_{\alpha^{2}}$ and the proposition. $\Box$
###### Corollary 4.7
Let $(\mathcal{A},\mu,\alpha,B)$ be a quadratic Hom-Novikov algebra and
$\alpha$ be an automorphism satisfying equation (4.4). Then
$(\mathcal{A},\alpha^{n}\circ\mu,\alpha^{n+1},B_{\alpha^{n+1}})$ is a
quadratic Hom-Novikov algebra, where
$B_{\alpha^{n+1}}(x,y)=B(\alpha^{n+1}(x),y)$, for all $n>0$ and
$x,y\in\mathcal{A}$.
Let $(\mathcal{A},\mu,\alpha)$ be a Hom-Novikov algebra, whose center is
denoted by $\mathcal{Z}(\mathcal{A})$ and defined by
$\displaystyle\mathcal{Z}(\mathcal{A})=\\{x\in\mathcal{A}|xy=yx=0,\ \mbox{for
all}\ y\in\mathcal{A}\\}.$
Let $(\mathcal{G},[\cdot,\cdot],\beta)$ be a Hom-Lie algebra. The lower
central series of $\mathcal{G}$ is defined as usual, i.e.,
$\mathcal{G}^{1}=\mathcal{G}$,
$\mathcal{G}^{i+1}=[\mathcal{G},\mathcal{G}^{i}]$ for all $i\geq 1$. We call
$\mathcal{G}$ is $i$-step nilpotent if $\mathcal{G}^{i+1}=0$. The center of
the Hom-Lie algebra is denoted by $\mathcal{C}(\mathcal{G})$ and defined by
$\displaystyle\mathcal{C}(\mathcal{G})=\\{x\in\mathcal{G}|[x,y]=0,\ \mbox{for
all}\ y\in\mathcal{G}\\}.$
###### Proposition 4.8
Let $(\mathcal{A},\mu,\alpha,B)$ be a quadratic Hom-Novikov algebra and
$HLie(\mathcal{A})$ be the sub-adjacent Hom-Lie algebra. If $\alpha$ is an
automorphism, then
$[HLie(\mathcal{A}),HLie(\mathcal{A})]\subseteq\mathcal{Z}(\mathcal{A}).$ As a
consequence, $HLie(\mathcal{A})$ is $2$-step nilpotent.
Proof. For all $x,y,z,w\in\mathcal{A}$, using equations (2.5), (2.6) and
(4.3), we have
$\displaystyle B(\alpha(x)[y,z],\alpha^{2}(w))$ $\displaystyle=$
$\displaystyle B(\alpha^{2}(x),[y,z]\alpha(w))$ $\displaystyle=$
$\displaystyle B(\alpha^{2}(x),(yz)\alpha(w)-(zy)\alpha(w))$ $\displaystyle=$
$\displaystyle B(\alpha^{2}(x),\alpha(y)(zw)-\alpha(z)(yw))$ $\displaystyle=$
$\displaystyle
B(\alpha(x)\alpha(y),\alpha(zw))-B(\alpha(x)\alpha(z),\alpha(yw))$
$\displaystyle=$ $\displaystyle B((xy)\alpha(z)-(xz)\alpha(y),\alpha^{2}(w))$
$\displaystyle=$ $\displaystyle 0.$
Using the symmetry of $B$ and equation (4.3), we have
$B([y,z]\alpha(w),\alpha^{2}(x))=B(\alpha(x)[y,z],\alpha^{2}(w))=0,$
which implies $[y,z]\subseteq\mathcal{Z}(\mathcal{A})$ since $\alpha$ is an
automorphism and $B$ is nondegenerate. Hence, we have
$[HLie(\mathcal{A}),HLie(\mathcal{A})]\subseteq\mathcal{Z}(\mathcal{A}).$
Obviously, $\mathcal{Z}(\mathcal{A})\subseteq\mathcal{C}(HLie(\mathcal{A}))$.
Then it follows that $HLie(\mathcal{A})$ is $2$-step nilpotent. $\Box$
§5. Hom-Novikov-Poisson Algebras
In this section, we discuss Hom-Novikov-Poisson algebras. We give different
ways to construct Hom-Novikov-Poisson algebras and provide some interesting
examples.
The following result (see also [References, Corollary 2.15])says that any Hom-
Novikov-Poisson algebra can be constructed from a Novikov-Poisson algebra and
an algebra endomorphism.
###### Proposition 5.1
Let $\big{(}\mathcal{A},\mu,\nu\big{)}$ be a Novikov-Poisson algebra such that
$\big{(}\mathcal{A},\nu\big{)}$ is a commutative associative algebra and
$\big{(}\mathcal{A},\mu\big{)}$ is a Novikov algebra. If
$\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ is an algebra homomorphism.
Then
$\mathcal{A}_{\alpha}=\big{(}\mathcal{A},\mu_{\alpha}=\alpha\circ\mu,\nu_{\alpha}=\alpha\circ\nu,\alpha\big{)}$
forms a Hom-Novikov-Poisson algebra.
Proof. It follows that $\big{(}\mathcal{A},\mu_{\alpha},\alpha\big{)}$ is a
Hom-Novikov algebra from Proposition 2.7 and
$\big{(}\mathcal{A},\nu_{\alpha},\alpha\big{)}$ is a Hom-associative algebra
from [References, Corollary 2.6]. Clearly,
$\big{(}\mathcal{A},\nu_{\alpha},\alpha\big{)}$ is commutative since $\alpha$
is an algebra homomorphism and $(\mathcal{A},\nu)$ is commutative. By
Definition 2.8, it suffices to check conditions (2.7) and (2.8). For all
$x,y,z\in\mathcal{A}$, Using equation (2.4), we have
$\displaystyle\nu_{\alpha}\big{(}\mu_{\alpha}(x,y),\alpha(z)\big{)}-\mu_{\alpha}\big{(}\alpha(x),\nu_{\alpha}(y,z)\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\nu\big{(}\alpha\circ\mu(x,y),\alpha(z)\big{)}-\alpha\circ\mu\big{(}\alpha(x),\alpha\circ\nu(y,z)\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\alpha\circ\nu\big{(}\mu(x,y),z\big{)}-\alpha\circ\alpha\circ\mu\big{(}x,\nu(y,z)\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\alpha\circ\Big{(}\nu\big{(}\mu(x,y),z\big{)}-\mu\big{(}x,\nu(y,z)\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\alpha\circ\Big{(}\nu\big{(}\mu(y,x),z\big{)}-\mu\big{(}y,\nu(x,z)\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\nu\big{(}\alpha\circ\mu(y,x),\alpha(z)\big{)}-\alpha\circ\mu\big{(}\alpha(y),\alpha\circ\nu(x,z)\big{)}$
$\displaystyle=$
$\displaystyle\nu_{\alpha}\big{(}\mu_{\alpha}(y,x),\alpha(z)\big{)}-\mu_{\alpha}\big{(}\alpha(y),\nu_{\alpha}(x,z)\big{)},$
which proves equation (2.8). Using equation (2.3), we have
$\displaystyle\mu_{\alpha}\big{(}\nu_{\alpha}(x,y),\alpha(z)\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\mu\big{(}\alpha\circ\nu(x,y),\alpha(z)\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\alpha\circ\mu\big{(}\nu(x,y),z\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\alpha\circ\nu\big{(}x,\mu(y,z)\big{)}$
$\displaystyle=$
$\displaystyle\nu_{\alpha}\big{(}\alpha(x),\mu_{\alpha}(y,z)\big{)},$
which proves equation (2.7) and the result. $\Box$
Suppose that $\big{(}\mathcal{A}_{1},\mu^{1},\nu^{1}\big{)}$ and
$\big{(}\mathcal{A}_{2},\mu^{2},\nu^{2}\big{)}$ are Novikov-Poisson algebras.
Define two operations $\nu^{1}\otimes\nu^{2}$ and $\mu^{1}\otimes\mu^{2}$ on
$\mathcal{A}_{1}\otimes\mathcal{A}_{2}$ by
$\displaystyle(\nu^{1}\otimes\nu^{2})(x_{1}\otimes x_{2},y_{1}\otimes y_{2})$
$\displaystyle=$
$\displaystyle\nu^{1}(x_{1},y_{1})\otimes\nu^{2}(x_{2},y_{2}),$ (5.1)
$\displaystyle(\mu^{1}\otimes\mu^{2})(x_{1}\otimes x_{2},y_{1}\otimes y_{2})$
$\displaystyle=$
$\displaystyle\mu^{1}(x_{1},y_{1})\otimes\nu^{2}(x_{2},y_{2})+\nu^{1}(x_{1},y_{1})\otimes\mu^{2}(x_{2},y_{2}),$
(5.2)
for all $x_{1},\ y_{1}\in\mathcal{A}_{1}$ and $x_{2},\
y_{2}\in\mathcal{A}_{2}$. Then
$\big{(}\mathcal{A}_{1}\otimes\mathcal{A}_{2},\mu^{1}\otimes\mu^{2},\nu^{1}\otimes\nu^{2}\big{)}$
forms a Novikov-Poisson algebra (cf.[21, 22]). Additionally, assume that
$\alpha$ and $\beta$ are algebra homomorphisms of
$\big{(}\mathcal{A}_{1},\mu^{1},\nu^{1}\big{)}$ and
$\big{(}\mathcal{A}_{2},\mu^{2},\nu^{2}\big{)}$, respectively. It follows from
Proposition 5.1 that
$\big{(}\mathcal{A}_{1},\mu^{1}_{\alpha},\nu^{1}_{\alpha},\alpha\big{)}$ and
$\big{(}\mathcal{A}_{2},\mu^{2}_{\beta},\nu^{2}_{\beta},\beta\big{)}$ are Hom-
Novikov-Poisson algebras. Define a linear map $\alpha\otimes\beta$ on
$\mathcal{A}_{1}\otimes\mathcal{A}_{2}$ by
$\displaystyle(\alpha\otimes\beta)(x\otimes y)=\alpha(x)\otimes\beta(y),\hskip
8.5359pt\mbox{for all}\ x\in\mathcal{A}_{1},\ y\in\mathcal{A}_{2}.$ (5.3)
For all $x_{1},y_{1}\in\mathcal{A}_{1}$ and $x_{2},y_{2}\in\mathcal{A}_{2}$,
we have
$\displaystyle(\alpha\otimes\beta)\big{(}\nu^{1}\otimes\nu^{2}(x_{1}\otimes
x_{2},y_{1}\otimes y_{2})\big{)}$ $\displaystyle=$
$\displaystyle(\alpha\otimes\beta)\big{(}\nu^{1}(x_{1},y_{1})\otimes\nu^{2}(x_{2},y_{2})\big{)}$
$\displaystyle=$
$\displaystyle\alpha\circ\nu^{1}(x_{1},y_{1})\otimes\beta\circ\nu^{2}(x_{2},y_{2})$
$\displaystyle=$
$\displaystyle\nu^{1}(\alpha(x_{1}),\alpha(y_{1}))\otimes\nu^{2}(\beta(x_{2}),\beta(y_{2}))$
$\displaystyle=$
$\displaystyle(\nu^{1}\otimes\nu^{2})\big{(}\alpha(x_{1})\otimes\beta(x_{2}),\alpha(y_{1})\otimes\beta(y_{2})\big{)}$
$\displaystyle=$
$\displaystyle(\nu^{1}\otimes\nu^{2})\big{(}\alpha\otimes\beta(x_{1}\otimes
x_{2}),\alpha\otimes\beta(y_{1}\otimes y_{2})\big{)}.$
Similarly, we have
$\displaystyle(\alpha\otimes\beta)\big{(}\mu^{1}\otimes\mu^{2}(x_{1}\otimes
x_{2},y_{1}\otimes
y_{2})\big{)}=(\mu^{1}\otimes\mu^{2})\big{(}\alpha\otimes\beta(x_{1}\otimes
x_{2}),\alpha\otimes\beta(y_{1}\otimes y_{2})\big{)}.$
Thus $\alpha\otimes\beta$ is an algebra homomorphism of
$\big{(}\mathcal{A}_{1}\otimes\mathcal{A}_{2},\mu^{1}\otimes\mu^{2},\nu^{1}\otimes\nu^{2}\big{)}$.
On the other hand, consider the following two operations on
$\mathcal{A}_{1}\otimes\mathcal{A}_{2}$ :
$\displaystyle(\nu^{1}_{\alpha}\otimes\nu^{2}_{\beta})(x_{1}\otimes
x_{2},y_{1}\otimes y_{2})$ $\displaystyle=$
$\displaystyle\nu^{1}_{\alpha}(x_{1},y_{1})\otimes\nu^{2}_{\beta}(x_{2},y_{2}),$
$\displaystyle(\mu^{1}_{\alpha}\otimes\mu^{2}_{\beta})(x_{1}\otimes
x_{2},y_{1}\otimes y_{2})$ $\displaystyle=$
$\displaystyle\mu^{1}_{\alpha}(x_{1},y_{1})\otimes\nu^{2}_{\beta}(x_{2},y_{2})+\nu^{1}_{\alpha}(x_{1},y_{1})\otimes\mu^{2}_{\beta}(x_{2},y_{2}),$
for all $x_{1},y_{1}\in\mathcal{A}_{1}$, $x_{2},y_{2}\in\mathcal{A}_{2}$.
Then, we have
$\displaystyle(\nu^{1}_{\alpha}\otimes\nu^{2}_{\beta})(x_{1}\otimes
x_{2},y_{1}\otimes y_{2})$ $\displaystyle=$
$\displaystyle\nu^{1}_{\alpha}(x_{1},y_{1})\otimes\nu^{2}_{\beta}(x_{2},y_{2})$
$\displaystyle=$
$\displaystyle\alpha\circ\nu^{1}(x_{1},y_{1})\otimes\beta\circ\nu^{2}(y_{1},y_{2})$
$\displaystyle=$
$\displaystyle\nu^{1}(\alpha(x_{1}),\alpha(y_{1}))\otimes\nu^{2}(\beta(x_{2}),\beta(y_{2}))$
$\displaystyle=$
$\displaystyle(\nu^{1}\otimes\nu^{2})\big{(}\alpha(x_{1})\otimes\beta(x_{2}),\alpha(y_{1})\otimes\beta(y_{2})\big{)}$
$\displaystyle=$
$\displaystyle(\nu^{1}\otimes\nu^{2})\big{(}\alpha\otimes\beta(x_{1}\otimes
x_{2}),\alpha\otimes\beta(y_{1}\otimes y_{2})\big{)}.$
Similarly, we have
$\displaystyle(\mu^{1}_{\alpha}\otimes\mu^{2}_{\beta})(x_{1}\otimes
x_{2},y_{1}\otimes
y_{2})=(\mu^{1}\otimes\mu^{2})\big{(}\alpha\otimes\beta(x_{1}\otimes
x_{2}),\alpha\otimes\beta(y_{1}\otimes y_{2})\big{)}.$
Now from the discussions above, we obtain
$\displaystyle\mu^{1}_{\alpha}\otimes\mu^{2}_{\beta}$ $\displaystyle=$
$\displaystyle(\alpha\otimes\beta)\circ(\mu^{1}\otimes\mu^{2})=(\mu^{1}\otimes\mu^{2})_{\alpha\otimes\beta},$
$\displaystyle\nu^{1}_{\alpha}\otimes\nu^{2}_{\beta}$ $\displaystyle=$
$\displaystyle(\alpha\otimes\beta)\circ(\nu^{1}\otimes\nu^{2})=(\nu^{1}\otimes\nu^{2})_{\alpha\otimes\beta}.$
Then, according to Proposition 5.1, we have the following result (see also
[References, Corollary 3.6]):
###### Theorem 5.2
With notations above. Let $\big{(}\mathcal{A}_{1},\mu^{1},\nu^{1}\big{)}$ and
$\big{(}\mathcal{A}_{2},\mu^{2},\nu^{2}\big{)}$ be two Novikov-Poisson
algebras. If $\alpha$ and $\beta$ are algebra homomorphisms of
$\big{(}\mathcal{A}_{1},\mu^{1},\nu^{1}\big{)}$ and
$\big{(}\mathcal{A}_{2},\mu^{2},\nu^{2}\big{)}$, respectively. Then
$\big{(}\mathcal{A}_{1}\otimes\mathcal{A}_{2},\mu^{1}_{\alpha}\otimes\mu^{2}_{\beta},\nu^{1}_{\alpha}\otimes\nu^{2}_{\beta},\alpha\otimes\beta\big{)}$
forms a Hom-Novikov-Poisson algebra.
We extend in the following the Yau’s Theorem (see [28]) to the Hom-Novikov-
Poisson algebra case. The following theorem gives a way to construct Hom-
Novikov-Poisson algebras from a commutative Hom-associative algebra along with
a derivation.
###### Theorem 5.3
Let $(\mathcal{A},\cdot,\alpha)$ be a commutative Hom-associative algebra and
$\partial$ be a derivation such that $\partial\alpha=\alpha\partial$. Define a
new operation on $\mathcal{A}$ by
$\displaystyle x\star y=x\cdot\partial(y),\hskip 8.5359pt\mbox{for all}\
x,y\in\mathcal{A}.$ (5.4)
Then $(\mathcal{A},\cdot,\star,\alpha)$ forms a Hom-Novikov-Poisson algebra.
Proof. It follows from [References, Theorem 1.2] that
$(\mathcal{A},\star,\alpha)$ is a Hom-Novikov algebra. We need to check the
two compatible conditions (2.7) and (2.8). For any $x,y,z\in\mathcal{A}$, we
have
$\displaystyle(x\cdot y)\star\alpha(z)$ $\displaystyle=$ $\displaystyle(x\cdot
y)\cdot\partial(\alpha(z))$ $\displaystyle=$ $\displaystyle(x\cdot
y)\cdot\alpha(\partial(z))$ $\displaystyle=$
$\displaystyle\alpha(x)\cdot(y\cdot\partial(z))$ $\displaystyle=$
$\displaystyle\alpha(x)\cdot(y\star z),$
which proves equation (2.7). Furthermore, we have
$\displaystyle(x\star y)\cdot\alpha(z)-\alpha(x)\star(y\cdot z)$
$\displaystyle=$
$\displaystyle(x\cdot\partial(y))\cdot\alpha(z)-\alpha(x)\cdot\partial(y\cdot
z)$ $\displaystyle=$
$\displaystyle(x\cdot\partial(y))\cdot\alpha(z)-\alpha(x)\cdot(\partial(y)\cdot
z+y\cdot\partial(z))$ $\displaystyle=$
$\displaystyle\alpha(x)\cdot(\partial(y)\cdot
z)-\alpha(x)\cdot\big{(}\partial(y)\cdot z+y\cdot\partial(z)\big{)}$
$\displaystyle=$ $\displaystyle-\alpha(x)\cdot\big{(}y\cdot\partial(z)\big{)}$
$\displaystyle=$ $\displaystyle-(x\cdot y)\cdot\alpha(\partial(z)).$
Similarly, we have
$\displaystyle(y\star x)\cdot\alpha(z)-\alpha(y)\star(x\cdot z)=-(y\cdot
x)\cdot\alpha(\partial(z)).$
Since $(\mathcal{A},\cdot,\alpha)$ is commutative, we get
$\displaystyle(x\star y)\cdot\alpha(z)-\alpha(x)\star(y\cdot z)=(y\star
x)\cdot\alpha(z)-\alpha(y)\star(x\cdot z),$
which completes the proof. $\Box$
The following result is due to Xu [22].
###### Proposition 5.4
Let $(\mathcal{A},\cdot,\ast)$ be a Novikov-Poisson algebra, such that
$(\mathcal{A},\cdot)$ is a commutative associative algebra and
$(\mathcal{A},\ast)$ is a Novikov algebra. Suppose that $(\mathcal{A},\cdot)$
contains an identity element $1$, that is, $1\cdot x=x\cdot 1=x$ for all
$x\in\mathcal{A}$. Define a map
$\partial:\mathcal{A}\longrightarrow\mathcal{A}$ by
$\displaystyle\partial(x)=1\ast x-(1\ast 1)\cdot x,\hskip 8.5359pt\mbox{for
all}\ x\in\mathcal{A}.$ (5.5)
Then $\partial$ is a derivation of $(\mathcal{A},\cdot)$.
In the following, we give some examples of Hom-Novikov-Poisson algebras by
using Theorem 5.3 and Proposition 5.4.
###### Example 5.5
Assume that $(\mathcal{A},\cdot,\ast)$ is a Novikov-Poisson algebra, where
$(\mathcal{A},\cdot)$ is a commutative associative algebra with unity $1$ and
$(\mathcal{A},\ast)$ is a Novikov algebra. Equation (2.3) implies that
$x\cdot(1\ast 1)=x\ast 1,\hskip 8.5359pt\mbox{for all}\ x\in\mathcal{A}.$
Since $(\mathcal{A},\cdot)$ is commutative, we have
$\displaystyle\partial(x)=1\ast x-x\ast 1,\hskip 8.5359pt\mbox{for all}\
x\in\mathcal{A},$ (5.6)
in which $\partial$ is defined by (5.5). Let $\alpha$ be an algebra
homomorphism of $(\mathcal{A},\cdot,\ast)$. Define a new multiplication on
$\mathcal{A}$ by
$\displaystyle x\bullet y=\alpha(x\cdot y),\hskip 8.5359pt\mbox{ for all}\
x,y\in\mathcal{A}.$
Then $(\mathcal{A},\bullet,\alpha)$ is a commutative Hom-associative algebra.
Obviously, $\alpha(1)=1$. As a consequence, $\alpha$ commutes with $\partial$.
Furthermore, $\partial$ is a derivation of $(\mathcal{A},\bullet)$ since it is
a derivation of $(\mathcal{A},\cdot)$ by Proposition 5.4. Now define a new
operation on $\mathcal{A}$ by
$x\star y=x\bullet\partial(y),\hskip 8.5359pt\mbox{for all}\
x,y\in\mathcal{A}.$
Then, thanks to Theorem 5.3, $(\mathcal{A},\bullet,\star,\alpha)$ is a Hom-
Novikov-Poisson algebra.
###### Example 5.6
Let the ground field be the complex field $\mathbb{C}$ and $\Delta$ be a
nonzero abelian subgroup of $\mathbb{C}$. Suppose that $f$ is a nontrivial
homomorphism of $\Delta$ into the additive group of $\mathbb{C}$. Consider a
vector space $\mathcal{A}$ with basis $\\{x_{a}|a\in\Delta\\}$. Fix an element
$q$ in $\Delta$. Define a multiplication on $\mathcal{A}$ by
$\displaystyle x_{a}\cdot x_{b}=x_{a+b+q},\hskip 8.5359pt\mbox{for all}\
a,b\in\Delta.$
Then $(\mathcal{A},\cdot)$ is a commutative associative algebra. Define a
linear map $\partial_{1}:\mathcal{A}\longrightarrow\mathcal{A}$ by
$\displaystyle\partial_{1}(x_{a})=f(a+q)x_{a},\hskip 8.5359pt\mbox{for all}\
a\in\Delta.$
It is easy to check that $\partial_{1}$ is a derivation of
$(\mathcal{A},\cdot)$. Then we obtain a Novikov-Poisson algebra
$(\mathcal{A},\cdot,\ast)$ with the operation $\ast$ defined by
$\displaystyle x_{a}\ast
x_{b}=x_{a}\cdot\partial_{1}(x_{b})=f(b+q)x_{a+b+q},\hskip 8.5359pt\mbox{for
all}\ a,b\in\Delta.$
Note that $x_{-q}$ is an identity element of $(\mathcal{A},\cdot)$. Define
another linear map $\partial_{2}:\mathcal{A}\longrightarrow\mathcal{A}$ by
$\displaystyle\partial_{2}(x_{a})=x_{-q}\ast x_{a}-x_{a}\ast x_{-q},\hskip
8.5359pt\mbox{for all}\ a\in\Delta.$
Then it follows from Proposition 5.4 and the discussions presented in Example
5.5 that $\partial_{2}$ is a derivation of $(\mathcal{A},\cdot)$.
Let $\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ be a linear map defined by
$\displaystyle\alpha(x_{a})=e^{a+q}x_{a},\hskip 8.5359pt\mbox{for all}\
a\in\Delta.$
Then $\alpha$ is an algebra homomorphism of $(\mathcal{A},\cdot,\ast)$ and
$\alpha(x_{-q})=x_{-q}$. Moreover, $\alpha$ commutes with $\partial_{2}$.
Define a new multiplication on $\mathcal{A}$ by
$\displaystyle x_{a}\bullet x_{b}=\alpha(x_{a}\cdot x_{b}),\hskip
8.5359pt\mbox{for all}\ a,b\in\Delta,$
which makes $(\mathcal{A},\bullet,\alpha)$ form a commutative Hom-associative
algebra. Also, $\partial_{2}$ is a derivation of $(\mathcal{A},\bullet)$.
Define a bilinear map $\star$ on $\mathcal{A}$ by
$\displaystyle x_{a}\star x_{b}=x_{a}\bullet\partial_{2}(x_{b}),\hskip
8.5359pt\mbox{for all}\ a,b\in\Delta,$
To be more precise, we have
$\displaystyle x_{a}\star x_{b}=e^{a+b+2q}f(b+q)x_{a+b+q},\hskip
8.5359pt\mbox{for all}\ a,b\in\Delta.$
According to Theorem 5.3, $(\mathcal{A},\bullet,\star,\alpha)$ is a Hom-
Novikov-Poisson algebra.
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|
arxiv-papers
| 2012-04-28T06:40:17 |
2024-09-04T02:49:30.322740
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lamei Yuan and Hong You",
"submitter": "Lamei Yuan",
"url": "https://arxiv.org/abs/1204.6373"
}
|
1204.6410
|
∎
11institutetext: Y. Kucukakca 22institutetext: Department of Physics, Akdeniz
University, 07058 Antalya, Turkey
22email: ykucukakca@gmail.com 33institutetext: U. Camci 44institutetext:
Department of Physics, Akdeniz University, 07058 Antalya, Turkey
44email: ucamci@akdeniz.edu.tr 55institutetext: İ. Semiz 66institutetext:
Department of Physics, Bogazici University, 34342 Bebek, Istanbul, Turkey
66email: semizibr@boun.edu.tr
# LRS Bianchi type I universes exhibiting Noether symmetry in the scalar-
tensor Brans-Dicke theory
Y. Kucukakca U. Camci İ. Semiz
(Received: date / Accepted: date)
###### Abstract
Following up on hints of anisotropy in the cosmic microwave background
radiation (CMB) data, we investigate locally rotational symmetric (LRS)
Bianchi type I spacetimes with non-minimally coupled scalar fields. To single
out potentially more interesting solutions, we search for Noether symmetry in
this system. We then specialize to the Brans-Dicke (BD) field in such a way
that the Lagrangian becomes degenerate (nontrivial) and solve the equations
for Noether symmetry and the potential that allows it. Then we find the exact
solutions of the equations of motion in terms of three parameters and an
arbitrary function. We illustrate with families of examples designed to be
generalizations of the well-known power-expansion, exponential expansion and
Big Rip models in the Friedmann-Robertson-Walker (FRW) framework. The
solutions display surprising variation, a large subset of which features late-
time acceleration as is usually ascribed to dark energy (phantom or
quintensence), and is consistent with observational data.
###### Keywords:
Bianchi type I spacetime; Noether symmetry; Brans-Dicke theory.
††journal: General Relativity and Gravitation
## 1 Introduction
The recent decade has witnessed the observational evidence for the cosmic
acceleration of the universe, which has become the central theme of modern
cosmology. This evidence is built up of observations of supernovae Type Ia
(SNe Ia) riess99 , cosmic microwave background radiation (CMB) netterfield02 ,
and large-scale structure of the Universe tegmark04 . In Einstein’s gravity,
this acceleration cannot be explained by normal matter or fields on Friedmann-
Robertson-Walker (FRW) metric background; therefore a mysterious cosmic fluid
with negative pressure, the so called _dark energy_ , is introduced.
Presently, understanding the nature of the dark energy is one of the main
problems in the research area of both theoretical physics and cosmology. A
simple example of dark energy is the cosmological constant, equivalent to a
fluid with the equation of state (EoS) parameter $w=-1$, where $w=p/\rho$ in
which $p$ is the pressure of dark energy, and $\rho$ its energy density.
However, the cosmological constant model is subject to the so-called _fine-
tuning_ and _coincidence problems_ copeland06 .
Dark energy, if it is a perfect fluid, must have EoS parameter $w<-1/3$. If
$-1/3>w>-1$, the dark energy is called the _quintessence_ cald98-sahni02 ; if
$w<-1$, dark energy is dubbed a _phantom fluid_ cald02 . In spite of the fact
that these models violate both the strong energy condition $\rho+3p>0$ and the
dominant energy condition $p+\rho>0$, and therefore may be physically
considered undesirable, the phantom energy is found to be compatible with
current data from SNe Ia observations, CMB anisotropy and the Sloan Digital
Sky Survey (SDSS) riess99 ; netterfield02 ; tegmark04 .
Dark energy behavior can be exhibited by possible new fundamental fields
acting on cosmological scales. The simplest such model is the single component
scalar field. Other alternatives in the literature include the Cardassian
expansion scenario Freese , the tachyon Sen00 , the quintom Li05 , the
k-essence Armen00 models and more.
Alternatively, modifications of Einstein’s gravity have been proposed to
explain the cosmic acceleration of the universe; among them $f(R)$ theories fr
. For example, quintessential behavior of the parameter $w$ can be achieved in
a geometrical way in higher order theories of gravity cap002 . Scalar fields
with various couplings and potentials can be put in by hand, or follow from
the model naturally.
One of the simplest modification of Einstein’s gravity is the Brans-Dicke (BD)
gravity theory brans , a well known example of a scalar-tensor theory which
represents the gravitational interaction using a scalar field in addition to
the metric field. This theory is parametrized by one extra constant parameter,
$W$. In the limit $W\rightarrow\infty$, the BD theory reduces to Einstein’s
gravity fm . The conditions, where the dynamics of a self-interacting BD field
can account for the accelerated expansion, have been considered in Ref.berto99
, where it was concluded that accelerated expanding solutions can be obtained
with a quadratic self-coupling of the BD field and a negative equation of
state (EoS) parameter. Astrophysical data indicate that EoS parameter $w$ lies
in an interval of negative values roughly centered around $-1$.
The majority of popular cosmological models, including all the ones referred
to above, use the cosmological principle, that is, they assume that the
universe is homogeneous and isotropic. On the other hand, there are hints in
the CMB temperature anisotropy studies that suggest that the assumption of
statistical isotropy is broken on the largest angular scales, leading to some
intriguing anomalies huterer . To provide predictions for the CMB
anisotropies, one may consider the homogeneous but anisotropic cosmologies
known as Bianchi type spacetimes, which include the isotropic and homogeneous
FRW models barrow . In this study we consider the simplest of these, the
locally rotationally symmetric (LRS) Bianchi type I spacetime as an
anisotropic background universe model; note that this spacetime is a
generalization of flat ($k=0$) FRW metric. Our aim is to investigate the
solutions of the field equations of scalar-tensor, in particular, BD theory of
gravity for the LRS Bianchi type I spacetime using the Noether symmetry
approach.
The Noether symmetry approach was introduced by De Ritis _et al._ ritis ;
demianski92 and Capozziello _et al._ capo93 ; capo00 to find preferred
solutions of the field equations and the dynamical conserved quantity. The
Noether theorem states that if the Lie derivative of a given Lagrangian $L$
dragging along a vector field ${\bf X}$ vanishes
$\pounds_{\bf X}L=0.$ (1)
then ${\bf X}$ is a symmetry for the dynamics, and it generates a conserved
current. Recently some exact solutions have been presented in the scalar
tensor theories following the Noether symmetry approach that allows the
potential to be chosen dynamically, restricting the arbitrariness in a
suitable way ritis ; demianski92 ; capo93 ; capo00 ; sanyal01 ; sanyal02 ;
sanyal03 ; modak04 ; dabrowski ; camci07 ; camci12 ; wei .
This paper is organized as follows. In the section 2, we present the field
equations in scalar-tensor theory for the LRS Bianchi type I spacetime. In
section 3, we search the Noether symmetry of the Lagrangian of scalar-tensor
theory for the LRS Bianchi type I spacetime, and for the BD case, find new
variables that include the cyclic one. In section 4 we derive the EoS of the
BD field. In section 5, we give the solutions of the field equations by using
new variables obtained in section 3. In section 6, we choose some examples
corresponding to the well-known solutions in the isotropic cases such as power
law, exponentially expanding and phantom/Big Rip models. We also make a
conformal transform to the so-called Einstein frame in Section 7 and discuss
the problem in that context. Finally, in section 8, we conclude with a brief
summary.
## 2 The Lagrangian and the field equations
The general form of the action that involves gravity non-minimally coupled
with a scalar field is given by sanyal03 , such that
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\int{d^{4}x\sqrt{-g}\left[F(\Phi)R-\frac{W(\Phi)}{\Phi}\Phi_{c}\Phi^{c}-U(\Phi)\right]}.$
(2)
Here $R$ is the Ricci scalar, $F(\Phi)$ and $W(\Phi)$ are generic functions
that describe the coupling, $U(\Phi)$ is the potential for the scalar field
$\Phi$, and $\Phi_{a}\equiv\Phi_{;a}$ stand for the components of the gradient
of the scalar field. Note that we use Planck units. For $F(\Phi)=1/2$ and
$W(\Phi)=\Phi/2$, the action reduces to the form of Einstein-Hilbert action
minimally coupled with a scalar field. The choice of $F(\Phi)$ and $W(\Phi)$
give us other gravity theories such as the BD theory, for which $F(\Phi)=\Phi$
and $W(\Phi)=$constant.
Variation of the general form of the action with respect to metric tensor
yields the field equations
$\displaystyle F(\Phi)G_{ab}=T^{\Phi}_{ab}$ (3)
where $G_{ab}=R_{ab}-\frac{1}{2}Rg_{ab}$ is the Einstein tensor,
$\displaystyle
T^{\Phi}_{ab}=\frac{W}{\Phi}\Phi_{a}\Phi_{b}-\frac{W}{2\Phi}g_{ab}\Phi_{c}\Phi^{c}-\frac{1}{2}g_{ab}U(\Phi)-g_{ab}\Box
F(\Phi)+F(\Phi)_{;ab}\qquad$ (4)
is the energy-momentum tensor of scalar field, and $\Box$ is the d’Alembert
operator. It is clear that $T^{\Phi}_{ab}$ includes the contributions from the
non-minimal coupling and the scalar field parts of the action varying with
respect to metric. In Ref.faraoni00 , it has been discussed that there are
three possible and inequivalent ways of writing the field equations,
corresponding to the ambiguity in the definition of the energy-momentum tensor
of non-minimally coupled scalar field. The variation with respect to $\Phi$
gives rise to the generalized Klein-Gordon equation governing the dynamics of
the scalar field
$\displaystyle
2\frac{W}{\Phi}\Box\Phi+RF^{\prime}(\Phi)+\left(\frac{W^{\prime}}{\Phi}-\frac{W}{\Phi^{2}}\right)\Phi_{c}\Phi^{c}-U^{\prime}(\Phi)=0,$
(5)
where the prime indicates the derivative with respect to $\Phi$. Note that
this equation is equivalent to the contracted Bianchi identity. It follows
from $G^{\,\,\,b}_{a\,;b}=0$ that using Eq.(3) together with Eq. (4), yields
$\displaystyle\left[\frac{1}{F}T_{a}^{\,\,\,b(\Phi)}\right]_{;b}=0\Leftrightarrow
T^{\,\,\,b(\Phi)}_{a\,\,\,;b}=-\frac{F_{;b}}{F}T_{a}^{\,\,\,b(\Phi)}$ (6)
From (4), we compute $T^{\,\,\,b(\Phi)}_{a\,\,\,;b}$ to get
$\displaystyle
T^{\,\,\,b(\Phi)}_{a\,\,\,;b}=\Phi_{a}\left[\left(\frac{W}{2\Phi}\right)^{\prime}\Phi_{c}\Phi^{c}+\frac{W}{\Phi}\Phi-\frac{1}{2}U^{\prime}(\Phi)\right]+F_{;b}R_{a}^{\,\,b}$
(7)
Using Eq.(4), Eq.(7) and $F_{;b}=F^{\prime}(\Phi)\Phi_{b}$ in Eq.(6), we have
$\displaystyle\Phi_{a}\left[2\frac{W}{\Phi}\Box\Phi+RF^{\prime}(\Phi)+\left(\frac{W^{\prime}}{\Phi}-\frac{W}{\Phi^{2}}\right)\Phi_{c}\Phi^{c}-U^{\prime}(\Phi)\right]=0.$
(8)
This is obviously the generalized Klein-Gordon equation for $\Phi_{a}\neq 0$.
The line element of the LRS Bianchi type I spacetime has the form
$ds^{2}=-dt^{2}+A^{2}dx^{2}+B^{2}\left(dy^{2}+dz^{2}\right),$ (9)
describing an anisotropic universe with equal expansion rate in two of the
three dimensions. The Ricci scalar of this spacetime is
$R=2\left[\frac{\ddot{A}}{A}+2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}+2\frac{\dot{A}\dot{B}}{AB}\right],$
(10)
where the dot represents differentiation with respect to $t$. For the metric
(9), the field and generalized Klein-Gordon equations can be obtained from
Eqs.(3) and (5) respectively
$\displaystyle\frac{\dot{B}^{2}}{B^{2}}+2\frac{\dot{A}\dot{B}}{AB}+\frac{F^{\prime}}{F}\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)\dot{\Phi}-\frac{1}{2F}\left[\frac{W}{\Phi}\dot{\Phi}^{2}+U\right]=0,$
(11) $\displaystyle
2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}+\frac{F^{\prime}}{F}\left[\ddot{\Phi}+2\frac{\dot{B}}{B}\dot{\Phi}\right]+\frac{1}{2F}\left(\frac{W}{\Phi}+2F^{\prime\prime}\right)\dot{\Phi}^{2}-\frac{U}{2F}=0,\quad$
(12)
$\displaystyle\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{F^{\prime}}{F}\left[\ddot{\Phi}+\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}\right)\dot{\Phi}\right]$
$\displaystyle\qquad\quad+\frac{1}{2F}\left(\frac{W}{\Phi}+2F^{\prime\prime}\right)\dot{\Phi}^{2}-\frac{U}{2F}=0,$
(13)
$\displaystyle\frac{\ddot{A}}{A}+2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}+2\frac{\dot{A}\dot{B}}{AB}-\frac{W}{F^{\prime}\Phi}\left[\ddot{\Phi}+\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)\dot{\Phi}\right]$
$\displaystyle\qquad\quad-\frac{\dot{\Phi}^{2}}{2F^{\prime}}\left(\frac{W^{\prime}}{\Phi}-\frac{W}{\Phi^{2}}\right)-\frac{U^{\prime}}{2F^{\prime}}=0$
(14)
where $F^{\prime}\neq 0$. The Lagrangian density of the LRS Bianchi type I
spacetime is
$\displaystyle
L=-2FA\dot{B}^{2}-4FB\dot{A}\dot{B}-2F^{\prime}B^{2}\dot{A}\dot{\Phi}-4F^{\prime}AB\dot{B}\dot{\Phi}$
$\displaystyle\qquad\quad+AB^{2}\left[\frac{W\dot{\Phi}^{2}}{\Phi}-U(\Phi)\right].$
(15)
Using such a Lagrangian, one may obtain the Euler-Lagrange equations as given
in (12)-(14). The _energy function_ , $E_{L}$, associated with the Lagrangian
(2) is found as
$\displaystyle E_{L}$ $\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{A}}\dot{A}+\frac{\partial
L}{\partial\dot{B}}\dot{B}+\frac{\partial L}{\partial\dot{\Phi}}\dot{\Phi}-L$
(16) $\displaystyle=$
$\displaystyle\frac{\dot{B}^{2}}{B^{2}}+2\frac{\dot{A}\dot{B}}{AB}+\frac{F^{\prime}}{F}\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)\dot{\Phi}-\frac{1}{2F}\left[\frac{W}{\Phi}\dot{\Phi}^{2}+U\right].$
Therefore, it is obvious that the (0,0)-field equation given by (11) is
equivalent to $E_{L}=0$.
## 3 Noether symmetry approach
In this section we seek the Noether symmetry of the the Lagrangian (15). The
configuration space of this Lagrangian is $Q=(A,B,\Phi)$, whose tangent space
is $TQ=(A,B,\Phi,\dot{A},\dot{B},\dot{\Phi})$. The existence of Noether
symmetry implies the existence of a vector field ${\bf X}$ such that
${\bf X}=\alpha\frac{\partial}{\partial A}+\beta\frac{\partial}{\partial
B}+\gamma\frac{\partial}{\partial\Phi}+\dot{\alpha}\frac{\partial}{\partial\dot{A}}+\dot{\beta}\frac{\partial}{\partial\dot{B}}+\dot{\gamma}\frac{\partial}{\partial\dot{\Phi}}$
(17)
where $\alpha,\beta$ and $\gamma$ are depend on $A,B$ and $\Phi$. Hence the
Noether equation given by (1) yields the following set of equations
$\displaystyle 2\frac{\partial\beta}{\partial
A}+B\frac{F^{\prime}}{F}\frac{\partial\gamma}{\partial A}=0,$ (18)
$\displaystyle\frac{\alpha}{2}+B\frac{\partial\alpha}{\partial
B}+A\frac{\partial\beta}{\partial
B}+A\frac{F^{\prime}}{F}\left(\frac{\gamma}{2}+B\frac{\partial\gamma}{\partial
B}\right)=0,$ (19)
$\displaystyle\frac{W}{\Phi}\left(\frac{\alpha}{2}+\beta\frac{A}{B}+A\frac{\partial\gamma}{\partial\Phi}\right)+\frac{A}{2}\left(\frac{W^{\prime}}{\Phi}-\frac{W}{\Phi^{2}}\right)\gamma$
$\displaystyle\qquad\quad-F^{\prime}\left(\frac{\partial\alpha}{\partial\Phi}+2\frac{A}{B}\frac{\partial\beta}{\partial\Phi}\right)=0,$
(20) $\displaystyle\beta+B\frac{\partial\alpha}{\partial
A}+A\frac{\partial\beta}{\partial A}+B\frac{\partial\beta}{\partial
B}+B\frac{F^{\prime}}{F}\left(\gamma+A\frac{\partial\gamma}{\partial
A}+\frac{B}{2}\frac{\partial\gamma}{\partial B}\right)=0,$ (21) $\displaystyle
2\frac{\partial\beta}{\partial\Phi}+\frac{F^{\prime}}{F}\left(2\beta+B\frac{\partial\alpha}{\partial
A}+B\frac{\partial\gamma}{\partial\Phi}+2A\frac{\partial\beta}{\partial
A}\right)$
$\displaystyle\qquad\quad+\frac{F^{\prime\prime}}{F}B\gamma-\frac{W}{F\Phi}AB\frac{\partial\gamma}{\partial
A}=0,$ (22)
$\displaystyle\frac{\partial\alpha}{\partial\Phi}+\frac{A}{B}\frac{\partial\beta}{\partial\Phi}+\frac{F^{\prime}}{F}\left(\alpha+\frac{A}{B}\beta+\frac{B}{2}\frac{\partial\alpha}{\partial
B}+A\frac{\partial\beta}{\partial
B}+A\frac{\partial\gamma}{\partial\Phi}\right)$
$\displaystyle\qquad\quad+\frac{F^{\prime\prime}}{F}A\gamma-\frac{W\,AB}{2F\Phi}\frac{\partial\gamma}{\partial
B}=0,$ (23) $\displaystyle(B\alpha+2A\beta)U+A\,B\,\gamma U^{\prime}=0.$ (24)
Leaving these equations as reference for future work, we now specialize to the
Brans-Dicke case, where $F=\Phi$ and $W$ is constant. We also require the
Hessian determinant,
$D=\Sigma\left|\frac{\partial^{2}L}{\partial\dot{Q}_{i}\partial\dot{Q}_{j}}\right|$,
to vanish in order to get nontrivial solutions. This condition reads for our
case as
$D=-\frac{16AB^{4}F}{\Phi}(3\Phi F^{\prime 2}+2WF)=0,$ (25)
which also determines a $W$ value. So we will be working with the coupling
functions
$F=\Phi,\qquad W=-\frac{3}{2}.$ (26)
Thus the Lagrangian (15) becomes degenerate, and the BD action has the form
$\displaystyle\mathcal{A_{BD}}$ $\displaystyle=$
$\displaystyle\int{d^{4}x\sqrt{-g}\left[\Phi
R+\frac{3}{2\Phi}\Phi_{c}\Phi^{c}-U(\Phi)\right]}$ (27)
which can be easily related the conformal relativity by defining new scalar
field $\varphi$ as $\Phi=\varphi^{2}/12$, transforming the action into
$\displaystyle\mathcal{A_{BD}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int{d^{4}x\sqrt{-g}\left[\frac{1}{6}\varphi^{2}R+\varphi_{c}\varphi^{c}-U(\varphi)\right]}.$
(28)
This action is of course conformally invariant, since the application of the
conformal transformation formulas together with the appropriate integration of
the boundary term gives the same form of this action (see Refs. fm and
dabrowski ).
Dabrowski _et al._ dabrowski have shown that the anisotropic non-zero spatial
curvature models of Bianchi types I, III and Kantowski-Sachs type are
admissible in $W=-3/2$ BD theory. They have given solutions of BD field
equations for these spacetimes without the potential $U(\Phi)$. In a previous
work camci07 , we studied the case
$F=\frac{\epsilon}{12}(\Phi-\Phi_{0})^{2},W=\frac{\Phi}{2}$ for the Bianchi
types I and III, and Kantowski-Sachs spacetimes, where $\epsilon$ is a
parameter depending the signature of metric, but did not study the qualitative
(acceleration etc.) behaviour of the universes in the solutions found.
In our present study we find the solutions of BD field equations with
potential $U(\Phi)$ for the LRS Bianchi type I spacetime, although we derive
the equations for a general function $W(\Phi)$. In this context, the solutions
of the above set of differential equations (18)-(24) for $\alpha,\beta$,
$\gamma$ and potential $U(\Phi)$ are obtained as
$\displaystyle\alpha=\left(AB\Phi^{3/2}\right)^{-1},\,\,\beta=\frac{1}{2}\left(A^{2}\Phi^{3/2}\right)^{-1},\,\,\gamma=-\left(A^{2}B\Phi^{1/2}\right)^{-1},$
$\displaystyle U(\Phi)=\lambda\Phi^{2}$ (29)
where $\lambda$ is a constant. The potential found indicates that for the
Noether symmetry to be present, the scalar field must be a (massive) free
field. The vector field ${\bf X}$ generating the Noether symmetry and
determining the dynamics of LRS Bianchi type I metric is given by
$\displaystyle{\bf
X}=\frac{1}{AB\Phi^{3/2}}\Big{[}\frac{\partial}{\partial{A}}+\frac{B}{A}\frac{\partial}{\partial{B}}-\frac{\Phi}{A}\frac{\partial}{\partial{\Phi}}-\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{3\dot{\Phi}}{2\Phi}\right)\frac{\partial}{\partial{\dot{A}}}$
$\displaystyle\qquad\qquad\qquad-\left(\frac{\dot{A}}{A}+\frac{3\dot{\Phi}}{4\Phi}\right)\frac{\partial}{\partial{\dot{B}}}+\left(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{1\dot{\Phi}}{2\Phi}\right)\frac{\partial}{\partial{\dot{\Phi}}}\Big{]}$
(30)
## 4 Equation of state
Using $F(\Phi)=\Phi$ and $W=-3/2$ in Eq. (4) we can derive the energy density
of the scalar field
$\displaystyle\rho^{\Phi}=-\frac{3}{4}\frac{\dot{\Phi}^{2}}{\Phi}+\frac{U(\Phi)}{2}-3H\dot{\Phi}$
(31)
and the directional pressure due to the scalar field
$\displaystyle
p_{i}^{\Phi}=-\frac{3}{4}\frac{\dot{\Phi}^{2}}{\Phi}-\frac{U(\Phi)}{2}+\ddot{\Phi}+\left(3H-H_{i}\right)\dot{\Phi},\quad
i=x,y,z$ (32)
for the $x,y$ and $z$ directions. Here $H_{i}$ represents the directional
Hubble parameters in the directions of $x,y$ and $z$ respectively, and may be
defined as
$\displaystyle H_{x}=\frac{\dot{A}}{A},\quad H_{y}=H_{z}=\frac{\dot{B}}{B}.$
(33)
The mean Hubble parameter for LRS Bianchi type I metric is given by
$\displaystyle H=\frac{\dot{a}}{a}=\frac{1}{3}\left(H_{x}+2H_{y}\right),$ (34)
where we define $a=(AB^{2})^{1/3}$ as the average scale factor of the
Universe. Also of interest is the volumetric deceleration parameter (for the
isotropic case, simply the ”deceleration parameter”), defined by
$q=-\frac{\ddot{a}a}{\dot{a}^{2}}\equiv-1-\frac{\dot{H}}{H^{2}}$ (35)
so that it is dimensionless.
The pressure is a vectorial quantity, as would be expected for anisotropic
expansion, and thus the EoS parameter of the scalar field for the LRS Bianchi
type I spacetime may be determined separately on each spatial axis:
$w_{i}^{\Phi}(t)=p_{i}^{\Phi}/\rho^{\Phi}$, i.e.
$\displaystyle
w_{i}^{\Phi}=\frac{p_{i}^{\Phi}}{\rho^{\Phi}}=\frac{\frac{3}{4}\frac{\dot{\Phi}^{2}}{\Phi}+\frac{U(\Phi)}{2}-\ddot{\Phi}-\left(3H-H_{i}\right)\dot{\Phi}}{\frac{3}{4}\frac{\dot{\Phi}^{2}}{\Phi}-\frac{U(\Phi)}{2}+3H\dot{\Phi}}$
(36)
and the average EoS parameter of scalar field for the LRS Bianchi type I
spacetime may be defined as
$\displaystyle
w^{\Phi}=\frac{p^{\Phi}}{\rho^{\Phi}}=\frac{1}{3}\left(w_{x}+2w_{y}\right)=\frac{\frac{3}{4}\frac{\dot{\Phi}^{2}}{\Phi}+\frac{U(\Phi)}{2}-\ddot{\Phi}-2H\dot{\Phi}}{\frac{3}{4}\frac{\dot{\Phi}^{2}}{\Phi}-\frac{U(\Phi)}{2}+3H\dot{\Phi}}$
(37)
where the $p^{\Phi}$ is the isotropic pressure Calogero .
In the anisotropic case, $w$ does not directly determine the sign of $q$ as it
does in the well-known (spatially) flat FRW cosmologies. To understand this,
consider the Raychaudhuri Equation gron
$\dot{\theta}+\frac{1}{3}\theta^{2}+\sigma_{\mu\nu}\sigma^{\mu\nu}-\omega_{\mu\nu}\omega^{\mu\nu}+\frac{\kappa}{2}(\rho+3p)-\Lambda=0$
(38)
where $\theta$ is the expansion scalar, $\sigma_{\mu\nu}$ is the shear tensor,
$\omega_{\mu\nu}$ is the vorticity tensor (descriptions/definitions also given
in gron ) and we take $\Lambda=0$. If comoving particles in a metric move
geodesically, the expansion scalar also gives the expansion rate of the space
itself, when we apply this equation to a swarm of such particles. This is the
case for the LRS Bianchi I metric, where $\theta$ turns out to be equal to
$3H$, therefore $\dot{\theta}+\frac{1}{3}\theta^{2}$ is $-3H^{2}q$. The
vorticity tensor vanishes for this case, and
$\sigma_{\mu\nu}\sigma^{\mu\nu}=\frac{2B(t)^{2}}{3A(t)^{2}}\left[\frac{d}{dt}\left(\frac{A(t)}{B(t)}\right)\right]^{2}.$
(39)
Therefore in the isotropic case $\sigma_{\mu\nu}\sigma^{\mu\nu}$ also vanishes
and since in that case $\rho\propto H^{2}$, we find that $q$ is proportional
to $1+3w$. This correspondence is obviously broken when there is anisotropy,
and therefore $w$ is not that useful a parameter.
## 5 The solutions from new coordinates and Lagrangian
The solutions of dynamical Eqs.(11)-(14) are not easy to evaluate in the
present form. In order to simplify these equations we can search for the
cyclic variable(s). In case of Noether symmetry, we should introduce new
variables instead of old variables, i.e. a point transformation
$\\{A,B,\Phi\\}\rightarrow\\{\mu,\nu,u\\}$ in which it is assumed that $\mu$
is the cyclic coordinate. A general discussion of this procedure has been
given in Ref.capo96 . The new variables $\\{\mu,\nu,u\\}$ satisfy the
following equations
$\displaystyle i_{\bf X}d\mu=1,\quad i_{\bf X}d\nu=0,\quad i_{\bf X}du=0$ (40)
where $i_{\bf X}$ is the interior product operator of ${\bf X}$. Using
Eq.(29), a solution of the above Eqs.(40) yield
$\displaystyle\mu=A^{2}B\Phi^{3/2},\quad\nu=B\Phi^{1/2},\quad u=A\Phi.$ (41)
The inverse transformation of these variables are
$\displaystyle A=\frac{\mu}{\nu u},\quad
B=\frac{(\mu\nu)^{1/2}}{u},\quad\Phi=\frac{\nu u^{2}}{\mu},$ (42)
from which it follows that
$\displaystyle a(t)=\frac{\mu^{2/3}}{u},$ (43)
for the average scale factor of the Universe defined by
$a=\left(AB^{2}\right)^{1/3}$. Considering the transformation of variables,
the coupling functions (26) and the potential $U(\Phi)$ given in (29), the
Lagrangian (15) becomes
$\displaystyle L=-2u^{-1}\dot{\mu}\dot{\nu}-\lambda u\nu^{2}$ (44)
which does not depend on $\mu$ (i.e. $\frac{\partial{L}}{\partial{\mu}}=0$),
as desired. This independence is preserved if we make further transformations
in the $u-\nu$ plane; and also is inherent in the procedure described in
Ref.capo96 . We used this freedom to get as simple a Lagrangian as possible.
The Euler-Lagrange equations relative to this Lagrangian are
$\displaystyle\dot{\nu}-l_{0}u=0,$ (45)
$\displaystyle\dot{\mu}\dot{\nu}-\frac{\lambda}{2}\nu^{2}u^{2}=0,$ (46)
$\displaystyle\frac{\ddot{\mu}}{\mu}-\frac{\dot{\mu}\dot{u}}{\mu
u}-\lambda\frac{u^{2}\nu}{\mu}=0,$ (47)
where $l_{0}$ is a constant of motion associated with the coordinate $\mu$.
Here $i_{\bf X}{\Theta_{L}}=a_{0}=-2l_{0}$ and $\Theta_{L}$ is the Cartan one-
form, and Eq.(46) is equivalent to the vanishing of the energy functional, as
noted before.
By substituting $u$ from Eq.(45) into Eq.(46) we get
$\mu(t)=l_{1}\nu^{3}+l_{2},$ (48)
where $l_{1}=\lambda/(6l_{0}^{2}),\,l_{0}\neq 0$, and $l_{2}$ is an
integration constant. The remaining equation (47) is identically satisfied
now. Inserting (45) and (48) into (42), we obtain the metric functions and
scalar field in terms of the arbitrarily specifiable $\nu$ (henceforth to be
called the seed function), and $u=\dot{\nu}/l_{0}$ as follows:
$\displaystyle
A(t)=l_{0}\frac{l_{1}\nu(t){{}^{3}}+l_{2}}{\nu(t)\dot{\nu}(t)},$ (49)
$\displaystyle
B(t)=\frac{l_{0}}{\dot{\nu}(t)}\sqrt{\nu(t)\left(l_{1}\nu(t){{}^{3}}+l_{2}\right)},$
(50)
$\displaystyle\Phi(t)=\frac{\nu(t)\dot{\nu}(t)^{2}}{l_{0}^{2}\left(l_{1}\nu(t){{}^{3}}+l_{2}\right)}.$
(51)
Thus, using (49) and (50), the LRS Bianchi type I metric (9) becomes
$ds^{2}=-dt^{2}+l_{0}^{2}\left(l_{1}\nu(t)^{3}+l_{2}\right)\frac{\nu(t)}{\dot{\nu}(t)^{2}}\left[\left(l_{1}+\frac{l_{2}}{\nu(t)^{3}}\right)dx^{2}+dy^{2}+dz^{2}\right].$
(52)
Then, the average scale factor of the Universe takes the form
$\displaystyle
a(t)=\frac{l_{0}}{\dot{\nu}(t)}\left(l_{1}\nu(t){{}^{3}}+l_{2}\right)^{2/3}.$
(53)
Also the case of isotropy can be seen, by requiring $A\propto B$, to
correspond to $l_{2}=0$.
In the general anisotropic case, the directional Hubble parameter in $x$
direction defined in Eq.(33) is found as
$\displaystyle
H_{x}=\frac{\dot{\mu}}{\mu}-\frac{\dot{\nu}}{\nu}-\frac{\dot{u}}{u}=-\frac{\ddot{\nu}}{\dot{\nu}}+\frac{\dot{\nu}}{\nu}\left[\frac{3l_{1}\nu^{3}}{l_{1}\nu{{}^{3}}+l_{2}}-1\right],$
(54)
the other directional Hubble parameters in the directions $y$ and $z$ are
similarly found as
$\displaystyle
H_{y}=H_{z}=\frac{1}{2}\left(\frac{\dot{\mu}}{\mu}+\frac{\dot{\nu}}{\nu}\right)-\frac{\dot{u}}{u}=-\frac{\ddot{\nu}}{\dot{\nu}}+\frac{\dot{\nu}}{2\nu}\left[\frac{3l_{1}\nu^{3}}{l_{1}\nu{{}^{3}}+l_{2}}+1\right],$
(55)
and the mean Hubble parameter defined in Eq.(34) is found as
$\displaystyle
H=\frac{2}{3}\frac{\dot{\mu}}{\mu}-\frac{\dot{u}}{u}=-\frac{\ddot{\nu}}{\dot{\nu}}+\frac{2l_{1}\nu^{2}\dot{\nu}}{l_{1}\nu{{}^{3}}+l_{2}}.$
(56)
At this point, we would like to emphasize that particular solutions can be
found, that is, the scale factors $A(t),B(t)$, the potential $\Phi(t)$ and the
average scale factor $a(t)$ can be obtained by specifying the function
$\nu(t)$. In the next section, we will find such examples.
## 6 Examples
To illustrate our family of solutions for the dynamics of LRS Bianchi type I
cosmologies containing a scalar field, we choose examples corresponding to the
well-known solutions in the isotropic case, that is, power-law models, i.e.
$a_{\rm isotropic}\propto t^{\eta}$, exponentially expanding models, i.e.
$a_{\rm isotropic}\propto e^{\chi t}$, and phantom/Big Rip models, i.e.
$a_{\rm isotropic}\propto(t_{c}-t)^{-\sigma}$, where $\eta,\chi$ and $\sigma$
are positive constants. We can find the form of each $\nu(t)$ by solving from
Eq.(53) after setting $l_{2}=0$.
The most general form we find for the power-law models is
$\nu(t)=\frac{C_{1}}{C_{2}-t^{\eta-1}}$, but for simplicity we use
$\nu(t)=a_{1}t^{n}$. For the exponential case, the most general form is
$\nu(t)=\frac{C_{3}}{C_{4}-e^{-\chi t}}$, but we use $\nu(t)=a_{2}e^{kt}$.
Similarly, for the phantom/Big Rip models, the most general form is
$\nu(t)=\frac{C_{5}}{C_{6}-(t_{c}-t)^{\sigma+1}}$, but we use
$\nu(t)=\frac{a_{3}}{(t_{c}-t)^{m}}$ ($n$, $k$ or $m$ are not necessarily
integer).
### 6.1 Power-law seed
When we use $\nu(t)=a_{1}t^{n}$, where $a_{1}$ and $n$ are nonzero constants,
the scale factors read
$A(t)=\frac{l_{0}}{a_{1}^{2}n}t^{1-2n}(l_{1}a_{1}^{3}t^{3n}+l_{2}),\;\;B(t)=\frac{l_{0}}{n\sqrt{a_{1}}}t^{1-n/2}\left[l_{1}a_{1}^{3}t^{3n}+l_{2}\right]^{1/2},$
(57)
the average scale factor is found from Eq.(53) as
$\displaystyle
a(t)=\frac{l_{0}}{a_{1}n}\left[l_{1}a_{1}^{3}t^{3n}+l_{2}\right]^{2/3}t^{1-n},$
(58)
the scalar field from Eq.(51) as
$\displaystyle\Phi(t)=\frac{a_{1}^{3}n^{2}t^{3n-2}}{l_{0}^{2}\left(l_{1}a_{1}^{3}t^{3n}+l_{2}\right)},$
(59)
and the mean Hubble parameter from Eq.(34) as
$\displaystyle
H=\frac{2l_{1}a_{1}^{3}nt^{3n-1}}{l_{1}a_{1}^{3}t^{3n}+l_{2}}+\frac{1-n}{t}.$
(60)
Inspection of the scale factors $A(t)$, $B(t)$ and $a(t)$ shows that if
$l_{1}$ or $l_{2}$ is zero, they will reduce to single powers of $t$. But also
in the general case where both $l_{1}$ and $l_{2}$ are nonzero, the scale
factors will approach one power of $t$ near zero and another near infinities.
These powers depends on the sign of $n$: For negative $n$, near $t=0$ we have
$A(t),B(t),a(t)\propto t^{1+n}$, near infinities we have $A(t)\propto
t^{1-2n}$, $B(t)\propto t^{1-n/2}$ and $a(t)\propto t^{1-n}$; for positive
$n$, the behaviors near zero and infinities are interchanged. Therefore, we
need to treat the cases of vanishing $l_{1}$ or $l_{2}$ separately from the
general case. These cases will be split into subcases, for example, according
to the value of $n$, etc. Each line in the Tables 1,2 and 3 shows the timeline
of an eventual subcase from $t\rightarrow-\infty$ to $t\rightarrow+\infty$,
which usually describes more than one universe, since the timeline may be
interrupted by singularities, shown by expressions in square brackets in the
lines of the tables. Since solutions cannot be taken valid through
singularities; each interval between singularities should be considered an
independent solution/universe. The universes not extending to the right or
left edge of a line have finite lifetime.
Obvious candidates for singularities are $t=0$ and $t\rightarrow\pm\infty$, in
light of the above discussion. Moreover, the $(l_{1}a_{1}^{3}t^{3n}+l_{2})$
terms in the scale factors can vanish at some finite $t$ value $t_{1}$, or
diverge at $t=0$, if both $l_{1}$ and $l_{2}$ are nonzero. The meaning of a
singularity not only depends on its mathematical behavior, but also on its
relative time-relation to the observers: If observers see (or calculate)
vanishing comoving volume in their finite past or finite future, they will
call that singularity a Big Bang (BB) or a Big Crunch (BC); This is shown as
”BC[0]BB” in the tables. Similarly, observers calculating diverging of the
comoving volume in their finite future will call that singularity a Big Rip
(BR), and the case of diverging comoving volume in the finite past we will
call it an Inverse Big Rip (iBR).
The square brackets should contain an ordered triple, where the first entry
shows the behavior of $A(t)$, the second entry the behavior of $B(t)$ and the
third, the behavior of $a(t)$. The behavior is indicated by the symbols $0,C$
or $\infty$ to denote that the relevant scale factor vanishes/goes to a finite
number/diverges at that point. To make the tables more compact, an ordered
pair (describing the behavior of $A(t)$ and $B(t)$; whenever these make the
behavior of $a(t)$ obvious) or a single entry (indicating that the behavior of
all three scale factors is the same) may be used. Finally, one should also
note that negative $t$ can only be considered if $n$ is a rational number with
an odd denominator, not for general $n$.
We discuss the three above-mentioned cases in order of increasing complexity:
First, we discuss the isotropic $l_{2}=0$ case, then the simple anisotropic
$l_{1}=0$ case, and finally the general anisotropic case where both $l_{1}$
and $l_{2}$ are nonzero.
#### 6.1.1 The $l_{2}=0$ (isotropic) case.
As stated after Eq.(53), in this case the LRS Bianchi type I spacetime becomes
isotropic, i.e. reduces to the well-known spatially flat FRW metric. The scale
factors become proportional to each other, and
$\displaystyle a(t)=a_{0}t^{1+n}$ (61)
where $a_{0}=a_{1}\frac{l_{0}l_{1}^{2/3}}{n}$. This means that $a(t)$ can
vanish or diverge at zero or infinity, and the critical $n$ values are -1 and
0, leading to the subcases shown in Table 1. The cases are
$n$ $t$ | $-\infty$ | | $0$ | | $+\infty$
---|---|---|---|---|---
$n<-1$ | [0]Z | | BR[$\infty$]iBR | | Z[0]
$n=-1$ | (C) | —static— | (C) | —static— | (C)
$-1<n<0$ | [$\infty$]I | ”decelerating” | BC[0]BB | decelerating | I[$\infty$]
$n\rightarrow 0$ | [$\infty$]I | linear | BC[0]BB | linear | I[$\infty$]
$n>0$ | [$\infty$]I | ”accelerating” | BC[0]BB | accelerating | I[$\infty$]
Table 1: Universes derived for $l_{2}=0$ (isotropic universes) where $\nu(t)$
is a power of $t$. Each line shows a timeline from $-\infty$ to $\infty$, but
may represent multiple universes delimited by singularities, shown by square
brackets. The expressions in the brackets refer to the scale factor $a(t)$.
Also, I: infinite scale factor at infinite future or past, BR: Big Rip, i:
Inverse, BC: Big Crunch, BB: Big Bang, Z: vanishing scale factor at infinite
future or past [Also see text and the note about negative time on page 6.1].
* $\bullet$
$n<-1$: Negative times111subject to the condition on $n$ mentioned on page 6.1
describe a universe that expands from a vanishingly small scale factor in the
infinite past, and then the scale factor diverges within a finite time: one
might call this a Zero-Big Rip (Z-BR) universe. Positive times describe
another universe, whose behavior is the time-reverse of the first, an iBR-Z
universe.
* $\bullet$
$n=-1$: This universe, being static and flat, is equivalent to Minkowski
space.
* $\bullet$
$-1<n<0$: For positive times, this case represents decelerating (spatially)
flat FRW universes, including the radiation-dominated ($n=-1/2$) and matter-
dominated ($n=-1/3$) expansions, in particular. For negative times1, however,
the universes contract from infinite scale factor in the infinite past to a
Big Crunch at $t=0$; so the universes represented on the last three lines of
Table 1 may be called Infinity-Big Crunch (I-BC) and Big Bang-infinity (BB-I)
universes. The I-BC universes are decelerating, too: The negative $\dot{a}(t)$
[for positive $a(t)$] becomes more negative with time.
* $\bullet$
$n\rightarrow 0$: This case can be analyzed with the help of an infinite scale
transformation [note that Eqs. (45)-(47) are satisfied for constant $\nu(t)$],
and describes universes linearly expanding (positive time) or contracting
(negative time1) with increasing time.
* $\bullet$
$n>0$: Positive times describe a universe that starts with a Big Bang and
expands forever with increasing speed; negative times1 describe a universe
with time-reversed behavior.
As is well-known (and mentioned above), in FRW cosmology the behavior of the
universe and the properties of the (possibly efective) fluid contained in the
universe are related. In fact in our $l_{2}=0$ case the Hubble parameter, the
deceleration and EoS parameters and the scalar field simplify to
$\displaystyle H=\frac{n+1}{t},$ (62) $\displaystyle q=-\frac{n}{n+1},$ (63)
$\displaystyle w=-\frac{3n+1}{3\left(n+1\right)},$ (64)
$\displaystyle\Phi(t)=\frac{6n^{2}}{\lambda}t^{-2}$ (65)
and we can confirm that $q$ is proportional to $1+3w$; in particular, for
$n=-1/2$ Eq.(64) gives $w=1/3$ and for $n=-1/3$, it gives $w=0$. For
$n\rightarrow 0$, the deceleration parameter vanishes. We know that empty
universes can have this property, and we see that the scalar field vanishes in
this case. For $n>0$, the effective fluid is equivalent to
quintessence/quiessence ($-1<w<-1/3$), approaching cosmological constant
($w=-1$) as $n\rightarrow\infty$. Only for $n<-1$ do we have phantom behavior
($w<-1$), and this is the case where a Big Rip appears (for negative times1,
at least).
The $n=-1$ case is particularly interesting. For this case, $q$ and $w$ seem
to diverge, while in Table 1 this subcase does not seem to be more problematic
than others. But, as $n$ approaches $-1$, the scale factor $a(t)$ approaches a
constant function, therefore $\dot{a}(t)$ and $\ddot{a}(t)$ approach zero.
Hence the deceleration parameter, which has division by $\dot{a}^{2}$ in its
definition, is not a good parameter to use in this limit. Also, since this
universe corresponds to Minkowski space, the stress-energy-momentum tensor
should vanish. Let us calculate the energy density and isotropic pressure,
using Eq.(65) and the potential expression in Eq.(29), in expressions (31) and
(32):
$\displaystyle\rho^{\Phi}=\frac{18}{\lambda}n^{2}(n+1)^{2}\,t^{-4},$ (66)
$\displaystyle p^{\Phi}=-\frac{18}{\lambda}n^{2}(n+1)(n+\frac{1}{3})\,t^{-4}.$
(67)
These expressions verify that $n=0$ corresponds to empty universes. It can
also be seen that both the density and pressure vanish for $n=-1$, although
the field does not vanish! Of course, this also makes $w$ meaningless in this
subcase.
#### 6.1.2 The $l_{1}=0$ case.
Since $l_{1}=\lambda/(6l_{0}^{2})$, in this case the potential
$U(\Phi)=\lambda\Phi^{2}$ vanishes, therefore this case is equivalent
considering the scalar field to be massless. Now $A(t)\propto t^{1-2n}$,
$B(t)\propto t^{1-n/2}$ and $a(t)\propto t^{1-n}$, so that the critical $n$
values are 1/2, 1 and 2; and we are led to Table 2 listing the subcases.
$n$ $t$ | $-\infty$ | | $0$ | | $+\infty$
---|---|---|---|---|---
$n<1/2$ | [$\infty,\infty,\infty$]I | | BC[0, 0, 0]BB | | I[$\infty,\infty,\infty$]
$n=1/2$ | [C, $\infty,\infty$]2I | | 2BC[C, 0, 0]2BB | | 2I[C,$\infty,\infty$]
$1/2<n<1$ | [0, $\infty$, $\infty$]IPa | | cBC[$\infty$, 0, 0]cBB | | IPa[0, $\infty$, $\infty$]
$n=1$ | [0, $\infty$, C]IPb | | BD[$\infty$, 0, C]iBD | | IPb[0, $\infty$, C]
$1<n<2$ | [0, $\infty$, 0]IPc | | cBR[$\infty$, 0, $\infty$]ciBR | | IPc[0, $\infty$, 0]
$n=2$ | [0, C, 0]1Z | | 1BR[$\infty$, C, $\infty$]1iBR | | 1Z[0, C, 0]
$n>2$ | [0, 0, 0]Z | | BR[$\infty,\infty,\infty$]iBR | | Z[0, 0, 0]
Table 2: Universes derived for $l_{1}=0$, where $\nu(t)$ is a power of $t$.
Each line shows a timeline from $-\infty$ to $\infty$, but may represent
multiple universes delimited by singularities, shown by square brackets. The
expressions in the brackets refer to $A(t)$, $B(t)$ and $a(t)$. Also, I:
infinite scale factors at infinite future or past, BR: Big Rip, i: Inverse,
BC: Big Crunch, BB: Big Bang, 1 or 2: only one or two of the spatial
dimensions, c: cigar-type, BD: Big Draw, IPa : ”Infinite pancake” of type a,
etc., Z: vanishing scale factors at infinite future or past [Also see text and
the note about negative time on page 6.1].
Let us note that
* $\bullet$
For $n=1/2$, we get universes evolving in the $y$ and $z$ dimensions only,
either from infinite scale factors to a BC, or from a BB to infinite scale
factors, in infinite time, possibly to be called 2I-2BC and 2BB-2I for two-
dimensional
* $\bullet$
For $1/2<n<1$, $A(t)$ diverges at $t=0$, while the other scale factors vanish.
This singularity is extremely anisotropic: As the universe contracts towards a
BC in the $y$ and $z$ directions, it actually expands infinitely in the $x$
direction! But this expansion is not enough to prevent the vanishing of a
comoving volume, and in this sense this event is a BB/BC. Singularities where
one dimension diverges while two shrink to zero are called ”cigar-type”, so we
might call this event a ”cBC/cBB”.
Again for $1/2<n<1$, $A(t)$ vanishes as $t\rightarrow\pm\infty$, while the
other scale factors do not. This type of singularity is also extremely
anisotropic, and is called a ”pancake” singularity. In this particular
subcase, the other scale factors diverge, so the singularity is called an
infinite pancake. This type of singularity appears in the next two subcases
too, but the behavior of the average scale factor $a(t)$ is diferent in each
subcase. So we classified the ”infinite pancakes” accordingly. (In roynarsin ,
a classification is made according to the behavior of the three separate scale
factors of a Bianchi I spacetime.)
* $\bullet$
For $n=1$, $A(t)$ diverges, $a(t)$ goes to a constant, $B(t)$ vanishes at
$t=0$. Although this $t=0$ event is also a cigar-type singularity, since the
comoving volume neither vanishes nor diverges, it is not a BB/BC or BR/iBR. We
suggest the names Big Draw (BD)222since the volume of a metal drawn to produce
a wire does not change. and Inverse Big Draw (iBD).
* $\bullet$
For $n=2$, the universe evolves in the $x$ dimension only, either from
vanishing scale factor to a BR, or from an iBR to vanishing scale factor, in
infinite time. For $n>2$, this occurs in all three dimensions.
For $l_{1}=0$, the mean Hubble parameter, the deceleration parameter, the EoS
parameter and the scalar field simplify respectively to
$\displaystyle H=\frac{1-n}{t},$ (68) $\displaystyle q=\frac{n}{1-n},$ (69)
$\displaystyle w=\frac{-5n+2}{3(n-2)},$ (70)
$\displaystyle\Phi(t)=\frac{a_{1}^{3}n^{2}}{l_{0}^{2}l_{2}}t^{3n-2}.$ (71)
There seem to be problems for $n=1$ and $n=2$, while in Table 2 these cases do
not seem to be more problematic than others. The apparent divergence of $q$ as
$n=1$ is understood as for the $n=-1$ line of Table 1: $a(t)$ becomes
constant. As discussed at the end of Section 4, the $w$ parameter is not very
useful when there is anisotropy333As a specific example, consider $n=7/5$,
which gives $w=25/9$ according to (70), which should give deceleration, but
gives $q=-7/2$, that is, acceleration according to (69)., but one might still
ask why it diverges for $n=2$. Let us again calculate the energy density and
isotropic pressure of the effective fluid for our subcase, recalling that the
potential vanishes. Then, using Eq.(71), we get
$\displaystyle\rho^{\Phi}=\frac{3a_{1}^{3}}{4l_{0}^{2}l_{2}}n^{2}(n-2)(3n-2)\,t^{3n-4},$
(72) $\displaystyle
p^{\Phi}=\frac{a_{1}^{3}}{4l_{0}^{2}l_{2}}n^{2}(2-5n)(3n-2)\,t^{3n-4}.$ (73)
explaining the divergence of $w$ by the vanishing of $\rho^{\Phi}$ for $n=2$.
We also see that the density and pressure both vanish (other than for the
trivial, isotropic $n=0$ case) for $n=2/3$, due to the constancy of the field
and vanishing of the potential.
#### 6.1.3 The case with both $l_{1}$ and $l_{2}$ nonzero.
For this subcase, the limiting behaviors of the scale factors mentioned on
page 6.1 show that they diverge near infinities for all $n$. Moreover, at late
time, the universe expands with acceleration, again for all $n$, as can be
seen from the asymptotic behavior mentioned after Eq.(60). But the behavior of
the scale factors near zero determines the critical values of $n$ as -1, 1/2,
1 and 2, leading to the subcases shown in Table 3. Outstanding features are
$n,t_{1}$ $t$ | $-\infty$ | | $-|t_{1}|$ | | $0$ | | $|t_{1}|$ | | $+\infty$
---|---|---|---|---|---|---|---|---|---
$n<-1$ | No $t_{1}$ | [$\infty$]I | | —B— | BR[$\infty$]iBR | —B— | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | BR[$\infty$]iBR | —B— | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | —B— | BR[$\infty$]iBR | | BC[0]BB | | I[$\infty$]
$n=-1$ | No $t_{1}$ | [$\infty$]I | | | | (C) | | —B—∗ | I[$\infty$]
| $\exists\;t_{1}$ | [$\infty$]I | | | | (C) | | BC[0]∗BB | | I[$\infty$]
$-1<n<0$ | No $t_{1}$ | [$\infty$]I | | | | BC[0]BB | | | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | BC[0]BB | | | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | | | BC[0]BB | | BC[0]BB | | I[$\infty$]
$n\rightarrow 0$ | [$\infty$]I | linear contraction | BC[0]BB | linear expansion | I[$\infty$]
$0<n<1/2$ | No $t_{1}$ | [$\infty$]I | | | | BC[0]BB | | | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | BC[0]BB | | | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | | | BC[0]BB | | BC[0]BB | | I[$\infty$]
$n=1/2$ | No $t_{1}$ | [$\infty$]I | | | | 2BC[C,0]2BB | | | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | 2BC[C,0]2BB | | | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | | | 2BC[C,0]2BB | | BC[0]BB | | I[$\infty$]
$1/2<n<1$ | No $t_{1}$ | [$\infty$]I | | | | cBC[$\infty$,0,0]cBB | | | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | cBC[$\infty$,0,0]cBB | | | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | | | cBC[$\infty$,0,0]cBB | | BC[0]BB | | I[$\infty$]
$n=1$ | No $t_{1}$ | [$\infty$]I | | | | BD[$\infty$,0,C]iBD | | | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | BD[$\infty$,0,C]iBD | | | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | | | BD[$\infty$,0,C]iBD | | BC[0]BB | | I[$\infty$]
$1<n<2$ | No $t_{1}$ | [$\infty$]I | | —B— | cBR[$\infty$,0,$\infty$]ciBR | —B— | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | cBR[$\infty$,0,$\infty$]ciBR | —B— | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | —B— | cBR[$\infty$,0,$\infty$]ciBR | | BC[0]BB | | I[$\infty$]
$n=2$ | No $t_{1}$ | [$\infty$]I | | —B— | 1BR[$\infty$,C,$\infty$]1iBR | —B— | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | 1BR[$\infty$,C,$\infty$]1iBR | —B— | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | —B— | 1BR[$\infty$,c,$\infty$]1iBR | | BC[0]BB | | I[$\infty$]
$n>2$ | No $t_{1}$ | [$\infty$]I | | —B— | BR[$\infty$]iBR | —B— | | I[$\infty$]
| $t_{1}<0$ | [$\infty$]I | | BC[0]BB | | BR[$\infty$]iBR | —B— | | I[$\infty$]
| $t_{1}>0$ | [$\infty$]I | | —B— | BR[$\infty$]iBR | | BC[0]BB | | I[$\infty$]
Table 3: Universes derived for $l_{1}\neq 0$ and $l_{2}\neq 0$, where $\nu(t)$
is a power of $t$. Each line shows a timeline from $-\infty$ to $\infty$, but
may represent multiple universes delimited by singularities, shown by square
brackets. A single expression in the brackets refers to all scale factors, two
expressions refer to $A(t)$ and $B(t)$, three expressions refer to $A(t)$,
$B(t)$ and $a(t)$. Also, I: infinite scale factors at infinite future or past,
B: Bounce, BR: Big Rip, i: Inverse, BC: Big Crunch, BB: Big Bang, 1 or 2: only
one or two of the spatial dimensions, c: cigar-type, BD: Big Draw [See text,
and also note about negative time on page 6.1. ∗Sign of bounce time or $t_{1}$
may be positive or negative].
* $\bullet$
The comoving volume diverges at $t=0$ for $n<-1$ and for $n>1$. Since it also
diverges as $t\rightarrow\pm\infty$, this means that it must go through a
minimum between two divergences — a Bounce (B). If $t_{1}$ does not exist,
there will be a Bounce each in the $t<0$ universe1 and the $t>0$ universe;
whereas if $t_{1}$ does exist, it will essentially bring one of the minima
down to zero, inserting a BC/BB, effectively splitting one of the universes
into two. Hence the first line of Table 3 has an Infinity-Bounce-Big Rip (I-B-
BR) universe and an iBR-B-I universe; the second line an I-BC universe, a
finite-lifetime BB-BC universe and an iBR-B-I universe; and so on.
* $\bullet$
The third line features an iBR-BC universe, i.e. a finite-lifetime universe
which starts with infinitely large scale factors that immediately contract to
finite values, and keep on contracting to a BC.
* $\bullet$
For $n=-1$, the scale factors become constant at $t=0$. Therefore the subcase
without $t_{1}$ features the only universe of the table without beginning1 or
end, a bouncing universe.
* $\bullet$
The $n\rightarrow 0$ case is the same as the corresponding case of Table 1. In
fact, it is isotropic.
* $\bullet$
For all cases with $n>0$, the singularity at $t=0$ is qualitatively the same
as in the corresponding $n$ value Table 2.
Interestingly, a wide range of cosmological possibilities, including late-time
acceleration, can be reproduced by using a power of $t$ as the seed function.
This even includes the isotropic case, where the time-dependence of the source
scalar field does not change with $n$: $\Phi(t)\propto t^{-2}$. This
surprising richness, including the possibility of vanishing energy-momentum
tensor while the field is nonzero, seems to be due to the coupling coefficient
selected by the Hessian determinant condition, and the potential selected by
the Noether symmetry approach.
### 6.2 Exponential seed
When we use $\nu(t)=a_{1}e^{kt}$ where $a_{2}$ and $k$ are non-zero constants,
in Eqs.(49)-(51), (53), (56) and (35); the scale factors, the scalar field,
the mean Hubble parameter and the deceleration parameter become
$\displaystyle
A(t)=\frac{l_{0}}{ka_{2}^{2}}\left(l_{1}a_{2}^{3}e^{3kt}+l_{2}\right)e^{-2kt},$
(74) $\displaystyle
B(t)=\frac{l_{0}}{k\sqrt{a_{2}}}\left(l_{1}a_{2}^{3}e^{3kt}+l_{2}\right)^{1/2}e^{-kt/2},$
(75) $\displaystyle
a(t)=\frac{l_{0}}{ka_{2}}\left(l_{1}a_{2}^{3}e^{3kt}+l_{2}\right)^{2/3}e^{-kt},$
(76)
$\displaystyle\Phi(t)=\frac{k^{2}a_{2}^{3}}{l_{0}^{2}}\frac{e^{3kt}}{l_{1}a_{2}^{3}e^{3kt}+l_{2}},$
(77) $\displaystyle
H=k\left[\frac{l_{1}a_{2}^{3}e^{3kt}-l_{2}}{l_{1}a_{2}^{3}e^{3kt}+l_{2}}\right],$
(78) $\displaystyle
q=-\frac{l_{1}^{2}a_{2}^{6}e^{6kt}+4l_{1}l_{2}a_{1}^{3}e^{3kt}+l_{2}^{2}}{\left(l_{1}a_{1}^{3}e^{3kt}-l_{2}\right)^{2}}.$
(79)
Unlike in Section 6.1, these scale factors are never singular at $t=0$,
however the $(l_{1}a_{2}^{3}e^{3kt}+l_{2})$ terms in the scale factors can
vanish at some finite $t$ value $t_{2}$, if $l_{1}$ and $l_{2}$ are both
nonzero. Since they do not diverge at finite time, there is no Big Rip.
In that case, the scale factors diverge exponentially at both infinities,
regardless of the signs of $l_{1}$, $l_{2}$ and $k$. Therefore, if $t_{2}$
does not exist, the universe is a bouncing universe, infinite in both time
directions. This solution is qualitatively similar to the one in the fourth
line of Table 3. If $t_{2}$ does exist, then the solution represents two
universes; one collapsing from infinite scale factors to a Big Crunch, one
expanding from a Big Bang to infinite scale factors. This solution is
qualitatively similar to the one in the fifth line of Table 3. For all
solutions (with nonzero $l_{1}$ and $l_{2}$), at late time, $H\rightarrow|k|$
and $q\rightarrow-1$, confirming the de Sitter-like asymptotic behavior of the
universe(s). For positive $k$, the scalar field $\Phi$ approaches a constant.
If either one of $l_{1}$ and $l_{2}$ is zero, the universe either expands or
contracts exponentially, depending on the sign of $k$ and on which constant
vanishes. In these cases, $q=-1$ at all times, and for $l_{2}=0$ (isotropic),
$\Phi$ is constant. A constant scalar field is the standard (simplest) way in
the literature for causing exponential expansion, and in the isotropic case
can be interpreted as creating a cosmological constant. For positive $k$, the
solution approaches isotropy at late times.
According to recent astrophysical data, the deceleration parameter of the
universe lies in the interval $-1.72<q<-0.58$ Melchiori . In all models of
this subsection at late time $q$ converges to $-1$, so in this sense these
models are consistent with the observational results.
### 6.3 Shifted-reversed power seed
We may also use $\nu(t)=\frac{a_{3}}{(t_{c}-t)^{m}}$, where $a_{3}$, $t_{c}$
and $m$ are non-zero constants, since this function is similar to the FRW Big
Rip models. Then Eqs.(49)-(51) become
$\displaystyle
A(t)=\frac{l_{0}}{a_{3}^{2}m}(t_{c}-t)^{1+2m}\left[l_{1}a_{3}^{3}(t_{c}-t)^{-3m}+l_{2}\right],$
(80) $\displaystyle
B(t)=\frac{-l_{0}}{m\sqrt{a_{3}}}(t_{c}-t)^{1+m/2}\left[l_{1}a_{3}^{3}(t_{c}-t)^{-3m}+l_{2}\right]^{1/2},$
(81)
$\displaystyle\Phi(t)=\frac{a_{3}^{3}m^{2}(t_{c}-t)^{-3m-2}}{l_{0}^{2}\left[l_{1}a_{3}^{3}(t_{c}-t)^{-3m}+l_{2}\right]}.$
(82)
The substitutions $m\rightarrow-n$ and $t\rightarrow t_{c}-t$ bring these
expressions into the same form as Eqs.(57) and (59). Therefore all solutions
from Section 6.1 apply, we just have to read the lines of the tables in that
section from right to left, and switch the sign of $n$ (and of course, shift
any ”i” designation to the other side of the singularity). So this seed does
not only give Big Rip models (see Ref.noj005 for detailed discussion).
### 6.4 Other seeds and parameter choices
Of course, $\nu(t)$ being a free function, an infinite number of choices is
possible, not to mention the freedom in the choices of $l_{1}$ and $l_{2}$.
Among these, we are naturally interested in those choices that give simple
metric functions $A(t)$ and $B(t)$.
One way of achieving this would be to set $l_{1}$ or $l_{2}$ equal to zero.
But $l_{2}=0$ takes us to isotropic solutions, which are not in line with the
emphasis of the present work. The choice $l_{1}=0$, although it will give
simple-looking solutions, limits us to the case of massless field, and there
seems to be no motivation for choosing a particular solution above others,
except possibly $\nu(t)=t$ (since the comoving volume stays constant), but
this is already covered in subsection 6.1.2.
One can also choose combinations to make both $\nu(t)$ and
$(l_{1}\nu^{3}+l_{2})$ simple. For example, choosing $\nu(t)=\cos^{2/3}(t)$,
$l_{0}=-2/3$, $l_{1}=-1$ and $l_{2}=1$ yields
$ds^{2}=-dt^{2}+\cos^{4/3}(t)\left[\tan^{2}(t)\,dx^{2}+dy^{2}+dz^{2}\right]$
(83)
a metric with scale factors
$\displaystyle A(t)=\frac{\sin(t)}{\cos^{1/3}(t)}$ (84) $\displaystyle
B(t)=\cos^{2/3}(t)$ (85) $\displaystyle
a(t)=\left[\sin(t)\cos(t)\right]^{1/3}$ (86)
i.e. a set of universes that start with a 1-D Big Bang and end with a cigar-
like Big Crunch (for $0\leq t\leq\pi/2$, in appropriate units of time) or
start with a cigar-like Big Bang and end with a 1-D Big Crunch (for $\pi/2\leq
t\leq\pi$).
Another solution,
$ds^{2}=-dt^{2}+\sinh^{4/3}t\left(\coth^{2}t\,dx^{2}+dy^{2}+dz^{2}\right)$
(87)
is mathematically similar, but describes universes with infinite lifetime, for
example the $t>0$ universe starts with a cigar-like Big Bang and approaches de
Sitter-like isotropically expanding universe at late time.
## 7 Solutions in Einstein frame
It is known that a scalar-tensor theory can be transformed to the so-called
Einstein frame, where the gravitational scalar field becomes minimally coupled
to curvature. The price to pay for this simplification is the equivalence
principle: Massive point particles do not follow geodesics any more, in
contrast to the Jordan frame, which we used so far in this work fm ; faraoni .
For the BD theory, the case $W=-3/2$ is special: We derived it from the
vanishing of the Hessian (25), it represents a fixed-point of the conformal
transformation faraoni , and marks the boundary beyond which ghosts appear
dabrowski . In the Einstein frame, this specialness is reflected in the
gravitational scalar field becoming non-dynamical. Hence, it would be
interesting to look at our solutions also in the Einstein frame.
The conformal transformation
$\displaystyle g_{ab}=\Omega^{-2}\tilde{g}_{ab},\quad
g^{ab}=\Omega^{2}\tilde{g}^{ab},$ (88)
takes us from the Jordan frame to the Einstein frame. Defining
$\Phi=e^{-\sigma}$, the $W=-3/2$ BD action given in Eq. (27) takes the form
$\displaystyle\mathcal{A_{BD}}$ $\displaystyle=$
$\displaystyle\int{d^{4}x\sqrt{-g}e^{-\sigma}\left[R+\frac{3}{2}\sigma_{c}\sigma^{c}-U_{1}(\sigma))\right]}$
(89)
where $U_{1}(\sigma)=U(\Phi)/\Phi$. It is easy to show that under a choice of
a conformal factor
$\displaystyle\Omega=e^{-\sigma/2},$ (90)
the above action (89) transforms into
$\displaystyle\mathcal{A_{E}}$ $\displaystyle=$
$\displaystyle\int{d^{4}x\sqrt{-\tilde{g}}\left[\tilde{R}-U_{2}(\sigma))\right]}.$
(91)
This is exactly the Einstein-Hilbert action with a potential
$U_{2}(\sigma)=e^{\sigma}U_{1}(\sigma)=U(\Phi)/\Phi^{2}$, and using the
potential obtained in (29) it takes the form $U_{2}(\sigma)=\lambda$, where
$\lambda$ is a constant and could be interpreted as the cosmological constant
in Einstein frame. In this case, the (transformed) scalar field $\sigma$ does
not appear in the action, hence the scalar field is non-dynamical as stated in
the beginning of this section; and the field equations take the form
$\displaystyle\tilde{R}_{ab}=\frac{\lambda}{2}\tilde{g}_{ab}.$ (92)
where the Ricci tensor $\tilde{R}_{ab}$ refers to the transformed metric
$\tilde{g}_{ab}$.
The LRS Bianchi I spacetime can be brought back to its original form
$ds^{2}=-d\tilde{t}^{2}+\tilde{A}^{2}dx^{2}+\tilde{B}^{2}\left(dy^{2}+dz^{2}\right),$
(93)
by a simple coordinate transformation after the conformal transformation. The
transformations of time coordinate and scale factors from the Jordan frame to
the Einstein frame are given by
$\displaystyle\tilde{t}=\int{\sqrt{\Phi}dt},\quad\tilde{A}=\sqrt{\Phi}A,\quad\tilde{B}=\sqrt{\Phi}B.$
(94)
For this spacetime in Einstein frame the field equations (92) give
$\displaystyle
2\frac{\tilde{A}^{\prime}\tilde{B}^{\prime}}{\tilde{A}\tilde{B}}+\left(\frac{\tilde{B}^{\prime}}{\tilde{B}}\right)^{2}=\frac{\lambda}{2},$
(95) $\displaystyle
2\frac{\tilde{B}^{\prime\prime}}{\tilde{B}}+\left(\frac{\tilde{B}^{\prime}}{\tilde{B}}\right)^{2}=\frac{\lambda}{2},$
(96)
$\displaystyle\frac{\tilde{A}^{\prime\prime}}{\tilde{A}}+\frac{\tilde{B}^{\prime\prime}}{\tilde{B}}+\frac{\tilde{A}^{\prime}\tilde{B}^{\prime}}{\tilde{A}\tilde{B}}=\frac{\lambda}{2},$
(97)
where the prime represents derivative with respect to tilted time coordinate
$\tilde{t}$. These equations can be solved exactly, giving
$\displaystyle\tilde{A}=c_{3}\left(\tilde{t}+c_{1}\right)^{-1/3},\quad\tilde{B}=c_{2}\left(\tilde{t}+c_{1}\right)^{2/3},$
(98)
for $\lambda=0$;
$\displaystyle\tilde{A}=c_{3}\sinh(k\tilde{t}+c_{1})\cosh^{-1/3}(k\tilde{t}+c_{1}),$
$\displaystyle\tilde{B}=c_{2}\cosh^{2/3}(k\tilde{t}+c_{1}),$ (99)
for $\lambda=8k^{2}/3>0$;
$\displaystyle\tilde{A}=c_{3}\sin(k\tilde{t}+c_{1})\cos^{-1/3}(k\tilde{t}+c_{1}),$
$\displaystyle\tilde{B}=c_{2}\cos^{2/3}(k\tilde{t}+c_{1}),$ (100)
for $\lambda=-8k^{2}/3<0$. We note here that the obtained scale factors given
in (98) represent the well known Kasner solution. We would like to check the
consistency between these solutions and those in the Jordan frame, (49)-(51).
For given $\lambda$, the Jordan frame solutions contain the arbitrary function
$\nu(t)$ and two arbitrary constants $l_{0}$ and $l_{2}$ ($l_{1}$ is not
independent). The Einstein frame solutions contain one arbitrary function
$\Phi(\tilde{t})$ (which does not appear in the metric, however) and three
arbitrary constants, $c_{1}$, $c_{2}$ and $c_{3}$. To show consistency, we
need to transform the Jordan frame solutions to the Einstein frame and show
that they agree with solution found in that frame.
Applying the transformations in (94) to (49) and (50), using (51), we get
$\displaystyle\tilde{A}=\sqrt{\frac{l_{1}\nu(t){{}^{3}}+l_{2}}{\nu(t)}},$
(101) $\displaystyle\tilde{B}=\nu(t)$ (102)
Using (102) in (51), we can write for $\Phi$ in the transformed coordinates
$\Phi(t)=\frac{\tilde{B}\dot{\tilde{B}}^{2}}{l_{0}^{2}\left(l_{1}\tilde{B}{{}^{3}}+l_{2}\right)},$
(103)
but using (94),
$\dot{\tilde{B}}=\frac{d\tilde{B}}{dt}=\frac{d\tilde{B}}{d\tilde{t}}\frac{d\tilde{t}}{dt}=\tilde{B}^{\prime}\sqrt{\Phi}\Longrightarrow\Phi=\frac{\tilde{B}\tilde{B}^{\prime
2}\Phi}{l_{0}^{2}\left(l_{1}\tilde{B}{{}^{3}}+l_{2}\right)},$ (104)
so $\Phi$ disappears from the equation: it cannot be determined in the
transformed coordinates. This was to be expected, since the scalar field is
absent from the action (91). The arbitrariness (or information) in $\nu(t)$ in
the Jordan frame has shifted to $\Phi(\tilde{t})$ in the Einstein frame.
Returning to (104) and using the solution found for $\tilde{B}(\tilde{t})$ ,
e.g. for positive $\lambda$,
$l_{0}^{2}\left(l_{1}c_{2}^{3}\cosh^{2}(k\tilde{t}+c_{1})+l_{2}\right)=c_{2}^{3}\frac{4}{9}k^{2}\sinh^{2}(k\tilde{t}+c_{1}),$
(105)
where it should be noted that $k^{2}=3\lambda/8=9l_{1}l_{0}^{2}/4$ (see after
eq.(48)). This will hold, if $c_{2}$ is chosen such that $l_{2}$ is equal to
$-l_{1}c_{2}^{3}$. This choice can be made, since obviously the set
$\\{c_{1},c_{2},c_{3}\\}$ cannot be independent of the set
$\\{l_{0},l_{2}\\}$: If two solutions describe the same physical reality, the
parameters (arbitrary constants) of one should be expressible in terms of the
parameters of the other, although information could hide in the arbitrary
function(s) in this case.
Similarly, (101) will agree with (99), if $c_{3}$ is properly chosen; but
$c_{1}$ is not related to $l_{0}$ or $l_{2}$, it comes from the integration in
the first term in (94). The same calculations can be made for the cases of
negative and vanishing $\lambda$, establishing consistency of Einstein frame
solutions with the Jordan frame solutions.
For example, we can transform the solutions of subsection 6.1.1 ($l_{2}=0$,
isotropic) as
$\sqrt{\Phi}=\frac{3n}{2kt}\Longrightarrow
t=t_{1}e^{\frac{2k}{3n}\tilde{t}},\;\;\tilde{A}=\frac{2ka_{1}}{3l_{0}}t_{1}^{n}e^{\frac{2k}{3}\tilde{t}},\;\;\tilde{B}=a_{1}t_{1}^{n}e^{\frac{2k}{3}\tilde{t}}$
(106)
and those of subsection 6.1.2 ($l_{1}=0$) as
$\displaystyle\sqrt{\Phi}=\left[\frac{n^{2}a_{1}^{3}}{l_{0}^{2}l_{2}}\right]^{1/2}t^{(3/2)n-1}\Longrightarrow$
$\displaystyle
t=t_{0}(\tilde{t}+\tilde{c_{1}})^{2/(3n)},\;\;\tilde{A}=A_{0}(\tilde{t}+\tilde{c_{1}})^{-1/3},\;\;\tilde{B}=B_{0}(\tilde{t}+\tilde{c_{1}})^{2/3},$
(107)
where
$t_{0}=\left[\frac{9l_{0}^{2}l_{2}}{4a_{1}^{3}}\right]^{1/(3n)},\;\;A_{0}=\left[\frac{2l_{2}}{3l_{0}}\right]^{1/3}\;\;{\rm
and}\;\;B_{0}=\left[\frac{9l_{0}^{2}l_{2}}{4}\right]^{1/3}.$ (108)
Both solutions (106) and (107) identically satisfy the Einstein frame
equations (95)-(97), if one recalls that $l_{1}=0$ implies vanishing of
$\lambda$.The constants $t_{1}$ in (106) and $\tilde{c_{1}}$ in (107)
correspond to the constants $c_{1}$ that appear in (98)-(100).
To summarize, the solutions we found in the Einstein frame, although
containing only three parameters, are the transformed versions of the
solutions in the Jordan frame, which contained an arbitrary function. This
correspondence between a solution for the metric in the Einstein frame and an
infinite number of metrics in the Jordan frame is possible, since the scalar
field, which determines the transformation between the two frames, is
arbitrary in the Einstein frame.
## 8 Concluding remarks
In this paper we have examined the scalar-tensor Brans-Dicke theory of gravity
for LRS Bianchi type I spacetime admitting Noether symmetry. This symmetry
approach is important because it provides us with a theoretical motivation to
select a region of the solution space (For other motivations, leading to
different regions of the solution space for the Bianchi type I spacetimes, see
e.g. Refs.rodrigues -calogero2 ). The Lagrangian density (15) of LRS Bianchi
type I becomes degenerate for $W(\Phi)=-3/2$, when the Brans-Dicke coupling
function $F(\Phi)=\Phi$ is used. This degeneracy is required for nontrivial
solutions, hence we use this value as the BD parameter. The existence of
Noether symmetry also restricts the form of potential $U(\Phi)$, and allows us
to find a transformation given by (42) in which the metric potentials and the
scalar field are stated in terms of new dynamical variables ($\mu,\nu,u$),
where the variable $\mu$ is cyclic. Under the transformation (42) the
Lagrangian (15) reduces to a new, simpler one (44).
We have obtained the new set of field equations (45)-(47) for the LRS Bianchi
type I spacetime by using these transformations. We have found the general
class of solutions of BD field equations with potential
$U(\Phi)=\lambda\Phi^{2}$ in the background of LRS Bianchi type I spacetime
exhibiting Noether symmetry. This solution family contains an arbitrary
function, called $\nu(t)$ in this work, and two arbitrary constants.
In Section 6, we gave examples using some simple forms of the seed function
$\nu(t)$; first as powers, then exponentials, powers of $(t_{c}-t)$, and some
others. The first three are chosen to be similar to popular models in FRW
cosmology. The solutions are shown concisely in tables, clearly showing the
relation between the behavior of the model universes and the parameters of the
seed function.
Because of the specialness of the value $W-3/2$ for the BD-theory when one
considers conformal transformations, we considered the problem also in the
Einstein frame in Section 7. For this value, the scalar field becomes non-
dynamical, taking on the arbitrary nature of the function $\nu(t)$ in the
Jordan frame. The solutions found in the Einstein frame are consistent with
those found in the Jordan frame; they can be transformed into each other.
The models found in Section 6 show a wide range of behaviors, featuring Big
Bangs, Big Crunches, Big Rips, Bounces, various singularities of the cigar or
pancake types, etc. While some of these solutions are of theoretical interest
only, there are many expanding-universe solutions with acceleration,
consistent with observational data. For example, all the solutions in Table 3
and all solutions in subsection 6.2 except the few strictly contracting
solutions, feature late-time acceleration.
Even the isotropic special case with power-seed function displays surprising
richness, despite the time-dependence of the scalar field being the same for
all powers. One solution is particularly interesting: It is Minkowski space,
containing a nontrivial scalar field. The particular nonminimal coupling and
the potential selected by the Hessian Determinant condition and the Noether
symmetry make this possible. Given the important guiding role of the concept
of symmetry in modern theoretical physics, we believe that the family of
solutions found and analyzed in this work constitute a potentially more
relevant set among all possible solutions for LRS Bianchi type-I cosmological
models containing a Brans-Dicke field.
## Acknowledgements
This work was supported by Akdeniz University, Scientific Research Projects
Unit.
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|
arxiv-papers
| 2012-04-28T14:23:43 |
2024-09-04T02:49:30.335328
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Kucukakca, U. Camci, \\.I. Semiz",
"submitter": "Ugur Camci",
"url": "https://arxiv.org/abs/1204.6410"
}
|
1204.6452
|
# Optimality of Graphlet Screening in High Dimensional Variable Selection
Jiashun Jin jiashun@stat.cmu.edu
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213, USA Cun-Hui Zhang cunhui@stat.rutgers.edu
Department of Statistics
Rutgers University
Piscataway, NJ 08854, USA Qi Zhang qizhang@stat.wisc.edu
Department of Biostatistics $\&$ Medical Informatics
University of Wisconsin-Madison
Madison, WI 53705, USA
###### Abstract
Consider a linear model $Y=X\beta+\sigma z$, where $X$ has $n$ rows and $p$
columns and $z\sim N(0,I_{n})$. We assume both $p$ and $n$ are large,
including the case of $p\gg n$. The unknown signal vector $\beta$ is assumed
to be sparse in the sense that only a small fraction of its components is
nonzero. The goal is to identify such nonzero coordinates (i.e., variable
selection).
We are primarily interested in the regime where signals are both rare and weak
so that successful variable selection is challenging but is still possible. We
assume the Gram matrix $G=X^{\prime}X$ is sparse in the sense that each row
has relatively few large entries (diagonals of $G$ are normalized to $1$). The
sparsity of $G$ naturally induces the sparsity of the so-called Graph of
Strong Dependence (GOSD). The key insight is that there is an interesting
interplay between the signal sparsity and graph sparsity: in a broad context,
the signals decompose into many small-size components of GOSD that are
disconnected to each other.
We propose Graphlet Screening for variable selection. This is a two-step
Screen and Clean procedure, where in the first step, we screen subgraphs of
GOSD with sequential $\chi^{2}$-tests, and in the second step, we clean with
penalized MLE. The main methodological innovation is to use GOSD to guide both
the screening and cleaning processes.
For any variable selection procedure $\hat{\beta}$, we measure its performance
by the Hamming distance between the sign vectors of $\hat{\beta}$ and $\beta$,
and assess the optimality by the minimax Hamming distance. Compared with more
stringent criterions such as exact support recovery or oracle property, which
demand strong signals, the Hamming distance criterion is more appropriate for
weak signals since it naturally allows a small fraction of errors.
We show that in a broad class of situations, Graphlet Screening achieves the
optimal rate of convergence in terms of the Hamming distance. Unlike Graphlet
Screening, well-known procedures such as the $L^{0}/L^{1}$-penalization
methods do not utilize local graphic structure for variable selection, so they
generally do not achieve the optimal rate of convergence, even in very simple
settings and even if the tuning parameters are ideally set.
The the presented algorithm is implemented as R-CRAN package ScreenClean and
in _matlab_ (available at
http://www.stat.cmu.edu/$\sim$jiashun/Research/software/GS-matlab/).
> Keywords: Asymptotic minimaxity, Graph of Least Favorables (GOLF), Graph of
> Strong Dependence (GOSD), Graphlet Screening (GS), Hamming distance, phase
> diagram, Rare and Weak signal model, Screen and Clean, sparsity.
## 1 Introduction
Consider a linear regression model
$Y=X\beta+\sigma z,\qquad X=X_{n,p},\qquad z\sim N(0,I_{n}).$ (1.1)
We write
$X=[x_{1},x_{2},\ldots,x_{p}],\qquad\mbox{and}\qquad
X^{\prime}=[X_{1},X_{2},\ldots,X_{n}],$ (1.2)
so that $x_{j}$ is the $j$-th design vector and $X_{i}$ is the $i$-th sample.
Motivated by the recent interest in ‘Big Data’, we assume both $p$ and $n$ are
large but $p\geq n$ (though this should not be taken as a restriction). The
vector $\beta$ is unknown to us, but is presumably sparse in the sense that
only a small proportion of its entries is nonzero. Calling a nonzero entry of
$\beta$ a signal, the main interest of this paper is to identify all signals
(i.e., variable selection).
Variable selection is one of the most studied problem in statistics. However,
there are important regimes where our understanding is very limited.
One of such regimes is the rare and weak regime, where the signals are both
rare (or sparse) and individually weak. Rare and weak signals are frequently
found in research areas such as Genome-wide Association Study (GWAS) or next
generation sequencing. Unfortunately, despite urgent demand in applications,
the literature of variable selection has been focused on the regime where the
signals are rare but individually strong. This motivates a revisit to variable
selection, focusing on the rare and weak regime.
For variable selection in this regime, we need new methods and new theoretical
frameworks. In particular, we need a loss function that is appropriate for
rare and weak signals to evaluate the optimality. In the literature, given a
variable selection procedure $\hat{\beta}$, we usually use the probability of
exact recovery $P(\hbox{\rm sgn}(\hat{\beta})\neq\hbox{\rm sgn}(\beta))$ as
the measure of loss (Fan and Li, 2001); $\hbox{\rm sgn}(\hat{\beta})$ and
$\hbox{\rm sgn}(\beta)$ are the sign vectors of $\hat{\beta}$ and $\beta$
respectively. In the rare and weak regime, the signals are so rare and weak
that exact recovery is impossible, and the Hamming distance between $\hbox{\rm
sgn}(\hat{\beta})$ and $\hbox{\rm sgn}(\beta)$ is a more appropriate measure
of loss.
Our focus on the rare and weak regime and the Hamming distance loss provides
new perspectives to variable selection, in methods and in theory.
Throughout this paper, we assume the diagonals of the Gram matrix
$G=X^{\prime}X$ (1.3)
are normalized to $1$ (and approximately $1$ in the random design model),
instead of $n$ as often used in the literature. The difference between two
normalizations is non-essential, but the signal vector $\beta$ are different
by a factor of $n^{1/2}$.
We also assume the Gram matrix $G$ is ‘sparse’ (aka. graph sparsity) in the
sense that each of its rows has relatively few large entries. Signal sparsity
and graph sparsity can be simultaneously found in the following application
areas.
* •
Compressive sensing. We are interested in a very high dimensional sparse
vector $\beta$. The goal is to store or transmit $n$ linear functionals of
$\beta$ and then reconstruct it. For $1\leq i\leq n$, we choose a
$p$-dimensional coefficient vector $X_{i}$ and observe
$Y_{i}=X_{i}^{\prime}\beta+\sigma z_{i}$ with an error $\sigma z_{i}$. The so-
called Gaussian design is often considered (Donoho, 2006a, b; Bajwa et al,
2007), where $X_{i}\stackrel{{\scriptstyle iid}}{{\sim}}N(0,\Omega/n)$ and
$\Omega$ is sparse; the sparsity of $\Omega$ induces the sparsity of
$G=X^{\prime}X$.
* •
Genetic Regulatory Network (GRN). For $1\leq i\leq n$,
$W_{i}=(W_{i}(1),\ldots,W_{i}(p))^{\prime}$ represents the expression level of
$p$ different genes of the $i$-th patient. Approximately,
$W_{i}\stackrel{{\scriptstyle iid}}{{\sim}}N(\alpha,\Sigma)$, where the
contrast mean vector $\alpha$ is sparse reflecting that only few genes are
differentially expressed between a normal patient and a diseased one (Peng et
al, 2009). Frequently, the concentration matrix $\Omega=\Sigma^{-1}$ is
believed to be sparse, and can be effectively estimated in some cases (e.g.,
Bickel and Levina (2008) and Cai et al (2010)), or can be assumed as known in
others, with the so-called “data about data” available (Li and Li, 2011). Let
$\hat{\Omega}$ be a positive-definite estimate of $\Omega$, the setting can be
re-formulated as the linear model
$(\hat{\Omega})^{1/2}Y\approx\Omega^{1/2}Y\sim N(\Omega^{1/2}\beta,I_{p})$,
where $\beta=\sqrt{n}\alpha$ and the Gram matrix $G\approx\Omega$, and both
are sparse.
Other examples can be found in Computer Security (Ji and Jin, 2011) and Factor
Analysis (Fan et al, 2011).
The sparse Gram matrix $G$ induces a sparse graph which we call the Graph of
Strong Dependence (GOSD), denoted by ${\cal G}=(V,E)$, where
$V=\\{1,2,\ldots,p\\}$ and there is an edge between nodes $i$ and $j$ if and
only the design vectors $x_{i}$ and $x_{j}$ are strongly correlated. Let
$S=S(\beta)=\\{1\leq j\leq p:\beta_{j}\neq 0\\}$ (1.4)
be the support of $\beta$ and ${\cal G}_{S}$ be the subgraph of ${\cal G}$
formed by all nodes in $S$. The key insight is that, there is an interesting
interaction between signal sparsity and graph sparsity, which yields the
subgraph ${\cal G}_{S}$ decomposable: ${\cal G}_{S}$ splits into many
“graphlet”; each “graphlet” is a small-size component and different components
are not connected (in ${\cal G}_{S}$).
While we can always decompose ${\cal G}_{S}$ in this way, our emphasis in this
paper is that, in many cases, the maximum size of the graphlets is small; see
Lemma 1 and related discussions.
The decomposability of ${\cal G}_{S}$ motivates a new approach to variable
selection, which we call Graphlet Screening (GS). GS is a Screen and Clean
method (Wasserman and Roeder, 2009). In the screening stage, we use
multivariate screening to identify candidates for all the graphlets. Let
$\hat{S}$ be all the nodes that survived the screening, and let ${\cal
G}_{\hat{S}}$ be the subgraph of GOSD formed by all nodes in $\hat{S}$.
Although $\hat{S}$ is expected to be somewhat larger than $S$, the subgraph
${\cal G}_{\hat{S}}$ is still likely to resemble ${\cal G}_{S}$ in structure
in the sense that it, too, splits into many small-size disconnected
components. We then clean each component separately to remove false positives.
The objective of the paper is two-fold.
* •
To propose a “fundamentally correct” solution in the rare and weak paradigm
along with a computationally fast algorithm for the solution.
* •
To show that GS achieves the optimal rate of convergence in terms of the
Hamming distance, and achieves the optimal phase diagram for variable
selection.
Phase diagram can be viewed as an optimality criterion which is especially
appropriate for rare and weak signals. See Donoho and Jin (2004) and Jin
(2009) for example.
In the settings we consider, most popular approaches are not rate optimal; we
explain this in Sections 1.1-1.3. In Section 1.4, we explain the basic idea of
GS and why it works.
### 1.1 Non-optimality of the $L^{0}$-penalization method for rare and weak
signals
When $\sigma=0$, Model (1.1) reduces to the “noiseless” model $Y=X\beta$. In
this model, Donoho and Stark (1989) (see also Donoho and Huo (2001)) reveals a
fundamental phenomenon on sparse representation. Fix $(X,Y)$ and consider the
equation $Y=X\beta$. Since $p>n$, the equation has infinitely many solutions.
However, a very sparse solution, if exists, is unique under mild conditions on
the design $X$, with all other solutions being much denser. In fact, if the
sparsest solution $\beta_{0}$ has $k$ elements, then all other solutions of
the equation $Y=X\beta$ must have at least $(\mathrm{rank}(X)-k+1)$ nonzero
elements, and $\mathrm{rank}(X)=n$ when $X$ is in a “general position”.
From a practical viewpoint, we frequently believe that this unique sparse
solution is the truth (i.e., Occam’s razor). Therefore, the problem of
variable selection can be solved by some global methods designed for finding
the sparsest solution to the equation $Y=X\beta$.
Since the $L^{0}$-norm is (arguably) the most natural way to measure the
sparsity of a vector, the above idea suggests that the $L^{0}$-penalization
method is a “fundamentally correct” (but computationally intractable) method
for variable selection, provided that some mild conditions hold (noiseless,
Signal-to-Noise Ratio (SNR) is high, signals are sufficiently sparse (Donoho
and Stark, 1989; Donoho and Huo, 2001)).
Motivated by this, in the past two decades, a long list of computationally
tractable algorithms have been proposed that approximate the solution of the
$L^{0}$-penalization method, including the lasso, SCAD, MC+, and many more
(Akaike, 1974; Candes and Tao, 2007; Efron et al, 2004; Fan and Li, 2001;
Schwarz, 1978; Tibshirani, 1996; Zhang, 2010, 2011; Zhao and Yu, 2006; Zou,
2006).
With that being said, we must note that these methodologies were built upon a
framework with four tightly woven core components: “signals are rare but
strong”, “the truth is also the sparsest solution to $Y=X\beta$”, “probability
of exact recovery is an appropriate loss function”, and “$L^{0}$-penalization
method is a fundamentally correct approach”. Unfortunately, when signals are
rare and weak, such a framework is no longer suitable.
* •
When signals are “rare and weak”, the fundamental uniqueness property of the
sparse solution in the noiseless case is no longer valid in the noisy case.
Consider the model $Y=X\beta+\sigma z$ and suppose that a sparse $\beta_{0}$
is the true signal vector. There are many vectors $\beta$ that are small
perturbations of $\beta_{0}$ such that the two models $Y=X\beta+\sigma z$ and
$Y=X\beta_{0}+\sigma z$ are indistinguishable (i.e., all tests are
asymptotically powerless). In the “rare and strong” regime, $\beta_{0}$ is the
sparsest solution among all such “eligible” solutions of $Y=X\beta+\sigma z$.
However, this claim no longer holds in the “rare and weak” regime and the
principle of Occam’s razor may not be as relevant as before.
* •
The $L^{0}$-penalization method is originally designed for “rare and strong”
signals where “exact recovery” is used to measure its performance (Donoho and
Stark, 1989; Donoho and Huo, 2001; Donoho, 2006a). When we must consider “rare
and weak” signals and when we use the Hamming distance as the loss function,
it is unclear whether the $L^{0}$-penalization method is still “fundamentally
correct”.
In fact, in Section 2.8 (see also Ji and Jin (2011)), we show that the
$L^{0}$-penalization method is not optimal in Hamming distance when signals
are rare and weak, even with very simple designs (i.e., Gram matrix is
tridiagonal or block-wise) and even when the tuning parameter is ideally set.
Since the $L^{0}$-penalization method is used as the benchmark in the
development of many other penalization methods, its sub-optimality is expected
to imply the sub-optimality of other methods designed to match its performance
(e.g., lasso, SCAD, MC+).
### 1.2 Limitation of Univariate Screening and UPS
Univariate Screening (also called marginal regression or Sure Screening (Fan
and Lv, 2008; Genovese et al, 2012)) is a well-known variable selection
method. For $1\leq j\leq p$, recall that $x_{j}$ is the $j$-th column of $X$.
Univariate Screening selects variables with large marginal correlations:
$|(x_{j},Y)|$, where $(\cdot,\cdot)$ denotes the inner product. The method is
computationally fast, but it can be seriously corrupted by the so-called
phenomenon of “signal cancellation” (Wasserman and Roeder, 2009). In our model
(1.1)-(1.3), the SNR associated with $(x_{j},Y)$ is
$\frac{1}{\sigma}\sum_{\ell=1}^{p}(x_{j},x_{\ell})\beta_{\ell}=\frac{\beta_{j}}{\sigma}+\frac{1}{\sigma}\sum_{\ell\neq
j}(x_{j},x_{\ell})\beta_{\ell}.$ (1.5)
“Signal cancellation” happens if SNR is significantly smaller than
$\beta_{j}/\sigma$. For this reason, the success of Univariate Screening needs
relatively strong conditions (e.g., Faithfulness Condition (Genovese et al,
2012)), under which signal cancellation does not have a major effect.
In Ji and Jin (2011), Ji and Jin proposed Univariate Penalized Screening (UPS)
as a refinement of Univariate Screening, where it was showed to be optimal in
the rare and weak paradigm, for the following two scenarios. The first
scenario is where the nonzero effects of variables are all positively
correlated: $(x_{j}\beta_{j})^{\prime}(x_{k}\beta_{k})\geq 0$ for all
$\\{j,k\\}$. This guarantees the faithfulness of the univariate association
test. The second scenario is a Bernoulli model where the “signal cancellation”
only has negligible effects over the Hamming distance of UPS.
With that being said, UPS attributes its success mostly to the cleaning stage;
the screening stage of UPS uses nothing but Univariate Screening, so UPS does
not adequately address the challenge of “signal cancellation”. For this
reason, we should not expect UPS to be optimal in much more general settings.
### 1.3 Limitations of Brute-force Multivariate Screening
One may attempt to overcome “signal cancellation” by multivariate screening,
with Brute-force Multivariate Screening (BMS) being the most straightforward
version. Fix an integer $1\leq m_{0}\ll p$. BMS consists of a series of
screening phases, indexed by $m$, $1\leq m\leq m_{0}$, that are increasingly
more ambitious. In Phase-$m$ BMS, we test the significance of the association
between $Y$ and any set of $m$ different design variables
$\\{x_{j_{1}},x_{j_{2}},\ldots,x_{j_{m}}\\}$, $j_{1}<j_{2}<\ldots<j_{m}$, and
retain all such design variables if the test is significant. The problem of
BMS is, it enrolls too many candidates for screening, which is both
unnecessary and unwise.
* •
(Screening inefficiency). In Phase-$m$ of BMS, we test about $\binom{p}{m}$
hypotheses involving different subsets of $m$ design variables. The larger the
number of hypotheses we consider, the higher the threshold we need to set for
the tests, in order to control the false positives. When we enroll too many
candidates for hypothesis testing, we need signals that are stronger than
necessary in order for them to survive the screening.
* •
(Computational challenge). Testing $\binom{p}{m}$ hypotheses is
computationally infeasible when $p$ is large, even when $m$ is very small
(say, $(p,m)=(10^{4},3)$).
### 1.4 Graphlet Screening: how it is different and how it works
Graphlet Screening (GS) uses a similar screening strategy as BMS does, except
for a major difference. When it comes to the test of significance between $Y$
and design variables $\\{x_{j_{1}},x_{j_{2}},\ldots,x_{j_{m}}\\}$,
$j_{1}<j_{2}<\ldots<j_{m}$, GS only carries out such a test if
$\\{j_{1},j_{2},\ldots,j_{m}\\}$ is a connected subgraph of the GOSD.
Otherwise, the test is safely skipped!
Fixing an appropriate threshold $\delta>0$, we let $\Omega^{*,\delta}$ be the
regularized Gram matrix:
$\Omega^{*,\delta}(i,j)=G(i,j)1\\{|G(i,j)|\geq\delta\\},\qquad 1\leq i,j\leq
p.$ (1.6)
The GOSD ${\cal G}\equiv{\cal G}^{*,\delta}=(V,E)$ is the graph where
$V=\\{1,2,\ldots,p\\}$ and there is an edge between nodes $i$ and $j$ if and
only if $\Omega^{*,\delta}(i,j)\neq 0$. See Section 2.6 for the choice of
$\delta$.
Remark. GOSD and ${\cal G}$ are generic terms which vary from case to case,
depending on $G$ and $\delta$. GOSD is very different from the Bayesian
conditional independence graphs (Pearl, 2000).
Fixing $m_{0}\geq 1$ as in BMS, we define
${\cal A}(m_{0})={\cal A}(m_{0};G,\delta)=\\{\mbox{all connected subgraphs of
${\cal G}^{*,\delta}$ with size $\leq m_{0}$}\\}.$ (1.7)
GS is a Screen and Clean method, consisting of a graphical screening step (GS-
step) and a graphical cleaning step (GC-step).
* •
$GS$-step. We test the significance of association between $Y$ and
$\\{x_{j_{1}},x_{j_{2}},\ldots,x_{j_{m}}\\}$ if and only if
$\\{j_{1},j_{2},\ldots,j_{m}\\}\in{\cal A}(m_{0})$ (i.e., graph guided
multivariate screening). Once $\\{j_{1},\ldots,j_{m}\\}$ is retained, it
remains there until the end of the GS-step.
* •
$GC$-step. The set of surviving nodes decompose into many small-size
components, which we fit separately using an efficient low-dimensional test
for small graphs.
GS is similar to Wasserman and Roeder (2009) for both of them have a screening
and a cleaning stage, but is more sophisticated. For clarification, note that
Univariate Screening or BMS introduced earlier does not contain a cleaning
stage and can be viewed as a counterpart of the GS-step.
We briefly explain why GS works. We discuss the GS-step and GC-step
separately.
Consider the GS-step first. Compared with BMS, the GS-step recruits far fewer
candidates for screening, so it is able to overcome the two major shortcomings
of BMS aforementioned: high computational cost and low statistical efficiency.
In fact, fix $K\geq 1$ and suppose ${\cal G}^{*,\delta}$ is $K$-sparse (see
Section 1.5 for the definition). By a well-known result in graph theory
(Frieze and Molloy, 1999),
$|{\cal A}(m_{0})|\leq Cm_{0}p(eK)^{m_{0}}.$ (1.8)
The right hand side is much smaller than the term $\binom{p}{m_{0}}$ as we
encounter in BMS.
At the same time, recall that $S=S(\beta)$ is the support of $\beta$. Let
${\cal G}_{S}\equiv{\cal G}_{S}^{*,\delta}$ be the subgraph of ${\cal
G}^{*,\delta}$ consisting all signal nodes. We can always split ${\cal
G}_{S}^{*,\delta}$ into “graphlets” (arranged lexicographically) as follows:
${\cal G}_{S}^{*,\delta}={\cal G}_{S,1}^{*,\delta}\cup{\cal
G}_{S,2}^{*,\delta}\ldots\cup{\cal G}_{S,M}^{*,\delta},$ (1.9)
where each ${\cal G}_{S,i}^{*,\delta}$ is a component (i.e., a maximal
connected subgraph) of ${\cal G}_{S}^{*,\delta}$, and different ${\cal
G}_{S,i}^{*,\delta}$ are not connected in ${\cal G}_{S}^{*,\delta}$. Let
$m_{0}^{*}=m_{0}^{*}(S(\beta),G,\delta)=\max_{1\leq i\leq M}|{\cal
G}_{S,i}^{*,\delta}|$ (1.10)
be the maximum size of such graphlets (note that $M$ also depends on
($S(\beta)$, $G$, $\delta$)).
In many cases, $m_{0}^{*}$ is small. One such case is when we have a Bernoulli
signal model.
###### Lemma 1
Fix $K\geq 1$ and $\epsilon>0$. If ${\cal G}^{*,\delta}$ is $K$-sparse and
$\hbox{\rm sgn}(|\beta_{1}|),\hbox{\rm sgn}(|\beta_{2}|),\ldots,\hbox{\rm
sgn}(|\beta_{p}|)$ are $iid$ from $\mathrm{Bernoulli}(\epsilon)$, then except
for a probability $p(e\epsilon K)^{m_{0}+1}$,
$m_{0}^{*}(S(\beta),G,\delta)\leq m_{0}$.
Lemma 1 is not tied to the Bernoulli model and holds more generally. For
example, it holds when $\\{\hbox{\rm sgn}(|\beta_{i}|)\\}_{i=1}^{p}$ are
generated according to certain Ising models (Ising, 1925).
We recognize that in order for the GS-step to be efficient both in screening
and in computation, it is sufficient that
$m_{0}\geq m_{0}^{*}.$ (1.11)
In fact, first, if (1.11) holds, then for each $1\leq\ell\leq M$, ${\cal
G}_{S,\ell}^{*,\delta}\in{\cal A}(m_{0})$. Therefore, at some point of the
screening process of the GS-step, we must have considered a significance test
between $Y$ and the set of design variables $\\{x_{j}:j\in{\cal
G}_{S,\ell}^{*,\delta}\\}$. Consequently, the GS-step is able to overcome the
“signal cancellations” (the explanation is a little bit long, and we slightly
defer it). Second, since $m_{0}^{*}$ is small in many situations, we could
choose a relatively small $m_{0}$ such that (1.11) holds. When $m_{0}$ is
small, as long as $K$ is small or moderately large, the GS-step is
computationally feasible. In fact, the right hand side of (1.8) is only larger
than $p$ by a moderate factor. See Section 2.2 for more discussion on the
computation complexity.
We now explain the first point above. The notations below are frequently used.
###### Definition 2
For $X$ in Models (1.1)-(1.2) and any subset ${\cal I}\subset\\{1,2,...,p\\}$,
let $P^{{\cal I}}=P^{{\cal I}}(X)$ be the projection from $\mathbb{R}^{n}$ to
the subspace spanned by $\\{x_{j}:j\in{\cal I}\\}$.
###### Definition 3
For an $n\times p$ matrix $A$ and sets ${\cal I}\subset\\{1,\ldots,n\\}$ and
${\cal J}\subset\\{1,\ldots,p\\}$, $A^{{\cal I},{\cal J}}$ is the $|{\cal
I}|\times|{\cal J}|$ sub-matrix formed by restricting the rows of $A$ to
${\cal I}$ and columns to ${\cal J}$.
When $p=1$, $A$ is a vector, and $A^{{\cal I}}$ is the sub-vector of $A$
formed by restricting the rows of $A$ to ${\cal I}$ . When ${\cal
I}=\\{1,2,\ldots,n\\}$ (or ${\cal J}=\\{1,2,\ldots,p\\}$), we write $A^{{\cal
I},{\cal J}}$ as $A^{\otimes,{\cal J}}$ (or $A^{{\cal I},\otimes}$). Note that
indices in ${\cal I}$ or ${\cal J}$ are not necessarily sorted ascendingly.
Recall that for each $1\leq\ell\leq M$, at some point of the GS-step, we must
have considered a significance test between $Y$ and the set of design
variables $\\{x_{j}:j\in{\cal G}_{S,\ell}^{*,\delta}\\}$. By (1.9), we rewrite
Model (1.1) as
$Y=\sum_{\ell=1}^{M}X^{\otimes,{\cal G}_{S,\ell}^{*,\delta}}\beta^{{\cal
G}_{S,\ell}^{*,\delta}}+\sigma z,\qquad z\sim N(0,I_{n}).$
The key is the set of matrices $\\{X^{\otimes,{\cal
G}_{S,\ell}^{*,\delta}}:1\leq\ell\leq M\\}$ are nearly orthogonal (i.e., for
any column $\xi$ of $X^{\otimes,{\cal G}_{S,k}^{*,\delta}}$ and any column
$\eta$ of $X^{\otimes,{\cal G}_{S,\ell}^{*,\delta}}$, $|(\xi,\eta)|$ is small
when $k\neq\ell$).
When we test the significance between $Y$ and $\\{x_{j},j\in{\cal
G}_{S,\ell}^{*,\delta}\\}$, we are testing the null hypothesis $\beta^{{\cal
G}_{S,\ell}^{*,\delta}}=0$ against the alternative $\beta^{{\cal
G}_{S,\ell}^{*,\delta}}\neq 0$. By the near orthogonality aforementioned,
approximately, $(X^{\otimes,{\cal G}_{S,\ell}^{*,\delta}})^{\prime}Y$ is a
sufficient statistic for $\beta^{{\cal G}_{S,\ell}^{*,\delta}}$, and the
optimal test is based on the $\chi^{2}$-test statistic $\|P^{{\cal
G}_{S,\ell}^{*,\delta}}Y\|^{2}$.
The near orthogonality also implies that significant “signal cancellation”
only happens among signals within the same graphlet. When we screen each
graphlet as a whole using the $\chi^{2}$-statistic above, “signal
cancellation” between different graphlets only has negligible effects. In this
way, GS-step is able to retain all nodes in ${\cal G}_{S,\ell}^{*,\delta}$ in
a nearly optimal way, and so overcome the challenge of “signal cancellation”.
This explains the first point.
Note that the GS-step consists of a sequence of sub-steps, each sub-step is
associated with an element of ${\cal A}(m_{0})$. When we screen ${\cal
G}_{S,\ell}^{*,\delta}$ as a whole, it is possible some of the nodes have
already been retained in the previous sub-steps. In this case, we implement
the $\chi^{2}$-test slightly differently, but the insight is similar. See
Section 2.1 for details.
We now discuss the GC-step. Let $\hat{S}$ be all the surviving nodes of the
GS-step, and let ${\cal G}_{\hat{S}}^{*,\delta}$ be the subgraph of ${\cal
G}^{*,\delta}$ formed by confining all nodes to $\hat{S}$. Similarly, we have
(a) the decomposition ${\cal G}_{\hat{S}}^{*,\delta}={\cal
G}_{\hat{S},1}^{*,\delta}\cup{\cal G}_{\hat{S},2}^{*,\delta}\ldots\cup{\cal
G}_{\hat{S},\hat{M}}^{*,\delta}$, (b) the near orthogonality between the
$\hat{M}$ different matrices, each is formed by $\\{x_{j}:j\in{\cal
G}_{\hat{S},\ell}^{*,\delta}\\}$. Moreover, a carefully tuned screening stage
of the GS ensures that most of the components ${\cal
G}_{\hat{S},\ell}^{*,\delta}$ are only small perturbations of their
counterparts in the decomposition of ${\cal G}_{S}^{*,\delta}={\cal
G}_{S,1}^{*,\delta}\cup{\cal G}_{S,2}^{*,\delta}\ldots\cup{\cal
G}_{S,M}^{*,\delta}$ as in (1.9), and the maximum size of ${\cal
G}_{\hat{S},\ell}^{*,\delta}$ is not too much larger than
$m_{0}^{*}=m_{0}^{*}(S(\beta),G,\delta)$. Together, these allow us to clean
${\cal G}_{\hat{S},\ell}^{*,\delta}$ separately, without much loss of
efficiency. Since the maximum size of ${\cal G}_{\hat{S},\ell}^{*,\delta}$ is
small, the computational complexity in the cleaning stage is moderate.
### 1.5 Content
The remaining part of the paper is organized as follows. In Section 2, we show
that GS achieves the minimax Hamming distance in the Asymptotic Rare and Weak
(ARW) model, and use the phase diagram to visualize the optimality of GS, and
to illustrate the advantage of GS over the $L^{0}/L^{1}$-penalization methods.
In Section 3, we explain that GS attributes its optimality to the so-called
Sure Screening property and the Separable After Screening property, and use
these two properties to prove our main result, Theorem 8. Section 4 contains
numeric results, Section 5 discusses more connections to existing literature
and possible extensions of GS, and Section 6 contains technical proofs.
Below are some notations we use in this paper. $L_{p}$ denotes a generic
multi-$\log(p)$ term that may vary from occurrence to occurrence; see
Definition 5. For a vector $\beta\in R^{p}$, $\|\beta\|_{q}$ denotes the
$L^{q}$-norm, and when $q=2$, we drop $q$ for simplicity. For two vectors
$\alpha$ and $\beta$ in $R^{p}$, $\alpha\circ\beta\in R^{p}$ denotes the
vector in $R^{p}$ that satisfies $(\alpha\circ\beta)_{i}=\alpha_{i}\beta_{i}$,
$1\leq i\leq p$; “$\circ"$ is known as the Hadamard product.
For an $n\times p$ matrix $A$, $\|A\|_{\infty}$ denotes the matrix
$L^{\infty}$-norm, and $\|A\|$ denotes the spectral norm (Horn and Johnson,
1990). Recall that for two sets ${\cal I}$ and ${\cal J}$ such that ${\cal
I}\subset\\{1,2,\ldots,n\\}$ and ${\cal J}\subset\\{1,2,\ldots,p\\}$,
$A^{{\cal I},{\cal J}}$ denotes the submatrix of $A$ formed by restricting the
rows and columns of $A$ to ${\cal I}$ and ${\cal J}$, respectively. Note that
the indices in ${\cal I}$ and ${\cal J}$ are not necessarily sorted in the
ascending order. In the special case where ${\cal I}=\\{1,2,\ldots,n\\}$ (or
${\cal J}=\\{1,2,\ldots,p\\}$), we write $A^{{\cal I},{\cal J}}$ as
$A^{\otimes,{\cal J}}$ (or $A^{{\cal I},\otimes}$). In the special case where
$n=p$ and $A$ is positive definite, $\lambda_{k}^{*}(A)$ denotes the minimum
eigenvalue of all the size $k$ principal submatrices of $A$, $1\leq k\leq p$.
For $X$ in (1.1), $P^{{\cal I}}$ denotes the projection to the column space of
$X^{\otimes,{\cal I}}$.
Recall that in Model (1.1), $Y=X\beta+\sigma z$. Fixing a threshold
$\delta>0$. Let $G=X^{\prime}X$ be the Gram matrix, and let
$\Omega^{*,\delta}$ be the regularized Gram matrix defined by
$\Omega^{*,\delta}(i,j)=G(i,j)1\\{|G(i,j)|\geq\delta\\}$, $1\leq i,j\leq p$.
Let ${\cal G}^{*,\delta}$ be the graph where each index in
$\\{1,2,\ldots,p\\}$ is a node, and there is an edge between node $i$ and node
$j$ if and only if $\Omega^{*,\delta}(i,j)\neq 0$. We let $S(\beta)$ be the
support of $\beta$, and denote ${\cal G}_{S}^{*,\delta}$ by the subgraph of
${\cal G}^{*,\delta}$ formed by all nodes in $S(\beta)$. We call ${\cal
G}^{*,\delta}$ the Graph of Strong Dependence (GOSD) and sometimes write it by
${\cal G}$ for short. The GOSD and ${\cal G}$ are generic notations which
depend on $(G,\delta)$ and may vary from occurrence to occurrence. We also
denote ${\cal G}^{\diamond}$ by the Graph of Least Favorable (GOLF). GOLF only
involves the study of the information lower bound. For an integer $K\geq 1$, a
graph ${\cal G}$, and one of its subgraph ${\cal I}_{0}$, we write ${\cal
I}_{0}\triangleleft{\cal G}$ if and only if ${\cal I}_{0}$ is a component of
${\cal G}$ (i.e., a maximal connected subgraph of ${\cal G}$), and we call
${\cal G}$ $K$-sparse if its maximum degree is no greater than $K$.
## 2 Main results
In Section 2.1, we formally introduce GS. In Section 2.2, we discuss the
computational complexity of GS. In Sections 2.3-2.6, we show that GS achieves
the optimal rate of convergence in the Asymptotic Rare and Weak model. In
Sections 2.7-2.8, we introduce the notion of phase diagram and use it to
compare GS with the $L^{0}/L^{1}$-penalization methods. We conclude the
section with a summary in Section 2.9.
### 2.1 Graphlet Screening: the procedure
GS consists of a GS-step and a GC-step. We describe two steps separately.
Consider the $GS$-step first. Fix $m_{0}\geq 1$ and $\delta>0$, recall that
${\cal G}^{*,\delta}$ denotes the GOSD and ${\cal A}(m_{0})$ consists of all
connected subgraphs of ${\cal G}^{*,\delta}$ with size $\leq m_{0}$.
* •
Initial sub-step. Let ${\cal U}_{p}^{*}=\emptyset$. List all elements in
${\cal A}(m_{0})$ in the ascending order of the number of nodes it contains,
with ties broken lexicographically. Since a node is thought of as connected to
itself, the first $p$ connected subgraphs on the list are simply the nodes
$1,2,\ldots,p$. We screen all connected subgraphs in the order they are
listed.
* •
Updating sub-step. Let ${\cal I}_{0}$ be the connected subgraph under
consideration, and let ${\cal U}_{p}^{*}$ be the current set of retained
indices. We update ${\cal U}_{p}^{*}$ with a $\chi^{2}$ test as follows. Let
$\hat{F}={\cal I}_{0}\cap{\cal U}_{p}^{*}$ and $\hat{D}={\cal
I}_{0}\setminus{\cal U}_{p}^{*}$, so that $\hat{F}$ is the set of nodes in
${\cal I}_{0}$ that have already been accepted, and $\hat{D}$ is the set of
nodes in ${\cal I}_{0}$ that is currently under investigation. Note that no
action is needed if $\hat{D}=\emptyset$. For a threshold
$t(\hat{D},\hat{F})>0$ to be determined, we update ${\cal U}_{p}^{*}$ by
adding all nodes in $\hat{D}$ to it if
$T(Y,\hat{D},\hat{F})=\|P^{{\cal
I}_{0}}Y\|^{2}-\|P^{\hat{F}}Y\|^{2}>t(\hat{D},\hat{F}),$ (2.12)
and we keep ${\cal U}_{p}^{*}$ the same otherwise (by default,
$\|P^{\hat{F}}Y\|=0$ if $\hat{F}=\emptyset$). We continue this process until
we finish screening all connected subgraphs on the list. The final set of
retained indices is denoted by ${\cal U}_{p}^{*}$.
See Table $1$ for a recap of the procedure. In the $GS$-step, once a node is
kept in any sub-stage of the screening process, it remains there until the end
of the $GS$-step (however, it may be killed in the $GC$-step). This has a
similar flavor to that of the Forward regression.
In principle, the procedure depends on how the connected subgraphs of the same
size are initially ordered, and different ordering could give different
numeric results. However, such differences are usually negligibly small.
Alternatively, one could revise the procedure so that it does not depend on
the ordering. For example, in the updating sub-step, we could choose to update
${\cal U}_{p}^{*}$ only when we finish screening all connected sub-graphs of
size $k$, $1\leq k\leq m_{0}$. While the theoretic results below continue to
hold if we revise GS in this way, we must note that from a numeric
perspective, the revision would not produce a very different result. For
reasons of space, we skip discussions along this line.
The GS-step uses a set of tuning parameters:
${\cal Q}\equiv\\{t(\hat{D},\hat{F}):\mbox{$(\hat{D},\hat{F})$ are as defined
in (\ref{DefineDF})}\\}.$
A convenient way to set these parameters is to let
$t(\hat{D},\hat{F})=2\sigma^{2}q\log p$ for a fixed $q>0$ and all
$(\hat{D},\hat{F})$. More sophisticated choices are given in Section 2.6.
The $GS$-step has two important properties: Sure Screening and Separable After
Screening (SAS). With tuning parameters ${\cal Q}$ properly set, the Sure
Screening property says that ${\cal U}_{p}^{*}$ retains all but a negligible
fraction of the signals. The SAS property says that as a subgraph of ${\cal
G}^{*,\delta}$, ${\cal U}_{p}^{*}$ decomposes into many disconnected
components, each has a size $\leq\ell_{0}$ for a fixed small integer
$\ell_{0}$. Together, these two properties enable us to reduce the original
large-scale regression problem to many small-size regression problems that can
be solved parallelly in the $GC$-step. See Section 3 for elaboration on these
ideas.
We now discuss the $GC$-step. For any $1\leq j\leq p$, we have either
$j\notin{\cal U}_{p}^{*}$, or that there is a unique connected subgraph ${\cal
I}_{0}$ such that $j\in{\cal I}_{0}\lhd{\cal U}_{p}^{*}$. In the first case,
we estimate $\beta_{j}$ as $0$. In the second case, for two tuning parameters
$u^{gs}>0$ and $v^{gs}>0$, we estimate the whole set of variables
$\beta^{{\cal I}_{0}}$ by minimizing the functional
$\|P^{{\cal I}_{0}}(Y-X^{\otimes,{\cal
I}_{0}}\xi)\|^{2}+(u^{gs})^{2}\|\xi\|_{0}$ (2.13)
over all $|{\cal I}_{0}|\times 1$ vectors $\xi$, each nonzero coordinate of
which $\geq v^{gs}$ in magnitude. The resultant estimator is the final
estimate of GS which we denote by
$\hat{\beta}^{gs}=\hat{\beta}^{gs}(Y;\delta,{\cal Q},u^{gs},v^{gs},X,p,n)$.
See Section 1.5 for notations used in this paragraph.
Table 1: Graphlet Screening Algorithm. $GS$-step: | List ${\cal G}^{*,\delta}$-connected submodels ${\cal I}_{0,k}$ with $|{\cal I}_{0,1}|\leq|{\cal I}_{0,2}|\leq\cdots\leq m_{0}$
---|---
| Initialization: ${\cal U}_{p}^{*}=\emptyset$ and $k=1$
| Test $H_{0}:{\cal I}_{0,k}\cap{\cal U}_{p}^{*}$ against $H_{1}:{\cal
I}_{0,k}$ with $\chi^{2}$ test (2.12)
| Update: ${\cal U}_{p}^{*}\leftarrow{\cal U}_{p}^{*}\cup{\cal I}_{0,k}$ if
$H_{0}$ rejected, $k\leftarrow k+1$
$GC$-step: | As a subgraph of ${\cal G}^{*,\delta}$, ${\cal U}_{p}^{*}$ decomposes into many components ${\cal I}_{0}$
| Use the $L^{0}$-penalized test (2.13) to select a subset ${\hat{\cal
I}}_{0}$ of each ${\cal I}_{0}$
| Return the union of ${\hat{\cal I}}_{0}$ as the selected model
Sometimes for linear models with random designs, the Gram matrix $G$ is very
noisy, and GS is more effective if we use it iteratively for a few times
($\leq 5$). This can be implemented in a similar way as that in Ji and Jin
(2011, Section 3). Here, the main purpose of iteration is to denoise $G$, not
for variable selection. See Ji and Jin (2011, Section 3) and Section 4 for
more discussion.
### 2.2 Computational complexity
If we exclude the overhead of obtaining ${\cal G}^{*,\delta}$, then the
computation cost of GS contains two parts, that of the GS-step and that of the
GC-step. In each part, the computation cost hinges on the sparsity of ${\cal
G}^{*,\delta}$. In Section 2.3, we show that with a properly chosen $\delta$,
for a wide class of design matrices, ${\cal G}^{*,\delta}$ is $K$-sparse for
some $K=K_{p}\leq C\log^{\alpha}(p)$ as $p\rightarrow\infty$, where $\alpha>0$
is a constant. As a result (Frieze and Molloy, 1999),
$|{\cal A}(m_{0})|\leq pm_{0}(eK_{p})^{m_{0}}\leq
Cm_{0}p\log^{m_{0}\alpha}(p).$ (2.14)
We now discuss two parts separately.
In the GS-step, the computation cost comes from that of listing all elements
in ${\cal A}(m_{0})$, and that of screening all connected-subgraphs in ${\cal
A}(m_{0})$. Fix $1\leq k\leq m_{0}$. By (2.14) and the fact that every size
$k$ ($k>1$) connected subgraph at least contains one size $k-1$ connected
subgraph, greedy algorithm can be used to list all sub-graphs with size $k$
with computational complexity $\leq Cp(K_{p}k)^{k}\leq Cp\log^{k\alpha}(p)$,
and screening all connected subgraphs of size $k$ has computational complexity
$\leq Cnp\log^{k\alpha}(p)$. Therefore, the computational complexity of the
GS-step $\leq Cnp(\log(p))^{(m_{0}+1)\alpha}$.
The computation cost of the $GC$-step contains the part of breaking ${\cal
U}_{p}^{*}$ into disconnected components, and that of cleaning each component
by minimizing (2.13). As a well-known application of the breadth-first search
(Hopcroft and Tarjan, 1973), the first part $\leq|{\cal U}_{p}^{*}|(K_{p}+1)$.
For the second part, by the SAS property of the $GS$-step (i.e., Lemma 16),
for a broad class of design matrices, with the tuning parameters chosen
properly, there is a fixed integer $\ell_{0}$ such that with overwhelming
probability, $|{\cal I}_{0}|\leq\ell_{0}$ for any ${\cal I}_{0}\lhd{\cal
U}_{p}^{*}$. As a result, the total computational cost of the $GC$-step is no
greater than $C(2^{\ell_{0}}\log^{\alpha}(p))|{\cal U}_{p}^{*}|n$, which is
moderate.
The computational complexity of GS is only moderately larger than that of
Univariate Screening or UPS (Ji and Jin, 2011). UPS uses univariate
thresholding for screening which has a computational complexity of $O(np)$,
and GS implements multivariate screening for all connected subgraphs in ${\cal
A}(m_{0})$, which has a computational complexity $\leq
Cnp(\log(p))^{(m_{0}+1)\alpha}$. The latter is only larger by a
multi-$\log(p)$ term.
### 2.3 Asymptotic Rare and Weak model and Random Design model
To analyze GS, we consider the regression model $Y=X\beta+\sigma z$ as in
(1.1), and use an Asymptotic Rare and Weak (ARW) model for $\beta$ and a
random design model for $X$.
We introduce the ARW first. Fix parameters $\epsilon\in(0,1)$, $\tau>0$, and
$a\geq 1$. Let $b=(b_{1},\ldots,b_{p})^{\prime}$ be the $p\times 1$ random
vector where
$b_{i}\stackrel{{\scriptstyle iid}}{{\sim}}\mathrm{Bernoulli}(\epsilon).$
(2.15)
We model the signal vector $\beta$ in Model (1.1) by
$\beta=b\circ\mu,$ (2.16)
where “$\circ$” denotes the Hadamard product (see Section 1.5) and
$\mu\in\Theta_{p}^{*}(\tau,a)$, with
$\Theta_{p}^{*}(\tau,a)=\\{\mu\in\Theta_{p}(\tau),\|\mu\|_{\infty}\leq
a\tau\\},\qquad\Theta_{p}(\tau)=\\{\mu\in\mathbb{R}^{p}:\,|\mu_{i}|\geq\tau,1\leq
i\leq p\\}.$ (2.17)
In this model, $\epsilon$ calibrates the sparsity level and $\tau$ calibrates
the minimum signal strength. We are primarily interested in the case where
$\epsilon$ is small and $\tau$ is smaller than the required signal strength
for the exact recovery of the support of $\beta$, so the signals are both rare
and weak. The constraint of $\|\mu\|_{\infty}\leq a\tau_{p}$ is mainly for
technical reasons (only needed for Lemma 16); see Section 2.6 for more
discussions.
We let $p$ be the driving asymptotic parameter, and tie $(\epsilon,\tau)$ to
$p$ through some fixed parameters. In detail, fixing $0<\vartheta<1$, we model
$\epsilon=\epsilon_{p}=p^{-\vartheta}.$ (2.18)
For any fixed $\vartheta$, the signals become increasingly sparser as
$p\rightarrow\infty$. Also, as $\vartheta$ ranges, the sparsity level ranges
from very dense to very sparse, and covers all interesting cases.
It turns out that the most interesting range for $\tau$ is
$\tau=\tau_{p}=O(\sqrt{\log(p)})$. In fact, when
$\tau_{p}\ll\sigma\sqrt{\log(p)}$, the signals are simply too rare and weak so
that successful variable selection is impossible. On the other hand, exact
support recovery requires $\tau\gtrsim\sigma\sqrt{2\log p}$ for orthogonal
designs and possibly even larger $\tau$ for correlated designs. In light of
this, we fix $r>0$ and calibrate $\tau$ by
$\tau=\tau_{p}=\sigma\sqrt{2r\log(p)}.$ (2.19)
Next, consider the random design model. The use of random design model is
mainly for simplicity in presentation. The main results in the paper can be
translated to fixed design models with a careful modification of the
notations; see Corollary 7 and Section 5.
For any positive definite matrix $A$, let $\lambda(A)$ be the smallest
eigenvalue, and let
$\lambda_{k}^{*}(\Omega)=\min\\{\lambda(A):\mbox{$A$ is a $k\times k$
principle submatrix of $\Omega$}\\}.$ (2.20)
For $m_{0}$ as in the GS-step, let $g=g(m_{0},\vartheta,r)$ be the smallest
integer such that
$g\geq\max\\{m_{0},(\vartheta+r)^{2}/(2\vartheta r)\\}.$ (2.21)
Fixing a constant $c_{0}>0$, introduce
${\cal M}_{p}(c_{0},g)=\\{\Omega:\mbox{$p\times p$ correlation matrix,
$\lambda_{g}^{*}(\Omega)\geq c_{0}$}\\}.$ (2.22)
Recall $X_{i}$ is the $i$-th row of $X$; see (1.2). In the random design
model, we fix an $\Omega\in{\cal M}(c_{0},g)$ ($\Omega$ is unknown to us), and
assume
$X_{i}\stackrel{{\scriptstyle iid}}{{\sim}}N(0,\frac{1}{n}\Omega),\qquad 1\leq
i\leq n.$ (2.23)
In the literature, this is called the Gaussian design, which can be found in
Compressive Sensing (Bajwa et al, 2007), Computer Security (Dinur and Nissim,
2003), and other application areas.
At the same time, fixing $\kappa\in(0,1)$, we model the sample size $n$ by
$n=n_{p}=p^{\kappa}.$ (2.24)
As $p\rightarrow\infty$, $n_{p}$ becomes increasingly large but is still much
smaller than $p$. We assume
$\kappa>(1-\vartheta),$ (2.25)
so that $n_{p}\gg p\epsilon_{p}$. Note $p\epsilon_{p}$ is approximately the
total number of signals. Condition (2.25) is almost necessary for successful
variable selection (Donoho, 2006a, b).
###### Definition 4
We call model (2.15)-(2.19) for $\beta$ the Asymptotic Rare Weak model
$\mathrm{ARW}(\vartheta,r,a,\mu)$, and call Model (2.23)-(2.25) for $X$ the
Random Design model $\mathrm{RD}(\vartheta,\kappa,\Omega)$.
### 2.4 Minimax Hamming distance
In many works on variables selection, one assesses the optimality by the
‘oracle property’, where the probability of non-exact recovery $P(\hbox{\rm
sgn}(\hat{\beta})\neq\hbox{\rm sgn}(\beta))$ is the loss function. When
signals are rare and weak, $P(\hbox{\rm sgn}(\hat{\beta})\neq\hbox{\rm
sgn}(\beta))\approx 1$ and ‘exact recovery’ is usually impossible. A more
appropriate loss function is the Hamming distance between $\hbox{\rm
sgn}(\hat{\beta})$ and $\hbox{\rm sgn}(\beta)$.
For any fixed $\beta$ and any variable selection procedure $\hat{\beta}$, we
measure the performance by the Hamming distance:
$h_{p}(\hat{\beta},\beta\bigl{|}X)=E\Bigl{[}\sum_{j=1}^{p}1\bigl{\\{}\hbox{\rm
sgn}(\hat{\beta}_{j})\neq\hbox{\rm
sgn}(\beta_{j})\bigr{\\}}\bigr{|}X\Bigr{]}.$
In the Asymptotic Rare Weak model, $\beta=b\circ\mu$, and
$(\epsilon_{p},\tau_{p})$ depend on $p$ through $(\vartheta,r)$, so the
overall Hamming distance for $\hat{\beta}$ is
$H_{p}(\hat{\beta};\epsilon_{p},n_{p},\mu,\Omega)=E_{\epsilon_{p}}E_{\Omega}\bigl{[}h_{p}(\hat{\beta},\beta\bigr{|}X)\bigr{]}\equiv
E_{\epsilon_{p}}E_{\Omega}\bigl{[}h_{p}(\hat{\beta},b\circ\mu\bigr{|}X)\bigr{]},$
where $E_{\epsilon_{p}}$ is the expectation with respect to the law of $b$,
and $E_{\Omega}$ is the expectation with respect to the law of $X$; see (2.15)
and (2.23). Finally, the minimax Hamming distance is
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)=\inf_{\hat{\beta}}\sup_{\mu\in\Theta_{p}^{*}(\tau_{p},a)}\bigl{\\{}H_{p}(\hat{\beta};\epsilon_{p},n_{p},\mu,\Omega)\bigr{\\}}.$
(2.26)
The Hamming distance is no smaller than the sum of the expected number of
signal components that are misclassified as noise and the expected number of
noise components that are misclassified as signal.
### 2.5 Lower bound for the minimax Hamming distance, and GOLF
We first construct lower bounds for “local risk” at different $j$, $1\leq
j\leq p$, and then aggregate them to construct a lower bound for the global
risk. One challenge we face is the least favorable configurations for
different $j$ overlap with each other. We resolve this by exploiting the
sparsity of a new graph to be introduced: Graph of Least Favorable (GOLF).
To recap, the model we consider is Model (1.1), where
$\beta\text{ is modeled by
}\mathrm{ARW}(\vartheta,r,a,\mu),\qquad\mbox{and}\qquad X\text{ is modeled by
}\mathrm{RD}(\vartheta,\kappa,\Omega).$ (2.27)
Fix $1\leq j\leq p$. The “local risk” at an index $j$ is the risk of
estimating the set of variables $\\{\beta_{k}:d(k,j)\leq g\\}$, where $g$ is
defined in (2.21) and $d(j,k)$ denotes the geodesic distance between $j$ and
$k$ in the graph ${\cal G}^{*,\delta}$. The goal is to construct two subsets
$V_{0}$ and $V_{1}$ and two realizations of $\beta$, $\beta^{(0)}$ and
$\beta^{(1)}$ (in the special case of $V_{0}=V_{1}$, we require $\hbox{\rm
sgn}(\beta^{(0)})\neq\hbox{\rm sgn}(\beta^{(1)})$), such that $j\in V_{0}\cup
V_{1}$ and
$\mbox{If $k\notin V_{0}\cup V_{1}$, $\beta^{(0)}_{k}=\beta_{k}^{(1)}$;
otherwise, $\beta^{(i)}_{k}\neq 0$ if and only if $k\in V_{i}$, $i=0,1$}.$
In the literature, it is known that how well we can estimate
$\\{\beta_{k}:d(k,j)\leq g\\}$ depends on how well we can separate two
hypotheses (where $\beta^{(0)}$ and $\beta^{(1)}$ are assumed as known):
$H_{0}^{(j)}:Y=X\beta^{(0)}+\sigma z\qquad\mbox{vs}.\qquad
H_{1}^{(j)}:Y=X\beta^{(1)}+\sigma z,\qquad z\sim N(0,I_{n}).$ (2.28)
The least favorable configuration for the local risk at index $j$ is the
quadruple $(V_{0},V_{1},\beta^{(0)},\beta^{(1)})$ for which two hypotheses are
the most difficult to separate.
For any $V\subset\\{1,2,\ldots,p\\}$, let $I_{V}$ be the indicator vector of
$V$ such that for any $1\leq k\leq p$, the $k$-th coordinate of $I_{V}$ is $1$
if $k\in V$ and is $0$ otherwise. Define
$B_{V}=\\{I_{V}\circ\mu:\;\mu\in\Theta_{p}^{*}(\tau_{p},a)\\},$ (2.29)
where we recall “$\circ$” denotes the Hadamard product (see Section 1.5).
Denote for short $\theta^{(i)}=I_{V_{0}\cup V_{1}}\circ\beta^{(i)}$, and so
$\beta^{(1)}-\beta^{(0)}=\theta^{(1)}-\theta^{(0)}$ and $\theta^{(i)}\in
B_{V_{i}}$, $i=0,1$. Introduce
$\alpha(\theta^{(0)},\theta^{(1)})=\alpha(\theta^{(0)},\theta^{(1)};V_{0},V_{1},\Omega,a)=\tau_{p}^{-2}(\theta^{(0)}-\theta^{(1)})^{\prime}\Omega(\theta^{(0)}-\theta^{(1)}).$
For the testing problem in (2.28), the optimal test is to reject $H_{0}^{(j)}$
if and only if
$(\theta^{(1)}-\theta^{(0)})^{\prime}X^{\prime}(Y-X\beta^{(0)})\geq
t\sigma\tau_{p}\sqrt{\alpha(\theta^{(0)},\theta^{(1)})}$ for some threshold
$t>0$ to be determined. In the ARW and RD models, $P(\beta_{k}\neq 0,\forall
k\in V_{i})\sim\epsilon_{p}^{|V_{i}|}$, $i=0,1$, and
$(\theta^{(0)}-\theta^{(1)})^{\prime}G(\theta^{(0)}-\theta^{(1)})\approx(\theta^{(0)}-\theta^{(1)})^{\prime}\Omega(\theta^{(0)}-\theta^{(1)})$,
since the support of $\theta^{(0)}-\theta^{(1)}$ is contained in a small-size
set $V_{0}\cup V_{1}$. Therefore the sum of Type I and Type II error of any
test associated with (2.28) is no smaller than (up to some negligible
differences)
$\epsilon_{p}^{|V_{0}|}\bar{\Phi}(t)+\epsilon_{p}^{|V_{1}|}\Phi\bigl{(}t-(\tau_{p}/\sigma)[\alpha(\theta^{(0)},\theta^{(1)})]^{1/2}\bigr{)},$
(2.30)
where $\bar{\Phi}=1-\Phi$ is the survival function of $N(0,1)$.
For a lower bound for the “local risk” at $j$, we first optimize the quantity
in (2.30) over all $\theta^{(0)}\in B_{V_{0}}$ and $\theta^{(1)}\in
B_{V_{1}}$, and then optimize over all $(V_{0},V_{1})$ subject to $j\in
V_{0}\cup V_{1}$. To this end, define
$\alpha^{*}(V_{0},V_{1})=\alpha^{*}(V_{0},V_{1};a,\Omega)$,
$\eta(V_{0},V_{1})=\eta(V_{0},V_{1};\vartheta,r,a,\Omega)$, and
$\rho_{j}^{*}=\rho_{j}^{*}(\vartheta,r,a,\Omega)$ by
$\alpha^{*}(V_{0},V_{1})=\min\bigl{\\{}\alpha(\theta^{(0)},\theta^{(1)};V_{0},V_{1},\Omega,a):\;\theta^{(i)}\in
B_{V_{i}},i=0,1,\hbox{\rm sgn}(\theta^{(0)})\neq\hbox{\rm
sgn}(\theta^{(1)})\\},$ (2.31)
$\eta(V_{0},V_{1})=\max\\{|V_{0}|,|V_{1}|\\}\vartheta+\frac{1}{4}\left[\left(\sqrt{\alpha^{*}(V_{0},V_{1})r}-\frac{\bigl{|}(|V_{1}|-|V_{0}|)\bigr{|}\vartheta}{\sqrt{\alpha^{*}(V_{0},V_{1})r}}\right)_{+}\right]^{2},$
and
$\rho_{j}^{*}(\vartheta,r,a,\Omega)=\min_{\\{(V_{0},V_{1}):j\in V_{1}\cup
V_{0}\\}}\eta(V_{0},V_{1}).$ (2.32)
The following shorthand notation is frequently used in this paper, which
stands for a generic multi-$\log(p)$ term that may vary from one occurrence to
another.
###### Definition 5
$L_{p}>0$ denotes a multi-$\log(p)$ term such that when $p\rightarrow\infty$,
for any $\delta>0$, $L_{p}p^{\delta}\rightarrow\infty$ and
$L_{p}p^{-\delta}\rightarrow 0$.
By (2.30) and Mills’ ratio (Wasserman and Roeder, 2009), a lower bound for the
“local risk” at $j$ is
$\displaystyle\qquad\qquad\sup_{\\{(V_{0},V_{1}):\;j\in V_{0}\cup
V_{1}\\}}\bigl{\\{}\inf_{t}\bigl{[}\epsilon_{p}^{|V_{0}|}\bar{\Phi}(t)+\epsilon_{p}^{|V_{1}|}\Phi\bigl{(}t-(\tau_{p}/\sigma)[\alpha^{*}(V_{0},V_{1})]^{1/2}\bigr{)}\bigr{]}\bigr{\\}}$
(2.33) $\displaystyle=$ $\displaystyle\sup_{\\{(V_{0},V_{1}):\;j\in V_{0}\cup
V_{1}\\}}\big{\\{}L_{p}\exp\bigl{(}-\eta(V_{0},V_{1})\cdot\log(p)\bigr{)}\bigr{\\}}=L_{p}\exp(-\rho_{j}^{*}(\vartheta,r,a,\Omega)\log(p)).$
(2.34)
We now aggregate such lower bounds for “local risk” for a global lower bound.
Since the “least favorable” configurations of $(V_{0},V_{1})$ for different
$j$ may overlap with each other, we need to consider a graph as follows.
Revisit the optimization problem in (2.31) and let
$(V_{0j}^{*},V_{1j}^{*})=\mathrm{argmin}_{\\{(V_{0},V_{1}):j\in V_{1}\cup
V_{0}\\}}\eta(V_{0},V_{1};\vartheta,r,a,\Omega).$ (2.35)
When there is a tie, pick the pair that appears first lexicographically.
Therefore, for any $1\leq j\leq p$, $V_{0j}^{*}\cup V_{1j}^{*}$ is uniquely
defined. In Lemma 22 of the appendix, we show that $|V_{0j}^{*}\cup
V_{1j}^{*}|\leq(\vartheta+r)^{2}/(2\vartheta r)$ for all $1\leq j\leq p$.
We now define a new graph, Graph of Least Favorable (GOLF), ${\cal
G}^{\diamond}=(V,E)$, where $V=\\{1,2,\ldots,p\\}$ and there is an edge
between $j$ and $k$ if and only if $(V_{0j}^{*}\cup V_{1j}^{*})$ and
$(V_{0k}^{*}\cup V_{1k}^{*})$ have non-empty intersections. Denote the maximum
degree of GOLF by $d_{p}({\cal G}^{\diamond})$.
###### Theorem 6
Fix $(\vartheta,\kappa)\in(0,1)^{2}$, $r>0$, and $a\geq 1$ such that
$\kappa>(1-\vartheta)$, and let ${\cal M}_{p}(c_{0},g)$ be as in (2.22).
Consider Model (1.1) where $\beta$ is modeled by $ARW(\vartheta,r,a,\mu)$ and
$X$ is modeled by $RD(\vartheta,\kappa,\Omega)$ and $\Omega\in{\cal
M}_{p}(c_{0},g)$ for sufficiently large $p$. Then as $p\rightarrow\infty$,
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)\geq
L_{p}\bigl{[}d_{p}({\cal
G}^{\diamond})\bigr{]}^{-1}\sum_{j=1}^{p}p^{-\rho_{j}^{*}(\vartheta,r,a,\Omega)}$.
A similar claim holds for deterministic design models; the proof is similar so
we omit it.
###### Corollary 7
For deterministic design models, the parallel lower bound holds for the
minimax Hamming distance with $\Omega$ replaced by $G$ in the calculation of
$\rho_{j}^{*}(\vartheta,r,a,\Omega)$ and $d_{p}({\cal G}^{\diamond})$.
Remark. The lower bounds contain a factor of $\bigl{[}d_{p}({\cal
G}^{\diamond})\bigr{]}^{-1}$. In many cases including that considered in our
main theorem (Theorem 8), this factor is a multi-$\log(p)$ term so it does not
have a major effect. In some other cases, the factor $\bigl{[}d_{p}({\cal
G}^{\diamond})\bigr{]}^{-1}$ could be much smaller, say, when the GOSD has one
or a few hubs, the degrees of which grow algebraically fast as $p$ grows. In
these cases, the associated GOLF may (or may not) have large-degree hubs. As a
result, the lower bounds we derive could be very conservative, and can be
substantially improved if we treat the hubs, neighboring nodes of the hubs,
and other nodes separately. For the sake of space, we leave such discussion to
future work.
Remark. A similar lower bound holds if the condition
$\mu\in\Theta^{*}_{p}(\tau_{p},a)$ of ARW is replaced by
$\mu\in\Theta_{p}(\tau_{p})$. In (2.31), suppose we replace
$\Theta_{p}^{*}(\tau_{p},a)$ by $\Theta_{p}(\tau_{p})$, and the minimum is
achieved at
$(\theta^{(0)},\theta^{(1)})=(\theta_{*}^{(0)}(V_{0},V_{1};\Omega),\theta_{*}^{(1)}(V_{0},V_{1};\Omega))$.
Let $g=g(m_{0},\vartheta,r)$ be as in (2.21) and define
$a_{g}^{*}(\Omega)=\max_{\\{(V_{0},V_{1}):|V_{0}\cup V_{1}|\leq
g\\}}\\{\|\theta_{*}^{(0)}(V_{0},V_{1};\Omega)\|_{\infty},\|\theta_{*}^{(1)}(V_{0},V_{1};\Omega)\|_{\infty}\\}.$
By elementary calculus, it is seen that for $\Omega\in{\cal M}_{p}(c_{0},g)$,
there is a a constant $C=C(c_{0},g)$ such that $a_{g}^{*}(\Omega)\leq C$. If
additionally we assume
$a>a_{g}^{*}(\Omega),$ (2.36)
then $\alpha^{*}(V_{0},V_{1})=\alpha^{*}(V_{0},V_{1};\Omega,a)$,
$\eta(V_{0},V_{1};\Omega,a,\vartheta,r)$, and
$\rho_{j}^{*}(\vartheta,r,a,\Omega)$ do not depend on $a$. Especially, we can
derive an alternative formula for $\rho_{j}^{*}(\vartheta,r,a,\Omega)$; see
Lemma 18 for details.
When (2.36) holds, $\Theta_{p}^{*}(\tau_{p},a)$ is broad enough in the sense
that the least favorable configurations
$(V_{0},V_{1},\beta^{(0)},\beta^{(1)})$ for all $j$ satisfy
$\|\beta^{(i)}\|_{\infty}\leq a\tau_{p}$, $i=0,1$. Consequently, neither the
minimax rate nor GS needs to adapt to $a$. In Section 2.6, we assume (2.36)
holds; (2.36) is a mild condition for it only involves small-size sub-matrices
of $\Omega$.
### 2.6 Upper bound and optimality of Graphlet Screening
Fix constants $\gamma\in(0,1)$ and $A>0$. Let ${\cal M}_{p}(c_{0},g)$ be as in
(2.22). In this section, we further restrict $\Omega$ to the following set:
${\cal M}_{p}^{*}(\gamma,c_{0},g,A)=\Bigl{\\{}\Omega\in{\cal
M}_{p}(c_{0},g):\,\sum_{j=1}^{p}|\Omega(i,j)|^{\gamma}\leq A,\,1\leq i\leq
p\Bigr{\\}}.$ (2.37)
Note that any $\Omega\in{\cal M}_{p}^{*}(\gamma,c_{0},g,A)$ is sparse in the
sense that each row of $\Omega$ has relatively few large coordinates. The
sparsity of $\Omega$ implies the sparsity of the Gram matrix $G$, since small-
size sub-matrices of $G$ approximately equal to their counterparts of
$\Omega$.
In GS, when we regularize GOSD (see (1.6)), we set the threshold $\delta$ by
$\delta=\delta_{p}=1/\log(p).$ (2.38)
Such a choice for threshold is mainly for convenience, and can be replaced by
any term that tends to $0$ logarithmically fast as $p\rightarrow\infty$.
For any subsets $D$ and $F$ of $\\{1,2,\ldots,p\\}$, define
$\omega(D,F;\Omega)=\omega(D,F;\vartheta,r,a,\Omega,p)$ by
$\omega(D,F;\Omega)=\min_{\xi\in\mathbb{R}^{|D|},\min_{i\in D}|\xi_{i}|\geq
1}\Big{\\{}\xi^{\prime}\bigl{(}\Omega^{D,D}-\Omega^{D,F}(\Omega^{F,F})^{-1}\Omega^{F,D}\bigr{)}\xi\Big{\\}},$
(2.39)
We choose the tuning parameters in the $GS$-step in a way such that
$t(\hat{D},\hat{F})=2\sigma^{2}q(\hat{D},\hat{F})\log p,$ (2.40)
where $q=q(\hat{D},\hat{F})>0$ satisfies (for short,
$\omega=\omega({\hat{D}},{\hat{F}};\Omega)$)
$\left\\{\begin{array}[]{ll}\sqrt{q_{0}}\leq\sqrt{q}\leq\sqrt{\omega
r}-\sqrt{\frac{(\vartheta+\omega r)^{2}}{4\omega
r}-\frac{|\hat{D}|+1}{2}\vartheta},&\ \mbox{$|\hat{D}|$ is odd \& $\omega
r/\vartheta>|\hat{D}|+(|\hat{D}|^{2}-1)^{1/2}$},\\\
\sqrt{q_{0}}\leq\sqrt{q}\leq\sqrt{\omega r}-\sqrt{\frac{1}{4}\omega
r-\frac{1}{2}|\hat{D}|\vartheta},&\ \mbox{$|\hat{D}|$ is even \& $\omega
r/\vartheta\geq 2|\hat{D}|$},\\\ \mbox{$q$ is a constant such that $q\geq
q_{0}$},&\ \mbox{otherwise}.\end{array}\right.$ (2.41)
We set the $GC$-step tuning parameters by
$u^{gs}=\sigma\sqrt{2\vartheta\log p},\qquad
v^{gs}=\tau_{p}=\sigma\sqrt{2r\log p}.$ (2.42)
The main theorem of this paper is the following theorem.
###### Theorem 8
Fix $m_{0}\geq 1$, $(\vartheta,\gamma,\kappa)\in(0,1)^{3}$, $r>0$, $c_{0}>0$,
$g>0$, $a>1$, $A>0$ such that $\kappa>1-\vartheta$ and (2.21) is satisfied.
Consider Model (1.1) where $\beta$ is modeled by $ARW(\vartheta,r,a,\mu)$, $X$
is modeled by $RD(\vartheta,\kappa,\Omega)$, and where $\Omega\in{\cal
M}_{p}^{*}(\gamma,c_{0},g,A)$ and $a>a_{g}^{*}(\Omega)$ for sufficiently large
$p$. Let $\hat{\beta}^{gs}=\hat{\beta}^{gs}(Y;\delta,{\cal
Q},u^{gs},v^{gs},X,p,n)$ be the Graphlet Screening procedure defined as in
Section 2.1, where the tuning parameters $(\delta,{\cal Q},u^{gs},v^{gs})$ are
set as in (2.38)-(2.42). Then as $p\rightarrow\infty$,
$\sup_{\mu\in\Theta_{p}^{*}(\tau_{p},a)}H_{p}(\hat{\beta}^{gs};\epsilon_{p},n_{p},\mu,\Omega)\leq
L_{p}\Big{[}p^{1-(m_{0}+1)\vartheta}+\sum_{j=1}^{p}p^{-\rho_{j}^{*}(\vartheta,r,a,\Omega)}\Big{]}+o(1)$.
Note that $\rho_{j}^{*}=\rho_{j}^{*}(\vartheta,r,a,\Omega)$ does not depend on
$a$. Also, note that in the most interesting range,
$\sum_{j=1}^{p}p^{-\rho_{j}^{*}}\gg 1$. So if we choose $m_{0}$ properly large
(e.g., $(m_{0}+1)\vartheta>1$), then
$\sup_{\mu\in\Theta_{p}^{*}(\tau_{p},a)}H_{p}(\hat{\beta}^{gs};\epsilon_{p},n_{p},\mu,\Omega)\leq
L_{p}\sum_{j=1}^{p}p^{-\rho_{j}^{*}(\vartheta,r,a,\Omega)}$. Together with
Theorem 6, this says that GS achieves the optimal rate of convergence,
adaptively to all $\Omega$ in ${\cal M}_{p}^{*}(\gamma,c_{0},g,A)$ and
$\beta\in\Theta_{p}^{*}(\tau_{p},a)$. We call this property optimal
adaptivity. Note that since the diagonals of $\Omega$ are scaled to $1$
approximately, $\kappa\equiv\log(n_{p})/\log(p)$ does not have a major
influence over the convergence rate, as long as (2.25) holds.
Remark. Theorem 8 addresses the case where (2.36) holds so
$a>a_{g}^{*}(\Omega)$. We now briefly discuss the case where
$a<a_{g}^{*}(\Omega)$. In this case, the set $\Theta_{p}^{*}(\tau_{p},a)$
becomes sufficiently narrow and $a$ starts to have some influence over the
optimal rate of convergence, at least for some choices of $(\vartheta,r)$. To
reflect the role of $a$, we modify GS as follows: (a) in the $GC$-step (2.13),
limit $\xi$ to the class where either $\xi_{i}=0$ or
$\tau_{p}\leq|\xi_{i}|\leq a\tau_{p}$, and (b) in the $GS$-step, replacing the
$\chi^{2}$-screening by the likelihood based screening procedure; that is,
when we screen ${\cal I}_{0}=\hat{D}\cup\hat{F}$, we accept nodes in $\hat{D}$
only when $h(\hat{F})>h({\cal I}_{0})$, where for any subset
$D\subset\\{1,2,\ldots,p\\}$,
$h(D)=\min\bigl{\\{}\frac{1}{2}\|P^{D}(Y-X^{\otimes,D}\xi)\|^{2}+\vartheta\sigma^{2}\log(p)|D|\big{\\}}$,
where the minimum is computed over all $|D|\times 1$ vectors $\xi$ whose
nonzero elements all have magnitudes in $[\tau_{p},a\tau_{p}]$. From a
practical point of view, this modified procedure depends more on the
underlying parameters and is harder to implement than is GS. However, this is
the price we need to pay when $a$ is small. Since we are primarily interested
in the case of relatively larger $a$ (so that $a>a_{g}^{*}(\Omega)$ holds), we
skip further discussion along this line.
### 2.7 Phase diagram and examples where $\rho_{j}^{*}(\vartheta,r,a,\Omega)$
have simple forms
In general, the exponents $\rho_{j}^{*}(\vartheta,r,a,\Omega)$ may depend on
$\Omega$ in a complicated way. Still, from time to time, one may want to find
a simple expression for $\rho_{j}^{*}(\vartheta,r,a,\Omega)$. It turns out
that in a wide class of situations, simple forms for
$\rho_{j}^{*}(\vartheta,r,a,\Omega)$ are possible. The surprise is that, in
many examples, $\rho_{j}^{*}(\vartheta,r,a,\Omega)$ depends more on the trade-
off between the parameters $\vartheta$ and $r$ (calibrating the signal
sparsity and signal strength, respectively), rather than on the large
coordinates of $\Omega$.
We begin with the following theorem, which is proved in Ji and Jin (2011,
Theorem 1.1).
###### Theorem 9
Fix $(\vartheta,\kappa)\in(0,1)$, $r>0$, and $a>1$ such that
$\kappa>(1-\vartheta)$. Consider Model (1.1) where $\beta$ is modeled by
$ARW(\vartheta,r,a,\mu)$ and $X$ is modeled by $RD(\vartheta,\kappa,\Omega)$.
Then as $p\rightarrow\infty$,
$\frac{\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)}{p^{1-\vartheta}}\gtrsim\left\\{\begin{array}[]{ll}1,&\qquad
0<r<\vartheta,\\\ L_{p}p^{-(r-\vartheta)^{2}/(4r)},&\qquad
r>\vartheta.\end{array}\right.$
Note that $p^{1-\vartheta}$ is approximately the number of signals. Therefore,
when $r<\vartheta$, the number of selection errors can not get substantially
smaller than the number of signals. This is the most difficult case where no
variable selection method can be successful.
In this section, we focus on the case $r>\vartheta$, so that successful
variable selection is possible. In this case, Theorem 9 says that a universal
lower bound for the Hamming distance is $L_{p}p^{1-(\vartheta+r)^{2}/(4r)}$.
An interesting question is, to what extend, this lower bound is tight.
Recall that $\lambda_{k}^{*}(\Omega)$ denotes the minimum of smallest
eigenvalues across all $k\times k$ principle submatrices of $\Omega$, as
defined in (2.20). The following corollaries are proved in Section 6.
###### Corollary 10
Suppose the conditions of Theorem 8 hold, and that additionally,
$1<r/\vartheta<3+2\sqrt{2}\approx 5.828$, and $|\Omega(i,j)|\leq
4\sqrt{2}-5\approx 0.6569$ for all $1\leq i,j\leq p$, $i\neq j$. Then as
$p\rightarrow\infty$,
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)=L_{p}p^{1-(\vartheta+r)^{2}/(4r)}$.
###### Corollary 11
Suppose the conditions of Theorem 8 hold. Also, suppose that
$1<r/\vartheta<5+2\sqrt{6}\approx 9.898$, and that
$\lambda_{3}^{*}(\Omega)\geq 2(5-2\sqrt{6})\approx 0.2021$,
$\lambda_{4}^{*}(\Omega)\geq 5-2\sqrt{6}\approx 0.1011$, and
$|\Omega(i,j)|\leq 8\sqrt{6}-19\approx 0.5959$ for all $1\leq i,j\leq p$,
$i\neq j$. Then as $p\rightarrow\infty$,
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)=L_{p}p^{1-(\vartheta+r)^{2}/(4r)}$.
In these corollaries, the conditions on $\Omega$ are rather relaxed. Somewhat
surprisingly, the off-diagonals of $\Omega$ do not necessarily have a major
influence on the optimal rate of convergence, as one might have expected.
Figure 1: Phase diagram for $\Omega=I_{p}$ (left), for $\Omega$ satisfying
conditions of Corollary 10 (middle), and for $\Omega$ satisfying conditions of
Corollary 11 (right). Red line: $r=\vartheta$. Solid red curve:
$r=\rho(\vartheta,\Omega)$. In each of the last two panels, the blue line
intersects with the red curve at $(\vartheta,r)=(1/2,[3+2\sqrt{2}]/2)$
(middle) and $(\vartheta,r)=(1/3,[5+2\sqrt{6}]/3)$ (right), which splits the
red solid curve into two parts; the part to the left is illustrative for it
depends on $\Omega$ in a complicated way; the part to the right, together with
the dashed red curve, represent $r=(1+\sqrt{1-\vartheta})^{2}$ (in the left
panel, this is illustrated by the red curve).
Note also that by Theorem 8, under the condition of either Corollaries 10 or
Corollary 11, GS achieves the optimal rate in that
$\sup_{\mu\in\Theta_{p}^{*}(\tau_{p},a)}H_{p}(\hat{\beta}^{gs};\epsilon_{p},n_{p},\mu,\Omega)\leq
L_{p}p^{1-(\vartheta+r)^{2}/(4r)}.$ (2.43)
Together, Theorem 9, Corollaries 10-11, and (2.43) have an interesting
implication on the so-called phase diagram. Call the two-dimensional parameter
space $\\{(\vartheta,r):0<\vartheta<1,r>0\\}$ the phase space. There are two
curves $r=\vartheta$ and $r=\rho(\vartheta,\Omega)$ (the latter can be thought
of as the solution of
$\sum_{j=1}^{p}p^{-\rho^{*}_{j}(\vartheta,r,a,\Omega)}=1$; recall that
$\rho_{j}^{*}(\vartheta,r,a,\Omega)$ does not depend on $a$) that partition
the whole phase space into three different regions:
* •
Region of No Recovery. $\\{(\vartheta,r):0<r<\vartheta,0<\vartheta<1\\}$. In
this region, as $p\rightarrow\infty$, for any $\Omega$ and any procedures, the
minimax Hamming error equals approximately to the total expected number of
signals. This is the most difficult region, in which no procedure can be
successful in the minimax sense.
* •
Region of Almost Full Recovery.
$\\{(\vartheta,r):\vartheta<r<\rho(\vartheta,\Omega)\\}$. In this region, as
$p\rightarrow\infty$, the minimax Hamming distance satisfies
$1\ll\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)\ll p^{1-\vartheta}$,
and it is possible to recover most of the signals, but it is impossible to
recover all of them.
* •
Region of Exact Recovery. In this region, as $p\rightarrow\infty$, the minimax
Hamming distance $\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)=o(1)$,
and it is possible to exactly recover all signals with overwhelming
probability.
In general, the function $\rho(\vartheta,\Omega)$ depends on $\Omega$ in a
complicated way. However, by Theorem 9 and Corollaries 10-11, we have the
following conclusions. First, for all $\Omega$ and $a>1$,
$\rho(\vartheta,\Omega)\geq(1+\sqrt{1-\vartheta})^{2}$ for all
$0<\vartheta<1$. Second, in the simplest case where $\Omega=I_{p}$,
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)=L_{p}p^{1-(\vartheta+r)^{2}/(4r)}$,
and $\rho(\vartheta,\Omega)=(1+\sqrt{1-\vartheta})^{2}$ for all
$0<\vartheta<1$. Third, under the conditions of Corollary 10,
$\rho(\vartheta,\Omega)=(1+\sqrt{1-\vartheta})^{2}$ if $1/2<\vartheta<1$.
Last, under the conditions of Corollary 11,
$\rho(\vartheta,\Omega)=(1+\sqrt{1-\vartheta})^{2}$ if $1/3<\vartheta<1$. The
phase diagram for the last three cases are illustrated in Figure 1. The blue
lines are $r/\vartheta=3+2\sqrt{2}$ (middle) and $r/\vartheta=5+2\sqrt{6}$
(right).
Corollaries 10-11 can be extended to more general situations, where
$r/\vartheta$ may get arbitrary large, but consequently, we need stronger
conditions on $\Omega$. Towards this end, we note that for any $(\vartheta,r)$
such that $r>\vartheta$, we can find a unique integer $N=N(\vartheta,r)$ such
that $2N-1\leq(\vartheta/r+r/\vartheta)/2<2N+1$. Suppose that for any $2\leq
k\leq 2N-1$,
$\lambda_{k}^{*}(\Omega)\geq\max_{\\{(k+1)/2\leq
j\leq\min\\{k,N\\}\\}}\Big{\\{}\frac{(r/\vartheta+\vartheta/r)/2-2j+2+\sqrt{[(r/\vartheta+\vartheta/r)/2-2j+2]^{2}-1}}{(2k-2j+1)(r/\vartheta)}\Big{\\}},$
(2.44)
and that for any $2\leq k\leq 2N$,
$\lambda_{k}^{*}(\Omega)\geq\max_{\\{k/2\leq
j\leq\min\\{k-1,N\\}\\}}\Big{\\{}\frac{(r/\vartheta+\vartheta/r)/2+1-2j}{(k-j)(r/\vartheta)}\Big{\\}}.$
(2.45)
Then we have the following corollary.
###### Corollary 12
Suppose the conditions in Theorem 8 and that in (2.44)-(2.45) hold. Then as
$p\rightarrow\infty$,
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)=L_{p}p^{1-(\vartheta+r)^{2}/(4r)}$.
The right hand sides of (2.44)-(2.45) decrease with $(r/\vartheta)$. For a
constant $s_{0}>1$, (2.44)-(2.45) hold for all $1<r/\vartheta\leq s_{0}$ as
long as they hold for $r/\vartheta=s_{0}$. Hence Corollary 12 implies a
similar partition of the phase diagram as do Corollaries 10-11.
Remark. Phase diagram can be viewed as a new criterion for assessing the
optimality, which is especially appropriate for rare and weak signals. The
phase diagram is a partition of the phase space
$\\{(\vartheta,r):0<\vartheta<1,r>0\\}$ into different regions where
statistical inferences are distinctly different. In general, a phase diagram
has the following four regions:
* •
An “exact recovery” region corresponding to the “rare and strong” regime in
which high probability of completely correct variable selection is feasible.
* •
An “almost full recovery” region as a part of the “rare and weak” regime in
which completely correct variable selection is not achievable with high
probability but variable selection is still feasible in the sense that with
high probability, the number of incorrectly selected variables is a small
fraction of the total number of signals.
* •
A “detectable” region in which variable selection is infeasible but the
detection of the existence of a signal (somewhere) is feasible (e.g., by the
Higher Criticism method).
* •
An “undetectable” region where signals are so rare and weak that nothing can
be sensibly done.
In the sparse signal detection (Donoho and Jin, 2004) and classification (Jin,
2009) problems, the main interest is to find the detectable region, so that
the exact recovery and almost full recovery regions were lumped into a single
“estimable” region (e.g., Donoho and Jin (2004, Figure 1)). For variable
selection, the main interest is to find the boundaries of the almost full
discovery region so that the detectable and non-detectable regions are lumped
into a single “no recovery” region as in Ji and Jin (2011) and Figure 1 of
this paper.
Variable selection in the “almost full recovery” region is a new and
challenging problem. It was studied in Ji and Jin (2011) when the effect of
signal cancellation is negligible, but the hardest part of the problem was
unsolved in Ji and Jin (2011). This paper (the second in this area) deals with
the important issue of signal cancellation, in hopes of gaining a much deeper
insight on variable selection in much broader context.
### 2.8 Non-optimaility of subset selection and the lasso
Subset selection (also called the $L^{0}$-penalization method) is a well-known
method for variable selection, which selects variables by minimizing the
following functional:
$\frac{1}{2}\|Y-X\beta\|^{2}+\frac{1}{2}(\lambda_{ss})^{2}\|\beta\|_{0},$
(2.46)
where $\|\beta\|_{q}$ denotes the $L^{q}$-norm, $q\geq 0$, and
$\lambda_{ss}>0$ is a tuning parameter. The AIC, BIC, and RIC are methods of
this type (Akaike, 1974; Schwarz, 1978; Foster and George, 1994). Subset
selection is believed to have good “theoretic property”, but the main drawback
of this method is that it is computationally NP hard. To overcome the
computational challenge, many relaxation methods are proposed, including but
are not limited to the lasso (Chen et al, 1998; Tibshirani, 1996), SCAD (Fan
and Li, 2001), MC+ (Zhang, 2010), and Dantzig selector (Candes and Tao, 2007).
Take the lasso for example. The method selects variables by minimizing
$\frac{1}{2}\|Y-X\beta\|^{2}+\lambda_{lasso}\|\beta\|_{1},$ (2.47)
where the $L^{0}$-penalization is replaced by the $L^{1}$-penalization, so the
functional is convex and the optimization problem is solvable in polynomial
time under proper conditions.
Somewhat surprisingly, subset selection is generally rate non-optimal in terms
of selection errors. This sub-optimality of subset selection is due to its
lack of flexibility in adapting to the “local” graphic structure of the design
variables. Similarly, other global relaxation methods are sub-optimal as well,
as the subset selection is the “idol” these methods try to mimic. To save
space, we only discuss subset selection and the lasso, but a similar
conclusion can be drawn for SCAD, MC+, and Dantzig selector.
For mathematical simplicity, we illustrate the point with an idealized
regression model where the Gram matrix $G=X^{\prime}X$ is diagonal block-wise
and has $2\times 2$ blocks
$G(i,j)=1\\{i=j\\}+h_{0}\cdot 1\\{\mbox{$|j-i|=1$, $max(i,j)$ is
even}\\},\;\;|h_{0}|<1,\;1\leq i,j\leq p.$ (2.48)
Using an idealized model is mostly for technical convenience, but the non-
optimality of subset selection or the lasso holds much more broadly than what
is considered here. On the other hand, using a simple model is sufficient
here: if a procedure is non-optimal in an idealized case, we can not expect it
to be optimal in a more general context.
At the same time, we continue to model $\beta$ with the Asymptotic Rare and
Weak model ARW$(\vartheta,r,a,\mu)$, but where we relax the assumption of
$\mu\in\Theta_{p}^{*}(\tau_{p},a)$ to that of $\mu\in\Theta_{p}(\tau_{p})$ so
that the strength of each signal $\geq\tau_{p}$ (but there is no upper bound
on the strength). Consider a variable selection procedure
$\hat{\beta}^{\star}$, where $\star=gs,ss,lasso$, representing GS, subset
selection, and the lasso (where the tuning parameters for each method are
ideally set; for the worst-case risk considered below, the ideal tuning
parameters depend on $(\vartheta,r,p,h_{0})$ but do not depend on $\mu$).
Since the index groups $\\{2j-1,2j\\}$ are exchangeable in (2.48) and the ARW
models, the Hamming error of $\beta^{\star}$ in its worst case scenario has
the form of
$\sup_{\\{\mu\in\Theta_{p}(\tau_{p})\\}}H_{p}(\hat{\beta}^{\star};\epsilon_{p},\mu,G)=L_{p}p^{1-\rho_{\star}(\vartheta,r,h_{0})}$.
We now study $\rho_{\star}(\vartheta,r,h_{0})$. Towards this end, we first
introduce
$\rho_{lasso}^{(3)}(\vartheta,r,h_{0})=\bigl{\\{}(2|h_{0}|)^{-1}[(1-h_{0}^{2})\sqrt{r}-\sqrt{(1-h_{0}^{2})(1-|h_{0}|)^{2}r-4|h_{0}|(1-|h_{0}|)\vartheta}]\bigr{\\}}^{2}$
and
$\rho_{lasso}^{(4)}(\vartheta,r,h_{0})=\vartheta+\frac{(1-|h_{0}|)^{3}(1+|h_{0}|)}{16h_{0}^{2}}\bigl{[}(1+|h_{0}|)\sqrt{r}-\sqrt{(1-|h_{0}|)^{2}r-4|h_{0}|\vartheta/(1-h_{0}^{2})}\bigr{]}^{2}$.
We then let
$\rho_{ss}^{(1)}(\vartheta,r,h_{0})=\left\\{\begin{array}[]{ll}2\vartheta,&\qquad\quad
r/\vartheta\leq 2/(1-h_{0}^{2})\\\
\,[2\vartheta+(1-h_{0}^{2})r]^{2}/[4(1-h_{0}^{2})r],&\qquad\quad
r/\vartheta>2/(1-h_{0}^{2})\end{array},\right.$
$\rho_{ss}^{(2)}(\vartheta,r,h_{0})=\left\\{\begin{array}[]{ll}2\vartheta,&\quad
r/\vartheta\leq 2/(1-|h_{0}|)\\\
2[\sqrt{2(1-|h_{0}|)r}-\sqrt{(1-|h_{0}|)r-\vartheta}]^{2},&\quad
r/\vartheta>2/(1-|h_{0}|)\end{array},\right.$
$\rho_{lasso}^{(1)}(\vartheta,r,h_{0})=\left\\{\begin{array}[]{ll}2\vartheta,&\qquad\qquad
r/\vartheta\leq 2/(1-|h_{0}|)^{2}\\\
\rho_{lasso}^{(3)}(\vartheta,r,h_{0}),&\qquad\qquad
r/\vartheta>2/(1-|h_{0}|)^{2}\end{array},\right.$
and
$\rho_{lasso}^{(2)}(\vartheta,r,h_{0})=\left\\{\begin{array}[]{ll}2\vartheta,&\qquad\qquad
r/\vartheta\leq(1+|h_{0}|)/(1-|h_{0}|)^{3}\\\
\rho_{lasso}^{(4)}(\vartheta,r,h_{0}),&\qquad\qquad
r/\vartheta>(1+|h_{0}|)/(1-|h_{0}|)^{3}\end{array}.\right.$
The following theorem is proved in Section 6.
###### Theorem 13
Fix $\vartheta\in(0,1)$ and $r>0$ such that $r>\vartheta$. Consider Model
(1.1) where $\beta$ is modeled by $ARW(\vartheta,r,a,\mu)$ and $X$ satisfies
(2.48). For GS, we set the tuning parameters $(\delta,m_{0})=(0,2)$, and set
$({\cal Q},u^{gs},v^{gs})$ as in (2.40)-(2.42). For subset selection as in
(2.46) and the lasso as in (2.47), we set their tuning parameters ideally
given that $(\vartheta,r)$ are known. Then as $p\rightarrow\infty$,
$\rho_{gs}(\vartheta,r,h_{0})=\min\bigl{\\{}\frac{(\vartheta+r)^{2}}{4r},\,\vartheta+\frac{(1-|h_{0}|)}{2}r,\,2\vartheta+\frac{\\{[(1-h_{0}^{2})r-\vartheta]_{+}\\}^{2}}{4(1-h_{0}^{2})r}\bigr{\\}},$
(2.49)
$\rho_{ss}(\vartheta,r,h_{0})=\min\bigl{\\{}\frac{(\vartheta+r)^{2}}{4r},\,\vartheta+\frac{(1-|h_{0}|)}{2}r,\,\rho_{ss}^{(1)}(\vartheta,r,h_{0}),\rho_{ss}^{(2)}(\vartheta,r,h_{0})\bigr{\\}},$
(2.50)
and
$\rho_{lasso}(\vartheta,r,h_{0})=\min\\{\frac{(\vartheta+r)^{2}}{4r},\,\vartheta+\frac{(1-|h_{0}|)r}{2(1+\sqrt{1-h_{0}^{2}})},\,\rho_{lasso}^{(1)}(\vartheta,r,h_{0}),\,\rho_{lasso}^{(2)}(\vartheta,r,h_{0})\bigr{\\}}.$
(2.51)
It can be shown that
$\rho_{gs}(\vartheta,r,h_{0})\geq\rho_{ss}(\vartheta,r,h_{0})\geq\rho_{lasso}(\vartheta,r,h_{0})$,
where depending on the choices of $(\vartheta,r,h_{0})$, we may have equality
or strict inequality (note that a larger exponent means a better error rate).
This fits well with our expectation, where as far as the convergence rate is
concerned, GS is optimal for all $(\vartheta,r,h_{0})$, so it outperforms the
subset selection, which in turn outperforms the lasso. Table 2 summarizes the
exponents for some representative $(\vartheta,r,h_{0})$. It is seen that
differences between these exponents become increasingly prominent when $h_{0}$
increase and $\vartheta$ decrease.
$\vartheta/r/h_{0}$ | .1/11/.8 | .3/9/.8 | .5/4/.8 | .1/4/.4 | .3/4/.4 | .5/4/.4 | .1/3/.2 | .3/3/.2
---|---|---|---|---|---|---|---|---
$\star=gs$ | 1.1406 | 1.2000 | 0.9000 | 0.9907 | 1.1556 | 1.2656 | 0.8008 | 0.9075
$\star=ss$ | 0.8409 | 0.9047 | 0.9000 | 0.9093 | 1.1003 | 1.2655 | 0.8007 | 0.9075
$\star=lasso$ | 0.2000 | 0.6000 | 0.7500 | 0.4342 | 0.7121 | 1.0218 | 0.6021 | 0.8919
Table 2: The exponents $\rho_{\star}(\vartheta,r,h_{0})$ in Theorem 13, where
$\star=gs,ss,lasso$.
As in Section 2.7, each of these methods has a phase diagram plotted in Figure
2, where the phase space partitions into three regions: Region of Exact
Recovery, Region of Almost Full Recovery, and Region of No Recovery.
Interestingly, the separating boundary for the last two regions are the same
for three methods, which is the line $r=\vartheta$. The boundary that
separates the first two regions, however, vary significantly for different
methods. For any $h_{0}\in(-1,1)$ and $\star=gs,ss,lasso$, the equation for
this boundary can be obtained by setting $\rho_{\star}(\vartheta,r,h_{0})=1$
(the calculations are elementary so we omit them). Note that the lower the
boundary is, the better the method is, and that the boundary corresponding to
the lasso is discontinuous at $\vartheta=1/2$. In the non-optimal region of
either subset selection or the lasso, the Hamming errors of the procedure are
much smaller than $p\epsilon_{p}$, so the procedure gives “almost full
recovery”; however, the rate of Hamming errors is slower than that of the
optimal procedure, so subset selection or the lasso is non-optimal in such
regions.
Subset selection and the lasso are rate non-optimal for they are so-called
one-step or non-adaptive methods (Ji and Jin, 2011), which use only one tuning
parameter, and which do not adapt to the local graphic structure. The non-
optimality can be best illustrated with the diagonal block-wise model
presented here, where each block is a $2\times 2$ matrix. Correspondingly, we
can partition the vector $\beta$ into many size $2$ blocks, each of which is
of the following three types (i) those have no signal, (ii) those have exactly
one signal, and (iii) those have two signals. Take the subset selection for
example. To best separate (i) from (ii), we need to set the tuning parameter
ideally. But such a tuning parameter may not be the “best” for separating (i)
from (iii). This explains the non-optimality of subset selection.
Seemingly, more complicated penalization methods that use multiple tuning
parameters may have better performance than the subset selection and the
lasso. However, it remains open how to design such extensions to achieve the
optimal rate for general cases. To save space, we leave the study along this
line to the future.
Figure 2: Phase diagrams for GS (top left), subset selection (top right), and
the lasso (bottom; zoom-in on the left and zoom-out on the right), where
$h_{0}=0.5$.
### 2.9 Summary
We propose GS as a new approach to variable selection. The key methodological
innovation is to use the GOSD to guide the multivariate screening. While a
brute-force $m$-variate screening has a computation cost of $O(p^{m}+np)$, GS
only has a computation cost of $L_{p}np$ (excluding the overhead of obtaining
the GOSD), by utilizing graph sparsity. Note that when the design matrix $G$
is approximately banded, say, all its large entries are confined to a diagonal
band with bandwidth $\leq K$, the overhead of GS can be reduced to $O(npK)$.
One such example is in Genome-Wide Association Study (GWAS), where $G$ is the
empirical Linkage Disequilibrium (LD) matrix, and $K$ can be as small as a few
tens. We remark that the lasso has a computational complexity of $O(npk)$,
where $k$, dominated by the number steps requiring re-evaluation of the
correlation between design vectors and updated residuals, could be smaller
than the $L_{p}$ term for GS (Wang et al, 2013).
We use asymptotic minimaxity of the Hamming distance as the criterion for
assessing optimality. Compared with existing literature on variable selection
where we use the oracle property or probability of exact support recovery to
assess optimality, our approach is mathematically more demanding, yet
scientifically more relevant in the rare/weak paradigm.
We have proved that GS achieves the optimal rate of convergence of Hamming
errors, especially when signals are rare and weak, provided that the Gram
matrix is sparse. Subset selection and the lasso are not rate optimal, even
with very simple Gram matrix $G$ and even when the tuning parameters are
ideally set. The sub-optimality of these methods is due to that they do not
take advantage of the ‘local’ graphical structure as GS does.
GS has three key tuning parameters: $q$ for the threshold level
$t(\hat{D},\hat{F})=2\sigma^{2}q\log p$ in the $GS$-step, and
$(u^{gs},v^{gs})=(\sigma\sqrt{2\vartheta\log p},\sigma\sqrt{2r\log p})$ in the
$GC$-step. While the choice of $q$ is reasonably flexible and a sufficiently
small fixed $q>0$ is usually adequate, the choice of $u^{gs}$ and $v^{gs}$ are
more directly tied to the signal sparsity and signal strength. Adaptive choice
of these tuning parameters is a challenging direction of further research. One
of our ideas to be developed in this direction is a subsampling scheme similar
to the Stability Selection (Meinsausen and Buhlmann, 2010). On the other hand,
as shown in our numeric results in Section 4, the performance of GS is
relatively insensitive to mis-specification of $(\epsilon_{p},\tau_{p})$; see
details therein.
## 3 Properties of Graphlet Screening, proof of Theorem 8
GS attributes the success to two important properties: the Sure Screening
property and the Separable After Screening (SAS) property.
The Sure Screening property means that in the $m_{0}$-stage $\chi^{2}$
screening, by picking an appropriate threshold, the set ${\cal U}_{p}^{*}$
(which is the set of retained indices after the GS-step) contains all but a
small fraction of true signals. Asymptotically, this fraction is comparably
smaller than the minimax Hamming errors, and so negligible. The SAS property
means that except for a negligible probability, as a subgraph of the GOSD,
${\cal U}_{p}^{*}$ decomposes into many disconnected components of the GOSD,
where the size of each component does not exceed a fixed integer. These two
properties ensure that the original regression problem reduces to many small-
size regression problems, and thus pave the way for the $GC$-step.
Below, we explain these ideas in detail, and conclude the section by the proof
of Theorem 8. Since the only place we need the knowledge of $\sigma$ is in
setting the tuning parameters, so without loss of generality, we assume
$\sigma=1$ throughout this section.
First, we discuss the $GS$-step. For short, write
$\hat{\beta}=\hat{\beta}^{gs}(Y;\delta,{\cal Q},u^{gs},v^{gs},X,p,n)$
throughout this section. We first discuss the computation cost of the
$GS$-step. As in Theorem 8, we take the threshold $\delta$ in ${\cal
G}^{*,\delta}$ to be $\delta=\delta_{p}=1/\log(p)$. The proof of the following
lemma is similar to that of Ji and Jin (2011, Lemma 2.2), so we omit it.
###### Lemma 14
Suppose the conditions of Theorem 8 hold, where we recall $\delta=1/\log(p)$,
and $\Omega^{*,\delta}$ is defined as in (1.6). As $p\rightarrow\infty$, with
probability $1-o(1/p^{2})$, $\|\Omega-\Omega^{*,\delta}\|_{\infty}\leq
C(\log(p))^{-(1-\gamma)}$, and ${\cal G}^{*,\delta}$ is $K$-sparse, where
$K\leq C(\log(p))^{1/\gamma}$.
Combining Lemma 14 and Frieze and Molloy (1999), it follows that with
probability $1-o(1/p^{2})$, ${\cal G}^{*,\delta}$ has at most
$p(Ce(\log(p))^{1/\gamma})^{m_{0}}$ connected subgraphs of size $\leq m_{0}$.
Note that the second factor is at most logarithmically large, so the
computation cost in the $GS$-step is at most $L_{p}p$ flops.
Consider the performance of the $GS$-step. The goal of this step is two-fold:
on one hand, it tries to retain as many signals as possible during the
screening; on the other hand, it tries to minimize the computation cost of the
$GC$-step by controlling the maximum size of all components of ${\cal
U}_{p}^{*}$. The key in the $GS$-step is to set the collection of thresholds
${\cal Q}$. The tradeoff is that, setting the thresholds too high may miss too
many signals during the screening, and setting the threshold too low may
increase the maximum size of the components in ${\cal U}_{p}^{*}$, and so
increase the computational burden of the $GC$-step. The following lemma
characterizes the Sure Screening property of GS, and is proved in Section 6.
###### Lemma 15
(Sure Screening). Suppose the settings and conditions are as in Theorem 8. In
the $m_{0}$-stage $\chi^{2}$ screening of the GS-step, if we set the
thresholds $t(\hat{D},\hat{F})$ as in (2.40), then as $p\rightarrow\infty$,
for any $\Omega\in{\cal M}_{p}^{*}(\gamma,c_{0},g,A)$,
$\sum_{j=1}^{p}P(\beta_{j}\neq 0,j\notin{\cal U}_{p}^{*})\leq
L_{p}[p^{1-(m_{0}+1)\vartheta}+\sum_{j=1}^{p}p^{-\rho_{j}^{*}(\vartheta,r,a,\Omega)}]+o(1)$.
Next, we formally state the SAS property. Viewing it as a subgraph of ${\cal
G}^{*,\delta}$, ${\cal U}_{p}^{*}$ decomposes into many disconnected
components ${\cal I}^{(k)}$, $1\leq k\leq N$, where $N$ is an integer that may
depend on the data.
###### Lemma 16
(SAS). Suppose the settings and conditions are as in Theorem 8. In the
$m_{0}$-stage $\chi^{2}$ screening in the GS-step, suppose we set the
thresholds $t(\hat{D},\hat{F})$ as in (2.40) such that $q(\hat{D},\hat{F})\geq
q_{0}$ for some constant $q_{0}=q_{0}(\vartheta,r)>0$. As
$p\rightarrow\infty$, under the conditions of Theorem 8, for any
$\Omega\in{\cal M}_{p}^{*}(\gamma,c_{0},g,A)$, there is a constant
$\ell_{0}=\ell_{0}(\vartheta,r,\kappa,\gamma,A,c_{0},g)>0$ such that with
probability at least $1-o(1/p)$, $|{\cal I}^{(k)}|\leq\ell_{0}$, $1\leq k\leq
N$.
We remark that a more convenient way of picking $q$ is to let
$\left\\{\begin{array}[]{ll}q_{0}\leq q\leq(\frac{\omega r+\vartheta}{2\omega
r})^{2}\omega r,&\qquad\mbox{$|\hat{D}|$ is odd \& $\omega
r/\vartheta>|\hat{D}|+(|\hat{D}|^{2}-1)^{1/2}$},\\\ q_{0}\leq
q\leq\frac{1}{4}\omega r,&\qquad\mbox{$|\hat{D}|$ is even \& $\omega
r/\vartheta\geq 2|\hat{D}|$},\end{array}\right.$ (3.52)
and let $q$ be any other number otherwise, with which both lemmas continue to
hold with this choice of $q$. Here, for short,
$\omega=\omega(\hat{D},\hat{F};\Omega)$. Note that numerically this choice is
comparably more conservative.
Together, the above two lemmas say that the $GS$-step makes only negligible
false non-discoveries, and decomposes ${\cal U}_{p}^{*}$ into many
disconnected components, each has a size not exceeding a fixed integer. As a
result, the computation cost of the following $GC$-step is moderate, at least
in theory.
We now discuss the $GC$-step. The key to understanding the $GC$-step is that
the original regression problem reduces to many disconnected small-size
regression problems. To see the point, define $\tilde{Y}=X^{\prime}Y$ and
recall that $G=X^{\prime}X$. Let ${\cal I}_{0}\lhd{\cal U}_{p}^{*}$ be a
component, we limit our attention to ${\cal I}_{0}$ by considering the
following regression problem:
$\tilde{Y}^{{\cal I}_{0}}=G^{{\cal I}_{0},\otimes}\beta+(X^{\prime}z)^{{\cal
I}_{0}},$ (3.53)
where $(X^{\prime}z)^{{\cal I}_{0}}\sim N(0,G^{{\cal I}_{0},{\cal
I}_{0}})\approx N(0,\Omega^{{\cal I}_{0},{\cal I}_{0}})$, and $G^{{\cal
I}_{0},\otimes}$ is a $|{\cal I}_{0}|\times p$ matrix according to our
notation. What is non-obvious here is that, the regression problem still
involves the whole vector $\beta$, and is still high-dimensional. To see the
point, letting $V=\\{1,2,\ldots,p\\}\setminus{\cal U}_{p}^{*}$, we write
$G^{{\cal I}_{0},\otimes}\beta=G^{{\cal I}_{0},{\cal I}_{0}}\beta^{{\cal
I}_{0}}+I+II$, where $I=\sum_{{\cal J}_{0}:{\cal J}_{0}\lhd{\cal
U}_{p}^{*},{\cal J}_{0}\neq{\cal I}_{0}}G^{{\cal I}_{0},{\cal
J}_{0}}\beta^{{\cal J}_{0}}$ and $II=G^{{\cal I}_{0},V}\beta^{V}$. First, by
Sure Screening property, $\beta^{V}$ contains only a negligible number of
signals, so we can think $II$ as negligible. Second, for any ${\cal
J}_{0}\neq{\cal I}_{0}$ and ${\cal J}_{0}\lhd{\cal U}_{p}^{*}$, by the SAS
property, ${\cal I}_{0}$ and ${\cal J}_{0}$ are disconnected and so the matrix
$G^{{\cal I}_{0},{\cal J}_{0}}$ is a small size matrix whose coordinates are
uniformly small. This heuristic is made precise in the proof of Theorem 8. It
is now seen that the regression problem in (3.53) is indeed low-dimensional:
$\tilde{Y}^{{\cal I}_{0}}\approx G^{{\cal I}_{0},{\cal I}_{0}}\beta^{{\cal
I}_{0}}+(X^{\prime}z)^{{\cal I}_{0}}\approx N(\Omega^{{\cal I}_{0},{\cal
I}_{0}}\beta^{{\cal I}_{0}},\Omega^{{\cal I}_{0},{\cal I}_{0}}),$ (3.54)
The above argument is made precise in Lemma 17, see details therein. Finally,
approximately, the $GC$-step is to minimize $\frac{1}{2}(\tilde{Y}^{{\cal
I}_{0}}-\Omega^{{\cal I}_{0},{\cal I}_{0}}\xi)^{\prime}(\Omega^{{\cal
I}_{0},{\cal I}_{0}})^{-1}(\tilde{Y}^{{\cal I}_{0}}-\Omega^{{\cal I}_{0},{\cal
I}_{0}}\xi)+\frac{1}{2}(u^{gs})^{2}\|\xi\|_{0}$, where each coordinate of
$\xi$ is either $0$ or $\geq v^{gs}$ in magnitude. Comparing this with (3.54),
the procedure is nothing but the penalized MLE of a low dimensional normal
model, and the main result follows by exercising basic statistical inferences.
We remark that in the $GC$-step, removing the constraints on the coordinates
of $\xi$ will not give the optimal rate of convergence. This is one of the
reasons why the classical subset selection procedure is rate non-optimal.
Another reason why the subset selection is non-optimal is that, the procedure
has only one tuning parameter, but GS has the flexibility of using different
tuning parameters in the $GS$-step and the $GC$-step. See Section 2.8 for more
discussion.
We are now ready for the proof of Theorem 8.
### 3.1 Proof of Theorem 8
For notational simplicity, we write
$\rho_{j}^{*}=\rho_{j}^{*}(\vartheta,r,a,\Omega)$. By Lemma 15,
$\sum_{j=1}^{p}P(\beta_{j}\neq 0,j\notin{\cal U}_{p}^{*})\leq
L_{p}[p^{1-(m_{0}+1)\vartheta}+\sum_{j=1}^{p}p^{-\rho_{j}^{*}}]+o(1).$ (3.55)
So to show the claim, it is sufficient to show
$\sum_{j=1}^{p}P(j\in{\cal U}_{p}^{*},\hbox{\rm sgn}(\beta_{j})\neq\hbox{\rm
sgn}(\hat{\beta}_{j}))\leq
L_{p}[\sum_{j=1}^{p}p^{-\rho_{j}^{*}}+p^{1-(m_{0}+1)\vartheta}]+o(1).$ (3.56)
Towards this end, let $S(\beta)$ be the support of $\beta$,
$\Omega^{*,\delta}$ be as in (1.6), and ${\cal G}^{*,\delta}$ be the GOSD. Let
${\cal U}_{p}^{*}$ be the set of retained indices after the GS-step. Note that
when $\hbox{\rm sgn}(\hat{\beta}_{j})\neq 0$, there is a unique component
${\cal I}_{0}$ such that $j\in{\cal I}_{0}\lhd{\cal U}_{p}^{*}$. For any
connected subgraph ${\cal I}_{0}$ of ${\cal G}^{*,\delta}$, let $B({\cal
I}_{0})=\\{\mbox{$k$: $k\notin{\cal I}_{0}$, $\Omega^{*,\delta}(k,\ell)\neq 0$
for some $\ell\in{\cal I}_{0}$, $1\leq k\leq p$}\\}$. Note that when ${\cal
I}_{0}$ is a component of ${\cal U}_{p}^{*}$, we must have $B({\cal
I}_{0})\cap{\cal U}_{p}^{*}=\emptyset$ as for any node in $B({\cal I}_{0})$,
there is at least one edge between it and some nodes in the component ${\cal
I}_{0}$. As a result,
$P(j\in{\cal I}_{0}\lhd{\cal U}_{p}^{*},B({\cal I}_{0})\cap
S(\beta)\neq\emptyset)\leq\sum_{{\cal I}_{0}:j\in{\cal I}_{0}}\sum_{k\in
B({\cal I}_{0})}P(k\notin{\cal U}_{p}^{*},\beta_{k}\neq 0),$ (3.57)
where the first summation is over all connected subgraphs that contains node
$j$. By Lemma 16, with probability at least $1-o(1/p)$, ${\cal G}^{*,\delta}$
is $K$-sparse with $K=C(\log(p))^{1/\gamma}$, and there is a finite integer
$\ell_{0}$ such that $|{\cal I}_{0}|\leq\ell_{0}$. As a result, there are at
most finite ${\cal I}_{0}$ such that the event $\\{j\in{\cal I}_{0}\lhd{\cal
U}_{p}^{*}\\}$ is non-empty, and for each of such ${\cal I}_{0}$, $B({\cal
I}_{0})$ contains at most $L_{p}$ nodes. Using (3.57) and Lemma 15, a direct
result is
$\sum_{j=1}^{p}P(j\in{\cal I}_{0}\lhd{\cal U}_{p}^{*},B({\cal I}_{0})\cap
S(\beta)\neq\emptyset)\leq
L_{p}[\sum_{j=1}^{p}p^{-\rho_{j}^{*}}+p^{1-(m_{0}+1)\vartheta}]+o(1).$ (3.58)
Comparing (3.58) with (3.56), to show the claim, it is sufficient to show that
$\sum_{j=1}^{p}P(\hbox{\rm sgn}(\beta_{j})\neq\hbox{\rm
sgn}(\hat{\beta}_{j}),j\in{\cal I}_{0}\lhd{\cal U}_{p}^{*},B({\cal I}_{0})\cap
S(\beta)=\emptyset)\leq
L_{p}[\sum_{j=1}^{p}p^{-\rho_{j}^{*}}+p^{1-(m_{0}+1)\vartheta}]+o(1).$ (3.59)
Fix $1\leq j\leq p$ and a connected subgraph ${\cal I}_{0}$ such that
$j\in{\cal I}_{0}$. For short, let $S$ be the support of $\beta^{{\cal
I}_{0}}$ and $\hat{S}$ be the support of $\hat{\beta}^{{\cal I}_{0}}$. The
event $\\{\hbox{\rm sgn}(\beta_{j})\neq\hbox{\rm
sgn}(\hat{\beta}_{j}),j\in{\cal I}_{0}\lhd{\cal U}_{p}^{*}\\}$ is identical to
the event of $\\{\hbox{\rm sgn}(\beta_{j})\neq\hbox{\rm
sgn}(\hat{\beta}_{j}),j\in S\cup\hat{S}\\}$. Moreover, Since ${\cal I}_{0}$
has a finite size, both $S$ and $\hat{S}$ have finite possibilities. So to
show (3.59), it is sufficient to show that for any fixed $1\leq j\leq p$,
connected subgraph ${\cal I}_{0}$, and subsets $S_{0},S_{1}\subset{\cal
I}_{0}$ such that $j\in S_{0}\cup S_{1}$,
$P(\hbox{\rm sgn}(\beta_{j})\neq\hbox{\rm
sgn}(\hat{\beta}_{j}),S=S_{0},\hat{S}=S_{1},j\in{\cal I}_{0}\lhd{\cal
U}_{p}^{*},B({\cal I}_{0})\cap S(\beta)=\emptyset)\leq
L_{p}[p^{-\rho_{j}^{*}}+p^{-(m_{0}+1)\vartheta}].$ (3.60)
We now show (3.60). The following lemma is proved in Ji and Jin (2011, A.4).
###### Lemma 17
Suppose the conditions of Theorem 8 hold. Over the event $\\{j\in{\cal
I}_{0}\lhd{\cal U}_{p}^{*}\\}\cap\\{B({\cal I}_{0})\cap
S(\beta)=\emptyset\\}$, $\|(\Omega\beta)^{{\cal I}_{0}}-\Omega^{{\cal
I}_{0},{\cal I}_{0}}\beta^{{\cal I}_{0}}\|_{\infty}\leq
C\tau_{p}(\log(p))^{-(1-\gamma)}$.
Write for short $\hat{M}=G^{{\cal I}_{0},{\cal I}_{0}}$ and $M=\Omega^{{\cal
I}_{0},{\cal I}_{0}}$. By definitions, $\hat{\beta}^{{\cal I}_{0}}$ is the
minimizer of the following functional
$Q(\xi)\equiv\frac{1}{2}(\tilde{Y}^{{\cal
I}_{0}}-\hat{M}\xi)^{\prime}\hat{M}^{-1}(\tilde{Y}^{{\cal
I}_{0}}-\hat{M}\xi)+\frac{1}{2}(u^{gs})^{2}\|\xi\|_{0}$, where $\xi$ is an
$|{\cal I}_{0}|\times 1$ vector whose coordinates are either $0$ or $\geq
v^{gs}$ in magnitude, $u^{gs}=\sqrt{2\vartheta\log(p)}$, and
$v^{gs}=\sqrt{2r\log(p)}$. In particular, $Q(\beta^{{\cal I}_{0}})\geq
Q(\hat{\beta}^{{\cal I}_{0}})$, or equivalently
$(\hat{\beta}^{{\cal I}_{0}}-\beta^{{\cal I}_{0}})^{\prime}(\tilde{Y}^{{\cal
I}_{0}}-\hat{M}\beta^{{\cal I}_{0}})\geq\frac{1}{2}(\hat{\beta}^{{\cal
I}_{0}}-\beta^{{\cal I}_{0}})^{\prime}\hat{M}(\hat{\beta}^{{\cal
I}_{0}}-\beta^{{\cal I}_{0}})+(|S_{1}|-|S_{0}|)\vartheta\log(p).$ (3.61)
Now, write for short $\delta=\tau_{p}^{-2}(\hat{\beta}^{{\cal
I}_{0}}-\beta^{{\cal I}_{0}})^{\prime}M(\hat{\beta}^{{\cal
I}_{0}}-\beta^{{\cal I}_{0}})$. First, by Schwartz inequality,
$[(\hat{\beta}^{{\cal I}_{0}}-\beta^{{\cal I}_{0}})^{\prime}(\tilde{Y}^{{\cal
I}_{0}}-\hat{M}\beta^{{\cal
I}_{0}})]^{2}\leq\delta\tau_{p}^{2}(\tilde{Y}^{{\cal
I}_{0}}-\hat{M}\beta^{{\cal I}_{0}})^{\prime}M^{-1}(\tilde{Y}^{{\cal
I}_{0}}-\hat{M}\beta^{{\cal I}_{0}})$. Second, by Lemma 17, $\tilde{Y}^{{\cal
I}_{0}}=w+M\beta^{{\cal I}_{0}}+rem$, where $w\sim N(0,M)$ and with
probability $1-o(1/p)$, $|rem|\leq C(\log(p))^{-(1-\gamma)}\tau_{p}$. Last,
with probability at least $(1-o(1/p))$,
$\parallel\hat{M}-M\parallel_{\infty}\leq
C\sqrt{\log(p)}p^{-[\kappa-(1-\vartheta)]/2}$. Inserting these into (3.61)
gives that with probability at least $(1-o(1/p))$,
$w^{\prime}M^{-1}w\geq\frac{1}{4}\biggl{[}\bigl{(}\sqrt{\delta
r}+\frac{(|S_{1}|-|S_{0}|)\vartheta}{\sqrt{\delta
r}}\bigr{)}_{+}\biggr{]}^{2}(2\log(p))+O((\log(p))^{\gamma})$. Since
$\gamma<1$, $O((\log(p))^{\gamma})$ is negligible. We note that
$w^{\prime}M^{-1}w\sim\chi_{|{\cal I}_{0}|}^{2}(0)$. Inserting this back to
(3.60), the left hand side $\leq\epsilon_{p}^{|S_{0}|}P(\chi_{|{\cal
I}_{0}|}^{2}(0)\geq[(\sqrt{\delta r}+(|S_{1}|-|S_{0}|)\vartheta/\sqrt{\delta
r})_{+}]^{2}(\log(p)/2))+o(1/p)$. Assume $\hbox{\rm
sgn}(\beta_{j})\neq\hbox{\rm sgn}(\hat{\beta}_{j})$, and fix all parameters
except $\delta$, $S_{0}$ and $S_{1}$. By arguments similar to the proof of
Lemma 18, the above quantity cannot achieve its maximum in the cases where
$S_{0}=S_{1}$. Hence we only need to consider the cases where $S_{0}\neq
S_{1}$. We also only need to consider the cases where
$\max(|S_{0}|,|S_{1}|)\leq m_{0}$, since the sum of the probabilities of other
cases is controlled by $p^{1-(m_{0}+1)\vartheta}$. The claim follows by the
definitions of $\rho_{j}^{*}$. $\Box$
## 4 Simulations
We conduct a small-scale simulation study to investigate the numerical
performance of Graphlet Screening and compare it with the lasso and the UPS.
The subset selection is not included for comparison since it is
computationally NP hard. We consider the experiments for both random design
and fixed design, where as before, the parameters $(\epsilon_{p},\tau_{p})$
are tied to $(\vartheta,r)$ by $\epsilon_{p}=p^{-\vartheta}$ and
$\tau_{p}=\sqrt{2r\log(p)}$ (we assume $\sigma=1$ for simplicity in this
section).
In random design settings where $p$ is not very large, we follow the spirit of
the refined UPS in Ji and Jin (2011) and propose the iterative Graphlet
Screening algorithm where we iterate Graphlet Screening for a few times ($\leq
5$). The main purpose for the iteration is to denoise the Gram matrix; see Ji
and Jin (2011, Section 3) for more discussion.
Even with the refinement as in Ji and Jin (2011, Section 3), UPS behaves
poorly for most examples presented below. Over close investigations, we find
out that this is due to the threshold choice in the initial $U$-step is too
low, and increasing the threshold largely increases the performance. Note that
the purpose of this step is to denoise the Gram matrix Ji and Jin (2011,
Section 3), not for signal retainment, and so a larger threshold helps.
In this section, we use this improved version of refined UPS, but for
simplicity, we still call it the refined UPS. With that being said, recall
that UPS is unable to resolve the problem of signal cancellation, so it
usually performs poorer than GS, especially when the effect of signal
cancellation is strong. For this reason, part of the comparison is between GS
and the lasso only.
The experiments with random design contain the following steps.
1. 1.
Fix $(p,\vartheta,r,\mu,\Omega)$ such that $\mu\in\Theta_{p}(\tau_{p})$.
Generate a vector $b=(b_{1},b_{2},\ldots,b_{p})^{\prime}$ such that
$b_{i}\stackrel{{\scriptstyle iid}}{{\sim}}\mathrm{Bernoulli}(\epsilon_{p})$,
and set $\beta=b\circ\mu$.
2. 2.
Fix $\kappa$ and let $n=n_{p}=p^{\kappa}$. Generate an $n\times p$ matrix with
$iid$ rows from $N(0,(1/n)\Omega)$.
3. 3.
Generate $Y\sim N(X\beta,I_{n})$, and apply the iterative Graphlet Screening,
the refined UPS and the lasso.
4. 4.
Repeat 1-3 independently, and record the average Hamming distances or the
Hamming ratio, the ratio of the Hamming distance and the number of the
signals.
The steps for fixed design experiments are similar, except for that $n_{p}=p$,
$X=\Omega^{1/2}$ and we apply GS and UPS directly.
GS uses tuning parameters $(m_{0},{\cal Q},u^{gs},v^{gs})$. We set $m_{0}=3$
for our experiments, which is usually large enough due to signal sparsity. The
choice of ${\cal Q}$ is not critical, as long as the corresponding parameter
$q$ satisfies (2.41), and we use the maximal ${\cal Q}$ satisfying (2.41) in
most experiments. Numerical studies below (e.g., Experiment $5a$) support this
point. In principle, the optimal choices of $(u^{gs},v^{gs})$ depend on the
unknown parameters $(\epsilon_{p},\tau_{p})$, and how to estimate them in
general settings is a lasting open problem (even for linear models with
orthogonal designs). Fortunately, our studies (e.g., Experiment 5b-5d) show
that mis-specifying parameters $(\epsilon_{p},\tau_{p})$ by a reasonable
amount does not significantly affect the performance of the procedure. For
this reason, in most experiments below, assuming $(\epsilon_{p},\tau_{p})$ are
known, we set $(u^{gs},v^{gs})$ as $(\sqrt{2\log(1/\epsilon_{p})},\tau_{p})$.
For the iterative Graphlet Screening, we use the same tuning parameters in
each iteration.
For the UPS and the refined UPS, we use the tuning parameters
$(u^{ups},v^{ups})=(u^{gs},v^{gs})$. For both the iterative Graphlet Screening
and the refined UPS, we use the following as the initial estimate:
$\hat{\beta}_{i}=\hbox{\rm sgn}(\tilde{Y}_{i})\cdot
1\\{|\tilde{Y}_{i}|\geq\tau_{p}\\}$, $1\leq i\leq p$, where
$\tilde{Y}=X^{\prime}Y$. The main purpose of initial estimate is to denoise
the Gram matrix, not for screening. We use glmnet package (Friedman et al,
2010) to perform lasso. To be fair in comparison, we apply the lasso with all
tuning parameters, and we report the Hamming error associated with the “best”
tuning parameter.
The simulations contain $6$ different experiments which we now describe
separately.
Experiment 1. The goal of this experiment is two-fold. First, we compare GS
with UPS and the lasso in the fixed design setting. Second, we investigate the
minimum signal strength levels $\tau_{p}$ required by these three methods to
yield exact recovery, respectively.
Fixing $p=0.5\times 10^{4}$, we let $\epsilon_{p}=p^{-\vartheta}$ for
$\vartheta\in\\{0.25,0.4,0.55\\}$, and $\tau_{p}\in\\{6,7,8,9,10\\}$. We use a
fixed design model where $\Omega$ is a symmetric diagonal block-wise matrix,
where each block is a $2\times 2$ matrix, with $1$ on the diagonals, and $\pm
0.7$ on the off-diagonals (the signs alternate across different blocks).
Recall the $\beta=b\circ\mu$. For each pair of $(\epsilon_{p},\tau_{p})$, we
generate $b$ as $p$ iid samples from $Bernoulli(\epsilon_{p})$, and we let
$\mu$ be the vector where the signs of $\mu_{i}=\pm 1$ with equal
probabilities, and $|\mu_{i}|\stackrel{{\scriptstyle
iid}}{{\sim}}0.8\nu_{\tau_{p}}+0.2h$, where $\nu_{\tau_{p}}$ is the point mass
at $\tau_{p}$ and $h(x)$ is the density of $\tau_{p}(1+V/6)$ with
$V\sim\chi_{1}^{2}$. The average Hamming errors across $40$ repetitions are
tabulated in Table 3. For all $(\vartheta,\tau_{p})$ in this experiment, GS
behaves more satisfactorily than the UPS, which in turn behaves more
satisfactorily than the lasso.
Suppose we say a method yields ‘exact recovery’ if the average Hamming error
$\leq 3$. Then, when $\vartheta=0.25$, the minimum $\tau_{p}$ for GS to yield
exact recovery is $\tau_{p}\approx 8$, but that for UPS and the lasso are much
larger ($\geq 10$). For larger $\vartheta$, the differences are less
prominent, but the pattern is similar.
The comparison between GS and UPS is particularly interesting. Due to the
block structure of $\Omega$, as $\vartheta$ decreases, the signals become
increasingly less sparse, and the effects of signal cancellation become
increasingly stronger. As a result, the advantage of GS over the UPS becomes
increasingly more prominent.
| $\tau_{p}$ | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---
$\vartheta=0.25$ | Graphic Screening | 24.7750 | 8.6750 | 2.8250 | 0.5250 | 0.1250
UPS | 48.5500 | 34.6250 | 36.3500 | 30.8750 | 33.4000
lasso | 66.4750 | 47.7000 | 43.5250 | 35.2500 | 35.0500
$\vartheta=0.40$ | Graphic Screening | 6.9500 | 2.1500 | 0.4000 | 0.0750 | 0.0500
UPS | 7.7500 | 4.0000 | 2.2000 | 2.7750 | 2.4250
lasso | 12.8750 | 6.8000 | 4.3250 | 3.7500 | 2.6750
$\vartheta=0.55$ | Graphic Screening | 1.8750 | 0.8000 | 0.3250 | 0.2250 | 0.1250
UPS | 1.8750 | 0.8000 | 0.3250 | 0.2250 | 0.1250
lasso | 2.5000 | 1.1000 | 0.7750 | 0.2750 | 0.1250
Table 3: Comparison of average Hamming errors (Experiment 1).
Experiment 2. In this experiment, we compare GS, UPS and the lasso in the
random design setting, and investigate the effect of signal cancellation on
their performances. We fix $(p,\kappa,\vartheta,r)=(0.5\times
10^{4},0.975,0.35,3)$, and assume $\Omega$ is blockwise diagonal. We generate
$\mu$ as in Experiment $1$, but to better illustrate the difference between
UPS and GS in the presence of signal cancellation, we generate the vector $b$
differently and allow it to depend on $\Omega$. The experiment contains $2$
parts, $2a$ and $2b$.
In Experiment 2a, $\Omega$ is the block-wise matrix where each block is $2$ by
$2$ matrix with $1$ on the diagonals and $\pm.5$ on the off diagonals (the
signs alternate on adjacent blocks). According to the blocks in $\Omega$, the
set of indices $\\{1,2,\ldots,p\\}$ are also partitioned into blocks
accordingly. For any fixed $\vartheta$ and
$\eta\in\\{0,0.01,0.02,0.03,0.04,0.05,0.06,0.1,0.2\\}$, we randomly choose
$(1-2p^{-\vartheta})$ fraction of the blocks (of indices) where $b$ is $0$ at
both indices, $2(1-\eta)p^{-\vartheta}$ fraction of the blocks where $b$ is
$0$ at one index and $1$ at the other (two indices are equally likely to be
$0$), $2\eta p^{-\vartheta}$ faction of the blocks where $b$ is $1$ on both
indices.
Experiment 2b has similar settings, where the difference is that (a) we choose
$\Omega$ to be a diagonal block matrix where each block is a $4$ by $4$ matrix
(say, denoted by $A$) satisfying $A(i,j)=1\\{i=j\\}+0.4\cdot
1\\{|i-j|=1\\}\cdot\hbox{\rm sgn}(6-i-j)+0.05\\{|i-j|\geq 2\\}\cdot\hbox{\rm
sgn}(5.5-i-j)$, $1\leq i,j\leq 4$, and (b) $(1-4p^{-\vartheta})$ is the
fraction of blocks where $b$ is nonzero in $k=0$ indices,
$4(1-\eta)p^{-\vartheta}$ is that for $k=1$, and $4\eta p^{-\vartheta}$ is
that for $k\in\\{2,3,4\\}$ in total. In a block where $\beta$ is nonzero at
$k$ indices, all configurations with $k=1$ are equally likely, and all those
with $k\in\\{2,3,4\\}$ are equally likely.
The average Hamming ratio results across $40$ runs for two Experiment 2a and
2b are reported in Figure 3, where UPS and GS consistently outperform the
lasso. Additionally, when $\eta$ is small, the effect of signal cancellation
is negligible, so UPS and GS have similar performances. However, when $\eta$
increases, the effects of signal cancellation grows, and the advantage of GS
over UPS becomes increasingly more prominent.
(a) Blockwise diagonal $\Omega$ in $2\times 2$ blocks
(b) Blockwise diagonal $\Omega$ in $4\times 4$ blocks
Figure 3: Hamming ratio results in Experiment 2
Through Experiment 1-2, the comparison of UPS and GS is more or less
understood. For this reason, we do not include UPS for study in Experiment
3-5, but we include UPS for study in Experiment 6 where we investigate
robustness of all three methods.
Experiment 3. In this experiment, we investigate how different choices of
signal vector $\beta$ affect the comparisons of GS and the lasso. We use a
random design model, and $\Omega$ is a symmetric tri-diagonal correlation
matrix where the vector on each sub-diagonal consists of blocks of
$(.4,.4,-.4)^{\prime}$. Fix $(p,\kappa)=(0.5\times 10^{4},0.975)$ (note
$n=p^{\kappa}\approx 4,000$). We let $\epsilon_{p}=p^{-\vartheta}$ with
$\vartheta\in\\{0.35,0.5\\}$ and let $\tau_{p}\in\\{6,8,10\\}$. For each
combination of $(\epsilon_{p},\tau_{p})$, we consider two choices of $\mu$.
For the first choice, we let $\mu$ be the vector where all coordinates equal
to $\tau_{p}$ (note $\beta$ is still sparse). For the second one, we let $\mu$
be as in Experiment $1$. The average Hamming ratios for both procedures across
$40$ repetitions are tabulated in Table 4.
$\tau_{p}$ | 6 | 8 | 10
---|---|---|---
Signal Strength | Equal | Unequal | Equal | Unequal | Equal | Unequal
$\vartheta=0.35$ | Graphic Screening | 0.0810 | 0.0825 | 0.0018 | 0.0034 | 0 | 0.0003
lasso | 0.2424 | 0.2535 | 0.1445 | 0.1556 | 0.0941 | 0.1109
$\vartheta=0.5$ | Graphic Screening | 0.0315 | 0.0297 | 0.0007 | 0.0007 | 0 | 0
lasso | 0.1107 | 0.1130 | 0.0320 | 0.0254 | 0.0064 | 0.0115
Table 4: Hamming ratio results of Experiment $3$, where “Equal” and “Unequal”
stand for the first and the second choices of $\mu$, respectively.
Experiment 4. In this experiment, we generate $\beta$ the same way as in
Experiment $1$, and investigate how different choices of design matrices
affect the performance of the two methods. Setting
$(p,\vartheta,\kappa)=(0.5\times 10^{4},0.35,0.975)$ and
$\tau_{p}\in\\{6,7,8,9,10,11,12\\}$, we use Gaussian random design model for
the study. The experiment contains $3$ sub-experiments $4a$-$4c$.
In Experiment $4a$, we set $\Omega$ as the symmetric diagonal block-wise
matrix, where each block is a $2\times 2$ matrix, with $1$ on the diagonals,
and $\pm 0.5$ on the off-diagonals (the signs alternate across different
blocks). The average Hamming ratios of $40$ repetitions are reported in Figure
4.
In Experiment $4b$, we set $\Omega$ as a symmetric penta-diagonal correlation
matrix, where the main diagonal are ones, the first sub-diagonal consists of
blocks of $(.4,.4,-.4)^{\prime}$, and the second sub-diagonal consists of
blocks of $(.05,-.05)^{\prime}$. The average Hamming ratios across $40$
repetitions are reported in Figure 4.
In Experiment $4c$, we generate $\Omega$ as follows. First, we generate
$\Omega$ using the function _sprandsym(p,K/p)_ in _matlab_. We then set the
diagonals of $\Omega$ to be zero, and remove some of entries so that $\Omega$
is $K$-sparse for a pre-specified $K$. We then normalize each non-zero entry
by the sum of the absolute values in that row or that column, whichever is
larger, and multiply each entry by a pre-specified positive constant $A$.
Last, we set the diagonal elements to be 1. We choose $K=3$ and $A=0.7$, draw
$5$ different $\Omega$ with this method, and for each of them, we draw
$(X,\beta,z)$ $10$ times independently. The average Hamming ratios are
reported in Figure 4. The results suggest that GS is consistently better than
the lasso.
Figure 4: $x$-axis: $\tau_{p}$. $y$-axis: Hamming ratios. Left to right:
Experiment $4a$, $4b$, and $4c$.
Experiment 5. In this experiment, we investigate how sensitive GS is with
respect to the tuning parameters. The experiment contains $4$ sub-experiments,
$5a$-$5d$. In Experiment $5a$, we investigate how sensitive the procedure is
with respect to the tuning parameter $q$ in ${\cal Q}$ (recall that the main
results hold as long as $q$ fall into the range given in (2.41)), where we
assume $(\epsilon_{p},\tau_{p})$ are known. In Experiment $5b$-$5d$, we mis-
specify $(\epsilon_{p},\tau_{p})$ by a reasonably small amount, and
investigate how the mis-specification affect the performance of the procedure.
For the whole experiment, we choose $\beta$ the same as in Experiment $1$, and
$\Omega$ the same as in Experiment $4b$. We use a fixed design model in
Experiment $5a$-$5c$, and a random design model in Experiment $5d$. For each
sub-experiment, the results are based on $40$ independent repetitions. We now
describe the sub-experiments with details.
In Experiment $5a$, we choose $\vartheta\in\\{0.35,0.6\\}$ and
$r\in\\{1.5,3\\}$. In GS, let $q_{max}=q_{max}(\hat{D},\hat{F})$ be the
maximum value of $q$ satisfying (2.41). For each combination of
$(\vartheta,r)$ and $(\hat{D},\hat{F})$, we choose
$q(\hat{D},\hat{F})=q_{max}(\hat{D},\hat{F})\times\\{0.7,0.8,0.9,1,1.1,1.2\\}$
for our experiment. The results are tabulated in Table 5, which suggest that
different choices of $q$ have little influence over the variable selection
errors. We must note that the larger we set $q(\hat{D},\hat{F})$, the faster
the algorithm runs.
$q(\hat{F},\hat{D})/q_{max}(\hat{F},\hat{D})$ | 0.7 | 0.8 | 0.9 | 1 | 1.1 | 1.2
---|---|---|---|---|---|---
$(\vartheta,r)=(0.35,1.5)$ | 0.0782 | 0.0707 | 0.0661 | 0.0675 | 0.0684 | 0.0702
$(\vartheta,r)=(0.35,3)$ | 0.0066 | 0.0049 | 0.0036 | 0.0034 | 0.0033 | 0.0032
$(\vartheta,r)=(0.6,1.5)$ | 0.1417 | 0.1417 | 0.1417 | 0.1417 | 0.1417 | 0.1417
$(\vartheta,r)=(0.6,3)$ | 0.0089 | 0.0089 | 0.0089 | 0.0089 | 0.0089 | 0.0089
Table 5: Hamming ratio results in Experiment $5a$.
In Experiment $5b$, we use the same settings as in Experiment $5a$, but we
assume $\vartheta$ (and so $\epsilon_{p}$) is unknown (the parameter $r$ is
assumed as known, however), and let $\vartheta^{*}$ is the misspecified value
of $\vartheta$. We take
$\vartheta^{*}\in\vartheta\times\\{0.85,0.925,1,1.075,1.15,1.225\\}$ for the
experiment.
In Experiment $5c$, we use the same settings as in Experiment $5b$, but we
assume $r$ (and so $\tau_{p}$) is unknown (the parameter $\vartheta$ is
assumed as known, however), and let $r^{*}$ is the misspecified value of $r$.
We take $r^{*}=r\times\\{0.8,0.9,1,1.1,1.2,1.3\\}$ for the experiment.
In Experiment $5b$-$5c$, we run GS with tuning parameters set as in Experiment
$1$, except $\vartheta$ or $r$ are replaced by the misspecified counterparts
$\vartheta^{*}$ and $r^{*}$, respectively. The results are reported in Table
6, which suggest that the mis-specifications have little effect as long as
$r^{*}/r$ and $\vartheta^{*}/\vartheta$ are reasonably close to $1$.
$\vartheta^{*}/\vartheta$ | 0.85 | 0.925 | 1 | 1.075 | 1.15 | 1.225
---|---|---|---|---|---|---
$(\vartheta,r)=(0.35,1.5)$ | 0.0799 | 0.0753 | 0.0711 | 0.0710 | 0.0715 | 0.0746
$(\vartheta,r)=(0.35,3)$ | 0.0026 | 0.0023 | 0.0029 | 0.0030 | 0.0031 | 0.0028
$(\vartheta,r)=(0.6,1.5)$ | 0.1468 | 0.1313 | 0.1272 | 0.1280 | 0.1247 | 0.1296
$(\vartheta,r)=(0.6,3)$ | 0.0122 | 0.0122 | 0.0139 | 0.0139 | 0.0130 | 0.0147
$r^{*}/r$ | 0.8 | 0.9 | 1 | 1.1 | 1.2 | 1.3
$(\vartheta,r)=(0.35,1.5)$ | 0.0843 | 0.0731 | 0.0683 | 0.0645 | 0.0656 | 0.0687
$(\vartheta,r)=(0.35,3)$ | 0.0062 | 0.0039 | 0.0029 | 0.0030 | 0.0041 | 0.0054
$(\vartheta,r)=(0.6,1.5)$ | 0.1542 | 0.1365 | 0.1277 | 0.1237 | 0.1229 | 0.1261
$(\vartheta,r)=(0.6,3)$ | 0.0102 | 0.0076 | 0.0085 | 0.0059 | 0.0051 | 0.0076
Table 6: Hamming ratio results in Experiment $5b$ (top) and in Experiment $5c$
(bottom).
In Experiment $5d$, we re-examine the mis-specification issue in the random
design setting. We use the same settings as in Experiment $5b$ and Experiment
$5c$, except for (a) while we use the same $\Omega$ as in Experiment $5b$, the
design matrix $X$ are generated according to the random design model as in
Experiment $4b$, and (b) we only investigate for the case of $r=2$ and
$\vartheta\in\\{0.35,0.6\\}$. The results are summarized in Table 7, which is
consistent with the results in $5b$-$5c$.
$\vartheta^{*}/\vartheta$ | 0.85 | 0.925 | 1 | 1.075 | 1.15 | 1.225
---|---|---|---|---|---|---
$(\vartheta,r)=(0.35,2)$ | 0.1730 | 0.1367 | 0.1145 | 0.1118 | 0.0880 | 0.0983
$(\vartheta,r)=(0.6,2)$ | 0.0583 | 0.0591 | 0.0477 | 0.0487 | 0.0446 | 0.0431
$r^{*}/r$ | 0.8 | 0.9 | 1 | 1.1 | 1.2 | 1.3
$(\vartheta,r)=(0.35,2)$ | 0.1881 | 0.1192 | 0.1275 | 0.1211 | 0.1474 | 0.1920
$(\vartheta,r)=(0.6,2)$ | 0.0813 | 0.0515 | 0.0536 | 0.0397 | 0.0442 | 0.0510
Table 7: Hamming ratio results in Experiment $5d$.
Experiment $6$. In this experiment, we investigate the robustness of all three
methods for the mis-specification of the linear model (1.1). We use the random
design setting as in Experiment $4b$, except that we fix
$(\vartheta,r)=(0.35,3)$. The experiment contains $3$ sub-experiments,
$6a$-$6c$, where we consider three scenarios where the linear model (1.1) is
in question: the presence of nonGaussianity, the presence of missing
predictors, and the presence of non-linearity, correspondingly.
In Experiment $6a$, we assume the noise vector $z$ in Model (1.1) is
nonGaussian, where the coordinates are iid samples from a $t$-distribution
with the same degree of freedom (df) (we assume that $z$ is normalized so each
coordinate has unit variance), where the df range in
$\\{3,4,5,6,7,8,9,10,30,50\\}$. Figure 5(a) shows how the Hamming ratios
(based on $40$ independent repetitions) change when the $df$ decreases. The
results suggest that all three methods are reasonably robust against
nonGaussianity, but GS continues to have the best performance.
In Experiment $6b$, we assume that the true model is $Y=X\beta+z$ where
$(X,\beta,z)$ are generated as in $4b$, but the model that is accessible to us
is a misspecified model where the some of the true predictors are missing. Fix
$\eta\in(0,1)$, and let $S(\beta)$ be the support of $\beta$. For each $i\in
S(\beta)$, we flip a coin that lands on head with probability $\eta$, and we
retain $i$ if and only if the coin lands on tail. Let $S^{*}\subset S(\beta)$
be the set of retained indices, and let $R=S^{*}\cup S^{c}$. The misspecified
model we consider is then $Y=X^{\otimes,R}\beta^{R}+z$.
For the experiment, we let $\eta$ range in
$0.02\times\\{0,1,2,3,4,5,6,7,8,9,10\\}$. The average Hamming ratios (based on
$40$ independent repetitions) are reported in Figure 5(b). The results suggest
that all three results are reasonably robust to missing predictors, with the
lasso being the most robust. However, as long as the proportion of true
predictors that are missing is reasonably small (say, $\eta\leq.1$), GS
continues to outperform UPS and the lasso.
In Experiment $6c$, for $i=1,\ldots,n$, the true model is an additive model in
the form of $Y_{i}=\sum_{j=1}^{p}f_{j}(X_{ij})\beta_{j}+z_{i}$, but what is
accessible to us is the linear model
$Y_{i}=\sum_{j=1}^{p}X_{ij}\beta_{j}+z_{i}$ (and thus misspecified; the true
model is non-linear). For experiment, we let $(X,\beta,z)$ be generated as in
$4b$, and $S(\beta)$ be the support of $\beta$. Fixing $\eta\in(0,1)$, for
each $i\in S(\beta)$, we flip a coin that lands on head with probability
$\eta$, and let $S_{nl}\subset S(\beta)$ be all indices of the heads. We then
randomly split $S_{nl}$ into two sets $S_{1}$ and $S_{2}$ evenly. For
$j=1,\ldots,p$, we define $f_{j}(x)=[\hbox{\rm sgn}(x)x^{2}\cdot 1\\{j\in
S_{1}\\}+(e^{\sqrt{n}x}-a_{j})\cdot 1\\{j\in S_{2}\\}+x\cdot 1\\{j\in
S_{nl}^{c}\\}]/c_{j}$, where $a_{j}$ and $c_{j}$ are constants such that
$\\{f_{j}(X(i,j))\\}_{i=1}^{n}$ has mean $0$ and variance $1/n$.
For the experiment, we let $\eta$ range in $.05\times\\{0,1,2,3,4,5,6,7,8\\}$.
The average Hamming ratios (based on $40$ independent repetitions) are
reported in Figure 5(c). The results suggest that all three methods are
reasonably robust to the presence of nonlinearity, and GS continues to
outperform UPS and the lasso when the degree of nonlinearly is moderate (say,
$\eta<.2$).
(a) nonGaussian noise
(b) Missing predictors
(c) Non-linear predictors
Figure 5: Hamming ratio results in Experiment $6$
## 5 Connection to existing literature and possible extensions
Our idea of utilizing graph sparsity is related to the graphical lasso
(Meinshausen and Buhlmann, 2006; Friedman et al, 2008), which also attempts to
exploit graph structure. However, the setting we consider here is different
from that in Meinshausen and Buhlmann (2006); Friedman et al (2008), and our
emphasis on precise optimality and calibration is also very different. Our
method allows nearly optimal detection of very rare and weak effects, because
they are based on careful analysis that has revealed a number of subtle high-
dimensional effects (e.g., phase transitions) that we properly exploit.
Existing methodologies are not able to exploit or capture these phenomena, and
can be shown to fail at the levels of rare and weak effects where we are
successful.
The paper is closely related to the recent work by Ji and Jin (Ji and Jin,
2011) (see also Fan and Lv (2008); Genovese et al (2012)), and the two papers
use a similar rare and weak signal framework and a similar random design
model. However, they are different in important ways, since the technical
devise developed in Ji and Jin (2011) can not be extended to the current
study. For example, the lower bound derived in this paper is different and
sharper than that in Ji and Jin (2011). Also, the procedure in Ji and Jin
(2011) relies on marginal regression for screening. The limitation of marginal
regression is that it neglects the graph structure of GOSD for the regularized
Gram matrix (1.5), so that it is incapable of picking variables that have weak
marginal correlation but significant joint correlation to $Y$. Correct
selection of such hidden significant variables, termed as the challenge of
signal cancellation (Wasserman and Roeder, 2009), is the difficulty at the
heart of the variable selection problem. One of the main innovation of GS is
that it uses the graph structure to guide the screening, so that it is able to
successfully overcome the challenge of signal cancellation.
Additionally, two papers have very different objectives, and consequently the
underlying analysis are very different. The main results of each of these two
papers can not be deduced from the other. For example, to assess optimality,
Ji and Jin (2011) uses the criterion of the partition of the phase diagram,
while the current paper uses the minimax Hamming distance. Given the
complexity of the high dimensional variable selection, one type of optimality
does not imply the other, and vice versa. Also, the main result in Ji and Jin
(2011) focuses on conditions under which the optimal rate of convergence is
$L_{p}p^{1-(\vartheta+r)^{2}/(4r)}$ for the whole phase space. While this
overlaps with our Corollaries 10 and 11, we must note that Ji and Jin (2011)
deals with the much more difficult cases where $r/\vartheta$ can get arbitrary
large; and to ensure the success in that case, they assume very strong
conditions on the design matrix and the range of the signal strength. On the
other hand, the main focus of the current paper is on optimal variable
selection under conditions (of the Gram matrix $G$ as well as the signal
vector $\beta$) that are as general as possible.
While the study in this paper has been focused on the Random Design model
RD$(\vartheta,\kappa,\Omega)$, extensions to deterministic design models are
straightforward (in fact, in Corollary 7, we have already stated some results
on deterministic design models), and the omission of discussion on the latter
is largely for technical simplicity and the sake of space. In fact, for models
with deterministic designs, since the likelihood ratio test in the derivation
of the lower bound matches the penalized MLE in the cleaning step of GS, the
optimality of GS follows from the Sure Screening and Separable After Screening
properties of GS. The proof of these properties, and therefore the optimality
of GS, follows the same line as those for random design as long as
$\max_{j}|\sum_{i}\beta_{i}G(i,j)I\\{\Omega^{*,\delta}(i,j)=0\\}|/\tau_{p}$ is
small. This last condition on $G$ holds when
$p^{1-\vartheta}\|G-\Omega\|_{\infty}=o(1)$ with a certain $\Omega\in{\cal
M}^{*}_{p}(\gamma,c_{0},g,A)$. Alternatively, this condition holds when
$p^{1-\vartheta}\|G-\Omega\|_{\infty}^{2}\log p=o(1)$ with $\Omega\in{\cal
M}^{*}_{p}(\gamma,c_{0},g,A)$, provided that $\hbox{\rm sgn}(\beta_{j})$ are
iid symmetric random variables as in Candes and Plan (2009).
In this paper, we assume the signal vector $\beta$ is independent of the
design matrix $X$, and that $\beta$ is modeled by a Bernoulli model through
$\beta=b\circ\mu$. Both assumptions can be relaxed. In fact, in order for GS
to work, what we really need is some decomposability condition similar to that
in Lemma 1, where except for negligible probabilities, the maximum size of the
graphlets $m_{0}^{*}=m_{0}^{*}(S(\beta),G,\delta)$ is small. In many
situations, we can show that $m_{0}^{*}$ does not exceed a fixed integer. One
of such examples is as follows. Suppose for any fixed integer $m\geq 1$ and
size-$m$ subset $S$ of $\\{1,2,\ldots,p\\}$, there are constants $C>0$ and
$d>0$ such that the conditional probability $P(\beta_{j}\neq 0,\forall j\in
S|X)\leq Cp^{-dm}$. In fact, when such a condition holds, the claim follows
since ${\cal G}^{*,\delta}$ has no more than $C(eK)^{m}$ size-$m$ connected
subgraphs if it is $K$-sparse. See the proof of Lemma 1 for details. Note that
when $\epsilon_{p}=p^{-\vartheta}$ as in the ARW, then the condition holds for
the Bernoulli model in Lemma 1, with $d=\vartheta$. Note also that the
Bernoulli model can be replaced by some Ising models.
Another interesting direction of future research is the extension of GS to
more general models such as logistic regression. The extension of the lower
bound in Theorem 6 is relatively simple since the degree of GOLF can be
bounded using the true $\beta$. This indicates the optimality of GS in
logistic and other generalized linear models as long as proper generalized
likelihood ratio or Bayes tests are used in both the $GS$\- and $GC$-steps.
## 6 Proofs
In this section, we provide all technical proofs. We assume $\sigma=1$ for
simplicity.
### 6.1 Proof of Lemma 1
When ${\cal G}_{S}^{*,\delta}$ contains a connected subgraph of size $\geq
m_{0}+1$, it must contain a connected subgraph with size $m_{0}+1$. By Frieze
and Molloy (1999), there are $\leq p(eK)^{m_{0}+1}$ connected subgraph of size
$m_{0}+1$. Therefore, the probability that ${\cal G}_{S}^{*,\delta}$ has a
connected subgraph of size $(m_{0}+1)$ $\leq
p(eK)^{m_{0}+1}\epsilon_{p}^{m_{0}+1}$. Combining these gives the claim.
$\Box$
### 6.2 Proof of Theorem 6
Write for short $\rho_{j}^{*}=\rho_{j}^{*}(\vartheta,r,a,\Omega)$. Without
loss of generality, assume
$\rho_{1}^{*}\leq\rho_{2}^{*}\leq\ldots\leq\rho_{p}^{*}$. We construct indices
$i_{1}<i_{2}<\ldots<i_{m}$ as follows. (a) start with $B=\\{1,2,\ldots,p\\}$
and let $i_{1}=1$, (b) updating $B$ by removing $i_{1}$ and all nodes $j$ that
are neighbors of $i_{1}$ in GOLF, let $i_{2}$ be the smallest index, (c)
defining $i_{3},i_{4},\ldots,i_{m}$ by repeating (b), and terminates the
process when no indices is left in $B$. Since each time we remove at most
$d_{p}({\cal G}^{\diamond})$ nodes, it follows that
$\sum_{j=1}^{p}p^{-\rho_{j}^{*}}\leq d_{p}({\cal
G}^{\diamond})\sum_{k=1}^{m}p^{-\rho_{i_{k}}^{*}}.$ (6.62)
For each $1\leq j\leq p$, as before, let $(V_{0j}^{*},V_{1j}^{*})$ be the
least favorable configuration, and let
$(\theta_{*j}^{(0)},\theta_{*j}^{(1)})=\mathrm{argmin}_{\\{\theta^{(0)}\in
B_{V_{0j}^{*}},\theta^{(1)}\in B_{V_{1j}^{*}},\hbox{\rm
sgn}(\theta^{(0)})\neq\hbox{\rm
sgn}(\theta^{(1)})\\}}\alpha(\theta^{(0)},\theta^{(1)};\Omega)$. By our
notations, it is seen that
$\rho_{j}^{*}=\eta(V_{0j}^{*},V_{1j}^{*};\Omega),\qquad\alpha^{*}(V_{0j}^{*},V_{1j}^{*};\Omega)=\alpha(\theta_{*j}^{(0)},\theta_{*j}^{(1)};\Omega).$
(6.63)
In case $(\theta_{*j}^{(0)},\theta_{*j}^{(1)})$ is not unique, pick one
arbitrarily. We construct a $p\times 1$ vector $\mu^{*}$ as follows. Fix
$j\in\\{i_{1},\cdots,i_{m}\\}$. For all indices in $V_{0j}^{*}$, set the
constraint of $\mu^{*}$ on these indices to be $\theta_{*j}^{(0)}$. For any
index $i\notin\cup_{k=1}^{m}V_{0i_{k}}^{*}$, set $\mu^{*}_{i}=\tau_{p}$. Since
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)\geq\inf_{\hat{\beta}}H_{p}(\hat{\beta};\epsilon_{p},n_{p},\mu^{*},\Omega)=\inf_{\hat{\beta}}\sum_{i=1}^{p}P(\hbox{\rm
sgn}(\hat{\beta}_{j})\neq\hbox{\rm sgn}(\beta_{j})),$ (6.64)
it follows that
$\mathrm{Hamm}_{p}^{*}(\vartheta,\kappa,r,a,\Omega)\geq\sum_{k=1}^{m}\sum_{j\in
V_{0i_{k}}\cup V_{1i_{k}}}P(\hbox{\rm sgn}(\hat{\beta}_{j})\neq\hbox{\rm
sgn}(\beta_{j})),$ (6.65)
where $\beta=b\circ\mu^{*}$ in (6.64)-(6.65). Combining (6.62) and (6.65), to
show the claim, we only need to show that for any $1\leq k\leq m$ and any
procedure $\hat{\beta}$,
$\sum_{j\in V_{0i_{k}}\cup V_{1i_{k}}}P(\hbox{\rm
sgn}(\hat{\beta}_{j})\neq\hbox{\rm sgn}(\beta_{j}))\geq
L_{p}p^{-\rho_{i_{k}}^{*}}.$ (6.66)
Towards this end, we write for short $V_{0}=V_{0i_{k}}$, $V_{1}=V_{1i_{k}}$,
$V=V_{0}\cup V_{1}$, $\theta^{(0)}=\theta_{*i_{k}}^{(0)}$, and
$\theta^{(1)}=\theta_{*i_{k}}^{(1)}$. Note that by Lemma 22,
$|V|\leq(\vartheta+r)^{2}/(2\vartheta r).$ (6.67)
Consider a test setting where under the null $H_{0}$,
$\beta=\beta^{(0)}=b\circ\mu^{*}$ and
$I_{V}\circ\beta^{(0)}=I_{V}\circ\theta^{(0)}$, and under the alternative
$H_{1}$, $\beta=\beta^{(1)}$ which is constructed by keeping all coordinates
of $\beta^{(0)}$ unchanged, except those coordinates in $V$ are perturbed in a
way so that $I_{V}\circ\beta^{(1)}=I_{V}\circ\theta^{(1)}$. In this
construction, both $\beta^{(0)}$ and $\beta^{(1)}$ are assumed as known, but
we don’t know which of $H_{0}$ and $H_{1}$ is true. In the literature, it is
known that $\inf_{\hat{\beta}}\sum_{j\in V}P(\hbox{\rm
sgn}(\hat{\beta}_{j})\neq\hbox{\rm sgn}(\beta_{j}))$ is not smaller than the
minimum sum of Type I and Type II errors associated with this testing problem.
Note that by our construction and (6.63), the right hand side is
$\alpha^{*}(V_{0},V_{1};\Omega)$. At the same time, it is seen the optimal
test statistic is
$Z\equiv(\theta^{(1)}-\theta^{(0)})^{\prime}X^{\prime}(Y-X\beta^{(0)})$. It is
seen that up to some negligible terms, $Z\sim
N(0,\alpha^{*}(V_{0},V_{1};\Omega)\tau_{p}^{2})$ under $H_{0}$, and $Z\sim
N(\alpha^{*}(V_{0},V_{1};\Omega)\tau_{p}^{2},\alpha^{*}(V_{0},V_{1};\Omega)\tau_{p}^{2})$
under $H_{1}$. The optimal test is to reject $H_{0}$ when $Z\geq
t[\alpha^{*}(V_{0},V_{1};\Omega)]^{1/2}\tau_{p}$ for some threshold $t$, and
the minimum sum of Type I and Type II error is
$\inf_{t}\bigl{\\{}\epsilon_{p}^{|V_{0}|}\bar{\Phi}(t)+\epsilon_{p}^{|V_{1}|}\Phi(t-[\alpha^{*}(V_{0},V_{1};\Omega)]^{1/2}\tau_{p})\bigr{\\}}.$
(6.68)
Here, we have used $P(H_{0})\sim\epsilon_{p}^{|V_{0}|}$ and
$P(H_{1})\sim\epsilon_{p}^{|V_{1}|}$, as a result of the Binomial structure in
$\beta$. It follows that $\sum_{j\in V}P(\hbox{\rm
sgn}(\hat{\beta}_{j})\neq\hbox{\rm
sgn}(\beta_{j}))\gtrsim\inf_{t}\bigl{\\{}\epsilon_{p}^{|V_{0}|}\bar{\Phi}(t)+\epsilon_{p}^{|V_{1}|}\Phi(t-[\alpha^{*}(V_{0},V_{1};\Omega)]^{1/2}\tau_{p})\bigr{\\}}$.
Using Mills’ ratio and definitions, the right hand side $\geq
L_{p}p^{-\eta(V_{0},V_{1};\Omega)}$, and (6.66) follows by recalling (6.63).
$\Box$
### 6.3 Proof of Corollaries 10, 11, and 12
When $a>a_{g}^{*}(\Omega)$, $\rho_{j}^{*}(\vartheta,r,a,\Omega)$ does not
depend on $a$, and have an alternative expression as follows. For any subsets
$D$ and $F$ of $\\{1,2,\ldots,p\\}$, let $\omega(D,F;\Omega)$ be as in (2.39).
Introduce $\rho(D,F;\Omega)=\rho(D,F;\vartheta,r,a,\Omega,p)$ by
$\rho(D,F;\Omega)=\frac{(|D|+2|F|)\vartheta}{2}+\left\\{\begin{array}[]{ll}\frac{1}{4}\omega(D,F;\Omega)r,&|D|\text{
is even},\\\
\frac{\vartheta}{2}+\frac{1}{4}\bigl{[}(\sqrt{\omega(D,F;\Omega)r}-\frac{\vartheta}{\sqrt{\omega(D,F;\Omega)r}})_{+}\bigr{]}^{2},&|D|\text{
is odd}.\end{array}\right.$ (6.69)
The following lemma is proved in Section 6.3.4.
###### Lemma 18
Fix $m_{0}\geq 1$, $(\vartheta,\kappa)\in(0,1)^{2}$, $r>0$, $c_{0}>0$, and
$g>0$ such that $\kappa>(1-\vartheta)$. Suppose the conditions of Theorem 6
hold, and that for sufficiently large $p$, (2.36) is satisfied. Then as
$p\rightarrow\infty$, $\rho_{j}^{*}(\vartheta,r,a,\Omega)$ does not depend on
$a$, and satisfies $\rho_{j}^{*}(\vartheta,r,a,\Omega)=\min_{\\{(D,F):{j\in
D\cup F},D\cap F=\emptyset,D\neq\emptyset,|D\cup F|\leq
g\\}}\rho(D,F;\Omega)$.
We now show Corollaries 10-12. Write for short $\omega=\rho(D,F;\Omega)$,
$T=r/\vartheta$, and $\lambda_{k}^{*}=\lambda_{k}^{*}(\Omega)$. The following
inequality is frequently used below, the proof of which is elementary so we
omit it:
$\omega\geq\lambda_{k}^{*}|D|,\qquad\mbox{where $k=|D|+|F|$}.$ (6.70)
To show these corollaries, it is sufficient to show for all subsets $D$ and
$F$ of $\\{1,2,\ldots,p\\}$,
$\rho(D,F;\Omega)\geq(\vartheta+r)^{2}/(4r),\qquad|D|\geq 1,$ (6.71)
where $\rho(D,F;\Omega)$ is as in (6.69). By basic algebra, (6.71) is
equivalent to
$\left\\{\begin{array}[]{ll}(\omega T+1/(\omega T)-2)1\\{\omega T\geq
1\\}\geq(T+1/T-2(|D|+2|F|)),&\qquad\mbox{$|D|$ is odd},\\\
\omega\geq\frac{2}{T}[(T+1/T)/2+1-(|D|+2|F|)],&\qquad\mbox{$|D|$ is
even}.\end{array}\right.$ (6.72)
Note that when $(|D|,|F|)=(1,0)$, this claim holds trivially, so it is
sufficient to consider the case where
$|D|+|F|\geq 2.$ (6.73)
We now show that (6.72) holds under the conditions of each of corollaries.
#### 6.3.1 Proof of Corollary 10
In this corollary, $1<(T+1/T)/2\leq 3$, and if either (a) $|D|+2|F|\geq 3$ and
$|D|$ is odd or (b) $|D|+2|F|\geq 4$ and $|D|$ is even, the right hand side of
(6.72) $\leq 0$, so the claim holds trivially. Therefore, all we need to show
is the case where $(|D|,|F|)=(2,0)$. In this case, since each off-diagonal
coordinate $\leq 4\sqrt{2}-5\equiv\rho_{0}$, it follows from definitions and
basic algebra that $\omega\geq 2(1-\rho_{0})=4(3-2\sqrt{2})$, and (6.72)
follows by noting that $\frac{2}{T}[(T+1/T)/2+1-(|D|+2|F|)]=(1-1/T)^{2}\leq
4(3-2\sqrt{2})$. $\Box$
#### 6.3.2 Proof of Corollary 11
In this corollary, $1<(T+1/T)/2\leq 5$. First, we consider the case where
$|D|$ is odd. By similar argument, (6.72) holds trivially when $|D|+2|F|\geq
5$, so all we need to consider is the case $(|D|,|F|)=(1,1)$ and the case
$(|D|,|F|)=(3,0)$. In both cases, $|D|+2|F|=3$. By (6.70), when $\omega T<1$,
there must be $T<1/\min(\lambda_{2}^{*},3\lambda_{3}^{*})$. By the conditions
of this corollary, it follows $T<(5+2\sqrt{6})/4<3+2\sqrt{2}$. When
$1<T<3+2\sqrt{2}$, there is $T+1/T-6<0$, and thus (6.72) holds for $\omega
T<1$. When $\omega T\geq 1$, (6.72) holds if and only if $\omega
T+\frac{1}{\omega T}-2\geq T+1/T-6$. By basic algebra, this holds if
$\omega\geq\frac{1}{4}\bigl{[}(1-1/T)+\sqrt{(1-1/T)^{2}-4/T}\bigr{]}^{2}.$
(6.74)
Note that the right hand side of (6.74) is a monotone in $T$ and has a maximum
of $(3+2\sqrt{2})(5-2\sqrt{6})$ at $T=(5+2\sqrt{6})$. Now, on one other hand,
when $(|D|,|F|)=(1,0)$, by (6.70) and conditions of the corollary, $\omega\geq
3\lambda_{3}^{*}>(3+2\sqrt{2})(5-2\sqrt{6})$. On the other hand, when
$(|D|,|F|)=(1,1)$, by basic algebra and that each off-diagonal coordinate of
$\Omega\leq\sqrt{1+(\sqrt{6}-\sqrt{2})}/(1+\sqrt{3/2})\equiv\rho_{1}$ in
magnitude, $\omega\geq 1-\rho_{1}^{2}=(3+2\sqrt{2})(5-2\sqrt{6})$. Combining
these gives (6.72).
We now consider the case where $|D|$ is even. By similar argument, (6.72)
holds when $|D|+2|F|\geq 6$, so all we need is to show is that (6.72) holds
for the following three cases: $(|D|,|F|)=(4,0),(2,1),(2,0)$. Equivalently,
this is to show that $\omega\geq\frac{2}{T}[(T+1/T)/2-3]$ in the first two
cases and that $\omega\geq\frac{2}{T}[(T+1/T)/2-1]$ in the last case.
Similarly, by the monotonicity of the right hand side of these inequalities,
all we need to show is $\omega\geq 4(5-2\sqrt{6})$ in the first two cases, and
$\omega\geq 8(5-2\sqrt{6})$ in the last case. Now, on one hand, using (6.70),
$\omega\geq 4\lambda_{4}^{*}$ in the first case, and $\omega\geq
2\lambda_{3}^{*}$ in the second case, so by the conditions of the corollary,
$\omega\geq 4(5-2\sqrt{6})$ in the first two cases. On the other hand, in the
last case, since all off-diagonal coordinates of $\Omega\leq
8\sqrt{6}-19\equiv\rho_{0}$ in magnitude, and $\omega\geq
2(1-\rho_{0})=8(5-2\sqrt{6})$. Combining these gives (6.72). $\Box$
#### 6.3.3 Proof of Corollary 12
Let $N$ be the unique integer such that $2N-1\leq(T+1/T)/2<2N+1$. First, we
consider the case where $|D|$ is odd. Note that when $|D|+2|F|\geq 2N+1$, the
right hand side of (6.72) $\leq 0$, so all we need to consider is the case
$|D|+2|F|\leq 2N-1$. Write for short $k=k(D,F)=|D|+|F|$ and
$j=j(D,F)=(|D|+2|F|+1)/2$. By (6.73), definitions, and that $|D|+2|F|\leq
2N-1$, it is seen that $2\leq k\leq 2N-1$ and $(k+1)/2\leq
j\leq\min\\{k,N\\}$. By the condition of the corollary,
$\lambda_{k}^{*}\geq\frac{(T+1/T)/2-2j+2+\sqrt{[(T+1/T)/2-2j+2]^{2}-1}}{T(2k-2j+1)}$.
Note that $|D|=2k-2j+1$. Combining these with (6.70) gives $\omega
T\geq(2k-2j+1)\lambda_{k}^{*}T\geq(T+1/T)/2-2j+2+\sqrt{[(T+1/T)/2-2j+2]^{2}-1}\geq
1$. and (6.72) follows by basic algebra.
We now consider the case where $|D|$ is even. Similarly, the right hand side
of (6.72) is negative when $|D|+2|F|\geq 2(N+1)$, so we only need to consider
the case where $|D|+2|F|\leq 2N$. Similarly, write for short
$k=k(D,F)=|D|+|F|$ and $j=(|D|+2|F|)/2$. It is seen that $2\leq k\leq 2N$ and
$k/2\leq j\leq\min\\{k-1,N\\}$. By the conditions of the corollary,
$\lambda_{k}^{*}\geq\frac{(T+1/T)/2+1-2j}{T(k-j)}$. Note that $|D|=k-j$. It
follows from (6.70) that $\omega\geq
2(k-j)\lambda_{k}^{*}\geq\frac{2}{T}[(T+1/T)/2+1-2j]$, and (6.72) follows.
$\Box$
#### 6.3.4 Proof of Lemma 18
Let sets $V_{0}$ and $V_{1}$ and vectors $\theta^{(0)}$ and $\theta^{(1)}$ be
as in Section 2.5, and let $V=V_{0}\cup V_{1}$. By the definition of
$\rho_{j}^{*}(\vartheta,r,a,\Omega)$,
$\rho_{j}^{*}(\vartheta,r,a,\Omega)=\min(I,II)$, where
$I=\min_{\\{(V_{0},V_{1}):j\in V_{1}\cup V_{0},V_{0}\neq
V_{1}\\}}\eta(V_{0},V_{1};\Omega)$ and $II=\min_{\\{V_{0}:j\in V_{0}\cup
V_{1},V_{0}=V_{1}\\}}\eta(V_{0},V_{1};\Omega)$. So to show the claim, it is
sufficient to show
$I=\min_{\\{(D,F):j\in D\cup F,D\cap F=\emptyset,D\neq\emptyset,|D\cup F|\leq
g\\}}\rho(D,F;\Omega),\qquad II\geq I.$ (6.75)
Consider the first claim in (6.75). Write for short
$F=F(V_{0},V_{1})=V_{0}\cap V_{1}$ and $D=D(V_{0},V_{1})=V\setminus F$. By the
definitions, $D\neq\emptyset$. The key is to show that when $|V_{0}\cup
V_{1}|\leq g$,
$\alpha^{*}(V_{0},V_{1};\Omega)=\omega(D,F;\Omega).$ (6.76)
Towards this end, note that by definitions,
$\alpha^{*}(V_{0},V_{1};\Omega)=\alpha(\theta_{*}^{(0)},\theta_{*}^{(1)})$,
where $(\theta_{*}^{(0)},\theta_{*}^{(1)})=\mathrm{argmin}_{\\{\theta^{(0)}\in
B_{V_{0}},\theta^{(1)}\in B_{V_{1}}\\}}\alpha(\theta^{(0)},\theta^{(1)})$. By
$a>a_{g}^{*}(\Omega)$ and the way $a_{g}^{*}(\Omega)$ is defined,
$(\theta_{*}^{(0)},\theta_{*}^{(1)})$ remains as the solution of the
optimization problem if we relax the conditions $\theta^{(i)}\in B_{V_{i}}$ to
that of $\theta^{(i)}=I_{V_{i}}\circ\mu^{(i)}$, where
$\mu^{(i)}\in\Theta_{p}(\tau_{p})$ (so that upper bounds on the signal
strengths are removed), $i=0,1$. As a result,
$\alpha^{*}(V_{0},V_{1};\Omega)=\min_{\\{\theta^{(i)}\in
I_{V_{i}}\circ\mu^{(i)},\mu^{(i)}\in\Theta_{p}(\tau_{p}),i=0,1,\\}}\alpha(\theta^{(0)},\theta^{(1)}).$
(6.77)
We now study (6.77). For short, write
$\xi=\tau_{p}^{-1}(\theta^{(1)}-\theta^{(0)})^{V}$,
$\Omega_{VV}=\Omega^{V,V}$,
$\xi_{D}=\tau_{p}^{-1}(\theta^{(1)}-\theta^{(0)})^{D}$, and similarly for
$\Omega_{DD}$, $\Omega_{DF}$, $\Omega_{FD}$, $\Omega_{FF}$, and $\xi_{F}$.
Without loss of generality, assume the indices in $D$ come first in $V$. It
follows
$\Omega_{VV}=\begin{pmatrix}\Omega_{DD}&\Omega_{DF}\\\
\Omega_{FD}&\Omega_{FF}\end{pmatrix},$
and
$\alpha(\theta^{(0)},\theta^{(1)})=\xi^{\prime}\Omega_{VV}\xi=\xi_{D}^{\prime}\Omega_{DD}\xi_{D}+2\xi_{D}^{\prime}\Omega_{DF}\xi_{F}+\xi_{F}^{\prime}\Omega_{FF}\xi_{F}.$
(6.78)
By definitions, it is seen that there is no constraint on the coordinates of
$\xi_{F}$, so to optimize the quadratic form in (6.76), we need to choose
$\xi$ is a way such that $\xi_{F}=-\Omega_{FF}^{-1}\Omega_{FD}\xi_{D}$, and
that $\xi_{D}$ minimizes
$\xi_{D}^{\prime}(\Omega_{DD}-\Omega_{DF}\Omega_{FF}^{-1}\Omega_{FD})\xi_{D}$,
where every coordinate of $\xi_{D}\geq 1$ in magnitude. Combining these with
(6.77) gives (6.76).
At the same time, we rewrite
$I=\min_{\\{(D,F):j\in D\cup F,D\neq\emptyset,D\cap
F=\emptyset\\}}\biggl{\\{}\min_{\\{(V_{0},V_{1}):V_{0}\cup V_{1}=D\cup
F,V_{0}\cap V_{1}=F\\}}\eta(V_{0},V_{1};\Omega)\biggr{\\}}.$ (6.79)
By similar arguments as in the proof of Lemma 22, the subsets $(V_{0},V_{1})$
that achieve the minimum of $\eta(V_{0},V_{1};\Omega)$ must satisfy
$|V_{0}\cup V_{1}|\leq g$. Using (6.76), for any fixed $D$ and $F$ such that
$|D\cup F|\leq g$, $D\neq\emptyset$ and $D\cap F=\emptyset$, the term in the
big bracket on the right hand side is $\min_{\\{(V_{0},V_{1}):V_{0}\cup
V_{1}=D\cup F,V_{0}\cap
V_{1}=F\\}}\\{\frac{(2|F|+|D|)\vartheta}{2}+\frac{\bigl{|}|V_{1}|-|V_{0}|\bigr{|}\vartheta}{2}+\frac{1}{4}[(\sqrt{\omega(D,F;\Omega)r}-\frac{\bigl{|}|V_{1}|-|V_{0}|\bigr{|}\vartheta}{\sqrt{\omega(D,F;\Omega)r}})_{+}]^{2}\\}$.
It is worth noting that for fixed $D$ and $F$, the above quantity is monotone
increasing with $\bigl{|}|V_{1}|-|V_{0}|\bigr{|}$. When $|D|$ is even, the
minimum is achieved at $(V_{0},V_{1})$ with $|V_{0}|=|V_{1}|$, and when $|D|$
is odd, the minimum is achieved at $(V_{0},V_{1})$ with
$\bigl{|}|V_{1}|-|V_{0}|\bigr{|}=1$, and in both cases, the minimum is
$\rho(D,F;\Omega)$. Inserting this to (6.79), it is seen that
$I=\min_{\\{(D,F):j\in D\cup F,D\cap F=\emptyset,D\neq\emptyset,|D\cup F|\leq
g\\}}\rho(D,F;\Omega),$ (6.80)
which is the first claim in (6.75).
Consider the second claim of (6.75). In this case, by definitions,
$V_{0}=V_{1}$ but $\hbox{\rm sgn}(\theta^{(0)})\neq\hbox{\rm
sgn}(\theta^{(1)})$. Redefine $D$ as the subset of $V_{0}$ where the signs of
the coordinates of $\theta^{(0)}$ do not equal to those of $\theta^{(1)}$, and
let $F=V\setminus D$. By definitions, it is seen that
$\alpha^{*}(V_{0},V_{0};\Omega)=4\alpha^{*}(F,V_{0};\Omega)$, where we note
$D\neq\emptyset$ and $F\neq V_{0}$. By the definition of
$\eta(V_{0},V_{1};\Omega)$, it follows that
$\eta(V_{0},V_{0};\Omega)\geq\eta(F,V_{0};\Omega)$, and the claim follows.
$\Box$
### 6.4 Proof of Lemma 15
Write for short $\rho_{j}^{*}=\rho_{j}^{*}(\vartheta,a,r,\Omega)$. To show the
claim, it is sufficient to show that for any fixed $1\leq j\leq p$,
$P(j\notin{\cal U}_{p}^{*},\beta_{j}\neq 0)\leq
L_{p}[p^{-\rho_{j}^{*}}+p^{-(m_{0}+1)\vartheta}+o(1/p)].$ (6.81)
Using Lemma 14 and (Ji and Jin, 2011, Lemma 3.1), there is an event $A_{p}$
that depends on $(X,\beta)$ such that $P(A_{p}^{c})\leq o(1/p)$ and that over
the event, $\Omega^{*,\delta}$ is $K$-sparse with $K=C(\log(p))^{1/\gamma}$,
$\|\Omega^{*,\delta}-\Omega\|_{\infty}\leq(\log(p))^{-(1-\gamma)}$,
$\|(X^{\prime}X-\Omega)\beta\|_{\infty}\leq
C\|\Omega\|\sqrt{2\log(p)}p^{-[(\kappa-(1-\vartheta)]/2}$, and for all subset
$B$ with size $\leq m_{0}$, $\|G^{B,B}-\Omega^{B,B}\|_{\infty}\leq
L_{p}p^{-\kappa/2}$. Recall that ${\cal G}^{*,\delta}$ is the GOSD and ${\cal
G}_{S}^{*,\delta}$ is the subgraph of the GOSD formed by the nodes in the
support of $\beta$, $S(\beta)=\\{1\leq j\leq p:\beta_{j}\neq 0\\}$. When
$\beta_{j}\neq 0$, there is a unique component ${\cal I}_{0}$ such that
$j\in{\cal I}_{0}\lhd{\cal G}_{S}^{*,\delta}$ ($A\lhd B$ means that $A$ is
component or maximal connected subgraph of $B$). Let $B_{p}$ be the event
$|{\cal I}_{0}|\leq m_{0}$. By Frieze Frieze and Molloy (1999), it is seen
that $P(B_{p}^{c}\cap A_{p})\leq L_{p}p^{-(m_{0}+1)\vartheta}$. So to show
(6.81), it is sufficient to show that
$P(j\notin{\cal U}_{p}^{*},j\in{\cal I}_{0}\lhd{\cal
G}_{S}^{*,\delta},A_{p}\cap B_{p})\leq L_{p}p^{-\rho_{j}^{*}}.$ (6.82)
Now, in the screening procedure, when we screen ${\cal I}_{0}$, we have ${\cal
I}_{0}=\hat{D}\cup\hat{F}$ as in (2.12). Since the event $\\{j\notin{\cal
U}_{p}^{*},j\in{\cal I}_{0}\lhd{\cal G}_{S}^{*,\delta}\\}$ is contained in the
event $\\{T(Y,\hat{D},\hat{F})<t(\hat{D},\hat{F})\\}$, $P(j\notin{\cal
U}_{p}^{*},j\in{\cal I}_{0}\lhd{\cal G}_{S}^{*,\delta},A_{p}\cap B_{p})\leq
P(T(Y,\hat{D},\hat{F})\leq t(\hat{D},\hat{F}),j\in{\cal I}_{0}\lhd{\cal
G}_{S}^{*,\delta},A_{p}\cap B_{p})$, where the right hand side does not exceed
$\sum_{({\cal I}_{0},D,F):\mbox{$j\in{\cal I}_{0}$ \& ${\cal I}_{0}=D\cup F$
is a partition}}P(T(Y,D,F)\leq t(D,F),j\in{\cal I}_{0}\lhd{\cal
G}_{S}^{*,\delta},A_{p}\cap B_{p});$ (6.83)
note that $({\cal I}_{0},D,F)$ do not depend on $z$ (but may still depend on
$(X,\beta)$). First, note that over the event $A_{p}$, there are at most
$(eK)^{m_{0}+1}$ ${\cal I}_{0}$ such that $j\in{\cal I}_{0}$ and $|{\cal
I}_{0}|\leq m_{0}$. Second, note that for each ${\cal I}_{0}$, there are only
finite ways to partition it to $D$ and $F$. Last, note that for any fixed $j$
and ${\cal I}_{0}$, $P(j\in{\cal I}_{0}\lhd{\cal
G}_{S}^{*,\delta})\leq\epsilon_{p}^{|{\cal I}_{0}|}$. Combining these
observations, to show (6.82), it is sufficient to show that for any such
triplet $({\cal I}_{0},D,F)$,
$\epsilon_{p}^{|{\cal I}_{0}|}P\bigl{(}T(Y,D,F)\leq t(D,F)\bigr{|}\\{j\in{\cal
I}_{0}\lhd{\cal G}_{S}^{*,\delta}\\}\cap A_{p}\cap B_{p}\bigr{)}\leq
L_{p}p^{-\rho_{j}^{*}}.$ (6.84)
We now show (6.84). Since $\lambda^{*}_{m_{0}}(\Omega)\geq C>0$, it follows
from the definition of $A_{p}$ and basic algebra that for any realization of
$(X,\beta)$ in $A_{p}\cap B_{p}$,
$\|(G^{{\cal I}_{0},{\cal I}_{0}})^{-1}\|_{\infty}\leq C.$ (6.85)
Recall that $\tilde{Y}=X^{\prime}Y$ and denote for short $y=(G^{{\cal
I}_{0},{\cal I}_{0}})^{-1}\tilde{Y}^{{\cal I}_{0}}$. It is seen that
$y=\beta^{{\cal I}_{0}}+w+rem,\qquad w\sim N(0,(G^{{\cal I}_{0},{\cal
I}_{0}})^{-1}),\qquad rem\equiv(G^{{\cal I}_{0},{\cal I}_{0}})^{-1}G^{{\cal
I}_{0},{\cal I}_{0}^{c}}\beta^{{\cal I}_{0}^{c}}.$ (6.86)
Since ${\cal I}_{0}$ is a component of ${\cal G}_{S}^{*,\delta}$,
$(\Omega^{*,\delta})^{{\cal I}_{0},{\cal I}_{0}^{c}}\beta^{{\cal
I}_{0}^{c}}=0$. Therefore, we can write $rem=(G^{{\cal I}_{0},{\cal
I}_{0}})^{-1}(I+II)$, where $I=(G^{{\cal I}_{0},{\cal
I}_{0}^{c}}-\Omega^{{\cal I}_{0},{\cal I}_{0}^{c}})\beta^{{\cal I}_{0}^{c}}$
and $II=[\Omega^{{\cal I}_{0},{\cal I}_{0}^{c}}-(\Omega^{*,\delta})^{{\cal
I}_{0},{\cal I}_{0}^{c}}]\beta^{{\cal I}_{0}^{c}}$. By the definition of
$A_{p}$, $\|I\|_{\infty}\leq C\sqrt{2\log(p)}p^{-[\kappa-(1-\vartheta)]/2}$,
and $\|II\|_{\infty}\leq\|\Omega-\Omega^{*,\delta}\|_{\infty}\|\beta^{{\cal
I}_{0}^{c}}\|_{\infty}\leq C\tau_{p}(\log(p))^{-(1-\gamma)}$. Combining these
with (6.85) gives $\|rem\|_{\infty}\leq C\tau_{p}(\log(p))^{-(1-\gamma)}$.
At the same time, let $y_{1}$, $w_{1}$, and $rem^{1}$ be the restriction of
$y$, $w$, and $rem$ to indices in $D$, correspondingly, and let
$H=[G^{D,D}-G^{D,F}(G^{F,F})^{-1}G^{F,D}]$. By (6.86) and direct calculations,
$T(Y,D,F)=y_{1}^{\prime}Hy_{1}$, $y_{1}\sim N(\beta^{D}+rem^{1},H^{-1})$, and
so $T(Y,D,F)$ is distributed as $\chi_{|D|}^{2}(\delta)$, where the non-
central parameter is
$(\beta^{D}+rem^{1})^{\prime}H(\beta^{D}+rem^{1})=\delta+O((\log(p))^{\gamma})$
and $\delta\equiv(\beta^{D})^{\prime}H\beta^{D}$. Since
$\lambda_{m_{0}}^{*}(\Omega)\geq C$, $\delta\geq C\tau_{p}^{2}$ and is the
dominating term. It follows that
$P(T(Y,D,F)\leq t(D,F)\bigr{|}\\{j\in{\cal I}_{0}\lhd{\cal
G}_{S}^{*,\delta}\\}\cap A_{p}\cap B_{p}\bigr{)}\lesssim
P\bigl{(}\chi_{|D|}^{2}(\delta)\leq t(D,F)\bigr{)}.$ (6.87)
Now, first, by definitions, $\delta\geq 2\omega(D,F;\Omega)r\log(p)$, so by
basic knowledge on non-central $\chi^{2}$,
$P(\chi_{|D|}^{2}(\delta)\leq t(D,F))\leq
P(\chi_{|D|}^{2}(2\omega(D,F;\Omega)r\log(p))\leq t(D,F)).$ (6.88)
Second, recalling $t(D,F)=2q\log(p)$, we have
$P(\chi_{|D|}^{2}(2\omega(D,F;\Omega)r\log(p))\leq t(D,F))\leq
L_{p}p^{-[(\sqrt{\omega(D,F;\Omega)r}-\sqrt{q})_{+}]^{2}}.$ (6.89)
Inserting (6.88)-(6.89) into (6.87) and recalling
$\epsilon_{p}=p^{-\vartheta}$,
$\epsilon_{p}^{|{\cal I}_{0}|}P(T(Y,D,F)\leq t(D,F)\bigr{|}\\{j\in{\cal
I}_{0}\lhd{\cal G}_{S}^{*,\delta}\\}\cap A_{p}\cap B_{p}\bigr{)}\leq
L_{p}p^{-(|{\cal
I}_{0}|\vartheta+[(\sqrt{\omega(D,F;\Omega)r}-\sqrt{q})_{+}]^{2})}.$ (6.90)
By the choice of $q$ and direct calculations,
$|{\cal
I}_{0}|\vartheta+[(\sqrt{\omega(D,F;\Omega)r}-\sqrt{q})_{+}]^{2}\geq\rho(D,F;\Omega)\geq\rho_{j}^{*},$
(6.91)
where $\rho(D,F;\Omega)$ as in (6.69). Combining (6.90)-(6.91) gives (6.84).
$\Box$
### 6.5 Proof of Lemma 16
In the screening stage, suppose we pick the threshold
$t(\hat{D},\hat{F})=2q\log(p)$ in a way such that there is a constant
$q_{0}(\vartheta,r,\kappa)>0$ such that $q=q(\hat{D},\hat{F})\geq
q_{0}(\vartheta,r,\kappa)>0$. Recall that ${\cal G}^{*,\delta}$ denotes the
GOSD. Let ${\cal U}_{p}^{*}$ be the set of retained indices. Viewing it as a
subgraph of ${\cal G}^{*,\delta}$, ${\cal U}_{p}^{*}$ decomposes into many
components ${\cal U}_{p}^{*}={\cal I}^{(1)}\cup{\cal I}^{(2)}\ldots\cup{\cal
I}^{(N)}$. Recall that $\tilde{Y}=X^{\prime}Y$. The following lemma is proved
below.
###### Lemma 19
Given that the conditions of Lemma 16 hold, there is a constant
$c_{1}=c_{1}(\vartheta,r,\kappa,\gamma,A)>0$ such that with probability at
least $1-o(1/p)$, for any component ${\cal I}_{0}\lhd{\cal U}_{p}^{*}$,
$\|\tilde{Y}^{{\cal I}_{0}}\|^{2}\geq 2c_{1}|{\cal I}_{0}|\log(p)$.
The remaining part of the proof is similar to that of Ji and Jin (2011, Lemma
2.3) so we omit it. We note that however Lemma 19 is new and needs a much
harder proof. $\Box$
#### 6.5.1 Proof of Lemma 19
First, we need some notations. Let ${\cal I}_{0}$ be a component of ${\cal
U}_{p}^{*}$, and let ${\cal I}_{0}^{(i)}$, $1\leq i\leq N_{0}$, be all
connected subgraphs with size $\leq m_{0}$, listed in the order as in the GS-
step, where $N_{0}$ is an integer that may depend on $(X,Y)$. For each $1\leq
i\leq N_{0}$, let ${\cal I}_{0}^{(i)}=\hat{D}^{(i)}\cup\hat{F}^{(i)}$ be the
exactly the same partition when we screen ${\cal I}_{0}^{(i)}$ in the
$m_{0}$-stage $\chi^{2}$-screening of the GS-step. In out list, we only keep
${\cal I}_{0}^{(i)}$ such that $\hat{D}^{(i)}\cap{\cal I}_{0}\neq\emptyset$.
Since ${\cal I}_{0}$ is a component of ${\cal U}_{p}^{*}$ and ${\cal
I}_{0}^{(i)}$ is a connected subgraph, it follows from the way that the
$\chi^{2}$-screening is designed and the definition of $\hat{D}^{(i)}$ that
${\cal I}_{0}^{(i)}\subset{\cal
I}_{0},\qquad\mbox{and}\qquad\hat{D}^{(i)}={\cal
I}_{0}^{(i)}\setminus(\cup_{j=1}^{i-1}{\cal I}_{0}^{(j)}),\qquad 1\leq i\leq
N_{0},$ (6.92)
and
$\mbox{${\cal
I}_{0}=\hat{D}^{(1)}\cup\hat{D}^{(2)}\ldots\cup\hat{D}^{(N_{0})}$ is a
partition},$ (6.93)
where $\hat{F}^{(1)}$ is empty.
Now, for each $1\leq i\leq N_{0}$, recall that as long as $G^{{\cal
I}_{0}^{(i)},{\cal I}_{0}^{(i)}}$ is non-singular, the $\chi^{2}$-test score
in GS is
$T(Y,\hat{D}^{(i)},\hat{F}^{(i)})=T(Y,\hat{D}^{(i)},\hat{F}^{(i)};{\cal
I}_{0}^{(i)},X,p,n)=(\tilde{Y}^{{\cal I}_{0}^{(i)}})^{\prime}(G^{{\cal
I}_{0}^{(i)},{\cal I}_{0}^{(i)}})^{-1}\tilde{Y}^{{\cal
I}_{0}^{(i)}}-(\tilde{Y}^{\hat{F}^{(i)}})^{\prime}(G^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}\tilde{Y}^{\hat{F}^{(i)}}$.
By basic algebra and direct calculations, it can be verified that
$T(Y,\hat{D}^{(i)},\hat{F}^{(i)})=\|W_{i}\|^{2}$, where
$W_{i}=W(\tilde{Y},\hat{D}^{(i)},\hat{F}^{(i)};{\cal I}_{0}^{(i)},X,p,n)$ is
defined as $W_{i}=V_{i}^{-1/2}y_{i}$, and for short,
$V_{i}=G^{\hat{D}^{(i)},\hat{D}^{(i)}}-G^{\hat{D}^{(i)},\hat{F}^{(i)}}(G^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}G^{\hat{F}^{(i)},\hat{D}^{(i)}}$,
$y_{i}=\tilde{Y}^{\hat{D}^{(i)}}-G^{\hat{D}^{(i)},\hat{F}^{(i)}}(G^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}\tilde{Y}^{\hat{F}^{(i)}}$.
At the same time, for a constant $\delta>0$ to be determined, define
$\tilde{\Omega}$ by $\tilde{\Omega}(i,j)=G(i,j)\cdot
1\\{|G(i,j)|\geq\delta\\}$. The definition of $\tilde{\Omega}$ is the same as
that of $\Omega^{*,\delta}$, except for that the threshold $\delta$ would be
selected differently. We introduce a counterpart of $W_{i}$ which we call
$W_{i}^{*}$,
$W_{i}^{*}=V_{i}^{-1/2}y_{i}^{*}.$ (6.94)
where
$y_{i}^{*}=\tilde{Y}^{\hat{D}^{(i)}}-\tilde{\Omega}^{\hat{D}^{(i)},\hat{F}^{(i)}}(\tilde{\Omega}^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}\tilde{Y}^{\hat{F}^{(i)}}$.
Let
$W^{*}=((W_{1}^{*})^{\prime},(W_{2}^{*})^{\prime},\ldots,(W_{N_{0}}^{*})^{\prime})^{\prime}$,
and define $|{\cal I}_{0}|\times|{\cal I}_{0}|$ matrices $H_{1}$ and $H_{2}$
as follows: $H_{1}$ is a diagonal block-wise matrix where the $i$-th block is
$V_{i}^{-1/2}$, and $H_{2}=\tilde{H}_{2}^{{\cal I}_{0},{\cal I}_{0}}$, where
$\tilde{H}_{2}$ is a $p\times p$ matrix such that for every component ${\cal
I}_{0}$ of ${\cal U}_{p}^{*}$, and $\hat{D}^{(i)}$ and $\hat{F}^{(i)}$ defined
on each component,
$\tilde{H}_{2}^{\hat{D}^{(i)},\hat{F}^{(i)}}=-(\tilde{\Omega})^{\hat{D}^{(i)},\hat{F}^{(i)}}[(\tilde{\Omega})^{\hat{F}^{(i)},\hat{F}^{(i)}}]^{-1}$,
$\tilde{H}_{2}^{\hat{D}^{(i)},\hat{D}^{(i)}}=I_{|\hat{D}^{(i)}|}$, and that
the coordinates of $\tilde{H}_{2}$ are zero elsewhere. Here $I_{k}$ stands for
$k\times k$ identity matrix. From the definitions, it is seen that
$W^{*}=H_{1}H_{2}\tilde{Y}^{{\cal I}_{0}}.$ (6.95)
Compared with $W_{i}$, $W_{i}^{*}$ is relatively easier to study, for it
induces column-sparsity of $H_{2}$. In fact, using (Ji and Jin, 2011, Lemma
2.2, 3.1), there is an event $A_{p}$ that depends on $(X,\beta)$ such that
$P(A_{p}^{c})\leq o(1/p^{2})$ and that over the event, for all subset $B$ with
size $\leq m_{0}$,
$\|G^{B,B}-\Omega^{B,B}\|_{\infty}\leq L_{p}p^{-\kappa/2}.$ (6.96)
The following lemma is proved below.
###### Lemma 20
Fix $\delta>0$ and suppose the conditions in Lemma 19 hold. Over the event
$A_{p}$, there is a constant $C>0$ such that each row and column of
$\tilde{H}_{2}$ has no more than $C$ nonzero coordinates.
We are now ready to show Lemma 19. To begin with, note that since we accept
$\hat{D}^{(i)}$ when we graphlet-screen ${\cal I}_{0}^{(i)}$ and
$|\hat{D}^{(i)}|\leq m_{0}$,
$\|W_{i}\|^{2}\geq 2(q_{0}/m_{0})|\hat{D}^{(i)}|\log(p).$ (6.97)
At the same time, by basic algebra,
$\|W_{i}-W_{i}^{*}\|\leq\|V_{i}^{-1/2}\|\|y_{i}-y_{i}^{*}\|$, and
$\|y_{i}-y_{i}^{*}\|\leq\|G^{\hat{D}^{(i)},\hat{F}^{(i)}}(G^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}-(\tilde{\Omega})^{\hat{D}^{(i)},\hat{F}^{(i)}}((\tilde{\Omega})^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}\|_{\infty}\cdot\|\tilde{Y}^{\hat{F}^{(i)}}\|$.
First, since $\lambda_{m_{0}}^{*}(\Omega)\geq C$, it is seen that over the
event $A_{p}$, $\|V_{i}^{-1/2}\|\leq C$. Second, by similar reasons, it is not
hard to see that except for probability $o(p^{-2})$,
$\|G^{\hat{D}^{(i)},\hat{F}^{(i)}}(G^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}-(\tilde{\Omega})^{\hat{D}^{(i)},\hat{F}^{(i)}}((\tilde{\Omega})^{\hat{F}^{(i)},\hat{F}^{(i)}})^{-1}\|_{\infty}\leq
C\delta^{1-\gamma}$, and $\|\tilde{Y}^{\hat{F}^{(i)}}\|\leq
C\sqrt{\log(p)}\leq C\tau_{p}$. Combining these gives
$\|W_{i}-W_{i}^{*}\|\leq C\delta^{1-\gamma}\tau_{p},$ (6.98)
Inserting this to (6.97), if we choose $\delta$ to be a sufficiently small
constant,
$\|W_{i}^{*}\|^{2}\geq\frac{1}{2}\|W_{i}\|^{2}\geq(q_{0}/m_{0})|\hat{D}^{(i)}|\log(p)$.
At the same time, by definitions, it follows from $\|V_{i}^{-1/2}\|\leq C$
that $\|H_{1}\|\leq C$. Also, since over the event $A_{p}$, each coordinate of
$H_{2}$ is bounded from above by a constant in magnitude, it follows from
Lemma 20 that $\|H_{2}\|\leq C$. Combining this with (6.93)-(6.95), it follows
from basic algebra that except for probability $o(p^{-2})$,
$(q_{0}/m_{0})|{\cal
I}_{0}|\log(p)\leq\|W^{*}\|^{2}\leq\|H_{1}H_{2}\tilde{Y}^{{\cal
I}_{0}}\|^{2}\leq C\|\tilde{Y}^{{\cal I}_{0}}\|^{2}$, and the claim follows
since $m_{0}$ is a fixed integer. $\Box$
#### 6.5.2 Proof of Lemma 20
By definitions, it is equivalent to show that over the event $A_{p}$, each row
and column of $\tilde{H}_{2}$ has finite nonzero coordinates. It is seen that
each row of $\tilde{H}_{2}$ has $\leq m_{0}$ nonzeros, so all we need to show
is that each column of $\tilde{H}_{2}$ has finite nonzeros.
Towards this end, we introduce a new graph $\tilde{{\cal G}}=(V,E)$, where
$V=\\{1,2,\ldots,p\\}$ and nodes $i$ and $j$ are connected if and only if
$|\tilde{\Omega}(i,j)|\neq 0$. This definition is the same as GOSD, except
that $\Omega^{*,\delta}$ is substituted by $\tilde{\Omega}$. It is seen that
over the event $A_{p}$, for any $\Omega\in{\cal M}_{p}^{*}(\gamma,c_{0},g,A)$,
$\tilde{{\cal G}}$ is $K$-sparse with $K\leq C\delta^{-1/\gamma}$. The key for
the proof is to show that for any $k\neq\ell$ such that
$\tilde{H}_{2}(k,\ell)\neq 0$, there is a path with length $\leq(m_{0}-1)$ in
$\tilde{{\cal G}}$ that connects $k$ and $\ell$.
To see the point, we note that when $\tilde{H}_{2}(k,\ell)\neq 0$, there must
be an $i$ such that $k\in\hat{D}^{(i)}$ and $\ell\in\hat{F}^{(i)}$. We claim
that there is a path in ${\cal I}_{0}^{(i)}$ (which is regarded as a subgraph
of $\tilde{{\cal G}}$) that connects $k$ and $\ell$. In fact, if $k$ and
$\ell$ are not connected in ${\cal I}_{0}^{(i)}$, we can partition ${\cal
I}_{0}^{(i)}$ into two separate sets of nodes such that one contains $k$ and
the other contains $\ell$, and two sets are disconnected. In effect, both the
matrix $\tilde{\Omega}^{\hat{D}^{(i)},\hat{D}^{(i)}}$ and
$\tilde{\Omega}^{\hat{D}^{(i)},\hat{F}^{(i)}}$ can be visualized as two by two
blockwise matrix, with off-diagonal blocks being $0$. As a result, it is seen
that $\tilde{H}_{2}(k,\ell)=0$. This contradiction shows that whenever
$\tilde{H}_{2}(k,\ell)\neq 0$, $k$ and $\ell$ are connected by a path in
${\cal I}_{0}^{(i)}$. Since $|{\cal I}_{0}^{(i)}|\leq m_{0}$, there is a path
$\leq m_{0}-1$ in $\tilde{{\cal G}}$ that connects $k$ and $\ell$ where
$k\neq\ell$.
Finally, since $\tilde{{\cal G}}$ is $K$-sparse with $K=C\delta^{-1/\gamma}$,
for any fixed $\ell$, there are at most finite $k$ connecting to $\ell$ by a
path with length $\leq(m_{0}-1)$. The claim follows. $\Box$
### 6.6 Proof of Theorem 13
Since $\sigma$ is known, for simplicity, we assume $\sigma=1$. First, consider
(2.49). By Theorem 8 and (6.69), $\rho_{gs}=\min_{\\{(D,F):D\cap
F=\emptyset,D\neq\emptyset,D\cup F\subset\\{1,2\\}\\}}\rho(D,F;\Omega)$, where
we have used that $G$ is a diagonal block-wise matrix, each block is the same
$2\times 2$ matrix. To calculate $\rho(D,F;\Omega)$, we consider three cases
(a) $(|D|,|F|)=(2,0)$, (b) $(|D|,|F|)=(1,0)$, (c) $(|D|,|F|)=(1,1)$. By
definitions and direct calculations, it is seen that
$\rho(D,F;\Omega)=\vartheta+[(1-|h_{0}|)r]/2$ in case (a),
$\rho(D,F;\Omega)=(\vartheta+r)^{2}/(4r)$ in case (b), and
$\rho(D,F;\Omega)=2\vartheta+[(\sqrt{(1-h_{0}^{2})r}-\vartheta/\sqrt{(1-h_{0}^{2})r})_{+}]^{2}/4$
in case (c). Combining these gives the claim.
Next, consider (2.50). Similarly, by the block-wise structure of $G$, we can
restrict our attention to the first two coordinates of $\beta$, and apply the
subset selection to the size $2$ subproblem where the Gram matrix is the
$2\times 2$ matrix with $1$ on the diagonals and $h_{0}$ on the off-diagonals.
Fix $q>0$, and let the tuning parameter $\lambda_{ss}=\sqrt{2q_{ss}\log(p)}$.
Define $f_{ss}^{(1)}(q)=\vartheta+[(\sqrt{r}-\sqrt{q})_{+}]^{2}$,
$f_{ss}^{(2)}(q)=2\vartheta+[(\sqrt{r(1-h_{0}^{2})}-\sqrt{q})_{+}]^{2}$, and
$f_{ss}^{(3)}(q)=2\vartheta+2[(\sqrt{r(1-|h_{0}|)}-\sqrt{q})_{+}]^{2}$, where
$x_{+}=\max\\{x,0\\}$. The following lemma is proved below, where the key is
to use Ji and Jin (2011, Lemma 4.3).
###### Lemma 21
Fix $q>0$ and suppose the conditions in Theorem 13 hold. Apply the subset
selection to the aforementioned size $2$ subproblem with
$\lambda_{ss}=\sqrt{2q\log(p)}$. As $p\rightarrow\infty$, the worst-case
Hamming error rate is $L_{p}p^{-f_{ss}(q)}$, where
$f_{ss}(q)=f_{ss}(q,\vartheta,r,h_{0})=\min\bigl{\\{}\vartheta+(1-|h_{0}|)r/2,q,f_{ss}^{(1)}(q),f_{ss}^{(2)}(q),f_{ss}^{(3)}(q)\bigr{\\}}$.
By direct calculations,
$\rho_{ss}(\vartheta,r,h_{0})=\max_{\\{q>0\\}}f_{ss}(\vartheta,r,h_{0})$ and
the claim follows.
Last, consider (2.51). The proof is very similar to that of the subset
selection, except for that we need to use Ji and Jin (2011, Lemma 4.1),
instead of Ji and Jin (2011, Lemma 4.3). For this reason, we omit the proof.
$\Box$
#### 6.6.1 Proof of Lemma 21
By the symmetry in (2.46)-(2.47) when $G$ is given by (2.48), we only need to
consider the case where $h_{0}\in[0,1)$ and $\beta_{1}\geq 0$. Introduce
events, $A_{0}=\\{\beta_{1}=\beta_{2}=0\\}$,
$A_{1}=\\{\beta_{1}\geq\tau_{p},\beta_{2}=0\\}$,
$A_{21}=\\{\beta_{1}\geq\tau_{p},\beta_{2}\geq\tau_{p}\\}$,
$A_{22}=\\{\beta_{1}\geq\tau_{p},\beta_{2}\leq-\tau_{p}\\}$,
$B_{0}=\\{\hat{\beta}_{1}=\hat{\beta}_{2}=0\\}$,
$B_{1}=\\{\hat{\beta}_{1}>0,\hat{\beta}_{2}=0\\}$,
$B_{21}=\\{\hat{\beta}_{1}>0,\hat{\beta}_{2}>0\\}$ and
$B_{22}=\\{\hat{\beta}_{1}>0,\hat{\beta}_{2}<0\\}$. It is seen that the
Hamming error
$=L_{p}(I+II+III),$ (6.99)
where $I=P(A_{0}\cap B_{0}^{c})$, $II=P(A_{1}\cap B_{1}^{c})$ and
$III=P(A_{21}\cap B_{21}^{c})+P(A_{22}\cap B_{22}^{c})$.
Let $H$ be the $2\times 2$ matrix with ones on the diagonals and $h_{0}$ on
the off-diagonals, $\alpha=(\beta_{1},\beta_{2})^{\prime}$, and
$w=(\tilde{Y}_{1},\tilde{Y}_{2})$, where we recall $\tilde{Y}=X^{\prime}Y$. It
is seen that $w\sim N(H\alpha,H)$. Write for short $\lambda=\sqrt{2q\log(p)}$.
Define regions on the plane of $(\tilde{Y}_{1},\tilde{Y}_{2})$,
$D_{0}=\\{\max(|\tilde{Y}_{1}|,|\tilde{Y}_{2}|)>\lambda\ or\
\tilde{Y}_{1}^{2}+\tilde{Y}_{2}^{2}-2h_{0}\tilde{Y}_{1}\tilde{Y}_{2}>2\lambda^{2}(1-h_{0}^{2})\\}$,
$D_{1}=\\{|\tilde{Y}_{1}|<\lambda\ ,\ \tilde{Y}_{1}<\tilde{Y}_{2}\ or\
|\tilde{Y}_{2}-h_{0}\tilde{Y}_{1}|>\lambda\sqrt{1-h_{0}^{2}}\\}$,
$D_{21}=\\{\tilde{Y}_{2}-h_{0}\tilde{Y}_{1}<\lambda\sqrt{1-h_{0}^{2}}\ or\
\tilde{Y}_{1}-h_{0}\tilde{Y}_{2}<\lambda\sqrt{1-h_{0}^{2}}\\}$ and
$D_{22}=\\{\tilde{Y}_{2}-h_{0}\tilde{Y}_{1}>-\lambda\sqrt{1-h_{0}^{2}}\ or\
\tilde{Y}_{1}-h_{0}\tilde{Y}_{2}>\lambda\sqrt{1-h_{0}^{2}}\ or\
\tilde{Y}_{1}^{2}+\tilde{Y}_{2}^{2}-2h_{0}\tilde{Y}_{1}\tilde{Y}_{2}<2\lambda^{2}(1-h_{0}^{2})\\}$.
Using (Ji and Jin, 2011, Lemma 4.3), we have
$B_{0}^{c}=\\{(\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\in D_{0}\\}$,
$B_{1}^{c}=\\{(\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\in D_{1}\\}$,
$B_{21}^{c}=\\{(\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\in D_{21}\\}$, and
$B_{22}^{c}=\\{(\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\in D_{22}\\}$. By direct
calculation and Mills’ ratio, it follows that for all
$\mu\in\Theta_{p}(\tau_{p})$,
$I=L_{p}\cdot(P(N(0,1)>\lambda)+P(\chi_{2}^{2}>2\lambda^{2}))=L_{p}\cdot
p^{-q},$ (6.100) $II\leq L_{p}\cdot
P(N((\tau_{p},h_{0}\tau_{p})^{\prime},H)\in D_{1})=L_{p}\cdot
p^{-\vartheta-\min[(\sqrt{r}-\sqrt{q})^{2},(1-h_{0})r/2,q]},$ (6.101)
and when $\beta_{1}=\tau_{p}$ and $\beta_{2}=0$, the equality holds in
(6.101). At the same time, note that over the event $A_{21}$, the worst case
scenario, is where $\beta_{1}=\beta_{2}=\tau_{p}$. In such a case,
$(\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\sim
N(((1+h_{0})\tau_{p},(1+h_{0})\tau_{p})^{\prime},H)$. Combining this with
Mills’ ratio, it follows that for all $\mu\in\Theta_{p}(\tau_{p})$,
$P(A_{21}\cap B_{21}^{c})=P((\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\in
D_{21})\leq L_{p}\cdot
p^{-2\vartheta-(\sqrt{r(1-h_{0}^{2})}-\sqrt{q})_{+}^{2}},$ (6.102)
and the equality holds when $\beta_{1}=\beta_{2}=\tau_{p}$. Similarly, note
that over the event $A_{22}$, in the worst case scenario,
$\beta_{1}=-\beta_{2}=\tau_{p}$. In such a case,
$(\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\sim
N(((1-h_{0})\tau_{p},-(1-h_{0})\tau_{p})^{\prime},H)$. Combining this with
Mills’ ratio, it follows that for all $\mu\in\Theta_{p}(\tau_{p})$,
$P(A_{22}\cap B_{22}^{c})=P((\tilde{Y}_{1},\tilde{Y}_{2})^{\prime}\in
D_{22})\leq L_{p}\cdot
p^{-2\vartheta-\min([(\sqrt{r(1-h_{0}^{2})}-\sqrt{q})_{+}]^{2},2\\{[\sqrt{r(1-h_{0})}-\sqrt{q}]_{+}\\}^{2})},$
(6.103)
and the equality holds when $\beta_{1}=-\beta_{2}=\tau_{p}$. Inserting
(6.100)-(6.103) into (6.99) gives the claim. $\Box$
### 6.7 Lemma 22 and the proof
###### Lemma 22
Let $(V_{0j}^{*},V_{1j}^{*})$ be defined as in (2.35). If the conditions of
Theorem 6 hold, then $\max\\{|V_{0j}^{*}\cup
V_{1j}^{*}|\\}\leq(\vartheta+r)^{2}/(2\vartheta r)$.
Proof. Let $V_{0}=\emptyset$ and $V_{1}=\\{j\\}$. It is seen that
$\alpha^{*}(V_{0},V_{1};\Omega)=1$, and
$\eta(V_{0},V_{1};\Omega)\leq(\vartheta+r)^{2}/(4r)$. Using this and the
definitions of $V_{0j}^{*}$ and $V_{1j}^{*}$,
$\max\\{|V_{0j}^{*}|,|V_{1j}^{*}|\\}\vartheta\leq(\vartheta+r)^{2}/(4r)$ and
the claim follows. $\Box$
Acknowledgments: We thank Pengsheng Ji and Tracy Ke for help and pointers. JJ
and QZ were partially supported by NSF CAREER award DMS-0908613. QZ was also
partially supported by the NIH Grant P50-MH080215-02. CHZ was partially
supported by the NSF Grants DMS-0906420, DMS-1106753 and NSA Grant
H98230-11-1-0205. The research was supported in part by the computational
resources on PittGrid.
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|
arxiv-papers
| 2012-04-29T03:57:18 |
2024-09-04T02:49:30.350552
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiashun Jin, Cun-Hui Zhang, Qi Zhang",
"submitter": "Qi Zhang",
"url": "https://arxiv.org/abs/1204.6452"
}
|
1204.6635
|
# The rare decay $B_{c}\rightarrow D_{s}^{*}\mu^{+}\mu^{-}$ in a family non-
universal $Z^{\prime}$ model
Lü Lin-Xia1 lvlinxia@sina.com Zhang Guo-Fang1 Wang Shuai-Wei1 Zhang Zhi-Qing2
1\. Physics and electronic engineering college, Nanyang Normal University,
Nanyang, Henan 473061, P.R.China
2\. Department of Physics, Henan University of Technology, Zhengzhou, Henan
450052, P.R.China
###### Abstract
Using the form factors calculated in the three-point QCD sum rules, we
calculate the new physics contributions to the physical observables of
$B_{c}\to D_{s}^{*}\mu^{+}\mu^{-}$ decay in a family non-universal
$Z^{\prime}$ model. Under the consideration of three cases of the new physics
parameters, we find that: (a) the $Z^{\prime}$ boson can provide large
contributions to the differential decay rates; (b) the forward-backward
asymmetry (FBA) can be increased by about $47\%$, $38\%$, and $110\%$ at most
in S1, S2, and extreme limit values (ELV), respectively. In addition, the zero
crossing can be shifted in all the cases; (c) when $\hat{s}>0.08$, the value
of $P_{L}$ can be changed from $-1$ in the Standard Model (SM) to $-0.5$ in
S1, $-0.6$ in S2, and $0$ in extreme limit values, respectively; (d) the new
physics corrections to $P_{T}$ will decrease the SM prediction about $25\%$
for the cases of S1 and S2, $100\%$ for the case of ELV.
Key words: Rare decays, Standard Model, a family non-universal $Z^{\prime}$
model
###### pacs:
13.20.He, 12.60.Cn, 12.15.Ji, 14.40.Nd
## I Introduction
The Standard Model (SM) of interactions among elementary particles is one of
the best verified physics theories up to now but there are many open
fundamental questions remain unanswered within the scope of the SM. High
energy physics experiments are designed to address the open questions through
the search of new physics (NP) using two complementary approaches. One is to
discover the new particles at the high energy Large Hadron Collider (LHC). The
other is to search for the effects of NP through measurements of flavor
physics reactions at lower energy scales and evidence of a deviation from the
SM prediction.
After the observation of the rare radiative decay $b\rightarrow s\gamma$
CLEO1995 , the flavor-changing neutral current (FCNC) transitions became more
attractive and since then many works about rare radiative, leptonic and
semileptonic decays have been intensively done in the $B_{u,d,s}$ system
Ali2005 . Among these decays, semileptonic decay channels are significant
because their branching ratios are relatively larger. These works will be more
perfect if similar studies for $B_{c}$, observed in 1998 by CDF Collaboration
CDF1998 , are also included.
The charmed $B_{c}$ meson is a ground state of two heavy quarks $b$ and $c$.
Because of the two heavy quarks, the decays of the $B_{c}$ meson are rather
different from $B_{u}/B_{d}/B_{s}$ mesons. Physicists therefore believe that
the $B_{c}$ physics must be very rich compared to the other $B$ mesons if the
statistics reaches high level CDF1998 ; Brambilla1 ; Brambilla2 . At LHC,
around $5\times 10^{10}$ $B_{c}$ events per year are expected Sun08 ;
Altarelli08 . The expected number events are motivating to work on the $B_{c}$
phenomenology and this possibility will provide facilities to study the
observables of rare $B_{c}$ decays such as branching ratios, forward-backward
asymmetry and polarization asymmetries.
The rare $B_{c}\rightarrow D_{s}^{*}l^{+}l^{-}$ decays are proceeded by FCNC
transition of $b\rightarrow sl^{+}l^{-}$, which are forbidden at the tree
level in the SM, and play an important role in the precision test of the SM.
Meanwhile, they offer a valuable possibility of an indirect search of NP for
their sensitivity to the gauge structure and new contributions. Up to now, the
possible new physics contributions to $B_{c}\rightarrow D_{s}^{*}l^{+}l^{-}$
decays have been studied extensively, for example, by using model independent
effective Hamiltonian Yilmaz2007 , in Supersymmetric models AAhmed , with
fourth generation effects IAhmed , and in single universal extra dimension
Yilmaz .
When concentrating on the exclusive $B_{c}\rightarrow D_{s}^{*}l^{+}l^{-}$
decays, one needs to know the form factors. As for $B_{c}\rightarrow
D_{s}^{*}$ transition, the form factors have been calculated using different
approaches, such as light front constituent quark models formfactor1 , a
relativistic constituent quark model formfactor2 , a relativistic quark model
formfactor3 , the Ward identities formfactor4 , in light cone QCD formfactor5
; formfactor6 , and QCD sum rules formfactor7 ; formfactor8 ; formfactor9 . In
this work, we will adopt the form factors calculated in the three-point QCD
sum rules formfactor9 to study the $Z^{\prime}$ effects on the observables
for $B_{c}\rightarrow D_{s}^{*}\mu^{+}\mu^{-}$ decay.
The general framework for non-universal $Z^{\prime}$ model has been developed
in Ref. Langacker . In this model, $Z^{\prime}$ gauge boson could be naturally
derived by adding additional $U(1)^{\prime}$ gauge symmetry. Non-universal
$Z^{\prime}$ couplings can induce FCNC $b\to s$ and $d$ transitions at tree
level. Its effects on $b\rightarrow s$ transition have received great
attention and been widely studied in the literature. The previous works in a
family non-universal $Z^{\prime}$ model boson redound to resolve many puzzles,
such as ”$\pi K$ puzzle” Barger1 ; Chang1 , anomalous $\bar{B}_{s}-B_{s}$
mixing phase Liu ; Chang2 and mismatch in $A_{FB}(B\to
K^{\ast}\mu^{+}\mu^{-})$ spectrum at low $q^{2}$ region Chang3 ; CDLv .
Motivated by this, we will study the effects of the $Z^{\prime}$ boson on the
rare decay $B_{c}\rightarrow D_{s}^{*}\mu^{+}\mu^{-}$.
This paper is organized as follows. In Section 2, we present the effective
Hamiltonian responsible for the $b\to sl^{+}l^{-}$ transition in both the SM
and the family non-universal $Z^{\prime}$ model. In this section we also
present the matrix element, and the expressions of various physical
observables in $Z^{\prime}$ model. In Section 3, we show the numerical results
of the observables for $B_{c}\to D^{*}_{s}\mu^{+}\mu^{-}$ decay in the SM and
$Z^{\prime}$ model. The final section is the summary.
## II Effective Hamiltonian, matrix elements and observables for $b\to
sl^{+}l^{-}$ decay
At quark level, the rare semileptonic decay $b\to sl^{+}l^{-}$ can be
described in terms of the effective Hamiltonian which is given by
Altmannshofer:2008dz ; Chetyrkin:1996vx
${\cal
H}_{{\text{eff}}}=-\frac{4\,G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}\sum_{i=1}^{10}C_{i}(\mu)O_{i}(\mu)\,.$
(1)
here the explicit expressions of $O_{i}$ can be found in Ref.
Altmannshofer:2008dz , in which
$\displaystyle
O_{9}=\frac{e^{2}}{g_{s}^{2}}(\bar{d}\gamma_{\mu}P_{L}b)(\bar{l}\gamma^{\mu}l)\,,\quad
O_{10}=\frac{e^{2}}{g_{s}^{2}}(\bar{d}\gamma_{\mu}P_{L}b)(\bar{l}\gamma^{\mu}\gamma_{5}l)\,.$
(2)
The Wilson coefficients $C_{i}$ can be expanded perturbatively Beneke:2001at ;
bobeth ; bobeth02 ; Huber:2005ig . The effective coefficients $C_{7,9}^{eff}$,
can be written as Altmannshofer:2008dz
$\displaystyle C_{7}^{\rm
eff}=\frac{4\pi}{\alpha_{s}}\,C_{7}-\frac{1}{3}\,C_{3}-\frac{4}{9}\,C_{4}-\frac{20}{3}\,C_{5}\,-\frac{80}{9}\,C_{6}\,,$
$\displaystyle C_{9}^{\rm eff}=\frac{4\pi}{\alpha_{s}}\,C_{9}+Y(\hat{s})\,,$
$\displaystyle C_{10}^{\rm eff}=\frac{4\pi}{\alpha_{s}}\,C_{10}\,,$ (3)
where the perturbative part $Y(q^{2})$ stands for the matrix element of four-
quark operators and is given by
$\displaystyle Y(q^{2})$ $\displaystyle=$ $\displaystyle
h(\hat{m_{c}},\hat{s})\big{(}\frac{4}{3}C_{1}+C_{2}+6C_{3}+60C_{5}\big{)}\,$
(4)
$\displaystyle+\frac{1}{2}h(1,\hat{s})\big{(}-7C_{3}-\frac{4}{3}C_{4}-76C_{5}-\frac{64}{3}C_{6}\big{)}\,$
$\displaystyle+\frac{1}{2}h(0,\hat{s})\big{(}-C_{3}-\frac{4}{3}C_{4}-16C_{5}-\frac{64}{3}C_{6}\big{)}\,$
$\displaystyle+\frac{4}{3}C_{3}+\frac{64}{9}C_{5}+\frac{64}{27}C_{6}\,.$
with $\hat{s}=q^{2}/m_{B_{c}}^{2}$, $\hat{m_{c}}=m_{c}/m_{B_{c}}$. We have
neglected the resonance contribution. For the detailed discussion of such
resonance effects, we refer to Refs. AAhmed ; IAhmed ; Yilmaz .
Exclusive decay $B_{c}\to D^{*}_{s}\mu^{+}\mu^{-}$ is described in terms of
matrix elements of the quark operators in the effective Hamiltonian over meson
states, which can be parameterized in terms of form factors. The matrix
elements of $B_{c}\to D^{*}_{s}$ transition are given by Ali-prd61
$\displaystyle\langle D_{s}^{*}(p)|(V-A)_{\mu}|B_{c}(p_{B_{c}})\rangle$
$\displaystyle=$
$\displaystyle-i\epsilon^{*}_{\mu}(m_{B_{c}}+m_{D_{s}^{*}})A_{0}(s)+i(p_{B_{c}}+p)_{\mu}(\epsilon^{*}p_{B_{c}})\,\frac{A_{+}(s)}{m_{B_{c}}+m_{D_{s}^{*}}}$
$\displaystyle+iq_{\mu}(\epsilon^{*}p_{B_{c}})\,\frac{2m_{D_{s}^{*}}}{s}A_{-}(s)\,+\epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}p_{B_{c}}^{\rho}p^{\sigma}\,\frac{2A_{V}(s)}{m_{B_{c}}+m_{D_{s}^{*}}}\,.$
(5)
and
$\displaystyle\langle
D_{s}^{*}(p)|\bar{s}\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b|{B_{c}}(p_{B_{c}})\rangle$
$\displaystyle=$ $\displaystyle
i\epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}p_{B_{c}}^{\rho}p^{\sigma}\,2T_{1}(s)$
(6)
$\displaystyle{}+T_{2}(s)\left\\{\epsilon^{*}_{\mu}(m_{B_{c}}^{2}-m_{{D_{s}^{*}}}^{2})-(\epsilon^{*}p_{B_{c}})\,(p_{B_{c}}+p)_{\mu}\right\\}$
$\displaystyle{}+T_{3}(s)(\epsilon^{*}p_{B_{c}})\left\\{q_{\mu}-\frac{s}{m_{B_{c}}^{2}-m_{{D_{s}^{*}}}^{2}}\,(p_{B_{c}}+p)_{\mu}\right\\}$
here $s=q^{2}$, $q_{\mu}=(p_{B_{c}}-p)_{\mu}$, and $\epsilon_{\mu}$ is
polarization vector of the vector meson $D_{s}^{*}$.
The dilepton invariant mass spectrum for $B_{c}\rightarrow
D_{s}^{*}l^{+}l^{-}$ decays can be expressed by Ali-prd61 ; liwenjun
$\frac{{\rm d}\Gamma}{{\rm
d}\hat{s}}=\frac{G_{F}^{2}\,\alpha^{2}\,m_{B_{c}}^{5}}{2^{10}\pi^{5}}\left|V_{ts}^{\ast}V_{tb}\right|^{2}\,{\hat{u}}(\hat{s})D$
(7)
where the explicit expression of $D$ is
$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{|A|^{2}}{3}\hat{s}{\lambda}(1+2\frac{\hat{m}_{l}^{2}}{\hat{s}})+|E|^{2}\hat{s}\frac{{\hat{u}}(\hat{s})^{2}}{3}\Bigg{.}$
(8)
$\displaystyle+\Bigg{.}\frac{1}{4\hat{m}_{D_{s}^{*}}^{2}}\left[|B|^{2}({\lambda}-\frac{{\hat{u}}(\hat{s})^{2}}{3}+8\hat{m}_{D_{s}^{*}}^{2}(\hat{s}+2\hat{m}_{l}^{2}))+|F|^{2}({\lambda}-\frac{{\hat{u}}(\hat{s})^{2}}{3}+8\hat{m}_{D_{s}^{*}}^{2}(\hat{s}-4\hat{m}_{l}^{2}))\right]\Bigg{.}$
$\displaystyle+\Bigg{.}\frac{{\lambda}}{4\hat{m}_{D_{s}^{*}}^{2}}\left[|C|^{2}({\lambda}-\frac{{\hat{u}}(\hat{s})^{2}}{3})+|G|^{2}\left({\lambda}-\frac{{\hat{u}}(\hat{s})^{2}}{3}+4\hat{m}_{l}^{2}(2+2\hat{m}_{D_{s}^{*}}^{2}-\hat{s})\right)\right]\Bigg{.}$
$\displaystyle-\Bigg{.}\frac{1}{2\hat{m}_{D_{s}^{*}}^{2}}\left[{\rm
Re}(BC^{\ast})({\lambda}-\frac{{\hat{u}}(\hat{s})^{2}}{3})(1-\hat{m}_{D_{s}^{*}}^{2}-\hat{s})\right.\Bigg{.}$
$\displaystyle+\left.\Bigg{.}{\rm
Re}(FG^{\ast})(({\lambda}-\frac{{\hat{u}}(\hat{s})^{2}}{3})(1-\hat{m}_{D_{s}^{*}}^{2}-\hat{s})+4\hat{m}_{l}^{2}{\lambda})\right]\Bigg{.}$
$\displaystyle-\Bigg{.}2\frac{\hat{m}_{l}^{2}}{\hat{m}_{D_{s}^{*}}^{2}}{\lambda}\left[{\rm
Re}(FH^{\ast})-{\rm
Re}(GH^{\ast})(1-\hat{m}_{D_{s}^{*}}^{2})\right]+\frac{\hat{m}_{l}^{2}}{\hat{m}_{D_{s}^{*}}^{2}}\hat{s}{\lambda}|H|^{2}\,,$
with $\hat{m}_{l}=m_{l}/m_{B_{c}}$, and
$\hat{m}_{D_{s}^{*}}=m_{D_{s}^{*}}/m_{B_{c}}$. The kinematic variables
$\hat{s}$ and $\hat{u}$ are the same as Ref. Ali-prd61 . The auxiliary
functions $A\,,B\,,C\,,E\,,F$ and $G$ which are combinations of the effective
Wilson coefficients in Eq. (II) and the form factors of $B_{c}\to D_{s}^{*}$
transition can be found in Refs. Ali-prd61 ; liwenjun . For the convenience of
the reader, we present these functions in the Appendix A.
The normalized forward-backward asymmetry (FBA) is defined as
${\cal A}_{FB}(\hat{s})=\int
d\hat{s}~{}\frac{\int^{+1}_{-1}dcos\theta\frac{d^{2}Br}{d\hat{s}dcos\theta}{\rm
Sign}(cos\theta)}{\int^{+1}_{-1}dcos\theta\frac{d^{2}Br}{d\hat{s}dcos\theta}}.$
(9)
According to this definition, the explicit expression of FBA is:
$\displaystyle\frac{d{\cal A}_{FB}}{d\hat{s}}D$ $\displaystyle=$
$\displaystyle\hat{u}(\hat{s})\hat{s}[Re(BE^{*})+Re(AF^{*})]\,.$ (10)
The lepton polarization can be defined as:
$\frac{d\Gamma(\hat{n})}{d\hat{s}}=\frac{1}{2}\big{(}\frac{d\Gamma}{d\hat{s}}\big{)}_{0}[1+(P_{L}\hat{e}_{L}+P_{N}\hat{e}_{N}+P_{T}\hat{e}_{T})\cdot\hat{n}]$
(11)
where the subscript $"0"$ stands for the unpolarized decay case. $P_{L}$ and
$P_{T}$ are the longitudinal and transverse polarization asymmetries in the
decay plane respectively, and $P_{N}$ is the normal polarization asymmetry in
the direction perpendicular to the decay plane.
The lepton polarization asymmetry $P_{i}$ can be derived by
$P_{i}(\hat{s})=\frac{d\Gamma(\hat{n}=\hat{e}_{i})/d\hat{s}-d\Gamma(\hat{n}=-\hat{e}_{i})/d\hat{s}}{d\Gamma(\hat{n}=\hat{e}_{i})/d\hat{s}+d\Gamma(\hat{n}=-\hat{e}_{i})/d\hat{s}}\;$
(12)
the results are
$\displaystyle P_{L}D$ $\displaystyle=$
$\displaystyle\sqrt{1-4\frac{\hat{m}^{2}_{l}}{\hat{s}}}\Bigg{\\{}\frac{2\hat{s}\lambda}{3}Re(AE^{*})+\frac{(\lambda+12\hat{s}\hat{m}^{2}_{D_{s}^{*}})}{3\hat{m}^{2}_{D_{s}^{*}}}Re(BF^{*})\Bigg{.}$
(13)
$\displaystyle\Bigg{.}-\frac{\lambda(1-\hat{m}^{2}_{D_{s}^{*}}-\hat{s})}{3\hat{m}^{2}_{D_{s}^{*}}}Re(BG^{*}+CF^{*})+\frac{\lambda^{2}}{3\hat{m}_{D_{s}^{*}}}Re(CG^{*})\Bigg{\\}},$
$\displaystyle P_{N}D$ $\displaystyle=$
$\displaystyle\frac{-\pi\sqrt{\hat{s}}\hat{u}(\hat{s})}{4\hat{m}_{D_{s}^{*}}}\Bigg{\\{}\frac{\hat{m}_{l}}{\hat{m}_{D_{s}^{*}}}\left[Im(FG^{*})(1+3\hat{m}^{2}_{D_{s}^{*}}-\hat{s})\right.\Bigg{.}$
(14)
$\displaystyle\Bigg{.}\left.+Im(FH^{*})(1-\hat{m}^{2}_{D_{s}^{*}}-\hat{s})-Im(GH^{*})\lambda\right]\Bigg{.}$
$\displaystyle\Bigg{.}+2\hat{m}_{D_{s}^{*}}\hat{m}_{l}[Im(BE^{*})+Im(AF^{*})]\Bigg{\\}},$
$\displaystyle P_{T}D$ $\displaystyle=$
$\displaystyle\frac{\pi\sqrt{\lambda}\hat{m}_{l}}{4\sqrt{\hat{s}}}\Bigg{\\{}4\hat{s}Re(AB^{*})+\frac{(1-\hat{m}^{2}_{D_{s}^{*}}-\hat{s})}{\hat{m}^{2}_{D_{s}^{*}}}\left[-Re(BF^{*})+(1-\hat{m}^{2}_{D_{s}^{*}})Re(BG^{*})+\hat{s}Re(BH^{*})\right]\Bigg{.}$
(15)
$\displaystyle\Bigg{.}+\frac{\lambda}{\hat{m}^{2}_{D_{s}^{*}}}[Re(CF^{*})-(1-\hat{m}^{2}_{D_{s}^{*}})Re(CG^{*})-\hat{s}Re(CH^{*})]\Bigg{\\}}.$
In the family non-universal $Z^{\prime}$ model, the flavor neutral currents
arise even at tree level owing to non-diagonal chiral coupling matrix.
Postulating that the couplings of right-handed quark flavors with $Z^{\prime}$
boson are diagonal, the $Z^{\prime}$ part of the effective Hamiltonian for
$b\to sl^{+}l^{-}$ transition is described by Liu
${\cal H}_{eff}^{Z^{\prime}}(b\to
sl^{+}l^{-})=-\frac{2G_{F}}{\sqrt{2}}V_{tb}V^{\ast}_{ts}\Big{[}-\frac{B_{sb}^{L}B_{ll}^{L}}{V_{tb}V^{\ast}_{ts}}(\bar{s}b)_{V-A}(\bar{l}l)_{V-A}-\frac{B_{sb}^{L}B_{ll}^{R}}{V_{tb}V^{\ast}_{ts}}(\bar{s}b)_{V-A}(\bar{l}l)_{V+A}\Big{]}+{\rm
h.c.}\,.$ (16)
To extract the $Z^{\prime}$ corrections to the Wilson coefficients, one can
reformulate Eq. (16) as
${\cal H}_{eff}^{Z^{\prime}}(b\to
sl^{+}l^{-})=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{\ast}_{ts}\left[\triangle
C_{9}^{\prime}O_{9}+\triangle C_{10}^{\prime}O_{10}\right]+{\rm h.c.}\,,$ (17)
with
$\displaystyle\triangle C_{9}^{\prime}(M_{W})$ $\displaystyle=$
$\displaystyle-\frac{g_{s}^{2}}{e^{2}}\frac{B_{sb}^{L}}{V_{ts}^{\ast}V_{tb}}S_{ll}^{LR}\,,$
$\displaystyle\triangle C_{10}^{\prime}(M_{W})$ $\displaystyle=$
$\displaystyle\frac{g_{s}^{2}}{e^{2}}\frac{B_{sb}^{L}}{V_{ts}^{\ast}V_{tb}}D_{ll}^{LR}\,.$
(18)
where $S_{ll}^{LR}=(B_{ll}^{L}+B_{ll}^{R})$,
$D_{ll}^{LR}=(B_{ll}^{L}-B_{ll}^{R})$ with $B_{sb}^{L}$ and $B_{ll}^{L,R}$
referring to the effective chiral $Z^{\prime}$ couplings to quarks and
leptons, respectively. The off-diagonal element $B_{sb}^{L}$ contains a new
weak phase and can be written as $|B_{sb}^{L}|e^{i\phi_{s}^{L}}$.
When we include the $Z^{\prime}$ contributions with the assumption of no
significant RG running effects between $M_{Z^{\prime}}$ and $M_{W}$ scales,
the Wilson coefficients can be written as
$C_{9,10}^{SM}(M_{W})\rightarrow C_{9,10}^{SM}(M_{W})+\triangle
C_{9,10}^{\prime}(M_{W})\;.$ (19)
After inclusion of the new contributions from $Z^{\prime}$ boson, the RG
evolution of the Wilson coefficients down to low scale is exactly the same as
in the SM.
## III Numerical results
In this section, we focus on the numerical calculations of the branching
ratios, forward-backward asymmetry and polarization asymmetries for
$B_{c}\rightarrow D_{s}^{*}\mu^{+}\mu^{-}$ decay. The input parameters which
are related to our analysis are summarized in Table 1.
Table 1: Default values of inputs parameters used in our numerical calculations. $m_{b}=4.8$ GeV, | $m_{B_{c}}=6.28$ GeV, | $m_{D_{s}^{*}}=2.112$ GeV, | $m_{\mu}=0.106$ GeV
---|---|---|---
$\left|V_{tb}V_{ts}^{*}\right|=0.041$, | $\alpha=1/137$, | $\tau_{B_{c}}=0.46\times 10^{-12}s$. |
For the form factors $A_{V}(s)$, $A_{0}(s)$, $A_{+}(s)$, $A_{-}(s)$,
$T_{1}(s)$, $T_{2}(s)$ and $T_{3}(s)$, we choose them derived by the three-
point QCD sum rules formfactor9 , in which the parametrization of the form
factors with respect to $q^{2}$ are as follows:
$F\left(q^{2}\right)=\frac{F\left(0\right)}{1+\alpha\hat{s}+\beta\hat{s}^{2}}.$
(20)
where the values of the parameters $F\left(0\right)$, $\alpha$ and $\beta$ are
listed in Table 2.
Table 2: $B_{c}\rightarrow D_{s}^{*}$ form factors in the QCD Sum Rules formfactor9 . $F(q^{2})$ | $\hskip 56.9055ptF(0)$ | $\hskip 56.9055pt\alpha$ | $\hskip 56.9055pt\beta$
---|---|---|---
$A_{V}\left(q^{2}\right)$ | $\hskip 65.44142pt0.54$ | $\hskip 56.9055pt-1.28$ | $\hskip 56.9055pt-0.230$
$A_{0}(q^{2})$ | $\hskip 65.44142pt0.30$ | $\hskip 56.9055pt-0.13$ | $\hskip 56.9055pt-0.180$
$A_{+}(q^{2})$ | $\hskip 65.44142pt0.36$ | $\hskip 56.9055pt-0.67$ | $\hskip 56.9055pt-0.066$
$A_{-}(q^{2})$ | $\hskip 56.9055pt-0.57$ | $\hskip 56.9055pt-1.11$ | $\hskip 56.9055pt-0.140$
$T_{1}(q^{2})$ | $\hskip 65.44142pt0.31$ | $\hskip 56.9055pt-1.28$ | $\hskip 56.9055pt-0.230$
$T_{2}(q^{2})$ | $\hskip 65.44142pt0.33$ | $\hskip 56.9055pt-0.10$ | $\hskip 56.9055pt-0.097$
$T_{3}(q^{2})$ | $\hskip 65.44142pt0.29$ | $\hskip 56.9055pt-0.91$ | $\hskip 65.44142pt0.007$
Table 3: The inputs parameters for the $Z^{\prime}$ couplings Chang2 ; Chang3 . | $|B_{sb}^{L}|(\times 10^{-3})$ | $\phi_{s}^{L}[^{\circ}]$ | $S^{LR}_{\mu\mu}(\times 10^{-2})$ | $D^{LR}_{\mu\mu}(\times 10^{-2})$
---|---|---|---|---
S1 | $1.09\pm 0.22$ | $-72\pm 7$ | $-2.8\pm 3.9$ | $-6.7\pm 2.6$
S2 | $2.20\pm 0.15$ | $-82\pm 4$ | $-1.2\pm 1.4$ | $-2.5\pm 0.9$
In the family non-universal $Z^{\prime}$ model, the $Z^{\prime}$ contributions
rely on four parameters $|B_{sb}^{L}|$, $\phi_{s}^{L}$, $S^{LR}_{\mu\mu}$ and
$D^{LR}_{\mu\mu}$. These parameters have been constrained from the well
measured decays by many groups Liu ; Chang2 ; Chang3 ; CDLv . $|B_{sb}^{L}|$
and $\phi_{s}^{L}$ have been strictly constrained by $\bar{B}_{s}-B_{s}$
mixing, $B\to\pi K^{(*)}$ and $\rho K$ decays. After taking into account
constraints from $\bar{B}_{d}\to X_{s}\mu\mu$, $K\mu\mu$ and $K^{*}\mu\mu$, as
well as $B_{s}\to\mu\mu$ decays, the bounds on $S^{LR}_{\mu\mu}$ and
$D^{LR}_{\mu\mu}$ are also obtained. For the sake of convenience, we recollect
their numerical results in Table 3, with S1 and S2 corresponding to two
fitting results of UTfit Collaboration for $\bar{B}_{s}-B_{s}$ mixing UTfit .
Recently, CDF, D0, and LHCb collaborations updated1 ; updated2 ; updated3
have updated the CP violation parameter $\phi_{s}$ in $B_{s}$ system. These
precise measurements will suppress the magnitude of $b-s-Z^{\prime}$ coupling
by about $10\%$, and have no effect on the new weak phase $\phi_{s}^{L}$.
However, the weak phase can be constrained by the data of $B\to\pi K^{(*)}$
and $\rho K$ decays and the results are consistent with the previous Refs.
Chang2 ; Chang3 . Indeed, the quantity that is directly related to the decay
studied here is the product of the couplings of $b-s-Z^{\prime}$ and
$\mu-\mu-Z^{\prime}$, and the updated experimental data of $B_{s}$ mixing have
less effect on it. According to the above analysis, we will adopt the inputs
parameters for the $Z^{\prime}$ couplings as in Table 3 in our theoretical
calculation. Meanwhile, we also choose the extreme values of S1 which are
named extreme limit values (ELV) to show the maximal effects of $Z^{\prime}$
contributions, and the ELV are
$|B_{sb}^{L}|=1.31\times
10^{-3}\,,\phi_{s}^{L}=-79^{\circ}\,,S^{LR}_{\mu\mu}=-6.7\times
10^{-2}\,,D^{LR}_{\mu\mu}=-9.3\times 10^{-2}\,.$ (21)
Using the input parameters given above, we obtain the results of the branching
ratios both in the SM and the family non-universal $Z^{\prime}$ model without
resonance contributions.
$\displaystyle Br(B_{c}\to D_{s}^{*}\mu^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}2.32^{+0.27}_{-0.26}\times
10^{-7}&{\rm(SM)},\\\ 3.36^{+0.38}_{-0.35}\times 10^{-7}&{\rm(S1)},\\\
2.80^{+0.32}_{-0.30}\times 10^{-7}&{\rm(S2)},\\\ 5.21^{+0.57}_{-0.54}\times
10^{-7}&{\rm(ELV)}.\end{array}\right.$ (26)
The theoretical errors are induced by the uncertainties of form factors. From
the numerical results, one can see that branching ratio for decay $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ is sensitive to the $Z^{\prime}$ contributions. With
respect to the central value of the SM prediction, the new physics
contributions in the family non-universal $Z^{\prime}$ model can provide an
enhancement about $45\%$, $21\%$, and $125\%$ for the case of S1, S2, and ELV,
respectively.
Figure 1: The $\hat{s}$ dependence of the differential decay rates
$dBr(B_{c}\to D_{s}^{*}\mu^{+}\mu^{-})/d\hat{s}$ both in the SM and the family
non-universal $Z^{\prime}$ model. The solid, dashed, dash-dotted, short-dashed
lines show the SM prediction, the theoretical results of S1, S2, and ELV,
respectively.
Fig. 1 shows the $\hat{s}$ dependence of the differential decay rates for
decay $B_{c}\to D_{s}^{*}\mu^{+}\mu^{-}$ both in the SM and the family non-
universal $Z^{\prime}$ model using the central values of the input parameters.
The solid line refers to the SM prediction, while the dashed, dash-dotted,
short-dashed curves correspond to the theoretical results of S1, S2, and ELV,
respectively. The $Z^{\prime}$ enhancements to the differential decay rate are
significant in almost the whole region of $\hat{s}$ and strongly depend on the
variation of NP parameters.
Figure 2: The FBA of decay $B_{c}\to D_{s}^{*}\mu^{+}\mu^{-}$ as a function
$\hat{s}$ both in the SM and the family non-universal $Z^{\prime}$ model.
The $\hat{s}$ dependence of forward-backward asymmetry for $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ decay is presented in Fig. 2. Compared to the SM
results, when including the NP effects from $Z^{\prime}$ boson, the FBA can be
increased by about $47\%$, $38\%$, and $110\%$ at most in S1, S2, and ELV,
respectively. It is easy to see that the zero crossing in $A_{FB}(B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-})$ also exists and $Z^{\prime}$ corrections can shift
$\hat{s}_{0}=0.075$ in the SM to $\hat{s}_{0}=0.104$ in S1, and
$\hat{s}_{0}=0.093$ in S2, respectively. As for the case of ELV, the
$Z^{\prime}$ effects on $A_{FB}(B_{c}\to D_{s}^{*}\mu^{+}\mu^{-})$ are more
significant and can lead zero crossing to vanish.
Figure 3: The longitudinal lepton polarization asymmetry of decay $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ as a function $\hat{s}$ both in the SM and the family
non-universal $Z^{\prime}$ model.
In Fig. 3, we plot the longitudinal lepton polarization asymmetry of decay
$B_{c}\to D_{s}^{*}\mu^{+}\mu^{-}$ as a function $\hat{s}$ both in the SM and
the family non-universal $Z^{\prime}$ model. After inclusion of the
$Z^{\prime}$ contributions, there are also apparent deviations in the values
of the $P_{L}(B_{c}\to D_{s}^{*}\mu^{+}\mu^{-})$ for all the cases in
$Z^{\prime}$ model from that of the SM predictions. When $\hat{s}>0.08$, the
value of the longitudinal polarization asymmetry can be changed from $-1$ in
the SM to $-0.5$ in S1, and $-0.6$ in S2, respectively. In the extreme case,
the $Z^{\prime}$ effects could flip the sign of the SM predictions when
$\hat{s}>0.006$ and the theoretical values might be close to zero in large
momentum region.
Figure 4: The transverse lepton polarization asymmetry of decay $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ as a function $\hat{s}$ both in the SM and the family
non-universal $Z^{\prime}$ model.
The transverse lepton polarization asymmetry of decay $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ as a function $\hat{s}$ both in the SM and the family
non-universal $Z^{\prime}$ model is given in Fig. 4. The new physics
corrections from $Z^{\prime}$ boson are small, and will decrease the SM
prediction about $25\%$ for the cases of S1 and S2 in low $\hat{s}$ region.
However, for the case of ELV, the decrease could be rather large and reach
$100\%$ of the SM predictions. In addition, the sign of $P_{T}$ will be
changed in low momentum region and its values approach to zero when
$\hat{s}>0.037$.
## IV Summary
In this paper, we calculated the $Z^{\prime}$ contributions to the branching
ratio, forward-backward asymmetry and polarization asymmetries for $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ decay in the family non-universal $Z^{\prime}$ model
by employing the effective Hamiltonian with the form factors calculated in the
three-point QCD sum rules.
In Section 2, we presented the theoretical framework of $b\to sl^{+}l^{-}$
transition including the effective Hamiltonian, matrix element and the
physical observables. In Section 3, we showed the numerical results of the
observables and made phenomenological analysis for $B_{c}\to
D^{*}_{s}\mu^{+}\mu^{-}$ decay in the SM and the family non-universal
$Z^{\prime}$ model.
As expected, the $Z^{\prime}$ contributions to the observables for $B_{c}\to
D_{s}^{*}\mu^{+}\mu^{-}$ decay could be significant in size. From the
numerical results, we found that:
* •
With respect to the SM prediction, the $Z^{\prime}$ contributions to the
differential decay rates are significant in almost the whole region of
$\hat{s}$ and strongly depend on the variation of NP parameters.
* •
The new physics enhancements to FBA could be large, and reach $47\%$, $38\%$,
and $110\%$ at most in S1, S2, and ELV, respectively. The zero crossing could
be shifted from $\hat{s}_{0}=0.075$ in the SM to $\hat{s}_{0}=0.104$ in S1,
and $\hat{s}_{0}=0.093$ in S2, respectively. As for the case of ELV, the
$Z^{\prime}$ effects could lead zero crossing to vanish.
* •
The values of $P_{L}$ deviated apparently from that of the SM predictions for
all the cases in $Z^{\prime}$ model. In high $\hat{s}$ region, the values of
$P_{L}$ could be changed from $-1$ in the SM to $-0.5$ in S1, $-0.6$ in S2,
and 0 in ELV, respectively.
* •
The new physics corrections to $P_{T}$ would decrease the SM prediction about
$25\%$ for the cases of S1 and S2 in low $\hat{s}$ region. However, for the
case of ELV, the decrease could be rather large and reach $100\%$ of the SM
predictions.
## Acknowledgments
One of the authors Lin-Xia Lü would like to thank Prof. Zhen-jun Xiao for his
valuable help. The work is supported by the National Natural Science
Foundation of China under Grant No. 10947020 and 11147004, and Natural Science
Foundation of Henan Province under Grant No. 112300410188.
## Appendix A: Auxiliary functions
The auxiliary functions are given as follows Ali-prd61 ; liwenjun :
$\displaystyle A(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{2}{1+\hat{m}_{D_{s}^{*}}}\widetilde{C}_{9}^{eff}(\hat{s})A_{V}(\hat{s})+\frac{4\hat{m}_{b}}{\hat{s}}\widetilde{C}_{7}^{eff}T_{1}(\hat{s}),$
(27) $\displaystyle B(\hat{s})$ $\displaystyle=$
$\displaystyle(1+\hat{m}_{D_{s}^{*}})\widetilde{C}_{9}^{eff}(\hat{s})A_{0}(\hat{s})+\frac{2\hat{m}_{b}}{\hat{s}}(1-\hat{m}^{2}_{D_{s}^{*}})\widetilde{C}_{7}^{eff}T_{2}(\hat{s}),$
(28) $\displaystyle C(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{1+\hat{m}_{D_{s}^{*}}}\widetilde{C}_{9}^{eff}(\hat{s})A_{+}(\hat{s})+\frac{2\hat{m}_{b}}{1-\hat{m}^{2}_{D_{s}^{*}}}\widetilde{C}_{7}^{eff}\left(T_{3}(\hat{s})+\frac{1-\hat{m}^{2}_{D_{s}^{*}}}{\hat{s}}T_{2}(\hat{s})\right),$
(29) $\displaystyle E(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{2}{1+\hat{m}_{D_{s}^{*}}}\widetilde{C}_{10}^{eff}A_{V}(\hat{s}),$
(30) $\displaystyle F(\hat{s})$ $\displaystyle=$
$\displaystyle(1+\hat{m}_{D_{s}^{*}})\widetilde{C}_{10}^{eff}A_{0}(\hat{s}),$
(31) $\displaystyle G(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{1+\hat{m}_{D_{s}^{*}}}\widetilde{C}_{10}^{eff}A_{+}(\hat{s}),$
(32) $\displaystyle H(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{2\hat{m}_{D_{s}^{*}}}{\hat{s}}\widetilde{C}_{10}^{eff}A_{-}(\hat{s}),$
(33)
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|
arxiv-papers
| 2012-04-30T13:53:50 |
2024-09-04T02:49:30.375156
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lu Lin-xia, Zhang Guo-fang, Wang Shuai-wei, Zhang Zhi-qing",
"submitter": "Linxia Lu",
"url": "https://arxiv.org/abs/1204.6635"
}
|
1204.6640
|
# Scattering problem for Klein-Gordon equation with cubic convolution
nonlinearity ††thanks: Supported by NSFC (10931007) and Zhejiang Provincial
NSF of China (Y6090158)
Ruying Xue
Department of Mathematics, Zhejiang University,
Hangzhou 310027, Zhejiang, P. R. China E-mail address: ryxue@zju.edu.cn
###### Abstract
The scattering problem for the Klein-Gordon equation with cubic convolution
nonlinearity is considered. Based on the Strichartz estimates for the
inhomogeneous Klein-Gordon equation, we prove the existence of the scattering
operator, which improves the known results in some sense.
Keywords: Asymptotic of solution; Klein-Gordon equation; scattering operator
Subject class: 35P25, 81Q05, 35B05
## 1 Introduction
This paper is concerned with the scattering problem for the nonlinear Klein-
Gordon equation of the form
$\left\\{\begin{array}[]{l}\partial_{t}^{2}u-\Delta
u+u=F_{\gamma}(u)\qquad(t,x)\in{R}\times{R}^{n}\\\
u|_{t=0}=f(x),\,\partial_{t}u|_{t=0}=g(x)\end{array}\right.$ (1.1)
where u is a real-valued or a complex-valued unknown function of $(t,x)\in
R\times R^{n}$. The nonlinearity is a cubic convolution term
$F_{\gamma}(u)=-(V_{\gamma}(x)\ast|u|^{2})u$ with $|V_{\gamma}(x)|\leq
C|x|^{-\gamma}$. Here, $0<\gamma<n$ and $\ast$ denotes the convolution in the
space variables. The term $F_{\gamma}(u)$ is an approximative expression of
the nonlocal interaction of specific elementary particles. The equation (1.1)
was studied by Menzala and Strauss in [1].
In order to define the scattering operator for (1.1), we first give some
Banach spaces. The usual Lebesgue space is given by $L^{p}=\\{\phi\in
S^{\prime}:\|\phi\|_{L^{p}}<+\infty\\}$, where the norm
$\|\phi\|_{L^{p}}=\\{\int_{R^{n}}|\phi(x)|dx\\}^{1/p}$ if $1\leq p<+\infty$
and $\|\phi\|_{L^{\infty}}=\sup_{x\in R^{n}}|\phi(x)|$ if $p=+\infty$. The
weighted Sobolev space $H^{\beta,k}_{p}$ is defined by
$H^{\beta,k}_{p}=\\{\phi\in S^{\prime}:\|\phi\|_{H^{\beta,k}_{p}}=\|\langle
x\rangle^{k}{\langle i\nabla\rangle}^{\beta}\phi\|_{L^{p}}<+\infty\\},$
with $\langle x\rangle=\sqrt{1+x^{2}}$ and ${\langle
i\nabla\rangle}=\sqrt{1-\bigtriangleup}$. We also write
$H^{\beta,k}=H^{\beta,k}_{2}$ and $H^{\beta}=H^{\beta,0}_{2}$ if it does not
cause a confusion. A Hilbert space $X^{\beta,k}$ is denoted by
$H^{\beta,k}\bigoplus H^{\beta-1,k}$. Let $X_{\rho}^{\beta,k}$ be a ball of a
radius $\rho>0$ with a center in the origin in the space $X^{\beta,k}$. The
scattering operator of (1.1) is defined as the mapping
$S:X_{\rho}^{\beta,k}\ni(f_{-},g_{-})\to(f_{+},g_{+})\in X^{\beta,0}$ if the
following condition holds:
For $(f_{-},g_{-})\in X_{\rho}^{\beta,k}$, there uniquely exists a time-global
solution $u\in C(R;H^{\beta})$ of (1.1), and data $(f_{+},g_{+})\in
X^{\beta,0}$ such that $u(t)$ approaches $u_{\pm}(t)$ in $H^{\beta}$ as
$t\to\pm\infty$, where $u_{\pm}(t)$ are solutions of linear Klein-Gordon
equations whose initial data are $(f_{\pm},g_{\pm})$, respectively.
We say that $(S,X^{\beta,k})$ is well-defined if we can define the scattering
operator $S:X_{\rho}^{\beta,k}\to X^{\beta,0}$ for some $\rho>0$. In [2],
Mochizuki prove that if $n\geq 3$, $\beta\geq 1$, $\gamma<n$ and
$2\leq\gamma\leq 2\beta+2$, then $(S,X^{\beta,0})$ is well-defined. Hidano [3]
see that if $n\geq 2$, $\beta\geq 1$, $4/3<\gamma<2$ and $k>1/3$, then
$(S,X^{\beta,k})$ is well-defined. By using the Strichartz estimate for pre-
admissible pair and the complex interpolation method for the weighted Sobolev
space, Hidano [4] shows that $(S,X^{\beta,k})$ is well-defined if $n\geq 2$,
$\beta\geq 1$, $4/3<\gamma<2$ and $k>(2-\gamma)/2$. Our aim of this article is
to show that $(S,X^{\beta,1})$ is well-defined if $n\geq 2$,
$1<\gamma<\min\\{\frac{2(n+1)}{n+2},\,\frac{3n-2}{n+2}\\},\frac{(n+2)(\gamma+1)}{4n}+\frac{1}{2}<\beta<\frac{(n+2)(\gamma+1)}{2n}.$
(1.2)
More precisely, we prove the following theorem.
###### Theorem 1.1
Let the function $V_{\gamma}(x)$ satisfy
$|V_{\gamma}(x)|\leq C|x|^{-\gamma},\,|\nabla V_{\gamma}(x)|\leq
C|x|^{-(1+\gamma)}.$
Assume that $n\geq 2$, $\gamma$ and $\beta$ satisfy (1.2). Then there exists a
positive number $\delta_{0}>0$ satisfying the following properties:
(1). For $(f,g)\in X^{\beta,1}$ with $\|(f,g)\|_{X^{\beta,1}}\leq\delta_{0}$,
there uniquely exist finial states $(f_{\pm},g_{\pm})\in X^{\beta,0}$ and a
solution $u(t)\in C(R;H^{\beta})$ of (1.1) such that $u(t)$ approaches
$u_{\pm}(t)$ in $X^{\beta,0}$ as $t\to\pm\infty$, where $u_{\pm}(t)$ are
solutions of the linear Klein-Gordon equation with initial data
$(f_{\pm},g_{\pm})$, respectively. Moreover, as $\pm t$ large enough we have
$\|(u(t),\partial_{t}u(t))-(u_{\pm}(t),\partial_{t}u_{\pm}(t))\|_{X^{\beta,0}}\leq
C\langle t\rangle^{-\delta}$
with $\delta=\frac{2n\beta}{n+2}-2>0$.
(2). For $(f_{-},g_{-})\in X^{\beta,1}$ with
$\|(f_{-},g_{-})\|_{X^{\beta,1}}\leq\delta_{0}$, there uniquely exists a
finial state $(f_{+},g_{+})\in X^{\beta,0}$ and a solution $u(t)\in
C(R;H^{\beta})$ of (1.1) such that $u(t)$ approaches $u_{\pm}(t)$ in
$X^{\beta,0}$ as $t\to\pm\infty$, where $u_{\pm}(t)$ are solutions of the
linear Klein-Gordon equation with initial data $(f_{\pm},g_{\pm})$,
respectively. Moreover, as $\pm t$ large enough we have
$\|(u(t),\partial_{t}u(t))-(u_{\pm}(t),\partial_{t}u_{\pm}(t))\|_{X^{\beta,0}}\leq
C\langle t\rangle^{-\delta}$
with $\delta=\frac{2n\beta}{n+2}-2>0$.
In this article we denote by $J_{\varepsilon}={\langle
i\nabla\rangle}x+i{\varepsilon}t\nabla$ ,
$L_{\varepsilon}=i{\partial}_{t}-{\varepsilon}{\langle i\nabla\rangle}$ and
$P=t\nabla+x\partial_{t}$ with ${\varepsilon}\in\\{+,-\\}$. For a given Banach
space with norm $\|\cdot\|$ and a vector $v=(v^{+},v^{-})$, denote by
$\|v\|=\|v^{+}\|+\|v^{-}\|,\,\|Pv\|=\|Pv^{+}\|+\|Pv^{-}\|,$
$\|Jv\|=\|J_{+}v^{+}\|+\|J_{-}v^{-}\|,\,\|Lv\|=\|L_{+}v^{+}\|+\|L_{-}v^{-}\|.$
We also denote by the space-time norm
$\|\phi\|_{L_{t}^{r}(I,L_{x}^{q})}=\|\|\phi(t)\|_{L_{x}^{q}(R^{n})}\|_{L_{t}^{r}(I)},$
where I is a bounded or unbounded time interval, and denote different positive
constants by the same letter C.
The rest of the article is organized as follows. In Section 2 we give some
preliminary calculations. Then Section 3 is devoted to the proof of Theorem
1.1.
## 2 Preliminaries
In this section, we prove some lemmas for our main results. Let
$w^{{\varepsilon}}=i\partial_{t}{\langle i\nabla\rangle}^{-1}u-\varepsilon u$
with ${\varepsilon}\in\\{+,-\\}$. Then the Klein-Gordon equation (1.1) can be
be rewritten as a system of equations
$\left\\{\begin{array}[]{l}L_{\varepsilon}w^{{\varepsilon}}={\langle
i\nabla\rangle}^{-1}F_{\gamma}(u)\\\
w^{\varepsilon}|_{t=0}=w_{0}^{\varepsilon}\end{array}\right.$ (2.1)
where $L_{\varepsilon}=i{\partial}_{t}-{\varepsilon}{\langle i\nabla\rangle}$,
$w_{0}^{\varepsilon}=i{\langle i\nabla\rangle}^{-1}g+{\varepsilon}f$. By the
fact that
$u=\frac{1}{2}(w^{+}-w^{-}),\partial_{t}u=-\frac{i}{2}{\langle
i\nabla\rangle}(w^{+}+w^{-}),$
we can rewrite the term $F_{\gamma}(u)$ as
$F_{\gamma}(u)=\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}(V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})w^{{\varepsilon}_{3}}$
with some constants $C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}$.
Denote $U_{\varepsilon}(t)\varphi=e^{-{\varepsilon}i{\langle
i\nabla\rangle}t}\varphi$ and for given $T\in R$,
$\Psi_{\varepsilon}[g]=\int_{T}^{t}U_{\varepsilon}(t-\tau){\langle
i\nabla\rangle}^{-1}g(\tau)d\tau,$
###### Lemma 2.1
Let $2\leq q<\frac{2n}{n-2}$, $\frac{2}{r}=\frac{n}{2}(1-\frac{2}{q})$. Then
for any time interval I and for any given $T\in I$ the following estimates are
true:
$\|\Psi_{\varepsilon}[g]\|_{L_{t}^{r}(I,L^{q})}\leq\|g\|_{L_{t}^{r\prime}(I,H_{q\prime}^{2\mu-1})},$
$\|\Psi_{\varepsilon}[g]\|_{L_{t}^{\infty}(I,L^{2})}\leq\|g\|_{L_{t}^{r\prime}(I,H_{q\prime}^{\mu-1})},$
and
$\|U_{\varepsilon}(t)\varphi\|_{L_{t}^{r}(I,L^{q})}\leq\|\varphi\|_{H^{\mu}},$
where $r^{\prime}=\frac{r}{r-1}$, $q^{\prime}=\frac{q}{q-1}$ and
$\mu=\frac{1}{2}(1+\frac{n}{2})(1-\frac{2}{q})$.
The proof of Lemma 2.1 is based on the duality argument along with the
$L^{p}-L^{q}$ time decay estimates. The similar result be found in [5].
###### Lemma 2.2
Assume $2\leq p<\frac{2n}{n-2}$ for $n\geq 3$ ($2\leq p<+\infty$ for $n=2$),
denote by $\alpha=(1+\frac{n}{2})(1-\frac{2}{p})$. The estimate is valid
$\|\phi\|_{L^{p}}\leq C\langle
t\rangle^{-\frac{n}{2}(1-\frac{2}{p})}(\|\phi\|_{H^{\alpha}}+\|J_{\varepsilon}\phi\|_{H^{\alpha-1}}),$
for all $t\in R$, provided that the right-hand side is finite.
This lemma comes from Lemma 2.1 in [5] and the fact that $\|\phi\|_{L^{p}}\leq
C\|\phi\|_{H^{\alpha}}$ when $p\geq 2$.
###### Lemma 2.3
Assume $|V_{\gamma}(x)|\leq|x|^{-\gamma}$ with $0<\gamma<n$,
${\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}$.
(1). For $1<r<+\infty$, $1<p_{1},p_{2}<+\infty$ and $p_{3}>r$ satisfying
$1+\frac{1}{r}=\frac{\gamma}{n}+\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}$,
we have
$\|(V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})w^{{\varepsilon}_{3}}\|_{L^{r}}\leq\|w^{{\varepsilon}_{1}}\|_{L^{p_{1}}}\|w^{{\varepsilon}_{2}}\|_{L^{p_{2}}}\|w^{{\varepsilon}_{3}}\|_{L^{p_{3}}}.$
(2). For $\rho>0$, $1<r<+\infty$, $1<p_{jk}<+\infty$ for $j,k\in\\{1,2\\}$ and
$p_{13},p_{23}>r$ satisfying
$1+\frac{1}{r}=\frac{\gamma}{n}+\frac{1}{p_{j1}}+\frac{1}{p_{j2}}+\frac{1}{p_{j3}}$,
we have
$\displaystyle\|(V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})w^{{\varepsilon}_{3}}\|_{H^{\rho}_{r}}$
$\displaystyle\leq$
$\displaystyle\|w^{{\varepsilon}_{1}}\|_{H^{\rho}_{p_{11}}}\|w^{{\varepsilon}_{2}}\|_{L^{p_{12}}}\|w^{{\varepsilon}_{3}}\|_{L^{p_{13}}}+\|w^{{\varepsilon}_{1}}\|_{L^{p_{12}}}\|w^{{\varepsilon}_{2}}\|_{H^{\rho}_{p_{11}}}\|w^{{\varepsilon}_{3}}\|_{L^{p_{13}}}$
$\displaystyle+\|w^{{\varepsilon}_{1}}\|_{L^{p_{21}}}\|w^{{\varepsilon}_{2}}\|_{L^{p_{22}}}\|w^{{\varepsilon}_{3}}\|_{H^{\rho}_{p_{23}}}$
Proof. To prove (1), put $\frac{1}{p_{4}}=\frac{1}{r}-\frac{1}{p_{3}}$. By the
Hölder inequality and the Hardy-Littlewood-Sobolev inequality, we have
$\displaystyle\|(V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})w^{{\varepsilon}_{3}}\|_{L^{r}}$
$\displaystyle\leq$
$\displaystyle\|V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}}\|_{L^{p_{4}}}\|w^{{\varepsilon}_{3}}\|_{L^{p_{3}}}$
$\displaystyle\leq$
$\displaystyle\|w^{{\varepsilon}_{1}}\|_{L^{p_{1}}}\|w^{{\varepsilon}_{2}}\|_{L^{p_{2}}}\|w^{{\varepsilon}_{3}}\|_{L^{p_{3}}}$
since $1+\frac{1}{p_{4}}=\frac{\gamma}{n}+\frac{1}{p_{1}}+\frac{1}{p_{2}}$.
To prove (2), we set $\frac{1}{r}=\frac{1}{p_{14}}+\frac{1}{p_{13}}$ and
$\frac{1}{r}=\frac{1}{p_{24}}+\frac{1}{p_{23}}$. For $\rho>0$, the generalized
Hölder inequality in [6] implies
$\|(V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})w^{{\varepsilon}_{3}}\|_{H^{\rho}_{r}}\leq\|V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}}\|_{H^{\rho}_{p_{14}}}\|w^{{\varepsilon}_{3}}\|_{L^{p_{13}}}+\|V\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}}\|_{L^{p_{24}}}\|w^{{\varepsilon}_{3}}\|_{H^{\rho}_{p_{23}}}$
By the generalized Hölder inequality and the Hardy-Littlewood-Sobolev
inequality, we have
$\displaystyle\|V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}}\|_{H^{\rho}_{p_{14}}}$
$\displaystyle\leq$ $\displaystyle\|V_{\gamma}\ast{\langle
i\nabla\rangle}^{\rho}(\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})\|_{L^{p_{14}}}\leq\|{\langle
i\nabla\rangle}^{\rho}(\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}})\|_{L^{p_{15}}}$
$\displaystyle\leq$
$\displaystyle\|w^{{\varepsilon}_{1}}\|_{H^{\rho}_{p_{11}}}\|w^{{\varepsilon}_{2}}\|_{L^{p_{12}}}+\|w^{{\varepsilon}_{2}}\|_{H^{\rho}_{p_{11}}}\|w^{{\varepsilon}_{1}}\|_{L^{p_{12}}}$
since $1+\frac{1}{p_{14}}=\frac{\gamma}{n}+\frac{1}{p_{15}}$ and
$\frac{1}{p_{15}}=\frac{1}{p_{11}}+\frac{1}{p_{12}}$. Similarly we have
$\|V_{\gamma}\ast\overline{w^{{\varepsilon}_{1}}}w^{{\varepsilon}_{2}}\|_{L^{p_{24}}}\leq\|w^{{\varepsilon}_{1}}\|_{L^{p_{21}}}\|w^{{\varepsilon}_{2}}\|_{L^{p_{22}}}.$
$\Box$
## 3 Proof of Theorem 1.1
For $1<\gamma<\min\\{\frac{2(n+1)}{n+2},\frac{3n-2}{n+2}\\}$, we choose
$\frac{(n+2)(\gamma+1)}{4n}+\frac{1}{2}<\beta<\frac{(n+2)(\gamma+1)}{2n},q=\left(\frac{2\beta}{n+2}+\frac{1}{2}-\frac{\gamma+1}{n}\right)^{-1},$
They satisfy
$1\leq\beta\leq 2,2<q<\frac{2n}{n+2(1-\gamma)},1<\gamma<\frac{3n\beta}{n+2}.$
Let $\mu=\frac{1}{2}(1+\frac{n}{2})(1-\frac{2}{q})$, we also have
$\mu+\beta-2\leq 0,\mu\leq\beta-1,\mbox{and}\quad 0<\mu\leq\frac{1}{2}.$
Let $r,p$ and $s$ be chosen as
$\frac{2}{r}=\frac{n}{2}(1-\frac{2}{q}),\frac{2}{p}+\frac{\gamma}{n}=2-\frac{2}{q},\frac{2}{s}=1-\frac{2}{r}.$
The proof of Theorem 1.1(1). Introduce the function space
$X=\\{v=(v^{+},v^{-})\in C(R;(L^{2}(R^{n}))^{2});\quad\|v\|_{X}<+\infty\\}$
with the norm
$\displaystyle\|v\|_{X}$ $\displaystyle=$
$\displaystyle\|v\|_{L_{t}^{\infty}(R,H^{\beta})}+\|v\|_{L_{t}^{r}(R,H^{\beta-\mu}_{q})}+\|\partial_{t}v\|_{L_{t}^{\infty}(R,H^{\beta-1})}+\|\partial_{t}v\|_{L_{t}^{r}(R,L^{q})}$
$\displaystyle+\|Pv\|_{L_{t}^{\infty}(R,H^{\beta-1})}+\|Pv\|_{L_{t}^{r}(R,L^{q})}+\|Jv\|_{L_{t}^{\infty}(R,H^{\beta-1})}.$
Denote by $X_{\rho}$ a ball of a radius $\rho>0$ with a center in the origin
in the space $X$. Let us consider the linearized version of (2.1)
$\left\\{\begin{array}[]{l}L_{\varepsilon}w^{{\varepsilon}}={\langle
i\nabla\rangle}^{-1}F_{\gamma}(v)\\\
w^{\varepsilon}|_{t=0}=w_{0}^{\varepsilon}\end{array}\right.$ (3.1)
with a given vecter $v=(v^{+},v^{-})\in X_{\rho}$, where
$F_{\gamma}(v)=\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}(V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}$
with some given constants
$C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}$. The integration of
the linearized Cauchy problem (3.1) with respect to time yields
$w^{\varepsilon}=U_{\varepsilon}(t)w_{0}^{\varepsilon}+\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}\Psi_{\varepsilon}((V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}).$
(3.2)
Taking the $L_{t}^{\infty}(R;H^{\beta})$-norm of (3.2), applying the Hölder
inequality, Lemma 2.1 and Lemma 2.3, we find
$\displaystyle\|w^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta})}$
$\displaystyle\leq$
$\displaystyle\|U_{\varepsilon}(t)w_{0}^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta})}+C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|\Psi_{\varepsilon}((V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}})\|_{L_{t}^{\infty}(R;H^{\beta})}$
(3.3) $\displaystyle\leq$
$\displaystyle\|w_{0}^{\varepsilon}\|_{H^{\beta}}+C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|(V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}\|_{L_{t}^{r^{\prime}}(R;H^{\beta+\mu-1}_{q^{\prime}})}$
$\displaystyle\leq$
$\displaystyle\|w_{0}\|_{H^{\beta}}+C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\left\|\|{v^{{\varepsilon}_{1}}}\|_{H^{\beta+\mu-1}_{q}}\|v^{{\varepsilon}_{2}}\|_{L^{p}}\|v^{{\varepsilon}_{3}}\|_{L^{p}}\right\|_{L_{t}^{r^{\prime}}(R)}$
$\displaystyle\leq$
$\displaystyle\|w_{0}\|_{H^{\beta}}+C\|{v}\|_{L_{t}^{r}(R;H^{\beta-\mu}_{q})}\|v\|^{2}_{L^{s}_{t}(R;L^{p})}$
$\displaystyle\leq$
$\displaystyle\|w_{0}\|_{H^{\beta}}+C\rho\|v\|^{2}_{L^{s}_{t}(R;L^{p})}$
since $p>2>q^{\prime}$, $q>2>q^{\prime}$, $\mu\leq\frac{1}{2}$ and
$2-\frac{2}{q}=\frac{\gamma}{n}+\frac{2}{p}$. Similarly, taking the
$L_{t}^{r}(R;H^{\beta-\mu}_{q})$ we obtain
$\displaystyle\|w^{\varepsilon}\|_{L_{t}^{r}(R;H^{\beta-\mu}_{q})}$
$\displaystyle\leq$
$\displaystyle\|U_{\varepsilon}(t)w_{0}^{\varepsilon}\|_{L_{t}^{r}(R;H^{\beta-\mu}_{q})}+C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|\Psi_{\varepsilon}((V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}})\|_{L_{t}^{r}(R;H^{\beta-\mu}_{q})}$
(3.4) $\displaystyle\leq$
$\displaystyle\|w_{0}^{\varepsilon}\|_{H^{\beta}}+C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|(V_{\gamma}\ast\
\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}\|_{L_{t}^{r^{\prime}}(R;H^{\beta+\mu-1}_{q^{\prime}})}$
$\displaystyle\leq$
$\displaystyle\|w_{0}^{\varepsilon}\|_{H^{\beta}}+C\|{v}\|_{L_{t}^{r}(R;H^{\beta+\mu-1}_{q})}\|v\|^{2}_{L^{s}_{t}(R;L^{p})}$
$\displaystyle\leq$
$\displaystyle\|w_{0}\|_{H^{\beta}}+C\rho\|v\|^{2}_{L^{s}_{t}(R;L^{p})}$
since $\mu\leq\frac{1}{2}$, $p>2>q^{\prime}$, $q>2>q^{\prime}$ and
$2-\frac{2}{q}=\frac{\gamma}{n}+\frac{2}{p}$. Applying the operator
$\partial_{t}$ to (3.1) we deduce that $\partial_{t}w^{\varepsilon}$ satisfies
the following system
$\left\\{\begin{array}[]{l}L_{\varepsilon}\partial_{t}w^{\varepsilon}={\langle
i\nabla\rangle}^{-1}\partial_{t}F_{\gamma}(v)\\\
\partial_{t}w^{\varepsilon}|_{t=0}=-i{\varepsilon}{\langle
i\nabla\rangle}w_{0}^{\varepsilon}-i{\langle
i\nabla\rangle}^{-1}F_{\gamma}(v)|_{t=0}\end{array}\right.$
with
$F_{\gamma}(v)=\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}.$
Then by integrating with respect to time,
$\partial_{t}w^{\varepsilon}=U_{\varepsilon}(t)(\partial_{t}w^{\varepsilon}|_{t=0})+\Psi_{\varepsilon}(\partial_{t}F_{\gamma}(v)).$
Taking the $L_{t}^{\infty}(R;H^{\beta-1})$-norm and $L^{r}_{t}(R,L^{q})$-norm,
applying the Hölder inequality and Lemma 2.1 we find that, since $\beta\geq
1$, $\mu\leq\beta-1$ and $\mu+\beta-2\leq 0$,
$\displaystyle\|\partial_{t}w^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta-1})}+\|\partial_{t}w^{\varepsilon}\|_{L_{t}^{r}(R;L^{q})}$
$\displaystyle\leq$
$\displaystyle\|\partial_{t}w^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}+\|\partial_{t}F_{\gamma}(v)\|_{L^{r^{\prime}}_{t}(R,H^{\mu+\beta-2}_{q^{\prime}})}$
$\displaystyle\leq$
$\displaystyle\|\partial_{t}w^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}$
$\displaystyle\quad+C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|(V_{\gamma}\ast(\overline{\partial_{t}v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}}+\overline{v^{{\varepsilon}_{1}}}\partial_{t}v^{{\varepsilon}_{2}}))v^{{\varepsilon}_{3}}+(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})\partial_{t}v^{{\varepsilon}_{3}}\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}$
$\displaystyle\leq$
$\displaystyle\|\partial_{t}w^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}+C\|\partial_{t}v\|_{L^{r}_{t}(R,L^{q})}\|v\|^{2}_{L^{s}_{t}(R,L^{p})}$
$\displaystyle\leq$
$\displaystyle\|\partial_{t}w^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}+C\rho\|v\|^{2}_{L^{s}_{t}(R,L^{p})}$
On the other hand, we have
$\|\partial_{t}w^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}\leq\|w^{\varepsilon}_{0}\|_{H^{\beta}}+\|F_{\gamma}(v)\|_{L^{\infty}_{t}(R,H^{\beta-2})},$
and for $p_{1}>2$ satisfying $\frac{3}{2}=\frac{\gamma}{n}+\frac{3}{p_{1}}$,
$\displaystyle\|F_{\gamma}(v)\|_{L^{\infty}_{t}(R,H^{\beta-2})}$
$\displaystyle\leq$ $\displaystyle
C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}})\|_{L_{t}^{\infty}(R;L^{2})}$
$\displaystyle\leq$ $\displaystyle
C\|v\|^{3}_{L_{t}^{\infty}(R;L^{p_{1}})}\leq
C\|v\|^{3}_{L_{t}^{\infty}(R;H^{\beta})}\leq C\rho^{3}$
since $\beta\leq 2$,$\gamma\leq 3\beta$ and $\|v\|_{L^{p_{1}}}\leq
C\|v\|_{H^{\beta}}$. Then
$\|\partial_{t}w^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta-1})}+\|\partial_{t}w^{\varepsilon}\|_{L_{t}^{r}(R;L^{q})}\leq
C\|w_{0}\|_{H^{\beta}}+C\rho^{3}+C\rho\|v\|^{2}_{L^{s}_{t}(R,L^{p})}.$ (3.5)
Notice that $P=t\bigtriangledown+x\partial_{t}$, $J_{\varepsilon}={\langle
i\nabla\rangle}x+i{\varepsilon}t\bigtriangledown$ and
$L_{\varepsilon}=i\partial_{t}-{\varepsilon}{\langle i\nabla\rangle}$. We get
$J_{\varepsilon}=i{\varepsilon}P-{\varepsilon}L_{\varepsilon},\,[L_{\varepsilon},P]=-i{\varepsilon}{\langle
i\nabla\rangle}^{-1}\bigtriangledown L_{\varepsilon},$ $[x,{\langle
i\nabla\rangle}]={\langle i\nabla\rangle}^{-1}\bigtriangledown,\,[P,{\langle
i\nabla\rangle}^{-1}]={\langle
i\nabla\rangle}^{-3}\bigtriangledown\partial_{t}$
and
$P((V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}})=(V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})P(v^{{\varepsilon}_{3}})+(t\nabla
V_{\gamma}\ast\overline{{v^{{\varepsilon}_{1}}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}.$
Applying the operator $P$ to (3.1) yields
$\left\\{\begin{array}[]{l}L_{\varepsilon}Pw^{\varepsilon}=i{\varepsilon}{\langle
i\nabla\rangle}^{-2}\nabla F_{\gamma}(v)-{\langle
i\nabla\rangle}^{-1}PF_{\gamma}(v)-{\langle
i\nabla\rangle}^{-3}\nabla\partial_{t}F_{\gamma}(v)\\\
Pw^{\varepsilon}|_{t=0}=x\partial_{t}w^{\varepsilon}|_{t=0}=x(-i{\varepsilon}{\langle
i\nabla\rangle}w_{0}^{\varepsilon}-i{\langle
i\nabla\rangle}^{-1}F_{\gamma}(v)|_{t=0})\end{array}\right.$
with
$PF_{\gamma}(v)=\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}C_{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})Pv^{{\varepsilon}_{3}}+(t\nabla
V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}.$
Integrating with respect to time, we get
$Pw^{\varepsilon}=U_{\varepsilon}(t)(Pw^{\varepsilon}|_{t=0})-\Psi_{\varepsilon}(i{\varepsilon}{\langle
i\nabla\rangle}^{-1}\nabla
F_{\gamma}(v))+\Psi_{\varepsilon}(PF_{\gamma}(v))+\Psi_{\varepsilon}({\langle
i\nabla\rangle}^{-2}\nabla\partial_{t}F_{\gamma}(v)).$ (3.6)
Taking the $L_{t}^{\infty}(R;H^{\beta-1})$-norm and the
$L^{r}_{t}(R,L^{q})$-norm of (3.6), applying the Hölder inequality and Lemma
2.1, we find
$\displaystyle\|Pw^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta-1})}+\|Pw^{\varepsilon}\|_{L^{r}_{t}(R,L^{q})}$
(3.7) $\displaystyle\leq$
$\displaystyle\|Pw^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}+\|{\langle
i\nabla\rangle}^{-1}\nabla
F_{\gamma}\|_{L^{r^{\prime}}_{t}(R,L^{q^{\prime}})}$
$\displaystyle\qquad+\|PF_{\gamma}(v)\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}+\|{\langle
i\nabla\rangle}^{-2}\nabla\partial_{t}F_{\gamma}(v)\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}$
$\displaystyle\leq$
$\displaystyle\|Pw^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}+\|F_{\gamma}(v)\|_{L^{r^{\prime}}_{t}(R,L^{q^{\prime}})}$
$\displaystyle\qquad+\|PF_{\gamma}(v)\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}+\|\partial_{t}F_{\gamma}(v)\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}$
since $\beta\geq 1$ and $\mu+\beta-2\leq 0$ and $\mu\leq\beta-1$. As in the
proof of (3.5) we deduce
$\displaystyle\|F_{\gamma}(v)\|_{L^{r^{\prime}}_{t}(R,L^{q^{\prime}})}+\|\partial_{t}F_{\gamma}(v)\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}$
(3.8) $\displaystyle\leq$ $\displaystyle
C\|v\|_{L^{r}_{t}(R,L^{q})}\|v\|^{2}_{L^{s}_{t}(R,L^{p})}+C\|\partial_{t}v\|_{L^{r}_{t}(R,L^{q})}\|v\|^{2}_{L^{s}_{t}(R,L^{p})}\leq
C\rho\|v\|^{2}_{L^{s}_{t}(R,L^{p})}.$
Let $p_{3}>2$ and $s_{3}>2$ satisfy
$\frac{3}{2}-\frac{1}{q}=\frac{\gamma+1}{n}+\frac{2}{p_{3}},1-\frac{1}{r}=\frac{2}{s_{3}}.$
The Hölder inequality and Lemma 2.3 imply
$\displaystyle\|PF_{\gamma}(v)\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}$ (3.9)
$\displaystyle\leq$ $\displaystyle
C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\left[\|(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})Pv^{{\varepsilon}_{3}})\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}+\|(t\nabla
V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}})\|_{L_{t}^{r^{\prime}}(R;L^{q^{\prime}})}\right]$
$\displaystyle\leq$ $\displaystyle
C\|Pv\|_{L^{r}_{t}(R,L^{q})}\|v\|^{2}_{L^{s}_{t}(R,L^{p})}+C\|v\|_{L^{\infty}_{t}(R,L^{2})}\|t^{1/2}v\|^{2}_{L^{s_{3}}_{t}(R,L^{p_{3}})}$
$\displaystyle\leq$ $\displaystyle
C\rho\|v\|^{2}_{L^{s}_{t}(R,L^{p})}+C\rho\|t^{1/2}v\|^{2}_{L^{s_{3}}_{t}(R,L^{p_{3}})},$
here we use the condition $\|\nabla V_{\gamma}\|\leq C|x|^{-(\gamma+1)}$. By
Lemma 2.2 we have
$\displaystyle\|v\|_{L^{s}_{t}(R,L^{p})}$ $\displaystyle\leq$ $\displaystyle
C\|\langle
t\rangle^{-\frac{n}{2}(1-\frac{1}{p})}\left(\|v\|_{H^{\alpha}}+\|Jv\|_{H^{\alpha-1}}\right)\|_{L^{s}_{t}(R)}$
(3.10) $\displaystyle\leq$ $\displaystyle
C\left(\|v\|_{L^{\infty}_{t}(R,H^{\beta})}+\|Jv\|_{L^{\infty}_{t}(R,H^{\beta-1})}\right)\leq
C\rho$
since $\alpha=(1+\frac{n}{2})(1-\frac{2}{p})\leq\beta$ and
$\frac{n}{2}(1-\frac{2}{p})>\frac{1}{s}$. Similarly,
$\displaystyle\|t^{1/2}v\|_{L^{s_{3}}_{t}(R,L^{p_{3}})}$ $\displaystyle\leq$
$\displaystyle C\|\langle
t\rangle^{-\frac{n}{2}(1-\frac{1}{p_{3}})+\frac{1}{2}}\left(\|v\|_{H^{\alpha_{3}}}+\|Jv\|_{H^{\alpha_{3}-1}}\right)\|_{L^{s_{3}}_{t}(R)}$
(3.11) $\displaystyle\leq$ $\displaystyle
C\left(\|v\|_{L^{\infty}_{t}(R,H^{\beta})}+\|Jv\|_{L^{\infty}_{t}(R,H^{\beta-1})}\right)\leq
C\rho,$
since $\alpha_{3}=(1+\frac{n}{2})(1-\frac{2}{p_{3}})\leq\beta$ and
$\frac{n}{2}(1-\frac{2}{p_{3}})>\frac{1}{s_{3}}$. Then we obtain, from
(3.7)-(3.11),
$\displaystyle\|Pw^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta-1})}+\|Pw^{\varepsilon}\|_{L^{r}_{t}(R,L^{q})}\leq\|Pw^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}+C\rho^{3},$
(3.12)
$\displaystyle\|w^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta})}+\|w^{\varepsilon}\|_{L_{t}^{r}(R;H^{\beta-\mu}_{q})}\leq\|w_{0}^{\varepsilon}\|_{H^{\beta}}+C\rho^{3},$
(3.13)
$\displaystyle\|\partial_{t}w^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta-1})}+\|\partial_{t}w^{\varepsilon}\|_{L_{t}^{r}(R;L^{q})}\leq\|w_{0}^{\varepsilon}\|_{H^{\beta}}+C\rho^{3},$
(3.14)
To estimate the term $\|Pw^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}$, we give some
estimates. It follows from the Sobolev embedding theorem that
$\displaystyle\|F_{\gamma}(v)\|_{L^{\infty}_{t}(R,L^{2})}$ $\displaystyle\leq$
$\displaystyle
C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})v^{{\varepsilon}_{3}}\|_{L^{\infty}_{t}(R,L^{2})}$
(3.15) $\displaystyle\leq$
$\displaystyle\|v\|^{3}_{L^{\infty}_{t}(R,L^{p_{5}})}\leq
C\|v\|^{3}_{L^{\infty}_{t}(R,H^{\beta})}\leq C\rho^{3},$
where $p_{5}=\frac{6n}{3n-2\gamma}$, which satisfies
$p_{5}\leq\frac{2n}{n-2\beta}$ because of $\gamma\leq 3\beta$. Using the
relation $x={\langle
i\nabla\rangle}^{-1}J_{\varepsilon}-i{\varepsilon}t{\langle
i\nabla\rangle}^{-1}\nabla$ we deduce
$\displaystyle\|xF_{\gamma}(v)\|_{L^{\infty}_{t}(R,L^{2})}\leq
C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\|(V_{\gamma}\ast\overline{v^{{\varepsilon}_{1}}}v^{{\varepsilon}_{2}})(xv^{{\varepsilon}_{3}})\|_{L^{\infty}_{t}(R,L^{2})}$
(3.16) $\displaystyle\leq$ $\displaystyle
C\sum_{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\in\\{+,-\\}}\left\|\|v^{{\varepsilon}_{1}}\|_{L^{p_{4}}}\|\|v^{{\varepsilon}_{2}}\|_{L^{p_{4}}}\left(\|{\langle
i\nabla\rangle}^{-1}J_{\varepsilon}v^{{\varepsilon}_{3}})\|_{L^{p_{4}}}+t\|{\langle
i\nabla\rangle}^{-1}\nabla
v^{{\varepsilon}_{3}})\|_{L^{p_{4}}}\right)\right\|_{L^{\infty}_{t}(R)}$
$\displaystyle\leq$ $\displaystyle
C\|v\|^{2}_{L^{\infty}_{t}(R,L^{p_{4}})}\|{\langle
i\nabla\rangle}^{-1}Jv\|_{L^{\infty}_{t}(R,L^{p_{4}})}+C\|t^{1/3}v\|^{3}_{L^{\infty}_{t}(R,L^{p_{4}})}$
$\displaystyle\leq$ $\displaystyle
C\|v\|^{2}_{L^{\infty}_{t}(R,H^{\beta})}\|Jv\|_{L^{\infty}_{t}(R,H^{\beta-1})}+C\left(\|v\|_{L^{\infty}_{t}(R,H^{\beta})}+\|Jv\|_{L^{\infty}_{t}(R,H^{\beta-1})}\right)^{3}$
$\displaystyle\leq$ $\displaystyle
C\left(\rho+\|Jv\|_{L^{\infty}_{t}(R,H^{\beta-1})}\right)^{3}\leq C\rho^{3},$
where $p_{4}=\frac{6n}{3n-2\gamma}$, which satisfies
$2<p_{4}\leq\frac{2n}{n-2\beta},\,\frac{n}{2}(1-\frac{2}{p_{4}})\geq\frac{1}{3},\,(1+\frac{n}{2})(1-\frac{2}{p_{4}})\leq\beta$
because of $1<\gamma\leq\frac{3n\beta}{n+2}$. Using the relation $[{\langle
i\nabla\rangle}^{\beta-1},x]=-(\beta-1){\langle
i\nabla\rangle}^{\beta-3}\nabla$ we deduce
$\displaystyle\|Pw^{\varepsilon}|_{t=0}\|_{H^{\beta-1}}\leq\|x{\langle
i\nabla\rangle}w_{0}^{\varepsilon}\|_{H^{\beta-1}}+\|x{\langle
i\nabla\rangle}^{-1}F_{\gamma}(v)\|_{L^{\infty}_{t}(R,H^{\beta-1})}$
$\displaystyle\leq$ $\displaystyle\|x{\langle
i\nabla\rangle}w_{0}^{\varepsilon}\|_{H^{\beta-1}}+\|{\langle
i\nabla\rangle}^{-1}xF_{\gamma}(v)\|_{L^{\infty}_{t}(R,H^{\beta-1})}+\|{\langle
i\nabla\rangle}^{-3}\nabla F_{\gamma}(v)\|_{L^{\infty}_{t}(R,H^{\beta-1})}$
$\displaystyle\leq$ $\displaystyle\|{\langle
i\nabla\rangle}^{\beta-1}x{\langle
i\nabla\rangle}w_{0}^{\varepsilon}\|_{L^{2}}+C\|xF_{\gamma}(v)\|_{L^{\infty}_{t}(R,L^{2})}+C\|F_{\gamma}(v)\|_{L^{\infty}_{t}(R,L^{2})}$
$\displaystyle\leq$ $\displaystyle\|\langle x\rangle{\langle
i\nabla\rangle}^{\beta}w_{0}^{\varepsilon}\|_{L^{2}}+C\rho^{3}+C\left(\rho+\|Jv\|_{L^{\infty}_{t}(R,H^{\beta-1})}\right)^{3}$
$\displaystyle\leq$ $\displaystyle\|w_{0}\|_{H^{\beta,1}}+C\rho^{3},$
which, combining with (3.12), yields
$\|Pw^{\varepsilon}\|_{L_{t}^{\infty}(R;H^{\beta-1})}+\|Pw^{\varepsilon}\|_{L^{r}_{t}(R,L^{q})}\leq\|w_{0}\|_{H^{\beta,1}}+C\rho^{3}.$
(3.17)
Notice that
$[L_{\varepsilon},x]=-{\varepsilon}{\langle
i\nabla\rangle}^{-1}\nabla,\,[x,{\langle i\nabla\rangle}^{-1}]=-{\langle
i\nabla\rangle}^{-3}\nabla.$
Then we deduce that $xw^{\varepsilon}$ satisfies
$L_{\varepsilon}(xw^{\varepsilon})=-{\varepsilon}{\langle
i\nabla\rangle}^{-1}\nabla w^{\varepsilon}-{\langle
i\nabla\rangle}^{-1}(xF_{\gamma}(v))+{\langle i\nabla\rangle}^{-1}\nabla
F_{\gamma}(v).$
Using $J_{\varepsilon}=i{\varepsilon}P-{\varepsilon}L_{\varepsilon}x$ and
(3.13) yields
$\|J_{\varepsilon}w^{\varepsilon}\|_{L^{\infty}_{t}(R,H^{\beta-1})}\leq\|Pw^{\varepsilon}\|_{L^{\infty}_{t}(R,H^{\beta-1})}+\|L_{\varepsilon}(xw^{\varepsilon})\|_{L^{\infty}_{t}(R,H^{\beta-1})},$
with
$\displaystyle\|L_{\varepsilon}(xw^{\varepsilon})\|_{L^{\infty}_{t}(R,H^{\beta-1})}$
$\displaystyle\leq$
$\displaystyle\|w^{\varepsilon}\|_{L^{\infty}_{t}(R,H^{\beta-1})}+\|{\langle
i\nabla\rangle}^{-2}F_{\gamma}(v)\|_{L^{\infty}_{t}(R,H^{\beta-1})}+\|{\langle
i\nabla\rangle}^{-1}(xF_{\gamma}(v))\|_{L^{\infty}_{t}(R,H^{\beta-1})}$
$\displaystyle\leq$
$\displaystyle\|w^{\varepsilon}\|_{L^{\infty}_{t}(R,H^{\beta})}+\|F_{\gamma}(v)\|_{L^{\infty}_{t}(R,L^{2})}+\|xF_{\gamma}(v)\|_{L^{\infty}_{t}(R,L^{2})}\leq
C\|w_{0}\|_{H^{\beta}}+C\rho^{3}.$
Then we get
$\|J_{\varepsilon}w^{\varepsilon}\|_{L^{\infty}_{t}(R,H^{\beta-1})}\leq
C\|w_{0}\|_{H^{\beta,1}}+C\rho^{3}.$ (3.18)
A combination of (3.12) with (3.13), (3.14), (3.17) and (3.18) yields
$\|w\|_{X}\leq C\|w_{0}\|_{H^{\beta,1}}+C\rho^{3}.$ (3.19)
Therefore the map $M:w=M(v)$ defined by the problem (3.1), transforms a ball
$X_{\rho}$ with a small radius $\rho=C\|w^{0}\|_{H^{\beta,1}}$ into itself.
Denote $\tilde{w}=M(\tilde{v})$, then in the same way as in the proof of
(3.19) we have
$\|M(v)-M(\tilde{v})\|_{X}\leq C\rho^{2}\|v-\tilde{v}\|_{X}.$
Thus $M$ is a contraction mapping in $X_{\rho}$ and so there exists a unique
solution $w=M(w)$ of (3.1) if the norm $\|w^{0}\|_{H^{\beta,1}}$ is small
enough.
To prove the asymptotic of the solution $w(t,x)$, we use the equation, for
$|t|>|t^{\prime}|$,
$U_{\varepsilon}(-t)w^{\varepsilon}(t)-U_{\varepsilon}(-t^{\prime})w^{\varepsilon}(t^{\prime})=\int_{t^{\prime}}^{t}U_{\varepsilon}(-\tau){\langle
i\nabla\rangle}^{-1}F_{\gamma}(w(\tau))d\tau.$
Taking the $H^{\beta}$-norm of this equation, using the similar proof of (3.3)
and (3.4), we deduce
$\|U_{\varepsilon}(-t)w^{\varepsilon}(t)-U_{\varepsilon}(-t^{\prime})w^{\varepsilon}(t^{\prime})\|_{H^{\beta}}\leq
C\rho^{2}\langle t^{\prime}\rangle^{-\delta}$
with $\delta=\frac{2n\beta}{n+2}-2>0$, since we have $\|w\|_{X}\leq\rho$ and
$\|\langle
t\rangle^{-\frac{n}{2}(1-\frac{1}{p})}\|^{2}_{L^{s}_{t}([t^{\prime},t])}\leq
C\langle t^{\prime}\rangle^{-\delta}.$
Then there uniquely exist finial states $w^{\varepsilon}_{\pm}\in H^{\beta}$
satisfying, for $\pm t$ large enough,
$\|w^{\varepsilon}(t)-U_{\varepsilon}(t)w^{\varepsilon}_{\pm}\|_{H^{\beta}}\leq
C\rho^{2}\langle t\rangle^{-\delta}.$
Set $u(t)=\frac{1}{2}(w^{+}(t)-w^{-}(t))$,
$f_{\pm}(x)=\frac{1}{2}(w_{\pm}^{+}-w_{\pm}^{-})$,
$g_{\pm}(x)=-\frac{i}{2}{\langle i\nabla\rangle}(w^{+}_{\pm}+w^{-}_{\pm})$ and
$u_{\pm}(t)=\frac{1}{2}(U_{+}(t)w^{+}_{\pm}-U_{-}(t)w^{-}_{\pm})$. Then $u(t)$
and $u_{\pm}(t)$ satisfy Theorem 1.1(1).
Proof of Theorem 1.1(2). For given $(f_{-},g_{-})\in X^{\beta,1}$ and
$v=\\{v^{+},v^{-})\in X_{\rho}$, we consider the linearized version of the
final state problem of (3.1)
$\left\\{\begin{array}[]{l}L_{\varepsilon}w^{\varepsilon}=-{\langle
i\nabla\rangle}^{-1}F_{\gamma}(v)\\\
\|U_{\varepsilon}(t)w^{\varepsilon}-w^{\varepsilon}_{-}(x)\|_{H^{\beta}}\to
0\,\mbox{ as}\,t\to\infty\end{array}\right.$
with $w^{\varepsilon}_{-}(x)=i{\langle
i\nabla\rangle}^{-1}g_{-}(x)-{\varepsilon}f_{-}(x)\in H^{\beta,1}$. The
integration with respective to time yields
$w^{\varepsilon}(t)=U_{\varepsilon}(t)w^{\varepsilon}_{-}+\int_{-\infty}^{t}U_{\varepsilon}(t-\tau){\langle
i\nabla\rangle}^{-1}F_{\gamma}(v(\tau))d\tau.$
In the same way as in the proof of Theorem 1.1(1), we find that, if
$\|(f_{-},g_{-})\|_{X^{\beta,1}}\leq\rho$ small, there uniquely exists a
global solution $w^{\varepsilon}(t)\in C(R,H^{\beta})$ and a final state
$w_{+}^{\varepsilon}\in H^{\beta}$ such that, as $t\to+\infty$,
$\|w^{\varepsilon}(t)-U_{\varepsilon}(t)w_{+}^{\varepsilon}\|_{H^{\beta}}\leq
C\langle t\rangle^{-\delta}$
with $\delta=\frac{2n\beta}{n+2}-2>0$. Set
$u(t)=\frac{1}{2}(w^{+}(t)-w^{-}(t))$,
$f_{+}(x)=\frac{1}{2}(w_{+}^{+}-w_{+}^{-})$, $g_{+}(x)=-\frac{i}{2}{\langle
i\nabla\rangle}(w^{+}_{+}+w^{-}_{+})$ and
$u_{+}(t)=\frac{1}{2}(U_{+}(t)w^{+}_{+}-U_{-}(t)w_{+}^{-})$. Then $u(t)$ and
$u_{+}(t)$ satisfy Theorem 1.1(2).
## References
* [1] G. Menzala and W. Strauss, On a wave equation with a cubic convolution, J. Differential Equations, 43 (1982), 93-105.
* [2] K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160.
* [3] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588.
* [4] K. Hidano, Small data scattering for the Klein-Gordon equation with a cubic convolution nonlinearity, Discrete Cont. Dyn. Sys., 15(2006), 973-981.
* [5] N. Hayashi, P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations 244(2008), 188-199.
* [6] G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential and Integral Equations, 4 (1991), 527-542.
* [7] S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), 631-641.
* [8] B. Marshall, W. Strauss, S. Wainger, $L^{p}-L^{q}$ estimates for the Klein-Gordon equation, J. Math. Pures Appl. 59(1980), 417-440.
|
arxiv-papers
| 2012-04-30T14:08:38 |
2024-09-04T02:49:30.382042
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruying Xue",
"submitter": "Ruying Xue",
"url": "https://arxiv.org/abs/1204.6640"
}
|
1205.0055
|
# The Minimum Variability Time Scale and its Relation to Pulse Profiles of
Fermi GRBs
G. A. MacLachlan1, A. Shenoy1, E. Sonbas2,3, K. S. Dhuga1, A. Eskandarian1, L.
C. Maximon1, and W. C. Parke1
1Department of Physics, The George Washington University, Washington, D.C.
20052, USA.
2University of Adiyaman, Department of Physics, 02040, Adiyaman, Turkey.
3NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.
E-mail: maclach@gwu.edu (GAM)
###### Abstract
We present a direct link between the minimum variability time scales extracted
through a wavelet decomposition and the rise times of the shortest pulses
extracted via fits of 34 Fermi GBM GRB light curves comprised of 379 pulses.
Pulses used in this study were fitted with log-normal functions whereas the
wavelet technique used employs a multiresolution analysis that does not rely
on identifying distinct pulses. By applying a corrective filter to published
data fitted with pulses we demonstrate agreement between these two independent
techniques and offer a method for distinguishing signal from noise.
###### keywords:
Gamma-ray bursts
## 1 Introduction
One approach for probing light curves which has received attention (Nemiroff,
2000; Norris et al., 2005; Hakkila & Nemiroff, 2009; Nemiroff, 2012) is to
express them as a series of displaced pulses, each with a parametric form.
There is an appeal to this approach because fitting routines are well-
understood and interpretations of rise time, decay time, full width at half
max, etc, are possible. On the other hand, one must make certain assumptions
when using the pulse-fitting procedure such as the choice of the functional
form to use for an individual pulse and the number of parameters to be
included in the fitting function. Moreover, light with high variability at low
power may show variations which are not statistically significant.
A complementary approach using a wavelet-based analysis of a set of both long
and short GRB light curves is discussed by MacLachlan et al. (2012) in which a
time scale, $\tau_{\beta}$, is identified that marks the transition from white
noise to a power law in the power density spectrum (a $f^{-\alpha}$ behavior).
It is argued that over time scales smaller than $\tau_{\beta}$ the light
curves appear stochastic and signal power is distributed uniformly. At time
scales larger than $\tau_{\beta}$, identifiable structures (such as pulses)
with signal power are no longer distributed uniformly over the periods of
light variation. For this reason $\tau_{\beta}$ is referred to as the _minimum
variability time scale_.
The analysis presented in (MacLachlan et al., 2012) is a non-parametric
approach to probing light curves for time scales. It makes no assumptions
about the nature of the structures in a given light curve that give rise to
the $f^{-\alpha}$ character. The technique, however, offers no firm connection
between $\tau_{\beta}$ and the constituent structures although it seems
reasonable to associate $\tau_{\beta}$ with the scale of the smallest emitting
structures present.
Results from an application of a log-normal pulse-fitting procedure to GRB
light curves have been published by Bhat et al. (2012). In this paper we make
a meta-analysis of the timing results presented by Bhat et al. (2012) compared
with the techniques of MacLachlan et al. (2012) for a set of 34 GRBs used in
both studies.
## 2 Analysis
We begin by considering the relation between $\tau_{\beta}$ and the pulse
parameters given in Table 3 of Bhat et al. (2012). The parameters with
temporal units in Table 3 are: _time-since-trigger_ , _rise time_ , _decay
time_ , and _FWHM_. In all, 34 GRBs comprising 379 pulses are considered here.
We note that rise time, decay time, and FWHM as presented in Table 3 of Bhat
et al. (2012) are tightly correlated and for the argument that follows are
interchangeable without affecting the conclusions. However, we use rise time
to make our argument because, as Bhat et al. (2012) noted, rise times are
observed to be shorter than decay time and FWHM (see Table 3 in Bhat et al.
(2012)). We considered only those light curves from NaI detectors and summed
over the energy acceptance as in Table 3 of Bhat et al. (2012) and in
MacLachlan et al. (2012).
In Fig. 1 we plot the rise time for all 379 pulses (34 GRBs) along the
vertical axis and $\tau_{\beta}$ along the horizontal. Note that for each GRB
for which one $\tau_{\beta}$ is computed, there is a possibility for multiple
pulses and therefore multiple rise times, hence the vertical columns of rise
times for a single value of $\tau_{\beta}$. For a given column of pulse times
the shortest pulse rise times are at the bottom and one finds larger rise
times by moving up the column. An equality line is also shown which is the
locus where $\tau_{\beta}$ equals rise time. Arguing as we do that
$\tau_{\beta}$ represents the minimum variability time scale the space in the
$\tau_{\beta}$-rise time plane below the equality line should be interpreted
as a structureless white noise region. If some method were capable of
discerning light curve structure in the region we define as white noise, then
our assertion of having found a minimum variability time scale will have been
disproven. Indeed, in Fig. 1 there are 27 pulses with rise times below the
equality line. The uncertainties accompanying these 27 rise times are small,
making their intrusion into the white noise region significant.
Table 1: Pulses with rise times smaller than $\tau_{\beta}$ but larger than bin widths. Pulse Number (#), rise (time), $\delta$rise (time), and bin width are taken from Table 3 of Bhat et al. (2012). The column labeled $\Delta$rise (time) is obtained by combining $\delta$rise and bin width in quadrature. The columns diff and % diff refer to the differences between $\tau_{\beta}$ and rise (time). GRB | Pulse # | $\tau_{\beta}$ [s] | $\delta\tau^{-}_{\beta}$ [s] | $\delta\tau^{+}_{\beta}$ [s] | rise [s] | $\delta$rise [s] | $\Delta$rise [s] | bin width [s] | diff [s] | % diff
---|---|---|---|---|---|---|---|---|---|---
080825593 | 17 | 0.0775 | 0.0138 | 0.0168 | 0.0660 | 0.0003 | 0.0200 | 0.0200 | 0.0115 | 17.4
080916009 | 9 | 0.2266 | 0.063 | 0.0872 | 0.1670 | 0.0022 | 0.1500 | 0.1500 | 0.0596 | 35.7
080916009 | 15 | 0.2266 | 0.063 | 0.0872 | 0.2190 | 0.0018 | 0.1500 | 0.1500 | 0.0076 | 3.47
080916009 | 20 | 0.2266 | 0.063 | 0.0872 | 0.1930 | 0.0016 | 0.1500 | 0.1500 | 0.0336 | 17.4
080916009 | 22 | 0.2266 | 0.063 | 0.0872 | 0.2260 | 0.0027 | 0.1500 | 0.1500 | 0.0006 | 0.265
080925775 | 10 | 0.1748 | 0.0425 | 0.0562 | 0.1710 | 0.0035 | 0.0501 | 0.0500 | 0.0038 | 2.22
081215784 | 1 | 0.0319 | 0.0043 | 0.005 | 0.0218 | 0.0020 | 0.0054 | 0.0050 | 0.0101 | 46.3
However, a closer inspection of Table 3 of Bhat et al. (2012) reveals that
there are 20 light pulses in Fig. 1 with rise times that are _smaller_ than
the smallest bin widths, in some cases smaller by factors of ten or a hundred.
Moreover, of those 20 pulses there are 16 pulses in Fig. 1 with full widths at
half max that are _smaller_ than the smallest bin widths and indeed those 16
all fall below the equality line. While it seems that inclusion of these
pulses in Table 3 is important for the sake of completeness, we question the
physical reality of these pulses. Note that in MacLachlan et al. (2012) all
light curves are binned at 200 microseconds. Fig. 2 shows the effect of
removing the 20 non-physical pulses.
Figure 1: Minimum variability time scale versus rise time in the Observer
frame. The rise times are taken from Table 3 of Bhat et al. (2012). The line
represents the locus where $\tau_{\beta}$ = rise time. We identify the area
below the line with white noise. The data are expected to press up against the
line from above but not to cross it. Figure 2: Minimum variability time scale
versus rise time in the Observer frame. We have removed all pulses (20) with
rise times smaller than the light curve bin width.
Note that in Fig. 2 the white noise region has been vacated by all but seven
points and none of the pulse rise times above the equality line have been
disturbed by the bin width cut. For the seven points that remain beneath the
equality line, we show in Table 1 that six are within one sigma of the
equality line.
We make one other point regarding the pulse rise times in Fig. 1 and Fig. 2,
in particular regarding the size of the uncertainties. Of the 379 pulse rise
times reported by Bhat et al. (2012) and used for this meta-analysis, 301 have
uncertainties smaller than the binning of the light curve, in some cases
hundreds or thousands of times smaller. We argue that a conservative estimate
of the uncertainties for the pulse rise times should be no smaller than a bin
width. Thus, we add in quadrature a bin width (as reported in Table 3 of Bhat
et al. (2012)) to the rise time uncertainties (also reported in Table 3 of
Bhat et al. (2012)) and plot the result in Fig. 3.
In Fig. 4 we plot only the smallest rise times for each GRB against
$\tau_{\beta}$. We argue that by rejecting pulse rise times smaller than light
curve bin widths and by folding rise time uncertainties with a bin width we
get strong evidence that Bhat et al. (2012) and MacLachlan et al. (2012) have
tracked the same physical observables over approximately three orders of
magnitude using independent methods.
Figure 3: Minimum variability time scale versus rise time in the Observer
frame as in Fig. 2. We have folded a single bin width into the rise time
uncertainties. Figure 4: Minimum variability time scale versus rise time in
the Observer frame as in Fig. 3 but with only smallest rise times included.
Note that the equality line between $\tau_{\beta}$ and rise time marks a
boundary between scaling processes and white noise and gives substance to the
interpretation of the minimum variability time scale.
## 3 Results and Discussion
For a large sample of short and long Fermi GBM bursts, MacLachlan et al.
(2012) used a technique based on wavelets to determine the minimum variability
time scale, $\tau_{\beta}$. The authors associate this time scale with a
transition from red-noise processes to parts of the power spectrum dominated
by white noise or random noise components. Accordingly, the authors note that
this time scale is the shortest resolvable variability time for physical
processes intrinsic to the GRB. In addition, histograms of the values of
$\tau_{\beta}$ for long and short GRBs were shown to exhibit a clear temporal
offset in the mean $\tau_{\beta}$ values for long and short GRBs.
In a separate analysis, using a particular functional form for pulse shapes,
Bhat et al. (2012) have extracted an extensive set of key pulse parameters
such as rise times, decay times, widths (FWHM), and times since trigger for a
host of bright GRBs detected by Fermi/GBM. Using the FWHM values, these
authors also reported a significant temporal offset between the mean values
for long and short GRBs.
Although neither group offers a concrete explanation for the temporal
difference between the distributions of long and short GRBs, it is noteworthy
that they arrive at a result which is quantitatively in good agreement with
one another, especially having used independent approaches. Both sets of
analyses also suggest scaling trends between characteristic timescales. In the
case of minimum variability timescales the trend is between $\tau_{\beta}$ and
the duration of the burst, typically denoted by $T_{90}$. For the pulse-shape
analysis, the trend is more readily evident and is demonstrated through a
number of positive correlations involving key parameters such as rise times,
decay times and FWHM times.
It is relatively straightforward to interpret the scaling trends in terms of
the internal shock model in which the basic units of emission are assumed to
be pulses that are produced via the collision of relativistic shells emitted
by the central engine. In the case of the pulse-fitting method this is
essentially the default assumption. Indeed, Quilligan et al. (2002) in their
study of the brightest BATSE bursts with $T_{90}>2$ sec were the first to
demonstrate this explicitly by identifying and fitting distinct pulses and
showing a strong positive correlation between the number of pulses and the
duration of the burst. More recent studies, Bhat & Guiriec (2011); Hakkila &
Cumbee (2008); Hakkila & Preece (2011) have provided further evidence for the
pulse paradigm view of the prompt emission in GRBs.
The wavelet analysis does not, however, rely on identifying distinct pulses
but instead uses the multiresolution capacity of the wavelet technique to
resolve the smallest temporal scale present in the prompt emission.
Nonetheless, as MacLachlan et al. (2012) have demonstrated, if the smallest
temporal scale is due to pulse emissions, then we can still get a measure of
the upper bound on the number of pulses in a given burst through the ratio
$T_{90}/\tau_{\beta}$. In the simple model in which a pulse is produced every
time two shells collide, the ratio $T_{90}/\tau_{\beta}$, should show a
correlation with the duration of the burst. Indeed, this correlation was
reported by MacLachlan et al. (2012).
The similar trends of scaling demonstrated by these two methods, not only
suggest the robustness of both methods, but also point, perhaps more
importantly, to an underlying interconnection between key parameters extracted
by the two techniques. In other words, the minimum variability time scale
extracted by the wavelet technique is directly related to key pulse time
parameters such as rise times (as depicted in Fig. 4), under suitably
controlled pulse-fitting methods.
## 4 CONCLUSIONS
Through a meta-analysis of results presented by Bhat et al. (2012) and by
MacLachlan et al. (2012), we have studied the relationship between key
parameters that describe the temporal properties of a sample of prompt-
emission light curves for long and short-duration GRBs detected by the
Fermi/GBM mission. We compare the minimum variability timescale extracted
through a technique based on wavelets, with the pulse-time parameters
extracted through a pulse-fitting procedure. Our main results are summarized
as follows:
a) Both methods indicate a temporal offset between short and long-duration
bursts. The quantitative agreement between the two methods is quite good.
b) Both methods point to scaling trends between characteristic timescales. In
the case of the pulse-fitting method the scaling appears to involve parameters
such as rise times, FWHM, and pulse intervals. For the wavelet technique, the
scaling involves a correlation between the minimum variability time scale and
the duration of the bursts.
c) By demonstrating a strong positive correlation between $\tau_{\beta}$ and
the rise time of the shortest fitted pulses, we provide for the first time, a
direct link between the shortest resolvable temporal structure in a GRB light
curve with that of a key pulse profile parameter.
d) By combining the two techniques, we have shown that one can arrive at a
much tighter demarcation of the boundary between the power spectrum domains
that separate red noise and white noise processes.
## 5 ACKNOWLEDGEMENTS
The NASA grant NNX11AE36G provided partial support for this work and is
gratefully acknowledged. The authors, in particular GAM and KSD, acknowledge
very useful discussions with Jon Hakkila and Narayan Bhat.
## References
* Bhat & Guiriec (2011) Bhat, P. N., & Guiriec, S. 2011, BASI, 39, 471
* Bhat et al. (2012) Bhat, P. N. et al. 2012, _ApJ._ , 744, 141 (online-version)
* Hakkila & Cumbee (2008) Hakkila, J. & Cumbee, R. 2008, arXiv:0901.3171v1 [astro-ph.HE].
* Hakkila & Nemiroff (2009) Hakkila, J. & Nemiroff, R. J. 2009, _ApJ._ , 705, 372
* Hakkila & Preece (2011) Hakkila, J. & Preese, R. 2011, _ApJ._ , 740, 104
* MacLachlan et al. (2012) MacLachlan, G. A., et al. 2012, arXiv:1201.4431v2 [astro-ph.HE]
* Nemiroff (2000) Nemiroff, R. J. et al. 2000, _ApJ._ , 544, 805
* Nemiroff (2012) Nemiroff, R. J. 2012, _MNRAS_ , 419, 1650
* Norris et al. (2005) Norris, J. P. et al. 2005, _ApJ._ , 627, 324
* Quilligan et al. (2002) Quilligan, F., et al. 2002, _A. & A._, 385, 377
|
arxiv-papers
| 2012-04-30T23:38:10 |
2024-09-04T02:49:30.400519
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. A. MacLachlan, A. Shenoy, E. Sonbas, K. S. Dhuga, A. Eskandarian,\n L. C. Maximon, and W. C. Parke",
"submitter": "Glen MacLachlan",
"url": "https://arxiv.org/abs/1205.0055"
}
|
1205.0146
|
# Phase transition and scaling behavior of topological charged black holes in
Hořava-Lifshitz gravity
Bibhas Ranjan Majhi
IUCAA, Post Bag 4, Ganeshkhind,
Pune University Campus, Pune - 411 007, India
and
Dibakar Roychowdhury
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
e-mail: bibhas@iucaa.ernet.ine-mail: dibakar@bose.res.in,
dibakarphys@gmail.com
###### Abstract
Gravity can be thought as an emergent phenomenon and it has a nice
“thermodynamic” structure. In this context, it is then possible to study the
thermodynamics without knowing the details of the underlying microscopic
degrees of freedom. Here, based on the ordinary thermodynamics, we investigate
the phase transition of the static, spherically symmetric charged black hole
solution with arbitrary scalar curvature $2k$ in Hořava-Lifshitz gravity at
the Lifshitz point $z=3$. The analysis is done using the canonical ensemble
frame work; i.e. the charge is kept fixed. We find (a) for both $k=0$ and
$k=1$, there is no phase transition, (b) while $k=-1$ case exhibits the second
order phase transition within the physical region of the black hole. The
critical point of second order phase transition is obtained by the divergence
of the heat capacity at constant charge. Near the critical point, we find the
various critical exponents. It is also observed that they sati sfy the usual
thermodynamic scaling laws.
## 1 Introduction
One of the fascinating solutions of the gravity theories is the black hole
space time. Pioneering works of Bekenstain [1], Hawking [2] and later on
Bardeen et al [3] illuminated the fact that all the laws of black hole
mechanics are identical to those of the ordinary thermodynamics with properly
identification of the respective quantities, like temperature, entropy, energy
etc. and hence they do act as the ordinary thermal system. People then started
thinking that black holes may play a crucial role to give idea on the quantum
nature of the gravity. But till now there is no such complete theory of
quantum gravity.
Several recent results strongly indicate the possibility that the field
equations of gravity have the same status as the equations of fluid mechanics
or elasticity. For a recent review, see [4]. One specific implementation of
this idea considers the field equations of the theory to be ‘emergent’ in a
well-defined sense, rather than use that term in a more speculative vein like
e.g., considering the space and time themselves to be emergent etc. The
evidence for such a specific interpretation comes from different facts like
the possibility of interpreting the field equation in a wide class of theories
as thermodynamic relations [5], the nature of action functional in
gravitational theories and their thermodynamic interpretation [6], the
possibility of obtaining the field equations from a thermodynamic extremum
principle [7], application of equipartition ideas to obtain the density of
microscopic degrees of freedom [8], the equivalence of Einstein’s field
equations to the Navier-Stokes equations near a null surface [9] etc. The
important fact in the emergent paradigm is that one does not need to know
about the details of the microscopic description of the theory. These lead to
the idea that one can study the different aspects of thermodynamics, in the
case of gravity, without any detail of the underlying microscopic structure.
Here we will adopt the same logic and study the phase transition of the black
hole solution in Hořava-Lifshitz theory following the prescription in ordinary
thermodynamics.
Studying the thermodynamics [10]-[27] as well as the critical behavior
[28]-[41] of various black hole solutions in the framework of usual Einstein
gravity has been a fascinating topic of research for the past few decades.
Apart from these, there also exist topological black hole solutions whose
thermodynamics has been studied in Einstein gravity [42, 43], Einstein-Gauss-
Bonnet gravity and dilaton gravity [44] as well as Lovelock gravity [45]. On
the other hand, inspired by the dynamical critical phenomena in usual
condensed matter systems, very recently P. Hořava proposed a UV complete
theory of gravity [46] that reduces to the usual Einstein gravity at large
scales. Since then a number of attempts have been made in order to understand
various aspects of this theory [47]-[49] including different cosmological
aspects [50]. Various black hole solutions were also found in [51, 52, 53].
The thermodynamics of these black holes have been studied in [52, 54, 55].
Although these attempts are self contained and rigorous, still there remains
some major questions which have not yet been attempted. Studying the critical
behavior of black holes is one of the important aspects, which we aim to
explore for Hořava-Lifshitz theory of gravity.
In ordinary thermodynamics, critical exponents plays a crucial role in order
to understand the singular behavior of various thermodynamic entities near the
critical point. In usual thermodynamic systems, it is customary to express the
singular behavior of various thermodynamic systems in terms of power laws
characterized by a set of static critical exponents which determines the
qualitative nature of the phase transition near the critical point. All these
exponents are not independent and are found to satisfy certain thermodynamic
scaling laws near the critical point [56]-[57]. It is generally observed that
for $d>4$, where $d$ is the spatial dimension of the system, the critical
exponents do not depend on the spatial dimension of the system, which reflects
the mean field characteristics of a theory. On the other hand if the
interaction between the constituent elements are short range then the critical
exponents are found to be depe ndent on $d$.
In this paper, based on a canonical framework (i.e; keeping the charge ($Q$)
of the black hole fixed [13]), we investigate the critical behavior of
topological charged black holes in Hořava-Lifshitz theory of gravity at the
Lifshitz point $z=3$. The black hole solution is given in [52]. Although all
the thermodynamic quantities were evaluated earlier [52], a detailed study of
the nature of phase transition is still lacking. Particularly the issue
regarding the scaling behavior of (charged) black holes has never been
investigated so far in the frame work of Hořava-Lifshitz theory of gravity. We
investigate the phase transition phenomena for all the three cases taking
$k=0,\pm 1$. We observe the following interesting features: $\bullet$ There is
no Hawking Page transition [10] for black hole with $k=0,1$. $\bullet$ For
$k=-1$, there is a upper bound in the value of the event horizon, above which
the temperature becomes negative. This indicates that above this critical
value of horizon, the black hole solution does not exist. We will call this
valid range as the physical region. $\bullet$ Within the physical region,
interesting phase structure could be observed for the hyperbolic charged black
holes ($k=-1$). In this particular case we observe the second order
transition. This is different from the usual one. In Einstein gravity, Hawking
page transition occurs for $k=1$ while there is no such transition for
$k=0,-1$. The critical point of second order phase transition is marked by the
divergence in the heat capacity at constant charge ($C_{Q}$). Finally, we
explicitly calculate all the static critical exponents associated with the
second order transition, and check the validity of thermodynamic scaling laws
near the critical point. Interestingly enough it is found that the hyperbolic
charged black holes ($k=-1$) in the Hořava-Lifshitz theory of gravity fall
under the same universality class to that with the black holes having
spherically symmetric topology ($k=1$) in the usual Einstein gravity. The
values of these critical exponents indeed suggest a universal mean field
behavior in black holes which valid in both the Einstein as well as Hořava-
Lifshitz theory of gravity [40]-[41].
The plan of the paper is as follows: We begin in section 2 by giving a brief
introduction of the black hole solution in Hořava-Lifshitz gravity and use it
in section 3 to study the different thermodynamic quantities as well as the
phase transition. In section 4, the critical exponents near the critical point
are being evaluated and a brief discussion on the validity of the ordinary
scaling laws is presented. Finally, we conclude in section 5.
## 2 Charged black hole space-time in Hořava-Lifshitz gravity
In this section a brief discussion on the black hole solution in Hořava-
Lifshitz gravity will be presented. For details, one can follow [52] where the
meaning of all the parameters are given explicitly. We will mainly concentrate
on the solutions at the Lifshitz point $z=3$, particularly, the static
spherically symmetric topological charged black holes solution
$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{k}^{2}~{},$ (1)
where, $d\Omega_{k}^{2}$ is the line element for a two dimensional Einstein
space with constant scalar curvature $2k$. Without loss of generality, one can
take $k=0,\pm 1$ respectively. The form of the metric coefficient, for the
detailed balance condition, is given by [52]
$f(r)=k+x^{2}-\sqrt{\alpha x-\frac{q^{2}}{2}}~{},$ (2)
where $x=\sqrt{-\Lambda}r$ and $\Lambda(=-\frac{3}{l^{2}})$ corresponds to the
negative cosmological constant. The physical mass and the charge ($Q$)
corresponding to the black hole solution are respectively given by,
$M=\frac{\kappa^{2}\mu^{2}\Omega_{k}\sqrt{-\Lambda}}{16}\alpha;\,\,\,\
Q=\frac{\kappa^{2}\mu^{2}\Omega_{k}\sqrt{-\Lambda}}{16}q$ (3)
where, $\alpha$, $q$ are the integration constants, $\Omega_{k}$ is the volume
of the two dimensional Einstein space and $\kappa$, $\mu$ are the constant
parameters of the theory. The event horizon is the solution of the equation
$f(r_{+})=0$.
## 3 Phase transition
In ordinary thermodynamics the phase transition is studied by the divergence
of relevant thermodynamic quantities. Here, the same technique will be
adopted. We shall first calculate the Hawking temperature, entropy and
specific heat of the black hole using a canonical ensemble frame work, which
means that we shall carry out our analysis keeping the total charge ($Q$) of
the black hole fixed [13]. Finally, a graphical analysis will be given to
study the phase transition. Before we proceed further, let us mention that in
the following analysis we re-scale our variables as
$M\rightarrow\frac{M}{\Omega_{k}}$, $S\rightarrow\frac{S}{\Omega_{k}}$,
$Q\rightarrow\frac{Q}{A\Omega_{k}\sqrt{-\Lambda}}$, where we have set $16\pi
A=1$ with $A=\frac{\kappa^{2}\mu^{2}}{16}$.
The Hawking temperature is calculated as
$T=\frac{f^{{}^{\prime}}(r_{+})}{4\pi}=\frac{\sqrt{-\Lambda}(3x_{+}^{4}+2kx_{+}^{2}-k^{2}-\frac{Q^{2}}{2})}{8\pi
x_{+}(k+x_{+}^{2})}~{},$ (4)
and after the above mentioned re-scaling the entropy is found to be
$S=\int\frac{dM}{T}=\left(\frac{x_{+}^{2}}{4}+\frac{k}{2}lnx_{+}\right)+S_{0}~{}.$
(5)
In the above, $S_{0}$ is the integration constant which must be fixed by the
physical consideration. To be precise, entropy is always determined upto some
additive constant. But in all physical considerations the difference is
important. Therefore in the present analysis we will consider only the
difference. However, the integration constant $S_{0}$ could be determined
using the usual thermodynamic prescription; i.e. determination of entropy in
the $T\rightarrow 0$ limit, which might be the entropy of an extremal black
hole.
Figure 1: Temperature ($T$) plot of topological black holes for $k=0,1$ with
respect to $x_{+}$ for $Q=5$ and $l=1$. Figure 2: Temperature ($T$) and
Entropy ($S$) plot of topological black holes for $k=-1$ with respect to
$x_{+}$ for $Q=5$ and $l=1$.
Before we proceed further, let us first try to analyze the behavior of Hawking
temperature ($T$) for different choices of $k$. First of all, for the case
$k=0,1$, we note that the Hawking temperature ($T$) is a monotonically
increasing function of horizon radius ($x_{+}$) (see fig. 1) which indicates
that the corresponding heat capacities ($C_{Q}$) are always positive definite.
Therefore these black holes are globally stable and there is no phase
transition.
Next consider the other case, $k=-1$. For this, the Hawking temperature (4)
reduces to the following form:
$T=\frac{\sqrt{-\Lambda}(3x_{+}^{2}+1)}{8\pi
x_{+}}-\frac{\sqrt{-\Lambda}Q^{2}}{16\pi x_{+}(x_{+}^{2}-1)}.$ (6)
It it interesting to note that the Hawking temperature ($T$) is always
positive in the range $0<x_{+}<1$. Furthermore, if we plot the entropy ($S$)
as a function of $x_{+}$, using (5), it shows that $S$ is also positive in
this range (see fig. 2). Interestingly enough this condition is found to be
valid for any value of the physical charge ($Q$) of the black hole. On the
other hand, we note that $T\rightarrow-\infty$ as $x_{+}\rightarrow 1$. This
is due to the fact that as $x_{+}\rightarrow 1$ the second term on the r.h.s
of (6) dominates over the first one which ultimately produces a large negative
temperature. This is a nonphysical situation and the corresponding black hole
solution does not exist for $x_{+}\geq 1$. Considering this fact, in the
present paper we carry out our analysis in the physical range $0<x_{+}<1$
where the temperature ($T$) of the black hole is finite as well as positive
definite.
In the above specified range ($0<x_{+}<1$) we observe a change in slope at
$x_{+}=x_{2}$ of the corresponding ($T-x_{+}$) plot (see fig. 2). This change
in slope signals a discontinuity at $x_{+}=x_{2}$ in the corresponding heat
capacity $C_{Q}$. It gives a indication of second order phase transition at
$x_{+}=x_{2}$, which we shall refer as the critical point of phase transition.
In order to make the discussion more transparent we next compute the
corresponding heat capacity ($C_{Q}$).
Using (4) and (5) the heat capacity is determined as
$\displaystyle C_{Q}$ $\displaystyle=$ $\displaystyle T\left(\frac{\partial
S}{\partial T}\right)_{Q}=T\frac{\left(\frac{\partial S}{\partial
x_{+}}\right)_{Q}}{\left(\frac{\partial T}{\partial x_{+}}\right)_{Q}}$ (7)
$\displaystyle=$
$\displaystyle\frac{(k+x_{+}^{2})^{2}(3x_{+}^{4}+2kx_{+}^{2}-k^{2}-\frac{Q^{2}}{2})}{6x_{+}^{6}+14kx_{+}^{4}+10k^{2}x_{+}^{2}+2k^{3}+Q^{2}(k+3x_{+}^{2})}~{}.$
Figure 3: Specific heat ($C_{Q}$) plot of topological black holes for $k=-1$
with respect to $x_{+}$ for $Q=5$ and $l=1$.
From the above figure (fig. 3), we see that specific heat ($C_{Q}$) indeed
suffers a discontinuity at $x_{+}=x_{2}$. This shows that there is a genuine
second order phase transition at $x_{+}=x_{2}$. For the phase 1 the heat
capacity ($C_{Q}$) is always found to be positive (see fig. 3) which means
that this phase is thermodynamically stable. On the other hand, $C_{Q}<0$ for
the phase 2 and therefore it is an unstable phase.
This scenario is completely different in the Einstein gravity, where we have
no Hawking Page transition for $k=0,-1$. Whereas, $k=1$ case exhibits Hawking
Page transition [42]-[44]. For the above phase structure one can further
investigate the critical behavior of the black hole in Hořava-Lifshitz theory
of gravity near the critical point $x_{2}$, which we aim to discuss in the
next section.
## 4 Critical phenomena and scaling laws
In this section, based on a thermodynamic approach, we explicitly calculate
various critical exponents (that is associated with the second order phase
transition at $x_{2}$) for the hyperbolic charged black holes (with $k=-1$) in
the fixed charge ($Q$) ensemble.
In the theory of phase transitions, critical exponents play an important role
in order to understand the singular behavior of various thermodynamic entities
near the critical point(s). In order to have a complete understanding of the
physics of phase transition phenomena, one generally introduces a set of
static critical exponents $(\alpha,\beta,\gamma,\delta,\varphi,\psi,\nu,\eta)$
which play a central role in the theory of critical phenomena, and studies the
so called thermodynamic scaling laws [56] near the critical point.
In order to find the critical exponents, we write near the critical point
$\displaystyle x_{+}=x_{2}(1+\Delta),$ (8)
where $|\Delta|<<1$. Defining $T_{2}\equiv T(x_{2})$, the Taylor expansion of
the temperature ($T$) about $x_{+}=x_{2}$ yields
$T=T_{2}+\Big{[}\left(\frac{\partial T}{\partial
x_{+}}\right)_{Q}\Big{]}_{x_{+}=x_{2}}(x_{+}-x_{2})+\frac{1}{2}\Big{[}\left(\frac{\partial^{2}T}{\partial
x_{+}^{2}}\right)_{Q}\Big{]}_{x_{+}=x_{2}}(x_{+}-x_{2})^{2}+{\textrm{ higher
order terms}}~{}.$ (9)
It has been shown earlier that at the critical point $x_{2}$, $C_{Q}$ diverges
(see Figure 3). Therefore (7) implies that the second term on the right hand
side of (9) vanishes. Hence neglecting the higher order terms and then using
(8) in (9) we obtain,
$\Delta=\sqrt{\frac{2}{D}}\frac{(T-T_{2})^{1/2}}{x_{2}}$ (10)
where,
$D=\left[\left(\frac{\partial^{2}T}{\partial
x_{+}^{2}}\right)_{Q}\right]_{x_{+}=x_{2}}=\frac{6x_{2}^{2}+3Q^{2}x_{2}^{2}-6x_{2}^{4}-6Q^{2}x_{2}^{4}-2-Q^{2}}{x_{2}^{3}(x_{2}^{2}-1)^{3}}.$
(11)
Now to find the critical exponent $\alpha$ which is defined by the standard
relation
$C_{Q}\sim|T-T_{2}|^{-\alpha}~{},$ (12)
we use the relation (8) in (7) and then keep the terms only linear in $\Delta$
to obtain
$\displaystyle
C_{Q}\simeq\frac{\mathcal{N}(x_{2},Q)}{\Delta(36x_{2}^{2}-56x_{2}^{4}+20x_{2}^{2}+6Q^{2}x_{2}^{2})}~{},$
(13)
with
$\mathcal{N}(x_{2},Q)=(x_{2}^{2}-1)^{2}(3x_{2}^{4}-2x_{2}^{2}-1-Q^{2}/2)~{}.$
(14)
Finally, using (10) in the above we find the behavior of $C_{Q}$ near the
critical point $x_{+}=x_{2}$:
$C_{Q}\simeq\frac{\mathcal{A}(x_{2},Q)}{(T-T_{2})^{1/2}}$ (15)
where,
$\mathcal{A}(x_{2},Q)=\frac{\mathcal{N}(x_{2},Q)\sqrt{D}x_{2}}{\sqrt{2}(36x_{2}^{2}-56x_{2}^{4}+20x_{2}^{2}+6Q^{2}x_{2}^{2})}~{}.$
(16)
A comparison of (15) with the standard relation (12) yields $\alpha=1/2$.
Next the determination of the critical exponent $\beta$ which is defined
through the relation (for a fixed value of charge $Q$) as,
$\Phi(x_{+})-\Phi(x_{2})\sim|T-T_{2}|^{\beta}$ (17)
will be done. Here the potential $\Phi(x)$ is given by
$\displaystyle\Phi(x)=\frac{Q}{x}~{}.$ (18)
To proceed, let us first Taylor expand $\Phi(x_{+})=\frac{Q}{x_{+}}$ about
$x_{+}=x_{2}$:
$\displaystyle\Phi(x_{+})$ $\displaystyle=$
$\displaystyle\Phi(x_{2})+\Big{[}\Big{(}\frac{\partial\Phi}{\partial
x_{+}}\Big{)}_{Q}\Big{]}_{x_{+}=x_{2}}(x_{+}-x_{2})+{\textrm{higher order
terms}}$ (19) $\displaystyle=$
$\displaystyle\Phi(x_{2})-\frac{Q}{x_{2}^{2}}(x_{+}-x_{2})+{\textrm{higher
order terms}}~{}.$
As before, neglecting the higher order terms and then using (8) we obtain,
$\Phi(x_{+})-\Phi(x_{2})=-\frac{Q}{x_{2}^{2}}\sqrt{\frac{2}{D}}(T-T_{2})^{1/2}~{},$
(20)
where in the final step (10) also has been used. This immediately determines
$\beta=1/2$.
To find out the exponent $\gamma$, associated with the divergence of the
inverse of the isothermal compressibility
$K_{T}^{-1}=\Big{[}Q\left(\frac{\partial\Phi}{\partial Q}\right)_{T}\Big{]}$
[32], we first find the explicit expression of $K_{T}^{-1}$. The thermodynamic
identity $\left(\frac{\partial\Phi}{\partial T}\right)_{Q}\left(\frac{\partial
T}{\partial Q}\right)_{\Phi}\left(\frac{\partial
Q}{\partial\Phi}\right)_{T}=-1$ yields
$\displaystyle\Big{(}\frac{\partial\Phi}{\partial
Q}\Big{)}_{T}=-\Big{(}\frac{\partial\Phi}{\partial
T}\Big{)}_{Q}\Big{(}\frac{\partial T}{\partial Q}\Big{)}_{\Phi}.$ (21)
In order to evaluate the right hand side of (21) first note that,
$\Big{(}\frac{\partial\Phi}{\partial
T}\Big{)}_{Q}=\frac{\Big{(}\frac{\partial\Phi}{\partial
x_{+}}\Big{)}_{Q}}{\Big{(}\frac{\partial T}{\partial x_{+}}\Big{)}_{Q}}.$ (22)
Also, from the functional relation,
$T=T(x_{+},Q)$ (23)
we find,
$\Big{(}\frac{\partial T}{\partial Q}\Big{)}_{\Phi}=\Big{(}\frac{\partial
T}{\partial x_{+}}\Big{)}_{Q}\Big{(}\frac{\partial x_{+}}{\partial
Q}\Big{)}_{\Phi}+\Big{(}\frac{\partial T}{\partial Q}\Big{)}_{x_{+}}.$ (24)
Right hand side of both (22) and (24) can be easily calculated using the
relations (4) and (18) which finally yields,
$K_{T}^{-1}=Q\left(\frac{\partial\Phi}{\partial
Q}\right)_{T}=\left(\frac{Q}{x_{+}}\right)\frac{6x_{+}^{6}+10x_{+}^{2}-14x_{+}^{4}-2+Q^{2}(x_{+}^{2}+1)}{6x_{+}^{6}+10x_{+}^{2}-14x_{+}^{4}-2+Q^{2}(3x_{+}^{2}-1)}.$
(25)
Then to obtain the near critical point expression, use (8) and next (10) in
the above. This yields,
$K_{T}^{-1}\simeq\frac{Q\sqrt{D}(6x_{2}^{6}+10x_{2}^{2}-14x_{2}^{4}-2+Q^{2}(x_{2}^{2}+1))}{\sqrt{2}(36x_{2}^{6}-56x_{2}^{4}+20x_{2}^{2}+6Q^{2}x_{2}^{2})}(T-T_{2})^{-1/2}.$
(26)
Comparing with the standard relation $K_{T}^{-1}\sim|T-T_{2}|^{-\gamma}$,
defined for the fixed value of charge ($Q$), we get $\gamma=1/2$.
The critical exponent $\delta$ is defined through the relation,
$\Phi(x_{+})-\Phi(x_{2})\sim|Q-Q_{2}|^{1/\delta}$ (27)
at constant temperature $T$. To find it, we first expand $Q(x_{+})$ in a
sufficiently small neighborhood of $x_{+}=x_{2}$ which yields,
$\displaystyle Q(x_{+})$ $\displaystyle=$ $\displaystyle
Q(x_{2})+\left[\left(\frac{\partial Q}{\partial
x_{+}}\right)_{T}\right]_{x_{+}=x_{2}}(x_{+}-x_{2})+\frac{1}{2}\left[\left(\frac{\partial^{2}Q}{\partial
x^{2}_{+}}\right)_{T}\right]_{x_{+}=x_{2}}(x_{+}-x_{2})^{2}$ (28)
$\displaystyle+$ $\displaystyle{\textrm{higher order terms}}~{}.$
Since $T=T(x_{+},Q)$ (see Eq. (4)), one finds for the fixed $T$
$\left(\frac{\partial Q}{\partial x_{+}}\right)_{T}=-\left(\frac{\partial
T}{\partial x_{+}}\right)_{Q}\left(\frac{\partial Q}{\partial
T}\right)_{x_{+}}~{}.$ (29)
Now at the critical point $x_{+}=x_{2}$, $C_{Q}$ diverges and so, as earlier,
$\Big{[}\left(\frac{\partial T}{\partial
x_{+}}\right)_{Q}\Big{]}_{x_{+}=x_{2}}=0$. Therefore, after neglecting the
higher order terms, (28) reduces to
$x_{+}-x_{2}=\sqrt{\frac{2}{M}}(Q(x_{+})-Q(x_{2}))^{1/2}$ (30)
where,
$M=\left[\left(\frac{\partial^{2}Q}{\partial
x^{2}_{+}}\right)_{T}\right]_{x_{+}=x_{2}}=\frac{18x_{2}^{8}+32x_{2}^{4}-44x_{2}^{6}-3Q^{2}x_{2}^{4}-4x_{2}^{2}-2-Q^{2}}{2Q^{2}x_{2}^{2}(x_{2}^{2}-1)^{2}}.$
(31)
Next since $\Phi(x_{+})=\frac{Q}{x_{+}}$, we have the functional relation
$\Phi=\Phi(x_{+},Q)$ and so
$\displaystyle\Big{(}\frac{\partial\Phi}{\partial
x_{+}}\Big{)}_{T}=\Big{(}\frac{\partial\Phi}{\partial
x_{+}}\Big{)}_{Q}+\Big{(}\frac{\partial\Phi}{\partial
Q}\Big{)}_{x_{+}}\Big{(}\frac{\partial Q}{\partial x_{+}}\Big{)}_{T}~{}.$ (32)
Therefore using (29) we find,
$\left[\left(\frac{\partial\Phi}{\partial
x_{+}}\right)_{T}\right]_{x_{+}=x_{2}}=\left[\left(\frac{\partial\Phi}{\partial
x_{+}}\right)_{Q}\right]_{x_{+}=x_{2}}=-\frac{Q}{x_{2}^{2}}.$ (33)
Finally, expanding $\Phi(x_{+})$ close to the critical point $x_{+}\sim x_{2}$
at constant $T$ and then using (30) and (33) we obtain,
$\Phi(x_{+})-\Phi(x_{2})\simeq-\frac{Q}{x_{2}^{2}}\sqrt{\frac{2}{M}}\Big{[}Q(x_{+})-Q(x_{2})\Big{]}^{1/2}~{}.$
(34)
Hence, the critical exponent is read off as $\delta=2$.
Now re-expressing (15) by using (10) in the following form:
$C_{Q}=\sqrt{\frac{2}{D}}\frac{{\cal{A}}}{x_{2}\Delta}=\sqrt{\frac{2}{D}}\frac{{\cal{A}}}{(x_{+}-x_{2})}~{},$
(35)
and then substituting the value of $x_{+}-x_{2}$ from (30) we
obtain111$Q(x_{2})=Q_{2}$.
$C_{Q}\sim\frac{1}{(Q(x_{+})-Q_{2})^{1/2}}~{}.$ (36)
Comparing this with the standard definition
$C_{Q}\sim{|Q(x_{+})-Q_{2}|^{-\varphi}}$ we find $\varphi=1/2$.
In the following the calculation, the critical exponent $\psi$, defined by
$S(x_{+})-S(x_{2})\sim|Q-Q_{2}|^{\psi}~{},$ (37)
will be found out. Expansion of the entropy for fixed charge $Q$ about the
critical point $x_{2}$ yields
$\displaystyle S(x_{+})=S(x_{2})+\Big{[}\Big{(}\frac{\partial S}{\partial
x_{+}}\Big{)}_{Q}\Big{]}_{x_{+}=x_{2}}(x_{+}-x_{2})+{\textrm{higher order
terms}}~{}.$ (38)
Then following the identical steps as earlier and using (30) we find
$S(x_{+})-S(x_{2})\sim\frac{(x_{2}^{2}-1)}{2x_{2}}\sqrt{\frac{2}{M}}(Q-Q_{2})^{1/2}~{},$
(39)
which yields $\psi=1/2$.
In the following table, we give the values of the critical exponents for the
present example:
Table 1: Various critical exponents and their values $Critical~{}Exponents$ | $\alpha$ | $\beta$ | $\gamma$ | $\delta$ | $\varphi$ | $\psi$
---|---|---|---|---|---|---
Values | 1/2 | 1/2 | 1/2 | 2 | 1/2 | 1/2
It may be interesting to mention that the critical exponents for the second
order phase transition in the case of hyperbolic black hole in Hořava-Lifshitz
gravity are exactly equal to those obtained earlier in [40] for the Einstein-
Born-Infeld gravity with $k=1$.
In the last part of the section, we discuss about the validity of the usual
thermodynamic scaling laws for our present situation. It should be mentioned
that in the realistic thermodynamic systems various critical exponents are
found to satisfy certain (scaling) relations among themselves, known as
thermodynamic scaling laws [56]-[57], which may be expressed as,
$\displaystyle\alpha+2\beta+\gamma=2,~{}~{}~{}\alpha+\beta(\delta+1)=2,~{}~{}~{}(2-\alpha)(\delta\psi-1)+1=(1-\alpha)\delta~{}~{}$
$\displaystyle~{}~{}\gamma(\delta+1)=(2-\alpha)(\delta-1),~{}~{}~{}\gamma=\beta(\delta-1),~{}~{}~{}\varphi+2\psi-\delta^{-1}=1.$
(40)
Since the critical exponents, we find here, are exactly equal to those
obtained in the usual Einstein-Born-Infeld gravity for $k=1$ [40], all the
above relations (40) are indeed satisfied for the present case, in spite of
difference in the nature of the black hole solution and the phase structure.
This can be checked easily by substituting the values from the table in (40).
In this sense, it may be possible that the black holes in both type of
theories fall under the same class such that their critical exponents are
identical.
Finally we are in a position to check the additional scaling laws for the
exponents $\nu$ and $\eta$ which are associated with the diverging nature of
correlation length and correlation function near the critical point. Since the
spatial dimension ($d$) of our theory is $\leq 4$, therefore it will be
natural to assume that the additional scaling relations,
$\gamma=\nu(2-\eta),~{}~{}~{}2-\alpha=\nu d$ (41)
will hold in general. This immediately determines the other exponents. Taking
$d=3$ and using the exponent values from table 1, we finally obtain,
$\nu=1/2,~{}~{}~{}\eta=1.$ (42)
In practical situation, the assumption that the scaling laws (41) are valid,
may not be true. Hence it must be checked that these are indeed valid or not.
Alternatively, one should find the above exponents by some alternative method.
## 5 Conclusions
It is now evident that gravity and thermodynamics are closely connected to
each other [1, 2, 3]. The repeated failure to quantize gravity led to a
parallel development where gravity is believed to be an emergent phenomenon
just like thermodynamics and hydrodynamics instead treating it as a
fundamental force [4]. The fundamental role of gravity is replaced by
thermodynamic interpretations leading to similar or equivalent results without
knowing the underlying microscopic details.
In this paper, we adopted the standard thermodynamic approach to explore the
phase structure of the topological charged black holes in Hořava-Lifshitz
gravity for the Lifshitz point $z=3$. These black holes were introduced
earlier in [52]. In [52] the authors have computed the general expression for
the Hawking temperature for these black holes. In spite of this attempt, till
date the issue regarding the nature of phase transition, particularly the
issue of critical phenomena, remains completely unexplored. In the present
paper, we attempted to provide an answer to all these questions. A number of
interesting features have been observed in this regard which have never been
explored so far to the best of our knowledge. These are as follows: $\bullet$
We showed that there exits a physical range $0<x_{+}<1$, where the black hole
solution exists. $\bullet$ Within this range, the second order phase
transition occurs. $\bullet$ The situation is not exactly identical to
Einstein theory. In Einstein theory, there is no Hawking Page phase transition
for $k=0,-1$ while $k=1$ case exhibits such phase transition. Here, only
$k=-1$ solution has phase transition. $\bullet$ Finally, the critical
exponents near the critical point of the phase transition were derived. This
was done again using the ordinary thermodynamic analogy. It may be noted that
the critical point has been marked by the discontinuity of the heat capacity
($C_{Q}$) which suggests that it is a second order phase transition. Moreover,
as a point of conformation, we also found that the critical exponent,
associated with the divergence in $C_{Q}$, is $1/2$ which simultaneously
indicates a mean field feature as well as a second order nature of the phase
transition. Interestingly, these are exactly equal to those of Einstein-Born-
Infeld theory [40]-[41] and also they satisfy the ordinary thermodynamic
scaling laws. This in fact suggests two remarkable facts: (1) black holes in
both the Einstein as well as Hořava-Lifshitz theory of gravity fall under the
same universality class, (2) there exists a universal mean field behavior in
both of these gravi ty theories.
It may be pointed out that the issues, dealt with the present paper, are
notoriously difficult to study. So far what we have done is completely based
on the ordinary thermodynamic analogy. But a simple analogy with the
thermodynamics may not be enough to draw safe conclusions. Also the
understanding of the thermodynamic interpretation of gravity is far from
complete since the arguments are more heuristic than concrete and depend upon
specific ansatz or assumptions. Therefore the results are suggestive rather
than definitive. Hence it is necessary to compute the critical exponents by
other alternative/direct procedures; e.g. correlation function technique of
the scalar modes, the AdS/CFT correspondence, re-normalization group scheme,
etc. which is beyond the scope of the present paper. In spite of this, we
still believe that the analysis, presented here, could illuminate the
underlying microscopic structure of the gravity.
Acknowledgement:
D.R would like to thank the Council of Scientific and Industrial Research (C.
S. I. R), Government of India, for financial help.
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|
arxiv-papers
| 2012-05-01T12:00:18 |
2024-09-04T02:49:30.407448
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi, Dibakar Roychowdhury",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1205.0146"
}
|
1205.0195
|
# Coherent Pattern Prediction in Swarms of Delay-Coupled Agents
Luis Mier-y-Teran-Romero, Eric Forgoston, and Ira B. Schwartz L. Mier-y-Teran-
Romero is a joint NIH postdoctoral fellow with the Johns Hopkins Bloomberg
School of Public Health, 615 North Wolfe Street, Baltimore, Maryland, 21205,
USA and the Nonlinear Systems Dynamics Section, Plasma Physics Division, Code
6792, U.S. Naval Research Laboratory, Washington, DC 20375, USA e-mail:
luis@nlschaos.nrl.navy.milE. Forgoston is with the Department of Mathematical
Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043,
USA e-mail: eric.forgoston@montclair.eduI.B. Schwartz is with the Nonlinear
Systems Dynamics Section, Plasma Physics Division, Code 6792, U.S. Naval
Research Laboratory, Washington, DC 20375, USA e-mail:
ira.schwartz@nrl.navy.mil
###### Abstract
We consider a general swarm model of self-propelling agents interacting
through a pairwise potential in the presence of noise and communication time
delay. Previous work [Phys. Rev. E 77, 035203(R) (2008)] has shown that a
communication time delay in the swarm induces a pattern bifurcation that
depends on the size of the coupling amplitude. We extend these results by
completely unfolding the bifurcation structure of the mean field
approximation. Our analysis reveals a direct correspondence between the
different dynamical behaviors found in different regions of the coupling-time
delay plane with the different classes of simulated coherent swarm patterns.
We derive the spatio-temporal scales of the swarm structures, and also
demonstrate how the complicated interplay of coupling strength, time delay,
noise intensity, and choice of initial conditions can affect the swarm. In
particular, our studies show that for sufficiently large values of the
coupling strength and/or the time delay, there is a noise intensity threshold
that forces a transition of the swarm from a misaligned state into an aligned
state. We show that this alignment transition exhibits hysteresis when the
noise intensity is taken to be time dependent.
###### Index Terms:
Autonomous agents, Delay systems, Pattern formation, Nonlinear dynamical
systems, Bifurcation
## I Introduction
The rich dynamic behavior of interacting multi-agent, or particle, systems has
been the focus of numerous recent studies. These multi-particle systems are
capable of self-organization, as shown by the various coherent conformations
with complex structure that they generate, even when the interactions are
short range and in the absence of a leader agent. The study of these
‘swarming’ or ‘herding’ systems has had many interesting biological
applications which have resulted in a better understanding of the spatio-
temporal patterns formed by bacterial colonies, fish, birds, locusts, ants,
pedestrians, etc. [1, 2, 3, 4, 5, 6, 7]. The mathematical study of these
swarming systems is also helpful in the understanding of oscillator
synchronization, as in the neural phenomenon of central pattern generators
[8]. The results of these studies have impacted and have been successfully
applied in the design of systems of autonomous, inter-communicating robotic
systems [9, 10, 11, 12], as well as mobile sensor networks [13].
It is possible to design swarming models for robotic motion planning,
consensus and cooperative control, and spatio-temporal formation. Pairwise
potentials for individual agents can be straightforwardly ported onto
autonomous vehicles. Furthermore, these pairwise interactions can be used in
conjunction with simple scalable algorithms to achieve multi-vehicle
cooperative motion [14]. Specific goals include: obstacle avoidance [11],
boundary tracking [15], environmental sensing [13, 16] and decentralized
target tracking [17].
An important problem is that of environmental consensus estimation. Here, the
individuals of the swarm communicate with each other through a network to
achieve asymptotically synchronous information about their environment [13].
Recently, consensus was extended to include time delayed communication among
agents [18].
Task allocation is another problem of interest involving robotic swarms. The
objective is to reallocate swarm robots to perform a set of tasks in parallel
and independently of one another in an optimal way. In order to make task
reallocation more realistic it is possible to consider a time delay that
arises from the amount of time required to switch between tasks [19].
Regardless of the design objective of a robotic swarm system, a comprehensive
theoretical analysis of the model must be performed in order to achieve
successful algorithm design.
Many different mathematical approaches have been utilized to study aggregating
agent systems. Some of these studies have treated the problem at a single-
individual level, using ordinary differential equations (ODEs) or delay
differential equations (DDEs) to describe their trajectories [20, 21, 22, 10].
An alternative method has been proposed by other researchers and consists of
using continuum models that consider averaged velocity and agent density
fields that satisfy partial differential equations (PDEs) [2, 3, 5, 6]. In
addition, authors also have studied the effects of noise on the swarm’s
behavior and have shown the existence of noise-induced transitions [23, 25].
The study of these systems has been enriched by tools from statistical physics
since both first and second order phase transitions have been found in the
formation of coherent states [28].
An additional effect that has recently been considered is that of
communication time delays between robots. Time delay models are common in many
areas of mathematical biology including population dynamics, neural networks,
blood cell maturation, virus dynamics and genetic networks [29, 30, 31, 32,
33, 34, 35, 36, 37]. In the context of swarming agents, it has been shown that
the introduction of a communication time delay may induce transitions between
different coherent states in a manner which depends on the coupling strength
between agents and the noise intensity [25]. Thus far, most of the work has
concentrated on the case of uniform time delays among agents [26]. However,
the practical engineering of multi-agent systems requires researchers to
consider the case in which time delays may vary due to data processing times,
problems in inter-agent communication, etc. The case of differing (and even
time-vary ing) time delays between agents may be treated similarly to the case
of a single delay by using a data buffer [27].
In this work, we carry out a detailed study of the bifurcation structure of
the mean field approximation used in [25] and investigate how the bifurcations
in the system are modified in the presence of noise. Section II contains the
swarm model, while Sec. III contains the derivation of the mean field
approximation. The bifurcation analysis of the mean field equation can be
found in Sec. IV, and Sec. V provides a comparison of the mean field analysis
with the nonlinear governing equations. In Sec. VI, we describe the effects of
noise on the swarm, and the conclusions are contained in Sec. VII.
## II Swarm Model
We consider a two-dimensional (2D) swarm that consists of $N$ identical self-
propelling individuals of unit mass that are mutually attracted to one another
in a symmetric fashion. Hence, the coupling of the agents occurs via a fully
connected graph. In addition, we consider the case in which the individuals
that comprise the swarm are communicating with each other in a stochastic
environment. Because of the finite communication times between individuals,
there is a time delay between interactions. Assuming that the communication
time between agents is constant and equal to $\tau>0$, the swarm dynamics is
described by the following governing equations:
$\displaystyle\dot{\mathbf{r}}_{i}=$ $\displaystyle\mathbf{v}_{i},$ (1a)
$\displaystyle\dot{\mathbf{v}}_{i}=$
$\displaystyle\left(1-|\mathbf{v}_{i}|^{2}\right)\mathbf{v}_{i}-\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau))+\boldsymbol{\eta}_{i}(t),$ (1b)
for $i=1,2\ldots,N$. The terms $\mathbf{r}_{i}$ and $\mathbf{v}_{i}$
respectively represent the 2D position and velocity of the $i$-th agent at
time $t$. The strength of the attraction is measured by the coupling constant
$a>0$. The self-propulsion and frictional drag forces on each agent is given
by the term $\left(1-|\mathbf{v}_{i}|^{2}\right)\mathbf{v}_{i}$. Therefore, in
the absence of coupling, agents tend to move on a straight line with unit
speed $|\mathbf{v}_{i}|=1$ as time goes to infinity. The term
$\boldsymbol{\eta}_{i}(t)=(\eta_{i}^{(1)},\eta_{i}^{(2)})$ is a 2D vector of
stochastic white noise with intensity equal to $D$ such that
$\langle\eta_{i}^{(\ell)}(t)\rangle=0$ and
$\langle\eta_{i}^{(\ell)}(t)\eta_{j}^{(k)}(t^{\prime})\rangle=2D\delta(t-t^{\prime})\delta_{ij}\delta_{\ell
k}$ for $i,j=1,2,\ldots N$ and $\ell,k=1,2$. It is the main objective of this
work to identify the possible swarm behaviors for different values of $a$ and
$\tau$.
The coupling between individuals arises from a time delayed, spring-like
potential. Hence, our equations of motion may be considered to be the first
term in a Taylor expansion of other more general time delayed potential
functions about an equilibrium point.
## III Mean Field Approximation
We can investigate the stability of the swarm system by deriving a mean field
approximation of the system. The derivation involves the consideration of
agent coordinates relative to the center of mass and the elimination of the
noise terms. The center of mass of the swarming system is given by
$\displaystyle\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t).$ (2)
The position of each individual can be decomposed into
$\displaystyle\mathbf{r}_{i}=\mathbf{R}+\delta\mathbf{r}_{i},\qquad
i=1,2\ldots,N,$ (3)
where $\delta\mathbf{r}_{i}$ is the vector from the center of mass to particle
$i$ and
$\displaystyle\sum_{i=1}^{N}\delta\mathbf{r}_{i}(t)=0.$ (4)
We substitute the ansatz given by Eq. (3) into the second order system that is
equivalent to Eqs. (1a)-(1b) with $D=0$. After simplification, one obtains
$\displaystyle\ddot{\mathbf{R}}+\delta\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)(\dot{\mathbf{R}}+\delta\dot{\mathbf{r}_{i}})$
$\displaystyle-\frac{a(N-1)}{N}\bigg{(}\mathbf{R}(t)-\mathbf{R}(t-\tau)+\delta\mathbf{r}_{i}(t)\bigg{)}$
$\displaystyle-\frac{a}{N}\delta\mathbf{r}_{i}(t-\tau),$ (5)
where we used the fact that Eq. (4) can be written as
$\displaystyle\delta\mathbf{r}_{i}(t-\tau)=-\mathop{\sum\limits_{j=1}^{N}}\limits_{i\neq
j}\delta\mathbf{r}_{j}(t-\tau).$ (6)
Summing Eq. (III) over $i$ and using Eq. (4), we find
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}-\frac{1}{N}\sum_{i=1}^{N}|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{R}}$
$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\left(2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}+|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\delta\dot{\mathbf{r}_{i}}$
$\displaystyle-a\frac{N-1}{N}\left(\mathbf{R}(t)-\mathbf{R}(t-\tau)\right).$
(7)
By inserting Eq. (III) into Eq. (III) it is possible to find the following
equation for $\delta\ddot{\mathbf{r}}_{i}$:
$\displaystyle\delta\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(\frac{1}{N}\sum_{j=1}^{N}|\delta\dot{\mathbf{r}}_{j}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{R}}$
$\displaystyle+\left(1-|\dot{\mathbf{R}}|^{2}-2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{i}-|\delta\dot{\mathbf{r}}_{i}|^{2}\right)\delta\dot{\mathbf{r}}_{i}$
$\displaystyle+\frac{1}{N}\sum_{j=1}^{N}\left(2\dot{\mathbf{R}}\cdot\delta\dot{\mathbf{r}}_{j}+|\delta\dot{\mathbf{r}}_{j}|^{2}\right)\
\delta\dot{\mathbf{r}}_{j}$
$\displaystyle-a\frac{N-1}{N}\delta\mathbf{r}_{i}-\frac{a}{N}\delta\mathbf{r}_{i}(t-\tau),$
(8)
for $i=1,2\ldots,N$.
Taken together, Eqs. (III) and (III) are equivalent to Eqs. (1a)-(1b) and they
merely involve a reconstruction of the original system that is written in
terms of particle coordinates $\mathbf{r}_{i}$ into this new system that is
written in terms of the center of mass $\mathbf{R}$ and coordinates relative
to the center of mass $\delta\mathbf{r}_{i}$. One can see that this mapping
has transformed the original $2N$ differential equations into $2N+2$
equations. Due to the relation given by Eq. (4), only $2N$ of the transformed
set of equations are independent. Therefore, there is no inconsistency between
the original and transformed equations.
By neglecting the fluctuation terms $\delta\mathbf{r}_{i}$ from Eq. (III) and
taking $N\rightarrow\infty$, we obtain the following heuristic mean field
approximation for the center of mass:
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{R}(t)-\mathbf{R}(t-\tau)\right),$
(9)
where we made the approximation $a\frac{N-1}{N}\approx a$ since we are
considering the large system size limit $N\to\infty$. We will address the
validity of neglecting the fluctuation terms in Section V.
## IV Bifurcations in the Mean Field Equation
Having derived a mean field equation, we continue by analyzing the bifurcation
structure. This bifurcation analysis will allow us to better understand the
behavior of the system in different regions of parameter space. Letting
$\mathbf{R}=(X,Y)$ and $\dot{\mathbf{R}}=(U,V)$, Eq. (9) may be written in
component form as
$\displaystyle\dot{X}$ $\displaystyle=U,$ (10a) $\displaystyle\dot{U}$
$\displaystyle=(1-U^{2}-V^{2})U-a(X-X(t-\tau)),$ (10b) $\displaystyle\dot{Y}$
$\displaystyle=V,$ (10c) $\displaystyle\dot{V}$
$\displaystyle=(1-U^{2}-V^{2})V-a(Y-Y(t-\tau)).$ (10d)
Regardless of the value of $a$ and $\tau$, Eqs. (10a)-(10d) have translational
invariant stationary solutions given by
$\displaystyle X=X_{0},\quad U=0,\quad Y=Y_{0},\quad V=0,$ (11)
where $X_{0}$ and $Y_{0}$ are two free parameters. In addition, Eqs.
(10a)-(10d) also have a three parameter family of uniformly translating
solutions given by
$\displaystyle X=U_{0}t+X_{0},\quad U=U_{0},\quad Y=V_{0}t+Y_{0},\quad
V=V_{0},$ (12)
which requires
$\displaystyle U_{0}^{2}+V_{0}^{2}=1-a\tau$ (13)
and is real-valued only when $a\tau\leq 1$. In the two-parameter space
$(a,\tau)$, the hyperbola $a\tau=1$ is in fact a pitchfork bifurcation curve
on which the uniformly translating states are born from the stationary state
$(X_{0},0,Y_{0},0)$. The pitchfork bifurcation curve can be seen in Fig. 1.
The other branch of the pitchfork bifurcation is an unphysical solution with
negative speed.
Linearizing Eqs. (10a)-(10d) about the stationary state, we obtain the
characteristic equation
$\displaystyle\left(a(1-e^{-\lambda\tau})-\lambda+\lambda^{2}\right)^{2}=0.$
(14)
It is sufficient to study the zeros of the function
$\displaystyle\mathcal{D}(\lambda)=a(1-e^{-\lambda\tau})-\lambda+\lambda^{2}=0,$
(15)
since the eigenvalues [see Eq. (14)] of the system given by Eqs. (10a)-(10d)
are obtained by duplicating those of Eq. (15).
We identify the Hopf bifurcations in the two parameter space $(a,\tau)$ by
letting the eigenvalue be purely imaginary. Our choice of $\lambda=i\omega$ is
substituted into Eq. (15), and one obtains
$\displaystyle a-\omega^{2}-i\omega=ae^{-i\omega\tau}.$ (16)
By taking the modulus of Eq. (16), one finds that $a$ at the Hopf point is
given by
$\displaystyle a_{H}^{2}=(a_{H}-\omega^{2})^{2}+\omega^{2}.$ (17)
If we consider the case when $\omega\neq 0$, then
$\displaystyle a_{H}=\frac{1+\omega^{2}}{2}.$ (18)
We substitute Eq. (18) into Eq. (16) and take the complex conjugate. This
allows us to obtain the following equation for $\tau$ at the Hopf point that
does not involve $a$:
$\displaystyle\frac{1-\omega^{2}}{1+\omega^{2}}+i\frac{2\omega}{1+\omega^{2}}=e^{i\omega\tau}.$
(19)
We isolate $\tau$ by equating the arguments of both sides, being careful to
use the branch of $\tan\theta$ in $(0,\pi)$ since the left hand side of the
equation above is on the upper complex plane for $\omega>0$. We then obtain a
family of Hopf bifurcation curves parameterized by $\omega$:
$\displaystyle a_{H}(\omega)$ $\displaystyle=\frac{1+\omega^{2}}{2},$ (20a)
$\displaystyle\tau_{H_{n}}(\omega)$
$\displaystyle=\frac{1}{\omega}\left(\arctan\left(\frac{2\omega}{1-\omega^{2}}\right)+2n\pi\right),\quad
n=0,1,\ldots$ (20b)
The first few members of the family of Hopf bifurcation curves are shown in
Fig. 1. It also is possible to eliminate the parameter $\omega$ in Eqs.
(20a)-(20b). Doing so, one obtains
Figure 1: (a) Hopf (blue) and pitchfork (red) bifurcation curves in
$(a$,$\tau)$ space. (b) A zoom-in of Fig. 1. Included is the saddle to node
transition curve (dashed black) and a number in each region (with boundaries
given by the solid curves) that indicates the number of eigenvalues with a
real part greater than zero.
$\displaystyle\tau_{H_{n}}(a)=$
$\displaystyle\frac{1}{\sqrt{2a-1}}\left(\arctan\left(\frac{\sqrt{2a-1}}{1-a}\right)+2n\pi\right),\quad
n=0,1,\ldots$ (21)
In spite of their appearance, the Hopf curves in Eqs. (20a)-(20b) and (IV) are
in fact continuous at $\omega=1$ and $a=1$, respectively [with the correct
branch of $\tan\theta$ in $(0,\pi)$]. Inspection of Eq. (20a), shows that the
Hopf frequency depends only on the value of $a$ for all members in the family.
The frequency equals one when $a=1$, and the frequency tends to infinity as
$a$ increases. Interestingly, only the first Hopf curve is defined at $a=1/2$
and has the value $\tau_{H_{0}}|_{a=1/2}=2$. The point ($a=1/2$, $\tau=2$)
which lies both on the first Hopf curve and on the pitchfork curve is a
Bogdanov-Takens (BT) point (the eigenvalues are zero), where $\omega=0$. None
of the other Hopf branches meet the pitchfork bifurcation curve since
$\tau\to\infty$ as $a\to 1/2$.
Figure 2: Real and Imaginary parts of the dominating eigenvalues as one moves
around the Bogdanov-Takens point $(a=1/2,\tau=2)$ in $(a,\tau)$ parameter
space. The eigenvalues shown are associated with the locations (a) $a=0.60$,
$\tau=2.0$, (b) $a=0.48$, $\tau=2.09$, (c) $a=0.40$, $\tau=2.01$, (d)
$a=0.53$, $\tau=1.90$, and (e) $a=0.55$, $\tau=1.91$. Refer to Fig. 1 to see
where each of the $(a,\tau)$ points lies in relation to the bifurcation
curves.
The pitchfork and Hopf bifurcation curves in the $(a,\tau)$ parameter space
were computed using a numerical continuation method [38]. These results (not
shown) are in perfect agreement with our analytical calculations. These
numerical continuation studies also allow for the determination of the number
of eigenvalues with real part greater than zero in different regions of the
$(a,\tau)$ parameter space. The results are shown in Fig. 1. In addition, our
numerical continuation analysis revealed node to focus transitions of the
steady state. These transitions occur at points where there are two real and
equal eigenvalues, i.e. where $\mathcal{D}(\lambda)=0$ and
$\mathcal{D}^{\prime}(\lambda)=0$, for real-valued $\lambda$. If
$\mathcal{D}^{\prime}(\lambda)=0$ then one can show that
$e^{-\tau\lambda}=\frac{1-2\lambda}{a\tau}$. Insertion of this relation into
$\mathcal{D}(\lambda)=0$ leads to
$\displaystyle\lambda^{2}-\left(1-\frac{2}{\tau}\right)\lambda+a-\frac{1}{\tau}=0,$
(22)
which has solutions
$\lambda=\frac{1}{2}\left[1-\frac{2}{\tau}\pm\sqrt{1+\frac{4}{\tau^{2}}-4a}\right]$.
For the roots to be repeated, we set the discriminant equal to zero and this
gives the following curve where the node-focus transitions occur:
$\displaystyle\tau=\frac{1}{\sqrt{a-1/4}}.$ (23)
Moreover, by inspecting the solutions to Eq. (22) one finds that the repeated
eigenvalues have positive real parts if $\tau>2$ and negative real parts if
$\tau<2$. In Figure 1, we show the pitchfork and Hopf bifurcation curves
overlaid with the node-focus transition curve given by Eq. (23).
As seen in Fig. 1, the pitchfork and first Hopf bifurcation curves, together
with the node-focus transition curve, split the area around the BT point into
five different regions. The behavior of the dominating eigenvalues (excluding
the one at the origin) in each of these five regions is shown in Figs. 2-2.
Starting at a point directly to the right of the BT point in $(a,\tau)$ space,
there is a pair of eigenvalues with positive real parts and non-zero imaginary
parts [Fig. 2]. Moving counter-clockwise, the eigenvalue pair collapse on the
positive real axis upon crossing the upper branch of the node-focus transition
curve [Fig. 2]. Continuing in the same direction, we observe two different
instances of eigenvalues crossing the origin: (i) first the smaller of the two
purely real and positive eigenvalues does so as the upper part of the
pitchfork bifurcation curve is crossed [Fig. 2] and (ii) then the remaining
purely real and positive eigenvalue crosses the origin as the lower part of
the pitchfork bifurcation curve is crossed [Fig. 2]. Finally, as the node-
focus transition curve is crossed, the two purely real and negative
eigenvalues coincide on the negative real axis and acquire non-zero imaginary
parts [Fig. 2]. Continuing upwards in parameter space, the complex pair of
eigenvalues crosses the imaginary axis as the Hopf bifurcation curve is
crossed, giving birth to a stable limit cycle.
## V Comparison of the Mean Field Analysis and the Full Swarm Equations
Figure 3: Regions in $(a,\tau)$ space with different dynamical behavior.
Our analysis of the deterministic mean field equations identified the
different dynamical behaviors that the approximation given by Eq. (9) exhibits
in different regions of the $(a,\ \tau)$ plane. However, the analysis does not
provide any information about how the swarm agents are distributed about the
center of mass. We neglect the stochastic terms in Eqs. (1a)-(1b) and use
extensive numerical simulations to identify some of the coherent structures
that the swarm adopts asymptotically in time:
* (i)
A translational state, in which all swarm particles have identical positions
and velocities and move uniformly in a straight line. The direction of motion
depends on the initial conditions. This behavior is only possible in region A
of Fig. 3. Moreover, the asymptotic convergence to this state requires that
all particles be located in close proximity and with aligned velocities at the
initial time. Hence, the basin of attraction is extremely small which causes
this state to be very sensitive to perturbations. This is discussed in more
detail below.
* (ii)
A ring state, in which the center of mass is stationary. The swarm agents
distribute themselves along the ring with roughly half of the agents moving
clockwise and half of the agents moving counter-clockwise. The final
stationary position of the center of mass and the particular behavior of each
individual in the swarm is dependent on the initial conditions. This behavior
is possible in regions A, B and C of Fig. 3.
* (iii)
A rotational state, in which all swarm agents collapse to the center of mass
and the latter rotates on a circular orbit. The direction of rotation depends
on the initial conditions. This behavior is only possible in region C of Fig.
3.
* (iv)
A degenerate rotational state, in which all swarm particles collapse to the
center of mass and the latter oscillates back and forth on a line. This
behavior is only possible in region C of Fig. 3. In addition, it requires that
the initial motion of all swarm particles be constrained to a line and so is
sensitive with respect to perturbations and noise.
The above list is not extensive and our simulations have revealed other time-
asymptotic patterns. However, all of these other patterns (and including the
translational state and the degenerate rotational state) require extreme
symmetry in the initial conditions and are very sensitive with respect to
perturbations and noise. Our numerical simulations suggest that only the ring
and the rotational state have a significant robustness with respect to
perturbations and noise.
The full system of equations predict a bistable behavior since the translating
and ring states are both possible in region A and C [Fig. 3], depending on the
initial conditions. The linear stability analysis of Section IV shows that the
mean field approximation fails to capture this bistable behavior.
The mean-field bifurcation results obtained here are of practical value since
they provide us with guidelines for selecting values for $a$ and $\tau$ that
will result in a particular coherent pattern asymptotically in time. In the
case of bistability, our numerical simulations strongly suggest that the
initial alignment of the agents’ velocities is critical in determining the
coherent state adopted. Specifically, to obtain the translating, rotating and
degenerate rotating states asymptotically in time (structures in which the
individuals’ velocities are perfectly aligned), one requires a high alignment
of the initial particles’ velocities; otherwise, the swarm will adopt the ring
state. However, how high an alignment is needed depends on the specific choice
of $(a,\tau)$. Our results indicate that it is easier to obtain aligned states
for larger values of the coupling constant $a$. Unfortunately, it is not
feasible to obtain analytic basin boundaries in this infinite dimensional
system. In principle, one may approximate such boundaries by performing
prohibitively extensive numerical simulations where the space of history
functions is restricted in some way. Therefore, the computation of basins of
attraction is outside the scope of this work and is left for future research.
For the non-degenerate and degenerate rotating states as well as for the
translating state, the approximation we made when neglecting the fluctuation
terms in Eq. (9) is entirely valid since in the noiseless case all agents
collapse to the center of mass. In the case of the ring structure, these
fluctuation terms are not necessarily small. However, in Eq. (III) all
fluctuation terms with the exception of the one containing the factor
$\frac{1}{N}\sum_{i=1}^{N}|\delta\dot{\mathbf{r}}_{i}|^{2}$ approximately
cancel out in the long time limit, due to the symmetry in the distribution of
the agents. The fluctuation term that remains becomes equal to one in the long
time limit. This has the effect of eliminating the self-propulsion of the
center of mass and what remains is solely cubic dissipation.
The following sub-sections contain detailed discussion regarding the spatio-
temporal scales of each coherent structure.
### V-A The Ring State
The analysis of Appendix A shows that the radius and angular frequency of the
swarm particles on the ring state is given by
$\displaystyle\rho_{j}=\frac{1}{\sqrt{a}},\qquad\dot{\theta}_{j}=\pm\sqrt{a},$
(24)
so that particles move at unit speed, $\rho_{j}\dot{\theta}_{j}=\pm 1$.
Figure 4: Comparison of numerical simulations (red circular markers) with the
analytical expressions (continuous blue curve) given by Eq. (24) for (a) the
radius and (b) the frequency of the ring state.(c) For each value of $a$, the
time delay was chosen as $\tau=1/\sqrt{a-1/4}$ (black circular markers).
We have numerically computed the radius and angular frequency for different
values of $a$ and $\tau$ within the region in which the mean field
approximation gives a stable stationary center of mass (Fig. 4). Figures 4-4
shows that there is excellent agreement between the numerical simulations and
the analytical result given by Eq. (24). It is worth noting that the condition
given by Eq. (29) and used to derive Eq. (24) is satisfied in the long time
limit in our simulations.
### V-B The Rotating State
We show in Appendix B that the circular orbit of the rotating state has radius
$\rho_{0}$ and frequency $\omega$ that satisfy the following relations:
$\displaystyle\omega^{2}=$ $\displaystyle a\cdot(1-\cos\omega\tau),$ (25a)
$\displaystyle\rho_{0}=$
$\displaystyle\frac{1}{|\omega|}\sqrt{1-a\frac{\sin\omega\tau}{\omega}}.$
(25b)
Figure 5: In $(a$,$\tau)$ space, we plot: Hopf (blue) and pitchfork (red)
bifurcation curves, and the curve $a\tau^{2}=2$ where the first limit cycle
ceases to exist by having its radius diverging to infinity (green).
Eqs. (25a)-(25b) can have as many solutions as desired by choosing $a$ and
$\tau$ large enough. However, a careful analysis reveals that the solutions to
Eqs. (25a)-(25b) are generated exactly along the Hopf curves of our previous
mean field analysis and represent the same limit cycles of that analysis [Fig.
1]. The expressions in Eqs. (25a)-(25b) thus determine the spatio-temporal
scales of these circular orbits beyond the Hopf curves where they are born.
Our analysis also shows that the circular limit cycle that is created on the
first member of the Hopf bifurcation curves persists to the left of the
pitchfork bifurcation curve and then ceases to exist as its radius diverges to
infinity on the curve $a\tau^{2}=2$ (Fig. 5). Moreover, numerical simulations
of the mean field equations reveal that both the translating state and the
rotating state are linearly stable for $(a,\tau)$ pairs inside the wedge
between the curve $a\tau^{2}=2$ and the pitchfork bifurcation curve $a\tau=1$
above the BT point.
Figures 6-6 show the excellent agreement between numerical simulations and the
analytical results given by Eqs. (25a)-(25b), for different values of $a$ and
$\tau$.
Figure 6: Comparison of numerical simulations (red circular markers) with the
analytical expressions (continuous blue curve) given by Eqs. (25a)-(25b) for
(a) the radius, (b) the period, and (c) the speed of the collapsed circular
orbit. (d) For each value of $a$, the time delay was chosen as
$\tau=\frac{2}{\sqrt{2a-1}}\arctan\left(\frac{\sqrt{2a-1}}{1-a}\right)$ (black
circular markers) to assure asymptotic time convergence to the collapsed
circular orbit state.
Interestingly, in Fig. 6 we note that in the asymptotic time limit the
collapsed agents move at a speed greater than one, the speed at which agents
would tend to move in the absence of coupling. This is explained by noting
that the ratio of the time delay to the period of oscillations is such that
the delayed position of the collapsed agents $\mathbf{R}(t-\tau)$ is ahead of
the present position $\mathbf{R}(t)$. The attraction that an individual
particle feels to the delayed position of the rest of the swarm forces the
whole system go faster.
### V-C The Degenerate Rotating State
A degenerate version of the rotating state is possible when the initial motion
of the swarm is restricted to a line, since in this case it follows from Eqs.
(1a)-(1b) that the swarm will remain on such a line for all times. As we show
in Appendix C, we may assume that the motion of the collapsed swarm occurs on
the $X=Y$ line of the center of mass coordinates and then use a finite Fourier
mode approximation of the ensuing dynamics. An approximation in terms of just
three modes gives
$\displaystyle X(t)=Y(t)=2c_{1}\cos\omega t+2|c_{3}|\cos(3\omega t+\phi_{3}),$
(26)
where $\omega$, $c_{1}$, $c_{3}$ and $\phi_{3}$ are obtained by solving Eqs.
(42a)-(42b) numerically.
Figure 7: Comparison of numerical simulation (red circular markers) with the
analytical expressions (continuous blue curve) given by Eqs. (42a)-(42b) for
(a) the amplitude, (b) period, and (c) the maximum speed of the collapsed
straight line orbit. (d) At each value of $a$, the time delay was chosen as
$\tau=\frac{2}{\sqrt{2a-1}}\arctan\left(\frac{\sqrt{2a-1}}{1-a}\right)$ (black
circular markers) to ensure asymptotic time convergence to the collapsed-
straight line orbit state.
Figures 7-7 show a comparison between our analytical results given by Eqs.
(42a)-(42b) and results obtained using numerical simulation for the amplitude,
period and maximum speed of oscillation for different values of $a$ and
$\tau$. There is excellent agreement in both amplitude and period between our
analysis and the numerical simulations [Figs. 7-7]. The agreement for the
speed of motion is very good as well, but the theoretical estimate is shifted
slightly with respect to the results from simulations [Fig. 7]. As in the
collapsed circular orbit, we note that the collapsed set of particles have a
maximum speed which exceeds one, the speed that individual, uncoupled
particles acquire in the long-time limit. As before, this effect arises from
the attraction that the current particle position $\mathbf{R}(t)$ feels
towards the delayed position $\mathbf{R}(t-\tau)$ when the latter lies in the
direction of motion of the collapsed particles.
## VI The Effects of Noise on the Swarm
In the absence of noise, the initial alignment of the swarm particles is
critical in determining the asymptotic behavior of the swarm (Sec. V). When
noise is introduced, the interplay of coupling strength, time delay and noise
intensity gives rise to very interesting behavior due to fluctuations in the
particles’ alignment. Specifically, our studies show that if the coupling
strength $a$ and/or the time delay $\tau$ are below a certain limit, then the
presence of noise promotes swarm transitions from aligned into misaligned
coherent states. More surprising, however, is that if the coupling strength
$a$ and/or the time delay $\tau$ are big enough, then there is a noise
intensity threshold that forces a transition in the swarm from misaligned into
aligned states. In addition, we show that for these high values of $a$ and/or
$\tau$, the system presents an interesting hysteresis phenomenon when the
noise intensity is time dependent.
For the purpose of these studies, we define the alignment of particle $j$ with
the rest of the swarm as the cosine of the angle between the velocity of
particle $j$ and the velocity of the swarm as a whole:
$\displaystyle\cos\theta_{j}=\frac{\dot{\mathbf{r}}_{j}\cdot\dot{\mathbf{R}}}{|\dot{\mathbf{r}}_{j}||\dot{\mathbf{R}}|}.$
(27)
Therefore the alignment of individual particles ranges from -1 to 1. A good
measure of the overall alignment of the swarm is furnished by the ensemble
average of these cosines given as
$\displaystyle\textrm{Mean swarm
alignment}=\frac{1}{N}\sum_{j=1}^{N}\cos\theta_{j}=\frac{1}{N}\sum_{j=1}^{N}\frac{\dot{\mathbf{r}}_{j}\cdot\dot{\mathbf{R}}}{|\dot{\mathbf{r}}_{j}||\dot{\mathbf{R}}|}.$
(28)
We first carry out a numerical simulation with coupling constant $a=0.5$ and
noise standard deviation $\sigma=0.05$ (noise intensity $D=0.00125$). At
$t=50$, a time delay of $\tau=0.5$ is turned on. These parameters correspond
to region A of Fig. 3. Initially, we place all particles at the origin and
align their velocities by choosing $\dot{x}_{j}=1$ and $\dot{y}_{j}=1$ for all
particles. We describe the behavior of the swarm by following the ensemble
averages of the particle distances to the center of mass [Fig. 8] and of the
particle alignment [Fig. 8] as functions of time. Before the time delay is
turned on at $t=50$, the swarm is in a translating state with particles
slightly spread out from the center of mass in a ‘pancake’ shape, as described
in [23], with an ensemble alignment close to one. Once the delay is turned on,
the translating state is broken up and the swarm converges to the ring state
in which the mean particle alignment is near zero. The radius of the ring
obtained in this numerical simulation matches the theoretical result [Eq.
(24)] that predicts a radius of $\frac{1}{\sqrt{a}}=\sqrt{2}\approx 1.41$. A
completely analogous situation ensues for parameters in region B of Fig. 3
(results not shown). In addition, in both cases the swarm will immediately
converge to the ring state if the swarm velocities are not sufficiently
aligned at time zero. We thus conclude that for these choices of $(a,\ \tau)$
pairs, the noise misaligns the particles’ velocities and forces a transition
into the ring state.
Figure 8: Time evolution of the ensemble average of (a) the particle distance
to the center of mass, and (b) the mean particle alignment showing how the
particle alignment breaks up due to the effects of noise. For long times the
swarm converges to a ring state. The parameter values of $a=0.5$ and
$\tau=0.5$ are associated with region A of Fig. 3. The time delay is turned on
at $t=50$ and the noise standard deviation is $\sigma=0.05$ ($D=0.00125$).
In contrast to the cases discussed above, for parameters in region C of Fig.
3, a sufficiently large noise intensity promotes transitions from misaligned
to aligned states. We show this by comparing the results of a series of
simulations for different values of the noise standard deviation $\sigma$. The
simulations are divided into two cases that differ only on the initial
conditions for the swarm particles. In all simulations, the coupling constant
$a=2$ and a time delay of $\tau=2$ is turned on at $t=50$. In the first case,
all particles start from the origin with identical velocities $\dot{x}_{j}=1$
and $\dot{y}_{j}=1$. In the second case, all swarm particles are initially
distributed uniformly on the unit square and are at rest.
In these simulations, the final state of the swarm may be visualized by
plotting the mean swarm alignment after transients have decayed ($t=300$) as a
function of noise intensity for the first case [Fig. 9] and the second case
[Fig. 9]. In the first case of simulations, the high initial alignment of
particles’ velocities forces the swarm to converge to a compact rotating state
independent of noise intensity. However, the rotational state is destroyed if
the noise standard deviation is bigger than $\sigma\approx 0.8$ [Fig. 9]. The
situation is more interesting and complex for the second set of simulations.
For low noise intensities ($\sigma\lesssim 0.26$) the low initial alignment of
the particles leads the swarm to converge to a ring state with near zero mean
alignment [Fig. 9]. A noise standard deviation just beyond the threshold of
$\sigma\approx 0.26$ displays an interesting effect.
Figure 9: Asymptotic value of the mean particle alignment for (a) particles
starting with perfectly aligned velocities at time zero and (b) for particles
distributed uniformly over the unit square and starting from rest for
different values of the noise standard deviation $\sigma$. The parameter
values of $a=2$ and $\tau=2$ (turned on at $t=50$) are associated with a
location in region C of Fig. 3.
As the $\sigma\approx 0.26$ threshold is crossed, the swarm transitions from
the ring state into the rotating state with high mean alignment. An
examination of the full simulation data reveals that the transition occurs as
an increasing group of particles gradually becomes aligned and eventually
absorbs all the remaining particles. A sufficient amount of noise is necessary
for this transition, since it allows each particles’ velocity vector to probe
many directions until finally enough of them become trapped in a ‘potential
well’ of alignment with other particles. As with the first case of simulations
a noise standard deviation bigger than $\sigma\approx 0.8$ breaks up the
rotating state. Figure 10 clearly shows the transition from the ring to the
compact, rotational state through the time evolution of the ensemble averages
of the particle distances to the center of mass and of the mean particle
alignment.
Further studies on the switching behavior between coherent states of the swarm
demonstrate that the system exhibits a hysteresis phenomena. With the swarm
system starting on the ring state with noise standard deviation of
$\sigma=0.24$, one can force a transition into the rotating state by
increasing the noise to $\sigma=0.26$. However, even if the noise is lowered
down to $\sigma=0.02$, the swarm remains in the rotating state with a high
velocity alignment [Figs. 11-11]. Nevertheless, it is possible to return the
swarm to the ring state if, once in the rotating state, the noise is raised to
very high amounts ($\sigma=1$) for a sufficient amount of time and then
dropped suddenly to a very low value ($\sigma=0.05$). The high noise levels
serve to completely misalign the particles’ velocities and allow them to
converge to the ring once the noise levels are below $\sigma\lesssim 0.26$.
Figure 10: Time evolution of the ensemble average of (a) the particle
distance to the center of mass, and (b) the mean particle alignment showing
how the swarm transitions from a ring state into a compact, rotational state
with alignment close to one. The parameter values of $a=2$ and $\tau=2$
(turned on at $t=50$) and $\sigma=0.4$ ($D=0.08$) are associated with region C
of Fig. 3. Particles are initially distributed uniformly over the unit square
and start from rest.
Figure 11: Time evolution of (a) mean particle alignment for example 1, (b)
noise standard deviation for example 1, (c) mean particle alignment for
example 2, and (d) noise standard deviation for example 2. The results show
how a time-dependent noise intensity may be used to force swarm transitions.
The parameter values of $a=2$ and $\tau=2$ (turned on at $t=10$) are
associated with region C of Fig. 3. Particles are initially distributed
uniformly over the unit square and start from rest.
## VII Conclusions
In this work we analyzed the dynamics of a self-propelling swarm where
individuals interact with a communication time delay in the presence of noise.
Using a mean field approximation in the deterministic case, we analytically
obtained the complete bifurcation picture in the parameter space of coupling
strength and communication time delay. This analysis shows how different
combinations of coupling strength and time delay induce the swarm to adopt
different coherent structures asymptotically in time. Our bifurcation studies
demonstrated the existence of a Bogdanov-Takens point, where the stationary
center of mass solution has a double zero eigenvalue, which is critical in
organizing the dynamics of the swarm.
The stable patterns that are possible for this system have several
applications for autonomous vehicles. More detailed applications for each
pattern are as follows: (1) the translational state may be used for target
tracking and group transport [11, 17]. (2) The ring state should prove useful
in terrain coverage and regional surveillance [39, 40]. (3) The rotating state
may be exploited in obstacle avoidance, boundary tracking and surveillance
[15, 11, 40]. In addition, we believe all three patterns are applicable to the
problem of environmental sensing [13, 16].
In numerical experiments with noise, we showed that the interplay of coupling
strength, time delay and noise intensity may give rise to interesting
switching behavior from one coherent structure to another. We found that if
the coupling strength $a$ and/or the time delay $\tau$ are below a certain
limit, then the presence of noise induces transitions from states in which the
alignment of the particles’ velocities is high into states with low alignment.
More surprising, however, is that if the coupling strength $a$ and/or the time
delay $\tau$ are big enough, then there is a noise intensity threshold that
forces a transition in the swarm from misaligned into aligned states. In
addition, by using a time-dependent noise intensity at these high values of
$a$ and/or $\tau$, we show that the system exhibits hysteresis since the
swarm’s transitions are not easily reversible. We note that analytical results
on the effects of noise on delay-coupled swarms are not easy to obtain. Two
examples relevant to our work are given in [23, 24], where the authors
investigate models similar to the one presented here but without time delay.
Realistic application of the model treated here to the motion of multi-robot
systems requires local repulsion among individuals to be taken into account.
We have simulated the swarm model with the addition of a repulsive inter-agent
potential of exponential form
$U_{ij}=c_{r}\exp{\left({-\frac{|\mathbf{r}_{i}-\mathbf{r}_{j}|}{L_{r}}}\right)}$.
These simulations demonstrate (results not shown) that the coherent patterns
we discussed in this article persist when the characteristic repulsion length
$L_{r}$ and repulsion strength $c_{r}$ between robots are small compared to
global attraction parameters. Stronger repulsion can destabilize the coherent
structures.
Recently, systems with non-uniform time delays have received much attention.
For example, the important question of synchronization in networks
communicating at randomly-distributed time delays has been recently
investigated [41, 42]. In practical applications, the case of differing (and
even time-varying) time delays between agents may be treated similarly to the
case of a single delay by using a data buffer [27]. The idea is to identify an
upper bound to the time delay ($\tau_{\textrm{max}}$) between all agent pairs
and then design the agents so that the actuation occurs when the data buffer
of size $\tau_{\textrm{max}}$ is full.
As part of our ongoing work, we are extending our investigations for the cases
in which: (_i_) the communication time delays vary between different pairs of
agents; and (_ii_) the communication graph is non-globally coupled. In
realistic settings, both of these cases may occur due to the effects of the
spatial distribution of agents such as signal travel times and imperfect
transmission arising, for example, from complex terrain topography or
component malfunction. In the case of communication delays that differ among
different pairs of agents (though constant in time), our preliminary results
show some patterns analogous to the ones observed here, but with much more
added complexity. The present investigation lays a good foundation on which to
base the study of these more complicated cases.
In summary, our results aid in understanding the stability of complex coherent
structures in swarming systems with time delayed communication and in the
presence of a noisy environment. Although our analytical and numerical results
were obtained using a model with linear, attractive interactions, our analysis
gives useful insight for the study of models with more general forms of time
delayed coupling between agents. Our results may prove to be valuable for the
control of man-made vehicles where actuation and communication are delayed, as
well as in understanding swarm alignment in biological systems.
## Appendix A Analysis of the Ring State
The swarm ring state is obtained when the center of mass is stationary. For
the solution $\mathbf{R}=$const. to satisfy Eq. (III) we require
$\displaystyle\sum_{i=1}^{N}\delta\dot{\mathbf{r}}_{i}^{2}\delta\dot{\mathbf{r}_{i}}=0.$
(29)
We simplify Eq. (III) by taking $\mathbf{R}=$const. and using Eq. (29) we
obtain
$\displaystyle\delta\ddot{\mathbf{r}}_{j}=$
$\displaystyle\left(1-\delta\dot{\mathbf{r}}_{j}^{2}\right)\delta\dot{\mathbf{r}}_{j}-a\delta\mathbf{r}_{j}-\frac{a}{N}\delta\mathbf{r}_{j}(t-\tau).$
(30)
We consider the large system size limit $N\rightarrow\infty$ and we drop the
delayed term. The resulting equations are simply ODEs and so the analysis
below shows that the ring orbit is not dependent on having time delays in the
system. Writing Eq. (30) in polar coordinates $\delta
x_{j}=\rho_{j}\cos{\theta_{j}}$ and $\delta y_{j}=\rho_{j}\cos{\theta_{j}}$,
we obtain
$\displaystyle\ddot{\rho}_{j}=$
$\displaystyle\left(1-\dot{\rho}_{j}^{2}-\rho_{j}^{2}\dot{\theta}_{j}^{2}\right)\dot{\rho}_{j}+\rho_{j}\dot{\theta}_{j}^{2}-a\rho_{j},$
(31a) $\displaystyle\rho_{j}\ddot{\theta}_{j}=$
$\displaystyle\left(1-\dot{\rho}_{j}^{2}-\rho_{j}^{2}\dot{\theta}_{j}^{2}\right)\rho_{j}\dot{\theta}_{j}-2\dot{\rho}_{j}\dot{\theta}_{j}.$
(31b)
Equations (31a)-(31b) have the trivial solution $\rho_{j}=0$ as well as a ring
solution:
$\displaystyle\rho_{j}=\frac{1}{\sqrt{a}},\qquad\dot{\theta}_{j}=\pm\sqrt{a},$
(32)
in which particles move at unit speed, $\rho_{j}\dot{\theta}_{j}=\pm 1$.
## Appendix B Analysis of the Rotating State
In the noiseless rotating state, all particles collapse to a point,
$\delta\mathbf{r}_{i}=0$, and the equation for the center of mass given by Eq.
(III) simplifies considerably to
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-\dot{\mathbf{R}}^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{R}(t)-\mathbf{R}(t-\tau)\right).$
(33)
We write $\mathbf{R}=(X,Y)$ and introduce polar coordinates
$X=\rho\cos{\theta}$ and $Y=\rho\sin{\theta}$ to obtain
$\displaystyle\ddot{\rho}=$
$\displaystyle\left(1-\dot{\rho}^{2}-\rho^{2}\dot{\theta}^{2}\right)\dot{\rho}+\rho\dot{\theta}^{2}-a\bigg{(}\rho-\rho_{\tau}\cos(\theta-\theta_{\tau})\bigg{)},$
(34a) $\displaystyle\rho\ddot{\theta}=$
$\displaystyle\left(1-\dot{\rho}^{2}-\rho^{2}\dot{\theta}^{2}\right)\rho\dot{\theta}-2\dot{\rho}\dot{\theta}+a\rho_{\tau}\sin(\theta-\theta_{\tau}),$
(34b)
where we’ve written $\rho_{\tau}\equiv\rho(t-\tau)$ and
$\theta_{\tau}\equiv\theta(t-\tau)$. Equations (34a)-(34b) have a circular
orbit solution, $\rho=\rho_{0}$ and $\theta=\omega t+\theta_{0}$, where
$\displaystyle\omega^{2}=$ $\displaystyle a\cdot(1-\cos\omega\tau),$ (35a)
$\displaystyle\rho_{0}=$
$\displaystyle\frac{1}{|\omega|}\sqrt{1-a\frac{\sin\omega\tau}{\omega}}.$
(35b)
and $\theta_{0}$ is obtained from the initial conditions. In the main text we
discuss the behavior of the solutions to Eqs. (35a)-(35b).
## Appendix C Analysis of the Degenerate Rotating State
When the motion of the whole swarm is initially constrained to a line, Eqs.
(1a)-(1b) dictate that the swarm will remain on this line for all times. If
the coupling parameter $a$ and/or the time delay $\tau$ are large enough, the
resulting motion is a degenerate form of the rotating solution in which the
swarm moves back and forth along a straight line.
In the case without noise all particles collapse to a point,
$\delta\mathbf{r}_{i}=0$, and the line along which motion occurs is arbitrary;
here we use $X=Y$. The problem reduces to analyzing a single delay equation
given by
$\displaystyle\ddot{X}=$
$\displaystyle\left(1-2\dot{X}^{2}\right)\dot{X}-a\left(X(t)-X(t-\tau)\right).$
(36)
We find a solution using Fourier analysis. We let
$\displaystyle X(t)=\sum_{n=-\infty}^{\infty}c_{n}e^{in\omega t},$ (37)
where the coefficients satisfy $c_{n}={c_{-n}}^{*}$ in order to ensure that
$X(t)$ is a real quantity. Substituting Eq. (37) into Eq. (36), we get for the
$n$-th mode
$\displaystyle-n^{2}\omega^{2}c_{n}$ $\displaystyle=in\omega c_{n}$
$\displaystyle+2i\omega^{3}\sum_{\ell,m\neq 0}c_{\ell}c_{m}c_{n-\ell-m}\ell
m(n-\ell-m)$ $\displaystyle-ac_{n}(1-e^{-in\omega\tau}),$ (38)
for $n=0,1,2,\dots$. The $n=0$ equation is
$\displaystyle\sum_{\ell,m\neq 0}c_{\ell}c_{m}c_{-\ell-m}\ell m(\ell+m)=0,$
(39)
which does not involve $c_{0}$. Unsurprisingly, $c_{0}$ is undetermined since
the position of the center of mass may be translated in space without
modifying the dynamics of the system.
We now approximate the motion of the center of mass by keeping the first three
modes. By appropriately choosing the time origin, we may take $c_{1}$ to be
purely real and positive. In contrast, $c_{2}$ and $c_{3}$ are complex
quantities which we write as $c_{i}=|c_{i}|e^{i\phi_{i}}$, for $i=2,3$. The
equations for the first three modes $n=1,2,3$ become
$\displaystyle-$ $\displaystyle\omega^{2}c_{1}=i\omega c_{1}$
$\displaystyle+2i\omega^{3}\left(-3c_{1}^{3}-36c_{2}^{2}c_{3}^{*}-54c_{1}|c_{3}|^{2}-24c_{1}|c_{2}|^{2}+9c_{1}^{2}c_{3}\right)$
$\displaystyle-ac_{1}(1-e^{-i\omega\tau}),$ (40a) $\displaystyle-$
$\displaystyle 4\omega^{2}c_{2}=2i\omega c_{2}$
$\displaystyle+2i\omega^{3}\left(-108c_{2}|c_{3}|^{2}-36c_{1}c_{2}^{*}c_{3}-24c_{2}|c_{2}|^{2}-12c_{2}c_{1}^{2}\right)$
$\displaystyle-ac_{2}(1-e^{-2i\omega\tau}),$ (40b) $\displaystyle-$
$\displaystyle 9\omega^{2}c_{3}=3i\omega c_{3}$
$\displaystyle+2i\omega^{3}\left(-18c_{3}c_{1}-72c_{3}|c_{2}|^{2}-81c_{3}|c_{3}|^{2}-12c_{2}^{2}c_{1}+c_{1}^{3}\right)$
$\displaystyle-ac_{3}(1-e^{-3i\omega\tau}).$ (40c)
In addition, the condition from Eq.(39) becomes
$\displaystyle 6(c_{2}c_{3}^{*}-c_{2}^{*}c_{3})-c_{1}(c_{2}-c_{2}^{*})=0.$
(41)
Separating Eqs. (40a)-(40c) and Eq. (41) into real and imaginary parts yields
a system of seven equations (since the real part of Eq. (41) is satisfied
automatically) for the six unknowns: $\omega$, $c_{1}$, $|c_{2}|$, $\phi_{2}$,
$|c_{3}|$ and $\phi_{3}$. These equations cannot be satisfied in general.
However, if $|c_{2}|=0$, then the equation for mode $n=2$ [Eq. (40b)] and Eq.
(41) are satisfied automatically, leaving four equations:
$\displaystyle-\omega^{2}c_{1}=$ $\displaystyle i\omega
c_{1}+2i\omega^{3}\left(-3c_{1}^{3}54c_{1}|c_{3}|^{2}+9c_{1}^{2}c_{3}\right)$
$\displaystyle-ac_{1}(1-e^{-i\omega\tau}),$ (42a)
$\displaystyle-9\omega^{2}c_{3}=$ $\displaystyle 3i\omega
c_{3}+2i\omega^{3}\left(-18c_{3}c_{1}-81c_{3}|c_{3}|^{2}+c_{1}^{3}\right)$
$\displaystyle-ac_{3}(1-e^{-3i\omega\tau}).$ (42b)
for the four unknowns $\omega$, $c_{1}$, $|c_{3}|$ and $\phi_{3}$. Equations
(42a)-(42b) may be solved numerically and permit one to approximate the motion
of the center of mass in the form
$\displaystyle X(t)=Y(t)=2c_{1}\cos\omega t+2|c_{3}|\cos(3\omega t+\phi_{3}).$
(43)
The frequency of the straight line orbit of the swarm center of mass is
approximately equal to the frequency of the circular orbit in Eq. (25a). In
addition, the amplitude of oscillation of the straight line orbit is
approximately equal to the radius of the circular orbit of Eq. (25b) divided
by a factor of $\sqrt{6}$.
## Acknowledgments
The authors gratefully acknowledge the Office of Naval Research for its
support. LMR and IBS are supported by Award Number R01GM090204 from the
National Institute Of General Medical Sciences. The content is solely the
responsibility of the authors and does not necessarily represent the official
views of the National Institute Of General Medical Sciences or the National
Institutes of Health. EF is supported by the Naval Research Laboratory (Award
No. N0017310-2-C007). We also extend our thanks to Kevin Lynch and M. Ani
Hsieh for reading early versions of the manuscript.
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|
arxiv-papers
| 2012-05-01T15:51:51 |
2024-09-04T02:49:30.415367
|
{
"license": "Public Domain",
"authors": "Luis Mier-y-Teran-Romero, Eric Forgoston, and Ira B. Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1205.0195"
}
|
1205.0243
|
arxiv-papers
| 2012-05-01T00:19:47 |
2024-09-04T02:49:30.425322
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andino Maseleno, Md. Mahmud Hasan",
"submitter": "Andino Maseleno",
"url": "https://arxiv.org/abs/1205.0243"
}
|
|
1205.0253
|
# The radius of baryonic collapse in disc galaxy formation
Susan A. Kassin,1 Julien Devriendt,2 S. Michael Fall,3 Roelof S. de Jong,4
Brandon Allgood,5,6 & Joel R. Primack5
1 Astrophysics Science Division, Goddard Space Flight Center, Code 665,
Greenbelt, MD 20771, USA
2 Sub-Department of Astrophysics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK
3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA
4 Astrophysikalisches Institut Potsdam (AIP), An der Sternwarte 16, 14482
Potsdam, Germany
5 Department of Physics, University of California, Santa Cruz,1156 High
Street, Santa Cruz, CA 95064, USA
6 currently at: Numerate Inc., 1150 Bayhill Drive, San Bruno, CA 94066, USA
NASA Postdoctoral Program FellowE-mail: susan.kassin@nasa.gov
###### Abstract
In the standard picture of disc galaxy formation, baryons and dark matter
receive the same tidal torques, and therefore approximately the same initial
specific angular momentum. However, observations indicate that disc galaxies
typically have only about half as much specific angular momentum as their dark
matter haloes. We argue this does not necessarily imply that baryons lose this
much specific angular momentum as they form galaxies. It may instead indicate
that galaxies are most directly related to the inner regions of their host
haloes, as may be expected in a scenario where baryons in the inner parts of
haloes collapse first. A limiting case is examined under the idealised
assumption of perfect angular momentum conservation. Namely, we determine the
density contrast $\Delta$, with respect to the critical density of the
Universe, by which dark matter haloes need to be defined in order to have the
same average specific angular momentum as the galaxies they host. Under the
assumption that galaxies are related to haloes via their characteristic
rotation velocities, the necessary $\Delta$ is $\sim 600$. This $\Delta$
corresponds to an average halo radius and mass which are $\sim 60$% and $\sim
75$%, respectively, of the virial values (i.e., for $\Delta=200$). We refer to
this radius as the radius of baryonic collapse $R_{BC}$, since if specific
angular momentum is conserved perfectly, baryons would come from within it. It
is not likely a simple step function due to the complex gastrophysics
involved, therefore we regard it as an effective radius. In summary, the
difference between the predicted initial and the observed final specific
angular momentum of galaxies, which is conventionally attributed solely to
angular momentum loss, can more naturally be explained by a preference for
collapse of baryons within $R_{BC}$, with possibly some later angular momentum
transfer.
###### keywords:
galaxies – formation, galaxies – evolution, galaxies – kinematics and
dynamics, galaxies – fundamental properties.
## 1 Introduction
In the standard picture of disc galaxy formation (e.g., Fall & Efstathiou,
1980; Dalcanton, Spergel, & Summers, 1997; Mo, Mao, & White, 1998), galaxies
consist of a dissipative baryonic component and a non-dissipative dark matter
component. Galaxies form hierarchially, and in this process, baryons and dark
matter acquire the same specific angular momentum ($j$) via tidal-torques.
This is because tidal-torques are most effective in the linear and the trans-
linear regimes, when baryons and dark matter are well-mixed. The dark matter
then collapses non-dissipatively, and the baryons dissipatively, likely with
some cloud-cloud collisions and possibly shocks (processes which are expected
to rearrange $j$ but not remove it). The baryons form rotating centrifugally-
supported discs at the centres of the potential wells. For a review of this
scenario see Fall (2002). This standard picture is able to correctly predict
galaxy properties such as scale-lengths and sizes if the baryons retain most
of their initial $j$. It has been extended to include additional physics
effects and larger samples of galaxies by e.g., White & Frenk (1991), Cole et
al. (1994), Somerville & Primack (1999), de Jong & Lacey (2000), Van den Bosch
(2001), Hatton et al. (2003), and Dutton (2009).
In order for this scenario to correctly predict galaxy properties, the baryons
must retain a large fraction of their initial angular momentum. However, early
numerical simulations of galaxy formation contradicted this expectation (Katz
& Gunn, 1991; Navarro & Benz, 1991; Navarro & White, 1994). They found a
factor of $\sim 30$ loss of angular momentum for simulated galaxies, and
referred to this as an “angular momentum catastrophe.” As simulations improved
over the years, it became clear that much of this catastrophe was actually a
numerical artifact: too little resolution and too much numerical viscosity
(see e.g., Governato et al., 2010; Brooks et al., 2011; Keres̆ et al., 2011;
Brook et al., 2011; Kimm et al., 2011, and references therein). Another
possible contribution to solving the angular momentum problem may be through
feedback effects which can delay baryons from falling onto discs (e.g., Weil,
Eke, & Efstathiou, 1998; Sommer-Larsen, Gelato, & Vedel, 1999; Eke,
Efstathiou, & Wright, 2000; Thacker & Couchman, 2001). With high numerical
resolution and some feedback, galaxy simulations are now at a stage where
angular momentum loss may be a relatively minor problem. In this paper, we
explore another option: that the discs of galaxies draw baryons mainly from
the inner parts of dark matter haloes. Some of the baryons in the outer parts
may have not yet collapsed onto the discs.
The angular momentum catastrophe prompted comparisons of the $j$ of simulated
haloes to that of observed galaxies. In these studies, the $j$ of dark matter
haloes is measured out to the virial radius, $R_{Vir}$, which is standardly
defined as $R_{\Delta=200}$, and is the effective radius at which the dark
matter ceases to collapse into the halo. Navarro & Steinmetz (2000) and
Burkert & D’Onghia (2004) found that observed galaxies have 45% and 70% of the
$j$ of their expected host haloes in simulations, respectively, under the
assumptions that galaxies can be related to simulated host haloes via
characteristic rotation velocities directly and via a scaling factor,
respectively. Recently, Dutton & van den Bosch (2012) found that the spin
parameters of observed galaxies are $\sim 60$% of those of simulated haloes.
These studies are consistent once differences in assumptions and
approximations are accounted for.
Studies which compare the total $j$ predicted for haloes by numerical
simulations to that observed for galaxies all assume that the effective outer
halo radius from which the baryons collapse (defined here as $R_{BC}$) is
equal to $R_{Vir}$. Because baryons in the inner parts of haloes will have
higher cooling rates and more frequent cloud-cloud collisions, it is
reasonable to expect that they form the galaxies, and that baryons from larger
radii are not captured. Although $R_{Vir}$ has traditionally been identified
with $R_{BC}$, these two radii are governed by different physics (dissipative
versus non-dissipative), and need not be related, as emphasized by Fall
(2002). The only requirement is that $R_{BC}$ must be interior to $R_{Vir}$,
since baryons cannot collapse from unvirialized regions. The purpose of this
paper is to determine the effect of relaxing the assumption that $R_{Vir}$ and
$R_{BC}$ are equal on the difference in $j$ between galaxies and haloes. We
assume for simplicity that the boundary between the collapsed and uncollapsed
baryons is a sharp one. In reality, it will be a gradual boundary because some
of the baryons in the halo within $R_{BC}$ might not collapse, and some
baryons outside of $R_{BC}$ might. Therefore, we regard $R_{BC}$ as the
effective boundary between these two regions.
In this paper, we ask the following question: If galaxies formed from all the
baryons in haloes out to $R_{BC}$, and beyond this radius the baryons remained
in the halo, what is the value of $R_{BC}$ required to match the $j$ of
galaxies? We address this question by comparing the $j$ observed for disc
galaxies with that of their expected dark matter haloes measured within a
range of halo radii. For disc galaxies, $j$ can be measured from observations
of surface brightness profiles and rotation curves. For dark matter haloes, we
must resort to numerical simulations.
This paper is organised as follows: In §2, we measure $j$ of dark matter
haloes in a cosmological dark matter-only simulation. We investigate its
dependence on the halo radius within which $j$ is measured and the halo radius
at which the rotation velocity is measured. The resulting predictions of dark
matter halo $j$ are compared to $j$ measured for a large observational sample
of local galaxies for which the completeness is known in §3. A discussion of
the results is in §4. We adopt a $\Lambda$CDM concordance universe
($\Omega_{m}=0.24,\Omega_{\Lambda}=0.76,h=H_{0}/[100$ km s-1
Mpc${}^{-1}]=0.73,\sigma_{8}=0.77,n=0.958),$ i.e., within one standard
deviation of both the WMAP 3 and 5 year best estimates (Spergel et al., 2007;
Dunkley et al., 2009). All logarithms are base ten.
## 2 N-body simulation of dark matter haloes
Figure 1: For simulated dark matter haloes at $z=0$, the relations between
$j_{\Delta}$ (for $\Delta=$ 200, 2000, and 20,000) and rotation velocities
$V_{200}$ and $V_{20,000}$ are shown. Individual haloes are plotted as grey
points, binned averages are shown as black triangles, and the rms scatter is
shown as black error bars. Contours in volume density are shown for 2 and
$20\times 10^{-5}$ haloes per 0.1 in log $j_{\Delta}$ and per 0.1 log $V$, per
Mpc3. The shapes of the distributions are similar for $j_{\Delta}$ whether
$V_{200}$ or $V_{20,000}$ is adopted. As $\Delta$ increases, the normalisation
of the relation between $j_{\Delta}$ and $V$ decreases, but the slope and
scatter do not change greatly. Similar relations are found for $V_{2000}$, but
are not shown to avoid redundancy.
To quantify the dependence of dark matter halo $j$ on how the outer radius of
a halo is defined, we look to a suite of cosmological N-body simulations of
dark matter haloes. These simulations include only dark matter and gravity
(i.e., neither baryons nor hydrodynamics). As discussed in the Introduction,
if the baryons in a given dark matter halo are initially distributed in the
same manner as the dark matter, and they later cool to form a disc while
conserving $j$, then the $j$ of the galaxy should be equal to that of the
virialized region of the dark matter halo. However, if baryons collapse
progressively from the inner to the outer parts of haloes, and they have not
finished collapsing (or, if some baryons never collapse), then galaxy $j$ may
be expected to reflect that of dark matter haloes within a given radius,
$R_{BC}$.
To predict the distribution of $j$ among dark matter haloes, a large N-body
simulation is needed which can model the acquisition of angular momentum for
even the slowest rotating galaxies in our sample (125 km s-1; §3). The
simulation we adopt is part of the Horizon Project suite (http://www.projet-
horizon.fr). This follows the evolution of a cubic cosmological volume of 100
$h^{-1}$ Mpc on a side (comoving) containing $\sim 134$ million dark matter
particles ($512^{3}$). It starts at $z=99$ and is evolved using the publicly
available treecode Gadget 2 (Springel, 2005) with a softening length of 5 h-1
kpc (co-moving). The adopted cosmology results in a dark matter particle mass
of $6.83\times 10^{8}M_{\odot}$. Dark matter haloes and the subhaloes they
contain are identified with the AdaptaHOP algorithm (Aubert et al., 2004). The
halo centres are positioned on the densest dark matter particle located in the
most massive substructure (see Tweed et al. 2009 for details). The total
number of haloes and subhaloes in the simulation volume at $z=0$ with more
than $100$ particles within $R_{200}$ and with circular velocities at this
radius which are greater than $100$ km s-1 is 9661.
The $j$ of a halo is measured within a range of radii as follows. First, the
halo is divided into 100 radial ellipsoidal shells, where the axis ratios of
the ellipsoid are obtained by computing the inertial tensor of all the
particles in the halo. Halo circular radii are defined as the cube root of the
radii of the three major axes of each ellipsoid. Next, the vector angular
momentum of the particles in each shell is calculated, and the angular momenta
of the shells is summed vectorially from the inner-most shell to the radii
specified before taking its modulus. The mass of a halo is measured in an
analogous manner, and $j$ is simply the angular momentum divided by the mass
within a given radius. Selected radii, $R_{\Delta}$, are defined by the
density of the haloes with respect to the critical density of the universe
($\Delta\equiv\bar{\rho}(r<R_{\Delta})/\rho_{\rm crit}$). Specific angular
momenta measured within these radii are defined as $j_{\Delta}$. Circular
velocities at these radii are $V_{\Delta}=(GM_{\Delta}/R_{\Delta})^{1/2}$,
where $M_{\Delta}$ and $R_{\Delta}$ are the mass and radius of the halo
defined by $\Delta$, and $G$ is the gravitational constant. The ranges of
$\Delta$, $R_{\Delta}/R_{200}$, and $M_{\Delta}/M_{200}$ probed are 50–20,000,
1.70–0.09, and 1.24–0.13, respectively.
Figure 2: These plots are the same as in Figure 1 for $V_{200}$, except here
observed galaxies are also shown ($V_{flat}$ of the galaxies is adopted as a
characteristic rotation velocity and is plotted on the horizontal axis).
Individual galaxies are plotted as green points, and contours in volume
density are shown in blue for 2 and $20\times 10^{-5}$ galaxies per 0.1 in log
$j_{\Delta}$ and per 0.1 in log $V_{200}$ per Mpc3. Binned averages for the
galaxies are shown as blue triangles, and the rms scatter is denoted by error
bars. The scatter in $j$ for galaxies is about half of that of the haloes.
Under the assumption that characteristic galaxy and halo rotation velocities
are equal (i.e., $V_{flat}=V_{200}$), galaxies have on average a factor of
$\sim 2$ less $j$ than haloes defined with $\Delta=200$, a factor of $\sim 2$
more $j$ than haloes defined with $\Delta=2000$, and a factor of $\sim 5.6$
more $j$ than haloes defined with $\Delta=20,000$.
In Figure 1, relations between halo $j_{\Delta}$ and $V_{\Delta}$ are
shown.111There is a drawback to a plot of $j$ versus $V$, namely both axes
incorporate factors of $V$, and a relation is expected by construction (e.g.,
Freeman, 1970). Because the local relation between galaxy $V$ and stellar mass
is tight (e.g., Bell & de Jong, 2001; Kassin, de Jong, & Weiner, 2006), there
is a similarly tight relation between between $j$ and stellar mass (e.g.,
Fall, 1983), which is not expected by construction. Halo $j$ is measured
within $R_{200},R_{2000}$, and $R_{20,000}$, and halo $V$ is measured at
$R_{200}$ and $R_{20,000}$. We do not show results for halo $V$ measured at
$R_{2000}$ since they do not differ significantly from those for $R_{200}$ or
$R_{20,000}$. The radii $R_{2000}$ and $R_{20,000}$ correspond to 34% and 9%
of $R_{200}$, respectively, on average Only haloes with more than 100
particles are retained, except for measurements of $j_{20,000}$ and
$V_{20,000}$ for which haloes with more than 50 particles are used. For these
50-particle haloes, the intrinsic relations remain the same, but the scatter
is increased slightly due to increased Poisson noise. The shapes of all the
distributions are similar in terms of slope and scatter, and are therefore
approximately independent of the radius for which $j$ or $V$ are measured. The
slope flattens slightly with increasing $\Delta$, and the scatter remains
about the same. We will quantify this in the following section. However, the
normalisation is strongly dependent on $\Delta$: it decreases by factors of
$\sim 3$ and $\sim 6$ for 10 and 100-fold increases in $\Delta$, respectively.
A decreasing normalisation with increasing $\Delta$ is a consequence of how
angular momentum is distributed in galactic haloes, with most of the angular
momentum located in the outer parts. As we increase $\Delta$, we exclude more
and more of the outer parts of the haloes, and the angular momenta decrease,
as illustrated by the simple analytic treatment in Fall (1983, Section 4). In
this paper, we quantify this decrease more precisely using numerical
simulations.
## 3 Comparison with observations of disc galaxies
Figure 3: The average difference between galaxy and halo log $j$, $<\rm log\
j_{\rm galaxies}$ \- log $j_{\Delta}>$, is shown as a function of $\Delta$,
$R_{\Delta}/R_{200}$, and $M_{\Delta}/M_{200}$, in panels a, b, and c,
respectively. Points demarcate discreet values, and solid lines simply connect
the points. There is no offset between galaxy and halo log $j$ for
$\Delta=578^{+34}_{-31}$, which corresponds to
$R_{BC}=R_{\Delta=578}/R_{200}=0.63^{+0.02}_{-0.01}$, and
$M_{\Delta=578}/M_{200}=0.74\pm 0.1$.
The goal of this section is to place measurements for disc galaxies on Figure
1. To do so, we need to (1) adopt a galaxy sample for which the completeness
is well-defined and which has the necessary data available to derive circular
velocities and $j$, and (2) relate galaxies to simulated host dark matter
haloes.
To address the first need, a large sample of 456 galaxies from Mathewson,
Ford, & Buchhorn (1992) and completeness measurements from de Jong & Lacey
(2000) are adopted. Details of this sample are given below. The large size of
and the data available for the sample necessitates simple estimates of $j$.
Therefore, we estimate $j$ as $2V_{\rm flat}r_{d}$, where $V_{\rm flat}$ is
the rotation velocity on the flat part of the rotation curve and $r_{d}$ is
the scale-length of the galaxy disc. This approximation is exact for an
exponential disc and a flat rotation curve. Uncertainties in estimates of $j$
are $\sim 15$%, which are dominated by errors in measurements of $r_{s}$
(mainly due to errors in sky background subtraction) and galaxy distances.
There are two minor effects on estimates of $j$, which we do not take into
account, but which work in opposite directions. On the one hand, galaxies have
rising rotation curves in their centres, and this causes the formula to
slightly overestimate $j$. On the other hand, most galaxies are expected to
have extended gas discs, but with very little mass, which would cause the
formula to slightly underestimate $j$.
The galaxy sample used is a sub-sample of the ESO-Uppsala Catalog of Galaxies
(Lauberts, 1982) which was selected by eye from photographic plates. It is
only incomplete for very late Hubble types (T $>6$, i.e., later than Scd; de
Jong & Lacey 2000). Values of $V_{\rm flat}$ were determined from optical and
radio observations. For the optical data, $V_{\rm flat}$ was defined as half
the difference between the maximum and minimum velocities of the H$\alpha$
rotation curves. For the radio data, $V_{\rm flat}$ was defined as half the
width of the HI profile between points where the intensity falls to 50% of the
highest values; these values were then corrected for dispersion and converted
to optical rotation velocities by multiplying by 1.03 and then subtracting 11
km s-1 (see §3.4 and Figure 5 of Mathewson, Ford, & Buchhorn, 1992). Disc half
light radii, which are the result of $I$-band bulge-disc decompositions from
de Jong & Lacey (2000), are converted to disc scale lengths by dividing by
1.679 (the exact ratio of the half-mass radius to the scale radius for a pure
exponential disc). Only those galaxies with rotation velocities greater than
125 km s-1 are used. This helps us to avoid galaxies with rotation curves
which do not flatten out at the radii measured. The distribution of galaxies
in $j$ versus $V_{\rm flat}$ does not differ significantly from the galaxy
sample commonly used in the literature (Courteau et al., 2007), but it has a
better completeness.
To address the second need, and relate galaxies to the dark matter haloes in
Figure 1, we assume for simplicity that the characteristic rotation velocity
of a galaxy (which we take to be $V_{flat}$) and that of its host halo at
$R_{200}$ are equal. For a massless disc in a Navarro, Frenk, & White (1996)
halo, $V_{c}$ at the location of the galaxy can be about half its value at
$R_{200}$. However, the self-gravity of the baryons is expected to increase
$V_{c}$ in the inner parts of haloes. The amount by which it increases is
difficult to calculate theoretically, so we look to observations. Dutton et
al. (2010) find a very small conversion factor between $V_{c}$ at $R_{200}$
and at the location of galaxy discs. In their analysis, Dutton et al. (2010)
combined dark halo masses measured from satellite kinematics and weak
gravitational lensing to show that $V_{2.2}\simeq V_{200}$ for
$V_{2.2}=90-260$ km s-1, where $V_{2.2}$ is the galaxy rotation velocity
measured at 2.2 $I$-band scale lengths. This equivalence is also consistent
with semi-analytic models of galaxy formation which require a similar ratio
between galaxy and halo velocities to simultaneously match the local Tully-
Fisher relation and galaxy luminosity function (e.g., Dutton & van den Bosch,
2009, and references therein).
In Figure 2, we compare the distribution of $j$ versus $V_{\rm flat}$ for
galaxies described in this section with the distributions of $j_{200}$,
$j_{2000}$, and $j_{20,000}$ versus $V_{200}$ for dark matter haloes from
Figure 1. As discussed above, it is assumed that haloes have the same rotation
velocities as the galaxies they host, so they can be directly compared in
Figure 2. The halo relations from Figure 1 for $V_{20,000}$ are not shown
because they are not significantly different from those for $V_{200}$. We fit
a linear relation to the galaxies using 100 bootstrap re-samplings and a
generalised least squares fitting routine (Weiner et al., 2006b), which gives
a slope of $2.5\pm 0.1$ rms. We also fit a linear relation to the haloes in
Figure 2 for $j_{200}$ versus $V_{200}$ for circular velocities which span the
velocity range of the galaxies, $125<V_{200}<315$ km s-1. This results in a
slope of $1.92\pm 0.02$ rms. The distribution of galaxies has a similar slope
to that of the haloes, as found by Fall (1983) and others (e.g., Mo, Mao, &
White, 1998; Navarro & Steinmetz, 2000), and approximately half the average
rms scatter (0.15 dex versus 0.27 dex). The lower scatter compared to the
haloes is related to the finding by de Jong & Lacey (2000) that the width of
the observed scale-radius distribution of galactic discs is narrower than that
expected from the distributions of halo spin parameters in cosmological
simulations. For the halo $j_{2000}$ versus $V_{200}$ and $j_{20,000}$ versus
$V_{200}$ relations, the slopes are $1.80\pm 0.02$ and $1.26\pm 0.02$,
respectively, and the average rms scatters are 0.28 and 0.26, respectively.
The slopes flatten slightly with increasing $\Delta$, but the scatter remains
constant to within errors. Given all the factors not included in our simple
picture, we consider it remarkable how similar the galaxy and halo slopes are.
The main result of this paper is encapsulated in the much larger difference in
normalisation between galaxies and haloes. We choose to measure this
difference at approximately the center of the distributions, at log
$V_{rot}$=2.35 ($V_{rot}=224$ km s-1). The average normalisation of the
galaxies is less than that of the haloes for $j_{200}$ by a factor of $\sim 2$
(0.30 dex), consistent with previous studies (e.g., Navarro & Steinmetz, 2000;
Dutton & van den Bosch, 2012). The average normalisation of the galaxies is
greater than that of the haloes for $j_{2000}$ and $j_{20,000}$ by factors of
$\sim 2$ (0.30 dex) and $\sim 5.6$ (0.75 dex), respectively.
We quantify the dependence of $j_{\Delta}$ on $\Delta$ as follows. We start by
measuring halo $j$ for a range of $\Delta$ and compare them with $j$ measured
for the galaxy sample, as in Figure 2. In Figure 3, we show the average
difference between galaxy and halo $j$ (measured at log $V_{rot}=2.35$) as a
function of $\Delta$, halo outer radius in terms of $R_{200}$ (i.e.,
$R_{\Delta}/R_{200}$), and halo mass in terms of $M_{200}$ (i.e.,
$M_{\Delta}/M_{200}$). The quantities $R_{\Delta}/R_{200}$ and
$M_{\Delta}/M_{200}$ are average values for all haloes with circular
velocities which span $125<V_{200}<315$ km s-1. The value of $\Delta$ at which
the average $j$ of galaxies and haloes match (i.e., $<{\rm log}\ j_{\rm
galaxies}-{\rm log}\ j_{\Delta}>\ =0$) is $578^{+34}_{-31}$. The halo
$j_{578}$ versus $V_{200}$ relation has a slope of $1.89\pm 0.03$ and an
average rms scatter of 0.28 dex over the velocity range of the galaxies. This
value of $\Delta$ corresponds to
$R_{BC}=R_{\Delta=578}/R_{200}=0.63^{+0.2}_{-0.1}$ and
$M_{\Delta=578}/M_{200}=0.74\pm 0.1$. We calculate these values by
interpolating the curves in Figure 3, and the errors by considering the
limiting case that galaxy $j$ values are systematically over and under-
estimated by the assumed measurement uncertainty. If baryons conserved $j$
perfectly during galaxy formation, then the collapse radius $R_{BC}$ is
$\simeq 63$% of the virial radius $R_{200}$. This portion of the haloes
contains on average 74% of their mass, and if baryons and dark matter are
initially well-mixed, the same percentage of the baryons. However, as
discussed in the next section, this radius and mass fraction are probably not
simple step functions; therefore we regard them as “effective” quantities.
## 4 Discussion
In this paper, we determine the extent to which the approximate factor of 2
discrepancy between the $j$ of galaxies and their expected host dark matter
haloes is sensitive to the conventional assumption that $R_{BC}=R_{200}$. This
difference in $j$ is usually attributed to loss of baryonic $j$ during galaxy
formation. However, there is no physical reason for the assumption that these
radii are equal to at least within a factor of $\sim 2$, as emphasized by Fall
(2002). This is because different physics governs each, namely dissipational
and dissipationless physics for $R_{BC}$ and $R_{Vir}$, respectively. The only
constraint on the relationship between these radii is that $R_{BC}$ must be
interior to $R_{Vir}$ since baryons cannot collapse from a region that is not
incorporated into the halo. A $R_{BC}$ which is interior to $R_{Vir}$ is a
natural expectation in the standard theory of galaxy formation where the inner
parts of haloes collapse first. As $R_{BC}$ decreases, the discrepancy between
the $j$ of galaxies and haloes is alleviated. We show that the discrepancy can
be explained entirely by a $R_{BC}$ which is $\sim 60$% of $R_{Vir}$.
To do so, we determine the value of $R_{BC}$ at which the $j$ of galaxies and
haloes match. This is done by comparing the distribution of $j$ observed for a
sample of local disc galaxies, for which the completeness is understood, to
that predicted for their host dark matter haloes from a dark matter-only
simulation of the Universe. It is assumed that galaxies and haloes can be
related directly via their rotation velocities. The necessary value of the
density contrast $\Delta$ needed to define the haloes which have the same
average $j$ as galaxies is $\sim 600$. This corresponds to an average
effective $R_{BC}$ which is $\sim 60$% of $R_{200}$, and an average halo mass
which is $\sim 75$% of $M_{200}$. Therefore, if galaxies formed from baryons
initially present in the inner parts of their host haloes and conserved $j$
perfectly, the baryons would come from within $R_{BC}$ and would comprise this
percentage of the baryons in the halo.
Even under the assumption of perfect conservation of $j$, $R_{BC}$ is not
likely a sharp boundary. The baryons which form the galaxy may only on average
come from within $R_{BC}$, with most material originating from smaller radii,
but some from more distant radii. In addition, the smaller scatter of the
galaxies in $j$ versus $V$ compared to that of the haloes may indicate a
mechanism by which only selected baryons form the disc, regulatory processes
which act upon the baryons, and/or haloes which form non-disc galaxies. This
is because, in our simple picture, the initial distribution of baryons in $j$
versus $V$ is expected to mirror that of the dark matter. Therefore, if only
selected baryons formed discs or regulatory processes acted upon them during
disc formation, it may be expected that the baryons which form the discs would
have a narrower distribution in $j$ versus $V$. In addition, since we compare
the predicted properties of dark matter haloes with those of disc galaxies,
not ellipicals which rotate slower than discs, it stands to reason that the
combined population of discs and ellipticals would be broader in $j$ versus
$V$ (Fall, 1983).
Eventually, it should be possible to compute $R_{BC}$ from hydrodynamical and
dark matter simulations of galaxy formation in a cosmological context. Current
simulations may have spatial and mass resolutions that are too coarse to model
accurately the complex processes expected to be at play, such as gas shocks,
cloud-cloud collisions, and a multiphase medium. These processes affect the
rate at which the baryons collapse, but they have may relatively little
influence on the angular momentum of the resulting galactic discs.
A number of phenomena can alter the $j$ of galaxies (see Fall 2002 and
Romanowsky & Fall 2012 for more complete discussions of these phenomena). For
example, torques exerted between the dark matter and the baryons could in
principle spin up the halo and spin down the disc. Minor mergers might also
affect the $j$ of galaxies. In addition, feedback from star-formation can
alter $j$ differently depending on how it varies with radius. Material in
outflows may be launched from inner or outer radii, or both. If material is
primarily removed from the inner or outer parts of galaxies, galaxy $j$ will
increase or decrease, respectively. If feedback is active but independent of
radius, then there would be no change in $j$. We expect some of these
phenomena to alter the $j$ of discs, but whether they have a major or a minor
effect on galaxy $j$ is still uncertain. In order to perform a more detailed
comparison of galaxies and haloes, we need a better understanding of the
processes of $j$ transfer in galaxy formation, and whether outflows can change
the $j$ of galaxies.
In summary, the difference between the predicted initial and the observed
final $j$ of galaxies, which is conventionally attributed solely to angular
momentum loss, hinges on the loosely motivated assumption that all the baryons
within $R_{Vir}$ collapse to form galaxies. There is no physical reason why
this has to be the case. If baryons in the inner parts of haloes collapse
first, as is expected, then the $j$ discrepancy between galaxies and haloes
can be fully explained by a collapse radius $R_{BC}$ which is $\sim 60$% of
the virial radius $R_{Vir}$. In the future, baryons from progressively larger
radii in the halo may collapse, and at some point in time $R_{BC}$ might equal
$R_{Vir}$. In reality, it may be that a combination of a preference of
collapse of the inner parts and some $j$ transfer between baryons and dark
matter is needed to solve the problem.
## Acknowledgments
This research was supported by an appointment to the NASA Postdoctoral Program
at NASA’s Goddard Space Flight Center, administered by Oak Ridge Associated
Universities through a contract with NASA. The research of JD is partly
supported by Adrian Beecroft, the Oxford Martin School and STFC. This work was
performed using code and simulations from the Horizon collaboration
(http://www.projet-horizon.fr). S.A.K. and J. D. are grateful to David S.
Graff. We thank Aaron Dutton and Aaron Romanowski for helpful comments on
drafts of this paper.
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|
arxiv-papers
| 2012-05-01T20:02:01 |
2024-09-04T02:49:30.429274
|
{
"license": "Public Domain",
"authors": "Susan A. Kassin, Julien Devriendt, S. Michael Fall, Roelof S. de Jong,\n Brandon Allgood, Joel R. Primack",
"submitter": "Susan Kassin",
"url": "https://arxiv.org/abs/1205.0253"
}
|
1205.0410
|
Mass spectra in $\mathbf{{\cal N}=1}$ SQCD with additional fields. I
Victor L. Chernyak
(e-mail: v.l.chernyak@inp.nsk.su)
Budker Institut of Nuclear Physics SB RAS and Novosibirsk State University,
630090 Novosibirsk, Russia
Considered is the ${\cal N}=1$ SQCD-like theory with $SU(N_c)$ colors and $0< N_F<2N_c$ flavors of light quarks $\,Q_i,{\ov Q}_j$ and with the additional $N^2_F$ colorless flavored fields $\Phi_{ij}$ with the large mass parameter $\mph\gg\la$. The mass spectra of this $\Phi$ - theory (and its dual variant, the $d\Phi$ - theory) are calculated at different values of $\mph/\la$ within the dynamical scenario which implies the (quasi)spontaneous breaking of chiral symmetry. It is shown that, under appropriate conditions, the seemingly heavy and dynamically irrelevant fields $\Phi$ `return back' and there appear two additional generations of light $\Phi$ - particles with small masses $\mu(\Phi)\ll\la$.
Also considered is the $X$ - theory which is the ${\cal N}=2$ SQCD with $SU(N_c)$ colors and $0< N_F<2N_c$ flavors of light quarks, broken down to ${\cal N}=1$ by the large mass parameter of the adjoint scalar field $X$, $\mu_X\gg\Lambda_2$. The tight interrelations between these $X$ and $\Phi$ - theories are described, in particular, the conditions under which they are equivalent.
1 Definitions
1.1 Direct $\mathbf{\Phi}$ - theory 3
1.2 Dual $\,\mathbf{d\Phi}$ - theory 4
2 Mass spectra at $\mathbf{N_F<N_c}$
2.1 Unbroken flavor symmetry 5
2.2 Broken flavor symmetry : $\mathbf{U(N_F)\ra U(n_1)\times U(n_2)}$ 7
3 Quark condensates and multiplicity of vacua at $\mathbf{N_c<N_F<2N_c}$ 11
4 Fions : one or three generations 14
5 Direct theory. Unbroken flavor symmetry
5.1 $\mathbf L$ - vacua 17
5.2 $\mathbf S$ - vacua 19
6 Dual theory. Unbroken flavor symmetry
6.1 $\mathbf L$ - vacua 20
6.2 $\mathbf S$ - vacua 23
7 Direct theory. Broken flavor symmetry. The region $\mathbf{\la\ll\mph\ll\mo}$
7.1 $\mathbf L$ - type vacua 25
7.2 ${\rm\bf br2}$ vacua 25
7.3 special vacua, $\mathbf{n_1=\nd,\,n_2=N_c}$ 29
8 Direct theory. Broken flavor symmetry. The region $\mathbf{\mo\ll\mph\ll\la^2/m_Q}$
8.1 $\rm\bf br1$ vacua, $DC_1-DC_2$ phase 30
8.2 $\rm\bf br1$ vacua, $Higgs_1-DC_2$ phase 32
8.3 $\rm\bf br1$ vacua, $Higgs_1-HQ_2$ phase 33
8.4 $\rm \bf br2$ and special vacua 34
9 Dual theory. Broken flavor symmetry. The region $\mathbf{\la\ll\mph\ll\mo}$
9.1 $\mathbf L$ - type vacua 34
9.2 ${\rm\bf br2}$ vacua 35
9.3 special vacua, $\mathbf{n_1=\nd,\, n_2=N_c}$ 38
10 Dual theory. Broken flavor symmetry. The region $\mathbf{\mo\ll\mph\ll\la^2/m_Q}$
10.1 $\rm\bf br1$ vacua, $DC_1-DC_2$ phase 40
10.2 $\rm\bf br1$ vacua, $HQ_1-DC_2$ phase 41
10.3 $\rm \bf br2$ and special vacua 43
11 Broken $\mathbf{{\cal N}=2}$ SQCD 43
12 Conclusions 46
Appendix A The RG flow in the $\mathbf \Phi$ - theory at $\mathbf{\mu>\la}$ 47
Appendix B There is no vacua with $\mathbf{\langle S\rangle=0}$ at $\mathbf{m_Q\neq 0}$
References 52
§ DEFINITIONS AND SOME GENERALITIES
1.1. Direct $\mathbf \Phi$ - theory
The field content of this direct ${\cal N}=1\,\,\,\Phi$ - theory includes $SU(N_c)$ gluons and $0< N_F<2N_c$ flavors of quarks ${\ov Q}_j, Q_i$. Besides, there is $N^2_F$ colorless but flavored fields $\Phi_{ji}$ (fions).
The Lagrangian at scales $\mu\gg\la$ (or at $\mu\gg\mu_H$ if $\mu_H\gg\la$, where $\mu_H$ is the next largest physical mass below $\mu^{\rm pole}_1(\Phi)\gg\la$, see the Appendix A ; $\nd=N_F-N_c$ , the exponents with gluons in the Kahler term K are implied here and everywhere below) looks as
K=1/f^2Tr (Φ^†Φ)+z(,μ)Tr( Q^†Q+(QQ) ) ,
W=-2π/α(μ,) S+W_Φ+W_Q ,
W_Φ=/2[Tr (Φ^2)-1/(Tr Φ)^2], W_Q=Tr Q(m_Q-Φ) Q, z_Q(,μ)∼(lnμ/)^N_c/≫1 .Here : $\mph$ and $m_Q$ are the mass parameters, $S=-W^{a}_{\beta}W^{a,\,\beta}/32\pi^2$ where $W^a_{\beta}$ is the gauge field strength, $a=1...N_c^2-1,\, \beta=1,2$, $\alpha(\mu,\la)=g^2(\mu,\la)/4\pi$ is the gauge coupling with its scale factor $\la$, $f$ is the Yukawa coupling, $a_f=N_cf^2/8\pi^2< 1$. This normalization of fields is used everywhere below in the main text. Besides, the perturbative NSVZ $\beta$-function for massless SUSY theories [1, 2] is used in this paper.
Therefore, finally, the $\Phi$-theory we deal with has the parameters : $N_c\,,\,0<N_F<2N_c\,,\,\mph$,
$\la,\, m_Q,\, f$, with the strong hierarchies $\mph\gg\la\gg m_Q$. Everywhere below in the text the mass parameter $\mph$ will be varied while $m_Q$ and $\la$ will stay intact.
The Konishi anomalies [3] for the $i$-th flavor look as (${\it i}=1\, ...\, N_F$)
⟨Φ_i⟩⟨∂W_Φ/∂Φ_i⟩=0 , ⟨m_Q,i^tot⟩⟨Q_i Q_i⟩=⟨S⟩ , ⟨m_Q, i^tot⟩=m_Q-⟨Φ_i⟩ , ⟨Φ_ij⟩=1/( ⟨Q_j Q_i ⟩-δ_ji1/N_cTr ⟨⟩) , ⟨Q_j Q_i ⟩=δ_ji⟨Q_i Q_i ⟩ ,
and, in cases with $\mu_H<\la$, $\langle m_{Q,l}^{\rm tot}\rangle$ is the value of the quark running mass at $\mu=\la$.
At all scales until the field $\Phi$ remains too heavy and non-dynamical, i.e. until its perturbative running mass $\mu_{\Phi}^{\rm pert}(\mu)>\mu$, it can be integrated out and the Lagrangian takes the form
K=z_Q(,μ)Tr(Q^†Q+QQ), W=-2π/α(μ,)S+W_Q , W_Q=m_QTr(Q Q)-1/2(Tr (QQ)^2-1/N_c(Tr Q Q )^2 ).
The Konishi anomalies for the Lagrangian (1.3) look as
⟨S⟩=⟨Q_i∂W_Q/∂Q_i⟩=m_Q⟨Q_i Q_i ⟩-
1/(∑_j⟨Q_i Q_j⟩⟨Q_j Q_i⟩-1/N_c⟨Q_i Q_i⟩⟨Tr Q Q ⟩)==⟨Q_i Q_i ⟩[ m_Q-1/( ⟨Q_i Q_i ⟩-1/N_c
⟨Tr Q Q ⟩) ] , i=1 ... N_F , ⟨S⟩=⟨λλ/32π^2⟩ ,
0=⟨Q_i∂W_Q/∂Q_i-Q_j∂W_Q/∂Q_j⟩=⟨Q_i Q_i -Q_j Q_j⟩[ m_Q-1/( ⟨Q_i Q_i+Q_j Q_j⟩-1/N_c⟨Tr Q Q ⟩) ] .It is most easily seen from (1.4) that there are only two types of vacua : a) the vacua with the unbroken flavor symmetry, b) the vacua with the spontaneously broken flavor symmetry, and the breaking is of the type $U(N_F)\ra U(n_1)\times U(n_2)$ only. In these vacua one obtains from (1.4)
⟨Q_1 Q_1+Q_2 Q_2-1/N_cTr Q Q⟩_br=m_Q, ⟨S⟩_br=1/⟨Q_1 Q_1 ⟩_br⟨Q_2 Q_2 ⟩_br, ⟨Q_1 Q_1 ⟩_br≠⟨Q_2 Q_2 ⟩_br ,⟨m^tot_Q,1⟩_br=m_Q-⟨Φ_1⟩_br=⟨Q_2 Q_2⟩_br/, ⟨m^tot_Q,2⟩_br=m_Q-⟨Φ_2⟩_br=⟨Q_1 Q_1⟩_br/ .
1.2. Dual $\mathbf {d\Phi}$ - theory
In parallel with the direct $\Phi$ - theory with $N_c<N_F<2N_c$ , we consider also the Seiberg dual variant [4, 5] (the $d\Phi$ - theory), with the dual Lagrangian at $\mu=\la$
K=1/f^2Tr Φ^†Φ+ Tr( q^†q + (qq) )+Tr M^†M/μ_2^2 , W= - 2π/α(μ=) s+W_M+W_q ,W_M=/2[Tr (Φ^2)-1/(Tr Φ)^2]+ Tr
M(m_Q-Φ), W_q= - 1/μ_1 Tr (q M q ) .
Here : the number of dual colors is ${\ov N}_c=(N_F-N_c)$ and $M_{ij}$ are the $N_F^2$ elementary mion fields, ${\ov a}(\mu)=\nd{\ov \alpha}(\mu)/2\pi=\nd{\ov g}^2(\mu)/8\pi^2$ is the dual running gauge coupling (with its scale parameter $\Lambda_q$), ${\ov s}=-{\rm \ov w}^{b}_{\beta}{\rm \ov w}^{b,\,\beta}/32\pi^2$, ${\rm \ov w}^b_{\beta}$ is the dual gluon field strength. The gluino condensates of the direct and dual theories are matched, $\langle{-\,\ov s}\rangle=\langle S\rangle=\lym^3$, as well as $\langle M_{ji}(\mu=\la)\rangle=\langle{\ov Q}_j Q_i (\mu=\la)\rangle$, and the scale parameter $\Lambda_q$ of the dual gauge coupling is taken as $|\Lambda_q|\sim\la$, see appendix in [7]. At $3/2<N_F/N_c<2$ this dual theory can be taken as UV free at $\mu\gg\la$, and this requires that its Yukawa coupling at
$\mu=\la,\, f(\mu=\la)=\mu_2/\mu_1$, cannot be larger than its gauge coupling ${\ov g}(\mu=\la)$, i.e. $\mu_2/\mu_1=O(1)$. The same requirement to the value of the Yukawa coupling follows from the conformal behavior of this theory at $3/2<N_F/N_c<2$ and $\mu\ll\la$, i.e. $f(\mu=\la)\simeq f_*=O(1)$. We consider below this dual theory at $\mu\leq\la$ only, where it claims to be equivalent to the direct $\Phi$ - theory. As was explained in [7], one has to take $\mu_1\sim\la$ at $\bd/\nd=O(1)$ in (1.6) to match the gluino condensates in the direct and dual theories. Therefore, $\mu_2\sim\mu_1\sim\la$ also.
Really, the fields $\Phi$ remain always too heavy and dynamically irrelevant in this $d\Phi$ - theory, so that they can be integrated out once and forever and, finally, we write the Lagrangian of the dual theory at $\mu=\la$ in the form
K= Tr( q^†q +(qq) )+Tr M^†M/^2 , W= - 2π/α(μ=) s+W_M+W_q ,W_M=m_QTr M -1/2[Tr (M^2)- 1/N_c(Tr M)^2 ] , W_q= - 1/ Tr (q M q ) .
The Konishi anomalies for the $i$-th flavor look here as (${\it i}=1\, ...\, N_F$)
⟨M_i⟩(⟨N_i⟩≡⟨q_i q_i(μ=)⟩)=⟨S⟩ ,
⟨N_i⟩/=m_Q-1/(⟨M_i-1/N_cTr M ⟩)=
⟨m_Q,i^tot⟩ .
In vacua with the broken flavor symmetry these can be rewritten as
⟨M_1+M_2-1/N_cTr M⟩_br=m_Q, ⟨S⟩_br=1/⟨M_1⟩_br⟨M_2⟩_br, ⟨M_1⟩_br≠⟨M_2⟩_br ,⟨N_1⟩_br/=⟨S⟩_br/⟨M_1⟩_br=⟨M_2⟩_br/=m_Q-1/(⟨M_1-1/N_cTr M⟩_br )=⟨m^tot_Q,1⟩_br ,
⟨N_2⟩_br/=⟨S⟩_br/⟨M_2⟩_br=⟨M_1⟩_br/=m_Q-1/(⟨M_2-1/N_cTr M⟩_br )=⟨m^tot_Q,1⟩_br .
Our purpose in this paper is to calculate the mass spectra in the two above theories, $\mathbf\Phi$ and $\mathbf{d\Phi}$. At present, to calculate the mass spectra in ${\cal N}=1$ SQCD-like theories, one has to rely on a definite dynamical scenario. Two different scenarios have been considered in [6, 7, 8] and the mass spectra were calculated in the standard direct ${\cal N}=1$ SQCD with the superpotential $W={\rm Tr}\,(\,{\ov Q}m_Q Q)$ and in its dual variant [4, 5]. It was shown that the direct theory and its Seiberg dual variant are not equivalent in both scenarios. In this paper we calculate the mass spectra in the $\mathbf\Phi$ and $\mathbf{d\Phi}$ theories within the scenario $\#1$.
We recall, see [6, 7], that this scenario implies that, at the appropriate conditions, the quarks are not higgsed ( i.e. $\langle{\ov Q}\rangle=\langle Q\rangle=0$ ) but form the coherent colorless diquark condensate (DC) and acquire the non-perturbative dynamical mass $\mc^2=\langle{\ov Q}Q\rangle$, and there appear light pseudo-Goldstone bosons $\Pi$ (pions) with masses $\mu(\Pi)\ll \mc$ .
§ MASS SPECTRA AT $N_F<N_C-1$
2.1. Unbroken flavor symmetry
There is $N_{\rm unbrok}=(2N_c-N_F)$ such vacua and all quarks are higgsed in all of them, but the hierarchies in the mass spectrum are parametrically different depending on the value of $\mph$ (see below). In any case, all $N_F^2$ fions are very heavy and dynamically irrelevant in these vacua at scales $\mu<\mu^{\rm pole}_{1}(\Phi)$ (see the Appendix A) and can be integrated out from the beginning.
All quarks are higgsed at the high scale $\mu=\mu_{\rm gl},\, \la\ll\mu_{\rm gl}\ll\mu^{\rm pole}_1(\Phi)$,
μ^2_gl=N_c g^2(μ=μ_gl)z_Q(,μ_gl)⟨Π⟩, ⟨Π⟩=⟨QQ(μ=)⟩, g^2=4πα,
where (in the approximation of leading logs, $C_F=(N_c^2-1)/2N_c\simeq N_c/2$)
2π/α(μ_gl)≃lnμ_gl/ , z_Q(,μ_gl)∼(α()/α(μ_gl))^2C_F/∼(lnμ_gl/)^N_c/≫1 ,
=3N_c-N_F .
Hence, after integrating out all heavy higgsed gluons and their superpartners at $\mu<\mu_{\rm gl}$ one remains with the $SU(N_c-N_F)$ pure Yang-Mills theory. Finally, after integrating out remained gluons at $\mu<\lym$ via the Veneziano-Yankielowicz (VY) procedure [9, 10] (see section 2 in [6] for more details), one obtains the Lagrangian of $N_F^2$ pions
K=z_Q(,μ_gl)2Tr √(Π^†Π) , W=-S+W_Π ,
S=( ^/Π )^1/N_c-N_F , W_Π=m_QTr Π-1/2[Tr (Π^2)- 1/N_c(Tr Π)^2 ],⟨Π_ij⟩=δ_ij ⟨Π⟩=δ_ij ⟨Q_1 Q_1(μ=)⟩, i,j=1 ... N_F .
It follows from (2.3) that depending on the value of $\mph/\la \gg 1$ there are two different regimes.
i) At $\la\ll\mph \ll \mo$ the term $m_Q{\rm Tr}({\ov Q}Q)$ in the superpotential (2.3) gives only a small correction and one obtains
⟨Π⟩_o∼^2(/)^N_c-N_F/2N_c-N_F≫^2 .
There are $(2N_c-N_F)$ such vacua, this agrees with [11].
To see that there are just $2N_c-N_F$ vacua and not less, one has to separate slightly all quark masses, $m_Q\ra m_Q^{(i)},\,i=1...N_F,\, 0<(\delta m_Q)^{ij}=(m_Q^{(i)}-m_Q^{(j)})\ll {\ov m}_Q$. All quark mass terms give only small power corrections to (2.4), but just these corrections show the $Z_{2N_c-N_F}$ multiplicity of vacua.
The masses of heavy gluons and their superpartners are given in (2.1) while from (2.3) the pion masses are
μ_o(Π)∼⟨Π⟩_o/z_Q(,μ_gl)∼/z_Q(,μ_gl)(/)^N_c/2N_c-N_F≫m_Q .
Besides, the scale of the gluino condensate of $SU(N_c-N_F)$ is
=⟨S⟩^1/3∼(^/⟨Π⟩_o)^1/3(N_c-N_F)∼(/)^N_F/3(2N_c-N_F) , μ_o(Π)≪≪≪μ_gl ,
and there is a large number of gluonia with the mass scale $\sim \lym$ (except for the case $N_F=N_c-1$ when the whole gauge group is higgsed and the non-perturbative superpotential in (2.3) originates from the instanton contribution).
ii) $(2N_c-N_F)$ vacua split into two groups of vacua with parametrically different mass spectra at $\mph\gg \mo$. There are $N_c$ SQCD vacua with $\langle \Pi \rangle_{\rm SQCD}\sim\la^2(\la/m_Q)^{(N_c-N_F)/N_c}$ differing by $Z_{N_c}$ phases (in these, the last term $\sim \Pi^2/\mph$ in the superpotential (2.3) can be neglected), and $(N_c-N_F)$ of nearly degenerate classical vacua with parametrically larger condensates $\langle \Pi \rangle_{\rm cl}\sim m_Q\mph$ (in these, the first non-perturbative quantum term $\sim S$ in the superpotential (2.3) gives only small corrections with $Z_{N_c-N_F}$ phases, but the multiplicity of vacua originates just from these small corrections). The properties of SQCD vacua have been described in detail in chapter 2 of [6], the pion masses are $\mu_{\rm SQCD}(\Pi)\sim m_Q/z_Q(\la,\mu^{SQCD}_{\rm gl})$ therein. In $(N_c-N_F)$ classical vacua the gluon and pion masses are given in (2.1) and (2.5) but now
⟨Π⟩_cl∼m_Q≫^2 , μ_cl(Π)∼m_Q/z_Q(,μ^cl_gl) ,
and in all vacua (except for the case $N_F=N_c-1$ ) there is a large number of gluonia with the mass scale
∼=⟨S ⟩^1/3∼(^/⟨Π⟩_SQCD)^1/3(N_c-N_F)∼(m_Q/)^N_F/3N_c in N_c SQCD vacua ,
∼∼(^/⟨Π⟩_cl)^1/3(N_c-N_F)∼(^2/m_Q)^N_F/3(N_c-N_F) in (N_c-N_F) classical vacua.
Finally, the change of regimes ${\bf i}\leftrightarrow {\bf ii}$ occurs at
(/)^N_c-N_F/2N_c-N_F∼m_Q /^2≫1 ∼(/m_Q)^2N_c-N_F/N_c .
2.2 Broken flavor symmetry : $U(N_F)\ra U(n_1)\times U(n_2)$
The quark condensates $\langle{\ov Q}_j Q_i\rangle\sim C_i\delta_{ij}$ split into two groups in these vacua with the spontaneously broken flavor symmetry : there are $1\leq n_1\leq [N_F/2]$ equal values $\langle\Pi_1\rangle=\langle{\ov Q}_1 Q_1\rangle$ and $n_2=(N_F-n_1)\geq n_1$ equal values $\langle\Pi_2\rangle=\langle{\ov Q}_2 Q_2\rangle\neq \langle{\ov Q}_1 Q_1\rangle$ (unless stated explicitly, here and everywhere below in the text it is implied that $1-(n_1/N_c),\,\, 1-(n_2/N_c)$ and $(2N_c-N_F)/N_c$ are all $O(1)$ ). And there will be two different phases, depending on the value of $\mph/\la \gg 1$ (see below).
2.2.1 At $\la\ll\mph\ll\mo$ all qualitative properties are similar to those for an unbroken symmetry. All quarks are higgsed at high scales $\mu_{\rm gl, 1}\sim \mu_{\rm gl, 2}\gg \la$ and the low energy Lagrangian has the form (2.3). The term $m_Q{\rm Tr} ({\ov Q}Q)$ in the superpotential in (2.3) gives only small corrections, while (1.5) can be rewritten here in the form
⟨Π_1+Π_2⟩_br=1/N_cTr ⟨Π⟩_br+m_Q≃1/N_c⟨n_1Π_1+n_2Π_2⟩_br (1-n_1/N_c)⟨Π_1⟩_br≃-(1-n_2/N_c)⟨Π_2⟩_br,⟨S⟩_br=(^/⟨Π_1⟩^n_1_br⟨Π_2⟩^n_2_br)^
1/N_c-N_F=⟨Π_1⟩_br⟨Π_2⟩_br/ ,
μ^2_gl, 1∼μ^2_gl, 2∼g^2(μ=μ_gl)z_Q(,μ_gl)
⟨Π_1 ⟩_br∼⟨Π_2⟩_br∼^2(/
)^N_c-N_F/2N_c-N_F .
The pion masses in this regime look as follows, see (2.3) : a) due to the spontaneous breaking of the flavor symmetry, $U(N_F)\ra U(n_1)\times U(n_2)$, there always will be $2n_1 n_2$ exactly massless particles and in this case these are the hybrids $\Pi_{12}$ and $\Pi_{21}$; b) other $n_1^2+n_2^2$ `normal' pions have masses as in (2.5).
There are
∑_n_1=1^n_1=[N_F/2](2N_c-N_F)C^ n_1_N_F , C^ n_1_N_F=N_F!/n_1! n_2!
such vacua (the factor $2N_c-N_F$ originates from $Z_{2N_c-N_F}$ (see the footnote 1) , for even $N_F$ the last term with $n_1=N_F/2$ enters (2.13) with the additional factor $1/2$, i.e. ${\ov C}^{\, n_1}_{N_F}$ differ from the standard $C^{\,n_1}_{N_F}$ in (2.13) only by ${\ov C}^{\,n_1={\rm k}}_{N_F=2{\rm k}}=C^{\,n_1={\rm k}}_{N_F=2{\rm k}}/2$ ), so that the total number of vacua
By convention, we ignore the continuous multiplicity of vacua due to the spontaneous flavor symmetry breaking. Another way, one can separate slightly all quark masses (see the footnote 1), so that all Nambu-Goldstone bosons will acquire small masses $O(\delta m_Q)\ll {\ov m}_Q$.
N_tot=( N_unbrok=2N_c-N_F)+N^tot_brok ,
this agrees with [11].
2.2.2 The change of the regime in these vacua with broken symmetry occurs at $\mo\ll\mph\ll{\tilde\mu}_{\Phi}$ , see (2.10),(2.20), when all quarks are still higgsed but there appears
a large hierarchy between the values of quark condensates at $\mph\gg\mo$ , see (1.5). Instead of $\langle \Pi_1 \rangle\sim \langle \Pi_2 \rangle $, they look now as:
a) $\rm{br}1$ - vacua
⟨Π_1 ⟩_br1≃(ρ_1=N_c/N_c-n_1) m_Q≫^2, ⟨Π_2 ⟩_br1≃^2(/m_Qρ_1 )^N_c-n_2/N_c-n_1(/ )^n_1/N_c-n_1≪⟨Π_1 ⟩_br1.
Unlike the mainly quantum $\langle\Pi\rangle_{\rm o}$ or mainly classical $\langle\Pi\rangle_{\rm cl}$ vacua with unbroken symmetry, these vacua are pseudo-classical : the largest value of the condensate $\langle \Pi_1 \rangle_{\rm br1}\sim m_Q\mph$ is classical while the smaller value of $\langle \Pi_2\rangle_{\rm br1}\sim\langle S\rangle_{\rm br1}/m_Q$ is of quantum origin, see (1.5). There are $N_{\rm br1}(n_1)=(N_c-n_1){\ov C}_{N_F}^{\, n_1}$ such vacua at given values of $n_1$ and $n_2$.
b) $\rm{br2}$ - vacua. These are obtained from (2.15) by $n_1\leftrightarrow n_2$ and there are $N_{\rm br2}(n_1)=(N_c-n_2){\ov C}_{N_F}^{\, n_1}$ such vacua. Of course, the total number of vacua, $N_{\rm brok}(n_1)=N_{\rm br1}(n_1)+N_{\rm br2}(n_1)=(2N_c-N_F){\ov C}_{N_F}^{\, n_1}$ remains the same at $\mph\lessgtr\mo$.
We consider $\rm br1$ vacua (all results in $\rm br2$ vacua can be obtained by $n_1\leftrightarrow n_2$). In the range
$\mo\ll\mph\ll {\tilde\mu}_{\Phi}$ (see below) where all quarks are higgsed finally, the masses of higgsed gluons look now as
μ^2_gl,1∼g^2(μ=μ_gl,1)z_Q(,μ_gl,1)⟨Π_1⟩≫μ^2_gl,2 .
The superpotential in the low energy Lagrangian of pions loooks as in (2.3), but the Kahler term of pions is different.
We write it in the form : $K\sim z_Q(\la,\mu_{\rm gl,1}){\rm Tr}\sqrt{\Pi^{\dagger}_z\Pi_z}$ . The $N_F\times N_F$ matrix $\Pi_z$ of pions looks as follows. Its $n_2\times n_2$ part consists of fields $z^{\prime}_Q(\mu_{\rm gl,1},\mu_{\rm gl,2})\Pi_{22}$, where $z^{\prime}_Q\ll 1$ is the perturbative logarithmic renormalization factor of ${\oq}_2,\, {\sq}_2$ quarks with unhiggsed colors which appears due to their additional RG evolution in the range of scales $\mu_{\rm gl,2}<\mu<\mu_{\rm gl,1}$, while at $\mu=\mu_{\rm gl,2}$ they are also higgsed. All other pion fields $\Pi_{11}, \Pi_{12}$ and $\Pi_{21}$ are normal. As a result, the pion masses look as follows. $2n_1n_2$ hybrid pions $\Pi_{12}$ and $\Pi_{21}$ are massless, while the masses of $n_1^2$ $\Pi_{11}$ and $n_2^2$ $\Pi_{22}$ are
μ(Π_11)∼m_Q/z_Q(,μ_gl,1) , μ(Π_22)∼m_Q/z_Q(,μ_gl,1)z^'_Q(μ_gl,1,μ_gl,2)≫μ(Π_11) .
Finally, the mass scale of gluonia from the unhiggsed $SU(N_c-N_F)$ group is $\sim \lym^{\rm (br1)}$ , where
⟩∼^3(/)^n_1/N_c-n_1(m_Q/)^n_2-n_1/N_c-n_1 .
2.2.3 At scales $\la\ll\mu<\mu_{\rm gl,1}\sim \langle \Pi_1 \rangle^{1/2}\sim (m_Q\mph)^{1/2}$ (ignoring logarithmic factors) the light degrees of freedom include the $SU(N_c-n_1)$ gluons and active quarks ${\oq}_2,\, {\sq}_2$ with unhiggsed colors and $n_2<(N_c-n_1)$ flavors, $n_1^2$ pions $\Pi_{11}$ and $2n_1 n_2$ hybrid pions $\Pi_{12}$ and $\Pi_{21}$ (in essence, these are ${\ov Q}_2,\, Q_2$ quarks with higgsed colors in this case). The scale factor ${\Lambda_1}$ of the gauge coupling in this lower energy theory is
Λ^b^'_o_1∼^/Π_11 , b^'_o=3(N_c-n_1)-n_2 , =3N_c-N_F .
The scale of the pole mass of ${\oq}_2,\, {\sq}_2$ quarks is $m_Q^{\rm pole}\sim m_Q$ , while the scale of $\mu_{\rm gl,2}$ is $\mu_{\rm gl,2}\sim \langle{\oq}_2{\sq}_2\rangle^{1/2}=\langle\Pi_2\rangle^{1/2}$ , with $\langle\Pi_2\rangle\ll \langle\Pi_1\rangle$ given in (2.15). Hence, the hierarchy at $\mo\ll\mph\ll{\tilde\mu}_{\Phi}$ looks as
$m_Q\ll {\Lambda_1}\ll\mu_{\rm gl, 2}\sim\langle\Pi_2\rangle^{1/2}$ and active ${\oq}_2,\, {\sq}_2$ quarks are also higgsed, while at $\mph\gg {\tilde\mu}_{\Phi}$ the hierarchy looks as $\langle\Pi_2\rangle^{1/2}\ll {\Lambda_1}\ll m_Q$ and the active quarks ${\oq}_2,\, {\sq}_2$ become too heavy and are in the $\rm HQ_2$ (heavy quark) phase. The phase changes at
⟨Π_2 ⟩^1/2 ∼m_Q∼⟨Λ_1⟩∼^(br1) μ̃_Φ∼(/m_Q)^ -n_1/n_1≫ .
Hence, we consider now this $Higgs_1-HQ_2$ phase realized at $\mph>{\tilde\mu}_{\Phi}$. For this it is convenient to retain all fields $\Phi$ although, in essence, they are too heavy and dynamically irrelevant. After integrating out all heavy higgsed gluons and ${\ov Q}_1, Q_1$ quarks, we write the Lagrangian at $\mu^2=\mu^2_{\rm gl,1}\sim N_c g^2(\mu=\mu_{\rm gl,1})z_Q(\la,\mu_{\rm gl,1})\langle\Pi_1\rangle$ in the form (see the Appendix A)
K=[ 1/f^2Tr(Φ^†Φ)+z_Q(,μ^2_gl,1)(K_Π+K__2)
K__2=Tr(^†_2 _2 +(_2_2 )) , K_Π= 2Tr√(Π^†_11Π_11)+K_hybr,K_hybr=Tr(Π^†_121/√(Π_11Π^†_11)Π_12+
Π_211/√(Π^†_11Π_11)Π^†_21),W=[-2π/α(μ_gl,1)]+/2[Tr (Φ^2) -1/(Tr Φ)^2]+Tr(_2m^tot__2_2)+W_Π,W_Π= Tr(m_QΠ_11+m^tot__2 Π_211/Π_11Π_12)-
Tr(Φ_11Π_11+Φ_12Π_21+Φ_21Π_12 ),
m^tot__2=(m_Q-Φ_22).In (2.21): $\oq_2,\, \sq_2$ and $\textsf V$ are the active ${\ov Q}_2, Q_2$ guarks and gluons with unhiggsed colors ($\textsf S$ is their field strength squared), $\Pi_{12}, \Pi_{21}$ are the hybrid pions (in essence, these are the ${\ov Q}_2, Q_2$ guarks with higgsed colors), $z_Q(\la,\mu^2_{\rm gl,1})$ is the corresponding perturbative logarithmic renormalization factor of massless quarks, see (2.2). Evolving now down in the scale and integrating $\oq_2,\, \sq_2$ quarks as heavy ones at $\mu<m^{\rm pole}_{\sq_2}$ and then unhiggsed gluons at $\mu<\lym^{(\rm br1)}$ one obtains the Lagrangian of pions and fions
W=(N_c-n_1)S+/2[Tr (Φ^2) -1/(Tr Φ)^2]+W_Π ,
S=[^m^tot__2/Π_11]^1/N_c-n_1 ,
We start with determining the masses of hybrids $\Pi_{12}, \Pi_{21}$ and $\Phi_{12}, \Phi_{21}$. They are mixed and their kinetic and mass terms look as
π_12+π^†_21π_21 ],
W_hybr=Tr(m_ϕϕ_12ϕ_21+m_ππ_12π_21-m_ϕπ(ϕ_12π_21+ϕ_21π_12)) ,m_ϕ=f^2, m_π=m_Q-⟨Φ_2⟩/z_Q=⟨Π_1⟩/z_Q∼m_Q/z_Q≪m_ϕ , z_Q=z_Q(,μ_gl,1) ,m_ϕπ=(f^2⟨Π_1⟩/z_Q)^1/2, m_ϕπ^2=m_ϕm_π .
Hence, the scalar potential looks as
V_S=|m|^2 |Ψ^(-)_12|^2+0|Ψ^(+)_12|^2 +(12→21), |m|=(|m_ϕ|+|m_π|) ,
Ψ^(-)_12=(c ϕ_12-s π_12 ), Ψ^(+)_12=(c π_12+s ϕ_12 ),
c=(|m_ϕ|/|m|)^1/2, s=(|m_π|/|m|)^1/2 .Therefore, the fields $\Psi^{(-)}_{12}$ and $\Psi^{(-)}_{21}$ are heavy, with the masses $|m|\simeq |m_{\phi}|$, while the fields $\Psi^{(+)}_{12}$ and $\Psi^{(+)}_{21}$ are massless. But the mixing is really parametrically small, so that the heavy fields are mainly $\phi_{12}, \phi_{21}$ while the massless ones are mainly $\pi_{12}, \pi_{21}$.
Everywhere below in the text we neglect mixing when it is small.
And finally from (2.22), the pole mass of pions $\Pi_{11}$ is
μ(Π_11)∼⟨Π_1⟩/z_Q(,μ_gl,1)∼m_Q/z_Q(,μ_gl,1) .
On the whole for this $Higgs_1-HQ_2$ phase the mass spectrum looks as follows at $\mph\gg{\tilde\mu}_{\Phi}$ . a) The heaviest are $n_1(2N_c-n_1)$ massive gluons and the same number of their scalar superpartners with the masses $\mu_{\rm gl,1}$, see (2.16), these masses originate from the higgsing of ${\ov Q}_1, Q_1$ quarks. b) There is a large number of 22-flavored hadrons made of weakly interacting and weakly confined non-relativistic $\oq_2, \sq_2$ quarks with unhiggsed colors (the string tension is $\sqrt\sigma\sim\lym^{(\rm br1)}\ll m^{\rm pole}_{\sq,2}$, see (2.18)), the scale of their masses is $m^{\rm pole}_{\sq,2}\sim m_Q/[z_Q(\la,\mu_{\rm gl,1})z^{\prime}_Q(\mu_{\rm gl,1}, m^{\rm pole}_{\sq,2})]$, where $z_Q\gg 1$ and $z^{\rm \prime}_Q\ll 1$ are the corresponding massless perturbative logarithmic renormalization factors. c) There are $n_1^2$ pions $\Pi_{11}$ with the masses (2.26), $\mu(\Pi_{11})\ll m^{\rm pole}_{\sq,2}$. d) There is a large number of gluonia made of gluons with unhiggsed colors, the scale of their masses is $\sim\lym^{(\rm br1)}$, see (2.18). e) The hybrids $\Pi_{12}, \Pi_{21}$ are massless.
All $N^2_F$ fions $\Phi_{ij}$ remain too heavy and dynamically irrelevant (see the footnote 3), their pole masses are
$\mu^{\rm pole}_1(\Phi)\sim f^2\mph\gg\mu_{\rm gl,1}$.
§ QUARK CONDENSATES AND MULTIPLICITY OF VACUA
AT $\MATHBF{N_C<N_F<2N_C}$
To obtain the numerical values of the quark condensates $\langle{\ov Q}_j Q_i\rangle=\delta_{ij}\langle{\ov Q} Q\rangle_i$ at $N_F>N_c$ (but only for this purpose), the simplest way is to use the known exact form of the non-perturbative contribution to the superpotential in the standard SQCD with the quark superpotential $m_Q{\rm Tr}({\ov Q}Q)$ and without the fions $\Phi$. It seems clear that at sufficiently large values of $\mph$ among the vacua of the $\Phi$-theory there should be $N_c$ vacua of SQCD in which, definitely, all fions $\Phi$ are too heavy and dynamically irrelevant. Therefore, they all can be integrated out and the exact superpotential can be written as ($\Pi_{ij}={\ov Q}
_j Q_i(\mu=\la),\, m_Q=m_Q(\mu=\la),\, \mph=\mph(\mu=\la)$, see Section 1 above and Sections 3 and 7 in [6])
W=-(Π/^)^1/+m_QTr Π-1/2
[Tr (Π^2)- 1/N_c(Tr Π)^2 ] .
Indeed, at sufficiently large $\mph$, there are $N_c$ vacuum solutions in (3.1) with the unbroken $SU(N_F)$ flavor symmetry. In these, the last term in (3.1) gives a small correction only and can be neglected and one obtains
⟨Π_ij⟩_SQCD≃δ_ij1/m_Q(^(SQCD))^3=δ_ij1/m_Q(^m_Q^N_F)^1/N_c .
Now, using the holomorphic dependence of the exact superpotential on the chiral fields $({\ov Q}_j Q_i)$ and the chiral parameters $m_Q$ and $\mph$, the exact form (3.1) can be used to find the values of the quark condensates $\langle{\ov Q}_j Q_i\rangle$ in all other vacua of the $\Phi$ - theory and at all other values of $\mph>\la$. It is worth recalling only that, in general, as in the standard SQCD [6, 7, 8]: a) (3.1) is not the superpotential of the genuine low energy Lagrangian describing lightest particles, it determines only the values of the vacuum condensates $\langle{\ov Q}_j Q_i\rangle$, b) and therefore, the notations in (3.1) do not imply literally that quarks are higgsed or form the diquark condensates, $\Pi_{ij}$ in (3.1) only have to be understood as convenient abbreviations for $({\ov Q}_j Q_i)$. (The genuine low energy Lagrangians in different vacua will be obtained below in sections 5-10, both in the direct and dual theories).
3.1 Vacua with the unbroken flavor symmetry $U(N_F)$. One obtains from (3.1) that at $\la\ll\mph\ll \mo$ there are two groups of such vacua with parametrically different values of condensates, $\langle{\ov Q}_j Q_i\rangle_L
=\delta_{ij}\langle{\ov Q} Q\rangle_L$ and $\langle{\ov Q}_j Q_i\rangle_S=\delta_{ij}\langle{\ov Q} Q\rangle_S$.
a) There are $(2N_c-N_F)$ L - vacua (see also the footnote 1) with
⟨(μ=)⟩_L=⟨Π_L⟩∼^2(/)^/2N_c-N_F≪^2 .
In these L (large) quantum vacua, the second term in the superpotential (3.1) gives numerically only a small correction.
b) There are $(N_F-N_c)$ classical S - vacua with
⟨(μ=)⟩_S=⟨Π_S⟩≃-/N_c m_Q .
In these S (small) vacua, the first non-perturbative term in the superpotential (3.1) gives only small corrections with $Z_{N_F-N_c}$ phases, but just these corrections determine the multiplicity of these $(N_F-N_c)$ nearly degenerate vacua. On the whole, there are
vacua with the unbroken flavor symmetry at $N_c<N_F<2N_c$.
One obtains from (3.1) that at $\mph\gg \mo$ the above $(2N_c-N_F)$ L - vacua and $(N_F-N_c)$ S - vacua degenerate into $N_c$ SQCD vacua (3.2).
The value of $\mo$ is determined from the matching
[⟨Π⟩_L∼^2(/)^/2N_c-N_F]∼[⟨Π⟩_S∼m_Q]∼[⟨Π⟩_SQCD∼^2(m_Q/)^/N_c] ∼(/m_Q)^2N_c-N_F/N_c≫ .
3.2 Vacua with the broken flavor symmetry $U(N_F)\ra U(n_1)\times U(n_2),\, n_1\leq [N_F/2]$. In these, there are $n_1$ equal condensates $\langle{\ov Q}_1Q_1(\mu=\la)\rangle=\langle\Pi_1
\rangle$ and $n_2\geq n_1$ equal condensates $\langle{\ov Q}_2 Q_2(\mu=\la)\rangle=\langle\Pi_2\rangle\neq
\langle\Pi_1\rangle$. The simplest way to find the values of quark condensates in these vacua is to use (1.5). We rewrite it here for convenience
⟨Π_1+Π_2-1/N_cTrΠ⟩_br=m_Q , ⟨S⟩_br=(⟨Π⟩_br=⟨Π_1⟩^n_1_br⟨Π_2⟩^n_2_br/^)^1/=⟨Π_1⟩_br⟨Π_2⟩_br/ .
Besides, the multiplicity of vacua will be shown below at given values of $n_1$ and $n_2\geq n_1$.
3.2.1 The region $\la\ll\mph\ll\mo$.
a) At $n_2\lessgtr N_c$, including $n_1=n_2=N_F/2$ for even $N_F$ but excluding $n_2=N_c$ , there are $(2N_c-N_F){\ov C}^{\,n_1}_{N_F}$ L - type vacua with the parametric behavior of condensates (see the footnote 1)
i.e. as in the L - vacua above but $\langle\Pi_1\rangle_{\rm Lt}\neq\langle\Pi_2\rangle_{\rm Lt}$ here.
b) At $n_2>N_c$ there are $(n_2-N_c)C^{n_1}_{N_F}$ $\rm br2$ - vacua with, see (3.7),
⟨Π_2⟩_br2∼m_Q , ⟨Π_1⟩_br2∼^2(/)^n_2/n_2-N_c(m_Q/)^N_c-n_1/n_2-N_c, ⟨Π_1⟩_br2/⟨Π_2⟩_br2∼(/)^N_c/n_2-N_c≪1 .
c) At $n_1=\nd,\, n_2=N_c$ there are $(2N_c-N_F)\cdot C^{n_1=\nd}_{N_F}$ `special' vacua with, see (3.7),
⟨Π_1⟩_spec=N_c/2N_c-N_F(m_Q) , ⟨Π_2⟩_spec∼^2(/)^/2N_c-N_F, ⟨Π_1⟩_spec/⟨Π_2⟩_spec∼(/)^N_c/2N_c-N_F≪1.
On the whole, there are ( $\theta(z)$ is the step function )
N_brok(n_1)=[(2N_c-N_F)+θ(n_2-N_c)(n_2-N_c)]C^ n_1_N_F=
=[(N_c-)+θ(-n_1)(-n_1)]C^ n_1_N_F ,( ${\ov C}^{\,n_1}_{N_F}$ differ from the standard $C^{\,n_1}_{N_F}$ only by ${\ov C}^{\,n_1={\rm k}}_{N_F=2{\rm k}}=C^{\,n_1={\rm k}}_{N_F=2{\rm k}}/2$, see (2.13) ) vacua with the broken flavor symmetry $U(N_F)\ra U(n_1)\times U(n_2)$, this agrees with [11].
3.2.2 The region $\mph\gg\mo$.
a) At all values of $n_2\lessgtr N_c$, including $n_1=n_2=N_F/2$ at even $N_F$ and the `special' vacua with $n_1=\nd,\, n_2=N_c$, there are $(N_c-n_1){\ov C}^{\,n_1}_{N_F}$ $\rm br1$ - vacua with, see (3.7),
⟨Π_1⟩_br1∼m_Q , ⟨Π_2⟩_br1∼^2(/)^n_1/N_c-n_1(/m_Q)^N_c-n_2/N_c-n_1 , ⟨Π_2⟩_br1/⟨Π_1⟩_br1∼(/)^N_c/N_c-n_1≪1 .
b) At $n_2<N_c$, including $n_1=n_2=N_F/2$, there are also $(N_c-n_2){\ov C}^{\,n_2}_{N_F}=(N_c-n_2){\ov C}^{\,n_1}_
{N_F}$ $\rm br2$ - vacua with, see (3.7),
⟨Π_2⟩_br2∼m_Q , ⟨Π_1⟩_br2∼^2(/)^n_2/N_c-n_2(/m_Q)^N_c-n_1/N_c-n_2 , ⟨Π_1⟩_br2/⟨Π_2⟩_br2∼(/)^N_c/N_c-n_2≪1 .
On the whole, there are
N_brok(n_1)=[(N_c-n_1)+θ(N_c-n_2)(N_c-n_2)]C^ n_1_N_F=
=[(N_c-)+θ(-n_1)(-n_1)]C^ n_1_N_F vacua. As it should be, the number of vacua at $\mph\lessgtr \mo$ is the same.
As one can see from the above, all quark condensates become parametrically the same at $\mph\sim\mo$. Clearly, this region $\mph\sim\mo$ is very special and most of the quark condensates change their parametric behavior and hierarchies at $\mph\lessgtr\mo$. For example, the br2 - vacua with $n_2<N_c\,,\,\,\langle\Pi_2 \rangle\sim m_Q\mph\gg\langle\Pi_1\rangle$ at $\mph\gg\mo$ evolve into the L - type vacua with $\langle
\Pi_2\rangle\sim\langle\Pi_1\rangle\sim \la^2 (\la/\mph)^{\nd/(2N_c-N_F)}$ at $\mph\ll\mo$, while the br2 - vacua with $n_2>N_c\,,\,\,\langle\Pi_2\rangle\sim m_Q\mph\gg\langle\Pi_1\rangle$ at $\mph\ll\mo$ evolve into the br1 - vacua with $\langle\Pi_1\rangle\sim m_Q\mph\gg\langle\Pi_2\rangle$ at $\mph\gg\mo$, etc. The exception is the special vacua with $n_1=\nd,\, n_2=N_c$ . In these, the parametric behavior $\langle\Pi_1 \rangle\sim m_Q\mph, \,\langle\Pi_2\rangle\sim \la^2(\la/\mph)^{\nd/(2N_c-N_F)}$ remains the same but the hierarchy is reversed at $\mph\lessgtr\mo\, :\, \langle\Pi_1\rangle/\langle\Pi_2\rangle\sim (\mph/\mo)^{N_c/(2N_c-N_F)}$.
The total number of all vacua at $N_c<N_F<2N_c$ is
N_tot=( N_unbrok=N_c )+( N_brok^tot=∑_n_1=1^[N_F/2]N_brok(n_1)
)=∑_k=0^N_c(N_c-k)C^ k_N_F ,
this agrees with [11]
But we disagree with their `derivation' in section 4.3. There is no their ${\cal N}_2$ vacua with $\langle M_{ii}\rangle\langle{\ov q}_i q_i\rangle/\la=\langle S\rangle=0,\,\, i=1,...N_F$ (no summation over $i$) in the dual theory at $m_Q\neq 0$. In all $N_{\rm tot}$ vacua in both direct and dual theories : $\langle\det M/\la^{\bo}\rangle^{1/\nd}=\langle\det {\ov Q}Q/\la^{\bo}\rangle^{1/\nd}=\langle S\rangle\neq 0$ at $m_Q\neq 0$ (see sections 5-10 below and the Appendix B). Really, the superpotential (4.48) contains all $N_{\rm tot}={\cal N}_1+{\cal N}_2$ vacua.
Comparing this with the number of vacua (2.13),(2.14) at $N_F<N_c$ it is seen that, for both $N_{\rm unbrok}$ and $N_{\rm brok}^{\rm tot}$ separately, the multiplicities of vacua at
$N_F<N_c$ and $N_F>N_c$ are not analytic continuations of each other.
The analog of (3.1) in the dual theory with $|\Lambda_q|=\la$, see (1.7), is obtained by the replacement $\Pi={\ov Q} Q(\mu=\la)\ra M(\mu=\la)$, so that $\langle M(\mu=\la)\rangle=\langle {\ov Q} Q(\mu=\la)\rangle$ in all vacua and multiplicities of vacua are the same.
§ FIONS: ONE OR THREE GENERATIONS
At $N_c<N_F<2N_c$ and in the interval of scales $\mu_H<\mu<\la$ ( $\mu_H$ is the largest physical mass in the quark-gluon sector), the quark and gluon fields are effectively massless. Because the quark renormalization factor $z_Q(\la,\mu\ll\la)=(\mu/\la)^{\gamma_Q>0}\ll 1$ decreases in this case in a power fashion with lowering energy due to the perturbative RG evolution, it is seen from (1.3) that the role of the 4-quark term $({\ov Q}Q)^2/\mph$ increases with lowering energy. Hence, while it is irrelevant at the scale $\mu\sim\la$ because $\mph\gg \la$, the question is whether it becomes dynamically relevant in the range of energies $\mu_H\ll\mu\ll \la$. For this, we estimate the scale $\mu_o$ where it becomes relevant in the massless theory (see section 7 in [6] for the perturbative strong coupling regime at $N_c<N_F<3N_c/2$ )
μ_o/1/z^2_Q(,μ_o)=μ_o/(/μ_o)^2γ_Q∼1 μ_o/∼(/)^1/(2γ_Q-1) ,
γ^conf_Q=/N_F μ^conf_o/∼(/)^N_F/3(2N_c-N_F) , γ^strong_Q=2N_c-N_F/
μ^strong_o/∼(/)^N_F/(5N_c-3N_F) .
Hence, if $\mu_H\ll\mu_o$, then at scales $\mu<\mu_o$ the four-quark terms in the superpotential (1.3) cannot be neglected any more and we have to account for them. For this, we have to reinstate the fion fields $\Phi$ and to use the Lagrangian (1.1) in which the Kahler term at $\mu_H<\mu\ll\la$ looks as
K=[z_Φ(,μ)/f^2Tr (Φ^†Φ)+z_Q(,μ)Tr(Q^†Q+(QQ))], z_Q(,μ)=(μ/)^γ_Q≪1.
We recall that even at those scales $\mu$ that the running perturbative mass of fions $\mu_{\Phi}(\mu)\gg \mu$ and so they are too heavy and dynamically irrelevant, the quarks and gluons remain effectively massless and active. Therefore, due to the Yukawa interactions of fions with quarks, the loops of still active light quarks (and gluons interacting with quarks)
still induce the running renormalization factor $z_{\Phi}(\la,\mu)$ of fions at all those scales where quarks are effectively massless, $\mu>\mu_H$. But, in contrast with a very slow logarithmic RG evolution at $N_F<N_c$ in section 2, the perturbative running mass of fions decreases now at $N_c<N_F<2N_c$ and $\mu<\la$ monotonically and very quickly with diminishing scale (see below), $\mph(\mu\ll \la)=\mph/f^2 z_\Phi(\la,\mu)\sim\mph(\mu/\la)^{|\gamma_{\Phi}|>1}\ll \mph$. Nevertheless, until $\mph(\mu)\gg \mu$, the fields $\Phi$ remain heavy and do not influence the RG evolution. But, when $\mu_H\ll\mu_o$ and $\mph(\mu)\sim\mph/z_{\Phi}(\la,\mu)$ is the main contribution to the fion mass
the cases when the additional contributions to the masses of fions from other perturbative or non-perturbative terms in the superpotential are not small in comparison with $\sim\mph/z_{\Phi}(\la,\mu)$ have to be considered separately
the quickly decreasing mass $\mph(\mu)$ becomes $\mu^{\rm pole}_2(\Phi)=\mph(\mu=\mu^{\rm pole}_2(\Phi))$ and $\mph(\mu<\mu^{\rm pole}_2(\Phi))< \mu$, so that : 1) there is a pole in the fion propagator at $p=\mu^{\rm pole}_2(\Phi)$, this is a second generation of fions (the first one is at $\mu^{\rm pole}_1(\Phi)\gg\la$, see Appendix A) ; 2) the fields $\Phi$ become effectively massless at $\mu<\mu^{\rm pole}_2(\Phi)$ and begin to influence the perturbative RG evolution. In other words, the seemingly `heavy' fields $\Phi$ return back, they become effectively massless and dynamically relevant. Here and below the terms `relevant' and `irrelevant' (at a given scale $\mu$ ) will be used in the sense of whether the running mass $\sim\mph/z_{\Phi}(\la,\mu\ll\la)$ of fions at a given scale $\mu$ is $<\mu$, so that they are effectively massless and participate actively in interactions at this scale , or they remain too heavy with the running mass $>\mu$ whose interactions at this scale give only small corrections.
It seems clear that the physical reason why the $4$-quark terms in the superpotential $(1.3)$ become relevant at scales $\mu<\mu_o$ is that the fion field $\Phi$ which was too heavy and so dynamically irrelevant at $\mu>\mu_o,\, \mph(\mu>\mu_o)>\mu$ , becomes effectively massless at $\mu<\mu_o,\, \mph(\mu<\mu_o)<\mu$ , and begins to participate actively in the RG evolution, i.e. it becomes relevant. In other words, the four quark term in (1.3) `remembers' about fions and signals about the scale below which the fions become effectively massless, $\mu_o=\mu^{\rm pole}_2(\Phi)$. This allows us to find the value of $z_{\Phi}(\la,\mu_o)$,
f^2/z_Φ(,μ_o)=μ_o , 1/f^2 z_Φ(,μ_o)=/μ_o=(μ_o/)^γ_Φ , γ_Φ=-2γ_Q .
The perturbative running mass $\mph(\mu)\sim\mph/z_{\Phi}(\la,\mu\ll\la)\ll\mph$ of fions continues to decrease strongly with diminishing $\mu$ at all scales $\mu_H<\mu<\la$ until quarks remain effectively massless, and becomes frozen only at
scales below the quark physical mass, when the heavy quarks decouple.
Hence, if $\mu_H\gg\mu_o$ , there is no pole in the fion propagator at the momenta $p<\la$ because the running fion mass is too large in this range of scales, $\mph(p>\mu_o)>p$. The fions remain dynamically irrelevant in this case at all momenta $p<\la$.
But when $\mu_H\ll\mu_o$, there will be not only the second generation of fions at $p=\mu^{\rm pole}_2(\Phi)=\mu_o$
but also a third generation at $p\ll\mu_o$. Indeed, after the heavy quarks decouple at momenta $p<\mu_H\ll\mu_o$ and the renormalization factor $z_{\Phi}(\la,\mu)$ of fions becomes frozen, $z_{\Phi}(\la,\mu<\mu_H)\sim z_{\Phi}(\la,\mu\sim
\mu_H)$, the frozen value $\mph(\mu<\mu_H)$ of the running fion mass is now $\mph(\mu\sim\mu_H)\ll p_H=\mu_H$. Hence, there is one more pole in the fion propagator at $p=\mu^{\rm pole}_3(\Phi)\sim \mph(\mu\sim\mu_H)\ll \mu_H$.
On the whole, in a few words for the direct theory (see the footnote 5 for reservations).
a) The fions remain dynamically irrelevant and there are no poles in the fion propagator at momenta $p<\la$ if $\mu_H\gg\mu_o$.
b) If $\mu_H\ll\mu_o\ll\la$, there are two poles in the fion propagator at momenta $p\ll\la$, $\mu^{\rm pole}_2(\Phi)\sim \mu_o$ and $\mu^{\rm pole}_3(\Phi)\sim \mph/z_{\Phi}(\la,\mu_H)\ll\mu^{\rm pole}_2(\Phi)$
(here and everywhere below in similar cases, - up to corrections due to possible nonzero decay widths of fions). In other words, the fions appear in three generations in this case (we recall that there is always the largest pole mass of fions $\mu^{\rm pole}_1(\Phi)\gg\la$, see the appendix A). Hence, the fions are effectively massless and dynamically relevant in the range of scales $\mu^{\rm pole}_3(\Phi)<\mu<\mu^{\rm pole}_2(\Phi)$.
Moreover, once the fions become effectively massless and dynamically relevant with respect to internal interactions, they begin to contribute simultaneously to the external anomalies ( the 't Hooft triangles in the external background fields).
The case $\mu_H\sim\mu_o$ requires additional information. The reason is that at scales $\mu\lesssim\mu_H$, in addition to the canonical kinetic term $\Phi^{\dagger}_R p^2\Phi_R$ (R=renormalized) of fions, there are also terms $\sim \Phi^{\dagger}_R p^2(p^2/\mu_H^2)^k\Phi_R$ with higher powers of momenta induced by loops of heavy quarks (and gluons). If $\mu_H\ll\mu_o$, then the pole in the fion propagator at $p=\mu^{\rm pole}_2(\Phi)=\mu_o$ is definitely there and, because $\mph(\mu=\mu_H)\ll\mu_H$, these additional terms are irrelevant in the region $p\sim\mph(\mu=\mu_H)\ll\mu_H$ and the pole in the fion propagator at $p=\mu^{\rm pole}_3(\Phi)=\mph(\mu=\mu_H)\ll\mu_H$ is also guaranteed. But $\mph(\mu\sim\mu_H)\sim\mu_H$ if $\mu_H\sim\mu_o$, and these additional terms become relevant. Hence, whether there is pole in the fion propagator in this case or not depends on all these terms.
Now, if $\mu_H<\mu_o$ so that the fions become relevant at $\mu<\mu_o$, the question is : what are the values of the quark and fion anomalous dimensions, $\gamma_Q$ and $\gamma_\Phi$, in the massless perturbative regime at $\mu_H<\mu<\mu_o$ ?
To answer this question, we use the approach used in [6] (see section 7). For this, we introduce first the corresponding massless Seiberg dual theory [5]. Our direct theory includes at $\mu_H<\mu<\mu^{\rm conf}_o$ not only the original effectively massless in this range of scales quark and gluon fields, but also the active $N_F^2$ fion fields $\Phi^j_i$ as they became now also effectively massless, so that the effective superpotential becomes nonzero and includes the Yukawa term ${\rm Tr}\,(\Phi{\ov Q}Q)$. Then, the massless dual theory with the same 't Hooft triangles includes only the massless qual quarks ${\ov q},\, q$ with $N_F$ flavors and the dual $SU(\nd=N_F-N_c)$ gluons. Further, one equates two NSVZ $\,{\widehat\beta}_{ext}$ - functions of the external baryon and $SU(N_F)_{L,R}$ - flavor vector fields in the direct and dual theories,
d/d lnμ 2π/α_ext=β_ext= -2π/α^2_ext β_ext= ∑_i T_i (1+γ_i) ,
where the sum runs over all fields which are effectively massless at scales $\mu_H<\mu<\mu_o$, the unity in the brackets is due to one-loop contributions while the anomalous dimensions $\gamma_i$ of fields represent all higher-loop effects. It is worth noting that these general NSVZ forms (4.4) of the external `flavored' $\beta$-functions are independent of the kind of massless perturbative regime of the internal gauge theory, i.e. whether it is conformal, or the strong coupling or the IR free one.
The effectively massless particles in the direct theory here are the original quarks $Q,\,{\ov Q}$ and gluons and, in addition, the fions $\Phi^j_i$, while in the dual theory these are the dual quarks $q,\, {\ov q}$ and dual gluons only.
For the baryon currents one obtains
N_F N_c ( B_Q=1 )^2 (1+γ_Q)=N_F ( B_q=N_c/
)^2 (1+γ_q) ,
while for the $SU(N_F)$ flavor currents one obtains
N_c (1+γ_Q)+N_F (1+γ_Φ)= (1+γ_q) .
Here, the left-hand sides are from the direct theory while the right-hand sides are from the dual one, $\gamma_Q$ and $\gamma_\Phi$ are the anomalous dimensions of the quark $Q$ and fion $\Phi$ , while $\gamma_q$ is the anomalous dimension of the dual quark $q$.
The massless dual theory is in the conformal regime at $3N_c/2<N_F<2N_c$ , so that $\gamma^{\rm conf}_q=\rm{{\ov b}_o}/N_F=(3\nd-N_F)/N_F$. Therefore, one obtains from (4.5),(4.6) that $\gamma^{\rm conf}_Q=\bo/N_F=(3N_c-N_F)/N_F$ and $\gamma^{\rm conf}_\Phi=-2\gamma_Q$, i.e. while only the quark-gluon sector of the direct theory behaves conformally at scales $\mu^{\rm conf}_o<\mu< \la$ where the fion fields $\Phi$ remain heavy and irrelevant, the whole theory including the fields $\Phi$ becomes conformal at scales $\mu_H<\mu< \mu^{\rm conf}_o$ where fions become effectively massless and relevant.
This does not mean that nothing changes at all after the fion field $\Phi$ begins to participate actively in the perturbative RG evolution at $\mu_H<\mu<\mu^{\rm conf}_o$. In particular, the frozen fixed point values of the gauge and Yukawa couplings $a^*$ and $a_{f}^*$ will change, and various appropriate Green functions will change their behavior, etc.
In the region $N_c<N_F<3N_c/2$ where the massless direct theory is in the strong coupling regime $a(\mu\ll\la)\gg 1$ (see section 7 in [6]), the massless dual theory is in the IR free logarithmic regime. Hence, $\gamma_q$ is logarithmically small at $\mu\ll\mu^{\rm strong}_o\ll\la,\, \gamma_q\ra 0$, see (4.1), and one obtains from (4.5),(4.6) in this case the values of $\gamma^{\rm strong}_Q(\mu_H\ll\mu\ll\mu^{\rm strong}_o\ll\la)$ and $\gamma^{\rm strong}_{\Phi}(\mu_H\ll\mu\ll\mu^{\rm strong}_o\ll\la)$
γ^strong_Q=2N_c-N_F/ , γ^strong_Φ=-(1+γ^strong_Q)=- N_c/ .
Therefore (within this approach, and in the absence of any other known way to obtain the values of $\gamma^{\rm strong}_Q$
and $\gamma^{\rm strong}_{\Phi}$ in the strong coupling regime), the quark anomalous dimension is $\gamma^{\rm strong}_Q=(2N_c-N_F)/\nd$ in the whole range $\mu_H<\mu<\la$, while the fion anomalous dimension $\gamma^{\rm strong}_{\Phi}$ changes when it becomes relevant, $\gamma^{\rm strong}_{\Phi}=-2\gamma^{\rm strong}_Q$ at $\mu^{\rm strong}_o<\mu<\la$ and $\gamma^{\rm strong}_{\Phi}=-(1+\gamma^{\rm strong}_Q)$ at $\mu_H<\mu<\mu^{\rm strong}_o$.
In the rest of this paper the mass spectra of the direct and dual theories will be considered within the conformal window $3N_c/2<N_F<2N_c$ only.
§ DIRECT THEORY. UNBROKEN FLAVOR SYMMETRY
As in the standard ${\cal N}=1$ SQCD with the superpotential $W=m_Q{\rm Tr}({\ov Q}Q)$, the results for the mass spectrum at $N_F>N_c$ of the theory with the Lagrangian (1.2) depend essentially on the used dynamical scenario. Two different scenarios $\#1$ [6, 7] and $\#2$ [8] have been considered previously. In this paper the mass spectra of the theory (1.2) will be calculated within the scenario $\#1$.
We recall that this scenario implies that when the scale of the quark condensate is in the range $\mq<\mc=|\langle{\ov Q}Q(\mu=\la)\rangle|^{1/2}<\la$ (and nothing prevents), the quarks are not higgsed but form the coherent colorless diquark condensate (DC) and acquire the dynamical mass $\mc$, and there appear light pseudo-Goldstone bosons $\Pi$ (pions) with masses $\mu(\Pi)\ll \mc$, see [6, 7].
5.1 L - vacua
In these $(2N_c-N_F)$ vacua at $\la\ll\mph\ll\mo$, we compare first the possible constituent and pole quark masses. As for the constituent mass, it looks as, see (3.3),
(μ^(L)_C)^2=⟨Π⟩_L=⟨QQ(μ=)⟩_L∼^2(/)^/2N_c-N_F≪^2 , ≪≪ .
In vacua with unbroken flavor symmetry all quark masses are equal, see (1.2),
⟨m^tot_Q⟩≡⟨m^tot_Q⟩(μ=)=m_Q+/N_c⟨Π⟩, ⟨Π⟩=⟨Q_1 Q_1(μ=)⟩ .
Then, the quark pole mass looks as
m^pole_Q=⟨m^tot_Q⟩/z_Q(,μ=m^pole_Q) , z_Q(,μ≪)=(μ/)^γ_Q=(/N_F)≪1 .
Now, in L-vacua with $\langle\Pi\rangle_L$ , see (5.2),
⟨m^tot_Q⟩_L/∼⟨Π⟩_L/∼(/)^N_c/2N_c-N_F, m^pole_Q,L/=(⟨m^tot_Q⟩_L/)^N_F/3N_c∼∼(/)^N_F/3(2N_c-N_F)∼^(L) , m^pole_Q,L/μ^(L)_C∼(/)^/6(2N_c-N_F)≪1 .
Therefore, all quarks are in the $DC$ (diquark condensate) phase. We check now whether the fions will be relevant in these L-vacua. For this, we have to compare $\mu^{\rm conf}_o$ in (4.1) with $\mu^{(L)}_C$ in (5.1),
μ^conf_o/μ^(L)_C∼(/)^/6(2N_c-N_F)≪1 .
Therefore, the fion fields remain dynamically irrelevant in these L-vacua, they can be integrated out and, at $\mu<\la$, we can start directly with the Lagrangian (1.3). Proceeding now as in section 3 of [6], i.e. integrating out first the heaviest constituent quarks at $\mu<\mu^{(L)}_C$ and then gluons at $\mu<\lym^{(L)}$, one obtains the Lagrangian of $N^2_F$ pions,
K=Tr √(Π^†Π) , W= - (Π/^)^1/+m_QTr Π-1/2
[Tr (Π^2)- 1/N_c(Tr Π)^2 ] .
The term with $m_Q$ in (5.6) can be neglected in these L-vacua at $\la\ll\mph\ll\mo$, and one obtains for the pion masses
μ_L(Π)∼⟨Π⟩_L/∼(/)^N_c/2N_c-N_F, μ_L(Π)/^(L)∼(/)^/3(2N_c-N_F)≪1 .
On the whole, the mass spectrum in these $(2N_c-N_F)$ L-vacua with the unbroken flavor symmetry looks as follows
at $\la\ll\mph\ll\mo$.
a) There is a large number of heaviest flavored hadrons made of non-relativistic and weakly confined (the string tension is ${\sqrt\sigma}\sim \lym^{(L)}\ll \mu^{(L)}_C$ ) constituent quarks with the masses $\mu^{(L)}_C=\langle{\ov Q}Q\rangle^{1/2}_L$ (5.1).
b) There is a large number of gluonia with the mass scale $\sim \lym^{(L)}=\langle S\rangle^{1/3}_L\ll \mu^{(L)}_C$ (5.4).
c) The lightest are $N_F^2$ pions with the masses $\mu_L(\Pi)$ in (5.7). The fions are dynamically irrelevant in these vacua.
The term with $m_Q$ cannot be neglected any more in (5.6) at $\mph\gtrsim \mo$ and all condensates and masses evolve to those in the standard SQCD, see section 3 in [6].
5.2 S - vacua
In these $(N_F-N_c)$ vacua at $\la\ll\mph\ll\mo$, the possible constituent and pole quark masses look as, see (1.2),(3.4),
(μ^(S)_C)^2=⟨Π⟩_S≃/N_c (m_Q) ,
⟨m^tot_Q⟩_S/∼⟨S⟩_S/⟨Π⟩_S∼(⟨Π⟩_S/^2)^N_c/∼(m_Q/^2)^N_c/ ,m^pole_Q,S/∼(m_Q/^2)^N_F/3, m^pole_Q,S/
∼(m_Q/^2)^/6≪1 , ≪≪ .Therefore, all quarks are in the $DC$-phase also. But the analog of (5.5) looks now as
μ^conf_o/∼(/m_Q)^1/2(/)^6N_c-N_F/6(2N_c-N_F) .
Hence, in these S-vacua, there appear two additional generations of fions at $\la\ll\mph<\mph^{(S,{\,\rm conf})}$,
^(S, conf)=(/m_Q)^3(2N_c-N_F)/6N_c-N_F ,
≪^(S, conf)≪ ,
while the fions remain too heavy and dynamically irrelevant at $\mph^{(S,{\,\rm conf})}<\mph<\mo$.
The conformal regime continues here down to the scale $\mu_H$, where $\mu_H$ is the largest physical mass in the quark-gluon sector and here it is $\mu_H=\mcs\sim (m_Q\mph)^{1/2}$. The RG evolution of the quark and fion fields becomes frozen at scales $\mu<\mcs$ because the heavy constituent quarks decouple. Proceeding as in [6] (see section 3), i.e. integrating out first constituent quarks as heavy ones at $\mu<\mcs$ and then gluons at $\mu<\lym^{(S)}$, one obtains the Lagrangian of the fion and pion fields
from now on we omit the constant $f=O(1)$ for simplicity
] ,
K=Tr √(Π^†Π)+z^(S)_Φ(,)Tr (Φ^†Φ) , z^(S)_Φ(,μ)∼(/μ)^2/N_F≫1 ,
W= - S+/2[Tr (Φ^2)-1/(Tr Φ)^2]+Tr (m_Q-Φ)Π , S=(Π/^)^1/.
The non-perturbative term with the determinant in (5.11) can be neglected in these S-vacua with $\langle \Pi\rangle_S
\sim m_Q\mph$ at $\la\ll\mo$, and one obtains for the masses of $\Phi$ and $\Pi$ (see the footnote 3)
μ_S(Φ)∼/z^(S)_Φ(,) , μ_S(Π)∼m_Q , μ_S(Φ)/
∼(/^(S, conf))^6N_c-N_F/2N_F .
Another way, because $\mu_S(\Phi)\gg\mu_S(\Pi)$, one can integrate out first in (5.11) the heavier fields $\Phi$ at scales $\mu<\mu_S(\Phi)$ and obtains
K_Π=Tr √(Π^†Π) , W_Π= - S -1/2[Tr (Π^2)-1/N_c(Tr Π)^2]+m_QTr Π .
Neglecting the term $\sim S$ with the determinant in these S-vacua, one obtains from (5.13) that the pion mass is $\mu_S(\Pi)\sim m_Q$.
We discuss now in more detail the fion masses in these S-vacua.
i) At $\mph>\mph^{(S,{\,\rm conf})}$ the running mass
$\mu_{\Phi}(\mu)=\mph/z_{\Phi}^{(S)}(\la,\mu)$ of fions remains larger than the scale $\mu$ in the whole range $\mcs<\mu<
\la$, until it becomes frozen at lower scales $\mu<\mcs$ at its value $\mph/z_{\Phi}^{(S)}(\la,\mcs)>\mcs$. Hence, there are no poles in the propagators of fions at all momenta $p<\la\ll\mu^{\rm pole}_1(\Phi)$ and the `mass' $\mu_S(\Phi)$ in (5.12) is not the observable pole mass but simply the limiting value of the mass term in the fion propagator at low momenta.
ii) The situation is qualitatively different at $\la<\mph<\mph^{(S,{\,\rm conf})}$. The running mass of fions
$\mu_{\Phi}(\mu)$ is $\mu_{\Phi}(\mu)>\mu$ at $\mu>\mu^{\rm conf}_o$ but it becomes $\mu_{\Phi}(\mu)<\mu$ at $\mu<\mu^{\rm conf}_o$. Therefore, there is a second pole in the fion propagator at $p=\mu^{\rm conf}_o$
( the first pole is at $\mu^{\rm pole}_1(\Phi)\sim\mph\gg\la$, see the Appendix A). Moreover, because the frozen value $\mu_{\Phi}(\mu=\mcs)$ of $\mu_{\Phi}(\mu)$ is $\mu_{\Phi}(\mu=\mcs)<\mcs$, there is a third pole in the fion propagator at $p=\mu_{\Phi}(\mu=\mcs)<\mcs$. Therefore, there are now three generations of fions at $\la<\mph<
\mph^{(S,{\,\rm conf})}$ with three different observable pole masses. In addition to the first universal generation with the very large pole mass $\mu^{\rm pole}_1(\Phi)\gg \la$, there are in this case two additional generations with parametrically smaller pole masses
μ^pole_2(Φ)=μ^conf_o∼(/)^N_F/3(2N_c-N_F)> ,
μ^pole_3(Φ)=μ_S(Φ)∼(/)^3N_c/N_F(m_Q/ )^/N_F<<μ^pole_2(Φ)≪ .
On the whole, the mass spectrum looks in these S-vacua as follows at $\la\ll\mph\ll\mo$.
a) There is a large number of flavored hadrons made of non-relativistic (and weakly confined, the string tension is $\sqrt{\sigma}\sim\lym^{(S)}\ll\mcs$ ) constituent quarks with the masses $\mcs\sim (m_Q\mph)^{1/2}$.
b) There is a large number of gluonia with the mass scale $\sim \lym^{(S)}\sim\la(m_Q\mph/\la^2)^{N_F/3\nd}
\ll\mcs$ .
c) The lightest are $N^2_F$ pions with the masses $\sim m_Q$.
d) Besides, at $\la<\mph<\mph^{(S,{\,\rm conf})}$, there are two additional generations of $N^2_F$ fions with the pole masses (5.14). The fions are effectively massless and dynamically relevant in the range of scales $\mu^{\rm pole}_3(\Phi)<\mu<\mu^{\rm pole}_2(\Phi)$. At $\mph>\mph^{(S,{\,\rm conf})}$ these additional poles in the fion propagator are absent and fions are dynamically irrelevant at $\mu<\la$.
Finally, all condensates and observable masses evolve to those in the standard SQCD at $\mph>\mo$, see section 3 in [6].
§ DUAL THEORY. UNBROKEN FLAVOR SYMMETRY
The Lagrangian of the dual theory [4, 5] at the scale $\mu=\la$ is given in (1.6).
From (1.6), the running mass of mions at $\mu\sim \la$ (and not too large $\mph$, see below) is $\sim \la^2/\mph\ll \la$ and only decreases at lower energies, so that mions are effectively massless at least in the interval of scales $\mu_H<\mu<\la$, where $\mu_H$ is the largest physical mass. Therefore, the regime is conformal at $\mu_H<\mu<\la$
and, with $|\Lambda_q|=\la$, both direct and dual theories enter the conformal regime simultaneously at $\mu<\la$.
6.1 L - vacua
The running mass $\mu_{q,L}\equiv \mu_{q,L}(\mu=\la)$ of dual quarks ${\ov q},\,q$ and their pole mass in these $(2N_c-N_F)$ dual L - vacua look as (we recall that $\langle M\rangle=\langle\Pi\rangle$ in all vacua, see section 1
and (3.3) )
μ_q,L/=⟨M⟩_L/^2∼(/)^/2N_c-N_F , μ^pole_q,L/=(μ_q,L/)^N_F/3∼(/)^N_F/3(2N_c-N_F) , ≪≪ ,
μ_q,L/=⟨M⟩_L/^2∼(m_Q/)^/N_c , μ^pole_q,L/=(μ_q,L/)^N_F/3∼(m_Q/)^N_F/3N_c , ≫.
The value of the dual quark constituent mass can be found from the Konishi anomaly (1.8)
()^2/^2=⟨N(μ=)⟩_L=⟨qq(μ=)⟩_L/^2=⟨ S⟩_L/⟨M⟩_L∼(/)^N_c/(2N_c-N_F) ,
≪≪ ,
m_Q/ , ≫ .Therefore,
μ^pole_q,L/∼(/)^2N_F-3N_c/6(2N_c-N_F)≪1 , ≪≪ , 3/2N_c<N_F<2N_c ,
and this shows that the dual quarks ${\ov q},\,q$ are in the DC phase where they acquire the large constituent mass
$\dl$ and $N_F^2$ lighter dual pions $N^i_j$ (nions) are formed, see section 5 in [6]. Hence, the regime is conformal at $\mu_H=\dl<\mu<\la$ , while the heavy constituent quarks decouple and the RG flow of mion fields becomes frozen at $\mu<\dl$. Therefore, proceeding as in [6] and integrating out the dual constituent quarks as heavy ones at $\mu<\dl$ and then the dual gluons at $\mu<\lym^{(L)}$ , one obtains the Lagrangian of mions and nions (all fields are always normalized at $\mu=\la,\,\,{\rm \bd}=3\nd-N_F=2N_F-3N_c$ )
K=[ z^(L)_M/^2 Tr ( M^† M) + Tr √(N^†N)] ,
z^(L)_M=(/)^2/N_F≫1 , ≪≪ ,
W=[1/Tr (- M N )+ N_c( N/Λ_Q^ )
^1/N_c+m_QTr M-1/2(Tr (M^2)- 1/N_c(Tr M)^2) ] .
The main contribution to the mion mass at $\la\ll\mph\ll \mo$ originates from the term $\sim M^2/\mph$ in (6.4)
μ_L(M)∼^2/z^(L)_M∼(/ )^/N_F(2N_c-N_F) ,
μ_L(M)/∼(/)^/2N_F≪1 .
The running mion mass $\mu_{M}(\mu)$ at the scale $\mu\sim \la$ is $\mu_{M}(\mu\sim\la)\sim\la^2/\mph\ll\la$, i.e. it is effectively massless and so dynamically relevant. With decreasing scale $\mu_{M}(\mu)$ decreases but more slowly than the scale $\mu$ itself, because $\gamma_M= - (2{\rm\bd}/N_F),\,\,|\gamma_M|<1$ at $3/2<N_F/N_c<2$ . Hence, if nothing prevents, the mion becomes too heavy and dynamically irrelevant at scales (compare with (4.1) )
μ_M(μ)=^2/z_M(μ)=^2/(μ/)^2/N_F>μ , i. e. at μ<μ^ conf_o=μ^conf_o=(/)^N_F/3(2N_c-N_F) ,
μ^conf_o/μ^L_C∼(/)^/6(2N_c-N_F)≪1 ,
μ^conf_o/μ_L(M)∼(/)^(2N_F-3N_c)^2/3N_F(2N_c-N_F)≪1 ,
<< .
Therefore, the hierarchy of masses at $\la<\mph<\mo$ looks as : ${\ov\mu}^L_C>{\mu}_L(M)>\mu^{\rm conf}_o$. Now, at scales $\mu<{\ov\mu}^L_C$ the mass ${\mu}_L(M)>\mu^{\rm conf}_o$ does not run any more. Hence, there is no pole in the mion propagator at $p\sim \mu^{\rm conf}_o$ but there is one at $p\sim {\mu}_L(M)\gg\mu^{\rm conf}_o$, so that ${\mu}_L(M)$ in (6.5) is the only pole mass of mions in this range of $\mph$ in these dual L - vacua.
The mions $M$ are much heavier than nions $N$ at $\la\ll\mph\ll\mo$ . Hence, they can be integrated out in (6.4) and one obtains
K=Tr √(N^†N) , W=N_c( N/Λ_Q^ )
^1/N_c+/2(Tr (N^2)-1/(Tr N)^2), N=(m_Q-N/) .
From (6.7), the nion mass looks in these dual L - vacua as
μ_L(N)∼⟨N ⟩_L ∼{
(/)^/(2N_c-N_F) at ≪≪
m_Q/ at ≪≪μ^ dSQCD_Φ
At $\mph\gtrsim\mo\sim \la(\la/m_Q)^{(2N_c-N_F)/N_c}$ the term $m_Q{\rm Tr}M$ cannot be neglected in (6.4) any more and the vacuum values $\langle M\rangle_L$ and $\langle N\rangle_L$ match those in the dual SQCD ( dSQCD ) :
$\langle M\rangle_{dSQCD}\sim \la^2(m_Q/\la)^{\nd/N_c}$ , $\langle N \rangle_{dSQCD}\sim m_Q\la$, and $z_M^{(L)}\sim z_M^{\rm dSQCD}\sim (\la/m_Q)^{\bd/N_F}$. But the mion and nion masses are not matched yet : ${\mu}^{\,\rm pole}_{L} (M)\gg {\mu}_{\rm dSQCD}(M)$ , while ${\mu}_{L} (N)\ll {\mu}_{\rm dSQCD}(N)$ at $\mph\sim\mo$. From (6.4), with increasing $\mph>\mo$ , the increasing mass ${\mu}_{L} (N)$ and decreasing mass ${\mu}^{\,\rm pole}_{L} (M)$ both match those in the dSQCD at $\mph\sim{\ov\mu}^{\,\rm dSQCD}_{\Phi}$ and stay then at their dSQCD values at $\mph>{\ov\mu}^{\,\rm dSQCD}_{\Phi}$ , see section 5 in [6] (and the hierarchies $\mu^{\rm conf}_o/{\ov\mu}^{L}_C\ll 1$ and $\mu^{\rm conf}_o/\mu^{\rm pole}_{L}(M)\ll 1$ are maintained at $\mph>\mo$) , where
μ_L (N)∼m_Qμ^ dSQCD_Φ/∼μ_ dSQCD(N)∼(m_Q/)^3/2N_F μ^ dSQCD_Φ∼(/m_Q)^/2N_F,
μ^ pole_L (M)∼^2/μ^ dSQCD_Φ[1/z_M^(L)∼(m_Q/)^/N_F]∼μ_ dSQCD(M)∼(m_Q/)^3/2N_F μ^ dSQCD_Φ∼(/m_Q)_.^/2N_F
On the whole, the mass spectrum in these $(2N_c-N_F)$ L - vacua of dSQCD look as follows.
a) There is a large number of heaviest flavored hadrons, mesons and baryons, made of non-relativistic and weakly confined (the string tesion is $\sqrt\sigma\sim\lym^{(L)}\ll \dl$) constituent dual quarks with the masses : $\dl\sim\la(\la/\mph)^{N_c/2(2N_c-N_F)}$ at $\la\ll \mph\ll\mo$ and $\dl\sim (m_Q\la)^{1/2}$ at $\mph\gg\mo$.
b) A large number of gluonia with the mass scale : $\lym^{(L)}=\langle S\rangle^{1/3}_L\sim\la(\la/\mph)
^{N_F/3(2N_c-N_F)}$ at $\la\ll \mph\ll\mo$, and $\lym^{(L)}\sim (\la^{\bo} m_Q^{N_F})^{1/3N_c}\sim \lym^{(\rm SQCD)}$ at
c) $N_F^2$ mions $M$ with the pole masses : $\mu^{\rm pole}_L(M)\sim\la(\la/\mph)^{\nd\bo/N_F(2N_c-N_F)}\ll\dl$ at $\la\ll \mph\ll{\ov\mu}^{\,\rm dSQCD}_{\Phi}$ , and $\mu^{\rm pole}_L(M)\sim \la\Bigl (m_Q/\la\Bigr )^{3\nd/2N_F}\sim \mu^{\,\rm pole}_{\rm dSQCD}(M)\ll\dl$ at $\mph\gg{\ov\mu}^{\,\rm dSQCD}_{\Phi}\gg\mo$.
d) $N_F^2$ nions $N$ with the masses (6.8) at $\la\ll\mph\ll{\ov\mu}^{\,\rm dSQCD}_{\Phi}$, and $\mu_L(N)\sim
\la\Bigl (m_Q/\la\Bigr )^{3\nd/2N_F}\\ \sim \mu_{\,\rm dSQCD}(N)$ at $\mph\gg{\ov\mu}^{\,\rm dSQCD}_{\Phi}$.
Therefore, the masses of constituent quarks and gluonia match those in dSQCD at $\mph>\mo$, while the masses of mions and nions match those in dSQCD at $\mph>{\ov\mu}^{\,\rm dSQCD}_{\Phi}\gg\mo$ only. The mions are dynamically relevant in the range of scales ${\mu}^{\,\rm pole}_L (M)<\mu<\la$ in these L-vacua at all values $\mph\gg \la$.
6.2 S - vacua
Proceeding as in the previous section 6.1, the running mass $\mu_{q,S}\equiv \mu_{q,S}(\mu=\la)$ of dual quarks ${\ov q},\,q$ and their pole mass in these $(N_F-N_c)$ dual S-vacua are
μ_q,S=⟨M⟩_S/∼m_Q/ , μ^pole_q,S/=(μ_q,S/)^N_F/3∼(m_Q/^2)^N_F/3 , ≪≪ .
μ_q,S/=⟨M⟩_S/^2∼(m_Q/)^/N_c , μ^pole_q,S/=(μ_q,S/)^N_F/3∼(m_Q/)^N_F/3N_c , ≫.The value of the dual quark constituent mass can be found from the Konishi anomaly (1.8)
()^2/^2=⟨N⟩_S=⟨qq(μ=)⟩_S/^2=⟨S⟩_S/⟨M⟩_S∼(m_Q/^2)^N_c/ , ≪≪ ,
m_Q/ , ≫ .Therefore,
μ^pole_q,S/∼(m_Q/^2)^(2N_F-3N_c)/6≪1 ,
≪≪ , 3/2N_c<N_F<2N_c ,
and this shows that dual quarks ${\ov q},\,q$ are here also in the DC phase where they acquire the constituent mass
$\ds$ and $N_F^2$ lighter dual pions $N^i_j$ (nions) are formed, see section 5 in [6].
The regime is conformal at $\ds<\mu<\la$, while the constituent dual quarks decouple at $\mu<\ds$ and the RG evolution of the mion renormalization factor $z_M^{(S)}(\mu)$ becomes frozen, $z_M^{(S)}=z_M^{(S)}(\mu<\ds)=z_M^{(S)}(\mu=\ds)=
(\la/\ds)^{2\bd/N_F}$. Hence, after integrating out dual constituent quarks at $\mu<\ds$ and dual gluons at $\mu<\lym^{(S)}$ one obtains the Lagrangian (6.4) with a replacement $z_M^{(L)}\ra z_M^{(S)}$ .
As for the low energy value of the mion mass in these $(N_F-N_c)$ dual S-vacua, it looks at $\la\ll\mph\ll\mo$ as
μ_S(M)∼^2/z_M^(S)∼^2/(m_Q/^2)^N_c/N_F .
But comparing the quark constituent mass $\ds$ with $\mu^{\rm conf}_o$ from (6.6) one obtains (compare with (5.10) )
/μ^conf_o≫1 ≫μ^ (S, conf)_Φ∼(/m_Q)^3(2N_c-N_F)/N_F+2N_c≫μ_Φ^(S, conf) ,
while ${\mu^{\rm conf}_o}\gg\ds$ at $\la\ll\mph\ll{\ov\mu}^{\,(S,{\,\rm conf})}_{\Phi}\ll\mo$.
Hence, the hierarchy of scales and masses looks here as $\la>\ds>\mu_S(M)>\mu^{\rm conf}_o$ at $\mph>{\ov\mu}^{\,(S,{\,\rm conf})}_{\Phi}$ only. This means that only in this region, as in the preceding section 6.1, the mass $\mu_S(M)$ in (6.13) is the mion pole mass and the mions are dynamically relevant at scales $\mu^{\rm pole}_S(M)<\mu<\la$.
The hierarchy of scales and masses looks as $\mu^{\rm conf}_o>\mu_S(M)>\ds$ at $\la\ll\mph\ll {\ov\mu}^{\,(S,{\,\rm conf})}_{\Phi}$. Therefore, there is no pole in the mion propagator at this frozen low energy value $p\sim\mu_S(M)$ of the mion mass term. But, because the running mion mass $\mu^{(S)}_M(\mu)$ behaves in this range of $\mph$ as
in the vicinity of $\mu^{\rm conf}_o$, it is $\mu^{(S)}_M(\mu)<\mu^{\rm conf}_o$ at $\mu>\mu^{\rm conf}_o$ and $\mu^{(S)}
_M(\mu)>\mu^{\rm conf}_o$ at $\mu<\mu^{\rm conf}_o$, so that there is a pole in the mion propagator at $p\sim \mu^{\rm conf}_o$ and, besides, the mions are effectively massless and dynamically relevant in the range of scales $\mu^{\rm pole}_S(M)=\mu^{\rm conf}_o<\mu<\la$.
In any case, the mions $M$ are much heavier than nions $N$ at $\la\ll\mph\ll{\ov\mu}^{\,\rm dSQCD}_{\Phi}$ . Hence, they can be integrated out and one obtains (6.7). From this, the mass of nions looks in these dual S - vacua as
μ_S(N)∼m_Q/ at ≪≪μ^ dSQCD_Φ .
On the whole, the mass spectrum in these $(N_F-N_c)$ S - vacua of dSQCD looks as follows.
a) There is a large number of flavored hadrons, mesons and baryons, made of non-relativistic and weakly confined (the string tesion is $\sqrt\sigma\sim\lym^{(S)}\ll \ds$) constituent dual quarks with the masses $\ds\sim\la
(m_Q\mph/\la^2)^{N_c/2\nd}$ at $\la\ll \mph\ll\mo$.
b) A large number of gluonia with the mass scale : $\lym^{(S)}=\langle S\rangle^{1/3}_S\sim\la(m_Q\mph/\la^2)
^{N_F/3\nd}$ at $\la\ll \mph\ll\mo$ and $\lym^{(S)}\sim (\la^{\bo} m_Q^{N_F})^{1/3N_c}\sim\lym^{(\rm SQCD)}$ at
c) $N_F^2$ mions $M$ with the pole masses : $\mu^{\rm pole}_S(M)\sim \mu^{\rm conf}_o\sim\la\Bigl (\la/\mph\Bigr )^{N_F/3(2N_c-N_F)}$ at $\la\ll\mph\ll{\ov\mu}^{\,(S,{\,\rm conf})}_{\Phi}\ll\mo$ and (6.13) at ${\ov\mu}^{\,(S,{\,\rm conf})}_{\Phi}\ll\mph\ll\mo$.
d) $N_F^2$ dual pions $N$ (nions) with the masses (6.16).
All condensates and mass spectra in these S - vacua and in the L - vacua from the preceding section 6.1 degenerate and become the same at $\mph\gg\mo\sim \la(\la/m_Q)^{(2N_c-N_F)/N_c}$.
§ DIRECT THEORY. BROKEN FLAVOR SYMMETRY.
THE REGION $\MATHBF {\LA<\MPH<\MO}$
7.1 L - type vacua
In these vacua
It will be implied everywhere below in the text that the numbers $n_1$ and $n_2$ are such that $1-(n_1/N_c),\, 1-(n_2/N_c)$ and $1-(N_F/2N_c)$ are all $O(1)$. The only exception considered explicitly below will be the special vacua with $n_2=N_c,\,n_1=\nd$.
and in this range of $\mph$, the term $m_Q{\ov Q}Q$ in the superpotential can be neglected and the parametric behavior is the same as in L - vacua in section 5.1. Therefore, all quarks are in the DC phase. The difference with previous calculations in section 5 in this region is that the flavor symmetry is broken spontaneously in a large number of these vacua with $\langle\Pi_1\rangle\neq\langle\Pi_2\rangle\,,\,\,\langle\Pi_1\rangle=\langle{\ov Q}_1Q_1\rangle\sim\langle
\Pi_2\rangle=\langle{\ov Q}_2 Q_2\rangle\gg (m_Q\mph)$, while the classical S - type vacua are absent.
Hence, proceeding as before and integrating out first all heaviest constituent quarks with the masses $\mu_{C,1}=\langle
\Pi_1\rangle^{1/2}\sim\mu_{C,2}=\langle\Pi_2\rangle^{1/2}\sim\langle\Pi\rangle_L^{1/2}\sim\la(\la/\mph )^{\nd/2(2N_c-N_F)}\gg (m_Q\mph)^{1/2}$ and then gluons at $\mu<\lym^{\rm (br)}\sim\lym^{(L)}$, one obtains the Lagrangian of $N_F^2$ pions
K=Tr √(Π^†Π) , W= - S+m_QTr Π-1/2[Tr (Π^2)- 1/N_c(Tr Π)^2
] ,S=(Π/^)^1/, (^(br))^3=⟨S⟩_br=(⟨Π_1⟩^n_1⟨Π_2⟩^n_2/^)^1/
=⟨Π_1⟩⟨Π_2⟩/ ,
⟨Π_1⟩+⟨Π_2⟩=m_Q+1/N_cTr ⟨Π⟩≃1/N_c
(n_1⟨Π_1⟩+n_2⟨Π_2⟩).From (7.1)
m_Q≪⟨Π_1⟩_Lt∼⟨Π_2⟩_Lt∼^2(/)^/2N_c-N_F≪^2 , ≪≪∼(/m_Q)^2N_c-N_F/N_c .
As a check of self-consistency, we have now instead of (5.2)
⟨m^tot_Q,1⟩=⟨Π_2⟩/ , ⟨m^tot_Q,2⟩=⟨Π_1⟩/∼⟨m^tot_Q,1⟩ ,
so that (5.4) is parametrically the same.
On the whole, the main qualitative difference in the mass spectra in comparison with the L - vacua $\langle\Pi\rangle_L$
with unbroken flavor symmetry in section 5.1 is that the hybrid pions $\Pi_{12}$ and $\Pi_{21}$ are Nambu-Goldstone particles here and are exactly massless.
7.2 br2 - vacua
In these vacua with $n_2>N_c,\, n_1<\nd<N_c$ , see (3.9),
⟨Π_2⟩_br2≃(ρ_2=-N_c/n_2-N_c)m_Q , ⟨Π_1⟩_br2∼^2(/)^n_2/n_2-N_c(m_Q/)^N_c-n_1/n_2-N_c
⟨Π_1⟩_br2/⟨Π_2⟩_br2∼(/)^N_c/n_2-N_c≪1 , ≪≪∼(/m_Q)^2N_c-N_F/N_c .To see what is the phase in this case we look at hierarchies of possible masses. The pole masses of quarks $Q_1$
and $Q_2$ look as
⟨m^tot_Q,1⟩=⟨Π_2⟩/∼m_Q , ⟨m^tot_Q,2⟩/⟨m^tot_Q,1⟩=⟨Π_1⟩/⟨Π_2⟩≪1 ,
m_Q,1^pole∼⟨m^tot_Q,1⟩/z_Q(, m^pole_Q,1)∼(m_Q/)^N_F/3N_c≫m_Q,2^pole , z_Q(,μ≪)=(μ/)^/N_F ,while their constituent masses look as
μ_C,2∼⟨Π_2⟩^1/2∼(m_Q)^1/2≫μ_C,1∼⟨Π_1⟩^1/2 .
The masses of gluons due to possible higgsing of quarks look as
μ_gl,2∼[⟨Π_2⟩z_Q(,μ_gl,2)]^1/2∼(⟨Π_2⟩/)^N_F/3≫μ_gl,1 .
Because from (7.5),(7.6)
m_Q,1^pole/μ_C,2∼(/)^1/2(m_Q/)^2N_F-3N_c/6N_c≪1 and μ_gl,2/m_Q,1^pole∼(/)^N_F/3≪1 ,
this shows that the quarks ${\ov Q}_2, Q_2$ are in the $DC_2$ - phase in the whole region $\la\ll\mph\ll\mo$.
But, as for the quarks ${\ov Q}_1, Q_1$,
μ_C,1/m_Q,1^pole∼(/)^n_2/2(n_2-N_c)(m_Q/)^δ/6N_c(n_2-N_c) , δ=[ n_1-(2 N_F-3N_c^2) ] , =2N_F-3N_c .
It follows from (7.9) that
a1) if $(2N_F\nd-3N_c^2)<0$, then $\delta>0$ and
μ_C,1/ m_Q,1^pole≫1 ≫μ_Φ ,
μ_Φ=(/m_Q)^δ>0/3n_2N_c≪ ,
so that at all $0<n_1<\nd$ the overall phase is $DC_1-DC_2$ at ${\widehat\mu}_{\Phi}\ll\mph\ll\mo$ and $HQ_1-DC_2$ at $\la\ll\mph\ll{\widehat\mu}_{\Phi}$ ;
a2) if $(2N_F\nd-3N_c^2)>0$, then $\mu_{C,1}\gg m_{Q,1}^{\rm pole}$ at $\delta<0$, i.e. the overall phase is $DC_1-DC_2$ at $0<n_1<n^o_1=(2N_F\nd-3N_c^2)/\bd$ only, but in the whole region $\la\ll\mph\ll\mo$ ;
a3) $\delta>0$ at $(2N_F\nd-3N_c^2)>0$, so that the phases is as in 'a1' above but now at $n^o_1<n_1<\nd$ only.
We start with the $DC_1-DC_2$ phase and recall that the largest constituent mass $\mu_{C,2}$ of ${\ov Q}_2, Q_2$ quarks is formed in this phase not at the scale $\mu\sim\mu_{C,2}$, but both constituent masses $\mu_{C,2}$ and $\mu_{C,1}$ are formed at the smaller scale $\mu\sim\mu_{C,1}$ [7]. Therefore, the RG flow of quarks and gluons is conformal down to $\mu\sim\mu_{C,1}\ll\mu_{C,2}$. , integrating out all constituent quarks at $\mu<\mu_{C,1}$ and then all gluons at $\mu<\lym^{(\rm br2)}\sim (m_Q\langle \Pi_1\rangle)^{1/3}$, the lower energy Lagrangian looks as
K=Tr [ z_Φ(,μ_C,1)(Φ^†Φ)+√(Π^†Π) ], z_Φ=
(/μ_C,1)^2/N_F=(^2/⟨Π_1⟩)^/N_F≫1 ,W=-S+/2[Tr (Φ^2) -1/(Tr Φ)^2]+Tr (m_Q-Φ)Π , S=(Π/^)^1/ ,
and one has to choose the br2 - vacua in (7.11).
To see whether fions are relevant or not in this $DC_1-DC_2$ phase, we compare $\mu_{C,1}$ and $\mu_o^{\rm conf}$, see (4.1),
⟩^1/2>1 ≪<^(relev)=(/m_Q)^ρ≪ ,
ρ=3(N_c-n_1)(2N_c-N_F)/2N_F(n_2-N_c)+3n_2(2N_c-N_F)>0 .Therefore, the fions are dynamically relevant in this $DC_1-DC_2$ phase at $\la\ll\mph<\mph^{\rm (relev)}$ and become irrelevant at $\mph>\mph^{\rm (relev)}$. Hence, at $\la\ll\mph\ll\mph^{\rm (relev)}$, there is the second generation of
all $N_F^2$ fions with the pole masses $\mu_{2}^{\rm pole}(\Phi_{ij})=\mu_o^{\rm conf}\gg\mu_{C,1}$ .
From (7.11), the mass terms of hybrids $\Phi_{12}, \Phi_{21}$ and $\Pi_{12}, \Pi_{21}$ look as in (2.23), but instead of (2.24) one has now
m_ϕ=/z_Φ(,μ_C,1) , m_π=⟨Π_1+Π_2⟩/ , m^2_ϕπ= m_ϕm_π .
The exact equality $m^2_{\phi\pi}= m_{\phi}m_{\pi}$ ensures that one of the two eigenvalues is zero. As one can check, the mixing $\phi_{12}\leftrightarrow\pi_{12}$ is parametrically small, so that the massless particles are mainly pions $\pi_{12},\pi_{21}$, while the heavy hybrids are mainly $\phi_{12},\phi_{21}$,
μ_3^pole(Φ_12)=μ_3^pole(Φ_21)∼/z_Φ(,μ_C,1)∼(⟨Π_1⟩/^2)^/N_F .
Further, the mixings $\phi_{11}\leftrightarrow\pi_{11}$ and $\phi_{22}\leftrightarrow\pi_{22}$ are also parametrically small and fions are much heavier than pions, with their third generation pole masses
μ_3^pole(Φ_11)≃μ_3^pole(Φ_22)∼/z_Φ(,μ_C,1)∼(⟨Π_1⟩/^2)^/N_F .
Hence, after integrating all fions one obtains the superpotential of pions
W= - S+m_QTr Π-1/2[Tr (Π^2)- 1/N_c(Tr Π)^2
] ,
and one has to choose br2 - vacua in (7.16). Then one obtains for the pion masses
μ(Π_11)≃μ(Π_22)∼⟨Π_2⟩/∼m_Q .
On the whole for the mass spectrum in this $DC_1-DC_2$ phase.
1) The heaviest are 22-flavored hadrons made of the constituent quarks ${\ov Q}_2, Q_2$ with the masses $\mu_{C,2}\sim (m_Q\mph)^{1/2}$.
2) The next mass scale is due to 11-flavored hadrons made of the constituent ${\ov Q}_1, Q_1$ quarks with the masses $\mu_{C,1}\sim\langle\Pi_1\rangle^{1/2}\ll\mu_{C,2}$.
3) The gluonia with the mass scale $\lym^{\rm (br2)}\sim ( m_Q\langle\Pi_1\rangle)^{1/3}\ll \mu_{C,1}$.
4) $n_1^2$ pions $\Pi_{11}$ and $n_2^2$ pions $\Pi_{22}$ with the masses $\sim m_Q\ll\lym^{\rm (br2)}$.
5) $2 n_1 n_2$ massless hybrids $\Pi_{12},\Pi_{21}$.
And finally, as for the fions.
a1) At $(2N_F\nd-3N_c^2)<0,\,n_1>0,\, \delta>0$ and when $\mph$ is in the interval ${\widehat\mu}_{\Phi}\ll\mph\ll\mu^{\rm (relev)}_{\Phi}$, see (7.10),(7.12), all $N_F^2$ fions appear at scales $\mu<\la$ in the two generations with the pole masses $\mu_{2}^{\rm pole}(\Phi_{ij})\sim\mu_o^{\rm conf}\gg\mu_{C,1}$ and $\mu_{3}^{\rm pole}(\Phi_{ij})\ll\mu_{C,1}$, see (7.14),(7.15). They are dynamically relevant then in the range of scales $\mu_{3}^{\rm pole}(\Phi_{ij})<\mu<\mu_{2}^{\rm pole}(\Phi_{ij})$. But there are no poles in the fion propagators at $\mu<\la$ and they become dynamically irrelevant at $\mu^{\rm (relev)}_{\Phi}\ll\mu_{\Phi}\ll\mo$.
a2) At $(2N_F\nd-3N_c^2)>0,\,n_1<n^o_1,\,\delta<0$ - the same as in 'a1' above but the fions are relevant now in a much wider interval $\la\ll\mph\ll\mu^{\rm (relev)}_{\Phi}$.
a3) At $(2N_F\nd-3N_c^2)>0,\, \delta>0$ - the same as in 'a1' above but at $ n_1>n^o_1$ only.
We consider now the $HQ_1-DC_2$ phase and proceed in this case similarly to that in section 4 of [7] where the standard SQCD in this phase was considered. The difference is due to fions and their Yukawa interactions with quarks. We recall also that the conformal regime is maintained in this phase in the range of scales $m^{\rm pole}_{Q,1}<\mu<\la$. After integrating out the constituent quarks ${\ov Q}_2, Q_2$ and the quarks ${\ov Q}_1, Q_1$ as heavy ones at $\mu<m^{\rm pole}_{Q,1}$, the Lagrangian takes the form
K=[ z_Φ(, m^pole_Q,1)Tr (Φ^†Φ)+Tr √(Π^†_22Π_22) ] , W=[ -2π/α(μ)S ]+W_Φ+W_Π ,
W_Φ=/2[Tr (Φ^2) -1/(Tr Φ)^2],
W_Π=-n_2(Π_22/^m^tot_Q,1)^1/n_2-N_c+Tr Π_22(m^tot_Q,2-Φ_211/m^tot_Q,1Φ_12),z_Φ(, m^pole_Q,1)=(/m^pole_Q,1)^2/N_F , m^tot_Q,1=(m_Q-Φ_11) , m^tot_Q,2=(m_Q-Φ_22) ,with pions $\Pi_{22}$ and $m^{\rm tot}_{Q,1}$ sitting on $\lym^{(\rm br2)}$ inside $\alpha(\mu)$ in (7.18). The pions $\Pi_{22}$ and all fions are frozen and do not evolve any more at $\mu<m^{\rm pole}_{Q,1}$. Further, after integrating out gluons at $\mu<\lym^{(\rm br2)}$ through the VY - procedure [9], the superpotential looks as
W=W_Φ-(n_2-N_c)(Π_22/^m^tot_Q,1)^1/n_2-N_c+Tr Π_22(m^tot_Q,2-Φ_211/m^tot_Q,1Φ_12) .
One obtains from the above for the masses of fions and pions in this $HQ_1-DC_2$ phase.
1) Because $m^{\rm pole}_{Q,1}\ll\mu_o^{\rm conf}$ in the region $\la\ll\mph\ll {\widehat\mu}_{\Phi}\ll\mo$
with this phase, there is the second generation of all $N_F^2$ fions with the pole masses $\mu_2^{\rm pole}(\Phi_{ij})\sim\mu_o^{\rm conf}$.
2) There is the third generation of $\Phi_{11}$ and $\Phi_{22}$ fions with the pole masses (the mixing of $\Phi_{22}$ and $\Pi_{22}$ is parametrically small)
μ_3^pole(Φ_11)∼μ_3^pole(Φ_22)∼/z_Φ(,m^pole_Q,1)∼(m_Q/)^2/3N_c≪m^pole_Q,1∼(m_Q/)^N_F/3N_c ,
and so the fions $\Phi_{11}$ and $\Phi_{22}$ are dynamically relevant in the range of scales $\mu_3^{\rm pole}(\Phi_{11})<\mu<\mu_o^{\rm conf}$.
3) The mass of $\Pi_{22}$ pions is $\mu(\Pi_{22})\sim m_Q\ll \mu_3^{\rm pole}(\Phi_{22})$.
4) The third generation hybrids are massless, $\mu_3^{\rm pole}(\Phi_{12})=\mu_3^{\rm pole}(\Phi_{21})=0$.
In addition, there are in a mass spectrum : the heaviest 22-flavored hadrons made of the constituent quarks ${\ov Q}_2, Q_2$ with the masses $\mu_{C,2}\sim (m_Q\mph)^{1/2}$, the next mass scale is due to 11-flavored hadrons made of ${\ov Q}_1, Q_1$ quarks with the masses $m^{\rm pole}_{Q,1}\ll\mu_{C,2}$ and, finally, there are gluonia with the mass scale $\sim\lym^{\rm (br2)}\sim (m_Q\langle\Pi_1\rangle)^{1/3}\ll m^{\rm pole}_{Q,1}$.
7.3 Special vacua, $\mathbf{n_2=N_c,\, n_1=\nd}$
In these vacua at $\la\ll\mph\ll\mo$, see (3.7),(3.10),
⟨Π_1⟩_spec=N_c/2N_c-N_F(m_Q) , ⟨Π_2⟩_spec=^2(/)^/2N_c-N_F , ⟨Π_1⟩_spec/⟨Π_2⟩_spec∼(/)^N_c/2N_c-N_F≪1
The most important possible masses look here as follows
μ_C,2∼⟨Π_2⟩^1/2_spec , μ_C,1∼⟨Π_1⟩^1/2_spec∼(m_Q)^1/2≪μ_C,2 ,
⟨m^tot_Q,1⟩=⟨Π_2⟩_spec/∼(/)^N_c/2N_c-N_F m^pole_Q,1∼(/)^N_F/3(2N_c-N_F)
≫m^pole_Q,2 ,μ^2_gl,2∼(a_*∼1)⟨Π_2⟩_spec(μ_gl,2/)^/N_F
μ_gl,2∼(/)^N_F/3(2N_c-N_F)∼m^pole_Q,1≫μ_gl,1 ,m^pole_Q,1/μ_C,2∼(/)^/6(2N_c-N_F)≪1 ,(μ_C,1/m^pole_Q,1)^2∼m_Q/(/)^
6N_c-N_F/3(2N_c-N_F)> 1 >μ^(DC)_Φ∼(/m_Q)^3(2N_c-N_F)/6N_c-N_F≪ ,
where $\mu_{C,2}$ is the possible constituent mass of ${\ov Q}_2, Q_2$ quarks and $\mu_{\rm gl,2}$ is the gluon mass due to their possible higgsing. Because $\mu_{\rm gl,2}\sim m^{\rm pole}_{Q,1}$ it is unclear beforehand whether the phase is $DC_2-HQ_1$ or $Higgs_2-HQ_1$. But an attempt to write the standard superpotential for the $DC_2-HQ_1$ phase shows that it will be singular at $n_2=N_c$ [7] and so the phase $DC_2-HQ_1$ cannot be realized in these special vacua (at least in a standard way). We assume here that the overall phase is $HQ_1-Higgs_2$ and the whole gauge group is higgsed at $\la\ll\mu\ll\mu^{(\rm DC)}_{\Phi}$, while the phase will be $DC_1-DC_2$ at $\mu^{(\rm DC)}_{\Phi}\ll\mu\ll\mo$.
We start with the $HQ_1-Higgs_2$ phase. Supposing that $m^{\rm pole}_{Q,1}=(\rm several)\mu_{\rm gl,2}$ and integrating out first the quarks ${\ov Q}_1, Q_1$ as heavy ones at $\mu<m^{\rm pole}_{Q,1}$ and then all higgsed gluons and their superpartners at $\mu<\mu_{gl,2}$, the Lagrangian takes the form
K=Tr [ z_Φ(Φ^†Φ)+ z_Q( 2√(Π^†_22Π_22 )+B^†_2 B_2+
B^ †_2B_2 )
] ,
z_Q=z_Q(,m^pole_Q,1)=(m^pole_Q,1/)^/N_F , z_Φ=z_Φ(,m^pole_Q,1)=1/z^2_Q ,W=W_non-pert+W_Φ+Tr Π_22(m^tot_Q,2-Φ_211/m^tot_Q,1Φ_12
), W_Φ=/2[Tr (Φ^2) -1/(Tr Φ)^2],
m^tot_Q,1=m_Q-Φ_11 , m^tot_Q,2=m_Q-Φ_22,where for the non-perturbative term we use the form proposed in [4]
W_non-pert=A[ 1-Π_22/λ^2N_c+B_2 B_2/λ^2 ] ,
⟨A⟩=⟨S⟩ , λ^2=(^m^tot_Q,1)^1/N_c, ⟨λ^2⟩=⟨Π_2⟩,
in which $A$ is the auxiliary field.
From (7.24),(7.25), the hybrids $\Phi_{12}, \Phi_{21}$ are massless, the baryons ${\ov B}_2,\, B_2$ are light
μ(B_2)=μ(B_2)∼m_Q/z_Q∼m_Q(/)^/3(2N_c-N_F)≪μ_gl,2 ,
while all other masses are parametrically $\sim\mu_{\rm gl,2}\sim m^{\rm pole}_{Q,1}$ (the pion masses increased due to their mixing with the fions). Besides, in particular, because $\mu_o^{\rm conf}\sim m^{\rm pole}_{Q,1}$ in these special vacua, there is no warranty that these nonzero masses of fions $\Phi_{11}$ and $\Phi_{22}$ are the pole masses. Maybe yes, but maybe not (see section 4).
On the whole, there are three scales in the mass spectrum : the hybrid fions $\Phi_{12}, \Phi_{21}$ are massless, the baryons have small masses (7.26), while all other masses are $\mu_{\rm gl,2}\sim m^{\rm pole}_{Q,1}\sim\la(\la/\mph)^{N_F/3(2N_c-N_F)}$ in these special vacua at $\la\ll\mph\ll\mu^{(\rm DC)}_{\Phi}$.
Now, we consider the phase $DC_1-DC_2$ with $\mu_{C,2}\gg\mu_{C,1}\gg\mu_o^{\rm conf}\sim m^{\rm pole}_{Q,1}$ in these special vacua at $\mu^{(\rm DC)}_{\Phi}\ll\mph\ll\mo$. We can proceed then as in this phase in section 7.2 above and to start directly with (7.11). But just because $\mu_{C,1}\gg\mu_o^{\rm conf}$, there are no poles in the fion propagators at all scales $\mu<\la$ and all fions are too heavy and dynamically irrelevant here. Hence, after integrating them out in (7.11), one obtains the superpotential (7.16). From this, the masses of $\Pi_{11}$ and $\Pi_{22}$ pions are as in (7.17), while the hybrids $\Pi_{12}$ and $\Pi_{21}$ are massless. We recall finally that the masses of constituent quarks are here $\mu_{C,2}\sim \langle\Pi_2\rangle_{\rm spec}^{1/2}\gg\mu_{C,1}$ and $\mu_{C,1}\sim\langle\Pi_1\rangle^{1/2}_{\rm spec}\gg m^{\rm pole}_{Q,1}$, and the mass scale of gluonia is $\lym^{(\rm spec)}=[\langle\Pi_1\rangle_{\rm spec}\langle\Pi_2\rangle_{\rm spec}/\mph]^{1/3}\sim\la\,[(m_Q/\la)(\la/\mph)^{\nd/(2N_c-N_F)}]^{1/3}\ll\mu_{C,1}$.
§ DIRECT THEORY. BROKEN FLAVOR SYMMETRY.
THE REGION $\MATHBF {\MO\LL\MPH\LL\LA^2/M_Q}$
8.1 br1 - vacua, $\mathbf{DC_1-DC_2}$ phase
In all L - type, br2 and special vacua the theory enters the region $\mph\sim (\rm several)\mo$ with all quarks in the $DC_1-DC_2$ phase and $\langle\Pi_1\rangle\sim\langle\Pi_2\rangle\sim\la^2(m_Q/\la)^{\nd/N_c}$, see sections 3 and 7. But there appears then a large hierarchy between the quark condensates with increasing $\mph$ at $\mph\gg\mo$ in these br1 - vacua
⟨Π_1⟩≃(ρ_1=N_c/N_c-n_1) m_Q≫⟨Π_2⟩∼^2(/)^n_1/N_c-n_1(m_Q/)^n_2-N_c/N_c-n_1 ,
i.e. the constituent masses $\mu_{C,1}$ of ${\ov Q}_1, Q_1$ quarks become parametrically larger than $\mu_{C,2}$ of ${\ov Q}_2, Q_2$ quarks.
We recall now once more the important feature of the $DC$ phase [7] : the $DC_1$ phase cannot be formed separately, i.e. the largest constituent mass $\mu_{C,1}=\langle\Pi_1\rangle^{1/2}\sim (m_Q\mph)^{1/2}$ of ${\ov Q}_1, Q_1$ quarks is not formed at the scale $\mu\sim\mu_{C,1}$ , but only at the appropriate lower scale $\mu_{\rm lower}$ below which all quark flavors become massive. Therefore, if nothing prevents, this lower scale in the case considered may be either the constituent mass $\mu_{C,2}=\langle\Pi_2\rangle^{1/2}$ of ${\ov Q}_2, Q_2$ quarks or their pole mass $m_{Q,2}^{\rm pole}$ (and both are parametrically smaller now than $\mu_{C,1}$ at $\mph\gg\mo$) , $\mu_{\rm lower}={\rm max}(\,\mu_{C,2}\,,\,m_{Q,2}^{\rm pole}\,)$.
But there is another competing mass $\mu_{\rm gl,1}$ , i.e. the mass of gluons due to possible higgsing of ${\ov Q}_1, Q_1$ quarks,
μ^2_gl,1∼a(μ=μ_gl,1)⟨Π_1⟩z_Q(,μ_gl,1), z_Q(, μ_gl,1)=( μ_gl,1/)^/N_F ,
a(μ)≡N_cα(μ)/2π, a_conf(μ=μ_gl,1≪)=a_*=O(1)<1 .
It is seen from
m_Q,2^pole/∼(m_Q/)^N_F/3N_c , m_Q,2^pole/μ_gl,1∼(/)^N_F/3≪1 ,
that $\mu_{\rm gl,1}\gg m_{Q,2}^{\rm pole}$ at $\mph\gg \mo$ , but as for $\mu_{C,2}$ , one obtains
μ_C,2=⟨Π_2⟩^1/2 , μ_gl,1/μ_C,2>1 at >μ^(higgs)_Φ , /≪μ^(higgs)_Φ/∼(/m_Q)^σ≪/m_Q ,
σ=(N_c-n_1)+3(2N_c-N_F)/2N_F(N_c-n_1)+3n_1>0 .
Therefore, when $\mu_{C,2}\gg\mu_{\rm gl,1}$ at $\mo\ll\mph\ll\mu^{\rm (higgs)}_{\Phi}$ , the quarks still are in the $DC_1-DC_2$ phase where both dynamical constituent masses $\mu_{C,1}$ and $\mu_{C,2}$ are formed simultaneously in the threshold region $[(\rm several)\mu_{C,2}>\mu>\mu_{C,2}/(\rm several)]\gg\mu_{\rm gl,1}$ . But at $\mph\gg\mu^{\rm (higgs)}_{\Phi}$, when $\mu_{\rm gl,1}\gg\mu_{C,2}$ , the quarks ${\ov Q}_1, Q_1$ are higgsed at the higher scale $\mu\sim \mu_{\rm gl,1}\gg\mu_{C,2}$ , before reaching the lower scale $\mu\sim\mu_{C,2}$ where the constituent masses $\mu_{C,1}$ and $\mu_{C,2}$ are formed (this variant was not considered in [7]). Hence, the phase will be either $Higgs_1-DC_2$ or $Higgs_1-HQ_2$, depending on the phase of ${\oq}_2, {\sq}_2$ quarks with unhiggsed colors (it is worth recalling here that, unlike the non-perturbative mechanism of the $DC_1$ condensate formation, the higgsing at $\mu\sim\mu_{\rm gl,1}$ operates separately, independently of what is going on at lower scales with ${\oq}_2, {\sq}_2$ quarks with unhiggsed colors).
Hence, the phase $DC_1-DC_2$ is maintained in the region $\mo<\mph<\mu^{\rm (higgs)}_{\Phi}$ in these br1 - vacua
and only the hierarchy of vacuum condensates $\langle\Pi_1\rangle$ and $\langle\Pi_2\rangle$ is different here in comparison with the section 7.1 above, $\langle\Pi_1\rangle\simeq\rho_1 m_Q\mph\gg \langle\Pi_2\rangle$. Therefore,
the Lagrangian of pions will be (7.1), $n_1^2$ pions $\Pi_{11}=({\ov Q}_1 Q_1)$ and $n_2^2$ pions $\Pi_{22}=({\ov Q}_2 Q_2)$ will have masses $\mu(\Pi_{11})= \mu(\Pi_{22})\sim m_Q$, while $2n_1n_2$ hybrids $\Pi_{12}=({\ov Q}_1 Q_2)$ and $\Pi_{21}=({\ov Q}_2 Q_1)$ will be massless. The masses of gluonia are $\sim \lym^{\rm (br1)}=\langle S\rangle^{1/3}_{\rm br1}$, as in (2.18). The hierarchy of nonzero masses looks here as : $\mu(\Pi_{11})\sim\mu(\Pi_{22})\ll\lym^{\rm (br1)}\ll\mu_{C,2}\ll\mu_{C,1}\ll \la$.
Finally, it remains to see that fions are dynamically irrelevant here. The renormalization factor $z_{\Phi}(\la,\mu)$ of fions becomes frozen at $z_{\Phi}(\la,\mu_{C,2})=(\la^2/\langle\Pi_2\rangle)^{\bo/N_F}\gg 1$, after all constituent quarks decouple at $\mu< \mu_{C,2}$ . Hence, the running mass of fions stops at the value $\mu_{\Phi}(\mu=\mu_{C,2})= \mph/z_{\Phi}(\la,\mu_{C,2})$. To see that they are irrelevant at $\la\ll\mph\ll\mu^{\rm (higgs)}_{\Phi}$ it is sufficient to check that $\mu_{C,2}>\mu^{\rm conf}_o$, see (4.1). This is fulfilled as $\mu_{C,2}>m_{Q,2}^{\rm pole}\sim\la(m_Q/\la)^{N_F/3N_c}$ here and $m_{Q,2}^{\rm pole}>\mu^{\rm conf}_o$ at $\mph>\mo$. Hence, there are no poles in the fion propagators at $\mu<\la$ and all fions remain dynamically irrelevant in this case.
8.2 br1 - vacua, $\mathbf{Higgs_1-DC_2}$ phase
At $\mo\ll\mu^{\rm higgs}_{\Phi}<\mph\ll\la^2/m_Q$ the quarks ${\ov Q}_1, Q_1$ are higgsed at $\mu\sim\mu_{\rm gl,1}\ll\la$ in the conformal regime at $a_{+}(\mu=\mu_{\rm gl,1})\sim a_{-}(\mu=\mu_{\rm gl,1})=a_*=O(1)<1$. The lower energy theory at $\mu<\mu_{\rm gl,1}$ has $N^{\,\prime}_c=(N_c-n_1)$ colors, $N^{\,\prime}_F=(N_F-n_1)=n_2$ flavors, ${\rm b}^{\prime}_o=(\bo-2n_1)$, and the new scale factor of its gauge coupling is
[Λ^'_Q(Π_11)]^b^'_o=z^n_2_Q(, μ_gl,1)^/Π_11 , Λ^'_Q=⟨Λ^'_Q(Π_11)⟩∼μ_gl,1 .
We consider first the case $n_1<\bo/2$ when the lower energy theory with $N^{\prime}_c=(N_c-n_1)$ colors and $\,N^{\prime}_F=(N_F-n_1)=n_2$ flavors and with unbroken flavor symmetry $U(n_2)$ remains in the conformal window with $3/2<N^{\prime}_F/N^{\prime}_c<3$ at $\mu<\mu_{\rm gl,1}$. Then the phase of ${\oq}_2, \sq_2$ quarks with unhiggsed colors is $DC_2$. Indeed, the new values of the pole and constituent masses of ${\oq}_2, {\sq}_2$ quarks with unhiggsed colors look now as ( $\gamma^{+}_Q=\bo/N_F>0,\, \gamma^{-}_Q={\rm b}^{\prime}_o/N^{\prime}_F>0$ )
m^pole_,2=(⟨m^tot_,2⟩=⟨Π_1⟩/∼m_Q )(z^+_Q(,μ_gl,1)z^-_Q(μ_gl,1, m^pole_,2)^-1 , (μ^ '_C,2)^2=⟨Π_2⟩z^+_Q(,μ_gl,1) ,z^+_Q=(μ_gl,1/)^γ^+_Q≪1, z^-_Q =( m^pole_,2/μ_gl,1)^γ^-_Q≪1 , m^pole_,2/μ^ '_C,2∼(/)^N_c(-2n_1)/6(N_c-n_1)≪1 .
Therefore, after the heaviest particles with the masses $\sim\mu_{\rm gl,1}$ have been integrated out, the Lagrangian at $\mu\sim \mu_{\rm gl,1}\ll \la$ takes the form (2.21), with a replacement of the logarithmic renormalization factor $z_Q(\la,\mu_{\rm gl,1}\gg\la)\gg 1$ and $z_{\Phi}\sim 1$ by the power-like ones, $z^+_Q(\la,\mu_{\rm gl,1}\ll\la)\ll 1$ and $z^+_{\Phi}(\la,\mu_{\rm gl,1}\ll\la)\gg 1$. Then, after integrating out the constituent $\oq_2, \sq_2$ quarks with the masses $\mu^{\,\prime}_{C,2}$ at $\mu<\mu^{\,\prime}_{C,2}$ and unhiggsed gluons at $\mu<\lym^{\rm (br1)}$ , the Lagrangian looks as
K=[z^+_Φ(,μ_gl,1)K_Φ+z^+_Q(,μ^2_gl,1)K_Π], z^+_Q=(μ_gl,1/)^γ^+_Q≪1, z^+_Φ=(1/z^+_Q)^2≫1 ,
K_Φ=Tr[ (Φ_11^†Φ_11+Φ_12^†Φ_12+Φ_21^†
Φ_21)+z^-_Φ(μ_gl,1,μ^ '_C,2 )Φ_22^†Φ_22 ], z^-_Φ=(μ_gl,1)/μ^ '_C,2)^2γ^-_Q≫1 ,K_Π=Tr [2 √(Π^†_11Π_11)+(Π^†_121/√(Π_11
√(Π^†_22Π_22) ] .W=-S+/2[Tr (Φ^2) -1/(Tr Φ)^2]+W_Π ,
S=(Π_11Π_22/^ )^1/ ,
W_Π= Tr[ (m_Q-Φ_11)Π_11+(m_Q-Φ_22)(Π_22+Π_211/Π_11Π_12
)-(Φ_12Π_21+Φ_21Π_12 ) ] .
We starts with the hybrids. The fions $\Phi_{12}$ are much heavier than $\Pi_{12}$ and mixing between them is
parametrically small. Neglecting it and integrating out $\Phi_{12}$, one obtains for the mass terms of $\Pi_{12}$ and
$\Pi_{21}$, see (1.5),
W_(Π)^hybr=( m_Q-⟨Φ_2⟩=⟨m_Q,2^tot⟩/⟨Π_1⟩
-1/ )Π_21Π_12=0 .
Further, the fions $\Phi_{11}$ and $\Phi_{22}$ are also much heavier than the pions $\Pi_{11},\Pi_{22}$. After integrating them out the superpotential of pions $\Pi_{11},\Pi_{22}$ looks as
W=-S+m_QTr Π-1/ [ Tr (Π^2_11+Π^2_22)-1/N_c(Tr Π)^2 ].
Hence, from (8.8) and (8.11), the pion masses look as
μ(Π_11)∼⟨Π_1⟩/z^+_Q∼m_Q/z^+_Q∼μ(Π_22) .
On the whole, the mass spectrum looks as follows.
a) The heaviest are $(2n_1N_c-n_1^2)$ higgsed gluons and their superpartners with the masses $\mu_{\rm gl,1}\sim(m_Q\mph)^{1/2}\ll\la$.
b) There is a large number of 22-flavored hadrons made of non-relativistic and weakly confined constituent quarks
${\oq}_2, {\sq}_2$ with the masses $\mu^{\,\prime}_{C,2}\ll\mu_{\rm gl,1}$ (the string tension is ${\sqrt\sigma}\sim \lym^{\rm (br1)}\ll\mu^{\,\prime}_{C,2}$).
c) $n^2_1$ pions $\Pi_{11}$ and $n^2_2$ pions $\Pi_{22}$ have masses $\mu(\Pi_{11})\sim\mu(\Pi_{22})\sim m_Q/z^+_Q\ll\lym^{\rm (br1)}$ ,
(μ(Π_11)/^(br1))^3∼(μ(Π_11)/μ^ '_C,2)^2∼(/)^N_c(-2n_1)/(N_c-n_1)≪1, ^(br1)∼(m_Q⟨Π_2⟩_br1)^1/3 .
d) The $2n_1n_2$ hybrids $\Pi_{12}$ and $\Pi_{21}$ are massless.
As one can check, all $N_F^2$ fions remain dynamically irrelevant in this region $\mph>\mo \,.$
8.3 br1 - vacua, $\mathbf{Higgs_1-HQ_2}$ phase
We consider now the case $\rm b^\prime_o<0$ , i.e. $N^\prime_F/N^\prime_c>3,\,\,n_1>\bo/2$ . Then (neglecting logarithmic factors), one has to replace $z^{-}_Q\ra 1$ in (8.7) at $\mu^{\rm (higgs)}_{\Phi}<\mph\ll\la^2/m_Q$ and obtains
μ^ '_C,2/m^pole_,2∼(/)^N_c(2n_1-)/2(N_c-n_1)≪1 .
This shows that ${\oq}_2, {\sq}_2$ quarks with unhiggsed colors are in the $HQ_2$ - phase and there is no lighter $\Pi_{22}$ pions. The lagrangian of pions takes now the form (2.22). This phase $Higgs_1-HQ_2$ is preserved also in the region $\mph\gg\la^2/m_Q$.
Finally, we comment in short on the behavior in the region $\mph\gg \la^2/m_Q$ when $\rm b^\prime_o>0$. Then
$\mu_{\rm gl, 1}\sim\langle\Pi_1\rangle^{1/2}\sim (m_Q\mph)^{1/2}\gg \la$ , i.e. the quarks ${\ov Q}_1, Q_1$ will be higgsed in the weak coupling region with the logarithmic RG flow, and (neglecting logarithmic factors) $(\Lambda^\prime_Q)^{\bo^\prime}\sim\la^\bo/\det \langle\Pi_{11}\rangle,\,\, \Lambda^{\prime}_Q\ll\la$ now. Hence, while $\Lambda^{\prime}_Q\sim\mu_{\rm gl,1}$ increased with increasing $\mph$ at $\mu_{\rm gl,1}\ll\la$, when $\mph$ becomes larger than $\la^2/m_Q$ and increases, $\Lambda^{\prime}_Q$ begins to decrease in a power-like fashion while the ratios $\mu^{\,\prime}_{C,2}/\Lambda^{\prime}_Q\,,\,\,m^{\rm pole}_{\sq, 2}/\Lambda^{\prime}_Q\,,\,\, m^{\rm pole}_{\sq, 2}/\mu^{\,\prime}_{C,2}$ are increasing with $\mph$. Until $\mph$ is not too large, $\mph<\mu^{\prime}_{\Phi}$ , the hierarchy $\Lambda^{\prime}_Q>\mu^{\,\prime}_{C,2}> m^{\rm pole}_{\sq, 2}$ is preserved and ${\oq}_2, {\sq}_2$ quarks stay in the $DC_2$ phase. But $m^{\rm pole}_{\sq,2}\sim\mu^{\,\prime}_{C,2}\sim\Lambda^{\prime}_Q\sim m_Q$ at $\mph\sim\mu^
{\prime}_{\Phi}\sim\la(\la/m_Q)^{(\bo-n_1)/n_1}\gg\la^2/m_Q$, and the hierarchy is reversed at $\mph>\mu^{\prime}_{\Phi}$, it becomes $m^{\rm pole}_{\sq, 2}>\mu^{\,\prime}_{C,2}>\Lambda^{\prime}_Q$ .
In this region $\mu^{\,\prime}_{C,2}\sim \mu_{\rm gl, 2}$ has the meaning of the gluon mass due to possible higgsing of ${\oq}_2, {\sq}_2$ quarks.
The phase is changed when $\mph$ becomes larger than $\mu^{\prime}_{\Phi},\, DC_2\ra HQ_2$, the ${\oq}_2, {\sq}_2$ quarks will be in the $HQ_2$ (heavy quark) phase and there will be no lighter $\Pi_{22}$ pions. The Lagrangian of pions $\Pi_{11}$ and hybrids $\Pi_{12},\,\Pi_{21}$ has the form (2.22). With further increasing $\mph$ this phase $Higgs_1-HQ_2$ stays intact.
8.4 br2 and special vacua
At $n_2<N_c$ there are also $\rm br2$ - vacua, see section 3. For these, their properties can be obtained by the replacement $n_1\leftrightarrow n_2$ in formulas of the preceding sections 8.1 and 8.3 . The only difference is that, because $n_2\geq N_F/2$ and so $2n_2>\bo$, there is no analog of the conformal regime at $\mu<\mu_{\rm gl,1}$ with $2n_1<\bo$ in section 8.2. I.e., see (8.5), at $\mph>{\mu}^{\rm higgs}_{\Phi}(n_1\leftrightarrow n_2)$ the lower energy theory at $\mu<\mu_{\rm gl,2}$ will be always in the perturbative IR free logarithmic regime and the overall phase will be $Higgs_2-HQ_1$.
As for the special vacua (see section 3), their properties can also be obtained with $n_1=\nd,\, n_2=N_c$
in formulas of the preceding sections 8.1-8.3 .
§ DUAL THEORY. BROKEN FLAVOR SYMMETRY.
THE REGION $\MATHBF {\LA<\MPH<\MO}$
9.1 L - type vacua.
Below in this section : $M_{11}$ is the $n_1\times n_1$ matrix, $\langle M_1\rangle=\langle M_{11}\rangle=\langle
\Pi_1\rangle$, $M_{22}$ and $N_{22}=({\ov q}_2 q_2)$ are $n_2\times n_2$ matrices, $\langle M_2\rangle=\langle M_{22}\rangle=\langle\Pi_2\rangle$ , and $M_{\rm hybr}$ includes $2n_1\times n_2$ $M_{12}$ and $M_{21}$ mesons (and the same for $N_F^2$ nions $N$ ).
Although $\langle M_1\rangle_{Lt}\neq\langle M_2\rangle_{Lt}$ are not equal now, but their parametric behavior in the region $\la\ll\mph\ll\mo$ is the same here as in the L - vacua with the unbroken flavor symmetry in section 6.1, $\langle M_1=\Pi_1\rangle_{Lt}\sim\langle M_2=\Pi_2
\rangle_{Lt}\sim \langle M\rangle_L$. I.e., the phase is $DC_1-DC_2,\, N_F^2$ nions $N_{ij}$ are formed and the Lagrangian is (6.4). All $N_F^2$ mions have masses (6.5) and are much heavier than nions. Hence, the Lagrangian of nions is (6.7). The masses of $n^2_1$ nions $N_{11}$ and $n^2_2$ nions $N_{22}$ are still $\mu(N_{11})\sim \mu(N_{22})\sim \la(\la/\mph)^{\nd/(2N_c-N_F)}$. But, due to a spontaneous breaking of the flavor symmetry, the masses of hybrid nions $N_{12}$ and $N_{21}$ differ qualitatively now from those in section 6.1. They are massless Nambu-Goldstone particles here.
9.2 br2 - vacua
The condensates of mions and dual quarks in these vacua with $n_2>N_c\,, n_1<\nd$ at $\la\ll\mph\ll\mo$ look as
⟨M_2⟩≃(ρ_2=-n_2-N_c/N_c)m_Q, ⟨M_1⟩∼^2
⟨M_1⟩/⟨M_2⟩∼(/)^N_c/n_2-N_c≪1 ,⟨N_1⟩=⟨q_1 q_1(μ=)⟩=⟨S⟩/⟨M_1⟩
=⟨M_2⟩/∼m_Q≫⟨N_2⟩ .
From these, some possible characteristic masses look as
μ_q,2=⟨M_2⟩/∼m_Q/, μ^pole_q,2∼(m_Q/^2)^N_F/3≫μ^pole_q,1 ,
μ^2_C,1∼⟨N_1⟩=⟨S⟩/⟨M_1⟩=⟨M_2⟩/∼m_Q , μ^pole_q,2/μ_C,1≪1 ,
(m_Q/)^N_c-n_1/n_2-N_c≪μ^2_C,1 ,μ_gl,1∼(⟨N_1⟩/^2)^N_F/3N_c∼(m_Q/)^N_F/3N_c≫μ_gl,2 ,μ_gl,1/μ^pole_q,2∼(/)^N_F/3≫1 , μ_gl,1/μ_C,1∼(m_Q/)^/6N_c≪1 , =2N_F-3N_c ,
(/m_Q)^δ/6N_c(n_2-N_c) , δ=n_1-(2 N_F-3N_c^2) ,
where $\mu^{\rm pole}_{q,1}$ and $\mu^{\rm pole}_{q,2}$ are the pole masses of ${\ov q}_1, q_1$ and ${\ov q}_2, q_2$ quarks, ${\ov\mu}_{C,1}$ and ${\ov\mu}_{C,2}$ are their constituent masses and ${\ov\mu}_{\rm gl,1},\, {\ov\mu}_{\rm gl,2}$ are the gluon masses due to possible higgsing of these quarks.
It is seen from (9.1)-(9.4) that the largest mass is ${\ov\mu}_{C,1}$. But we recall that it does not work by itself
[7], it is important what is the next mass. It follows from (9.1)-(9.4) that :
a1) if $(2 N_F\nd -3N_c^2)<0$, then $\delta>0$ and at all $0<n_1<\nd$ the phase is $DC_1-DC_2$ in the region ${\widehat{\ov\mu}}_{\Phi}\ll\mph\ll\mo$ and $Higgs_1-HQ_2$ in the region $\la\ll\mph\ll {\widehat{\ov\mu}}_{\Phi}$, see (7.10),
μ_Φ=(/m_Q)^δ >0/3N_c^2≫μ_Φ ;
a2) if $(2 N_F\nd -3N_c^2)>0$ and $0<n_1<{\ov n}^o_1=n^o_1=(2 N_F\nd -3N_c^2)/\bd$, then $\delta<0$ and the next mass
is ${\ov\mu}_{C,2}$ , this means that the overall phase is $DC_1-DC_2$ in the whole region $\la\ll\mph\ll\mo$ ;
a3) if $(2 N_F\nd -3N_c^2)>0$ and $\delta>0$, then the phases is as in 'a1' but at $n^o_1<n_1<\nd$ only.
Besides, when ${\ov q}_1, q_1$ quarks are higgsed, the lower energy theory at $\mu<{\ov\mu}_{\rm gl,1}$ remains in the conformal regime at $0<n_1<\bd/2$ in the case 'a2' and at $n^o_1<n_1<\bd/2$ in the case 'a3', and enters the IR free regime at $n_1>\bd/2$ in the case 'a2' and at $n_1>{\rm max}\,[\, n^o_1,\,\bd/2\,]$ in the case 'a3'.
We consider first the $DC_1-DC_2$ phase. The quarks acquire the constituent masses ${\ov\mu}_{C,1}$ and ${\ov\mu}_{C,2}$ and $N_F^2$ dual pions (nions) are formed at the scale $\mu\sim{\ov\mu}_{C,2}\ll{\ov\mu}_{C,1}$. After integrating out all constituent quarks at $\mu<{\ov\mu}_{C,2}$ and then all gluons at $\mu<\lym^{(\rm br2)}$, the Lagrangian of nions $N$ and mions $M$ looks as in (6.4), with a replacement ${\ov\mu}^{L}_{C}\ra{\ov\mu}_{C,2}$. All mions are much heavier here than nions and mixing between them is small (and is neglected). Hence, after integrating out all mions one obtains (6.7) and the nion masses look as ( at $\la\ll\mph\ll\mo$ for 'a2' and ${\widehat{\ov\mu}}_{\Phi}\ll\mph\ll\mo$ for 'a1' and 'a3')
μ(N_11)∼μ(N_22)∼m_Q/ , μ(N_12)=μ(N_21)=0 .
As for the mion running masses, they all become frozen at $\mu<{\ov\mu}_{C,2}$ at the value
μ(M_ij)∼^2/z_M(,μ_C,2)≫μ(N) , z_M(,μ_C,2)=(/μ_C,2)^2/N_F .
To see whether mions are dynamically relevant or not in this $DC_1-DC_2$ phase, we compare ${\ov\mu}_{C,2}$ and $\mu_o^{\rm conf}$, see (9.2),(6.6),
(/)^ρ≫1 , ρ=2n_1 N_F-(2N_F^2+N_c N_F-6 N_c^2)/3(n_2-N_c)(2N_c-N_F)>0 .
It is seen from (9.8) that $\mu(M)$ in (9.7) are not the pole masses of mions, the hierarchies look here as $\mu_o^{\rm conf}\gg\mu(M)\gg{\ov\mu}_{C,2}$. The mion propagators have poles at $p=\mu^{\rm pole}(M)\sim\mu_o^{\rm conf}$ only and so the mions are dynamically relevant only at scales $\mu_o^{\rm conf}<\mu<\la$.
On the whole, in addition to the mions and nions, the mass spectrum in this $DC_1-DC_2$ phase includes : a) a large number of hadrons made of the weakly confined (the string tension is ${\sqrt\sigma}\sim\lym^{(\rm br2)}\ll{\ov\mu}_{C,2}
\ll{\ov\mu}_{C,1}$) constituent dual quarks with the masses ${\ov\mu}_{C,1}$ and ${\ov\mu}_{C,2}$,
b) a large number of gluonia made of $SU(\nd)$ dual gluons with the mass scale $\sim\lym^{(\rm br2)}=(\langle M_1\rangle\langle M_2\rangle/\mph)^{1/3}$.
We consider now the $Higgs_1-HQ_2$ phase. The largest physical mass here is ${\ov\mu}_{\rm gl,1}$, see (9.2). The lower energy theory at $\mu<{\ov\mu}_{\rm gl,1}$ has $\nd^{\,\prime}=\nd-n_1$ dual colors and $N_F^{\prime}=N_F-n_1=n_2>N_c$ lighter $\ov{\textsf{q}}_2$, $\textsf{q}_2$ quarks with unhiggsed colors. It is in the conformal regime at $2<N_F^{\prime}/\nd^{\,\prime}<3$, i.e. at $n_1<\bd/2$, and in the IR free one at $N_F^{\prime}/\nd^{\,\prime}>3,\, n_1>\bd/2$.
We start with $2<N_F^{\prime}/\nd^{\,\prime}<3$. After integrating out all heaviest higgsed gluons and their superpartners at $\mu<{\ov\mu}_{\rm gl,1}$, the Lagrangian of remained lighter degrees of freedom looks as
K=z_MTr [M^†M/^2]+z_q Tr [2√(N^†_11N_11)+K_hybr+ ^†_2_2+(_2_2) ] ,
K_hybr=(N^†_121/√(N_11 N^†_11)N_12+
N_211/√(N^†_11N_11)N^†_21),z_q=z_q (,μ_gl,1)=(μ_gl,1/)^/N_F , z_M=z_M (,μ_gl,1)=1/z^2_q ,W=[-2π/α(μ)]+W_M -W_MN-1/Tr(_2 M_22_2) ,
W_M=m_QTr M-1/2(Tr (M^2)- 1/N_c(Tr M)^2) ,W_MN=1/Tr ( M_11N_11+N_12M_21+N_21M_12+M_22N_211/N_11N_12
) ,where $N_{11}$ are the dual pions (nions) due to higgsing of ${\ov q}_1, q_1$ quarks (besides, they are sitting inside ${\ov\alpha}(\mu)$), $\ov{\textsf{S}}$ is the gauge field strength squared of unhiggsed dual gluons, $\ov{\textsf{q}}_2$ and $\textsf{q}_2$ are the ${\ov q}_2, q_2$ quarks with unhiggsed colors and $N_{12}, N_{21}$ are the hybrid nions (in essence, these are the ${\ov q}_2, q_2$ quarks with higgsed colors).
The quarks $\ov{\textsf{q}}_2, \textsf{q}_2$ are in the $HQ_2$ phase. After integrating them out at $\mu<{\ov\mu}^{\rm pole}_{q,2}$ and then unhiggsed gluons at $\lym^{(\rm br2)}$ the lower energy Lagrangian is
K=z_M/^2Tr [ M^†_11 M_11+M^†_12M_12+M^†_21M_21+z^'_M M^†_22 M_22]+z_q Tr [2√(N^†_11N_11)+K_hybr] ,W=(-n_1)(^( M_22/)/N_11)^1/(-n_1)+W_M - W_MN ,
μ_gl,1 ∼(m_Q/)^N_F/3N_c ,
z^'_M=z^'_M(μ_gl,1,μ^pole_q,2)=(μ_gl,1/μ^pole_q,2)^2^'/N_F^' .The factor $z^{\prime}_M$ in (9.11) appears due to the additional evolution of $\ov{\textsf{q}}_2, \textsf{q}_2$ quarks
and $M_{22}$ mions in the range of scales ${\ov\mu}^{\rm pole}_{q,2}<\mu<{\ov\mu}_{\rm gl,1}$, while ${\ov\mu}^{\rm pole}_{q,2}$ is the pole mass of $\ov{\textsf{q}}_2, \textsf{q}_2$ quarks in this $Higgs_1-HQ_2$ phase
μ^pole_q,2=μ_q,2/z_q z^'_q , z^'_q=z^'_q(μ_gl,1,
'≪1 .
From (9.11) : i) the low energy values of the mion running masses $M_{11}$ are much larger than those of nions $N_{11}$ and the mixing among them is small; ii) the same for the hybrids $M_{12}$ and $N_{12}$. Hence, one obtains from (9.11) for the masses
μ(M_11)∼μ(M_12)∼μ(M_21)∼^2/z_M , μ(M_22)∼^2/z_M z^'_M ,
μ(N_11)∼⟨N_1⟩/z_q^2∼(m_Q/^2)^2/3N_c , μ(N_12)=μ(N_21)=0 .
To see whether (9.13) are the mion poles masses or not it is sufficient to check the hierarchies $\mu_o^{\rm conf}\gg
\mu(M_{11})\gg{\ov\mu}_{\rm gl,1},\,\,\mu(M_{11})\gg\mu(M_{22})\gg{\ov\mu}^{\rm pole}_{q,2}$. Therefore, the values of the running mion 'masses' in (9.13) are not their poles masses, these are simply the low energy limiting values of mass terms in their propagators. The propagators of all mions have poles at $p=\mu^{\rm pole}(M)\sim\mu_o^{\rm conf}$ only and so they are dynamically relevant only in the range of scales $\mu_o^{\rm conf}<\mu<\la$.
On the whole, in addition to the mions and nions, the mass spectrum in this $Higgs_1-HQ_2$ phase includes : a) the heaviest $n_1(2\nd-n_1)$ higgsed dual gluons and the same number of their superpartners, b) a large number of hadrons made of the unhiggsed $\ov{\textsf{q}}_2, \textsf{q}_2$ quarks with the mass scale $\sim {\ov\mu}^{\rm pole}_{q,2}$ (9.12), c) a large number of gluonia made of the unhiggsed $SU(\nd-n_1)$ dual gluons with the mass scale $\sim\lym^{(\rm br2)}=(\langle M_1\rangle\langle M_2\rangle/\mph)^{1/3}$.
We consider now the case $N_F^{\prime}/\nd^{\,\prime}>3$ where the theory at $\mu<{\ov\mu}_{\rm gl,1}$ is in the
logarithmic IR free regime at scales ${\ov\mu}^{\rm pole}_{q,2}<\mu<{\ov\mu}_{\rm gl,1}$. The Lagrangians will be as in (9.9)-(9.11) and only $z^\prime_M$ in (9.11) and ${\ov\mu}^{\rm pole}_{q,2}$ in (9.12) will be different. Neglecting logarithmic factors, one can replace $z^\prime_M\ra 1$ in (9.11) and $z^\prime_q\ra 1$ in (9.12). The nion masses will be
as in (9.14) while $\mu(M_{22})$ in (9.13) will be $\mu(M_{22})\sim\mu(M_{11})$.
9.3 Special vacua, $\mathbf{n_1=\nd,\, n_2=N_c}$
The most important possible masses look here as follows,
⟨M_1⟩_spec=N_c/2N_c-N_F(m_Q) , ⟨M_2⟩_spec=^2(/)^/(2N_c-N_F)≫⟨M_1⟩_spec ,
μ_q,2=⟨M_2⟩/ , μ^pole_q,2∼(⟨M_2⟩/^2)^N_F/3∼(/)^N_F/3(2N_c-N_F)≫μ^pole_q,1 ,μ^2_C,1∼⟨N_1⟩=⟨M_2⟩/∼^2(/)^N_c/2N_c-N_F≫μ^2_C,2∼m_Q , μ^pole_q,2/μ_C,1∼(/)^/6(2N_c-N_F)≪1 ,μ_gl,1∼(⟨N_1⟩/^2)^N_F/3N_c∼μ^pole_q,2≫μ_gl,2 ,and, see (7.23),
(μ_C,2/μ^pole_q,2)^2∼m_Q/(/)^2N_F/3(2N_c-N_F)> 1 >μ^(DC)_Φ∼(/m_Q)^3(2N_c-N_F)/2N_F≫μ^(DC)_Φ ,
where ${\ov\mu}_{C,1}$ is the possible constituent mass of ${\ov q}_1, q_1$ quarks and ${\ov\mu}_{\rm gl,1}$ is the gluon mass due to their possible higgsing. Because ${\ov\mu}_{\rm gl,1}\sim \mu^{\rm pole}_{q,2}$ it is unclear beforehand whether the phase is $DC_1-HQ_2$ or $Higgs_1-HQ_2$.
But taking $\bd/\nd\ll 1$ and using the results from [8] we obtain $\mu^{\rm pole}_{q,2}/{\ov\mu}_{\rm gl,1}\sim
\exp \{3\nd/14\bd\}\gg 1$.
But an attempt to write the standard superpotential for the $DC_1-HQ_2$ phase shows that it will be singular at $n_1=\nd$ [7] and, similarly to the special vacua in the direct theory in section 7.3, we assume here that
the overall phase will be $Higgs_1-HQ_2$ and the whole dual gauge group will be higgsed at $\la\ll\mu\ll{\ov\mu}^{(\rm DC)}_{\Phi}$, while the phase will be $DC_1-DC_2$ at ${\ov\mu}^{(\rm DC)}_{\Phi}\ll\mu\ll\mo$.
We start with the $Higgs_1-HQ_2$ phase and proceed as in the section 7.3. I.e., after integrating out first the quarks ${\ov q}_2, q_2$ as heavy ones at $\mu<\mu^{\rm pole}_{q,2}$ and then all higgsed dual gluons and their superpartners at $\mu<{\ov\mu}_{\rm gl,1}$, the Lagrangian takes the form
K=Tr [ z_MM^†M/^2+ z_q(√(N^†_11 N_11)+b^†_1 b_1+
b^ †_1 b_1) ] , z_q=z_q(,μ^pole_q,2)=(μ^pole_q,2/)^/N_F , z_M=z_M(,μ^pole_q,2)=1/z^2_q ,W=W_non-pert-W_M -1/Tr N_11(M_11-M_121/M_22M_21) ,
W_M=1/2[Tr (M^2) -1/N_c(Tr M)^2]+m_QTr M ,where the non-perturbative term looks here as
W_non-pert=A [ 1-N_11/λ^2+b_1 b_1/λ^2
] , ⟨A⟩=⟨S⟩=⟨M_1⟩⟨M_2⟩/ ,
λ^2=(^M_22/)^1/ , ⟨λ^2⟩=
⟨N_1⟩=⟨m^tot_Q,1⟩=⟨M_2⟩/ ,and $\ov A$ is the auxiliary field.
From (9.17),(9.18) : the hybrids $M_{12}, M_{21}$ are massless, the baryons ${\ov b}_1,\, b_1$ are light
μ(b_1)=μ(b_1)∼⟨M_1⟩/z_q∼m_Q(/)^/3(2N_c-N_F)≪μ_gl,1 ,
while all other masses are $\sim{\ov\mu}_{\rm gl,1}\sim \mu^{\rm pole}_{q,2}$ (the nion masses increased due to their mixing with the mions). Besides, in particular, because $\mu_o^{\rm conf}\sim \mu^{\rm pole}_{q,2}$ in these special vacua, there is no warranty that these nonzero masses of mions $M_{11}$ and $M_{22}$ are the pole masses. Maybe so but maybe not (see section 4).
On the whole, there are three scales in the mass spectrum : the hybrid mions $M_{12}, M_{21}$ are massless, the baryon masses are (9.19), while all other masses are $\sim {\ov\mu}_{\rm gl,1}\sim\mu^{\rm pole}_{q,2}\sim\\
\sim\la(\la/\mph)^{N_F/3(2N_c-N_F)}$ in these special vacua at $\la\ll\mph\ll{\ov\mu}^{(\rm DC)}_{\Phi}$.
Now, we consider the phase $DC_1-DC_2$ with ${\ov\mu}_{C,1}\gg{\ov\mu}_{C,2}\gg\mu_o^{\rm conf}\sim \mu^{\rm pole}_{q,2}$ in these special vacua at ${\ov\mu}^{(\rm DC)}_{\Phi}\ll\mph\ll\mo$. We can proceed then as in this phase in section 9.2 above and to start directly with the Lagrangian (6.4). Because ${\ov\mu}_{C,2}\gg\mu_o^{\rm conf}$, there are poles in the mion propagators at $ p=\mu^{\rm pole}(M_{ij})\ll {\ov\mu}_{C,2}\ll\la$,
μ^pole(M_ij)∼^2/z_M(,μ_C,2)∼^2/(m_Q/)^/N_F≫μ_o^conf , z_M(,μ_C,2)=(/m_Q)^/N_F ,
and all mions are dynamically relevant here in the range of scales $\mu^{\rm pole}(M_{ij})<\mu<\la$. Hence, after integrating them out in (6.4) one obtains the Lagrangian (6.7). From this, the masses of $N_{11}$ and $N_{22}$ nions are $\mu(N_{11})\sim\mu(N_{22})\sim\la(\la/\mph)^{\nd/(2N_c-N_F)}$, while the hybrids $N_{12}$ and $N_{21}$ are massless. We recall finally that the masses of constituent quarks are here ${\ov\mu}_{C,1}\sim (\langle M_2\rangle_{\rm spec}/\mph)^{1/2}\gg{\ov\mu}_{C,2}$ and ${\ov\mu}_{C,2}\sim (\langle M_1\rangle_{\rm spec}/\mph)^{1/2}\gg \mu^{\rm pole}_{q,2}$, and the mass scale of gluonia is $\lym^{(\rm spec)}=[\langle M_1\rangle_{\rm spec}\langle M_2\rangle_{\rm spec}/\mph]^{1/3}\sim\la\,[(m_Q/\la)(\la/\mph)^{\nd/(2N_c-N_F)}]^{1/3}\gg\mu(N_{11})\sim\mu(N_{22})$.
§ DUAL THEORY. BROKEN FLAVOR SYMMETRY.
THE REGION $\MATHBF {\MO\LL\MPH\LL\LA^2/M_Q}$
10.1 br1 - vacua, $\mathbf{DC_1-DC_2}$ phase
We recall, see (8.1), that the condensates of mions and dual quarks in this vacua are
⟨M_1⟩∼m_Q , ⟨M_2⟩∼^2(/)^n_1/N_c-n_1(m_Q/)^n_2-N_c/N_c-n_1 , ⟨M_2⟩/⟨M_1⟩∼(/)^N_c/N_c-n_1≪1 ,⟨N_2⟩=⟨q_2 q_2(μ=)⟩=⟨M_1⟩/∼m_Q≫⟨N_1⟩ ,and so some potentially relevant masses look here as
μ_C,2^ 2∼⟨N_2⟩=⟨M_1⟩/∼m_Q≫μ_C,1^ 2∼⟨N_1⟩ , μ^pole_q,1∼(m_Q/^2)^N_F/3≫μ^pole_q,2 ,
μ_gl,2∼(⟨M_1⟩/)^N_F/3N_c∼(m_Q/)^N_F/3N_c≫μ_gl,1 , μ_gl,2/μ^pole_q,1∼(/)^N_F/3≪1 .
From (10.1)
μ^pole_q,1/μ_C,1>1 at >μ_Φ,1=(/m_Q)^σ , σ=(N_c-n_1)+3(2N_c-N_F)/2N_F(N_c-n_1)+3N_c>0 ,
μ^pole_q,1/μ_C,2>1 at >μ_Φ,2=(/m_Q)^/2N_F , ≪μ_Φ,1≪μ_Φ,2≪^2/m_Q .
Therefore, the mass hierarchies in the region $\mo\ll\mph\ll{\ov\mu}_{\Phi,1}$ look as ${\ov \mu}_{C,2}\gg{\ov \mu}_{C,1}\gg\omp$ and the phase is $DC_1-DC_2$. In the regions ${\ov\mu}_{\Phi,1}\ll\mph\ll{\ov\mu}_{\Phi,2}$
and ${\ov\mu}_{\Phi,2}\ll\mph\ll\la^2/m_Q$ the mass hierarchies look, respectively, as ${\ov \mu}_{C,2}\gg
\omp\gg{\ov \mu}_{C,1}$ and $\omp\gg{\ov \mu}_{C,2}\gg{\ov \mu}_{C,1}$ and the phase is $HQ_1-DC_2$.
We start with the $DC_1-DC_2$ phase and recall that the largest constituent mass ${\ov\mu}_{C,2}$ is formed not at the scale $\mu\sim {\ov\mu}_{C,2}$ but at the lower scale $\mu\sim {\ov\mu}_{C,1}$, see [7]. Hence, after integrating out simultaneously all dual quarks at $\mu < {\ov\mu}_{C,1}$ and then the dual gluons at $\mu<\lym^{\rm (br1)}$, one obtains the Lagrangian (6.4) with the only difference that the factor $z_M$ is now $z_M=(\la/{\ov\mu}_{C,1})^{2\bd/N_F}$. As one can check, all $N_F^2$ mions with masses $\mu(M)\sim (\la^2/z_M\mph)$ are still much heavier than all $N_F^2$ nions and so the Lagrangian of nions is (6.7). The hybrid nions $N_{12}$ and $N_{21}$ are massless, while the masses of $n_1^2$ nions $N_{11}$ and $n_2^2$ nions $N_{22}$ look now as $\mu(N_{11})\sim \mu(N_{22})\sim m_Q\mph/\la$ . Because ${\ov\mu}_{C,1}\gg\mu^{\rm conf}_o$ at $\mo<\mph<{\ov\mu}_{\Phi,1}$, the above mion masses are their pole masses and the mions are dynamically relevant in the range of scales $\mu^{\rm pole}(M)\sim (\la^2/z_M\mph)<\mu<\la$.
Besides, there is in the mass spectrum : a) a large number of hadrons made of dual constituent quarks with the masses ${\ov\mu}_{C,2}$ and ${\ov\mu}_{C,1},\,\, {\ov\mu}_{C,2}\gg{\ov\mu}_{C,1}\gg\mu^{\rm pole}(M)$, b) a large number of gluonia with the mass scale $\sim\lym^{(\rm br1)}=(\langle M_1\rangle_{\rm br1}\langle M_2\rangle_{\rm br1}/\mph)^{1/3}\sim (m_Q\langle M_2\rangle_{\rm br1})^{1/3}\ll{\ov\mu}_{C,1}$ made of dual $SU(\nd)$ gluons.
10.2 br1 - vacua, $\mathbf{HQ_1-DC_2}$ phase
10.2.1 The region ${\ov\mu}_{\Phi,1}\ll\mph\ll{\ov\mu}_{\Phi,2}\,,\,\, {\ov \mu}_{C,2}>\omp >{\ov \mu}_{C,1}$ . Here, the largest dynamical mass ${\ov\mu}_{C,2}$ is formed not at the scale $\mu\sim{\ov \mu}_{C,2}$ but at the lower scale $\mu\sim \omp\ll {\ov\mu}_{C,2}$ , see [7]. Hence, proceeding as in [7] (see section 4), i.e. integrating out the constituent quarks ${\ov q}_2, q_2$ and the quarks ${\ov q}_1, q_1$ as heavy ones at $\mu<\omp$ and then all $SU(\nd)$ dual gluons at $\mu<\lym^{\rm (br1)}$, the Lagrangian of $N_F^2$ mions $M$ and $n_2^2$ nions $N_{22}$ takes now the form
K=[ z_M(,)/^2Tr (M^†M)+Tr √(N^†_22 N_22) ] ,
W=(N_c-n_1)S-W_NM+W_M , S=(N_22/^(M_11/) )^1/N_c-n_1 ,
W_NM=Tr N_22/(M_22-M_211/M_11M_12) , W_M=m_QTr M-1/2(Tr (M^2)-1/N_c(Tr M)^2),z_q(,)=(/)^/N_F , z_M=z_M(,)=1/z^2_q(,) .
From (10.3),(10.4) the masses of $n_1^2$ mions $M_{11}$ , $n_2^2$ mions $M_{22}$ and $n_2^2$ nions $N_{22}$ look here as
μ^pole(M_11)∼μ^pole(M_22)∼^2/z_M(,)≫μ(N_22)∼m_Q/ ,
while $2n_1n_2$ hybrid mions $M_{12}$ and $M_{21}$ are massless.
On the whole, the mass spectrum in this region includes : a) a large number of 22-flavored hadrons made of non-relativistic weakly confined constituent quarks ${\ov q}_2,\, q_2$ with the masses ${\ov\mu}_{C,2}\gg\omp$ (the string tension is ${\sqrt\sigma}\sim \lym^{\rm (br1)}\ll\omp\ll {\ov\mu}_{C,2}$) , b) a large number of 11-flavored hadrons made of non-relativistic weakly confined quarks ${\ov q}_1,\, q_1$ with the masses $\omp\gg \lym^{(\rm br1)}$, c) corresponding heavy hybrid hadrons with the masses $\sim({\ov\mu}_{C,2}+\omp)$, d) a large number of gluonia with the mass scale $\sim\lym^{(\rm br1)}$, e) $M_{11},\, M_{22}$ mions and $N_{22}$ nions with the masses (10.5) , f) the hybrid mions $M_{12}\,,\, M_{21}$ are massless.
10.2.2 The region ${\ov\mu}_{\Phi,2}\ll\mph\ll\la^2/m_Q,\,\, {\ov \mu}_{C,2}\ll\omp$. Here, the largest physical mass is $\omp$. Hence, unlike the constituent quarks ${\ov q}_2,\, q_2$ with the soft non-perturbative dynamical masses ${\ov\mu}_{C,2}$ above, the quarks ${\ov q}_1,\, q_1$ with the hard perturbative masses $\omp$ can now be integrated out at $\mu<\omp$ independently of other degrees of freedom. After integrating them out, the lower energy theory at $\mu<\omp$ has $N^{\,\prime}_F=(N_F-n_1)=n_2$ flavors, $\nd$ colors, ${\rm\ov b}^{\,\prime}_o=(3\nd-N^{\,\prime}_F)
=(3\nd-n_2)$, and the new scale factor $\Lambda^{\prime}_q$ of its gauge coupling is
[Λ^'_q(M_11)]^b^ '_o=z^n_2_q(,)^(M_11/) , Λ^'_q=⟨Λ^'_q (M_11)⟩∼ .
Because $3/2<N^{\,\prime}_F=(N_F-n_1)/\,\nd <3$ at $n_1<\bo/2$, this lower energy theory will be in the conformal regime with the anomalous dimensions of ${\ov q}_2,\, q_2$ quarks and $M_{22}$ mions (the mions $M_{11}$ and hybrids $M_{12}, M_{21}$ do not evolve any more at $\mu<\omp$)
γ_q^ '=γ_q^ ', conf=^ '/N_F^ '=+n_1/N_F-n_1 , γ_M^ '=γ_M^ ', conf= -2γ_q^ ', conf ,
while, because $1<N^{\,\prime}_F=(N_F-n_1)/\,\nd <3/2$ at $\bo/2<n_1\leq N_F/2$, it will be in the strong coupling regime
with the anomalous dimensions
γ_q^ '=γ_q^ ', str=2-n_2/n_2-=2-n_2/N_c-n_1 , γ_M^'=γ_M^ ', str=-(1+γ_q^ ', str)=-/N_c-n_1 .
Instead of (10.1), the constituent and pole masses of the ${\ov q}_2,\, q_2$ quarks look now as, see (10.4),
μ^ '_C,2=[ z_q(,)(⟨N_2⟩∼m_Q)]^1/2 , μ̃_q,2^pole=⟨M_2⟩/z_q(,) ( /μ̃_
q,2^pole )^γ^'_q ,(μ̃_q,2^pole/μ^ '_C,2)_conf∼(/)^N_c(n_1+)/6(N_c-n_1)≪1 , (μ̃_q,2^pole/μ^ '_C,2)_str∼(/)^N_c/2≪1 ,
so that the lower energy theory is, on the whole, in the $HQ_1-DC_2$ phase in any case.
After integrating out the heaviest ${\ov q}_1, q_1$ quarks as heavy ones at $\mu<\omp$ the Lagrangian looks as, see (10.3),(10.4),
K=[ z_M(,)/^2Tr (M^†M)+z_q(,)(q_2^†q_2+q_2^†q_2) ] ,
W=[-2π/α(μ)s ]-1/Tr q_2(M_22-M_211/M_11M_12)q_2+W_M ,with $\Lambda^{\prime}_q(M_{11})$ of the gauge coupling given in (10.6).
Therefore, after integrating then out the constituent ${\ov q}_2,\, q_2$ quarks at $\mu<{\ov \mu}^{\,\prime}_{C,2}$ and, finally, all $SU(\nd)$ gluons at $\mu<\lym^{({\rm br1})}$ , the Lagrangian of mions $M$ and nions $N_{22}$ has the superpotential as in (10.3) while the Kahler term is
K=z_M(,)/^2 Tr ( M_11^†M_11+M_12^†M_12+M_21^†M_21
+z^'_M(,μ^ '_C,2) M_22^†M_22)+
+z_q(,)Tr √(N^†_22 N_22) , z^'_M(,μ^ '_C,2)=(μ^ '_C,2/)^γ^'_M≫1 .
From (10.11), the mass of mions $M_{11}$ is
(m_Q/^2)^2/3∼(/)^N_c/ ,
while the masses of mions $M_{22}$ and nions $N_{22}$ look as
μ(M_22)∼μ(N_22)∼(z_q(,) m_Q/z_M^'(,μ^ '_C,2))^1/2 .At $2n_1<\bo$ this is
μ(M_22)∼μ(N_22)∼(/)^3N_c/n_2 ,
and at $\bo<2n_1\leq N_F$ this is
μ(M_22)∼μ(N_22)∼(/)^N_c(n_2+N_c-n_1)/4(N_c-n_1) .
And finally, the hybrid mions $M_{12}$ and $M_{21}$ are massless.
On the whole, the mass spectrum includes in this case. a) A large number of heaviest 11-flavored hadrons made of weakly confined quarks ${\ov q}_1,\, q_1$ with the masses $\omp$, see (10.1), (the string tension is ${\sqrt\sigma}\sim \lym^{\rm (br1)}\ll {\ov\mu}_{C,2}\ll\omp$) . b) A large number of 22-flavored hadrons made of non-relativistic and weakly confined quarks ${\ov q}_2,\, q_2$ with the constituent masses ${\ov\mu}^{\,\prime}_{C,2}$ (10.9). c) Corresponding heavy hybrid hadrons with the masses $\sim(\omp+{\ov\mu}^{\,\prime}_{C,2})$. d) A large number of gluonia with the mass scale $\sim\lym^{(\rm br1)}$ made of dual $SU(\nd)$ gluons. e) $n_1^2$ mions $M_{11}$ with the masses (10.12). f) $n_2^2$ mions $M_{22}$ and $n_2^2$ nions $N_{22}$ with the masses (10.13),(10.14).
g) And finally, the hybrid mions $M_{12}\,,\, M_{21}$ are massless.
10.3 br2 and special vacua.
At $n_2<N_c$ there are also $\rm br2$ - vacua, see section 3. For these, all their properties can be obtained by the replacement $n_1\leftrightarrow n_2$ in formulas of the preceding sections 10.1-10.2 . The only difference is that, because $n_2\geq N_F/2$, there is no analog of the conformal regime with $n_1<\bo/2$ at $\mph>{\ov\mu}_{\Phi,2}$ and
$\mu<\omp$, i.e. at $\mph>{\ov\mu}_{\Phi,2}$ the lower energy theory at $\mu<\omp$ is always in the strong coupling regime.
As for the special vacua (see section 3), all their properties can also be obtained with $n_1=\nd,\, n_2=N_c$
in formulas of the preceding sections 10.1-10.2 .
§ BROKEN $\MATHCAL{N}=2$ SQCD
We consider now ${\cal N}=2$ SQCD with $N_c$ colors, $N_F$ flavors of light quarks, the scale factor $\Lambda_2$ of the gauge coupling, and with ${\cal N}=2$ broken down to ${\cal N}=1$ by the large mass parameter $\mu_X\gg \Lambda_2$ of the adjoint field $X=X^a\lambda^a,\, {\rm Tr}(\lambda^a \lambda^b)=\delta^{ab}/2$. At very high scales $\mu\gg\mu_X$ the Lagrangian looks as (the exponents with gluons are implied in the Kahler term K)
K=1/g^2(μ,Λ_2)Tr (X^†X)+Tr (Q^†Q+Q Q) , W=-2π/α(μ,Λ_2)S+μ_X Tr (X^2)+√(2) Tr (QX
Q)+ m Tr (QQ).
The running mass of $X$ is $\mu_X(\mu)=g^2(\mu)\mu_X$, so that at scales $\mu<\mu^{\rm pole}_X=g^2(\mu^{\rm pole}_X)\mu_X$ the field $X$ decouples from the dynamics and the RG evolution becomes those of ${\cal N}=1$ SQCD. The matching of ${\cal N}=2$ and ${\cal N}=1$ couplings at $\mu=\mu^{\rm pole}_X$ looks as ($\Lambda_2$ and $\la$ are the scale factors of ${\cal N}=2$ and ${\cal N}=1$ gauge couplings, $\la$ is held fixed when $\mu_X\gg\la$ is varied, ${\rm b}_2=2N_c-N_F\,,\,\bo=3N_c-N_F$)
2π/α(μ=μ^pole_X,Λ_2)=2π/α(μ=μ^pole_X,) ,
μ_X≫≫Λ_2 ,b_2lnμ^pole_X/Λ_2
=lnμ^pole_X/-N_Flnz_Q(,μ^pole_X)+N_cln1/g^2(μ=μ^pole_X) ,
^=Λ^b_2_2μ^N_c_X/z^N_F_Q (,μ^pole_X)=
z^N_F_Q (μ^pole_X,)Λ^b_2_2μ^N_c_X , z_Q(,μ=μ^pole_X)∼(lnμ^pole_X/)^N_c/≫1.
Although the field $X$ becomes too heavy and does not propagate any more at $\mu<\mu^{\rm pole}_X$, the loops of light quarks and gluons which are still active at $\la<\mu<\mu^{\rm pole}_X$ if the next largest physical mass $\mu_H$ is below $\la$ and at $\mu_H<\mu<\mu^{\rm pole}_{X}$ if $\mu_H>\la$, induce him a non-trivial logarithmic renormalization factor $z_X(\mu^{\rm pole}_X,\mu<\mu^{\rm pole}_X)\ll 1$.
Therefore, finally, at scales $\la\ll\mu\ll\mu^{\rm pole}_X$ if $\mu_H<\la$ and at $\mu_H\ll\mu\ll\mu^{\rm pole}
_X$ if $\mu_H>\la$, the Lagrangian of the broken ${\cal N}=2$ - theory with $0<N_F<2N_c$ can
be written as
K=z_X(μ^pole_X,μ)/g^2(μ^pole_X) Tr (X^†X)+z_Q(μ^pole_X,μ) Tr (Q
^†Q+QQ) ,
W=-2π/α(μ,)S+μ_X Tr (X^2)+√(2) Tr (Q X Q)+ m Tr
(QQ) .z_X(μ^pole_X,μ)∼( ln (μ/)/ln (μ^pole_X/) )^
b_2/≪1 ,
z_Q(μ^pole_X,μ)=z_Q(μ^pole_X,)z_Q(,μ), z_Q(,μ)∼(lnμ/)^N_c/≫1 .
In all cases when the field $X$ remains too heavy and dynamically irrelevant, it can be integrated out in (11.3) and one obtains
K=z_Q(μ^pole_X,μ) Tr (Q^†Q+QQ) ,
W_Q=-2π/α(μ,)S+ m Tr(Q Q)-1/2μ_X(Tr (QQ)^2-1/N_c(Tr Q Q )^2 ).
Now we redefine the normalization of the quarks fields
Q=1/z^1/2_Q(μ^pole_X,) Q , Q=1/z^1/2_Q(μ^pole_X,
) Q ,
K=z_Q(,μ)Tr( Q^†Q+(QQ) ), W= -2π/α(μ,)S+W_Q ,
W_Q=m/z_Q(μ^pole_X,) Tr(Q Q)-1/2 z^2_Q(μ^pole_X,)μ_X(Tr
(QQ)^2-1/N_c(Tr Q Q )^2 ).
Comparing this with (1.3) and choosing
m/z_Q(μ^pole_X,)=m_Q≪ , z^2_Q(μ^pole_X,)μ_X=≫it is seen that with this matching the $\Phi$ - theory and the broken ${\cal N}=2$ SQCD will be equivalent.
Therefore, until both $X$ and $\Phi$ fields remain dynamically irrelevant, all results obtained above for the $\Phi$ - theory will be applicable to the broken ${\cal N}=2$ SQCD as well. Besides, the $\Phi$ and $X$ fields remain dynamically irrelevant in the same region of parameters, i.e. at $N_F<N_c$ and at $\mu_H>\mu_o$ if $N_F>N_c$ , see (4.1).
Moreover, some general properties of both theories such as the multiplicity of vacua with unbroken or broken flavor symmetry and the values of vacuum condensates of corresponding chiral superfields (i.e. $\langle{\ov Q}_j Q_i\rangle$ and $\langle S\rangle$, see section 3) are the same in these two theories, independently of whether the fields $\Phi$ and $X$ are irrelevant or relevant.
Nevertheless, once the fields $\Phi$ and $X$ become relevant, the phase states, the RG evolution, the mass spectra etc., become very different in these two theories. The properties of the $\Phi$ - theory were described in detail above in the text. In general, once $X$ becomes sufficiently light and dynamically relevant, the dynamics of the broken ${\cal N}=2$ SQCD with $\mu_X\gg\la$ becomes complicated (we expect that the field $X$ will be higgsed, with $\mu_{\rm gl}\sim \mu_o$, see (4.1) ) and is outside the scope of this paper.
Finally, we trace now a transition to the slightly broken ${\cal N}=2$ theory with small $\mu_X\ll\Lambda_2$
and fixed $\Lambda_2$. For this, we write first the appropriate form of the effective superpotential obtained from (11.7),(11.8)
W^eff_Q=-S+m/z_Q(μ^pole_X,) Tr(Q Q)-1/2 z^2_Q(μ^pole_X,)μ_X(Tr (QQ)^2-1/N_c(Tr Q Q )^2 ),S=( QQ/^ )^1/ , ^=z^N_F_Q(μ^pole_X,)Λ^b_2_2μ^N_c_X
and restore now the original normalization of the quark fields $\ov{\textbf{Q}}, \textbf{Q}$ appropriate for the slightly broken ${\cal N}=2$ theory with varying $\mu_X\ll\Lambda_2$ and fixed $\Lambda_2$, see (11.6),
W^eff_Q=-S+m Tr(Q Q)-1/2 μ_X(Tr (QQ)^2
-1/N_c(Tr Q Q )^2 ), S=(QQ/Λ^b_2_2μ^N_c_X )^1/
One can obtain now from (11.11) the values of the quark condensates $\langle\ov{\textbf{Q}}_j \textbf{Q}_i\rangle$ at fixed $\Lambda_2$ and small $\mu_X\ll\Lambda_2$. Clearly, in comparison with $\langle{\ov Q}_j Q_i\rangle$ in section 3, the results for $\langle\ov{\textbf{Q}}_j \textbf{Q}_i\rangle$ are obtained by the replacement : $m_Q\ra m\,,\, \mph\ra\mu_X\,,\,\la^{\bo}\ra\Lambda^{b_2}_2\mu^{N_c}_X$, while the multiplicities of vacua are the same. From (11.11), the dependence of $\langle\ov{\textbf{Q}}_j \textbf{Q}_i\rangle$ and $\langle S\rangle$ on $\mu_X$ is trivial in all vacua, $\sim \mu_X$.
With the above replacements, the expressions for $\langle{\ov Q}_j Q_i\rangle$ in section 3 in the region $\la\ll\mph\ll\mo$ correspond here to the hierarchy $m\ll\Lambda_2$, while those in the region $\mph\gg\mo$ correspond here to $m\gg\Lambda_2$. In the language of [12] used in [11] (see sections 6-9 therein), the correspondence between the $r$ - vacua [12, 11] of the slightly broken ${\cal N}=2$ theory with $0< \mu_X/\Lambda_2\ll 1,\,\, 0< m/\Lambda_2\ll 1$ and our vacua in section 3 looks as
This correspondence is based on comparison of multiplicities of our vacua at $\mph\ll\mo$ described in section 3 and those of $r$ - vacua at $m\ll\Lambda_2$ and $\mu_X\ll\Lambda_2$ as these last are given in [11].
: a) $r=n_1$, b) our L - vacua with the unbroken or the L - type ones with spontaneously broken flavor symmetry correspond, respectively, to the first group of vacua of the non-baryonic branches with $r=0$ and $r\geq 1,\, r\neq\nd$ in [11] , c) our S - vacua with the unbroken flavor symmetry and $\rm br2$ - vacua with the spontaneously broken flavor symmetry correspond to the first type from the second group of vacua of the baryonic branches with, respectively, $r=0$ and $1\leq r<\nd$ in [11], d) our special vacua with $n_1=\nd,\, n_2=N_c$ correspond to the second type of vacua from this group, see [11].
§ CONCLUSIONS
The mass spectra and phase states of the ${\cal N}=1$ SQCD-like $\Phi$ - theory (and its dual variant, the $d\Phi$ - theory) with additional colorless flavored fion fields $\Phi_{ij}$ were described above in the text in some details, within the dynamical scenario $\#1$ (see [6, 7] for more details about this scenario with the coherent colorless diquark condensate).
In comparison with the standard ${\cal N}=1$ SQCD with the superpotential $W=m_Q{\rm Tr}({\ov Q}Q)$ and the only small parameter $m_Q/\la\ll 1$ which serves as the infrared regulator, the $\Phi$ - theory includes two independent competing small parameters which serve as infrared regulators, $m_Q/\la\ll 1$ and $\la/\mph\ll 1$, see (1.3). Due to this the dynamics of this theory is much richer. Two main qualitatively new elements in this $\Phi$ - theory are :
a) the appearance of a large number of vacua with the spontaneously broken vectorial flavor symmetry, $U(N_F)\ra U(n_1)\times U(n_2)$,
b) in a number of cases with $N_F>N_c$, due to their interactions with the light quarks, the seemingly heavy and dynamically irrelevant fion fields $\Phi$ `return back ' and there appear two additional generations of light $\Phi$ - particles, see the section 4.
This is not a purpose of these conclusions to repeat in a shorter form all results obtained above in the main text for the phase states and mass spectra of the direct and dual theories at different values of $\mph/\la\gg 1$. We only point out here that, similarly to the ordinary SQCD [6, 7, 8], the direct $\Phi$ -theory and its dual variant, the $d\Phi$ - theory, are also not equivalent (at least, within the scenario $\#1$ considered in this paper).
We will try only to formulate here in a few words the most general qualitative property of SQCD-like theories which emerged from the studies in [6, 7, 8] and in this paper. This is the extreme sensitivity of their dynamical behavior in the IR region of momenta, of their mass spectra and even of the phase states, to the values of small parameters in the Lagrangian which serve as infrared regulators.
The $\Phi$-theory with $\mph\gg\la$ considered in this paper is tightly connected with the $X$-theory which is the ${\cal N}=2$ SQCD broken down to ${\cal N}=1$ by the large mass parameter $\mu_X\gg\Lambda_2$ of the adjoint fields $X$. The multiplicity of vacua and the numerical values of the quark and gluino condensates, $\langle{\ov Q}_j Q_i\rangle$ and $\langle S\rangle$, are the same in both theories (under the appropriate matching of parameters, see the section 11). Moreover, in all those cases when the fields $\Phi$ are dynamically irrelevant in the $\Phi$-theory, the fields $X$ are also dynamically irrelevant in the $X$-theory and these two theories are completely equivalent. But even in these cases, this does not mean that these two theories are simply equivalent to the ordinary SQCD with small unimportant corrections. First, the whole physics in a large number of additional vacua with the spontaneously broken flavor symmetry is completely different. And second, even in vacua with the unbroken flavor symmetry, these theories evolve to the standard SQCD with small corrections not at $\mph=(\rm{several})\la$ as one can naively expect, but only at parametrically large values of $\mph/\la$ (and, besides, these values differ parametrically in the direct and dual theories, see sections 5 and 6).
But when the fields $\Phi$ and $X$ become dynamically relevant the phase states, the mass spectra, etc. become very different in the $\Phi$ and $X$ - theories.
We have described also in section 11 the connections between the values of the quark condensates in different vacua in the strongly broken ${\cal N}=2$ SQCD with large varying $\mu_X\gg\la$ and fixed $\la$ with those in the slightly broken ${\cal N}=2$ SQCD with small varying $\mu_X\ll\Lambda_2$ and fixed $\Lambda_2$.
This work was supported in part by Ministry of Education and Science of the Russian Federation and RFBR grant 12-02-00106-a.
§ THE RG FLOW IN THE $\MATHBF \PHI$ - THEORY AT $\MATHBF{\MU>\LA}$
A.1 We first consider the $\Phi$ - theory at $N_c<N_F<2N_c$ where it is taken as UV-free. We start with the canonically normalized Kahler term $K$ at the very high scale $\mu\sim \mu_{\rm UV}$ and the running couplings and mass parameters
K=Tr (Φ^†Φ)+Tr( Q^†Q+
( QQ) ) , W=-2π/α(μ)S+W_Φ+W_Q , W_Φ=(μ)/2[Tr (Φ^2)-1/(Tr Φ)^2] , W_Q=- f(μ)Tr ( QΦQ)+Tr
( Q m_ Q(μ) Q) .
Now, instead of running parameters, we introduce $\mu$-independent ones, $\la,\,\, \mph$ and $m_Q$ ($\mph\gg\la$ and $m_Q\ll\la$ in the main text),
1/a(μ)=2π/N_cα(μ)=/N_clnμ/-N_F/N_clnz_Q(,μ)+ln1/a(μ)+C_a , =3N_c-N_F ,
a_f(μ)=N_c f^2(μ)/2π=a_f=N_c f^2/2π/z_Φ(,μ)z^2_Q(,μ) , (μ)≡f^2/z_Φ(,μ) , m_Q(μ)≡m_Q/z_Q(,μ) ,where $z_Q(\la,\mu\gg\la)\gg 1$ and $z_{\Phi}(\la,\mu)$ are the perturbative renormalization factors (logarithmic in this case) in the theory with all fields massless, $a_f$ is taken as $a_f\sim 1/(\rm several)$ and $C_a$ is also $O(1)$ (it will be omitted for simplicity). Therefore, after redefinitions of the quark and $\Phi$ fields,
the Lagrangian at the very high scale can be rewritten as
K=z_Φ(,μ)1/f^2Tr (Φ^†Φ)+z_Q(,μ)Tr( Q^†Q+(QQ)
) ,
W_Φ=/2[Tr (Φ^2)-1/(Tr Φ)^2] , W_Q=-Tr ( QΦQ)+Tr ( Qm_Q Q) .
From (A.2)
d a_f(μ)/dlnμ=β_f=-a_f(μ)(2γ_Q(μ)+γ_Φ(μ)), γ_Q=dlnz_Q(μ)/dlnμ, γ_Φ=dlnz_Φ(μ)/dlnμ .
In the approximation of leading logarithms at large $\mu$
γ_Q(μ)≃2 C_F/N_ca(μ)-N_F/N_ca_f(μ), γ_Φ(μ)≃-a_f(μ) , 2 C_F/N_c=N_c^2-1/N_c^2≃1 .
From (A.4),(A.5), there is the UV free solution
a(μ)≃N_c/1/ln(μ/) , a_f(μ)∼a_f(1/ln(μ/))^2N_c/≪a(μ), 1<2N_c/<2 ,
z_Q(,μ)∼(lnμ/)^N_c/≫1, z_Φ(,μ)∼1 .
It is seen from (A.6) that the Yukawa coupling $a_f(\mu)$ is parametrically small in comparison with the gauge coupling $a(\mu)$ and, up to small corrections, it has no effect on the RG evolution at large $\mu$.
The first physical mass parameter which influences the RG flow with lowering the scale $\mu$ is $\mu^{\rm pole}_{1}(\Phi)=\mph(\mu=\mu^{\rm pole}_{1}(\Phi)\,)=f^2\mph/z_{\Phi}(\la,\mu^{\rm pole}_{1}(\Phi))\sim f^2\mph\gg\la$, so that $\mu_{\Phi}(\mu)$ becomes $\mu_{\Phi}(\mu)\sim f^2\mph>\mu$ at $\mu<\mu^{\rm pole}_{1}(\Phi)$ and the fields $\Phi$ become too heavy. They do not propagate any more and do not influence the RG evolution until $\mu_{\Phi}(\mu)>\mu$. Nevertheless, the anomalous dimension $\gamma_{\Phi}(\mu)$ remains small but nonzero even at $\mu<\mu^{\rm pole}_{1}(\Phi)$ due to loops of still active light quarks (and gluons interacting with quarks) and, instead of (A.5), the anomalous dimensions look at $\mu<\mu^{\rm pole}_{1}(\Phi)$ as
γ_Q(μ)≃a(μ), γ_Φ(μ)≃-a_f(μ) ,
while (A.6),(A.7) remain the same. Hence, although the heavy fields $\Phi_{ij}$ decouple at $\la<\mu<\mu^{\rm pole}_{1}(\Phi)$, the RG flow remains parametrically the same because their role even at $\mu>\mu^{\rm pole}_{1}(\Phi)$ was small.
Therefore, finally, at scales $\la<\mu<\mu^{\rm pole}_{1}(\Phi)$ if there is no physical masses $\mu_H>\la$ and at
$\mu_H<\mu<\mu^{\rm pole}_{1}(\Phi)$ if $\mu_H>\la$, the Lagrangian of the $\Phi$ - theory with $N_c<N_F<2N_c$ light flavors can be written as
K=1/f^2Tr (Φ^†Φ)+z_Q(,μ)Tr( Q^†Q+(QQ) ) , W=-2π/α(μ,)S+W_Φ+W_Q , W_Φ=/2[Tr (Φ^2)-1/(Tr Φ)^2] , W_Q=Tr ( Q m^tot_Q Q) , m^tot_Q=m_Q-Φ ,
with $z_Q(\la,\mu)$ given in (A.7).
A.2 We consider now the case $1\leq N_F<N_c$ . Although the $\Phi$ - theory is not UV free in this case and requires UV completion at $\mu>\mu_{\rm UV}$, the RG flow at $\mu_H<\mu\ll\mu_{\rm UV}$ is very specific (see below, the quarks are really higgsed in this case at $\mu_H=\mu_{\rm gl},\,\,\la\ll\mu_{\rm gl}\ll\mph\ll\mu_{UV}$, see section 2). We take from the beginning $a_f$ in (A.2) to be sufficiently small, $a_f\ll 1$, and calculate the behavior of $a(\mu)$ and $a_f(\mu)$ at $\la\ll\mu\ll\mu_{UV}$ in the massless theory which follows from their definitions in (A.2). Then, by definition, in the theory with $\la\ll\mu^{\rm pole}_1(\Phi)\ll\mu_{UV}$, the behavior of $a(\mu)$ and $a_f(\mu)$ at $\mu^{\rm pole}_1(\Phi)\ll\mu\ll\mu_{UV}$ will be the same while, in general, it can be different at $\mu<\mu^{\rm pole}_1(\Phi)$.
There is the same solution (A.6) also at $1\leq N_F<N_c$, with a difference that $2/3<2N_c/\bo<1$ now and $a_f\ll 1$. Hence, starting with $\mu>\la,\,\, a_f(\mu)$ begins first to decrease with increasing $\mu$, but more slowly now than $a(\mu)\sim\ln^{-1}(\mu/\la)$. Due to this, $\beta_f(\mu)$ in (A.4) changes a sign at $\mu\sim\ov\mu$,
a_f(μ)∼a(μ) lnμ/∼(1/a_f)^/N_c-N_F
≫1, a_f(μ)∼1/ln(μ/)∼(a_f)^/N_c-N_F≪a_f≪1
and then $a_f(\mu)$ begins to grow
a_f(μ>μ)∼1/ln(μ_UV/μ) , ln(μ_UV/μ)∼(1/a_f)^/N_c-N_F≫1
with further increasing $\mu>{\ov\mu}$. Therefore, $z_{\Phi}(\la,\mu<{\ov\mu})\sim 1$ in the massless theory.
For our purposes in section 2 it will be sufficient to have $\mu_{\rm gl}\ll\mu^{\rm pole}_1(\Phi)\sim a_f\mph\ll{\ov\mu}\ll\mu_{\rm UV}$. This leads to a sufficiently weak logarithmic restriction
1/a_f≫(ln/)^N_c-N_F/, 0<N_c-N_F/<1/3 ,
and then $z_{\Phi}(\la,\mu<\mu^{\rm pole}_1(\Phi))$ remains $\sim 1$ also in the $\Phi$ - theory with massive fields
§ THERE IS NO VACUA WITH $\MATHBF{\LANGLE S\RANGLE=0}$ AT $\MATHBF{M_Q\NEQ 0}$
The purpose of this appendix is to show that the gluino condensate $\langle S\rangle\neq 0$ at $m_Q\neq 0$ in all vacua with the broken flavor symmetry, $U(N_F)\ra U(n_1)\times U(n_2)$, in both the direct and dual theories.
1 . Direct theory
We assume that there is at $N_c<N_F<2N_c$ a large number of additional vacua with either $1\leq n_1\leq N_c-1$ components $\langle{\ov Q}_1Q_1=\Pi_1\rangle=0$, or $n_2\geq n_1$ components $\langle{\ov Q}_2Q_2=\Pi_2\rangle=0$. Even in this case the relations at $\mu=\la$
⟨Π_1+Π_2⟩-1/N_cTr ⟨Π⟩=m_Q, ⟨S⟩=1/⟨Π_1⟩⟨Π_2⟩, ⟨Π_1⟩≠⟨Π_2⟩ ,
⟨m^tot_Q,1⟩=⟨m_Q-Φ_1⟩=⟨Π_2⟩/ , ⟨m^tot_Q,2⟩=⟨m_Q-Φ_2⟩=⟨Π_1⟩/ ,following from the Konishi anomalies (1.2),(1.4) remain valid. Therefore, one obtains from (B.1) that either
⟨Π_2⟩=0 , ⟨Π_1⟩=N_c/N_c-n_1 m_Q , ⟨S⟩=0 , 1≤n_1≤N_c-1 ,
⟨m^tot_Q,1⟩=⟨Π_2⟩/=0 , ⟨m^tot_Q,2⟩=⟨Π_1⟩/=N_c/N_c-n_1 m_Q ,or
⟨Π_1⟩=0 , ⟨Π_2⟩=N_c/N_c-n_2 m_Q ,
⟨S⟩=0 , n_2≠N_c ,
⟨m^tot_Q,2⟩=⟨Π_1⟩/=0 , ⟨m^tot_Q,1⟩=
⟨Π_2⟩/=N_c/N_c-n_2 m_Q in these vacua. We will show below that this assumption is not self-consistent. I.e., we will start with (B.2) or (B.3) and calculate then explicitly $\langle S\rangle\neq 0$ in these vacua. For this, using a holomorphic dependence of $\langle S\rangle$ on $\mph$, it will be sufficient to calculate $\langle S\rangle\neq 0$ in some range of most convenient values of $\mph$. Hence, we take $m_Q\mph\sim \la^2$.
In vacua (B.2) with $\langle\Pi_2\rangle=0,\, \langle\Pi_1\rangle\sim m_Q\mph\sim \la^2$ the quarks ${\ov Q}_1,\, Q_1$ are higgsed with $\langle{\ov Q}_1\rangle=\langle Q_1\rangle\sim \la$. At $n_1<N_c-1$ the lower energy theory at $\mu<\la$ contains $SU(N_c-n_1)$ unbroken gauge symmetry with the scale factor of the gauge coupling $(\Lambda^\prime)^{\bo^\prime}\sim\la^{\bo}/\det \Pi_{11},\, \langle\Lambda^\prime\rangle\sim\la$, $n^2_1$ pions $\Pi_{11}$ and ${\ov Q}_2,\, Q_2$ quarks with zero condensate and the running mass $\langle m^{\rm tot}_{Q,2}\rangle=\langle m_Q-\Phi_2\rangle=\langle\Pi_1\rangle/\mph\sim m_Q$ at $\mu=\la$. For this reason, the variants with the $DC_2$ or $Higgs_2$ phases of these quarks are excluded, they will be always in the heavy quark $HQ_2$ - phase. At all $n_1<N_c-1$, proceeding as in [6, 7, 8], i.e. lowering the scale down to $\mu<m^{\rm pole}_{Q,2}\sim m_Q/z_Q(\la,m^{\rm pole}_{Q,2})$ and integrating out ${\ov Q}_2, Q_2$ quarks as heavy particles, there remains the pure $SU(N_c-n_1)$ Yang-Mills theory (and $n^2_1$ pions $\Pi_{11}$) with the scale factor of its gauge coupling
and, finally, with the Lagrangian of the form (2.22) at $\mu<\langle\lym\rangle$. From (B.4)
⟨S⟩=⟨^3⟩∼^3(/)^n_1/N_c-n_1(m_Q/)^n_2-n_1/N_c-n_1≠0 .
At $n_1=N_c-1$ the gauge group will be broken completely and (B.4) originates from the instanton contribution.
The vacua (B.3) with $\langle\Pi_1\rangle=0$ are considered the same way and one obtains (B.4),(B.5) with the replacement
$n_1\leftrightarrow n_2$. (In vacua (B.3) the cases with $n_2>N_c$ are excluded from the beginning as the rank of $\langle Q_2\rangle$ is $\leq N_c$ and the unbroken $U(n_2)$ flavor symmetry cannot be maintained; the case $n_2=N_c$ is also excluded as $\langle\Pi_1\rangle\neq 0$ in this case, see (B.1) ). Hence, only the cases with $n_2\leq N_c-1$ remain).
On the whole, the assumption about the existence of additional vacua (B.2) or (B.3) with $\langle\Pi_1\rangle=0$ or $\langle\Pi_2\rangle=0$ at $N_c<N_F<2N_c$ is not self-consistent.
2 . Dual theory
The dual analog of (B.1)-(B.3) looks as, see (1.8),
⟨M_1+M_2⟩-1/N_cTr ⟨M⟩=m_Q, ⟨S⟩=1/⟨M_1⟩⟨M_2⟩, ⟨M_1⟩≠⟨M_2⟩ ,
By assumption, there is a large number of additional vacua with either
⟨M_2⟩=0 , ⟨M_1⟩=N_c/N_c-n_1 m_Q , ⟨S⟩=0 ,
1≤n_1≤N_c-1 ,
⟨N_1⟩=⟨m^tot_Q,1⟩=⟨M_2⟩/=0 , ⟨N_2⟩=⟨m^tot_Q,2⟩=⟨M_1⟩/=N_c/N_c-n_1 m_Q ,or
⟨M_1⟩=0 , ⟨M_2⟩=N_c/N_c-n_2 m_Q , ⟨S⟩=0 ,
n_2≠N_c ,
⟨N_2⟩=⟨m^tot_Q,2⟩=⟨M_1⟩/=0 , ⟨N_1⟩=⟨m^tot_Q,1⟩=⟨M_2⟩/=N_c/N_c-n_2 m_Q .In this case, it is more convenient for our purposes to choose the regions $\la\ll\mph\ll\mo$ at $3N_c/2<N_F<2N_c$ and $\la\ll\mph\ll\la(\la/m_Q)^{1/2}$ at $N_c<N_F<3N_c/2$.
We start with (B.7). It is not difficult to check that in these ranges of $\mph$ and at all $N_c<N_F<2N_c$ the largest mass is ${\ov\mu}_{\rm gl,2}\gg \mu^{\rm pole}_{q,1}$ due to higgsing of ${\ov q}_2, q_2$ quarks. Hence, in these (B.7) vacua, the cases with $n_2>\nd$ are excluded from the beginning as the rank of $\langle q_2\rangle$ is $\leq \nd$ and the unbroken $U(n_2)$ flavor symmetry cannot be maintained. But this excludes all such vacua as $n_1\leq N_c-1$ and $n_2=N_F-n_1\geq
\nd+1$.
Therefore, there remain only (B.8) vacua. In these, in the above ranges of $\mph$, the largest mass is ${\ov\mu}_{\rm gl,1}\gg \mu^{\rm pole}_{q,2}$ due to higgsing of ${\ov q}_1, q_1$ quarks. Hence, one obtains from similar considerations that $n_1\leq\nd-1$ (the case $n_1=\nd$ is also excluded from (B.6),(B.8) ). And similarly, because their condensate $\langle N_2\rangle=0$, the quarks ${\ov q}_2, q_2$ will be always in the heavy quark $HQ_2$ - phase only. Hence, at all $n_1<N_c-1$, proceeding as in [6, 7, 8], i.e. integrating out first higgsed gluons and ${\ov q}_1, q_1$ quarks at $\mu<{\ov\mu}_{\rm gl,1}$, then ${\ov q}_2, q_2$ quarks with unhiggsed colors at $\mu<\mu^{\rm pole}_{q,2}$ and, finally,
unhiggsed gluons at $\mu<\langle\lym\rangle$, one obtains the low energy Lagrangian of the form (9.11) with
^3=(^(M_22/)/N_11)^1/(-n_1) , ⟨S⟩=⟨^3⟩∼^3(/)^n_2/n_2-N_c(m_Q/)^n_2-n_1/n_2-N_c≠0 .
At $n_1=\nd-1$ the dual gauge group will be broken completely and (B.9) originates from the instanton contribution.
On the whole, the assumption about the existence of additional vacua (B.7) or (B.8) with $\langle M_1\rangle=0$ or $\langle M_2\rangle=0$ at $N_c<N_F<2N_c$ is also not self-consistent.
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K. Konishi, Anomalous supersymmetry transformation of some composite operators in SQCD,
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N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories,
Phys. Rev. D 49 (1994) 6857, hep-th/9402044
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N. Seiberg, Electric - magnetic duality in supersymmetric nonabelian gauge theories,
Nucl. Phys. B 435 (1995) 129, hep-th/9411149
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Equal quark masses, JETP 110 (2010) 383, arXiv : 0712.3167 [hep-th]
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arXiv : 0805.2299 [hep-th]
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V.L. Chernyak, On mass spectrum in SQCD and problems with the Seiberg duality.
Another scenario, JETP 114 (2012) 61, arXiv: 0811.4283 [hep-th]
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G. Veneziano, S. Yankielowicz, An effective Lagrangian for the pure ${\cal N}=1$ supersymmetric Yang-Mills theory, Phys. Lett. B 113 (1982) 231
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T. Taylor, G. Veneziano, S. Yankielowicz, Supersymmetric QCD and its massless limit: an effective Lagrangian analysis, Nucl. Phys. B 218 (1983) 493
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|
arxiv-papers
| 2012-05-02T12:42:15 |
2024-09-04T02:49:30.443374
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Victor L. Chernyak",
"submitter": "Victor Chernyak",
"url": "https://arxiv.org/abs/1205.0410"
}
|
1205.0434
|
# Na2IrO3 as a molecular orbital crystal
I. I. Mazin Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
Harald O. Jeschke Institut für Theoretische Physik, Goethe-Universität
Frankfurt, 60438 Frankfurt am Main, Germany Kateryna Foyevtsova Institut für
Theoretische Physik, Goethe-Universität Frankfurt, 60438 Frankfurt am Main,
Germany Roser Valentí Institut für Theoretische Physik, Goethe-Universität
Frankfurt, 60438 Frankfurt am Main, Germany D. I. Khomskii II.
Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937 Köln,
Germany
###### Abstract
Contrary to previous studies that classify Na2IrO3 as a realization of the
Heisenberg-Kitaev model with dominant spin-orbit coupling, we show that this
system represents a highly unusual case in which the electronic structure is
dominated by the formation of quasi-molecular orbitals (QMOs), with
substantial quenching of the orbital moments. The QMOs consist of six atomic
orbitals on an Ir hexagon, but each Ir atom belongs to three different QMOs.
The concept of such QMOs in solids invokes very different physics compared to
the models considered previously. Employing density functional theory
calculations and model considerations we find that both the insulating
behavior and the experimentally observed zigzag antiferromagnetism in Na2IrO3
naturally follow from the QMO model.
###### pacs:
75.10.-b,75.10.Jm,71.70.Ej,71.15.Mb
High interest in the recently synthesized hexagonal iridates felner ;
kobayashi ; gegenwart is due to the hypothesis Jackeli2009 ; Shitade2009
that the electronic structure in these materials is dominated by the spin-
orbit (SO) interaction. In this case, the Ir $t_{2g}$ bands are most naturally
described by relativistic atomic orbitals with the effective angular moment,
$j_{\rm eff}=3/2$ and $j_{\rm eff}=1/2.$ In this approximation, the splitting
between the 3/2 and 1/2 states is larger than their dispersion. The upper band
$j_{\rm eff}=1/2$ is half filled and Ir atoms can be described as localized
($j_{\rm eff}=1/2,$ $M=1$ $\mu_{\rm B}$) magnetic moments note1 with the
exchange interaction strongly affected by SO coupling. In particular, this
picture leads to a very appealing framework known as Heisenberg-Kitaev model
jackeli ; trebst , with highly nontrivial physical properties. However,
experimental evidence for the $j_{\rm eff}$ scenario is lacking jeff .
In this Letter, based on ab initio density functional theory (DFT)
calculations and model considerations, we show that this picture does not
apply to the actual Na2IrO3. Instead, this system represents a highly unusual
case where the formation of electronic structure is dominated by quasi-
molecular orbitals (QMOs), which involve six Ir atoms arranged in a hexagon.
What distinguishes this picture from molecular solids is that there is no
associated spatial clusterization, but each Ir atom (via its three $t_{2g}$
orbitals) participates in three different QMOs, yet in the first approximation
there is no inter-QMO hopping the thus formed bands are dispersionless.
Such an electronic structure calls for a new approach. There is no known
recipe for handling its magnetic properties, or adding Coulomb correlations,
for instance. While we will not present a complete theory of spin dynamics and
correlations in the QMO framework, we will outline the general directions and
most important questions, in the expectation that this will stimulate more
theoretical and experimental work and eventually generate more insight. Yet,
the key observable features of Na2IrO3: small magnetic moment, unusual zigzag
antiferromagnetism, and Mott-enhanced insulating behavior, are naturally
consistent with the QMO framework.
The main crystallographic element of Na2IrO3 (see SI) is an Ir4+ (5$d^{5}$)
honeycomb layer with a Na1+ ion located at its center. Each Ir is surrounded
by an O octahedron, squeezed along the cubic [111] (hexagonal $z$) axis.
Therefore, Ir $d$-states are split into an upper $e_{g}$ doublet and a lower
$t_{2g}$ triplet. The [111] squeezing further splits the $t_{2g}$ levels into
a doublet and singlet; initially this effect was neglected Jackeli2009 ;
jackeli ; trebst , however, it was later included KimKim ; trig1 (and
overestimated) to explain the observed deviations from the Heisenberg-Kitaev
model.
In the previous works, after identifying the $t_{2g}-e_{g}$ splitting it was
assumed that the energy scales are $W$ $<$ ($J_{H}$, $\lambda$) $<$ $U$, where
$W\sim 4t$ is the $d$-electron band width, $t$ the effective hopping
parameter, $J_{H}$ the Hund’s rule coupling, $\lambda$ the SO parameter, and
$U$ the on-site Coulomb repulsion. In this limit, the electrons are localized
and the system is a Mott insulator. While $\lambda$ $\sim$ 0.4-0.5 eV for $5d$
ions, the bandwidth for 5$d$ orbitals is 1.5-2 eV and $U\sim 1-2$ eV,
$J_{H}\sim 0.5$ eV, reduced compared to typical $U\sim 3-5$ eV and $J_{H}\sim
0.8-0.9$ eV for $3d$ electrons. Many-body renormalization may narrow the bands
by a factor $(m^{\ast}/m);$ however, given that in Ir $U\sim W$, it is
unrealistic to expect a large renormalization. Therefore, the usual starting
point $W<(J_{H},\lambda)<U$ is not valid here, rather, the system is close to
an itinerant regime. $I.e.$, one cannot justify reducing the description of
Na2IrO3 (and possibly other iridates) to an effective $j=1/2$ model, decoupled
from the other $j_{eff}$ states.
Figure 1: (color online) Electronic structure of the non-magnetic Na2IrO3 for
the experimentally determined Radu crystal structure. The calculations were
performed with the full potential local orbital (FPLO) basis using the
generalized gradient approximation (see SI). The solid purple and dotted green
lines refer to calculations without and with SO interaction, respectively.
Note that the Fermi levels (shown by the horizontal dotted lines) are not
aligned.
Thus, the first step (usually skipped) is to understand the non-relativistic
band structure. We have therefore performed DFT calculations (see SI)
initially without SO effects (see Fig. 1, solid purple lines). Inverting the
band structure results (see SI), we obtained the corresponding tight-binding
Hamiltonian. The leading channel (by far) is the nearest neighbor (NN)
O-assisted hopping between unlike orbitals (see Fig. 2). This was also
correctly identified previously Jackeli2009 ; Shitade2009 . There are three
different types of NN Ir-Ir bonds; for one (we name it $xy$ bond) (see Fig. 3)
this hopping is only allowed between $d_{xz}$ and $d_{yz}$ orbitals, for the
next ($xz$) between $d_{yz}$ and $d_{xy}$ orbitals and for the third bond
($yz$) between $d_{xy}$ and $d_{xz}$. In our calculations this hopping,
$t_{1}^{\prime}$ (the prime indicates that the hopping is $via$ O) is about
270 meV. Perturbatively, this term is proportional to
$t_{pd\pi}^{2}/(E_{t_{2g}}-E_{p}),$ where $p$ stands for the O $p$ states.
Ref. Shitade2009 pointed out another (next nearest neighbors, NNN) O-assisted
term, which we find to be $\sim 75$ meV. Jackeli and Khalliulin Jackeli2009
invoked another NN hopping process, between like orbitals pointing directly to
each other. Despite the short Ir-Ir distance, these matrix elements are
surprisingly small, $\lesssim 30$ meV. Finally, some authors KimKim ; trig1
addressed the trigonal squeeze, which creates non-zero matrix elements between
the same-site $t_{2g}$ orbitals.
Figure 2: (color online) Most relevant O $p$-assisted hopping paths in
idealized Na2IrO3 structure. For each of the three Ir-Ir bond types only
hopping between two particular $t_{2g}$ orbitals is possible. The same holds
for the second and third nearest neighbor hopping via O $p$ and Na $s$
orbitals. Ir-Ir bonds are color coded as follows: $xy$ bonds are shown by blue
lines, $xz$ bonds by green, and $yz$ bonds by red ones.
The main feature of the calculated non-relativistic band structure (see Fig.
1) is formation of a singly degenerate (not counting spins) band state at
$\sim-1.2$ eV, a doubly degenerate one at $-0.7$ eV, and a three-band manifold
between $-0.3$ and $0.2$ eV. This clear separation, of the order of 0.3 eV,
cannot be related to the trigonal squeeze, as this can only split the 6
$t_{2g}$ bands (there are two Ir per cell) into a doublet and quartet.
In order to understand this, we start with the dominant hopping, the NN
O-assisted $t_{1}^{\prime}$. Let us consider an electron on a given Ir site in
a particular orbital state, say, $d_{xz}.$ The site has three NN neighbors. As
discussed above, this electron can hop, with the amplitude $t_{1}^{\prime},$
to a neighboring state of $d_{yz}$ symmetry, located at a particular NN site.
From there, it can hop further into a $d_{xy}$ state on the next site, and so
on (see Figs. 2 and 3). At each site, the electron has only one bond along
which it can hop. Following the electron around, we see that after six hops it
returns to the same state and site from where it started. This means that in
the NN $t_{1}^{\prime}$ approximation every electron is fully localized within
6 sites forming a hexagon. Such a state could be called a molecular orbital,
except that there are no spatially separated molecules on which electrons are
localized. Each Ir belongs to three hexagons, and each Ir-Ir bond to two.
Thus, three different $t_{2g}$ orbitals on each Ir site belong to three
different “quasi-molecular” orbitals (QMO) and these QMOs are fully localized
in this approximation (Fig. 3).
Figure 3: (color online) (a) Schematic plot of a Ir6Na hexagon. We use the
same color coding as in Fig. 2, $xy$ bonds are shown by blue lines and
$d_{xy}$ orbitals by blue dots, etc. (b) A quasi-molecular composite orbital
on a given hexagon. (c) Three neighboring quasi-molecular orbitals.
Six QMOs localized on a particular hexagon form six levels, listed in Table 1,
grouped into the lowest B1u singlet, the highest A1g singlet, and two doublets
E1g and E2u. The energy separation between the lowest and the highest level is
$4t_{1}^{\prime},$ which is close to the calculated total non-relativistic
$t_{2g}$ band width.
Table 1: Six quasi-molecular orbitals formed by the six $t_{2g}$ atomic orbitals on a hexagon. $(\omega=\exp(i\pi/3))$. Note that $t_{1}^{\prime}$$>$ 0 and $t_{2}^{\prime}$$<$0 Symmetry | Eigenenergy | Eigenvector(s)
---|---|---
$A_{1g}$ | $2(t_{1}^{\prime}+t_{2}^{\prime})$ | $(1,1,1,1,1,1)$
$E_{2u}$ | $t_{1}^{\prime}-t_{2}^{\prime}$ | $(1,\omega,\omega^{2},-1,\omega^{4},\omega^{5})$
(twofold) | | $(1,\omega^{5},\omega^{4},-1,\omega^{2},\omega)$
$E_{1g}$ | $-t_{1}^{\prime}-t_{2}^{\prime}$ | $(1,\omega^{2},\omega^{4},1,\omega^{2},\omega^{4})$
(twofold) | | $(1,\omega^{4},\omega^{2},1,\omega^{4},\omega^{2})$
$B_{1u}$ | $-2(t_{1}^{\prime}+t_{2}^{\prime})$ | $(1,-1,1,-1,1,-1)$
We now add the O-assisted NNN hopping $t_{2}^{\prime}$. Here there are several
such paths. However, the dominant hopping takes advantage of the diffuse Na
$s$ orbital (see Fig. 2), and is proportional to
$t_{pd\pi}^{2}t_{sp}^{2}/(E_{t_{2g}}-E_{p})^{2}(E_{t_{2g}}-E_{s})<0$. It
connects unlike NNN $t_{2g}$ orbitals that belong to the same QMO, and
therefore retains the complete localization of individual QMOs. It does shift
the energy levels though, as shown in Table 1. The upper singlet and doublet
get closer and the lower bands move apart providing the average energy
separations of $\sim 0.5$, $\sim 0.6$, and $\sim 0.1$ eV among the calculated
non-relativistic subbands (at $|t_{1}^{\prime}/t_{2}^{\prime}|=2$ the upper
two levels merge; in reality, $|t_{1}^{\prime}/t_{2}^{\prime}|\approx 3.3)$.
Given that the subband widths are 0.2–0.3 eV, obviously, the upper doublet and
singlet merge to form one three-band manifold.
Several effects contribute to the residual dispersion of the QMO subbands. The
trigonal splitting plays a role, albeit smaller than often assumed: the
trigonal hybridization is $\Delta\approx 25$ meV (the splitting being
$3\Delta$). This may seem surprising, given the large distortion of the O
octahedra, however,in triangular layers several factors of different signs
contribute to $\Delta,$ and strong cancellations are not uncommonDevina .
Trigonal splitting, combined with various NN and NNN hoppings not accounted
for above, all of them on the order of 20 meV, trigger subband dispersions of
200–300 meV (see SI for further discussion).
We shall now address the SO interaction. The corresponding bands and density
of states (DOS) are shown in Fig. 1. The lowest two subbands hardly exhibit
any SO effect, even though the spin-orbit parameter $\lambda$ in Ir is $\sim$
0.4-0.5 eV, larger than both the subband widths and subband separation.
However, a simple calculation shows that not only are the orbital momentum
matrix elements between the QMOs on the same hexagon zero (this follows from
the quenching of the orbital momentum in the QMO states), but they also vanish
between the like QMOs, located at the neighboring hexagons, such as
$B_{1u}-B_{1u}.$ Furthermore, at $\Gamma$ the matrix elements between the two
lowest subbands, $B_{1u}$ and $E_{1g},$ vanish because of different parities;
away from the $\Gamma$ point the effect of SO increases, in the first
approximation, as
$F(\mathbf{k})=\sin^{2}\mathbf{kA}+\sin^{2}\mathbf{kB}+\sin^{2}\mathbf{kC}$,
where A, B and C are the three vectors connecting the centers of the hexagons,
as can be worked out by applying the ${\bf L}\cdot{\bf S}$ operator to the
corresponding QMOs.
The situation becomes more complex in the upper manifold, where three bands,
$A_{1g}$ and two $E_{2u}$, come very close. Even though the diagonal matrix
elements, as well as nondiagonal elements at $\Gamma$ still vanish, the fact
that $A_{1g}$ and $E_{2u}$ are nearly degenerate in energy induces a
considerable SO effect at all other k-points (which grows linearly with $k,$
as $\sqrt{F(\mathbf{k)}}).$ Note that deviations from the minimal model
($t_{1}^{\prime},t_{2}^{\prime}$) and SO coupling with the lower $E_{1g}$
states also affect the bands at $k=0$. We also remind that the orbital moment
of the individual electronic states can only be finite if the QMOs mix (which
is the case), and the direction of the orbital moment is different in
different parts of the Brillouin zone: along one of the three cardinal in-
plane directions it is parallel to the cubic $x$, along another to $y$, $etc$.
Since the spin moment tends to be parallel to the orbital moment, SO is
competing with the Hund’s rule coupling and suppresses the tendency to
magnetism.
Let us now discuss the effect of the Hubbard correlations. It was initially
conjectured that Na2IrO3 was a Mott insulator. This seems counterintuitive,
since similar $4d$ Ru and Rh compounds are correlated metals, and more diffuse
5$d$ orbitals have a smaller Hubbard $U\sim 1.5-2$ eV and stronger
hybridization. It is hard to justify that this $U$ can drive a 5/6 filled band
of a similar width into an insulating state. Recently another, more logical
concept has gained currency: on the LDA level Na2IrO3 is a semimetal, barely
missing being a semiconductor, and a small Hubbard $U$ just helps to enhance
the already (spin-orbit driven) existing gap. Indeed, in our calculations the
minimal gap is $-$8 meV, but the average direct gap is 150 meV, consistent
with the optical absorptionDirk . The minimal direct (optical) gap is 50 meV,
so it is plausible that it is somewhat enhanced by correlation effects.
In order to include the effect of an onsite Hubbard $U$ in the QMO basis, a
$U_{\rm QMO}\sim U/6$ has to be applied to each QMOGunnarsson , with a
residual Coulomb repulsion between neighboring QMOs, $V_{\rm QMO}\sim
U/18=U_{\rm QMO}/3$ (note that two QMOs overlap on two sites). Overall, we
expect that the effect of the Coulomb repulsion in our system is similar to
that in a single-site two-orbital Hubbard model at half filling (the upper QMO
band is half-filled) and $U_{\rm QMO}\approx W\approx 150-200$ meV. In this
case, since $U_{\rm QMO}$ does not compete with one-electron hopping any more,
one should expect that the gap will be enhanced by a considerable fraction of
$U_{\rm QMO}$, which is consistent with the experiment. Thus, Hubbard
correlations are of no qualitative importance, and only moderately enhance the
existing gap.
Since all electrons are fully delocalized over six sites, any model assuming
localized spins (whether Heisenberg or Kitaev) is difficult to justify. On the
other hand, the QMOs are not magnetically rigid objects and neighboring QMOs
overlap on 2 out of 6 sites, which makes a model with magnetic moments
localized on QMOs equally unsuitable Comment_Fe .
We will consider therefore magnetism in the itinerant approach. In the non-
relativistic case, the non-magnetic DOS shows a high peak at $E_{F}$ due to
$E_{2u}$ and $A_{1g}$ merging and rather flat band dispersion (see Fig. 1).
Such a system is unstable against ferromagnetism (FM) and the peak is easily
split gaining exchange energy (1 $\mu_{\rm B}/$Ir) with little loss of kinetic
energy. The resulting FM state is half-metallic (Fig. 4) (see SI).
Figure 4: Non-relativistic non-magnetic (purple) and ferromagnetic (orange)
density of states (DOS) of Na2IrO3 calculated with the FPLO basis.
Turning on the SO interaction has a drastic effect on magnetism. SO competes
with the Hund’s rule that favors all onsite orbitals to be collinear. The spin
moment is then reduced from $1$ $\mu_{\rm B}$ to $\approx 0.4$ $\mu_{\rm
B}$/Ir for ferro-, and $\approx 0.2$ $\mu_{\rm B}$/Ir for the zigzag and
stripe antiferromagnetic (AFM) arrangements (see SI). The orbital moment is
parallel to the spin one, reminiscent of the $j_{\rm eff}=1/2$ state, and is
roughly equal in magnitude and not twice larger, as it should be for $j_{\rm
eff}=1/2$. The energy gain for the FM case drops to a few meV/Ir gain , and
the zigzag pattern evolves as the most favorable AFM state.
Qualitatively, two closely competing ground states emerge from the
relativistic DFT calculations: ferromagnetic and zigzag. In the context of an
itinerant picture, we can argue as follows. SO creates a pseudogap at the
Fermi level in the non-magnetic calculations (see Fig. 1). This gains one-
electron energy and any AFM arrangement that destroys this pseudogap incurs a
penalty. From the three considered AFM states only zigzag preserves (even
slightly enhances) the pseudogap (see SI). That gives this state an immediate
energetical advantage and leads to the energy balance described above. Two
notes are in place: first, all the above holds in LDA+U calculations with a
reasonable atomic $U$ (we have checked $U$ up to 3.8 eV). The role of $U$ in
these systems - as stated previously- is merely enhancing the existing SO-
driven gap. Second, if the DOS indeed plays a decisive role in magnetic
interactions, it is unlikely that they can be meaningfully mapped onto a
short-range exchange model, Heisenberg or otherwise.
Summarizing, our DFT calculations demonstrate that Na2IrO3 is close to an
itinerant regime. The electronic structure of this system is naturally
described on the basis of quasi-molecular orbitals centered each on its own
hexagon. This makes this, and similar materials rather unique. Proceeding from
this description one can understand the main properties of Na2IrO3, including
its unique zigzag magnetic ordering with small magnetic moment.
However, the main goal of our work is not a complete understanding of the
magnetic properties of Na2IrO3. We realize that this understanding is still
incomplete and that full explanation of the weak antiferromagnetism, as well
as of the magnetic response in this compound remains a challenge. Rather, we
lay out the framework in which this challenge has to be resolved. We
demonstrate that both the simplified (but correct) TB model proposed in
previous studies Jackeli2009 ; Shitade2009 , and full ab initio calculations
provide a framework that is best described by the quasi-molecular orbitals.
This is an as yet unexplored concept (as opposed to molecular orbitals or
atomic orbitals), and there are many open questions about how to treat
correlations, magnetic response etc. in this framework, however, it appears to
be the only way to reduce the full 12 atomic orbitals ($t_{2g}$ or their
relativitsic combinations) problem to a smaller subspace (3$\times 2=6$) QMOs.
I.I.M. acknowledges many stimulating discussions with Radu Coldea and his
group, and with Alexey Kolmogorov, and is particularly thankful to Radu Coldea
for introducing him to the world of quasihexagonal iridates. H.O.J., R.V. and
D.Kh. acknowledge support by the Deutsche Forschungsgemeinschaft through
grants SFB/TR 49 and FOR 1346 (H.O.J. and R.V.) and SFB 608 and FOR 1346
(D.Kh.). H.O.J. acknowledges support by the Helmholtz Association via
HA216/EMMI.
## References
* (1) I. Felner and I.M. Bradaric, Physica B 311, 195 (2002).
* (2) H. Kobayashi, M. Tabuchi, M. Shikano, H. Kageyama and R. Kanno, J. Mater. Chem. 13, 957 (2003).
* (3) Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).
* (4) G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
* (5) A. Shitade, H. Katsura, J. Kuneš, X.-L. Qi, S.-C. Zhang, and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).
* (6) Note that in this case the spin magnetic moment is 1/3 $\mu_{\rm B}$ and the orbital moment is 2/3 $\mu_{\rm B}.$
* (7) J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 105, 027204 (2010).
* (8) J. Reuther, R. Thomale, and S. Trebst, Phys. Rev. B 84, 100406(R) (2011).
* (9) While in recent XAS measurements (J.P. Clancy et al. arXiv:1205.6540) a non-zero branching ratio was reported in Na2IrO3, this observation only indicates that there is a substantial correlation between the spin and the orbital moments direction, $<{\bf L}\cdot{\bf S}>\neq 0$, but this fact per se does not tell us that relativistic atomic orbitals $j_{\rm eff}=1/2,3/2$ form a good basis for describing the electronic structure. As we show in the Letter, because of an accidental degeneracy of the three top molecular orbitals, the effect of SO is substantial, but are not necessarily described in terms of a particular $j_{\rm eff}$.
* (10) C. H. Kim, H.S. Kim, H. Jeong, H. Jin, and J. Yu, Phys. Rev. Lett. 108, 106401 (2012).
* (11) S. Bhattacharjee, S.-S. Lee and Y. B. Kim, New J. Phys. 14, 073015 (2012).
* (12) S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor, Phys. Rev. Lett. 108, 127204 (2012).
* (13) D. Pillay, M. D. Johannes, I. I. Mazin, O. K. Andersen, Phys. Rev. B 78, 012501 (2008).
* (14) R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C.N. Veenstra, J.A. Rosen, Yogesh Singh, P. Gegenwart, D. Stricker, J.N. Hancock, D. van der Marel, I.S. Elfimov, A. Damascelli, arXiv:1204.4471 (unpublished).
* (15) Compare to Mott-Hubbard transition in fullerides: O. Gunnarsson. Alkali-doped fullerides: narrow-band solids with unusual properties. World Scientific, 2004.
* (16) Spin dynamics can be in principle always mapped onto a localized spin model of a sufficient range, but this can be a dangerous exercise: compare the Fe pnictides, where such mapping led to an unphysically drastic temperature dependence of the Heisenberg exchange parameters, until it was realized that proper mapping requires a strong biquadratic term (A.L. Wysocki, K. D. Belashchenko, and V.P. Antropov, Nature Physics 7, 485 (2011)).
* (17) Incidentally, the ferromagnetic state in relativistic calculations acquires substantial anisotropy. Depending on the polarization direction, the energy gain over the nonmagnetic state varies by a factor of two. This is consistent with the experimentally observed substantial anisotropy of the uniform susceptibility in the paramagnetic state gegenwart .
## I Supplementary Information
We performed density functional theory (DFT) calculations considering various
full potential all electron codes, such as WIEN2k S (1), ELK S (2), and FPLO S
(3) using the generalized gradient approximation functional in its PBE form S
(4), and verified that the results agree reasonably well among different
codes. Such comparison is particularly important because the codes implement
the spin-orbit coupling in slightly different ways, employing usually
unimportant, but in principle unequal approximations. In the non-relativistic
calculations the core electrons were treated fully relativistically and the
valence electrons non-relativistically (scalar relativistic approximation). In
the fully relativistic calculations, i.e. with inclusion of spin-orbit
coupling, all electrons were treated fully relativistically. We considered the
C$2/m$ crystal structure as given in Ref. S, 5 and shown in Fig. S1.
Figure S1: Crystal structure of Na2IrO3 in the cubic setting. The hexagonal
direction is along the [111] direction in this setting. Ir, O and Na atoms are
shown as grey, magenta, and yellow spheres, respectively. The three
inequivalent Ir-Ir bonds are labeled according to their cubic directions.
We used projective Wannier functions as implemented in the FPLO basis S (6) to
determine a tight-binding (TB) representation for the Ir $5d$ bands. In Figure
S2 we show the DFT band structure together with the bands corresponding to the
Wannier representation and the TB bands derived from this representation.
Figure S2: Non-relativistic non-magnetic band structure of Na2IrO3 (red
symbols) shown together with the Wannier bands (yellow) and the tight-binding
bands (blue).
In Fig. S3 we present the projective Wannier functions for the $5d$ orbitals
of one Ir site. The Wannier functions exhibit the typical shape of the $5d$
functions at the Ir site. Besides, they show a clear asymmetry due to Na as
well as tails on the O sites.
Figure S3: Projective Wannier functions for five of the ten Ir $5d$ bands,
together with a structure showing the perspective.
In order to analyze the contribution to the non-relativistic band structure of
the various tight-binding hopping parameters and its relation to the quasi-
molecular orbital (QMO) picture, we present in Fig. S4 the band structure that
results if we restrict the tight-binding Hamiltonian to first neighbors (top
left), up to second nearest neighbors (top right), up to third nearest
neighbors (bottom left), and without restriction (bottom right). One can see
that already the second neighbors model provides a good semiquantitative
description of the band formation.
Figure S4: Band structure of Na2IrO3 (red symbols) shown together with the
tight-binding models that include only nearest neighbors (top left), up to
next nearest neighbors (top right), up to third nearest neighbors (bottom
left) and neighbors up to 16 Å (bottom right).
In the next Figure S5 we show the tight-binding band structures within the QMO
model. In these calculations we have included the on-site trigonal splitting
(the top left panel), adding the nearest neighbors $t_{1}^{\prime}$ hopping
(top right), then the second nearest neighbors $t_{2}^{\prime}$ hopping
(bottom left) and, finally, including also the third nearest neighbors hopping
between the like orbital, which also proceeds through Na and does not take an
electron out of the corresponding QMO (bottom right). The small dispersion
that arises for nearest neighbors is due to deviations from the perfect
octahedral environment of iridium. Upon inclusion of second nearest neighbors,
as mentioned in the main text, the upper doublet and singlet merge to form one
three-band manifold.
Figure S5: Band structure of Na2IrO3 (red symbols) shown together with the
tight-binding models that involve only parameters compatible with the quasi-
molecular orbitals. Only on-site parameters (top left), up to nearest
neighbors (top right), up to second nearest neighbors (bottom left) and up to
third nearest neighbors (bottom right). Figure S6: Density of states of
Na2IrO3 projected onto the six quasi-molecular orbitals given in Table [1] of
the main text for (a) a nonrelativistic and (b) a relativistic calculation.
In Figure S6, we show projections of the total density of states of Na2IrO3
onto the quasi-molecular orbitals specified in Table [1] of the main text.The
eigenvector matrix
$U=\begin{pmatrix}1&1&1&1&1&1\\\
1&\omega&\omega^{2}&-1&\omega^{4}&\omega^{5}\\\
1&\omega^{5}&\omega^{4}&-1&\omega^{2}&\omega\\\
1&\omega^{2}&\omega^{4}&1&\omega^{2}&\omega^{4}\\\
1&\omega^{4}&\omega^{2}&1&\omega^{4}&\omega^{2}\\\
1&-1&1&-1&1&-1\end{pmatrix}$
(with $\omega=\exp(i\pi/3)$) is a unitary transformation that rotates the
atomic Ir $t_{2g}$ orbitals into the QMO orbital space. $E_{1g}$ and $E_{2u}$
states are perfectly degenerate in the nonrelativistic case (Figure S6 (a)).
When spin-orbit coupling is turned on (Figure S6 (b)), interestingly, the
three upper bands are no more equivalent in this sense, with the central band
being mostly $A_{1g}$, and the other two mostly $E_{2u}$. Importantly, there
is hardly any mixing between the lower three bands and the upper three bands,
emphasizing the fact that the low-energy physics is nearly exclusively defined
by the upper three QMOs, and their mutual interaction, whether with or without
spin-orbit. At the same time, one can, alternatively, project the same bands
onto the relativistic orbitals, $j_{eff}=1/2$ and $j_{eff}=3/2$, and, as
observed beforeS (7), the upper two bands have more $j_{eff}=1/2$ character
than $j_{eff}=3/2$ character, but, for instance, at the $Gamma$ point, only
slightly so (more at some other points). Thus, even though the SO effects are
considerable, they are not strong enough to reduce the problem to a two
$j_{eff}=1/2$ model.
The magnetic patterns considered in our non-relativistic and fully
relativistic calculations are shown in Fig. S7.
The ferromagnetic state shows in the absence of SO an energy gain of nearly 80
meV per Ir with respect to the non-magnetic solution and about half this value
against competing antiferromagnetic states (zigzag and stripy phases); the
simple Néel state is much higher in energy. Inclusion of SO changes the
energetics considerably, as described in the main text, with the zigzag
antiferromagnetic ordering becoming competitive with the ferromagnetic one,
and lower in energy than the stripy phase. We deliberately do not discuss the
calculated energies in detail, because the energy differences involved are on
the order of one meV per atom, which is beyond the accuracy of the density
functional theory itself, and on the border of the technical accuracy of
existing band structure codes.
Figure S7: Three antiferromagnetic patterns considered in this paper: (a)
zigzag, (b) stripy, and (c) Néel.
In Fig. S8 we show the density of states for some magnetic orderings
considered in our fully relativistic calculations. Note that the zigzag
ordering preserves the nonmagnetic pseudogap at the Fermi level, while the
stripy ordering destroys it.
Figure S8: Density of states, spin-orbit included, for two competing magnetic
patterns compared with that for the nonmagnetic state.
Finally some considerations about the Hubbard $U$ are at place. In fact, there
are two ways of defining $U$ in this case. As usually, the actual value of $U$
depends on which orbitals it is being applied to. For instance, it is well
known that in Fe pnictides the appropriate value of $U$ acting on the Wannier
functions combining Fe $d$ and As $p$ states is more than twice smaller that
that acting on actual atomic $d$ orbitals since the screening effects change
depending on the basis of active states considered. In molecular solids, such
as fullerides, the atomic value of $U$ often appears completely irrelevant,
and the physically meaningful value of $U$ is the (much smaller) energy of
Coulomb repulsion of two electrons placed on two molecular orbitals. In the
case of Na2IrO3 one has a choice of using an atomic $U\sim$ 1.5-2 eV,
realizing that the results will be strongly affected by the fact that
electrons are localized not on individual ions, but on individual QMOs, or of
constructing $U$ in the QMO basis. The former way is readily available in such
formalisms as LDA+U but it may be a poor choice for the description of a
system based on quasi-molecular orbitals.
## References
* S (1) P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz 2001 WIEN2k, An Augmented PlaneWave+LocalOrbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universität Wien, Austria).
* S (2) http://elk.sourceforge.net/
* S (3) K. Koepernik and H. Eschrig, Phys. Rev. B 59. 1743 (1999); http://www.FPLO.de
* S (4) J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77 3865 (1996).
* S (5) S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell, P. G. Radaelli, Yogesh Singh, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, J. Taylor, Phys. Rev. Lett. 108, 127204 (2012).
* S (6) H. Eschrig and K. Koepernik, Phys. Rev. B 80, 104503 (2009).
* S (7) A. Shitade, H. Katsura, J. Kuneš, X.-L. Qi, S.-C. Zhang, and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).
|
arxiv-papers
| 2012-05-02T13:59:48 |
2024-09-04T02:49:30.461483
|
{
"license": "Public Domain",
"authors": "I. I. Mazin, H. O. Jeschke, K. Foyevtsova, R. Valenti, and D. I.\n Khomskii",
"submitter": "Harald O. Jeschke",
"url": "https://arxiv.org/abs/1205.0434"
}
|
1205.0489
|
# Phase Diagram of Ba1-xKxFe2As2
S. Avci Materials Science Division, Argonne National Laboratory, Argonne, IL
60439-4845, USA O. Chmaissem Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439-4845, USA Physics Department, Northern Illinois
University, DeKalb, IL 60115, USA D. Y. Chung S. Rosenkranz Materials
Science Division, Argonne National Laboratory, Argonne, IL 60439-4845, USA E.
A. Goremychkin Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439-4845, USA ISIS Neutron and Muon Source, Rutherford Appleton
Laboratory, Didcot, OX11 0QX, United Kingdom J.-P. Castellan I. S. Todorov
J. A. Schlueter H. Claus Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439-4845, USA A. Daoud-Aladine D. D. Khalyavin
ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11
0QX, United Kingdom M. G. Kanatzidis Materials Science Division, Argonne
National Laboratory, Argonne, IL 60439-4845, USA Department of Chemistry,
Northwestern University, Evanston, IL 60208-3113, USA R. Osborn Materials
Science Division, Argonne National Laboratory, Argonne, IL 60439-4845, USA
SAvci@anl.gov
###### Abstract
We report the results of a systematic investigation of the phase diagram of
the iron-based superconductor, Ba1-xKxFe2As2, from $x=0$ to $x=1.0$ using high
resolution neutron and x-ray diffraction and magnetization measurements. The
polycrystalline samples were prepared with an estimated compositional
variation of $\Delta x\lesssim 0.01$, allowing a more precise estimate of the
phase boundaries than reported so far. At room temperature, Ba1-xKxFe2As2
crystallizes in a tetragonal structure with the space group symmetry of
$I4/mmm$, but at low doping, the samples undergo a coincident first-order
structural and magnetic phase transition to an orthorhombic (O) structure with
space group $Fmmm$ and a striped antiferromagnet (AF) with space group
$F_{c}mm^{\prime}m^{\prime}$. The transition temperature falls from a maximum
of 139 K in the undoped compound to 0 K at $x=0.252$, with a critical exponent
as a function of doping of 0.25(2) and 0.12(1) for the structural and magnetic
order parameters, respectively. The onset of superconductivity occurs at a
critical concentration of $x=0.130(3)$ and the superconducting transition
temperature grows linearly with $x$ until it crosses the AF/O phase boundary.
Below this concentration, there is microscopic phase coexistence of the AF/O
and superconducting order parameters, although a slight suppression of the
AF/O order is evidence that the phases are competing. At higher doping,
superconductivity has a maximum $T_{c}$ of 38 K at $x=0.4$ falling to 3 K at
$x=1.0$. We discuss reasons for the suppression of the spin-density-wave order
and the electron-hole asymmetry in the phase diagram.
## I Introduction
There is now an extensive body of research into the origin of
superconductivity in the iron-based superconductors demonstrating the
importance of the subtle interplay of their electronic properties with
crystalline structure [1,2]. These compounds all contain a common structural
motif, namely a square planar net of iron atoms tetrahedrally coordinated with
pnictogens or chalcogens producing Fe${}_{2}X_{2}$ layers, in which $X$ = As
or Se/Te in the highest $T_{c}$ compounds. They are separated by buffer layers
comprising, for example, rare earth oxides in the so-called 1111 systems, such
as those based on LaFeAsO, or alkaline earths in the so-called 122 systems,
such as those based on BaFe2As2. Like the cuprate superconductors, the buffer
layers can act as charge reservoirs, controlling the carrier concentration and
inducing superconductivity in the iron planes by the introduction of
aliovalent dopants, e.g., LaFeAsO1-xFx [3,4] and Ba1-xKxFe2As2 [5,6], but it
is also possible to dope the pnictogen or chalcogen sites, e.g., BaFe2As2-xPx
[7,8], or the iron planes themselves by substituting other transition metal
ions, e.g., BaFe2-xCoxAs2 and BaFe2-xNixAs2 [9,10].
In this article, we report on a systematic investigation of the phase diagram
of Ba1-xKxFe2As2 from $x=0$ to $x=1.0$ using high resolution neutron and x-ray
diffraction combined with bulk characterization. In spite of the diversity of
doping strategies employed to modify the superconducting properties of the
iron-based superconductors, their phase diagrams show remarkable similarities.
There is typically an undoped “parent” compound that is antiferromagnetic
rather than superconducting[1]. These are fully compensated metals, whose
electronic properties are dominated by multiple iron-derived $d$-bands near
the Fermi level with approximately equal concentrations of hole and electron
carriers [11]. The Fermi surfaces consist of quasi-two-dimensional cylinders,
with two or three hole pockets at the Brillouin zone centers and two electron
pockets at the $M$-points on the zone boundaries, i.e., along the direction of
the nearest-neighbor iron-iron bonds [12]. All the Fermi surface pockets have
similar radii in the undoped compounds, making their electronic structure
particularly susceptible to magnetic instabilities resulting from a nesting of
the disconnected hole and electron Fermi surfaces [13]. Since the nesting
wavevector corresponds to the antiferromagnetic wavevector observed by neutron
diffraction [14-16], it is plausible that the magnetism can be explained by a
purely itinerant model of spin density waves, and ab initio density functional
theory does indeed predict the correct magnetic structure [17], although there
is an ongoing debate about the strength of electron correlations [18,19].
The magnetic structure breaks the tetragonal symmetry with an in-plane
wavevector of either (0,$\pi$) or ($\pi$,0) in the unfolded Brillouin zone
with one iron atom per unit cell. With a finite magnetoelastic coupling, this
would induce an orthorhombic structural transition, which is usually observed
to occur at the same temperature as magnetic order in the parent compounds
[20,21]. However, the addition of both hole and electron charge carriers
through chemical substitution suppresses both transitions. A variety of
scenarios are possible in Ginzburg-Landau treatments of the magnetoelastic
coupling [22,23]. The two phase transitions could be first- or second-order
and occur simultaneously or separately. In most of the iron-based compounds,
the two transition temperatures split with doping [24] and there is a report
of a split transition in Ba1-xKxFe2As2 as well [25]. However, this is
inconsistent with our previously reported neutron diffraction data, which
shows unambiguously that they are coincident and first-order for all $x$
before they are both suppressed at $x\lesssim 0.3$ [6]. Unusually, the two
order parameters, magnetic and structural, are proportional to each other,
apparently indicating a biquadratic coupling that is usually only observed at
a tetracritical point [26,27], not over an extended range of compositions.
Possible explanations for this observation will be discussed in the
conclusions.
Superconductivity emerges before the complete suppression of the
antiferromagnetic/orthorhombic (AF/O) phase and coexists at low-doping levels.
The nature of the competition between AF/O order and superconductivity is a
central question in understanding iron-based superconductivity [28,29]. There
were earlier reports based on local probes that, in Ba1-xKxFe2As2, the
coexistence region is characterized by a mesoscopic phase separation into AF/O
and superconducting droplets [30,31], but our previously reported diffraction
data are only consistent with a microscopic phase coexistence [6], a
conclusion since supported by muon spin rotation ($\mu$SR) experiments [32],
suggesting that the earlier reports may be due to compositional fluctuations
within the samples.
Figure 1: (Color online) Structure of BaFe2As2, which crystallizes in a
tetragonal ThCr2Si2-type structure with the space group symmetry of $I4/mmm$.
Potassium substitutes onto the barium sites.
One of the main reasons for studying Ba1-xKxFe2As2 is that superconductivity
extends up to much higher hole-doping levels, with 0.5 holes/Fe atom, than in
the electron-doped superconductors produced by transition metal substitutions.
In the case of BaFe2-xCoxAs2, superconductivity vanishes at only 0.12
electrons/Fe atom [10]. Furthermore, the maximum $T_{c}$ with hole-doping is
38 K, significantly higher than the maximum $T_{c}$ of $\sim 25$ K obtained
with electron doping. This electron-hole asymmetry in the phase diagram has
been attributed to an enhanced Fermi surface nesting in the hole-doped
compounds, consistent with angle-resolved photoemission spectroscopy (ARPES)
data and band structure calculations [33]. This explanation is also supported
by the evolution of resonant spin excitations, which become incommensurate due
to the mismatch in hole and electron Fermi surface volumes when $T_{c}$ starts
to fall [34]. On the other hand, there is also a strong correlation between
$T_{c}$ and internal structural parameters such as the Fe-As-Fe bond angles
[35,36]. These are known to have an influence on the band structure and the
degree of moment localization, but their role in optimizing superconductivity
and the implications for the gap symmetry is a matter of debate [37,38].
There have been two previous reports of the doping dependence of this series
in addition to our own brief report, which are all in qualitative agreement
[6,36,39]. At room temperature, all members of the Ba1-xKxFe2As2 series
crystallize in a tetragonal structure with the space group symmetry of
$I4/mmm$ (Fig. 1), while low-doped samples also exhibit a low temperature
phase transition to an orthorhombic structure with space group $Fmmm$ [5].
Superconducting samples at higher doping have a maximum $T_{c}$ of 38 K and
remain tetragonal at all measured temperatures down to 1.7 K. However, there
are significant discrepancies in the published reports concerning the critical
dopant concentrations defining the onset of superconductivity and the
suppression of the AF/O phase, with the latter varying from $x\sim 0.3$ [36]
to $\sim 0.4$ [39]. As already mentioned, there have also been disagreements
about the nature of the competition between the three ordered phases at low
doping. We believe that these discrepancies are due to uncertainties in the
actual composition of the synthesized samples, since it is well-known that
potassium is particularly volatile. Controlling the inhomogeneity to within
acceptable limits in order to improve the accuracy of the various phase
boundaries has been a key goal of this work and we estimate that we have been
able to make samples in which $\Delta x<0.01$. We have performed neutron and
x-ray diffraction studies of the magnetic and structural order using high-
resolution powder diffractometers so that the systematic variation of the
lattice parameters and internal structural parameters can be used to estimate
the degree of uncertainty in the average composition and its variation within
the samples.
$x$ | $x$ | $x$ | $x$ | $T_{c}$ (K) | $T_{N}$ (K) | $T_{N}$ (K) | $T_{s}$ (K) | Magn. Moment ($\mu_{B}$) | $\delta\times 10^{3}$
---|---|---|---|---|---|---|---|---|---
| (fitted) | (nominal) | (ICP) | | (magnetization) | (neutron) | | |
0 | | 0 | 0 | | 139(1) | 139.0(1) | 138.17(6) | 0.756(36) | 3.92(4)
0.1 | 0.097 | 0.1 | 0.094(2) | | 136(1) | 136.5(3) | 136.02(8) | 0.741(21) | 3.68(3)
0.125 | 0.126 | 0.125 | 0.114(2) | | 130(2) | | 128.29(6) | 0.697(29) | 3.49(4)
0.15 | 0.15 | 0.15 | 0.139(1) | 4 | 122(2) | 122.1(3) | 122.09(7) | 0.702(21) | 3.35(4)
0.175 | 0.172 | 0.175 | 0.159(2) | 10 | 113(2) | 113.9(1) | 112.1(6) | 0.683(24) | 3.14(5)
0.2 | 0.202 | 0.2 | 0.184(2) | 17 | 100(2) | 102.0(1) | 102.00(2) | 0.652(46) | 2.76(7)
0.21 | 0.209 | 0.24 | | 18 | | 96.0(1) | 96.0(3) | 0.610(32) | 2.59(3)
0.22 | 0.225 | 0.22 | | 23.5 | | 93.97(1) | 93.93(1) | 0.550(22) | 2.20(9)
0.24 | 0.237 | 0.26 | | 26 | | 79.9(1) | 80.0(2) | 0.572(29) | 2.00(3)
0.25 | 0.249 | 0.24 | | 28.5 | | 74.9(1) | 74.8(8) | 0.456(22) | 1.43(8)
0.28 | | 0.28 | | 34 | | | | |
0.3 | | 0.3 | 0.312(4) | 36 | | | | |
0.4 | | 0.4 | | 38 | | | | |
0.5 | | 0.5 | 0.476(8) | 34 | | | | |
0.7 | | 0.7 | 0.675(3) | 20 | | | | |
0.9 | | 0.9 | 0.892(1) | 7 | | | | |
1 | | 1 | 1.00(2) | 3 | | | | |
Table 1: Structural and magnetic phase diagram of Ba1-xKxFe2As2. The nominal
value of x represents the starting composition given by the Ba/Fe ratio. The
fitted value is determined by smoothing the variation in the a-axis lattice
parameter from $0.1\leq x\leq 0.25$ using a power law function. The first
column is the value of $x$ used in the text and figures. The superconducting
transition temperature is determined from magnetization measurements. $T_{N}$
and $T_{s}$ are determined by magnetization and neutron diffraction data as
described in the text. The magnetic moments and orthorhombic order parameters,
$\delta=(a-b)/(a+b)$, are determined from the low-temperature Rietveld
refinements.
In this article, we present the results of Rietveld refinements for the entire
series and use this analysis along with bulk measurements to produce a
comprehensive magnetic and structural phase diagram that provides insight into
the nature of the phase competition that underlies iron-based
superconductivity. Our results show that there is a steeper decrease in
$T_{c}$ and hence a narrower region of phase coexistence of the AF/O order
with superconductivity than previous reports. After a description of our
experimental results, we combine our findings with results reported in the
literature on the electron-doped BaFe2-xCoxAs2 series in order to elucidate
the origin of the electron-hole asymmetry in the phase diagram.
## II Experimental Details
The synthesis of homogeneous single phase Ba1-xKxFe2As2 samples is known to be
particularly delicate due to unfavorable kinetics, high vapor pressures, and a
significant difference in the chemical reactivity of K and Ba with FeAs that
may result in stabilizing other binary by-products. For this work, the
synthesis and properties of our samples were optimized by the systematic
examination of all reasonable combinations of reaction parameters, e.g.,
purity of the starting materials, reaction containers, temperature, and
duration of heating, etc. Our final samples were produced according to the
following procedure: Handling of all materials was performed in a nitrogen-
filled glove box. Raw materials (BaAs/KAs/Fe2As) were prepared by heating
elemental mixtures at 400∘C, 600∘C, and 850∘C, respectively. The
stoichiometric mixture of these starting precursors for a desired composition
was thoroughly ground to ensure uniform and homogeneous and subsequently
annealed at 1050∘C in a Nb tube sealed in a quartz tube. The closed metal
tubes are needed to eliminate any chemical loss that may otherwise result from
the evaporation of K and As. A large number of high quality samples were
synthesized covering the full phase diagram $0\leq x\leq 1$ with special
emphasis given to the $0.1\leq x\leq 0.25$ range in which the potassium
content was incremented in very small amounts with $\Delta x=0.025$. The
deliberate synthesis of samples with finely tuned K content was necessary in
order to carefully investigate the rapid suppression of magnetism with
increasing K and to elucidate the nature of phase coexistence with
superconductivity within the same sample. Samples with coarse K increments
would otherwise lead to inconclusive results.
Initial characterization of the samples was performed by x-ray diffraction,
magnetization measurements, and Inductively Coupled Plasma (ICP) elemental
analysis. For select samples, neutron powder diffraction experiments were
performed on the HRPD ($x=0$, 0.1, 0.125, 0.15, 0.175, 0.2, 0.21, 0.24, 0.3,
0.5 and 1) and Wish diffractometers ($x=0.22$ and 0.25) at the ISIS Pulsed
Neutron and Muon Source (Rutherford Appleton Laboratory) and on beamline
11-BM, ($x=0.28$) at the Advanced Photon Source (Argonne National Laboratory).
The resolution, $\Delta d/d$, at 2 Åis 0.001 for HRPD and 0.002 for Wish. For
diffraction experiments, the samples were sealed under vacuum for shipment and
re-opened just before the measurements in a helium environment to prevent air
exposure. The nuclear and magnetic structures together with interatomic bond-
lengths and bond-angles were determined by the Rietveld refinement technique
using the comprehensive General Structure Analysis System (GSAS) software
suite [40] and the associated graphical user interface (EXPGUI) [41]. Minute
traces of no more than 0.5-3% by weight of FeAs and Fe2As impurity phases were
observed in some of the samples, too small to affect the analysis.
Our neutron diffraction results show that the synthesis methods has reduced
the compositional uncertainty significantly, which we estimate to be $\Delta
x\lesssim 0.01$. In previously reported phase diagrams, it is not totally
clear whether the potassium contents were nominal or actual measured values.
Because of its volatility, it is always necessary to add excess potassium, so
the eventual stoichiometry is largely governed by the Ba/Fe ratio. Thus, any
perceived discrepancy with other work, such as Ref. [39] is probably due to
differences in the handling and control of the volatile potassium and arsenic
constituents. In this article, we have used a number of methods to
characterize the sample compositions, including direct measurements of the
stoichiometry from ICP analysis. To produce the final compositions used in our
phase diagrams, we started with the nominal $x$ values determined from the
starting Ba/Fe ratio and then smoothed the variation in the a-axis lattice
parameter from $0.1\leq x\leq 0.25$ using a power law function (see Table 1).
In most cases, the agreement with the nominal value was better than 0.005,
with just two samples requiring a significant shift of 0.02 and 0.03,
respectively. In all cases, the adjustments also improved the consistency of
other measurements, such as the variations in transition temperatures and
order parameters. The first column of Table 1 shows the value of $x$ that we
have used in the text and figure labels, derived by rounding the fitted values
to the nearest 0.005, although the fitted values were used in the plots and in
numerical analyses of the doping dependence, e.g., the fit of $T_{c}$ vs $x$.
For $x\geq 0.28$, where the precise composition is not so critical, we have
used the nominal values.
Figure 2: (Color online) Variation of lattice constants, $a$, $b$ and $c$,
with temperature for $x=0$, 0.1, 0.21 and 0.3. The lattice constants, $a$ and
$b$, in the orthorhombic phase are divided by $\sqrt{2}$.
We are also able to monitor fluctuations in composition within a single sample
because HRPD has sufficiently high resolution that it is sensitive to
distributions of the lattice parameter and internal strains caused by
compositional gradients and even particle size broadening [42]. The
diffraction peaks from the (220) reflection shown in Fig. 1 of Ref. [6] show
that there is no change in the linewidths and lineshapes of the diffraction
peaks from the undoped compound up to $x=0.24$, consistent with a high-degree
of compositional homogeneity ($\Delta x\lesssim 0.01$). The regular spacing
between the peak positions is in agreement with the fixed steps in $x$.
## III Results
### III.1 Structural Phase Diagram
Figure 3: (Color online) Temperature dependence of the (110) peak in the
vicinity of structural transition temperature $T_{s}$. The bold curve shows
the peak at the estimated $T_{s}$. For $0\leq x\leq 0.24$, the other curves
show the peak at intervals of 2 K around $T_{s}$. For $x=0.3$, the peak is
shown in 20 K intervals between 1.7 K and 120 K.
In the undoped BaFe2As2 compound, there is a structural phase transition at
139 K from the tetragonal ThCr2Si2-type structure with the space group
symmetry $I4/mmm$ to an orthorhombic $\beta$-SrRh2As2-type structure with
space group $Fmmm$ [5]. The structure of both the tetragonal and orthorhombic
phases of Ba1-xKxFe2As2 can be described as a stack of edge-sharing Fe2As2
layers separated by layers of (Ba,K) ions (Fig. 1). The (Ba,K) ions occupy
crystallographic positions that are tetrahedrally coordinated with four
arsenic anions.
With potassium doping, the tetragonal to orthorhombic structural transition
temperature, $T_{s}$, decreases until it is fully suppressed for $x>0.25$. In
Fig. 2, we show the lattice parameters as a function of temperature for $x=0$,
0.1, 0.21 and 0.3. Below $x=0.3$, a significant orthorhombic splitting of the
basal plane $a$ and $b$ lattice parameters is observed. The evolution of the
orthorhombic order parameter defined by $\delta=(a-b)/(a+b)$ is discussed
later. Although the transitions appear to be continuous, we observed small but
sharp volume anomalies at all the structural phase transitions (Fig. 3 in Ref.
[6]) showing that they are weakly first-order in character over the entire
phase diagram.
Figure 4: (Color online) Variation of lattice constants and volume in
Ba1-xKxFe2As2 with $x$ at 1.7 K. Solid lines are guide to the eye. The lattice
constants ($a$ and $b$) and the volume in the orthorhombic phase are divided
by $\sqrt{2}$ and 2, respectively, from those for the $Fmmm$ space group.
The structural transition temperatures in Table 1 were determined by fitting a
power law, $\delta\propto(T_{s}-T)^{\beta}/T_{s}$, close to the transition
temperature, yielding exponents mostly in the range $\beta\sim 0.13$ to 0.2.
Apart from $x=0$, where $\beta=0.129(3)$, in reasonable agreement with Ref.
[27], the exponents will be modified by compositional fluctuations and so are
not reliable estimates of the critical behavior. The exponents at $x=0.21$ and
0.24 were anomalously high ($\beta=0.25$ and 0.30, respectively), which could
reflect a slightly greater degree of compositional variation within those
samples. As a check on these values of $T_{s}$, we used the peak profiles
close to the transition temperature. Fig. 3 shows the temperature dependence
of the tetragonal (110) Bragg peak for $x=0$, 0.1, 0.15, 0.2, 0.24 and 0.3 in
the vicinity of the structural transition. This reflection splits into two
(022) and (202) orthorhombic peaks below $T_{s}$. Close to the transition, the
two peaks cannot be resolved but we can determine $T_{s}$ from the temperature
dependence of the full width at half maximum (FWHM) as they merge. In the
high-temperature phase, the FWHM of $\sim 0.0037(3)$Åis independent of the
composition. These two methods of determining $T_{s}$ agreed within the
errors.
Figure 5: (Color online) Variation of Fe-Fe and Fe-As bond lengths with $x$ at 1.7 K and with temperature for different K substitutions. Blue triangles represent the Fe-As bonds. Black square and red circle symbols represent the Fe-Fe bond lengths merging at $T_{s}$. Solid lines are guides to the eye. $x$ | $a$(Å) | $b$(Å) | $c$(Å) | Vol(Å3) | $z$As | Fe-As(Å) | Fe-Fe(Å) | Fe-Fe(Å) | As-Fe-As(∘) | As-Fe-As(∘) | As-Fe-As(∘)
---|---|---|---|---|---|---|---|---|---|---|---
0 | 5.6157(2) | 5.5718(2) | 12.9424(4) | 404.970(45) | 0.35375(3) | 2.3905(1) | 2.8078(1) | 2.7859(1) | 108.071(3) | 108.718(3) | 111.648(2)
0.1 | 5.5997(1) | 5.5587(1) | 13.0031(4) | 404.755(36) | 0.35405(3) | 2.3919(1) | 2.7998(1) | 2.7793(1) | 108.356(7) | 108.961(7) | 111.110(15)
0.125 | 5.5940(2) | 5.5551(2) | 13.0243(5) | 404.745(52) | 0.35405(3) | 2.3918(2) | 2.7970(1) | 2.7775(1) | 108.438(8) | 109.011(8) | 110.977(16)
0.15 | 5.5890(2) | 5.5517(2) | 13.0404(5) | 404.634(46) | 0.35399(3) | 2.3905(1) | 2.7945(1) | 2.7758(1) | 108.464(3) | 109.013(3) | 110.947(3)
0.175 | 5.5842(2) | 5.5492(2) | 13.0563(6) | 404.598(55) | 0.35386(3) | 2.3901(2) | 2.7921(1) | 2.7746(1) | 108.522(9) | 109.037(9) | 110.864(17)
0.2 | 5.5767(4) | 5.5460(4) | 13.0736(9) | 404.353(82) | 0.35372(5) | 2.3885(1) | 2.7883(2) | 2.7730(2) | 108.578(5) | 109.030(5) | 110.815(4)
0.21 | 5.5750(1) | 5.5462(2) | 13.0749(4) | 404.288(36) | 0.35404(2) | 2.39073(4) | 2.7875(1) | 2.7731(1) | 108.678(2) | 109.102(2) | 110.640(2)
0.22 | 5.5706(5) | 5.5461(4) | 13.0803(12) | 404.126(112) | 0.3543(7) | 2.3925(6) | 2.7855(3) | 2.7733(3) | 108.798(21) | 109.155(21) | 110.47(4)
0.24 | 5.5672(2) | 5.5449(2) | 13.0888(4) | 404.051(39) | 0.35394(3) | 2.3894(2) | 2.7836(1) | 2.7724(1) | 108.750(7) | 109.078(7) | 110.592(14)
0.25 | 5.5636(4) | 5.5476(4) | 13.1027(10) | 404.157(90) | 0.35398(6) | 2.3900(5) | 2.7810(2) | 2.7732(2) | 108.846(17) | 109.073(16) | 110.50(3)
0.3 | 3.9165(2) | 3.9165(2) | 13.1614(5) | 201.877(23) | 0.35383(4) | 2.3878(3) | 2.7694(2) | 2.7694(2) | 109.090(28) | | 110.237(14)
0.5 | 3.8893(2) | | 13.3242(6) | 201.554(20) | 0.35376(4) | 2.3859(3) | 2.7501(1) | | 109.615(11) | | 109.185(21)
1 | 3.8251(2) | | 13.7846(5) | 201.691(40) | 0.35314(4) | 2.3833(4) | 2.7047(1) | | 110.855(12) | | 106.708(23)
Table 2: Results of Rietveld refinements for Ba1-xKxFe2As2 from neutron powder
diffraction data collected on HRPD at 1.7 K. For $x<0.3$, the space group is
$Fmmm$, in which $a\neq b$, there are two inequivalent Fe-Fe bond distances
and three inequivalent As-Fe-As bond angles. For $x\geq 0.3$, the space group
is $I4/mmm$, in which $a=b$, there is one Fe-Fe bond distance and two
inequivalent As-Fe-As bond angles
Fig. 4 shows the evolution of the lattice parameters and unit cell volume at
1.7 K as a function of $x$. Barium substitution by the smaller potassium
cations reduces the in-plane $a$\- and $b$-lattice parameters while
significantly lengthening the out-of-plane $c$-axis. However, the c-axis
enhancement is not large enough to fully compensate for the shrinking basal
plane axes and the unit cell volume gradually decreases in magnitude upon
increasing the K content until $x\sim 0.5$. The reason for the non-monotonic
behavior of the unit cell volume at high dopant levels is not understood and
will require further investigation.
The behavior of the Fe-Fe and Fe-As interatomic distances (at 1.7 K) are
presented in Fig. 5. The Fe-Fe distances mimic the in-plane lattice parameters
both as a function of K content and of temperature. The similarity in behavior
is explained by the fact that Fe atoms occupy special rigid positions along
the long edges of the lattice. Six As-Fe-As bond-angles can be identified in
each FeAs4 tetrahedron. In the tetragonal $I4/mmm$ structure, these angles can
be grouped into two independent angles: two equivalent angles, $\alpha_{1}$,
and four equivalent smaller ones, $\alpha_{2}$. In the orthorhombic $Fmmm$
structure below $T_{N}$, the angle $\alpha_{1}$ remains unaffected but the
angle $\alpha_{2}$ splits into two pairs of equivalent angles
$\alpha^{\prime}_{2}$ and $\alpha^{\prime\prime}_{2}$ with an angular
separation of $\sim 0.5-0.6^{\circ}$. A sketch showing the different angles is
displayed in Fig. 6. As shown in the same panel, the angle $\alpha_{1}$
increases linearly and continuously throughout the whole phase diagram.
However, starting from BaFe2As2, the angles $\alpha^{\prime}_{2}$ and
$\alpha^{\prime\prime}_{2}$ increase with increasing K until reaching a
critical composition below $x\sim 0.3$ beyond which the structural transitions
are suppressed and the two angles merge into $\alpha_{2}$, which continues to
increase with higher K contents. The refined values of the lattice parameters,
bond lengths and bond angles at 1.7 K are shown in Table 2.
Figure 6: (Color online) Variation of As-Fe-As bond angles with $x$ at 1.7 K
and with temperature for different K substitutions. The top panel shows
$\alpha_{1}$, $\alpha^{\prime}_{2}$ and $\alpha^{\prime\prime}_{2}$ in
orthorhombic setting. Blue circles represent $\alpha_{1}$, red triangles and
black squares represent $\alpha^{\prime}_{2}$ and $\alpha^{\prime\prime}_{2}$
merging into one $\alpha_{2}$ at $T_{s}$. Solid lines are guides to the eye.
### III.2 Magnetic Phase Diagram
Neutron powder diffraction data reveal the presence of weak magnetic Bragg
reflections that appear below the structural phase transition for all of the
orthorhombic samples. The magnetic peaks shown in Fig. 7, located at 2.45 Åand
3.43 Å, were indexed as 121 and 103 in agreement with the widely reported
antiferromagnetic spin density wave (SDW) ground state [15,16,39,43]. As with
the structural transitions, the antiferromagnetic transition (Néel)
temperatures as a function of doping were determined by power-law fits to
temperature variation of the magnetic moment (see Table 1). These coincide
with the orthorhombic transition for all values of $x$. It is important to
realize that the structural and magnetic orders are identified in the same
measurements, the first from the splitting of the nuclear Bragg peaks and the
second by the intensity of the magnetic Bragg peaks. This means that the
conclusion that the two transitions are coincident does not depend on the
accuracy of the thermometry. As shown in Ref. [6], the two order parameters
determined in this way are directly proportional to each other at all
temperatures over the entire phase diagram, an unusual result that we will
discuss in more detail in the discussion.
Figure 7: (Color online) Neutron diffraction at 1.7 K with the magnetic Bragg
peaks at a $d$-spacing of 2.45 Åand 3.43 Åindicated by the arrows. They are
absent above $T_{N}$, for $x=0$, 0.1, 0.21, 0.22 and 0.25. At $x=0.3$, no
magnetic peaks are observed.
As a further check on our data, the values of $T_{N}$ determined by neutron
diffraction are in excellent agreement with those determined by peaks in the
temperature derivative of the magnetization (Fig. 8 and Table 1). These
magnetization peaks decrease in magnitude because of the progressive
attenuation of the magnetic signal due to increasing K content until they are
no longer detected for $x\geq 0.21$.
A full analysis of the magnetic structure was performed using the allowed
subgroup magnetic symmetries of $Fmmm$. All possible models were tested but
only the magnetic space group $F_{c}mm^{\prime}m^{\prime}$ resulted in a
proper fit to the data. Removal of the time reversal symmetry from two of the
mirror planes resulted in an antiferromagnetic arrangement of the magnetic
moments with a magnetic wave vector Q = (1,0,1); that is, the Fe magnetic
moments are antiferromagnetically coupled in the $x$ and $z$ directions and
ferromagnetically coupled along the $y$ axis. This model is consistent with
similar results previously reported for the parent BaFe2As2 material
[15,16,44].
Figure 8: (Color online) SQUID magnetization measurements for $x=0$, 0.125,
0.15, 0.175, 0.2 and 0.24 in 2 kG applied magnetic field. Insets are the first
derivatives of the magnetization curves, d$M$/d$T$ (10-5), used to determine
the Néel temperatures given by the arrows. For $x>0.2$, the magnetization
anomaly at $T_{N}$ is too weak to be detected. Figure 9: (Color online)
Dependence of the neutron diffraction intensity for Ba1-xKxFe2As2 at 1.7 K
with $x$. The magnetic Bragg peaks are shown by the arrows. The solid lines
represent the calculated intensity of the Rietveld refinement.
Rietveld refinements of both the atomic and magnetic structures were performed
simultaneously as a function of temperature and doping, allowing the magnetic
moment to be defined in absolute units by normalization of the intensity of
the magnetic Bragg peaks to the structural Bragg peaks. The neutron data
displayed in Fig. 9 were collected at 1.7 K and normalized to the sample mass
and exposure time (measured in beam pulses). The figure qualitatively shows
the intensities of the magnetic (121) and (103) Bragg reflections to remain
roughly unchanged for the $x=0$ and 0.1 samples followed by a monotonic
decrease upon increasing the K content until they nearly vanish at $x=0.24$.
The refinements show that the magnetic moment drops from $\mu=0.75\mu_{B}$ for
the parent BaFe2As2 material to $0.46\mu_{B}$ for $x\sim 0.25$ (see Table 2).
No magnetic peaks are observed beyond this value.
Figure 10: (Color online) Dependence of the square of the magnetic moment,
$\mu^{2}$ (red stars), and the orthorhombic order parameter,
$\delta=(a-b)/(a+b)$ (blue circles), at 1.7 K on the potassium concentration,
$x$. The inset shows a comparison of $T_{N}$ and $\mu^{2}$ vs $x$, showing
that $T_{N}\propto\mu^{2}$. Solid lines in the main panel and the inset are a
fit to $(1-x/x_{c})^{2\beta}$ with $\beta=0.125$.
The doping dependence of the two order parameters making up the AF/O phase is
shown in Fig. 10. We compare $\delta$ and $\mu^{2}$ vs $x$, showing that they
are directly proportional over the entire range of AF/O order. We have not
been able to measure any samples between $0.25\leq x\leq 0.28$, but a power
law fit to $\delta$ and $\mu^{2}$ close to $x_{c}$, i.e., to
$(1-x/x_{c})^{2\beta}$, gives a critical concentration of 0.252 with an
exponent of $\beta=0.125(1)$ for the structural and magnetic order parameters.
The inset to Fig. 10 shows that $T_{N}$ is also proportional to $\mu^{2}$. In
a mean-field model, $T_{N}$ scales as $J\mu^{2}$, where $J$ is the effective
interionic exchange interaction, so this result would seem to indicate that
$J$ is approximately independent of $x$ over this range.
### III.3 Superconductivity and Phase Coexistence
Figure 11: (Color online) SQUID magnetization measurements (zero field-cooled)
in 0.1 G magnetic field for (a) $x=0.15$ (solid squares), 0.175 (open
squares), 0.21 (solid circles), 0.22 (open circles), 0.25 (solid triangles),
$x=0.3$ (open triangles), and (b) 0.5 (solid triangles), 0.7 (solid circles),
0.9 (solid squares) showing well-defined superconducting transitions.
Magnetization values are normalized to mass of the samples. (c)
Superconducting transition temperatures (onset $T_{c}$) of the underdoped
compounds. Solid line represents the linear regression showing that the
critical concentration for superconductivity is 0.130(3).
Bulk superconductivity is observed for all samples with $x\geq 0.15$. Zero-
field magnetization data shows that superconducting transition temperatures
peaks at $\sim 38$ K for $x=0.4$ before it slowly decreases to 3 K for the end
member KFe2As2 (Fig. 11). By comparing the samples’ magnetic moment with that
of a Sn-powder sample of similar volume, we estimate that these samples are
bulk superconductors with a volume fraction of at least 80%. The uncertainty
is due to variations in the demagnetization factor between samples. All the
superconducting transitions are well-defined and sharp, even close to the
critical concentration where $T_{c}$ is varying rapidly with $x$. Even low
levels of compositional inhomogeneity associated with the uneven distribution
of Ba/K ions would be revealed in magnetic susceptibility measurements by a
broad or stepped-like transition from the normal state to the superconducting
state, so this is further evidence of the sample quality. The increase in
$T_{c}$ at low doping is approximately linear with dopant concentration up to
$x\sim 0.25$, so we have estimated the critical concentration for the onset of
superconductivity using linear regression to be $x_{c}=0.130(3)$.
In Ref. [6], we discussed the behavior of the order parameters below $T_{c}$.
We observed a small reduction in both the magnetic and structural order
parameters of approximately 5% at $x=0.21$ and 0.24, without seeing evidence
of additional phases. In a scenario in which the sample divides into separate
mesoscopic regions of AF/O phase and superconducting phase, this would imply
that 95% of the sample remains in the AF/O phase and only 5% becomes
superconducting. This is inconsistent with the magnetization measurements
showing bulk superonductivity. While it is not possible to rule out the
presence of other phases, the results indicate that there is microscopic phase
coexistence of magnetism and superconductivity, with the reduction in the AF/O
order parameters being due to a competition with the superconducting order
parameter. This competition has been discussed extensively by Fernandes et al
[45], who show that there should be an additional phase boundary, with a
positive slope vs $x$, between the coexistence region and the region of purely
superconducting phase. We have not yet identified any anomalies corresponding
to the phase line below $T_{c}$, so we assume that it rises steeply with $x$.
## IV Discussion
The overall phase diagram of Ba1-xKxFe2As2 is shown in Fig. 12. We first
discuss the nature of the spin-density wave order and orthorhombic order.
Unlike the electron-doped compounds, where the two transitions split within
increased doping, the two transitions are coincident and first-order in
Ba1-xKxFe2As2. We reported the first-order character of the transition by the
observation of volume anomalies at $T_{s}$ [6]. Similar volume anomalies were
also observed by Tegel et al [46] in unsubstituted SrFe2As2 and EuFe2As2 but,
because of the small magnitude of these anomalies, the authors suggested that
the structural phase transition may be second-order. However, other authors
reported first-order transitions in polycrystalline SrFe2As2 [20] and single
crystals of CaFe2As2 [21] and BaFe2As2 [44]. In the latter reference, a first-
order-like hysteresis was obtained for the intensities of the (101) Bragg peak
when measured on cooling and warming. However, no such hysteresis was observed
by Wilson et al [27], when examining their BaFe2As2 single crystal. The
systematic observation of volume anomalies across the phase diagram is
unambiguous evidence that, at least in this system, all the transitions are
first-order, although weakly first-order with extremely small hysteresis.
Figure 12: (Color online) Phase diagram of Ba1-xKxFe2As2 with the
superconducting critical temperatures, $T_{c}$ (circles), the Néel
temperatures, $T_{N}$ (stars), and the structural transition temperatures,
$T_{s}$ (squares).
Phenomenological theory of magnetoelastic coupling predicts the possibility of
simultaneous first-order transitions that are driven by a linear-quadratic
term in the Ginzburg-Landau expansion [22,23] (see also Ref. [47]). This is
the lowest-order term allowed by symmetry. If the magnetic transition were to
occur at higher temperature than the structural transition (in the absence of
any competition), the magnetic order would drive the structural order in a
simultaneous first-order transition. The converse would produce two split
transitions as seen in most of the iron-based compounds [24]. There is a
report of a split transition in Ba1-xKxFe2As2 based on nuclear magnetic
resonance results [25], but this is a local probe, which cannot necessarily
identify compositional fluctuations. As we have already discussed, the neutron
measurements, which represent true averages over the bulk, are quite
unambiguous that the two transitions are simultaneous, although there could be
some rounding of the transitions at higher doping from small compositional
fluctuations.
The two order parameters are directly proportional to each other as a function
of temperature, which seems to indicate an unusual biquadratic coupling in the
Ginzburg-Landau expansion, rather than a linear-quadratic coupling. This is
usually only observed at a tetracritical point [26,27] where two phase
boundaries intersect, whereas our observations extend over a range of
compositions. There are a number of possible reasons for this. The most
intriguing and exotic idea is that the AF/O order parameters are both
secondary to another order parameter and directly driven by it. This would be
the case in, for example, valley density wave theory, in which a mother
density-wave drives both the magnetic and charge-density-wave orders [48]. A
second explanation is provided by the recent theoretical work of A.
Nevidomskyy [49], which uses a microscopic Kugel-Khomskii model to produce a
biquadratic spin-orbital term in the free energy. A subtle, but ultimately
more conventional explanation, is that the coupling is linear-quadratic after
all, but the proximity to a first-order transition produces a temperature
dependence that is approximately equivalent to biquadratic coupling to first
order [29]. This is in the context of a theory in which Ising-nematic order,
produced by an itinerant model of Fermi surface nesting, drives the structural
transition. Support for this explanation is provided by Fig. 10, where the
doping dependence of the magnetic and structural order parameters at low
temperature indicates a linear-quadratic coupling. Whatever the eventual
explanation, it is clear that this result is key to understanding the nature
of the normal state and the role of nematic order in the eventual
superconductivity.
The strong coupling between AF/O order parameters persists into the regime of
phase coexistence with superconductivity. We have already argued that
Ba1-xKxFe2As2 is characterized by microscopic phase coexistence because
mesoscopic phase separation would result in a significant decrease in the
volume fraction of the AF/O phase below $T_{c}$. The consensus in favor of
microscopic phase coexistence has existed for some time in the electron-doped
superconductors, such as BaFe2-xCoxAs2 [24], where the phase boundary below
$T_{c}$ to a non-magnetic, purely superconducting region has also been
identified. Theoretically, this behavior is consistent with unconventional
$s_{\pm}$ pairing of the Cooper pairs suggesting that itinerant long range
magnetism and superconductivity may coexist and compete for the same electrons
[45]. However, the idea of microscopic phase coexistence was more
controversial in Ba1-xKxFe2As2 because of local probe measurements that seemed
to indicate a phase separation into mesoscopic regions of magnetism and
superconductivity [30,31]. Since the most recent $\mu$SR data are also
consistent with microscopic phase coexistence [32], it appears that the
earlier reports may have been due to compositional fluctuations close to the
phase boundaries and that microscopic phase coexistence has now been
confirmed.
Figure 13: (Color online) (Top panel) Magnetic and structural phase diagram of
electron-doped Ba(Fe1-xCox)2As2 and hole-doped Ba1-xKxFe2As2 with the
superconducting critical temperatures, $T_{c}$ (squares), Néel temperatures,
$T_{N}$ (stars) and structural transition temperatures, $T_{s}$ (circles). The
$x$-axis is normalized to the charge carrier per iron atom. Data for the
electron-doped side where the transition temperatures are represented with
open symbols are taken from Ref [50]. The error bars for $T_{N}$ and $T_{s}$
values in the hole-doped side are within the symbols. The dashed line
enveloping the superconducting dome represents the Lindhard function taken
from Ref [33]. (Bottom panel) Charge carrier dependence of the As-Fe-As bond
angles for both electron- and hole-doping. Solid triangles represent the
results of our neutron diffraction study at 1.7 K for the hole-doped
Ba1-xKxFe2As2. At this temperature one of the As-Fe-As angles splits due to
orthorhombic distortion below $x=0.3$. Therefore, we took the average of these
two splitting angles. The As-Fe-As bond angle data for the electron doped side
is taken from Ref [51]. Solid lines are guide to the eye.
Finally, we discuss the electron-hole asymmetry in the phase diagram, shown in
Fig. 13, where we have added data from the literature [50,51] to allow a
comparison with the more commonly studied electron-doped superconductors. In
this phase diagram, the $x$-axis is normalized to the number of charge
carriers per Fe atom. Neupane et al have recently suggested that this
asymmetry is due to differences in the effective masses of the hole and
electron pockets [33]. This is justified by ARPES data that show that hole
doping can be well described within a rigid band approximation [52]. An ab
initio calculation of the Lindhard function of the non-interacting
susceptibility at the Fermi surface nesting wavevector shows exactly this
asymmetry, with a peak at $x\sim 0.4$ where the maximum $T_{c}$ occurs. Our
recent inelastic neutron scattering measurements of the resonant spin
excitations that are also sensitive to Fermi surface nesting have shown a
similar correlation between the strength of superconductivity and the mismatch
in the hole and electron Fermi surface volumes [34], that is responsible for
the fall of the Lindhard function at high $x$. An overall envelope may be
drawn (dashed line in Fig. 13) to encompass both the hole and electron
superconducting domes of the phase diagram. If anything, the Lindhard function
underestimates the asymmetry, predicting a larger superconducting dome on the
electron-doped side. We attribute this behavior to the fact that the iron
arsenide layers remain intact in the potassium substituted series, whereas Co
substitution for Fe disturbs the contiguity of the FeAs4 tetrahedra and
interferes with superconductivity in these layers.
Interestingly, the maximum overall $T_{c}$ also correlates with the perfect
tetrahedral angle of $\sim 109.5^{\circ}$ as demonstrated in the bottom panel
of Fig. 13. In the plot, average $<$As-Fe-As$>$ bond angles for our
K-substituted series have been extracted from the Rietveld refinements. The
As-Fe-As bond angles for BaFe2-xCoxAs2 are extracted from the literature [51].
The continuity of the bond angles across the electron-doped and hole-doped
sides of the phase diagram is remarkable and the crossing of the two
independent angles at $x\sim 0.4$ to yield a perfect tetrahedron and maximum
$T_{c}$ is clear. This has been remarked before in other systems [35,53]. It
is possible that these two apparently distinct explanations for the maximum
$T_{c}$ are two sides of the same coin. In a theoretical analysis of the 1111
compounds [38], it has been suggested that the pnictogen height is important
in controlling the energies of different orbital contributions to the
$d$-bands and so affect the strength of the interband scattering that produces
superconductivity.
We now turn our attention to the SDW region of the phase diagram. While it is
clear that spin-density-wave order has to be suppressed in order to allow
superconductivity to develop, it is not immediately clear what is responsible
for the suppression. Both the strength of magnetic interactions and
superconductivity, at least in an itinerant model, depend on the same Lindhard
function [54], the former on the peak in the susceptibility at the magnetic
wavevector, and the latter on an integral over the Fermi surfaces. It would
seem therefore that the magnetic transition temperature should also peak at
$x\sim 0.4$. One intriguing reason why it would peak at $x=0$ is because
magnetic order is more sensitive to disorder-induced suppression of the peak
susceptibility whereas superconductivity is more robust. There is some support
for this idea from the observation that isoelectronic doping produces a
similar suppression of magnetic order than seen with hole-doping [51]. On the
other hand, Kimber et al succeeded in rendering the parent BaFe2As2 material
to exhibit zero resistance at 30.5 K by the application of significant
external pressures up to 5.5 GPa [55], i.e., without introducing disorder, but
they remarked that superconductivity needs to be confirmed by other bulk
measurement techniques. Interestingly, the authors also correlate the induced
$T_{c}$ with approaching a perfect tetrahedron angle of 109.5o similar to our
observations for $x\sim 0.4$.
In summary, we have synthesized high quality samples covering the full phase
diagram of Ba1-xKxFe2As2. Using high resolution neutron powder diffraction and
SQUID magnetization measurements, we have investigated the effects of
potassium substitution on superconductivity, structural transformation and
magnetic ordering. Our measurements allowed the construction of a detailed
magnetic and structural phase diagram, which displays a narrower phase
coexistence region than the previous reports. Moreover, neutron diffraction
and the SQUID magnetization data confirmed that magnetic and structural
transitions are coincident with first order transitions. Additionally, we
determined the effects of temperature and substitution on the various internal
atomic and structural parameters. Our results confirm the importance of
obtaining precise structural parameters across the whole phase diagram as a
way of providing insight into the nature of the phase competition that
underlies iron-based superconductivity.
This work was supported by the Materials Sciences and Engineering Division of
the Office of Basic Energy Sciences, Office of Science, U.S. Department of
Energy, under contract No. DE-AC02-06CH11357.
## References
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|
arxiv-papers
| 2012-05-02T16:56:17 |
2024-09-04T02:49:30.471662
|
{
"license": "Public Domain",
"authors": "Sevda Avci, Omar Chmaissem, Duck-Young Chung, Stephan Rosenkranz,\n Eugene A. Goremychkin, John-Paul Castellan, Ilya S. Todorov, John A.\n Schlueter, Helmut Claus, Aziz Daoud-Aladine, Dmitry D. Khalyavin, Mercouri G.\n Kanatzidis and Raymond Osborn",
"submitter": "Ray Osborn",
"url": "https://arxiv.org/abs/1205.0489"
}
|
1205.0676
|
# Effective representations of Hecke-Kiselman monoids of type $A_{n}$
###### Abstract.
We prove effectiveness of certain representations of Hecke-Kiselman monoids of
type $A_{n}$ constructed by Ganyushkin and Mazorchuk and also construct
further classes of effective representations for these monoids. As a
consequence the effective dimension of monoids of type $A_{n}$ is determined.
We also show that odd Fibonacci numbers $F_{2n+1}$ appear as the cardinality
of certain bipartite HK-monoids and count the number of multiplicity free
elements in any HK-monoid of type $A_{n}$.
## 1\. Introduction
Let $S$ be a monoid and $\mathcal{R}$ an integral commutative domain. A
(finite dimensional) linear representation of $S$ over $\mathcal{R}$ is a
homomorphism $\varphi$ from $S$ to the semigroup $\mathrm{Mat}_{n\times
n}(\mathcal{R})$ of $n\times n$ matrices over $\mathcal{R}$. The
representation $\varphi$ is called effective if different elements of $S$ are
represented by different matrices. Note that a faithful representation of the
semigroup algebra $\mathcal{R}S$ induces an effective representation of $S$,
but the converse is false in general. The least $n\in\mathbb{N}$ such that $S$
has an effective representation in $\mathrm{Mat}_{n\times n}(\mathcal{R})$ is
called the _effective dimension_ (of $S$ over $\mathcal{R}$) and is denoted by
$\mathrm{eff}.\dim_{\mathcal{R}}(S)$. For finite semigroups the problem of
determining $\mathrm{eff}\dim_{\mathcal{R}}(S)$ is effectively computable when
$\mathcal{R}$ is an algebraically closed or a real closed field, see [8], but
giving the answer as a closed formula is usually very hard.
C. Kiselman defined in [6] a monoid generated by three operators $c,l,m$ with
origins in convexity theory and showed that it has presentation
(1.1) $K=\langle c,l,m:c^{2}=c,l^{2}=l,m^{2}=m\\\
clc=lcl=lc,cmc=mcm=mc,lml=mlm=ml\rangle.$
In [6] it was shown that $K$ has $2^{3}$ idempotents and that $K$ has an
effective representation by (nonnegative) integer valued $3\times 3$-matrices.
O. Ganyushkin and V. Mazorchuk generalized $K$ to a series of monoids $K_{n}$,
called _Kiselman monoids_ (unpublished) given by the following presentation:
$K_{n}=\langle c_{1},c_{2},\cdots,c_{n}:c_{i}^{2}=c_{i}\forall
i,c_{i}c_{j}c_{i}=c_{j}c_{i}c_{j}=c_{i}c_{j}\forall i\leq j\rangle.$
It was showed by G. Kudryavtseva and V. Mazorchuk in [7] that $K_{n}$ is a
finite monoid with $2^{n}$ idempotents and that $K_{n}$ has an effective
representation by (nonnegative) integer valued $n\times n$-matrices. The proof
of the latter fact is technically rather involved.
Let $\Gamma=(V(\Gamma),E(\Gamma))$ be a graph representing a disjoint union of
simply laced Dynkin diagrams, $W_{\Gamma}$ the corresponding Weyl group and
$\mathcal{H}_{q}(W_{\Gamma})$, where $q\in\mathcal{R}$, the Hecke algebra of
$W_{\Gamma}$. By specializing $q=0$ we obtain an algebra which is isomorphic
to the monoid algebra of the so-called $0$-Hecke monoid $\mathcal{H}_{\Gamma}$
which has the following presentation:
(1.2) $\mathcal{H}_{\Gamma}=\langle v\in V(\Gamma):v^{2}=v\forall v,\\\
vwv=wvw\text{ for all }\\{v,w\\}\in E(\Gamma),vw=wv\text{ for all
}\\{v,w\\}\not\in E(\Gamma)\rangle.$
O. Ganyushkin and V. Mazorchuk proposed in [3] a common generalization for
$K_{n}$ and $\mathcal{H}_{\Gamma}$ by introducing the so-called Hecke-Kiselman
monoids which are defined as follows.
###### Definition 1.1.
Let $\Gamma=\big{(}V(\Gamma),E(\Gamma)\big{)}$ be a simple directed graph. The
_Hecke-Kiselman monoid_ $HK_{\Gamma}$ of $\Gamma$ is the quotient of the free
monoid $\big{(}V(\Gamma)\big{)}^{*}$ by the following relations:
1. (I)
$x^{2}=x$ for all $x\in V(\Gamma)$.
2. (II)
If there is no edge between $x$ and $y$, then $xy=yx$.
3. (III)
If there is an edge from $x$ to $y$ but no edge from $y$ to $x$, then
$xyx=yxy=xy$.
4. (IV)
If there are edges in both directions between $x$ and $y$, then $xyx=yxy$.
We will refer to the above relations (including $x^{2}=x$) as _edge
relations_. Since there is no risk of confusion we will use $x\in\Gamma$ as a
shorthand for $x\in V(\Gamma)$.
It is often convenient to think of a pair of edges $x\to y$ and $y\to x$ in
$\Gamma$ as a single undirected edge. In order to simplify statements this
identification is done for the rest of the paper. Both Kiselman and $0$-Hecke
monoids are special cases of Hecke-Kiselman monoids. The $0$-Hecke monoid
corresponds to the underlying undirected graph of a Dynkin diagram. The
Kiselman monoid $K_{n}$ corresponds to $\Gamma$ for which
$V(\Gamma)=\\{1,2\cdots,n\\}$ and edge set $E(\Gamma)=\\{(i,j)|i<j\\}$. O.
Ganyushkin and V. Mazorchuk proved that the semigroups $HK_{\Gamma}$ and
$HK_{\Gamma^{\prime}}$ are isomorphic if and only if $\Gamma$ and
$\Gamma^{\prime}$ are isomorphic graphs.
###### Definition 1.2.
A simple directed graph $\Gamma$ is of _type $A_{n}$_ if its underlying
undirected graph is the Dynkin diagram $A_{n}$ for some $n\in\mathbb{N}$. A
graph of type $A_{n}$ with exactly one sink and one source (which may
coincide) and no unoriented edges is called _linearly ordered_. A _canonical
order_ on the vertices of a graph of type $A_{n}$ is one where neighboring
vertices have indices that differ by 1. The canonical order on a linearly
ordered graph is from a source to a sink, i.e.
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0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{$\textstyle{v_{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
We will say that $HK_{\Gamma}$ is of type $A_{n}$ if $\Gamma$ is of type
$A_{n}$. The monoids $HK_{\Gamma}$ of type $A_{n}$ naturally appear as monoid
of projection functors as defined by A.-L. Grensing in [4, 9], see also some
further development in [5].
In view of the above the following questions arise naturally:
1. (1)
Does $HK_{\Gamma}$ have $2^{n}$ idempotents, where $n=|V(\Gamma)|$ is the
number of vertices? If not, then how many?
2. (2)
Does $HK_{\Gamma}$ have an effective representation by $n\times n$\- matrices
over some $\mathcal{R}$? over $\mathbb{Z}$?
3. (3)
Is $HK_{\Gamma}$ finite? Can we calculate its cardinality?
These are the questions which we address in the present paper for various
families of graphs. After the first version of the present paper appeared, R.
Aragona and A. D’Andrea addressed the cardinality problem in the case $\Gamma$
is small (has at most four vertices) and discovered a nontrivial example of a
graph with an unoriented edge for which the cardinality of the Hecke-Kiselman
monoid is infinite, see [2]. In this paper we study only graphs for which
there are no unoriented edges, or, equivalently, there is at most one oriented
edge between any pair of vertices.
The paper is organized as follows: In Section 2 we collected some basic
notation and definitions from the combinatorics of words. Section 3 some
preliminary results on linear representations of some Hecke-Kiselman monoids.
In Section 4 we discuss effective representations and cardinalities of the
Hecke-Kiselman monoids considered in Section 3. In Section 5 we investigate
obstructions to generalize our methods to further classes of Hecke-Kiselman
monoids.
All representations in this paper are linear. We set
$\underline{n}=\\{1,2,\cdots,n\\}$.
## 2\. Words, content and canonical projections
If $\Gamma$ is a disjoint union of $\Gamma_{1}$ and $\Gamma_{2}$, then it is
easy to see that
$HK_{\Gamma}=HK_{\Gamma_{1}}\oplus HK_{\Gamma_{2}}.$
Thus we only need to study connected graphs.
We will use bold fonts to denote a word
$\mathbf{w}\in\big{(}V(\Gamma)\big{)}^{*}$. Elements in $HK_{\Gamma}$ are
equivalence classes of words and are denoted by brackets: $[\mathbf{w}]\in
HK_{\Gamma}$. The empty word is denoted by $\varepsilon$. We will make use of
two binary relations on $\big{(}V(\Gamma)\big{)}^{*}$:
$\mathbf{w}\sim\mathbf{w}^{\prime}:\iff[\mathbf{w}]=[\mathbf{w}^{\prime}],\text{
and}$
$\mathbf{w}\approx\mathbf{w}^{\prime}:\iff\big{(}\mathbf{w}=\mathbf{xyz}\text{
and }\mathbf{w}^{\prime}=\mathbf{xy^{\prime}z},\text{ where
}\mathbf{y}=\mathbf{y^{\prime}}\text{ is an edge relation.}\big{)}$
Note that $\sim$ is an equivalence relation while $\approx$ is not. Moreover,
$\sim$ is the transitive closure of $\approx$. Hence the statement
$\mathbf{w}\approx\mathbf{w}^{\prime}$ is stronger than the statement
$\mathbf{w}\sim\mathbf{w}^{\prime}$.
For each word $\mathbf{w}$ we may define the content
$\mathfrak{c}(\mathbf{w})\subset V(\Gamma)$ as the set of vertices that appear
at least once in $\mathbf{w}$. Note that the equality
$\mathfrak{c}(\mathbf{w})=\mathfrak{c}(\mathbf{w}^{\prime})$ holds for each of
the edge relations $\mathbf{w}=\mathbf{w}^{\prime}$. Since
$\mathfrak{c}(\mathbf{xyz})=\mathfrak{c}(\mathbf{x})\cup\mathfrak{c}(\mathbf{y})\cup\mathfrak{c}(\mathbf{z})$,
this implies $\mathfrak{c}(\mathbf{w})=\mathfrak{c}(\mathbf{w}^{\prime})$ when
$\mathbf{w}\approx\mathbf{w}^{\prime}$, and, by transitivity, when
$\mathbf{w}\sim\mathbf{w}^{\prime}$. Thus we may define the content
$\mathfrak{c}([\mathbf{w}])$ for each element in $[\mathbf{w}]\in
HK_{\Gamma}$. Note that
$\mathfrak{c}([\mathbf{w}][\mathbf{w}^{\prime}])=\mathfrak{c}([\mathbf{w}])\cup\mathfrak{c}([\mathbf{w}^{\prime}])$.
In particular $\mathfrak{c}([\mathbf{w}])=\mathfrak{c}([\mathbf{w}^{\prime}])$
implies $\mathfrak{c}([\mathbf{w}])=\mathfrak{c}([\mathbf{ww^{\prime}}])$ and
the set of elements with a fixed content form a subsemigroup. In this way we
obtain a semilattice decomposition of $HK_{\Gamma}$.
Let $\Gamma^{\prime}\subset\Gamma$ be a subgraph. Then $HK_{\Gamma^{\prime}}$
is a submonoid of $HK_{\Gamma}$ in the natural way. Furthermore, sending each
$x\in V(\Gamma^{\prime})$ to itself and each $x\in
V(\Gamma\setminus\Gamma^{\prime})$ to $\varepsilon$ extends uniquely to a
homomorphism $p:HK_{\Gamma}\to HK_{\Gamma^{\prime}}$ which is obviously
surjective and hence will be called the canonical projection. Therefore
$HK_{\Gamma^{\prime}}$ is a retract of $HK_{\Gamma}$.
## 3\. Preliminary results
Let $\mathcal{R}$ be an integral domain and $W=\bigoplus_{v\in
V(\Gamma)}\mathcal{R}v$ the formal vector space over $\mathcal{R}$ with basis
$V(\Gamma)$. Let $f:E(\Gamma)\to\mathcal{R}\setminus\\{0\\}$ be a function (we
will call it a weight function). For notational reasons we denote the weight
on the edge from $x$ to $y$ by $f_{xy}$ (assuming it exists). Set
$\theta_{x}^{f}(y)=\begin{cases}y,&x\neq y;\\\ \sum_{z\to
x}f_{zx}z,&x=y;\end{cases}$
and extend this by linearity to an endomorphism of $W$. Define the map
$R_{f}:V(\Gamma)\to End_{\mathcal{R}}(W)$ by $x\mapsto\theta_{x}^{f}$. This
uniquely extends to a homomorphism $R_{f}:\big{(}V(\Gamma)\big{)}^{*}\to
End_{\mathcal{R}}(W)$ using the fact that $V(\Gamma)$ is a set of free
generators of $\big{(}V(\Gamma)\big{)}^{*}$. The endomorphisms
$\theta_{x}^{f}$ will be called _atomic_. If the weight function is fixed in
advance, we sometimes omit it from the notation and write simply $\theta_{x}$
for $\theta_{x}^{f}$. This construction generalizes the construction of
“linear integral representations” in [3], and the following statement
generalizes [3, Proposition 7].
###### Theorem 3.1.
$R_{f}$ induces a well-defined homomorphism $HK_{\Gamma}\to
End_{\mathcal{R}}(W)$.
###### Proof.
We have to check that $R_{f}$ respects the edge relations. We start with the
relation $x^{2}=x$. Since $\Gamma$ is simple, an edge $z\to x$ implies $z\neq
x$. Thus
$\theta_{x}\circ\theta_{x}(y)=\begin{cases}y,&x\neq y;\\\ \sum_{z\to
x}f_{zx}\theta_{x}(z),&x=y\end{cases}=\begin{cases}y,&x\neq y;\\\ \sum_{z\to
x}f_{zx}z,&x=y\end{cases}=\theta_{x}(y).$
Relation $xy=yx$ for $x$ and $y$ different and non-adjacent.
$\theta_{x}\circ\theta_{y}(z)=\begin{cases}\theta_{x}(z),&y\neq z;\\\
\sum_{\omega\to y}f_{\omega
y}\theta_{x}(\omega),&y=z\end{cases}=\begin{cases}z,&x,y\neq z;\\\
\sum_{\omega\to x}f_{\omega x}\omega,&x=z;\\\ \sum_{\omega\to y}f_{\omega
y}\omega,&y=z.\end{cases}$
As the right hand side is symmetric in $x$ and $y$, so is the left hand side.
Relation $xyx=yxy=xy$ when there is a directed edge from $x$ to $y$.
$\theta_{x}\circ\theta_{y}\circ\theta_{x}(z)=\begin{cases}\theta_{x}\circ\theta_{y}(z),&x\neq
z;\\\ \sum_{\omega\to x}f_{\omega
x}\theta_{x}\circ\theta_{y}(\omega),&x=z\end{cases}=\begin{cases}z,&x,y\neq
z;\\\ \sum_{\omega\to y}f_{\omega y}\theta_{x}(\omega),&y=z;\\\
\sum_{\omega\to x}f_{\omega x}\omega,&x=z.\end{cases}$
Since there is a directed edge from $x$ to $y$ we have
$\sum_{\omega\to y}f_{\omega y}\theta_{x}(\omega)=\sum_{\omega\to
y,x\neq\omega}f_{\omega y}\omega+f_{xy}\sum_{\omega\to x}f_{\omega x}\omega.$
On the other hand
$\theta_{x}\circ\theta_{y}(z)=\begin{cases}\theta_{x}(z),&y\neq z;\\\
\sum_{\omega\to y}f_{\omega
y}\theta_{x}(\omega),&y=z\end{cases}=\begin{cases}z,&x,y\neq z;\\\
\sum_{\omega\to x}f_{\omega x}\omega,&x=z;\\\ \sum_{\omega\to y}f_{\omega
y}\theta_{x}(\omega),&y=z;\end{cases}$
This implies $xyx=xy$. Observe that $\theta_{x}\circ\theta_{y}(z)$ has no $y$
component for any $z$. Thus, by definition $\theta_{y}$ acts as identity on
$\theta_{x}\circ\theta_{y}(z)$ and we get $xy=yxy$. ∎
In the case $\Gamma$ is linearly ordered of type $A_{n}$, $f$ is constantly
equal to $1$ and $\mathbb{Z}\subset\mathcal{R}$ we have that $R_{f}$ is
effective, as proved in [3]. We denote a constant function
$E(\Gamma)\to\mathcal{R}$ with value $c$ simply by $c\in\mathcal{R}$ and the
corresponding representation by $R_{c}$.
###### Lemma 3.2.
Let $\Gamma$ be linearly ordered of type $A_{n}$. Then any choice of $f$ gives
an effective representation $R_{f}$ of $HK_{\Gamma}$.
It is important to recall that all values of $f$ are non-zero by definition.
###### Proof.
For a linearly ordered $\Gamma$ of type $A_{n}$ each vertex of $\Gamma$ is a
target of at most one arrow. Therefore the definition of $R_{f}$ implies that
for any $v,x\in V(\Gamma)$ the linear transformation $R_{f}([v])$ maps $x$ to
a scalar multiple of some other vertex, say $y$. By induction on the length of
a word it follows that for any $x\in V(\Gamma)$ and $[\mathbf{w}]\in
HK_{\Gamma}$ there exists $v\in V(\Gamma)$ and
$c_{x,f,\mathbf{w}}\in\mathcal{R}$ such that
$R_{f}([\mathbf{w}])(x)=c_{x,f,\mathbf{w}}y$. Certainly, $c_{x,f,\mathbf{w}}$
depends on $f$. However, we claim that
(3.1) $c_{x,f,\mathbf{w}}\neq 0\quad\text{ implies that }\quad
c_{x,f^{\prime},\mathbf{w}}\neq 0\quad\text{ for any other }\quad f^{\prime},$
or, in other words, that the fact that $c_{x,f,\mathbf{w}}$ is non-zero does
not depend on $f$. Claim (3.1) and effectiveness of $R_{1}$ established in [3]
imply the claim of our lemma.
To prove claim (3.1) assume that $c_{x,f,\mathbf{w}}\neq 0$. Let
$w=w_{k}w_{k-1}\cdots w_{1}$ and set $y_{0}=x$. For $i=1,\dots,k$ define
recursively $y_{i}$ as the unique vertex of $\Gamma$ such that
$R_{f}([w_{i}w_{i-1}\cdots w_{1}])(x)=c_{i}y$ for some non-zero
$c_{i}\in\mathcal{R}$. This is well-defined as $c_{x,f,\mathbf{w}}\neq 0$. We
have $c_{k}=c_{x,f,\mathbf{w}}\neq 0$. The definition of $R_{f}$ and the fact
that $\mathcal{R}$ is a domain imply that for $f^{\prime}$ we will have that
$R_{f^{\prime}}([w_{i}w_{i-1}\cdots w-1])(x)=c^{\prime}_{i}y$ for some non-
zero $c^{\prime}_{i}\in\mathcal{R}$. In particular, $c^{\prime}_{k}\neq 0$ and
the claim follows. ∎
###### Remark 3.3.
We note that the representation $R_{f}$ is not effective if $\Gamma$ contains
even a single unoriented edge. Let $\Gamma$ be a graph containing the vertices
$v$ and $w$ with an unoriented edge in between. To simplify notation, let
$A=\sum_{z\to v,z\neq w}f_{zv}z\text{ and }B=\sum_{z\to w,z\neq v}f_{zw}z.$
From the definition of $R_{f}$ we have
$\theta_{v}\circ\theta_{w}\circ\theta_{v}(v)=f_{wv}\theta_{v}\circ\theta_{w}(w+A)=f_{wv}A+f_{wv}f_{vw}\theta_{v}(v+B)=$
$f_{wv}A+f_{wv}f_{vw}B+f_{wv}^{2}f_{vw}A+f_{wv}^{2}f_{vw}w,\text{ while}$
$\theta_{w}\circ\theta_{v}\circ\theta_{w}(v)=\theta_{w}\circ\theta_{v}(v)=f_{wv}\theta_{w}(w+A)=f_{wv}A+f_{wv}f_{vw}B+f_{wv}f_{vw}v.$
As $v$ and $w$ are linearly independent, we must have $f_{wv}f_{vw}=0$. Since
$\mathcal{R}$ is an integral domain, this implies $f_{wv}=0$ or $f_{vw}=0$.
But $f$ only takes nonzero values by definition. If we allow $f$ to be zero on
an edge $v\to w$ then straightforward calculations show that
$R_{f}[wv]=R_{f}[vwv]$, showing that $f$ is not effective.
We will say that a word $\mathbf{w}$ is _multiplicity free with respect to a
vertex $v$_ if $v$ appears at most once in $\mathbf{w}$. A word $\mathbf{w}$
is called _multiplicity free_ if it is multiplicity free with respect to every
vertex. An element $[\mathbf{w}]\in HK_{\Gamma}$ is called _multiplicity free_
if $[\mathbf{w}]$ contains a multiplicity free word. For $A\subset V(\Gamma)$
define $\mathcal{MF}_{A}\subset\big{(}V(\Gamma)\big{)}^{*}$ as the set of
words which are multiplicity free with respect to all vertices in $A$. Let
$\mathcal{S}=\mathcal{S}_{\Gamma}\subset V(\Gamma)$ denote the set of all
sources and sinks. Multiplicity free words are easier to handle because they
allow us to speak about _the_ position of a single vertex (assuming it does).
Recall that a subword of a word $v_{1}v_{2}\dots v_{k}$ is a word of the form
$v_{i_{1}}v_{i_{2}}\dots v_{i_{j}}$ where $1\leq i_{1}<i_{2}<\dots<i_{j}\leq
k$.
###### Lemma 3.4.
Let $A\subset\mathcal{S}$ and $\mathbf{w}\in\big{(}V(\Gamma)\big{)}^{*}$. Then
$[\mathbf{w}]\cap\mathcal{MF}_{A}$ contains a subword of $\mathbf{w}$.
Furthermore, if $\mathbf{w}\sim\mathbf{w}^{\prime}$ are both in
$\mathcal{MF}_{A}$, then there exist a series of words
$\mathbf{w}_{i}\in[\mathbf{w}]\cap\mathcal{MF}_{A}$, such that
$\mathbf{w}=\mathbf{w}_{1}\approx\mathbf{w}_{2}\approx\cdots\approx\mathbf{w}_{k}=\mathbf{w}^{\prime}.$
###### Proof.
If $\mathbf{w}$ is already in $\mathcal{MF}_{A}$, we take
$\mathbf{w}^{\prime}=\mathbf{w}$ and we are done. Assume that
$\mathbf{w}=\mathbf{w}_{1}a\mathbf{w}_{2}a\mathbf{w}_{3}$ is a word with at
least two occurrences of $a\in A$, and that $a$ is a source (if $a$ is a sink,
a similar argument works with all words reversed). Because of our restrictions
on $\Gamma$, for any $x\in V(\Gamma)$ we have one of the following:
1. (1)
$x=a$ and $ax=aa=aaa=axa$;
2. (2)
there is a directed edge from $a$ to $x$ and $ax=axa$;
3. (3)
$a$ and $x$ are nonadjacent and $ax=aax=axa$.
Note that in all cases we have the relation $ax=axa$. Therefore, for
$\mathbf{w}_{2}=x_{1}x_{2}\cdots x_{k}$ we have
$\mathbf{w}=\mathbf{w}_{1}ax_{1}x_{2}\cdots
x_{k}a\mathbf{w}_{3}\sim\mathbf{w}_{1}ax_{1}ax_{2}\cdots
x_{k}a\mathbf{w}_{3}\sim\cdots\sim\mathbf{w}_{1}ax_{1}ax_{2}a\cdots
ax_{k}a\mathbf{w}_{3}\sim$ $\mathbf{w}_{1}ax_{1}ax_{2}a\cdots
ax_{k}\mathbf{w}_{3}\sim\cdots\sim\mathbf{w}_{1}ax_{1}x_{2}\cdots
x_{k}\mathbf{w}_{3}=\mathbf{w}_{1}a\mathbf{w}_{2}\mathbf{w}_{3}.$
Thus we have lowered the multiplicity of $a$ in $\mathbf{w}$ while leaving all
other vertices untouched. Continuing in this fashion for all $a\in A$ gives us
the desired subword of $\mathbf{w}$ contained in
$[\mathbf{w}]\cap\mathcal{MF}_{A}$. This proves the first claim of the lemma.
Given two words $\mathbf{w}\sim\mathbf{\tilde{w}}$ there is, by definition, a
sequence of words $\mathbf{w}_{i}\in[\mathbf{w}]$, $i\in\underline{k}$, such
that
$\mathbf{w}=\mathbf{w}_{1}\approx\mathbf{w}_{2}\approx\cdots\approx\mathbf{w}_{k}=\mathbf{\tilde{w}}.$
Let $\psi:\big{(}V(\Gamma)\big{)}^{*}\to\mathcal{MF}_{A}$ be the function
which removes all superfluous occurrences of all $a\in A$ as described above.
We want to show that if $\mathbf{a}\approx\mathbf{b}$, then
$\psi(\mathbf{a})\approx\psi(\mathbf{b})$ or
$\psi(\mathbf{a})=\psi(\mathbf{b})$ (in which case we may shorten the
sequence). By definition, $\mathbf{a}\approx\mathbf{b}$ means that
$\mathbf{a}=\mathbf{a}_{l}\mathbf{c}\mathbf{a}_{r}$ and
$\mathbf{b}=\mathbf{a}_{l}\mathbf{d}\mathbf{a}_{r}$, where
$\mathbf{c}=\mathbf{d}$ is an edge relation. Depending on the letters which
appear in the edge relation, we consider different cases.
Case 1. Relation $x^{2}=x,xyx=yxy=xy$ or $xy=yx$ for $x,y\not\in A$. In this
case the application of $\psi$ affects neither $\mathbf{c}$ nor $\mathbf{d}$.
Assume that $a\in A$ is a source which appears in exactly one of the words
$\mathbf{a}_{l}$ or $\mathbf{a}_{r}$. Then the application of $\psi$ to both
$\mathbf{a}$ and $\mathbf{b}$ deletes all but the leftmost occurrences of $a$.
If $a\in A$ is a source which appears in both $\mathbf{a}_{l}$ and
$\mathbf{a}_{r}$, then the application of $\psi$ to both $\mathbf{a}$ and
$\mathbf{b}$ deletes all occurrences of $a$ in $\mathbf{a}_{r}$ and all but
the leftmost occurrences of $a$ in $\mathbf{a}_{l}$. Similarly one considers
the case when $a\in A$ is a sink. It follows that
$\psi(\mathbf{a})\approx\psi(\mathbf{b})$ in this case.
Case 2. Relations $a^{2}=a,ax=xa$ and $axa=xax=ax$ where $a\in A$ and
$x\not\in A$. We assume that $a$ is a source (the case when $a$ is a sink is
done similarly). If $a$ appears in $\mathbf{a}_{l}$, then $\psi$ deletes all
$a$ in $\mathbf{a}_{r}$, $\mathbf{c}$ and $\mathbf{d}$ (and leaves just the
leftmost occurrences of $a$ in $\mathbf{a}_{l}$). Note that our edge relations
become $\varepsilon=\varepsilon$, $x=x$ and $x=x^{2}$, respectively. It
follows that in this case $\psi(\mathbf{a})=\psi(\mathbf{b})$ or
$\psi(\mathbf{a})\approx\psi(\mathbf{b})$. If $a$ does not appear in
$\mathbf{a}_{l}$, then $\psi$ deletes all $a$ in $\mathbf{a}_{r}$ and leaves
the leftmost occurrences of $a$ in $\mathbf{c}$ and $\mathbf{d}$. Note that
our edge relations become $a=a$, $ax=xa$ and $ax=xax$, respectively. It
follows that in this case we have either
$\psi(\mathbf{a})\approx\psi(\mathbf{b})$ or
$\psi(\mathbf{a})=\psi(\mathbf{b})$, depending on the edge relation.
Case 3. Relation $ab=ba$ where $a,b\in A$. Similarly to the above, the
application of $\psi$ does the same thing to the subword $\mathbf{a}_{l}$ of
both $\mathbf{a}$ and $\mathbf{b}$, it does the same thing to the subword
$\mathbf{a}_{r}$ of both $\mathbf{a}$ and $\mathbf{b}$, and it maps $ab=ba$ to
either $ab=ba$ or $a=a$ or $b=b$ or $\varepsilon=\varepsilon$, depending on
whether $a$ or $b$ appear in $\mathbf{a}_{l}$ (if they are sources) or in
$\mathbf{a}_{r}$ (if they are sinks). In all cases we get that either
$\psi(\mathbf{a})\approx\psi(\mathbf{b})$ or
$\psi(\mathbf{a})=\psi(\mathbf{b})$.
Case 4. Relation $aba=bab=ab$, where $a,b\in A$. Here $a$ is a source and $b$
is a sink. Similarly to the above, the application of $\psi$ does the same
thing to the subword $\mathbf{a}_{l}$ of both $\mathbf{a}$ and $\mathbf{b}$
and it does the same thing to the subword $\mathbf{a}_{r}$ of both
$\mathbf{a}$ and $\mathbf{b}$. Depending on the appearance of $a$ in
$\mathbf{a}_{l}$ and $b$ in $\mathbf{a}_{r}$, the relation is either mapped to
$ab=ab$ or to $a=a$ or to $b=b$ or to $\varepsilon=\varepsilon$. Again, in all
cases we get that either $\psi(\mathbf{a})\approx\psi(\mathbf{b})$ or
$\psi(\mathbf{a})=\psi(\mathbf{b})$. The claim follows. ∎
###### Lemma 3.5.
Let $\Gamma$ be a simple directed graph with at most one edge between any pair
of vertices. Let $\Gamma^{\prime}\subset\Gamma$ be a full subgraph and let
$\mathbf{w}$ be a word such that
$\mathfrak{c}[\mathbf{w}]=V(\Gamma^{\prime})$. Then exactly one of the
following alternatives takes place:
1. (1)
$\Gamma^{\prime}$ contains an oriented cycle and $[\mathbf{w}]^{k}$ are
pairwise distinct for all $k\in\mathbb{N}$.
2. (2)
$\Gamma^{\prime}$ contains no oriented cycles and
$[\mathbf{w}]^{|V(\Gamma^{\prime})|}$ equals the zero element in
$HK_{\Gamma^{\prime}}$.
###### Proof.
Assume first that $\Gamma^{\prime}$ contains an oriented cycle $C$ (which then
necessarily has length at least $3$). Using $p:HK_{\Gamma}\to HK_{C}$ it is
enough to prove the first claim under the assumption
$\Gamma=\Gamma^{\prime}=C$ (since if $p([\mathbf{w}]^{k})$ are pairwise
distinct then $[\mathbf{w}]^{k}$ are pairwise distinct as well).
Let the vertices in $C$ be enumerated by $\mathbb{Z}_{n}$, such that there is
a directed edge from $v_{i}$ to $v_{i+1}$ for all $i\in\underline{n}$. To
separate elements, we choose the representation $R_{2}$ (with the ground ring
$\mathcal{R}=\mathbb{Z}$) and prove that the images $R_{2}([\mathbf{w}^{k}])$
are pairwise different. Recall that each $\theta_{i}$ maps $v_{i}$ to
$2v_{i-1}$ and $v_{j}$ to $v_{j}$ for $j\neq i$. It follows that there is a
transformation $t:V(C)\to V(C)$ and a set of nonnegative integers
$m_{1},\cdots,m_{n}$ such that $R_{2}([\mathbf{w}])(v_{i})=2^{m_{i}}v_{t(i)}$.
Moreover, $\mathfrak{c}([\mathbf{w}])=V(C)$ implies that $m_{i}\geq 1$ for
each $i$. We define the sequence $n_{i}$ by
$R_{2}([\mathbf{w}])(v_{t^{i-1}(1)})=2^{n_{i}}v_{t^{i}(1)}$, and
$\overline{n_{i}}=\sum_{j=1}^{i}n_{i}$. Then
$R_{2}([w^{k}])(v_{1})=2^{n_{1}}R_{2}([w^{k-1}])(v_{t(1)})=\cdots=2^{\overline{n_{k}}}v_{t^{k}(1)}.$
Since the exponent is strictly increasing in $k$, it follows that $[w^{k}]$
are pairwise distinct.
Assume now that $\Gamma$ contains no oriented cycles. In this case
$HK_{\Gamma}$ is a quotient of Kiselman’s semigroup and hence the second claim
follows from [7, Lemma 12]. ∎
###### Theorem 3.6.
There is a bijection between idempotents in $HK_{\Gamma}$ and full subgraphs
of $\Gamma$ which do not contain any oriented cycles.
###### Proof.
Let $[\mathbf{w}]\in\Gamma$ be an idempotent and $\Gamma^{\prime}$ be the full
subgraph whose set of vertices is $\mathfrak{c}[\mathbf{w}]$. Then
$\Gamma^{\prime}$ contains no oriented cycles by the first claim of the
previous lemma.
Conversely, let $\Gamma^{\prime}$ be a full subgraphs of $\Gamma$ which does
not contain any oriented cycles. Then $HK_{\Gamma^{\prime}}$ is a quotient of
Kiselman’s semigroup, in particular, $HK_{\Gamma^{\prime}}$ is finite and
contains a unique idempotent of maximal content, namely the zero element, see
[7]. The claim follows. ∎
## 4\. Main results
In every graph of type $A_{n}$ there are sinks and sources where the direction
of the edges changes. Thus we have small linearly ordered pieces of the graph,
glued together in sources an sinks. Since we know a lot about effective
representations for the pieces (see the previous paragraph), we would like to
know what the gluing does to representations.
###### Definition 4.1.
Given a graph $\Gamma$ and a vertex $a\in\Gamma$ the _source graph_ $S_{a}$ is
the full subgraph of $\Gamma$ with the vertex set
$V(S_{a})=\\{v\in\Gamma|\text{ there exists a directed path from }v\text{ to
}a\\}.$
###### Lemma 4.2.
For a vertex $a\in\Gamma$ let $p_{a}:HK_{\Gamma}\to HK_{S_{a}}$ be the
projection morphism. Then for any $[\mathbf{w}]\in HK_{\Gamma}$ we have the
equality $R_{f}([\mathbf{w}])(a)=R_{f}(p_{a}([\mathbf{w}]))(a)$.
###### Proof.
Let $x\in V(\Gamma)$ and $v\in S_{a}$. From the definition of $R_{f}$ we have
the equality $R_{f}([x])(v)=\sum_{w\in S_{a}}c_{w}w$ for some constants
$c_{v}$. Therefore the linear span $L$ of all $v\in S_{a}$ is invariant with
respect to the action of $HK_{\Gamma}$. Furthermore, for any $x\in
V(\Gamma)\setminus S_{a}$ we have $R_{f}([x])(v)=v$ for all $v\in S_{a}$,
which means that $R_{f}([x])$ acts as the identity on $L$. It follows that the
actions of $R_{f}([\mathbf{w}])$ and $R_{f}(p_{a}([\mathbf{w}]))$ on $L$
coincide. ∎
If $\Gamma=\cup_{i\in I}\Gamma_{i}$ is a disjoint union of full subgraphs
which pairwise do not have any common edges and $f^{i}$ is a collection of
weight functions, then there is a unique weight function $f$ on $\Gamma$ whose
restriction to $\Gamma_{i}$ coincides with $f^{i}$ for every $i$. We will call
$f$ the _extension_ of the $\\{f^{i}:i\in I\\}$.
A full subgraph $\Gamma^{\prime}\subset\Gamma$ will be called path complete if
every oriented path in $\Gamma$ which starts from a vertex of
$\Gamma^{\prime}$ and ends at a vertex of $\Gamma^{\prime}$ is contained
entirely in $\Gamma$. For example, if $\Gamma=\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 59.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
124.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
94.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 124.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{3}$}}}}}}}\ignorespaces}}}}\ignorespaces$, then the
full subgraph with vertices $1$ and $2$ is path complete while the full
subgraph with vertices $1$ and $3$ is not path complete.
Let $\Gamma^{\prime}$ be a path complete subgraph of $\Gamma$ and $f$ a weight
function. Let $f^{\prime}$ denote the restriction of $f$ to $\Gamma^{\prime}$
(then we have the corresponding representation $R_{f^{\prime}}$ of
$HK_{\Gamma^{\prime}}$). Consider the representation $R_{f}$ of $HK_{\Gamma}$
on $W=\bigoplus_{v\in\Gamma}\mathcal{R}v$. Let $\Gamma^{\prime\prime}$ be the
full subgraph of $\Gamma$ whose set of vertices consists of all vertices of
$\Gamma$ from which there is an oriented (but maybe trivial) path to a vertex
of $\Gamma^{\prime}$. Clearly, $\Gamma^{\prime}$ is a subgraph of
$\Gamma^{\prime\prime}$. Finally, let $\Gamma^{\prime\prime\prime}$ be the
full subgraph of $\Gamma$ whose set of vertices coincides with the set of all
vertices of $\Gamma^{\prime\prime}$ which do not belong to $\Gamma^{\prime}$.
Then both $X=\bigoplus_{v\in\Gamma^{\prime\prime}}\mathcal{R}v$ and
$Y=\bigoplus_{v\in\Gamma^{\prime\prime\prime}}\mathcal{R}v$ are invariant
under the action of $HK_{\Gamma}$ (the fact that $Y$ is invariant follows from
the path completeness of $\Gamma^{\prime}$). Let $\rho$ be the corresponding
representation of $HK_{\Gamma}$ on $X/Y$.
###### Proposition 4.3.
The representations $\rho$ and $R_{f^{\prime}}\circ p$ of $HK_{\Gamma}$ are
isomorphic.
###### Proof.
Define the map $\Phi:X/Y\to\bigoplus_{v\in\Gamma^{\prime}}\mathcal{R}v$ as the
unique $\mathcal{R}$-linear map which sends $v+Y$ for $v\in
V(\Gamma^{\prime})$ to $v$. This is obviously linear and bijective. The fact
that it is a homomorphism of $HK_{\Gamma}$-modules follows directly from the
definitions. ∎
This theorem says that, for every word $\mathbf{w}\in(V(\Gamma))^{*}$, the
minor of the matrix of $R_{f}([\mathbf{w}])$ corresponding to the basis
vectors $\\{v:v\in\Gamma^{\prime}\\}$ coincides with the matrix
$R_{f^{\prime}}\circ p([\mathbf{w}])$. This fails if $\Gamma^{\prime}$ is not
path complete. For example, if we let $\Gamma=\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.64294pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-5.64294pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.64294pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.64294pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
63.9346pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
63.9346pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{c}$}}}}}}}\ignorespaces}}}}\ignorespaces$
and $\Gamma^{\prime}$ be the full subgraph with vertices $a$ and $c$, then for
$\mathbf{w}=bc$ and $f\equiv 1$ we have $R_{1}([bc])(c)=a$ while
$R_{1^{\prime}}\circ p([bc])(c)=0$.
###### Proposition 4.4.
Assume that $\Gamma$ is a union of two full subgraphs, $\Gamma_{1}$ and
$\Gamma_{2}$ such that $V(\Gamma_{1})\cap V(\Gamma_{2})=\\{a\\}$ for some
$a\in\mathcal{S}_{\Gamma}$. Then the map $p=(p_{1},p_{2}):HK_{\Gamma}\to
HK_{\Gamma_{1}}\times HK_{\Gamma_{2}}$, where $p_{i}:HK_{\Gamma}\to
HK_{\Gamma_{i}}$, $i=1,2$, are projection morphisms, is injective.
###### Proof.
Let $\Gamma^{\prime}$ denote the full subgraph of $\Gamma$ with vertices
$V(\Gamma)\setminus\\{a\\}$ and define $\Gamma^{\prime}_{1}$ and
$\Gamma^{\prime}_{2}$ similarly. Since $a\in\mathcal{S}_{\Gamma}$, by Lemma
3.4 every element $[\mathbf{w}]\in HK_{\Gamma}$ contains at least one word
$\mathbf{w}^{\prime}$ which is multiplicity free with respect to $a$, i.e.
$\mathbf{w}^{\prime}=\mathbf{w}_{1}\alpha\mathbf{w}_{2}$ where
$\alpha\in\\{\varepsilon,a\\}$ and $\mathbf{w}_{1},\mathbf{w}_{2}\in
HK_{\Gamma^{\prime}}$. Moreover, $\Gamma^{\prime}$ is a disjoint union of
$\Gamma^{\prime}_{1}$ and $\Gamma^{\prime}_{2}$, so we may assume that
$\mathbf{w}_{1}=\mathbf{x}_{1}\mathbf{y}_{1},\mathbf{w}_{2}=\mathbf{x}_{2}\mathbf{y}_{2}$
for some $\mathbf{x}_{1},\mathbf{x}_{2}\in HK_{\Gamma^{\prime}_{1}}$ and
$\mathbf{y}_{1},\mathbf{y}_{2}\in HK_{\Gamma^{\prime}_{2}}$. We then have
$p([\mathbf{x}_{1}\mathbf{y}_{1}\alpha\mathbf{x}_{2}y_{2}])=([\mathbf{x}_{1}\alpha\mathbf{x}_{2}],[\mathbf{y}_{1}\alpha\mathbf{y}_{2}])$.
Let now $[\tilde{\mathbf{w}}]\in HK_{\Gamma}$ be such that
$p([\mathbf{w}])=p([\tilde{\mathbf{w}}])$ and assume that
$\tilde{\mathbf{w}}=\tilde{\mathbf{x}}_{1}\tilde{\mathbf{y}}_{1}\tilde{\alpha}\tilde{\mathbf{x}}_{2}\tilde{\mathbf{y}}_{2}$
is a decomposition as above. Then
$\tilde{\mathbf{x}}_{1}\alpha\tilde{\mathbf{x}}_{2}\sim\mathbf{x}_{1}\alpha\mathbf{x}_{2}$
and
$\tilde{\mathbf{y}}_{1}\alpha\tilde{\mathbf{y}}_{2}\sim\mathbf{y}_{1}\alpha\mathbf{y}_{2}$,
which, in particular, implies $\tilde{\alpha}=\alpha$. Note that all elements
from $HK_{\Gamma^{\prime}_{1}}$ commute with all elements from
$HK_{\Gamma^{\prime}_{2}}$ (since there are no edges between
$\Gamma^{\prime}_{1}$ and $\Gamma^{\prime}_{2}$). Therefore
$\tilde{\mathbf{x}}_{1}\tilde{\mathbf{y}}_{1}\tilde{\alpha}\tilde{\mathbf{x}}_{2}\tilde{\mathbf{y}}_{2}\sim\tilde{\mathbf{x}}_{1}\tilde{\mathbf{y}}_{1}\tilde{\alpha}\tilde{\mathbf{y}}_{2}\tilde{\mathbf{x}}_{2}\sim\tilde{\mathbf{x}}_{1}\mathbf{y}_{1}\alpha\mathbf{y}_{2}\tilde{\mathbf{x}}_{2}\sim\\\
\sim\mathbf{y}_{1}\tilde{\mathbf{x}}_{1}\alpha\tilde{\mathbf{x}}_{2}\mathbf{y}_{2}=\mathbf{y}_{1}\tilde{\mathbf{x}}_{1}\tilde{\alpha}\tilde{\mathbf{x}}_{2}\mathbf{y}_{2}\sim\mathbf{y}_{1}\mathbf{x}_{1}\alpha\mathbf{x}_{2}\mathbf{y}_{2}\sim\mathbf{x}_{1}\mathbf{y}_{1}\alpha\mathbf{x}_{2}\mathbf{y}_{2}.$
Hence $\tilde{\mathbf{w}}\sim\mathbf{w}$ and the claim follows. ∎
###### Theorem 4.5.
Let $\Gamma$, $\Gamma_{1}$ and $\Gamma_{2}$ be as in Proposition 4.4. Let
$f_{1}$ and $f_{2}$ be weight functions for $\Gamma_{1}$ and $\Gamma_{2}$,
respectively and $f$ be the extension of $\\{f_{1},f_{2}\\}$ to $\Gamma$. Then
the representation $R_{f}$ of $HK_{\Gamma}$ is effective if and only if the
representations $R_{f_{1}}$ of $HK_{\Gamma_{1}}$ and $R_{f_{2}}$ of
$HK_{\Gamma_{2}}$ are effective.
###### Proof.
Let $\Gamma^{\prime}$, $\Gamma^{\prime}_{1}$ and $\Gamma^{\prime}_{2}$ be as
in the proof of Proposition 4.4. We start with the “only if” part. Both
$\Gamma_{1}$ and $\Gamma_{2}$ are path complete, which implies that
$R_{f}[\mathbf{w}](v)=\begin{cases}R_{f_{i}}p_{i}[\mathbf{w}](v),&\text{ if
}v\in\Gamma_{i}^{\prime}\\\
R_{f_{1}}p_{1}[\mathbf{w}](a)+R_{f_{2}}p_{2}[\mathbf{w}](a),&\text{ if
}v=a\text{ and }a\in\mathfrak{c}[\mathbf{w}]\\\ 0&\text{otherwise}\end{cases}$
Now assume that $R_{f_{1}}$ is noneffective, i.e. there exists
$[\mathbf{w}_{1}]\neq[\mathbf{w}_{2}]\in HK_{\Gamma_{2}}$ such that
$R_{f_{1}}[\mathbf{w}_{1}]=R_{f_{1}}[\mathbf{w}_{2}]$. Then
$p_{2}[\mathbf{w}_{1}]=p_{2}[\mathbf{w}_{2}](=[a]\text{ or }[\varepsilon])$,
and
$R_{f}[\mathbf{w}_{1}](v)=\begin{cases}R_{f_{1}}p_{1}[\mathbf{w}_{1}](v),&\text{
if }v\in\Gamma_{1}^{\prime}\\\ R_{f_{2}}p_{2}[\mathbf{w}_{1}](v),&\text{ if
}v\in\Gamma_{2}^{\prime}\\\
R_{f_{1}}p_{1}[\mathbf{w}_{1}](a)+R_{f_{2}}p_{2}[\mathbf{w}_{1}](a),&\text{ if
}v=a\text{ and }a\in\mathfrak{c}[\mathbf{w}]\\\
0&\text{otherwise}\end{cases}=$
The “if” part follows by combining Propositions 4.3 and 4.4 (note again that
both $\Gamma_{1}$ and $\Gamma_{2}$ are path connected in $\Gamma$). Assume
that the representations $R_{f_{1}}$ of $HK_{\Gamma_{1}}$ and $R_{f_{2}}$ of
$HK_{\Gamma_{2}}$ are effective and consider the representation $R_{f}$ of
$HK_{\Gamma}$. Let $[\mathbf{w}]\neq[\mathbf{w}^{\prime}]$ be two elements of
$HK_{\Gamma}$. Assume $R_{f}([\mathbf{w}])=R_{f}([\mathbf{w}^{\prime}])$. Then
Propositions 4.3 and the arguments from the first part of the proof imply that
$R_{f_{1}}(p_{1}([\mathbf{w}]))=R_{f_{1}}(p_{1}([\mathbf{w}^{\prime}]))$ and
$R_{f_{2}}(p_{2}([\mathbf{w}]))=R_{f_{2}}(p_{2}([\mathbf{w}^{\prime}]))$.
Since both $R_{f_{1}}$ and $R_{f_{2}}$ are effective, we get
$p_{1}([\mathbf{w}])=p_{1}([\mathbf{w}^{\prime}])$ and
$p_{2}([\mathbf{w}])=p_{2}([\mathbf{w}^{\prime}])$. Now from Proposition 4.4
we get $[\mathbf{w}]=[\mathbf{w}^{\prime}]$, a contradiction. The claim
follows. ∎
This statement can now be iterated as follows. Assume that $\Gamma$ is a union
of full subgraphs, $\displaystyle\Gamma=\bigcup_{i=1}^{n}\Gamma_{i}$, where
$n>1$, such that each pair of different subgraphs does not have any common
edges and, moreover, we assume that for every $k=2,3,\dots,n$ there is
$a_{k}\in\mathcal{S}_{\Gamma}$ such that
$V(\Gamma_{k})\cap\big{(}\bigcup_{i=1}^{k-1}V(\Gamma_{i})\big{)}=\\{a_{k}\\}.$
In this case we will say that $\Gamma$ satisfies the _gluing condition_.
For example, let $\Gamma$ be of type $A_{n}$. If we define $\Gamma_{i}$ to be
the maximal connected linearly ordered full subgraphs of $\Gamma$, then
$\Gamma$ satisfies the gluing condition with respect to these subgraphs as
illustrated below (here $\Gamma_{i}$ is a subgraph with vertices between
$a_{i}$ and $a_{i+1}$, where $a_{0}$ is the leftmost vertex and $a_{k+1}$ is
the rightmost vertex):
(4.1)
---
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{k}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{1}}$$\textstyle{\Gamma_{2}}$$\textstyle{\cdots}$$\textstyle{\Gamma_{k}}$
This implies the following corollary which answers [3, Question 8].
###### Corollary 4.6.
Let $\Gamma$ be of type $A_{n}$. Then the representation $R_{1}$ of
$HK_{\Gamma}$ is effective.
###### Proof.
This follows from Theorem 4.5 and [3, Subsection 3.2] by induction on $n$ (the
parameter in (4.1)). ∎
As a byproduct of the proof of Proposition 4.4 we get formulas for certain
cardinalities. An element $[\mathbf{w}]$ is said to have _maximal content_ if
$\mathfrak{c}[\mathbf{w}]=V(\Gamma)$. The subsemigroup consisting of elements
with maximal content is denoted by $\mathfrak{m}(\Gamma)$.
###### Theorem 4.7.
Assume that $\displaystyle\Gamma=\bigcup_{i=1}^{k}\Gamma_{i}$ satisfies the
gluing condition. Then
$|\mathfrak{m}(\Gamma)|=\prod_{i=1}^{k}|\mathfrak{m}(\Gamma_{i})|.$
###### Proof.
By induction, it is enough to prove the claim for
$\Gamma=\Gamma_{1}\cup\Gamma_{2}$. All we need to show is that the restricted
map
$p:\mathfrak{m}(\Gamma)\to\mathfrak{m}(\Gamma_{1})\times\mathfrak{m}(\Gamma_{2})$
is a bijection. Since we know that it is injective (by Proposition 4.4), we
only have to establish its surjectivity. Let
$([\mathbf{x}],[\mathbf{y}])\in\mathfrak{m}(\Gamma_{1})\times\mathfrak{m}(\Gamma_{2})$.
By definition, $a\in\mathfrak{c}[\mathbf{x}]$ and
$a\in\mathfrak{c}[\mathbf{y}]$, so there are words
$\mathbf{x}\sim\mathbf{x}_{1}a\mathbf{x}_{2},\mathbf{y}\sim\mathbf{y}_{1}a\mathbf{y}_{2}$
which are multiplicity free with respect to $a$. Then
$[\mathbf{x}_{1}\mathbf{y}_{1}a\mathbf{x}_{2}\mathbf{y}_{2}]$ is the preimage
of $([\mathbf{x}],[\mathbf{y}])$. The claim follows. ∎
The Catalan numbers $C_{n}=\frac{1}{n+1}{\binom{2n}{n}}$ are the cardinalities
of the HK-monoids of linearly ordered graphs, see [3, Theorem 1(vi)]. They can
also be used to calculate the cardinality of any HK-monoid of type $A_{n}$.
Let $\displaystyle\Gamma=\bigcup_{i=1}^{k}\Gamma_{i}$ be a graph of type
$A_{n}$ as in (4.1) or its opposite. For a subset $Q\subset\\{2,3,\cdots,k\\}$
and $i\in\\{2,3,\cdots,k-1\\}$ set
$\delta_{i}(Q)=\begin{cases}1,&\text{ if }a_{i}\in Q;\\\
0,&\text{otherwise};\end{cases}$
$c_{i}(Q)=\begin{cases}C_{l_{i}-1},&\delta_{i-1}(Q)=\delta_{i}(Q)=0;\\\
C_{l_{i}}-C_{l_{i}-1},&\delta_{i-1}(Q)+\delta_{i}(Q)=1;\\\
C_{l_{i}+1}-2C_{l_{i}}+C_{l_{i}-1},&\delta_{i-1}(Q)=\delta_{i}(Q)=1;\end{cases}$
$c_{1}(Q)=\begin{cases}C_{l_{1}},&\delta_{2}(Q)=0;\\\
C_{l_{1}+1}-C_{l_{1}},&\delta_{2}(Q)=1;\end{cases}\quad
c_{k}(Q)=\begin{cases}C_{l_{k}},&\delta_{k}(Q)=0;\\\
C_{l_{k}+1}-C_{l_{1}},&\delta_{k}(Q)=1.\end{cases}$
###### Corollary 4.8.
Let $\Gamma$ be as above. Then we have
1. (i)
$\displaystyle|\mathfrak{m}(\Gamma)|=\prod_{i=1}^{k}C_{l_{i}}$,
2. (ii)
$\displaystyle|HK_{\Gamma}|=\sum_{Q\subset\underline{k}\setminus\\{1\\}}\prod_{i=1}^{k}c_{i}(Q)$,
3. (iii)
if $\Gamma=\mathcal{S}_{\Gamma}$, that is,
$\Gamma=a\rightarrow b\leftarrow c\rightarrow d\leftarrow\cdots x\quad\text{
or }\quad\Gamma=a\leftarrow b\rightarrow c\leftarrow d\rightarrow\cdots x,$
then $|HK_{\Gamma}|=F_{2n+1}$ is the $(2n+1)$-th Fibonacci number (where
$F_{1}=F_{2}=1)$.
###### Proof.
To prove the first claim we need to know $\mathfrak{m}(\Gamma_{i})$ and then
apply Theorem 4.7. Since $\Gamma_{i}$ is linearly ordered, the elements of
$HK_{\Gamma_{i}}$ are in bijection with order preserving and order decreasing
transformations on a set with $l_{i}+1$ elements, see [3, Theorem 1(vii)].
This bijection restricts to a bijection between $\mathfrak{m}(\Gamma_{i})$ and
transformations $\tau$ such that $\tau(j)<j$ for all $j\neq 1$. Changing $j$
in the domain to $j-1$ gives a bijection between $\mathfrak{m}(\Gamma_{i})$
with order preserving and order decreasing transformations on a set with
$l_{i}$ elements and hence $|\mathfrak{m}(\Gamma_{i})|=C_{l_{i}}$ by [3,
Theorem 1(vi) and (vii)].
For the second claim we have to work more. Assume that
$\Gamma=\Gamma_{1}\cup\Gamma_{2}$ satisfies the gluing condition with
$a\in\mathcal{S}_{\Gamma}$ as the common vertex. Then $HK_{\Gamma}$ splits
into two subsets (in fact subsemigroups) $A_{\Gamma}:=\\{[\mathbf{w}]\in
HK_{\Gamma}|a\in\mathfrak{c}[\mathbf{w}]\\}$ and
$B_{\Gamma}:=\\{[\mathbf{w}]\in
HK_{\Gamma}|a\not\in\mathfrak{c}[\mathbf{w}]\\}$. Similarly,
$HK_{\Gamma_{i}}=A_{\Gamma_{i}}\cup B_{\Gamma_{i}}$ for $i=1,2$. The function
$p$ in Proposition 4.4 restricts to bijections
$p:A_{\Gamma}\to A_{\Gamma_{1}}\times A_{\Gamma_{2}}\quad\text{ and }\quad
p:B_{\Gamma}\to B_{\Gamma_{1}}\times B_{\Gamma_{2}}.$
By the multiplicative principle we have
$|A_{\Gamma}|=|A_{\Gamma_{1}}|\cdot|A_{\Gamma_{2}}|$ and
$|B_{\Gamma}|=|B_{\Gamma_{1}}|\cdot|B_{\Gamma_{2}}|$.
For $Q\subset\\{a_{2},a_{3},\dots,a_{k}\\}$ and $i=2,\dots,k-1$ define
$X_{i}(Q):=\\{[\mathbf{w}]\in
HK_{\Gamma_{i}}:\mathfrak{c}([\mathbf{w}])\cap\\{a_{i},a_{i+1}\\}=A\cap\\{a_{i},a_{i+1}\\}\\}$
and set
$\displaystyle X_{1}(Q):=\\{[\mathbf{w}]\in
HK_{\Gamma_{1}}:\mathfrak{c}([\mathbf{w}])\cap\\{a_{1}\\}=A\cap\\{a_{1}\\}\\}$
$\displaystyle X_{k}(Q):=\\{[\mathbf{w}]\in
HK_{\Gamma_{k}}:\mathfrak{c}([\mathbf{w}])\cap\\{a_{k}\\}=A\cap\\{a_{k}\\}\\}.$
The the above discussion implies
(4.2)
$|HK_{\Gamma}|=\sum_{Q\subset\underline{k}\setminus\\{1\\}}\prod_{i=1}^{k}|X_{i}(Q)|.$
To compute cardinalities of $X_{i}(Q)$ we have to consider several case. We
start with the case $i\neq 1,k$.
1. (1)
Assume that $i\not\in Q$ and $i+1\not\in Q$. If
$a_{i},a_{i+1}\not\in\mathfrak{c}[\mathbf{w}_{i}]$, then $[\mathbf{w}_{i}]$
can be thought of as living in the smaller HK-monoid
$HK_{\Gamma_{i}\setminus\\{a_{i},a_{i+1}\\}}$, and vice versa. Since
$\Gamma_{i}$ was assumed to be linearly ordered, it will be the case for
$\Gamma_{i}\setminus\\{a_{i},a_{i+1}\\}$ as well. Note that all $\Gamma_{i}$
have length $l_{i}\geq 2$ (if $\Gamma$ has at least two vertices). Therefore
$|HK_{\Gamma_{i}\setminus\\{a_{i},a_{i+1}\\}}|=C_{l_{i}-1}$.
2. (2)
Assume that exactly one of $i,i+1$ is in $Q$. Without loss of generality
assume $i\in A$. Similarly as above we count the number of elements that _do
not_ contain $a_{i+1}$. That cardinality is $C_{l_{i}}$. However, we need to
exclude the elements that do not contain $a_{i}$, leaving us with exactly
$C_{l_{i}}-C_{l_{i}-1}$.
3. (3)
If none of $i,i+1$ is in $Q$, we use the inclusion exclusion formula to get
$C_{l_{i}+1}-2(C_{l_{i}}-C_{l_{i}-1})-C_{l_{i}-1}=C_{l_{i}+1}-C_{l_{i}}+C_{l_{i}-1}$
elements.
When $i=1$ or $i=k$, we get the following two cases:
1. (1)
Elements that do not contain $a=a_{2}$ or $a_{k}$. There are $C_{l_{i}}$ such
elements.
2. (2)
Elements that do contain $a$. By exclusion there are $C_{l_{i}+1}-C_{l_{i}}$
such elements.
Now the second claim of our theorem follows from (4.2) and the definition of
$c_{i}(A)$.
We will prove the last claim by showing that it satisfies the same recursion
formula as the odd Fibonacci numbers and has the same initial values. Note
that if the number of vertices $n$ is fixed there are only two possibilities
for a graph of type $A_{n}$ to have alternating sinks or sources (the first
vertex can either be a sink or a source). However, these graphs are opposite
to each other, so the cardinalities of the corresponding HK-monoids have to
agree by [3, Theorem 1(v)]. Let $\mathcal{A}_{n}$ be the graph of type $A_{n}$
with alternating sinks and sources whose vertices are $v_{i}$,
$i\in\underline{n}$, and we assume that $v_{1}$ is a source. Let
$f_{n}=|HK_{\mathcal{A}_{n}}|$. The first two $\mathcal{A}_{n}$ are
$\mathcal{A}_{0}=$ the empty graph, and $\mathcal{A}_{1}=v_{1}$. This gives
$f_{0}=|\\{\varepsilon\\}|=1=F_{2\cdot 0+1}$ and
$f_{1}=|\\{\varepsilon,[v_{1}]\\}|=2=F_{2\cdot 1+1}$. The Fibonacci numbers
satisfy the recursion formula
$F_{n+2}=F_{n+1}+F_{n}=2F_{n}+F_{n-1}=3F_{n}-F_{n-2}.$
Thus we want to show that $f_{n+1}=3f_{n}-f_{n-1}$. We can separate the
elements of $HK_{\mathcal{A}_{n+1}}$ into two cases. Either the content of an
element contains $v_{n}$ or it does not.
1. (1)
Assume $v_{n}\not\in\mathfrak{c}[\mathbf{w}]$, then $[\mathbf{w}]$ equals an
element from $HK_{\mathcal{A}_{n-1}}$ multiplied with either $\varepsilon$ or
$v_{n+1}$. Thus there are $f_{n-1}\cdot 2=2f_{n-1}$ such elements.
2. (2)
Assume $v_{n}\in\mathfrak{c}[\mathbf{w}]$, then $[\mathbf{w}]$ is a product of
an element from $HK_{\mathcal{A}_{n}}$ _containing_ $v_{n}$ with an element
from $HK_{\mathcal{B}}$ _containing_ $v_{n}$, where $\mathcal{B}$ is the full
subgraph of $\mathcal{A}_{n+1}$ with vertices $\\{v_{n},v_{n+1}\\}$. We have
$(f_{n}-f_{n-1})\cdot 3=3f_{n}-3f_{n-1}$ such elements.
This implies $f_{n+1}=3f_{n}-f_{n-1}$ and completes the proof of our theorem.
∎
The sequence $F_{2n+1}$ was guessed with the help of [1] and has been found
independently by Grensing [4].
###### Corollary 4.9.
Let $\Gamma$ be of type $A_{n}$. Then the number of multiplicity free elements
in $HK_{\Gamma}$ is the Fibonacci number $F_{2n+1}$. In particular, it does
not depend on the orientation of edges in $\Gamma$.
###### Proof.
Lemma 3.4 tells us that every element $[\mathbf{w}]$ contains a word
$\mathbf{w}^{\prime}$ which is multiplicity free _with respect to every source
and sink_. Since $\mathcal{A}_{n}$ has only sources and sinks every element of
$HK_{\mathcal{A}_{n}}$ is in fact multiplicity free. Since
$|HK_{\mathcal{A}_{n}}|=F_{2n+1}$ by the previous theorem, we need to show
that there is a bijection between multiplicity free elements of $HK_{\Gamma}$
and $HK_{\mathcal{A}_{n}}$. Let $\mathbf{w}$ be a multiplicity free word and
assume that $\Gamma$ is enumerated canonically. Then for each
$i\in\underline{n-1}$ exactly one of the following holds.
1. (1)
$v_{i+1}\not\in\mathfrak{c}[\mathbf{w}]$.
2. (2)
$v_{i}\not\in\mathfrak{c}[\mathbf{w}]$ but
$v_{i+1}\in\mathfrak{c}[\mathbf{w}]$.
3. (3)
Both $v_{i},v_{i+1}\in\mathfrak{c}[\mathbf{w}]$ and $v_{i}$ appears before
$v_{i+1}$ in $\mathbf{w}$.
4. (4)
Both $v_{i},v_{i+1}\in\mathfrak{c}[\mathbf{w}]$ and $v_{i+1}$ appears before
$v_{i}$ in $\mathbf{w}$.
We claim that these properties do not depend on the choice of a _multiplicity
free_ element $\mathbf{w}^{\prime}\in[\mathbf{w}]$. For the first two
properties the claim is obvious. Let $\Gamma_{i}$ be the complete subgraph of
$\Gamma$ whose vertices are $v_{i},v_{i+1}$, and consider the map
$p:\big{(}V(\Gamma)\big{)}^{*}\to\big{(}V(\Gamma_{i})\big{)}^{*}$ defined by
deletion of vertices not in $\Gamma_{i}$. It is clear that
1. (a)
If $\mathbf{w}$ is multiplicity free, then so is $p(\mathbf{w})$.
2. (b)
If $\mathbf{w}\sim\mathbf{w}^{\prime}$ then $p(\mathbf{w})\sim
p(\mathbf{w}^{\prime})$.
This means that under $p$ all multiplicity free words that contain both
$v_{i}$ and $v_{i+1}$ are mapped to $v_{i}v_{i+1}$ or $v_{i+1}v_{i}$. However,
there is an edge between $v_{i}$ and $v_{i+1}$, so $v_{i}v_{i+1}\not\sim
v_{i+1}v_{i}$, proving our claim for the third and the fourth properties.
We will show that multiplicity free elements are uniquely defined by the
relative positions of $v_{i}$ and $v_{i+1}$ for each $i$. Set
$\mathbf{w}_{1}=\begin{cases}v_{1}&\text{ if
}v_{1}\in\mathfrak{c}[\mathbf{w}];\\\ \varepsilon&\text{ if
}v_{1}\not\in\mathfrak{c}[w];\end{cases}$
$\mathbf{w}_{i+1}=\begin{cases}\mathbf{w}_{i}&\text{ if
}v_{i+1}\not\in\mathfrak{c}[\mathbf{w}];\\\ \mathbf{w}_{i}v_{i+1}&\text{ if
}v_{i}\not\in\mathfrak{c}[\mathbf{w}]\text{ or }v_{i}\text{ appears to the
left of }v_{i+1}\text{ in }\mathbf{w};\\\ v_{i+1}\mathbf{w}&\text{ if
}v_{i+1}\text{ appears to the left of }v_{i}\text{ in }\mathbf{w}.\end{cases}$
Let $\mathfrak{M}\subset HK_{\Gamma}$ denote the set of all multiplicity free
elements and define the map $\phi:\mathfrak{M}\to HK_{\mathcal{A}_{n}}$ by
$\phi([\mathbf{w}])=[\mathbf{w}_{n}]$, where $\mathbf{w}$ is multiplicity free
and $\mathbf{w}_{n}$ is defined from $\mathbf{w}$ by the above above. Note
that $\phi$ is well-defined because it only uses invariant properties. Since
$\phi:HK_{\mathcal{A}_{n}}\to HK_{\mathcal{A}_{n}}$ is the identity and the
multiplicity free _words_ are the same for all $\Gamma$ (over the same
vertices) it follows that $\phi$ is surjective for any domain. To show that
$\phi$ is injective it suffices to show that $\mathbf{w}\sim\mathbf{w}_{n}$
(in $HK_{\Gamma}$) for each multiplicity free $\mathbf{w}$. We show this by
deforming $\mathbf{w}$ into $\mathbf{w}_{n}$. Clearly $\mathbf{w}$ can be
factorized as $\mathbf{w}=\mathbf{x}_{1}\mathbf{w}_{1}\mathbf{y}_{1}$, where
$v_{1}\not\in\mathfrak{c}([\mathbf{x}_{1}]),\mathfrak{c}([\mathbf{y}_{1}])$.
Assume that $\mathbf{w}\sim\mathbf{x}_{i}\mathbf{w}_{i}\mathbf{y}_{i}$ where
$\\{v_{1},\cdots,v_{i}\\}\cap\big{(}\mathfrak{c}([\mathbf{x}_{i}])\cup\mathfrak{c}([\mathbf{y}_{i}])\big{)}=\emptyset$.
Note that $\mathbf{w}_{i}$ commutes with every vertex _except_ $v_{i+1}$.
1. (1)
If $v_{i+1}\not\in\mathfrak{c}([\mathbf{w}])$, then
$\mathbf{w}_{i+1}=\mathbf{w}_{i}$.
2. (2)
If $v_{i}\not\in\mathfrak{c}([\mathbf{w}])$ but
$v_{i+1}\in\mathfrak{c}([\mathbf{w}])$ then $\mathbf{w}_{i}$ commutes with
every vertex and we may move it so that it ends up just to the left of
$v_{i+1}$.
3. (3)
If both $v_{i},v_{i+1}\in\mathfrak{c}([\mathbf{w}])$ and $v_{i}$ appears
before $v_{i+1}$ in $\mathbf{w}$ then $\mathbf{w}_{i}$ commutes with every
vertex in $\mathbf{y}_{i}$ preceding $v_{i+1}$. Hence we may move
$\mathbf{w}_{i}$ so that it ends up just to the left of $v_{i+1}$.
4. (4)
If both $v_{i},v_{i+1}\in\mathfrak{c}([\mathbf{w}])$ and $v_{i+1}$ appears
before $v_{i}$ in $\mathbf{w}$ we move $\mathbf{w}_{i}$ so that it ends up
just to the right of $v_{i+1}$.
In all cases we find that
$\mathbf{w}\sim\mathbf{x}_{i}\mathbf{w}_{i}\mathbf{y}_{i}\sim\mathbf{x}_{i+1}\mathbf{w}_{i+1}\mathbf{y}_{i+1}$
for some $\mathbf{x}_{i+1}$ and $\mathbf{y}_{i+1}$ such that
$\\{v_{1},\cdots,v_{i},v_{i+1}\\}\cap\big{(}\mathfrak{c}([\mathbf{x}_{i+1}])\cup\mathfrak{c}([\mathbf{y}_{i+1}])\big{)}=\emptyset$.
Proceeding inductively, we get
$\mathbf{w}\sim\mathbf{x}_{n}\mathbf{w}_{n}\mathbf{y}_{n}$ such that
$\mathfrak{c}([\mathbf{x}_{n}])=\mathfrak{c}([\mathbf{y}_{n}])=\emptyset$, so
$\mathbf{w}\sim\mathbf{w}_{n}$. ∎
To illustrate the process that turns $\mathbf{w}$ into $\mathbf{w}_{n}$ let
$\mathbf{w}=cfadb$ (here the order of letters is
$a=v_{1},b=v_{2},c=v_{3},d=v_{4},e=v_{5},f=v_{6}$).
$\mathbf{x}_{1}\mathbf{w}_{1}\mathbf{y}_{1}=(cf)(a)(db)$ | $\mathbf{x}_{2}\mathbf{w}_{2}\mathbf{y}_{2}=(cfd)(ab)()$ | $\mathbf{x}_{3}\mathbf{w}_{3}\mathbf{y}_{3}=()(cab)(fd)$
---|---|---
$\mathbf{x}_{4}\mathbf{w}_{4}\mathbf{y}_{4}=(f)(cabd)()$ | $\mathbf{x}_{5}\mathbf{w}_{5}\mathbf{y}_{5}=(f)(cabd)()$ | $\mathbf{x}_{6}\mathbf{w}_{6}\mathbf{y}_{6}=()(cabdf)()$
As a consequence of the previous corollary we have $F_{2n+1}\leq|HK_{\Gamma}|$
for any $\Gamma$. We also have $|HK_{\Gamma}|\leq C_{n+1}$ by [3, Theorem
1(vi)]. It seems plausible that more sources and sinks corresponds to a
smaller monoid. The following theorem makes this idea more precise.
###### Theorem 4.10.
Let $\Gamma=\Gamma_{1}\cup\Gamma_{2}$ be an edge disjoint union of graphs such
that $\Gamma_{1}\cap\Gamma_{2}=\\{a\\}$ is a source or a sink. Let
$\overleftarrow{\Gamma_{2}}$ be the graph obtained from $\Gamma_{2}$ by
reversing the direction of all edges and let
$\tilde{\Gamma}=\Gamma_{1}\cup\overleftarrow{\Gamma_{2}}$. Then
$|HK_{\Gamma}|\leq|HK_{\tilde{\Gamma}}|$ and the equality holds if and only if
$a$ is isolated in at least one of $\Gamma_{1},\Gamma_{2}$.
###### Proof.
For any HK-monoid $HK_{\Gamma^{\prime}}$ and $a\in V(\Gamma^{\prime})$ define
$HK_{\Gamma^{\prime}}^{0}=\\{[\mathbf{w}]\in
HK_{\Gamma^{\prime}}|a\not\in\mathfrak{c}[\mathbf{w}]\\},$
$HK_{\Gamma^{\prime}}^{1}=\\{[\mathbf{w}]\in
HK_{\Gamma^{\prime}}|a\in\mathfrak{c}[\mathbf{w}]\text{ and
}[\mathbf{w}]\text{ is multiplicity free with respect to }a\\},$
$HK_{\Gamma^{\prime}}^{2}=\\{[\mathbf{w}]\in
HK_{\Gamma^{\prime}}|[\mathbf{w}]\text{ is not multiplicity free with respect
to }a\\}.$
Clearly $HK_{\Gamma^{\prime}}$ is a disjoint union of
$HK_{\Gamma^{\prime}}^{0}$, $HK_{\Gamma^{\prime}}^{1}$ and
$HK_{\Gamma^{\prime}}^{2}$.
Because $|HK_{\Gamma^{\prime}}|=|HK_{\overleftarrow{\Gamma^{\prime}}}|$ by [3,
Theorem 1(v)], it follows that
$|HK_{\Gamma}^{0}|=|HK_{\Gamma_{1}\setminus\\{a\\}}||HK_{\Gamma_{2}\setminus\\{a\\}}|=|HK_{\Gamma_{1}\setminus\\{a\\}}||HK_{\overleftarrow{\Gamma_{2}\setminus\\{a\\}}}|=|HK_{\tilde{\Gamma}}^{0}|.$
To see that $|HK_{\Gamma}^{1}|=|HK_{\tilde{\Gamma}}^{1}|$ observe that any
element $[\mathbf{w}]$ in $HK_{\Gamma}^{1}$ has a word of the form
$\mathbf{w}=\mathbf{x}_{1}\mathbf{y}_{1}a\mathbf{x}_{2}\mathbf{y}_{2}$ for
some $\mathbf{x}_{1},\mathbf{x}_{2}\in
HK_{\Gamma_{1}\setminus\\{a\\}},\mathbf{y}_{1},\mathbf{y}_{2}\in
HK_{\Gamma_{2}\setminus\\{a\\}}$. Similarly for $HK_{\tilde{\Gamma}^{1}}$.
This is true because there are no edges between $\Gamma_{1}\setminus\\{a\\}$
and $\Gamma_{2}\setminus\\{a\\}$. Let $\phi:HK_{\Gamma}^{1}\to
HK_{\tilde{\Gamma}}^{1}$ be defined by
$\phi[\mathbf{x}_{1}\mathbf{y}_{1}a\mathbf{x}_{2}\mathbf{y}_{2}]=\mathbf{x}_{1}\overleftarrow{\mathbf{y}_{2}}a\mathbf{x}_{2}\overleftarrow{\mathbf{y}_{1}}$,
where $\overleftarrow{\mathbf{y}}$ is the reverse of $\mathbf{y}$. Note that
$\phi$ is a bijection if it is well-defined. To see that it is well-defined,
assume
$\mathbf{x}_{1}\mathbf{y}_{1}a\mathbf{x}_{2}\mathbf{y}_{2}\sim\mathbf{x}_{1}^{\prime}\mathbf{y}_{1}^{\prime}a\mathbf{x}_{2}^{\prime}\mathbf{y}_{2}^{\prime}$.
By taking the right projection morphisms (onto $HK_{\Gamma_{1}}$ and
$HK_{\Gamma_{2}}$, respectively) we obtain
$\mathbf{x}_{1}a\mathbf{x}_{2}\sim\mathbf{x}_{1}^{\prime}a\mathbf{x}_{2}^{\prime}$
and
$\mathbf{y}_{1}a\mathbf{y}_{2}\sim\mathbf{y}_{1}^{\prime}a\mathbf{y}_{2}^{\prime}$.
Note that
$\mathbf{y}\sim\mathbf{y}^{\prime}\iff\overleftarrow{\mathbf{y}}\sim\overleftarrow{\mathbf{y}^{\prime}}$.
We have:
(4.3)
$\phi(\mathbf{w})=\phi(\mathbf{x}_{1}\mathbf{y}_{1}a\mathbf{x}_{2}\mathbf{y}_{2})=\mathbf{x}_{1}\overleftarrow{\mathbf{y}_{2}}a\mathbf{x}_{2}\overleftarrow{\mathbf{y}_{2}}\sim\mathbf{x}_{1}\overleftarrow{\mathbf{y}_{2}}a\overleftarrow{\mathbf{y}_{2}}\mathbf{x}_{2}=\mathbf{x}_{1}\overleftarrow{\mathbf{y}_{1}a\mathbf{y}_{2}}\mathbf{x}_{2}\sim\\\
\mathbf{x}_{1}\overleftarrow{\mathbf{y}_{1}^{\prime}a\mathbf{y}_{2}^{\prime}}\mathbf{x}_{2}=\mathbf{x}_{1}\overleftarrow{\mathbf{y}_{2}^{\prime}}a\overleftarrow{\mathbf{y}_{2}^{\prime}}\mathbf{x}_{2}\sim\overleftarrow{\mathbf{y}_{2}^{\prime}}\mathbf{x}_{1}a\mathbf{x}_{2}\overleftarrow{\mathbf{y}_{2}^{\prime}}\sim\overleftarrow{\mathbf{y}_{2}^{\prime}}\mathbf{x}_{1}^{\prime}a\mathbf{x}_{2}^{\prime}\overleftarrow{\mathbf{y}_{2}^{\prime}}\sim\mathbf{x}_{1}^{\prime}\overleftarrow{\mathbf{y}_{2}^{\prime}}a\mathbf{x}_{2}^{\prime}\overleftarrow{\mathbf{y}_{2}^{\prime}}=\phi(\mathbf{w}^{\prime}).$
This implies that $|HK_{\Gamma}^{1}|=|HK_{\tilde{\Gamma}}^{1}|$.
Finally, because $a\in\Gamma$ is a source or a sink, the set $HK_{\Gamma}^{2}$
is empty. Thus we have established the inequality
$|HK_{\Gamma}|\leq|HK_{\tilde{\Gamma}}|$. This inequality is an equality
precisely if and only if $HK_{\tilde{\Gamma}}^{2}$ is empty. If there exist
$b,c$ such that $b\to a\to c$, then $[abca]\in HK_{\tilde{\Gamma}}^{2}$. This
can only happen if $a$ is connected to some $b^{\prime}\in HK_{\Gamma_{1}}$
and some $c^{\prime}\in HK_{\Gamma_{2}}$. On the other hand, if $a$ is not
connected to any other vertex in either $\Gamma_{1}$ or $\Gamma_{2}$, then $a$
is a source or a sink in $\tilde{\Gamma}$ as well, implying that
$HK_{\tilde{\Gamma}}^{2}$ is empty. This completes the proof. ∎
## 5\. Limits of the method
The main idea of taking smaller graphs, whose HK-monoids have known effective
representations, and gluing them together has a few limitations
1. (1)
We only know effective representations for a few types of graphs _that
themselves are not reached in this way_. So far we have the linearly ordered
graphs of type $A_{n}$ and the graphs $\kappa_{n}$ of the Kiselman monoids.
2. (2)
When we glue two subgraphs together we are limited to gluing sinks to sinks
and sources to sources.
3. (3)
When we glue two subgraphs together the intersection must consist of one
vertex. In particular this prevents us from forming loops.
We may enlarge the set of “building blocks” by considering the following
family of graphs:
|
---|---
$\textstyle{Z_{n}=}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdots}$$\textstyle{v_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdots}$$\textstyle{v_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{b}$
We can get a complete description of the elements in $HK_{Z_{n}}$ by observing
that, since $a$ is a source, any element contains a word with at most one $a$.
The subwords before and after $a$ do not contain $a$ and could therefore be
thought of as living in $\big{(}V(Z_{n}\setminus\\{a\\})\big{)}^{*}$. However,
in $Z_{n}\setminus\\{a\\}$ every vertex is a sink or a source, so we may
assume that there is at most one $v_{i}$ on each side of $a$. A similar
argument holds for $b$. Thus each element $[\mathbf{w}]$ in $HK_{Z_{n}}$
contains a word $\mathbf{w}^{\prime}$ which falls into exactly one of the
following types:
1. (1)
We have $a,b\not\in\mathfrak{c}(\mathbf{w}^{\prime})$ and
$\mathbf{w}^{\prime}$ multiplicity free. Since all $v_{i}$ commute with each
other we may take $\mathbf{w}^{\prime}=v_{i_{1}}v_{i_{1}}\cdots v_{i_{j}}$
with $i_{1}<i_{2}<\cdots<i_{j}$.
2. (2)
We have
$a\in\mathfrak{c}[\mathbf{w}^{\prime}],b\not\in\mathfrak{c}[\mathbf{w}^{\prime}]$
and $\mathbf{w}^{\prime}=\mathbf{w}_{1}a\mathbf{w}_{2}$ is multiplicity free.
Similarly as in the first case, $\mathbf{w}_{1},\mathbf{w}_{2}$ may be taken
with internally increasing order.
3. (3)
We have
$a\not\in\mathfrak{c}[\mathbf{w}^{\prime}],b\in\mathfrak{c}[\mathbf{w}^{\prime}]$
and $\mathbf{w}^{\prime}=\mathbf{w}_{1}b\mathbf{w}_{2}$ is multiplicity free.
As in the second case, $\mathbf{w}_{1},\mathbf{w}_{2}$ may be taken with
internally increasing order.
4. (4)
We have $a,b\in\mathfrak{c}[\mathbf{w}^{\prime}]$ and
$\mathbf{w}^{\prime}=\mathbf{w}_{1}a\mathbf{w}_{2}b\mathbf{w}_{3}$. We may
take $\mathbf{w}^{\prime}$ such that
$\mathfrak{c}(w_{1})\cap\mathfrak{c}(w_{2})=\emptyset$ and
$\mathfrak{c}(w_{2})\cap\mathfrak{c}(w_{3})=\emptyset$ and
$\mathbf{w}_{1},\mathbf{w}_{2}$ and $\mathbf{w}_{3}$ are internally
increasing.
5. (5)
We have $a,b\in\mathfrak{c}[\mathbf{w}^{\prime}]$ and
$\mathbf{w}^{\prime}=\mathbf{w}_{1}b\mathbf{w}_{2}a\mathbf{w}_{3}$ with the
same restrictions as in the previous case and, additionally,
$\mathbf{w}_{2}\neq\varepsilon$.
The proof of the next theorem shows that all these elements are, in fact,
different.
###### Theorem 5.1.
Let $\Gamma=Z_{n}$ and let $f$ be defined by $f_{av_{i}}=1,f_{v_{i}b}=2^{i}$
for all $i$. Then $R_{f}$ is an effective representation of $HK_{Z_{n}}$.
###### Proof.
First we note that $R_{f}([\mathbf{w}])(x)=x$ if and only if
$x\not\in\mathfrak{c}(\mathbf{w})$. This means that $R_{f}$ distinguishes
elements with different contents from each other. Assume first that the
content of $\mathbf{w}$ satisfies $a\not\in\mathfrak{c}(\mathbf{w})$ or
$b\not\in\mathfrak{c}(\mathbf{w})$. Then $[\mathbf{w}]$ belongs to some
$HK_{\Gamma^{\prime}}$ for which $\Gamma^{\prime}$ is a path connected
subgraph of $\Gamma$ and is a finite edge-disjoint union of graphs of type
$A_{2}$. Since $\Gamma$ has no oriented cycles, there is an
$HK_{\Gamma^{\prime}}$-subquotient of $R_{f}$ isomorphic, as a vector space,
to $\oplus_{v\in V(\Gamma^{\prime})}\mathcal{R}v$ with the induced action
(i.e. everything outside this space is treated as zero). From the results of
the previous section it follows that the restriction of $R_{f}$ to the action
of $HK_{\Gamma^{\prime}}$ on $\oplus_{v\in V(\Gamma^{\prime})}\mathcal{R}v$ is
effective and hence separates different elements with content
$\mathfrak{c}(\mathbf{w})$.
If $a,b\in\mathfrak{c}(\mathbf{w})$ but some
$v_{i}\not\in\mathfrak{c}(\mathbf{w})$, then we can use induction on $n$.
Without loss of generality we may assume $i=n$. In this case $[\mathbf{w}]$
belongs to $HK_{Z_{n-1}}$. Considering the induced action of $HK_{Z_{n-1}}$ on
$\oplus_{v\in V(Z_{n-1})}\mathcal{R}v$ and taking the quotient modulo the
invariant subspace $\mathcal{R}v_{n}$, we may use the induction hypothesis and
conclude that $R_{f}$ separates elements with such content. Notre that the
basis of the induction is the case of a graph of type $A_{3}$ which is dealt
with using the results from the previous section.
Finally, it remains to show that $R_{f}$ separates elements of the maximal
content, that is elements described in the last two cases (4) and (5) before
the theorem with the condition
$\mathfrak{c}(\mathbf{w}_{1})\cup\mathfrak{c}(\mathbf{w}_{2})\cup\mathfrak{c}(\mathbf{w}_{3})=\\{v_{3},v_{4},\dots,v_{n}\\}$.
A direct computation shows that the element
$\mathbf{w}_{1}a\mathbf{w}_{2}b\mathbf{w}_{3}$ from (4) acts on the basis
elements of $\oplus_{v\in V(\Gamma)}\mathcal{R}v$ as follows:
$\displaystyle b\mapsto 0;\quad\quad
a\mapsto\sum_{v_{i}\not\in\mathfrak{c}(\mathbf{w}_{1})}f_{av_{i}}v_{i}+\sum_{v_{i}\in\mathfrak{c}(\mathbf{w}_{1})}f_{av_{i}}f_{v_{i}b}b;$
$\displaystyle\mathfrak{c}(\mathbf{w}_{3})\ni v_{i}\mapsto
0;\quad\quad\mathfrak{c}(\mathbf{w}_{2})\ni v_{i}\mapsto
f_{v_{i}b}b;\quad\quad\mathfrak{c}(\mathbf{w}_{1})\setminus\mathfrak{c}(\mathbf{w}_{3})\ni
v_{i}\mapsto f_{v_{i}b}b.$
Similarly, the element $\mathbf{w}_{1}b\mathbf{w}_{2}a\mathbf{w}_{3}$ from (5)
acts as follows:
$\displaystyle b\mapsto 0;\quad\quad
a\mapsto\sum_{v_{i}\not\in\mathfrak{c}(\mathbf{w}_{1})\setminus\mathfrak{c}(\mathbf{w}_{2})}f_{av_{i}}v_{i}+\sum_{v_{i}\in\mathfrak{c}(\mathbf{w}_{1})\setminus\mathfrak{c}(\mathbf{w}_{2})}f_{av_{i}}f_{v_{i}b}b;$
$\displaystyle\mathfrak{c}(\mathbf{w}_{3})\ni v_{i}\mapsto
0;\quad\quad\mathfrak{c}(\mathbf{w}_{2})\ni v_{i}\mapsto
0;\quad\quad\mathfrak{c}(\mathbf{w}_{1})\setminus\mathfrak{c}(\mathbf{w}_{3})\ni
v_{i}\mapsto f_{v_{i}b}b.$
Comparing these two formulae it is easy to see that different words act
differently, in particular, they define different elements in $HK_{Z_{n}}$. ∎
We can describe the graphs whose HK-monoids we have constructed effective
representations for. We do it in three steps.
1. (1)
Pick a forest $\Gamma^{\prime}$ whose edges are undirected.
2. (2)
Direct every edge such that every vertex in $\Gamma^{\prime}$ becomes either a
source or a sink. Call the resulting graph $\Gamma^{\prime\prime}$
3. (3)
Replace every edge in $\Gamma^{\prime\prime}$ independently with one of the
following:
1. (a)
a linearly ordered graph of type $A_{n},n\geq 1$,
2. (b)
a "Kiselman graph" $\kappa_{n},n\geq 2$,
3. (c)
or some $Z_{n}$ for $n\geq 4$,
matching source to source and sink to sink. Call the resulting graph $\Gamma$.
By choosing any weights on the linearly ordered parts, weights according to
[7] on the "Kiselman graph"-parts and weights according to the previous
theorem on the $Z_{n}$-parts one obtains an effective representation of
$HK_{\Gamma}$ over any ground ring $\mathcal{R}\supset\mathbb{Z}$.
###### Example.
A graph that we can reach with the three step algorithm
1. (1)
Pick a forest:
| |
---|---|---
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
2. (2)
Direct the edges so that vertices become _only_ sources and sinks:
| |
---|---|---
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
3. (3)
Replace edges with linearly ordered $A_{n}$-graphs, graphs $\kappa_{n}$ or
graphs $Z_{n}$:
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
## References
* [1] The on-line encyclopedia of integer sequences.
* [2] R. Aragona and A. D’Andrea. Hecke-kiselman monoids of small cardinality. Semigroup Forum, 86(1):32–40, 2013.
* [3] O. Ganyushkin and V. Mazorchuk. On kiselman quotients of 0-hecke monoids. International Electronic Journal of Algebra, 10:174–191, 2011.
* [4] A.-L. Grensing. Monoid algebras of projection functors. Journal of Algebra, 369(1):16–41, 2012.
* [5] A.-L. Grensing and V. Mazorchuk. Categorification of the Catalan monoid. Preprint arXiv:1211.2597. Semigroup Forum (to appear).
* [6] C. Kiselman. A semigroup of operators in convexity theory. Transactions of the American Mathematical Society, 354(5):2035–2053, 2002.
* [7] G. Kudryavtseva and V. Mazorchuk. On kiselman’s semigroup. Yokohama Math. J., 55(1):21–46, 2009.
* [8] V. Mazorchuk and B. Steinberg. Effective dimension of finite semigroups. Journal of Pure and Applied Algebra, 216(12):2737–3753, Dec. 2012\.
* [9] A.-L. Paasch. Monoidalgebren von Projektionsfunktoren. PhD thesis, Bergischen Universität Wuppertal, 2011.
|
arxiv-papers
| 2012-05-03T11:21:19 |
2024-09-04T02:49:30.484252
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Love Forsberg",
"submitter": "Love Forsberg",
"url": "https://arxiv.org/abs/1205.0676"
}
|
1205.0827
|
# Data Analysis of Gravitational Wave Signals from Millisecond Pulsars
Fernanda G. Oliveira Rubens M. Marinho Jr and Jaziel G. Coelho Instituto
Tecnológico de Aeronáutica, Praça Marechal Eduardo Gomes 50, São José dos
Campos, SP 12228-900, Brazil Nadja S. Magalhaes Universidade Federal de São
Paulo, DCET, Rua São Nicolau 210, Diadema, SP 09913-030, Brazil
(Day Month Year; Day Month Year)
###### Abstract
The present work is devoted to the detection of monochromatic gravitational
wave signals emitted by pulsars using ALLEGRO’s data detector. We will present
the region (in frequency) of millisecond pulsars of the globular cluster 47
Tucanae (NGC 104) in the band of detector. With this result it was possible to
analyse the data in the frequency ranges of the pulsars J1748-2446L and
J1342+2822c, searching for annual Doppler variations using power spectrum
estimates for the year 1999. We tested this method injecting a simulated
signal in real data and we were able to detect it.
###### keywords:
Data analysis; Gravitational Waves; Pulsar
PACS numbers: 11.25.Hf, 123.1K
## 1 Introduction
The focus of this work was to analyse ALLEGRO’s data[[1]] for the year 1999
taking into account the effect due to the orbital motion of the Earth for
specific frequencies, $891.0$ Hz and $923.4$ Hz, that correspond to the
pulsars located in 47 Tucanae (NGC 104) named 1748-2446L and J1342+2822c,
respectively. This analysis was based on estimates of power spectrum of the
data using averaged modified periodograms[[2]] which reinforce the presence of
peaks due to monochromatic signals.
## 2 The Characteristic Amplitude of a Pulsar’s Gravitational Waves
Pulsars with non-axisymmetric rotation are expected to emit monochromatic
gravitational wave signals (MGW). The amplitude of gravitational waves (GW)
emitted by a rotating neutron star (NS) can be expressed in terms of the NS
rotation period $P$, the distance to the Earth $r$, the moment of inertia $I$
and the ellipticity $\epsilon$ resulting from the distortion process as[[3]]:
$h_{c}=4.21\times 10^{-4}\left(\frac{\rm ms}{P}\right)^{2}\left(\frac{\rm
kpc}{r}\right)\left(\frac{I}{10^{38}\rm
kgm^{2}}\right)^{2}\left(\frac{\epsilon}{10^{-6}}\right).$ (1)
Its value depends on the physical mechanism that makes the star non-
axisymmetric and is highly uncertain. The values of $h_{c}$ resulting from Eq.
(1) are show in the Figure 1 for millisecond pulsars in 47 Tucanae.
## 3 Determination of the Observation Time
In the present analysis we are interested only in the annual Doppler shift
that a monochromatic, continuous gravitational wave signal should experience,
so we need to choose an observation time, $\Delta t$, such that the diurnal
Doppler shift, $\Delta\nu_{d}$, remains in the same frequency bin $\Delta
f=1/\Delta t$. The minimum size of this bin, $\Delta f_{min}$, corresponds to
the maximum diurnal Doppler shift, $\Delta\nu_{dmax}=\Delta f_{min}=1/\Delta
t_{max}$. The maximum diurnal Doppler shift happens when the Earth and the
star are in the line of the nodes[4]:
$\Delta\nu_{dmax}=\nu_{s}\frac{2wr}{c},$ (2)
where $w$ and $r$ are the angular velocity of rotation and the radius of the
Earth, respectively. The annual Doppler shift in a full year of observation
is,
$\Delta\nu_{a}=\nu_{s}\frac{2R\Omega}{c},$ (3)
where $R$ is the average radius of the Earth’s orbit around the Sun and
$\Omega$ is its angular velocity in this orbital motion. The values of the
Doppler shifts obtained for the pulsars radiating with frequency $\nu_{s}$ are
given in Table 1.
Table 1: Doppler shifts and observation times for two pulsars in 47 Tucanae
expected to emit gravitational waves in the frequencies $\nu_{s}$. Pulsar
$\nu_{s}$ $\Delta\nu_{dmax}$ $\Delta\nu_{a}$ $\Delta t_{max}$ PSR J1748-2446L
891.0 Hz 2.8 mHz $0.1770$ Hz 362 s PSR J1342+2822c 923.4 Hz 2.9 mHz $0.1835$
Hz 350 s
In order to eliminate the daily Doppler shift we used $\Delta t=300$ s for the
observation time in our data analysis.
Figure 1: The figure shows the gravitational strain $h_{EC}$ (EC stands for
Energy Conservation or spin-down limit) for the known pulsars in 47 Tucanae
(NGC 104). The strain $h_{EC}$ is compared to the strain-noise spectrum of the
detector ALLEGRO (in units of $h/\sqrt{\rm Hz}$).
## 4 The Strain-Noise Sensitivity of ALLEGRO
In Figure 1 we present the region of MGW signals from pulsars in the strain-
noise spectrum of ALLEGRO. This figure shows the gravitational strain $h_{EC}$
for the known pulsars in the band of ALLEGRO from the ATNF catalog. This
quantity was derived assuming that all observed spin-down is due to energy
loss caused by emission of gravitational radiation (and no other braking
mechanisms[[5]]). The strain for the pulsars is compared to the noise
sensitivity curve (in units of $h/\sqrt{Hz}$) for the ALLEGRO detector.
Figure 2: Upper plot: Variation of the power spectrum for PSR J1748-2446L.
Lower plot: Variation of the power spectrum for the simulated signal added to
the data.
Figure 3: Upper plot: Variation of the power spectrum for PSR J1342+2822c.
Lower plot: Variation of the power spectrum for the simulated signal added to
the data.
## 5 The Data Analysis
The goal of the analysis was to search for an annual Doppler shift in
ALLEGRO’s data. For this we stablished an observation time $\Delta t=300$ s so
that the frequency of a possible observed signal would not change from one bin
to another during the day. We have taken the power spectral density for 266
days of the year 1999. We fixed our attention on the bins that contained the
frequencies 891.0 Hz and 923.4 Hz that would correspond to GW radiated
respectively by the pulsars J1748-2446L and J1342+2822c, looking for an excess
energy in these bins during those 266 days. We have chosen these two pulsars
because their radiated frequencies were near the frequencies where the
detector is the most sensitive[[1]]. We simulated a GW signal with
dimensionless amplitude $h=2.6\times 10^{-17}$ and added it to ALLEGRO’s data.
The results of this analysis are shown in Figures 2 and 3.
## 6 Conclusions
In this analysis we are not able to identify any Doppler modulation in real
data, as seen from the upper plots in Figures 2 and 3. However, it was
possible to test our data analysis procedure for detection of monochromatic
gravitational wave signals since we were able to notice the simulated signal
buried in the noise (lower plots in Figures 2 and 3). The calculation of the
detection probability using the Neyman-Pearson criterion will be the subject
of a forthcoming paper.
## References
* [1] E. Mauceli., P. M. McHugh., W.O. Hamilton., W.W. Johnson and A. Morse. Phys. Rev. D 54, 1264 (1996).
* [2] F. G. Oliveira et al., Inter. Journ. of Mod. Phys. D 19, 1293 (2010).
* [3] E. Gourgoulhon and S. Bonazzola, Gravitational Waves from isolated neutron stars. ArXiv:astro-ph/9605150v1, (1996).
* [4] F. G. Oliveira et al., Nuclear Physics. B, Proceedings Supplement 199, 353 (2010).
* [5] G. Santostasi, Upper and lower limits on the Crab pulsar’s astrophysical parameters set from gravitational wave observations by LIGO: braking index and energy considerations. ArXiv:gr-qc/0807.2485v1, (2008).
* [6] P. D. Welch, IEEE Trans. and Audio Electroacoust., AU-15, 70 (1970).
|
arxiv-papers
| 2012-05-03T22:14:17 |
2024-09-04T02:49:30.499109
|
{
"license": "Public Domain",
"authors": "F. G. de Oliveira, R. M. Marinho Jr, J. G. Coelho and N. Magalhaes",
"submitter": "Jaziel Goulart Coelho",
"url": "https://arxiv.org/abs/1205.0827"
}
|
1205.0833
|
Version: August 27, 2024
# Epitaxial (111) Films of Cu, Ni, and CuxNiy on $\alpha-$Al2O3(0001) for
Graphene Growth by Chemical Vapor Deposition
David L. Miller, Mark W. Keller, Justin M. Shaw Magnetics Group,
Electromagnetics Division Ann N. Chiaramonti, Robert R. Keller Nanoscale
Reliability Group, Materials Reliability Division National Institute of
Standards and Technology, Boulder, CO 80305 david.miller@nist.gov,
mark.keller@nist.gov Official contribution of the National Institute of
Standards and Technology; not subject to copyright in the United States
###### Abstract
Films of (111)-textured Cu, Ni, and CuxNiy were evaluated as substrates for
chemical vapor deposition of graphene. A metal thickness of $400\,\mathrm{nm}$
to $700\,\mathrm{nm}$ was sputtered onto a substrate of $\alpha-$Al2O3(0001)
at temperatures of $250\,\mathrm{{}^{\circ}C}$ to $650\,\mathrm{{}^{\circ}C}$.
The films were then annealed at $1000\,\mathrm{{}^{\circ}C}$ in a tube
furnace. X-ray and electron backscatter diffraction measurements showed all
films have (111) texture but have grains with in-plane orientations differing
by $60^{\circ}$. The in-plane epitaxial relationship for all films was
$[110]_{\textrm{metal}}$||$[10\bar{1}0]_{\textrm{Al}{}_{2}\textrm{O}_{3}}$.
Reactive sputtering of Al in O2 before metal deposition resulted in a single
in-plane orientation over 97 % of the Ni film but had no significant effect on
the Cu grain structure. Transmission electron microscopy showed a clean
Ni/Al2O3 interface, confirmed the epitaxial relationship, and showed that
formation of the $60^{\circ}$ twin grains was associated with features on the
Al2O3 surface. Increasing total pressure and Cu vapor pressure during
annealing decreased the roughness of Cu and and CuxNiy films. Graphene grown
on the Ni(111) films was more uniform than that grown on polycrystalline
Ni/SiO2 films, but still showed thickness variations on a much smaller length
scale than the distance between grains.
## 1 Introduction
Uniform growth of graphene over wafer-scale areas is a critical enabling step
for the commercial realization of various electronic, photonic, mechanical,
and other devices based upon the superlative properties of graphene [1].
Considerable progress has been made in the growth of monolayer and few-layer
graphene by chemical vapor deposition (CVD) on transition metals, particularly
Cu and Ni [2, 3]. Typically, the metal surface is heated to
$1000\,\mathrm{{}^{\circ}C}$ and exposed to a hydrocarbon gas such as methane
that decomposes catalytically to provide a source of C atoms on the surface.
In the case of Cu, growth occurs almost entirely on the surface and is limited
to a single layer for a wide range of hydrocarbon partial pressure. In the
case of Ni, growth involves C dissolution into the film at high temperature
and then precipitation to the surface as the film cools, which can result in
both monolayer and multilayer graphene [4]. Alloys of Cu and Ni offer control
over graphene thickness by adjusting the C solubility in the film [5]. The Cu
or Ni can be etched away to transfer the graphene layer to an insulating
substrate for further device fabrication steps [6].
Commercially available polycrystalline foils of Cu and Ni are commonly used
for CVD growth of graphene because they can be annealed to have millimeter
grain sizes. The best CVD graphene films to date, in terms of graphene grain
size and carrier mobility, have been achieved on foils [7], but single crystal
films offer several advantages. First, the hexagonal symmetry and small
lattice mismatch of the (111) surface should provide a more ideal template for
graphene growth than the (100), (110), or higher index surfaces. This idea is
supported by recent experiments showing less rotational disorder in graphene
grown on Cu(111) than on Cu(100) [8, 9]. Second, films typically provide a
much smoother growth surface because foils have roughness due to the rolling
process that is not fully removed by annealing. The experiments in [9] show
how surface defects on Cu(111) can cause rotational disorder either at
nucleation or during subsequent growth of graphene islands. Third, the
diffusion of C into and out of Ni is different at grain boundaries than in the
interior of grains, so better growth control and better uniformity are
expected if grain boundaries are eliminated [10, 11]. Lastly, films are
supported by a flat, rigid substrate, helping to simplify the process of
graphene transfer to other materials. Thus we anticipate that high quality
Cu(111) and Ni(111) films will enable CVD growth of graphene having improved
properties compared to graphene grown on polycrystalline foils.
A suitable substrate for the metal films must, at a minimum, promote epitaxial
growth with (111) texture and be physically and chemically stable under
graphene CVD conditions. For wafer-scale production of graphene, the substrate
should be commercially available in wafer form at a reasonable cost. The
ability to reuse substrate wafers after metal etching to release the graphene
layer is also desirable. Currently, the material that appears to best satisfy
these requirements is $\alpha-$Al2O3(0001), which is widely used by
manufacturers of radio-frequency electronics and light-emitting diodes in the
form of wafers with diameters up to $200\,\mathrm{mm}$. Given the advantages
of single crystal Cu(111) and Ni(111) films described above, achieving such
films on Al2O3 is an important step toward commercial graphene devices.
Previous investigations of Cu deposited on $\alpha-$Al2O3(0001), mainly in
ultra-high vacuum environments, found a variety of growth behaviors depending
on how the Al2O3 surface was prepared. Kelber _et al._ [12] summarized several
earlier results, discussed them in terms of multiple Al2O3(0001) surface
terminations and multiple Cu ionization states, and emphasized the role of a
hydroxyl layer bonded to the Al2O3 surface. The last point is particularly
relevant for the work presented here. When Al2O3 is exposed to ambient air,
water decomposes to form a hydroxyl (OH) layer [13] that is difficult to
remove. While bombardment with 200 eV to 1000 eV Ar ions can remove all but
$\sim 0.1\,\mathrm{monolayer}$ of OH, annealing in O2 at temperatures $\geq
800\,\mathrm{{}^{\circ}C}$ is required to recrystallize the damaged surface
[14, 15]. In typical film deposition chambers where such treatments are not
available, an OH layer is likely present on Al2O3 substrates and may interfere
with epitaxial growth. This may be why previous Cu films deposited on
Al2O3(0001) [16, 17, 18] were not single crystals but consisted of (111)
grains having in-plane orientations differing by $60^{\circ}$. While Ni on
$\alpha-$Al2O3(0001) has not been the subject of extensive experimental
studies, the OH layer can also be expected to interfere with epitaxy in this
case. As with Cu, previous Ni films on Al2O3(0001) showed in-plane
orientations differing by $60^{\circ}$ [11].
Epitaxy and film adhesion are relevant to graphene growth for several reasons.
For Cu, which has a bulk melting point of $1084\,\mathrm{{}^{\circ}C}$,
evaporation can cause roughness or even complete loss of the Cu film [19].
Previous work on Cu [18] showed that epitaxial and highly textured films with
large grains survive graphene CVD conditions better than polycrystalline films
with small grains. Ni evaporation under growth conditions is negligible, since
the bulk melting point is $1453\,\mathrm{{}^{\circ}C}$, but Ni films can be
damaged by the annealing in H2 which typically precedes graphene growth [20].
An additional consideration for Ni is that films thinner than
$100\,\mathrm{nm}$ are desirable in order to limit the total amount of C that
is available to precipitate to the surface, since $\approx 35\,\mathrm{nm}$ of
Ni at $1000\,\mathrm{{}^{\circ}C}$ can absorb enough C to form a monolayer
[21]. Such thin Ni films, even when epitaxial and highly textured, can be
damaged by graphene CVD conditions [11]. Given the various factors summarized
here, the properties and limitations of Cu and Ni films must be considered,
alongside the kinetics and thermodynamics of graphene formation from a
hydrocarbon precursor, when optimizing a recipe for graphene growth by CVD.
This paper reports a study of sputtered Cu, Ni, and Cu/Ni films that have been
exposed to typical graphene CVD conditions. We used x-ray diffraction (XRD)
and electron backscatter diffraction (EBSD) to measure crystallinity, atomic
force microscopy (AFM) to measure surface morphology, transmission electron
microscopy (TEM) to image the metal/substrate interface, and optical
microscopy to show film properties over large areas. We also report results
for graphene growth by CVD on these films, characterized by scanning electron
microscopy (SEM) and Raman spectroscopy.
## 2 Film Preparation
Wafers of $\alpha$-Al2O3(0001) were prepared by annealing at
$1000\,\mathrm{{}^{\circ}C}$ in O2 at atmospheric pressure for 24 h to remove
scratches due to polishing and give atomically flat terraces. Chips of
$5\,\mathrm{mm}\textrm{ x }6\,\mathrm{mm}$ were cleaned by ultrasonic
agitation in acetone and isopropanol, mounted on a resistively heated Cu puck,
and placed in a cryopumped vacuum system with a base pressure below
$10^{-5}\,\mathrm{Pa}$ $(10^{-7}\,\mathrm{Torr})$. Metal films were deposited
by dc magnetron sputtering from 76 mm targets of 99.99+ % Cu or Ni in
$0.67\,\mathrm{Pa}$ $(5\,\mathrm{mTorr})$ of Ar. Sputtering powers between
$50\,\mathrm{W}$ and $200\,\mathrm{W}$, corresponding to deposition rates
calibrated using a profilometer of $0.3\,\mathrm{nm/s}$ to $1\,\mathrm{nm/s}$,
produced no significant differences in film properties. For some films, a
“seed layer” of Al2O3 was deposited immediately before the metal by reactive
sputtering from a 76 mm target of 99.999 % Al in a mixture of 40 % O2 in Ar
(both 99.999 %) at a total pressure of $0.67\,\mathrm{Pa}$
$(5\,\mathrm{mTorr})$ and a power of $50\,\mathrm{W}$. The target-to-substrate
distance in all cases was 10 cm.
The use of a reactively sputtered seed layer to improve film adhesion and
epitaxy was motivated by a desire to remove OH without using ion bombardment
and high temperature annealling. Sputtering with high O2 content causes
“resputtering” of the substrate due to bombardment by O– ions generated at the
target and accelerated across the plasma sheath [22]. This results in a low
net deposition rate, $<0.01\,\mathrm{nm/s}$ in our case, and should remove the
OH layer from the substrate. The TEM results discussed below show that the
reactive sputtering did not damage the surface. For the films reported here,
this step was done at a substrate temperature of $650\,\mathrm{{}^{\circ}C}$
to promote crystallinity, but the technique was also successful at
$500\,\mathrm{{}^{\circ}C}$.
Metal film properties were strongly influenced by the substrate temperature
during sputtering. The deposition temperature $T_{\textrm{d}}$ reported here
was measured using a thermocouple clamped to the side of the puck facing away
from the sputter gun. For fixed $T_{\textrm{d}}$ and without a seed layer, the
films did not adhere to the Al2O3 substrate above $\approx
400\,\mathrm{{}^{\circ}C}$ for Cu and $\approx 600\,\mathrm{{}^{\circ}C}$ for
Ni. Most films reported here were deposited at
$T_{\textrm{d}}=275\,\mathrm{{}^{\circ}C}$ for Cu and
$T_{\textrm{d}}=475\,\mathrm{{}^{\circ}C}$ for Ni. Cu/Ni bilayer films were
formed by depositing $50\,\mathrm{nm}$ of Ni and then $350\,\mathrm{nm}$ of
Cu. These films formed a homogeneous alloy upon annealing (see below) and we
refer to them as “Cu-Ni” in the rest of the paper. When the Al2O3 seed layer
was used, better film adhesion allowed higher $T_{\textrm{d}}$. These films
were deposited using a linear ramp of $T_{\textrm{d}}$ during deposition, from
$275\,\mathrm{{}^{\circ}C}$ to $400\,\mathrm{{}^{\circ}C}$ for Cu and from
$475\,\mathrm{{}^{\circ}C}$ to $600\,\mathrm{{}^{\circ}C}$ for Ni.
In order to evaluate survivability, the films were annealed in a hot-wall tube
furnace before characterization. The annealing conditions were similar to
those used for graphene CVD, but without the hydrocarbon precursor: duration
10 min to 15 min, temperature of $1000\,\mathrm{{}^{\circ}C}$, flowing 5 % H2
in Ar at a total pressure of $270\,\mathrm{Pa}$ to $5300\,\mathrm{Pa}$
($2\,\mathrm{Torr}$ to $40\,\mathrm{Torr}$). The temperature of the furnace
was measured using a thermocouple mounted just outside the quartz tube.
## 3 Film Characterization
The overall crystalline structure of the films without a seed layer, as
characterized by XRD using Cu $K\alpha$ radiation with a spot size of several
mm2, is shown in Fig. 1. The $\theta-2\theta$ curves in Fig. 1a, with
intensity plotted on a logarithmic scale, show a peak at $41.6{}^{\circ}$ from
the Al2O3(0001) substrate and a peak corresponding to (111) orientation of the
metal film. These peaks are located at $43.4{}^{\circ}$ for Cu,
$43.8{}^{\circ}$ for Cu-Ni, and $44.6{}^{\circ}$ for Ni. The peaks between
$90{}^{\circ}$ and $100{}^{\circ}$ are higher-order reflections, and the lack
of any other peaks indicates all films are exclusively (111) textured. The
rocking curves in Fig. 1b were fit with a gaussian function, yielding a full-
width-at-half-maximum of 0.334° for Cu, 0.533° for Cu-Ni, and 0.393° for Ni.
The single peak observed for the Cu-Ni alloy film indicates complete mixing of
the two metals during annealing. Its position corresponds to an alloy
composition of Cu70Ni30, whereas the Cu and Ni film thicknesses predict a
composition of Cu87Ni13. We attribute this difference to the preferential
evaporation of Cu during annealing.
Figure 1: Xray diffraction data for $400\,\mathrm{nm}$ films of Cu, Cu-Ni, and
Ni, deposited without Al2O3 seed layer, after annealing at
$1000\,\mathrm{{}^{\circ}C}$ and $270\,\mathrm{Pa}$ ($2\,\mathrm{Torr}$). (a)
$\theta-2\theta$ curves, plotted on a logarithmic intensity scale, show that
all films are exclusively (111) textured. (b) Rocking curves with Gaussian
fits yielding full-width-at-half-maximum of 0.334° for Cu, 0.533° for Cu-Ni,
and 0.393° for Ni. (c, d, e) (111) pole figures plotted on a logarithmic color
scale. The six innermost spots come from the substrate and the other spots
come from the metal film. The six outer spots of roughly equal intensity
indicate each film contains two families of (111) grains differing by an in-
plane rotation of $60^{\circ}$.
Figures 1c, d, and e show pole figures measured at the peak of the (111)
reflection and plotted on a logarithmic color scale. In each pole figure, the
six innermost spots come from the substrate, the other spots come from the
metal film, and the rotational alignment of the two sets of spots indicates
the film is epitaxially aligned with the substrate. The measured epitaxial
relationship for all films can be expressed as
$(111)_{\textrm{metal}}$||$(0001)_{\textrm{Al}{}_{2}\textrm{O}_{3}}$ and
$[110]_{\textrm{metal}}$||$[10\bar{1}0]_{\textrm{Al}{}_{2}\textrm{O}_{3}}$,
consistent with prior results for several fcc metals on Al2O3 [23]. For a
(111) film with a single in-plane orientation, the pole figure would show only
three spots. The six spots with similar intensity shown in Fig. 1c indicate
roughly half the Cu film is rotated in-plane by $60^{\circ}$ with respect to
the rest of the film. The second set of spots appears with lower intensity in
Figs. 1c, d, indicating that only a minority of the Cu-Ni and Ni films are
rotated by $60^{\circ}$. Comparing the integrated intensities for the two sets
of spots, we estimate the fraction of the film composed of these $60^{\circ}$
twin domains to be 50 % for Cu, 15 % for Cu-Ni, and 25 % for Ni. The Cu-Ni
film also shows faint spots in between the six stronger spots, indicating that
a small minority (< 1 %) of the film is rotated in-plane by $\pm$30°. Overall,
the XRD characterization shows that all films are (111) textured but have some
degree of in-plane rotational disorder. This is not unexpected because there
are two equivalent orientations for a cubic film deposited on a hexagonal
substrate.
Figure 2: Optical and atomic force microscopy images showing the morphology of
the same films as in Fig. 1. Top row (a, b, c) shows optical images
($50\,\mathrm{\mu m}$ scale bar, shown in (c)) and bottom row (d, e, f) shows
AFM images ($10\,\mathrm{\mu m}$ scale bar, shown in (f)) Dark spots in the
optical images indicate areas of dewetting. The Cu surface becomes rough,
while the Ni surface remains much smoother.
The morphology of the films without a seed layer is shown in Fig. 2. The Cu
film shows a rough surface in both optical and AFM images. The black spots in
the optical image are areas of the film that have dewetted from the substrate,
and these spots increase in size and density for longer annealing times. Both
the roughness and the dewetting indicate the Cu film is only marginally stable
at $1000\,\mathrm{{}^{\circ}C}$. Such instability has been noted previously
[24, 19] and can be attributed to the rapid evaporation of Cu when heated near
its bulk melting temperature in rough vacuum. This is discussed further below.
The Cu-Ni film is smoother than the Cu film and shows a lower density of
dewetting spots. The Ni film is smoothest of all and has the fewest dewetting
spots. The AFM image of the Ni shows areas of several $\mu\textrm{m}^{2}$ that
are atomically smooth (rms roughness $0.2\,\mathrm{nm}$ to $0.4\,\mathrm{nm}$
for a $1\,\mathrm{\mu m}\times 1\,\mathrm{\mu m}$ area). The EBSD
characterization discussed below suggests that the dark lines in the optical
images for Cu-Ni and Ni are boundaries between the $60^{\circ}$ twin domains
revealed in the XRD data. These grain boundaries are wider for the Cu-Ni film
than for the Ni film, and are obscured by roughness for the Cu film. In the
AFM images for Cu-Ni and Ni, the grain boundaries appear as trenches. These
trenches, which do not appear in the as-deposited films, likely form when
grain boundaries move during annealing. These morphological characterizations
show that the Ni film clearly survives the annealing better than the Cu film,
as expected from the higher melting point of Ni.
Figure 3: Electron backscatter diffraction data for the same films as in Fig.
1 and for a polycrystalline Ni foil. (a, b, c) Orientation maps using a color
scale that represents in-plane direction only: blue and green regions differ
by $60^{\circ}$. The scale bar shown in (c) is $50\,\mathrm{\mu m}$. (e, f, g)
(111) pole figures with a logarithmic color scale. The Cu film shows equal
amounts of each orientation, while the Cu-Ni and Ni films show a majority of
one orientation in the regions shown. (d) Orientation map for a
polycrystalline Ni foil after annealing to $1000\,\mathrm{{}^{\circ}C}$. Wedge
color scale represents out-of-plane orientation. Scale bar is
$400\,\mathrm{\mu m}$. (h) (111) pole figure for the polycrystalline Ni foil.
The micro-crystalline structure of the films can be characterized by EBSD, as
shown in Fig. 3 for films without a seed layer. Because the films are fully
(111) textured, we use a color scale for in-plane direction only: blue and
green regions differ by a $60^{\circ}$ rotation about the [111] axis. The
upper images are therefore spatial maps of the $60^{\circ}$ twin domains
revealed in the XRD data. The lower images are (111) pole figures with a
logarithmic color scale. The spatial map for the Cu film shows a broad
distribution of grain sizes, and the pole figure shows equal intensity for
both in-plane orientations. For the Cu-Ni film, both the spatial map and the
pole figure indicate that one orientation dominates in the region shown here.
Comparing the orientation map of Fig. 3b and the optical image of Fig. 2b, we
conclude that the dark lines enclosing small areas in the optical image are
boundaries between $60^{\circ}$ twin domains. A less pronounced preference for
one in-plane orientation is apparent in the spatial map and pole figure of the
Ni film. For comparison with our (111) films, Figs. 3d, h show EBSD data for a
polycrystalline Ni foil, also annealed at $1000\,\mathrm{{}^{\circ}C}$. In
this case we use a color scale for out-of-plane orientation. The foil shows a
broad distribution of grain sizes and orientations.
Figure 4: Crystallinity and morphology of $400\,\mathrm{nm}$ Ni film deposited
with Al2O3 seed layer, after annealing at $1000\,\mathrm{{}^{\circ}C}$ and
$270\,\mathrm{Pa}$ ($2\,\mathrm{Torr}$). Optical image in (a) and EBSD map in
(b) ($50\,\mathrm{\mu m}$ scale bars) show fewer grains than the corresponding
images for Ni without the seed layer (Figs. 2c and 3c, respectively). AFM
image in (c) shows an ordered surface and shallower trenches between grains
than in Fig. 2f ($10\,\mathrm{\mu m}$ scale bar). XRD (111) pole figure in
(d), plotted on a logarithmic intensity scale, shows a single in-plane
orientation over 97 % of the film. Line profile in (e), taken along the black
line in (c), shows steps $\approx 1\,\mathrm{\mu m}$ wide and $\approx
5\,\mathrm{nm}$ high.
We now turn to the effect of the Al2O3 seed layer and other measures that
improved the metal films in terms of epitaxy and/or survivability under
graphene CVD conditions. The seed layer improved the adhesion of all films, as
indicated by a reduced density of dewetting areas after annealing. For the Cu
films, the seed layer had little effect on crystallinity: the XRD pole figure
(not shown) has six spots of similar intensity, indicating roughly equal
amounts of each $60^{\circ}$ twin orientation, as in the data of Fig. 1c. For
the Ni films, the seed layer increased grain size and reduced $60^{\circ}$
twin formation, as shown in Fig. 4. The second set of spots in the XRD pole
figure of Fig. 4d is faint even on a logarithmic scale, and from their
integrated intensity we estimate that only 3 % of the film area is composed of
$60^{\circ}$ twin domains. The optical image in Fig. 4a shows fewer lines due
to grain boundaries, and the lines are less distinct than in Fig. 2c. Finally,
the AFM image in Fig. 4c shows the detailed morphology of the improved Ni
film. The surface shows a regular series of atomically flat terraces, also
seen in the profile of Fig. 4e, that are $\approx 1\,\mathrm{\mu\text{m}}$
wide and separated by steps $\approx 5\,\mathrm{\text{nm}}$ tall. This profile
corresponds to a slope of 0.3°, consistent with the $\pm 0.5^{\circ}$ miscut
specification of the Al2O3 wafer. The trenches between $60^{\circ}$ twin
domains are shallower than for Ni without the seed layer, which is consistent
with less grain boundary motion during annealing due to the higher deposition
temperature.
Figure 5: Cross-sectional transmission electron microscopy of Ni on Al2O3. (a,
b) Without seed layer. (c, d, e) With seed layer. In (a), a $60^{\circ}$ twin
boundary in the Ni is seen at the edge of an extra (0003) Al2O3 lattice fringe
that may indicate a dislocation or step at the surface. The focused probe
diffraction pattern corresponding to the region of the Ni twin boundary is
shown in (b), where the subscript “T” labels spots from one of the twin
grains. In (c), the lattice fringes of the film with the seed layer are
continuous and smooth, indicating the surface was not damaged by the
resputtering process. Selected area diffraction patterns are shown for the Ni
in (d) and for the Al2O3 in (e).
To further investigate the effect of the seed layer, TEM was performed on a
cross-section of the Ni/Al2O3 interface from unannealed samples with and
without the seed layer. The bottom half of each image in Figs. 5a,c shows
Al2O3(0003) lattice fringes, which are kinematically forbidden but dynamically
allowed and are observed due to double diffraction. Ni lattice fringes are not
resolved because the distance between Ni(111) planes, 0.203 nm, is beyond the
0.28 nm point resolution of the microscope. For the Ni film without the seed
layer, Fig. 5a, the last Al2O3 fringe shows a lateral intensity modulation
that could indicate a fractional-unit-cell island or dislocation near the
surface. At the edge of the extra fringe, there is a vertical grain boundary
in the Ni (darker region near the center of the image). The focused probe
diffraction pattern corresponding to this region, shown in Fig. 5b, indicates
both grains share the [111] out-of-plane and [110] in-plane directions. This
can only occur for a twin boundary with a $60^{\circ}$ in-plane rotation about
the [111] direction. The presence of a Ni grain boundary at the edge of an
extra (0003) Al2O3 lattice fringe was not uncommon, so the extra fringe is
likely correlated with the mechanism for grain boundary formation. Thus the
TEM results independently confirm the EBSD results and provide a microscopic
view of the origin of the $60^{\circ}$ twin domains.
For the Ni film grown with the seed layer, we found far fewer grain boundaries
for an equivalent length of cross-sectional interface. Fig. 5c shows a typical
image for this film. The Al2O3 lattice fringes are continuous and smooth,
indicating the reactive sputtering produced a seed layer with an undamaged
surface. Selected area electron diffraction patterns show the Ni (Fig. 5d) and
the Al2O3 (Fig. 5e) are highly ordered.
While the Al2O3 seed layer did not significantly affect the amount of
$60^{\circ}$ twin domains or other properties of the Cu films, the roughness
of these films was strongly affected by the annealing conditions. The rough
surface shown in Fig. 2a, produced by annealing for 15 min at
$1000\,\mathrm{{}^{\circ}C}$ in a total pressure of $270\,\mathrm{Pa}$
($2\,\mathrm{Torr}$), offers a poor substrate for graphene growth. Increasing
the total pressure during annealing reduced the overall roughness of Cu and
Cu-Ni films, as shown in Table 1. We attribute this improvement to the reduced
evaporation rate of Cu at higher pressure. Table 1 also shows that roughness
was further decreased by placing a Cu foil $\approx 1\,\mathrm{\text{mm}}$
above the surface of the Cu film. This improvement is presumably due to a
higher vapor pressure of Cu near the film, and is likely related to the
improvements seen in graphene CVD when the Cu foil was folded and crimped to
make an enclosure [25]. With both higher total pressure and a foil cover, a
Cu(111) film annealed at $1065\,\mathrm{{}^{\circ}C}$ for 15 min showed a
roughness only slightly larger than that of the Ni film in Fig. 4c. Thus thin
Cu(111) films on Al2O3 can survive graphene CVD conditions, even those
approaching the Cu melting point, if the total pressure is high enough and
there is a nearby source of Cu vapor.
Film | Thickness (nm) | Temp. (${}^{\circ}\textrm{C}$) | Press. (Torr) | Foil cover | Roughness (nm)
---|---|---|---|---|---
Cu | 400 | 1000 | 2 | No | 27
Cu | 700 | 1000 | 20 | No | 9.5
Cu | 700 | 1000 | 40 | No | 6.8
Cu | 700 | 1000 | 20 | Yes | 5.8
Cu | 700 | 1065 | 40 | Yes | 5.0
Cu-Ni | 350/50 | 1000 | 2 | No | 18
Cu-Ni | 350/50 | 1000 | 20 | No | 13
Ni | 400 | 1000 | 2 | No | 3.3
Table 1: RMS roughness of metal films on Al2O3(0001) under different annealing
conditions. Annealing time was 15 min in all cases. All roughness values were
measured from a $30\,\mathrm{\mu m}\times 30\,\mathrm{\mu m}$ AFM image.
## 4 Graphene Growth and Characterization
Graphene was grown using CH4 as the precursor gas in a hot-wall tube furnace
with an inner diameter of 80 mm and an overall length of 1.5 m. The tube was
evacuated using a scroll pump (base pressure of $2.5\,\mathrm{Pa}$) and all
gases were 99.99+ % pure. For growth on Ni, the films were annealed at
$1000\,\mathrm{{}^{\circ}C}$ in a flow of $27\,\mathrm{\mu mol/s}$
($36\,\mathrm{sccm}$) of Ar and $27\,\mathrm{\mu mol/s}$ of H2 for 15 min,
cooled to $900\,\mathrm{{}^{\circ}C}$, and then exposed to $27\,\mathrm{\mu
mol/s}$ of H2 and $27\,\mathrm{\mu mol/s}$ of CH4. Growth conditions were
maintained for 30 min to ensure full C saturation, and the film was cooled at
$4\,\mathrm{{}^{\circ}C/min}$ in the same gas flows used for growth. A total
pressure of $2700\,\mathrm{Pa}$ ($20\,\mathrm{Torr}$) was maintained
throughout the process. For growth on Cu, the films were annealed at
$1035\,\mathrm{{}^{\circ}C}$ in $370\,\mathrm{\mu mol/s}$
($500\,\mathrm{sccm}$) of Ar and $7.4\,\mathrm{\mu mol/s}$
($10\,\mathrm{sccm}$) of H2 for 15 min. A flow of $27\,\mathrm{\mu mol/s}$
($36\,\mathrm{sccm}$) of CH4 was then added to the Ar and H2, growth lasted 4
min, and the film was cooled at approximately $(50\textrm{ to
100)}\,\mathrm{{}^{\circ}C/min}$ in the same gas flows used for growth. A
total pressure of $5300\,\mathrm{Pa}$ ($40\,\mathrm{Torr}$) was maintained
throughout the process.
Figure 6: Scanning electron microscopy (SEM) images of multilayer graphene
grown on Ni films. (a, c, e) $400\,\mathrm{nm}$ epitaxial Ni(111) on Al2O3.
(b, d) $400\,\mathrm{nm}$ polycrystalline Ni on SiO2. Scale bars are
$40\,\mathrm{\mu m}$ for (a) and (b), $4\,\mathrm{\mu m}$ for (c) and (d), and
$10\,\mathrm{\mu m}$ for (e). (f) Raman spectroscopy indicates there is at
least a monolayer of graphene at all points. Darker regions correspond to
thicker growth.
Scanning electron microscopy (SEM) at low accelerating voltage (5 kV) was used
to image graphene on the Ni and Cu surfaces. Figure 6 compares growth on
$400\,\mathrm{nm}$ films of epitaxial Ni(111) on Al2O3 (Fig. 6a,c,e) and
polycrystalline Ni on SiO2 (Fig. 6b,d). The graphene films were grown
simultaneously to ensure identical treatment. We attribute variations in the
grey-scale to variations in graphene thickness, with darker areas presumably
corresponding to thicker graphene. Although contrast variations from the
polycrystalline Ni film itself could exist due to channeling, we expect this
effect to be minimal because the images were taken with an in-lens detector.
The most striking feature in these images is the different relationship
between graphene thickness changes and Ni grain boundaries for the two films.
For the polycrystalline Ni film, most changes in thickness occur at grain
boundaries, leading to mostly faceted edges for regions of a given thickness
(Fig. 6d). For the Ni(111) film, the graphene thickness varies dramatically on
a length scale much smaller than the size of the grains shown in the EBSD data
of Fig. 4b. Thus while graphene growth on Ni(111) is indeed more uniform than
growth on polycrystalline Ni, it is not nearly as uniform as one would expect
if thickness variations are simply caused by faster precipitation of C at
grain boundaries than within the interior of grains. This suggests there is an
additional source of thickness variations that remains to be identified.
Similar thickness variations occured for a variety of growth parameters.
The higher resolution SEM images in Fig. 6c,e show detailed features of the
thickness variations that are worth mentioning. In some places, thicker
graphene follows the boundary between a minority grain and the surrounding
majority grain (red arrows in Fig. 6e). This has been reported previously for
growth on Ni [11] and is likely due either to faster precipitation of C or to
trapping of C atoms at these boundaries. In other places, the boundaries of
minority grains have no change in graphene thickness (green arrow in Fig. 6e).
In many of the thicker graphene regions, straight, bright lines may indicate
buckling of the graphene from differential contraction during cooling (yellow
arrow in Fig. 6c). Some of the thinnest graphene regions show a network of
bright lines that may indicate a discontinuous mosaic of graphene patches
(blue arrows in Fig. 6c). The variety of growth features on a substrate of
such high quality highlights the need for a better understanding of how
graphene grows on Ni.
Raman spectroscopy using an excitation wavelength of 633 nm was done to verify
the existence of graphene and as a rough characterization of defects. The
thickest graphene ($\gtrsim 4$ layers) on Ni films appears darker in an
optical microscope. Raman spectra with a spot size of $\approx 1\,\mathrm{\mu
m}\times 3\,\mathrm{\mu m}$ were taken in between these darker patches to
preferentially sample the thinnest regions. Fig. 6f shows typical spectra,
which have the _G_ and _G_ ’ peaks characteristic of few-layer graphene [26].
No places were found that did not have these peaks, indicating the films were
fully covered by graphene. The lack of a significant _D_ peak in the Raman
spectra for graphene on Ni indicates there are few defects.
The graphene grown on Cu is primarily monolayer, as shown by the Raman
spectrum in Fig. 6e, and therefore SEM images (not shown) reveal little about
the growth. The large _D_ peak is likely due to a high density of graphene
nucleation sites, resulting in a large number of graphene grain boundaries.
When graphene growth is interrupted before a complete layer forms, the
nucleation sites can be seen directly in SEM images [7]. Nucleation sites were
separated by roughly $1\,\mathrm{\mu m}$, much less than the distance between
Cu grain boundaries in Fig. 3a. As with Cu foils [7], we expect that
optimization of growth conditions can improve the quality of graphene on
Cu(111) films.
## 5 Conclusion
The metal films reported here have many of the properties needed for wafer-
scale growth of graphene by CVD. They can be deposited epitaxially with a pure
(111) texture on commercially available $\alpha-$Al2O3(0001) substrates. The
films fall short of being single crystals only due to grain boundaries between
(111) regions that differ by an in-plane rotation of $60^{\circ}$. For Ni, a
film that is 97 % single orientation and has a smooth, well ordered surface
can be achieved by reactive sputtering of an Al2O3 seed layer immediately
before metal deposition. This process, which probably helps by removing the OH
layer typically found on Al2O3 after exposure to air, does not require ion
bombardment or temperatures above $650\,\mathrm{{}^{\circ}C}$ and thus can be
implemented in many commercial sputtering systems if they are fitted with a
basic substrate heater. Cu films do not show significant improvement from the
seed layer process, but the roughness of these films caused by evaporation
under graphene CVD conditions can be reduced by a combination of higher total
pressure and proximity of a Cu foil. Cu-Ni films deposited as bilayers become
homogeneous alloys when annealed at $1000\,\mathrm{{}^{\circ}C}$,
demonstrating the possibility of tuning C solubility through alloy composition
and thereby controlling graphene thickness. The evaporation of Cu during
annealing can have a significant effect on the final alloy composition.
Further improvements are needed to reach the goal of wafer-scale, single
crystal thin films of Cu, Ni, and Cu-Ni alloys. Optimization of the reactive
sputtering process and the use of a sputtering system with lower base pressure
may result in fewer grain boundaries than demonstrated here. The improvements
found in the Ni films may be transferable to Cu films through the use of a few
nanometers of Ni deposited before the Cu. Another possible route is the use of
MgO substrates. In recent work, both Cu [27] and Ni [28] deposited on MgO(111)
showed a (111) texture without apparent in-plane rotational disorder, and the
films did not noticeably degrade under graphene CVD conditions. These
measurements may have been less sensitive to in-plane rotations than the XRD
pole figures presented here, but if these films do contain a small fraction
that is rotated in-plane, they may also benefit from a reactively sputtered
seed layer. Unfortunately, wafers of crystalline MgO are not widely available.
Independent of progress on film quality, there is clearly room for improvement
in our understanding of graphene CVD on metal substrates, be they foils, films
or ingots. The work presented here implies that even a single crystal metal
film free of grain boundaries may not be sufficient to achieve the graphene
uniformity needed for commercial applications. The identification of other
mechanisms for graphene nonuniformity remains an important topic of research.
## 6 Acknowledgments
We thank Dustin Hite and David Pappas for advice on metal film growth,
Lawrence Robins and Angela Hight-Walker for assistance with Raman
spectroscopy, and Stephen Russek for helpful discussions about magnetron
sputtering.
## References
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|
arxiv-papers
| 2012-05-03T22:49:16 |
2024-09-04T02:49:30.505214
|
{
"license": "Public Domain",
"authors": "David L. Miller, Mark W. Keller, Justin M. Shaw, Ann N. Chiaramonti,\n Robert R. Keller",
"submitter": "Mark Keller",
"url": "https://arxiv.org/abs/1205.0833"
}
|
1205.0897
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-114 LHCb-PAPER-2012-009 May 4, 2012
Measurement of the $D^{+}_{s}-D^{-}_{s}$ production asymmetry in 7 TeV $pp$
collisions
The LHCb collaboration†††Authors are listed on the following pages.
Heavy quark production in 7 TeV centre-of-mass energy $pp$ collisions at the
LHC is not necessarily flavour symmetric. The production asymmetry, $A_{\rm
P}$, between $D_{s}^{+}$ and $D_{s}^{-}$ mesons is studied using the
$\phi\pi^{\pm}$ decay mode in a data sample of 1.0 fb-1 collected with the
LHCb detector. The difference between $\pi^{+}$ and $\pi^{-}$ detection
efficiencies is determined using the ratios of fully reconstructed to
partially reconstructed $D^{*\pm}$ decays. The overall production asymmetry in
the $D_{s}^{\pm}$ rapidity region 2.0 to 4.5 with transverse momentum larger
than 2 GeV is measured to be $A_{\rm P}=(-0.33\pm 0.22\pm 0.10)$%. This result
can constrain models of heavy flavour production.
Submitted to Physics Letters B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G.
Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C.
Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-
Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35,
G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S.
Hansmann-Menzemer11, N. Harnew52, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M.
Kucharczyk20,23,35,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36,
D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42,
R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J.
Lefrançois7, O. Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9,
M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac
Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35,
J. Magnin1, S. Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev27, E.
Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R. McNulty12, M.
Meissner11, M. Merk38, J. Merkel9, S. Miglioranzi35, D.A. Milanes13, M.-N.
Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski23, R.
Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B.
Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I.
Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M.
Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A. Nomerotski52,35, A.
Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48,
O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P.
Owen50, B.K. Pal53, J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35,
A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N. Patrick46, C.
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Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
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Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
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S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
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V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T. Skwarnicki53, N.A.
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D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
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Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Production of charm and bottom hadrons at the LHC in 7 TeV $pp$ collisions is
quite prolific. The bottom cross-section in the pseudorapidity region between
2 and 6 is about 80 $\rm\,\upmu b$ [1], and the charm cross-section is about
30 times higher [2]. In $pp$ collisions the production rates of charm and
anti-charm particles need not be the same. While production diagrams are
flavour symmetric, the hadronization process may prefer antiparticles to
particles or vice versa. Figure 1 gives an example of $c\overline{c}$
production via gluon fusion. If the quarks that contribute to charm meson
production are created in an independent fragmentation process, equal numbers
of $D$ and $\overline{D}$ will be produced. On the other hand, if they combine
with valence quarks in beam protons, the $\overline{c}$-quark can form a
meson, while the $c$-quark can form a charmed baryon. Therefore, we may expect
a small excess of $D_{s}^{-}$ over $D_{s}^{+}$ mesons. However, there are
other subtle QCD effects that might contribute to a charm meson production
asymmetry [3, 4, *Norrbin:2000zc]; we note for $b$ quarks the asymmetries are
estimated to be at the 1% level [6], and we would expect them to be smaller
for $c$ quarks, although quantitative predictions are difficult. Another
conjecture is that any asymmetries might be reduced as particles are produced
at more central rapidities.
Figure 1: Production of $c\overline{c}$ quark pairs in a $pp$ collision via
gluons.
Measurements of $C\\!P$ violating asymmetries in charm and bottom decays are
of prime importance. These can be determined at the LHC if production and
detection asymmetries are known. The measurement of asymmetries in flavour
specific modes usually involves detection of charged hadrons, and thus
requires the relative detection efficiencies of $\pi^{+}$ versus $\pi^{-}$ or
$K^{+}$ versus $K^{-}$ to be determined. While certain asymmetry differences
can be determined by cancelling the detector response differences to
positively and negatively charged hadrons [7], more $C\\!P$ violating modes
can be measured if the relative detection efficiencies can be determined.
In this Letter we measure the production asymmetry,
$A_{\rm
P}=\frac{\sigma(D^{+}_{s})-\sigma(D^{-}_{s})}{\sigma(D^{+}_{s})+\sigma(D^{-}_{s})},$
(1)
where $\sigma(D^{-}_{s})$ is the inclusive prompt production cross-section. We
use $D^{\pm}_{s}\rightarrow\phi\pi^{\pm}$ decays, where $\phi\rightarrow
K^{+}K^{-}$. Since $D^{\pm}_{s}\rightarrow\phi\pi^{\pm}$ is Cabibbo favoured,
no significant $C\\!P$ asymmetry is expected [8, 9]. Assuming it to be
vanishing, $A_{\rm P}$ is determined after correcting for the relative
$D_{s}^{+}$ and $D_{s}^{-}$ detection efficiencies. Since the final states are
symmetric in kaon production, this requires only knowledge of the relative
$\pi^{+}$ and $\pi^{-}$ detection efficiencies,
$\epsilon(\pi^{+})/\epsilon(\pi^{-})$.
## 2 Data sample and detector
The data sample is obtained from $1.0\;\text{fb}^{-1}$ of integrated
luminosity, collected with the LHCb detector [10] using $pp$ collisions at a
center-of-mass energy of 7 TeV. The detector is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$.111We work in units with $c$=1. Charged
hadrons are identified using two ring-imaging Cherenkov (RICH) detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a muon system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. Approximately 40% of the data was taken with the
magnetic field directed away from the Earth (up) and the rest down. We exploit
the fact that certain detection asymmetries cancel if data from different
magnet polarities are combined.
Events are triggered by the presence of a charm hadron decay. The hardware
trigger requires at least one hadronic transverse energy deposit of
approximately 3 GeV. Subsequent software triggers and selection criteria
require a subset of tracks to not point to a primary $pp$ collision vertex
(PV), and form a common vertex.
## 3 Measurement of relative pion detection efficiency
In order to measure $\epsilon(\pi^{+})/\epsilon(\pi^{-})$, we use the decay
sequence $D^{*+}\rightarrow\pi^{+}_{s}D^{0}$, $D^{0}\rightarrow
K^{-}\pi^{+}\pi^{+}\pi^{-}$, and its charge-conjugate decay, where
$\pi^{+}_{s}$ indicates the “slow” pion coming directly from the $D^{*+}$
decay. Assuming that the $D^{*\pm}$ comes from the PV, there are sufficient
kinematic constraints to detect this decay even if one pion from the $D^{0}$
decay is missed. We call these “partially” reconstructed decays. We can also
“fully” reconstruct this decay. The ratio of fully to partially reconstructed
decays provides a measurement of the pion reconstruction efficiency. We
examine $D^{*+}$ and $D^{*-}$ candidate decays separately, and magnet up data
separately from magnet down data. The latter is done to test for any possible
left-right detector asymmetries. In both cases the missing pion’s charge is
required to be opposite of that of the detected kaon.
Kaon and pion candidates from candidate $D^{0}$ decays are required to have
transverse momentum, $\mbox{$p_{\rm T}$}>400$ MeV, and a track quality fit
with $\chi^{2}$ per number of degrees of freedom (ndf) $<3$, keeping more than
99% of the good tracks. The distance of closest approach of track candidates
to the PV is called the impact parameter (IP). A restrictive requirement is
imposed on the IP $\chi^{2}$, which measures whether the track is consistent
with coming from the PV, to be greater than 4. In addition both particles must
be identified in the RICH. For the $\pi^{+}_{s}$, the $p_{\rm T}$ requirement
is lowered to 250 MeV, with both IP $<$ 0.3 mm and IP $\chi^{2}\,<\,4$ being
required. Further tight restrictions are placed on $D^{0}$ candidates. The
candidate tracks from the $D^{0}$ decay must fit to a common vertex with
$\chi^{2}$/ndf$\,<\,$6, the $D^{0}$ candidates must have a flight distance of
at least 4 mm from the PV and have a flight distance $\chi^{2}>\,$120\. We
require $1.4\,<\,m(K^{-}\pi^{+}\pi^{-})\,<\,1.7$ GeV, and that the invariant
mass of the $\pi^{+}\pi^{-}$ candidates must be within $\pm$200 MeV of the
$\rho(770)$ mass, to improve the signal to background ratio.
We select partially reconstructed right-sign (RS) $D^{0}$ candidates by
examining the mass difference $\Delta
m_{\text{prt}}=m(\pi^{+}_{s}K^{-}\pi^{+}\pi^{-})-m(K^{-}\pi^{+}\pi^{-})$.
Wrong-sign (WS) candidates are similarly selected but by requiring that the
charge of the kaon be the same as that of the $\pi_{s}^{+}$. Figure 2 shows
distributions of $\Delta m_{\text{prt}}$ for magnet up data. Note that the
yield of WS events is reduced due to a prescale factor applied in the
selection.
Figure 2: Distributions of mass differences in partial reconstruction for (a)
RS $m(\pi^{+}_{s}K^{-}\pi^{+}\pi^{-})-m(K^{-}\pi^{+}\pi^{-})$ and (b) WS
$m(\pi^{+}_{s}K^{+}\pi^{+}\pi^{-})-m(K^{+}\pi^{+}\pi^{-})$ candidates, for
magnet up data. The (green) dotted line shows the signal, the (red) dashed
line the background, and the (blue) solid line the total. The fit shapes are
defined in Appendix A.
In order to determine the size of the signals above the background we perform
simultaneous binned maximum likelihood fits to the RS and WS distributions.
The parametrization of the signal probability density function (PDF) is given
in Appendix A. The signal and background PDFs are identical for RS and WS
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ events, only
the absolute normalizations are allowed to differ. We also include a “signal”
term in the fit to WS events to account for the doubly-Cabibbo-suppressed
(DCS) signals. The ratio of the DCS signal in WS events to the signal in RS
events is fixed to that obtained in the mass difference fit in full
reconstruction.
Using momentum and energy conservation and knowledge of the direction of the
$D^{0}$ flight direction, the inferred three-momentum of the missing pion,
$\vec{P}_{\\!\rm inf}$, is reconstructed using a kinematic fitting technique
[11]. Our resolution on inferred pions may be determined from the the fully
reconstructed $D^{*\pm}$ sample, by removing one detected pion whose three-
momentum is well known, and treating the track combination as if it was
partially reconstructed. We then have both detected and inferred momentum, and
thus a measurement of the missing pion momentum resolution distribution,
$\Delta P/P=(P_{\text{detected}}-P_{\text{inf}})/P_{\text{detected}}$. For
further study, we take only combinations with good inferred resolution by
accepting those where ${P}_{\\!\rm inf}$ divided by its calculated uncertainty
is greater than two and also where the transverse component of ${P}_{\\!\rm
inf}$ divided by its uncertainty is greater than 2.5; this eliminates about
37% of the sample.
In our sample of partially reconstructed events, we subsequently look for
fully reconstructed decays by searching for the missing track. Candidate
tracks must have $p>2$ GeV, $\mbox{$p_{\rm T}$}>300$ MeV and be identified as
a pion in the RICH. They also must form a vertex with the other three tracks
from the decay with a vertex fit $\chi^{2}$/ndf$<6$, and have a four-track
invariant mass within 30 MeV of the $D^{0}$ mass peak. Certain areas of the
detector near its edges preferentially find only one charge or the other
depending on magnet polarity. We remove fully reconstructed candidates where
the detected pion projects to these regions, discarding 3% of the candidates.
The mass difference for fully reconstructed combinations, $\Delta
m_{\text{full}}=m(\pi^{+}_{s}K^{-}\pi^{+}\pi^{-}\pi^{+})-m(K^{-}\pi^{+}\pi^{-}\pi^{+})$
is shown in Fig. 3, for both RS and WS cases. Only $D^{*+}$ data in the magnet
up configuration are shown. The shape of the mass difference signal PDF is
described in Appendix B.
Figure 3: Distributions of the mass difference $\Delta m_{\text{full}}$ for
(a) RS and (b) WS events using magnet up data. The (green) dotted line shows
the signal, the (red) dashed line the background, and the (blue) solid line
the total. The fit shapes are defined in Appendix B.
In order to extract the signal yields, we perform a binned maximum likelihood
fit to the $D^{*+}$ and $D^{*-}$ events, both RS and WS, simultaneously. Table
1 lists the signal yields for partial reconstruction, $N_{\rm prt}$, and full
reconstruction, $N_{\rm full}$. The efficiency ratios are derived from ratios
of RS yields,
$\epsilon(\pi^{+})=N_{\text{full}}(D^{0}\pi^{+}_{s})/N_{\text{prt}}(D^{0}\pi^{+}_{s})$
and $\epsilon(\pi^{-})=N_{\text{full}}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}_{s})/N_{\text{prt}}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}_{s})$. (The absolute
efficiency inferred from these yields includes geometric acceptance effects.)
The $p$ and $p_{\rm T}$ spectra of the $\pi^{\pm}$ used for the efficiency
measurement are shown in Fig 4.
Table 1: Event yields for partial and full reconstruction. Category | Magnet up | Magnet down
---|---|---
$N_{\text{prt}}(D^{0}\pi^{+}_{s})$ | 460,005$\pm$890 | 671,638$\pm$1020
$N_{\text{prt}}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}_{s})$ | 481,823$\pm$873 | 694,268$\pm$1035
$N_{\text{full}}(D^{0}\pi^{+}_{s})$ | 207,504$\pm$465 | 299,629$\pm$ 570
$N_{\text{full}}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}_{s})$ | 219,230$\pm$478 | 308,344$\pm$ 579
Figure 4: Distribution of fully reconstructed signal candidates for magnet up
data as a function of pion (a) $p$ and (b) $p_{\rm T}$.
The ratios of pion detection efficiencies are 0.9914$\pm$0.0040 and
1.0045$\pm$0.0034 for magnet up and magnet down, respectively, with
statistical uncertainties only. To obtain the efficiency ratio as a function
of momentum we need to use the inferred momentum of the missing pion. Because
of finite resolution it needs to be corrected. This is accomplished through an
unfolding matrix estimated using the fully reconstructed sample by comparing
the measured momentum of a found pion that is then ignored and its momentum
inferred using the kinematic fit. The efficiency ratio is shown as a function
of momentum in Fig. 5. Most systematic uncertainties cancel in the efficiency
ratio, however, some small residual effects remain. To assess them we change
the signal and background PDFs in full and partial reconstruction by
eliminating, in turn, each of the small correction terms to the main
functions. The full fit is then repeated. Each change in the efficiency ratios
are between 0.01$-$0.02%. We also change the amount of DCS decays by the
measured uncertainty in the branching fraction. This also gives a 0.020%
change. The total systematic error is 0.045%. Furthermore, the entire
procedure was checked using simulation.
Figure 5: Relative detection efficiency in bins of detected pion momentum:
(red) circles represent data taken with magnet polarity up and (blue) squares
show data taken with magnet polarity down. Only statistical errors are shown.
Although we correct relative pion efficiencies as a function of $p$, it is
possible that there also is a $p_{\rm T}$ dependence that would have an effect
if the $p_{\rm T}$ distributions of the $D^{*\pm}$ and $D_{s}^{\pm}$ were
different. The efficiency ratios for different slices are shown in Fig. 6. For
a fixed $p$ interval there is no visible $p_{\rm T}$ dependence.
Figure 6: Relative efficiency averaged over magnet up and magnet down samples
versus pion $p$ and $p_{T}$.
The relative pion efficiencies are consistent with being independent of $p$
and $p_{\rm T}$. The tracking acceptance does depend, however, on the
azimuthal production angle of the particles, $\varphi$. This is mostly because
tracks can be swept into the beam pipe and not be detected by the downstream
tracking system. Therefore, for purposes of the production asymmetry analysis
we determine $\epsilon(\pi^{+})/\epsilon(\pi^{-})$ as a function of $\varphi$
in two momentum intervals: $2-20$ GeV, and above 20 GeV. The r.m.s. resolution
on the inferred $\varphi$ is 0.25 rad, much smaller than the $\pi$/4 bin size.
The correction factors are shown in Fig. 7. The average correction for magnet
up and magnet down is consistent with unity. Thus any residual biases in the
$D_{s}^{\pm}$ yields due to $\pi^{+}/\pi^{-}$ asymmetries will also cancel in
the average.
Figure 7: Azimuthal angle distribution of
$\epsilon(\pi^{+})/\epsilon(\pi^{-})$ for magnet up data (red circles) and
magnet down data (blue squares), and their average (black diamonds) for (a)
pion momentum $2<p<20$ GeV and (b) $p>20$ GeV.
## 4 $D^{\pm}_{s}$ production asymmetry
The decay $D_{s}^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}$ is used with the
invariant mass of the $K^{+}K^{-}$ required to be within $\pm$20 MeV of the
$\phi$ mass. Events are triggered at the hardware level by requiring that
either the $K^{+}$ or the $K^{-}$ deposits more than 3 GeV of transverse
energy in the hadron calorimeter. Subsequent software triggers are required to
select both $\phi$ decay products.
To select a relatively pure sample of $D_{s}^{\pm}\rightarrow
K^{+}K^{-}\pi^{\pm}$ candidates each track is required to have $\chi^{2}$/ndf
$<$4, $\mbox{$p_{\rm T}$}>\,$300 MeV, IP $\chi^{2}>4$, and be identified in
the RICH. All three candidate tracks from the $D_{s}^{\pm}$ have
$\mbox{$p_{\rm T}$}>2$ GeV, and must form a common vertex that is detached
from the PV. The $\chi^{2}$ requiring all three tracks to come from a common
origin must be $<8.33$, this decay point must be at least 100 $\mu$m from the
PV, and the significance of the detachment must be at least 10 standard
deviations. The $D_{s}^{\pm}$ candidates’ momentum vector must also point to
the PV, which reduces contamination from $b$-hadron decays to the few percent
level. We remove signal candidates with pions which pass through the detector
areas with large inherent asymmetries, as we did to measure the relative pion
efficiencies.
Figure 8 shows the invariant mass distributions for (a)
$K^{+}\kern-1.60004ptK^{-}\pi^{+}$ and (b) $K^{+}K^{-}\pi^{-}$ candidates for
data taken with magnet polarity down. We perform a binned maximum likelihood
fit to extract the signal yields. The fitting functions for both $D^{\pm}$ and
$D_{s}^{\pm}$ signals are triple Gaussians where all parameters are allowed to
vary, except two of the Gaussians are required to have the same mean. The
background function is a second order polynomial. The numbers of $D_{s}^{\pm}$
events obtained from the fits are listed in Table 2.
Figure 8: Invariant mass distributions for (a) $K^{+}\kern-1.60004ptK^{-}\pi^{+}$ and (b) $K^{-}K^{+}\pi^{-}$ candidates, when $m(K^{+}K^{-})$ is within $\pm$20 MeV of the $\phi$ mass, for the dataset taken with magnet polarity down. The shaded areas represent signal, the dashed line the background and the solid curve the total. Table 2: Fitted numbers of $D_{s}^{\pm}$ events for both magnet up and down data. | Magnet up | Magnet down
---|---|---
$D_{s}^{+}$ | 152,696$\pm$448 | 230,860$\pm$514
$D_{s}^{-}$ | 154,209$\pm$438 | 233,266$\pm$549
The rapidity of the $D^{+}_{s}$ is defined as
$y=\frac{1}{2}\ln\frac{E+p_{z}}{E-p_{z}},$ (2)
where $E$ and $p_{z}$ are the energy and $z$ component of the $D^{\pm}_{s}$
momentum. We measure the production asymmetry $A_{\rm P}$ as a function of
both $D^{\pm}_{s}$ $y$ and $p_{\rm T}$. In each $y$ or $p_{\rm T}$ bin we
extract the efficiency corrected ratio of yields by applying corrections as a
function of azimuthal angle in the two pion momentum intervals defined
previously. Magnet up and down data are treated separately. The $y$ and
$p_{\rm T}$ distributions are shown in Fig. 9. Here sidebands in $KK\pi$ mass
have been used to subtract the background, where the sidebands are defined as
between $30-70$ MeV above and below the peak mass value of 1969 MeV. As this
interval is twice as wide as the signal peak, we weight these events by a
factor of 1/2.
Figure 9: (a) $D^{\pm}_{s}$ rapidity distribution (b) $D^{\pm}_{s}$ $p_{T}$
distribution for background subtracted magnet up data. The statistical
uncertainty on the number of events in each bin is smaller than the line
thickness.
Figure 10 shows $A_{\rm P}$ as a function of either $y$ or $p_{\rm T}$. The
error bars reflect only the statistical uncertainties, which includes both the
statistical errors on $\epsilon(\pi^{+})/\epsilon(\pi^{-})$ and the
$D^{\pm}_{s}$ yields; the error bars are partially correlated, the
uncertainties from the $D_{s}^{\pm}$ yields are about half the size of those
shown. The values in $p_{\rm T}$ and $y$ intervals are listed in Table 3.
Table 3: $A_{\rm P}$ (%) shown as a function of both $y$ and $p_{\rm T}$. $p_{\rm T}$ (GeV) | $y$
---|---
| $2.0-3.0$ | $3.0-3.5$ | $3.5-4.5$
$2.0-~{}\,6.5$ | $0.2\pm 0.5$ | $-0.7\pm 0.5$ | $-0.9\pm 0.4$
$6.5-~{}\,8.5$ | $-0.3\pm 0.4$ | $~{}~{}\,0.1\pm 0.5$ | $-1.2\pm 0.5$
$8.5-25.0$ | $0.2\pm 0.3$ | $-0.3\pm 0.5$ | $-1.0\pm 0.8$
An average asymmetry in this $y$ and $p_{\rm T}$ region can be derived by
weighting the asymmetry in each bin by the production yields. Thus we take the
asymmetry in each $y$ and $p_{\rm T}$ interval, weight by the measured event
yields divided by the reconstruction efficiencies. The resulting integrated
production asymmetry $A_{\rm P}$ is $(-0.20\pm 0.34)$%, and $(-0.45\pm
0.28)$%, for magnet up and magnet down samples, respectively. The errors are
statistical only. Averaging the two results, giving equal weight to each to
cancel any residual systematic biases, gives
$A_{\rm P}=(-0.33\pm 0.13\pm 0.18\pm 0.10)\%,$
where the first uncertainty is statistical from the $D_{s}^{\pm}$ yields, the
second statistical due to the error on the efficiency ratio and the third
systematic.
Figure 10: Observed production asymmetry $A_{\rm P}$ as a function of (a) y,
and (b) $p_{\rm T}$. The errors shown are statistical only.
The systematic uncertainty on $A_{\rm P}$ has several contributions.
Uncertainties due the background shape in the $D^{\pm}_{s}$ mass fit are
evaluated using a higher order polynomial function, that gives a $0.06\%$
change. Statistical uncertainty on MC efficiency adds 0.06%. Constraining the
signal shapes of the $D_{s}^{+}$ and $D_{s}^{-}$ to be the same makes a 0.04%
difference. Possible changes in detector acceptance during magnet up and
magnet down data taking periods are estimated to contribute 0.03%. The
systematic uncertainty from the pion efficiency ratio contributes 0.02%.
Differences in the momentum distributions of $K^{-}$ and $K^{+}$ that arise
from interference with an S-wave component under the $\phi$ peak can introduce
a false asymmetry [12]. For our relatively high momentum $D_{s}^{\pm}$ mesons
this is a 0.02% effect. Contamination from $b$ decays causes a negligible
effect. Adding all sources in quadrature, the overall systematic uncertainty
on $A_{\rm P}$ is estimated to be $0.10\%$.
## 5 Conclusions
We have developed a method using partially and fully reconstructed $D^{*\pm}$
decays to measure the relative detection efficiencies of positively and
negatively charged pions as a function of momentum. Applying this method to
$D_{s}^{\pm}$ mesons produced directly in $pp$ collisions, _i.e._ not
including those from decays of $b$ hadrons, we measure the overall production
asymmetry in the rapidity region 2.0 to 4.5, and $\mbox{$p_{\rm T}$}>2$ GeV as
$A_{\rm
P}=\frac{\sigma(D^{+}_{s})-\sigma(D^{-}_{s})}{\sigma(D^{+}_{s})+\sigma(D^{-}_{s})}=(-0.33\pm
0.22\pm 0.10)\%.$ (3)
The asymmetry is consistent with being independent of $p_{\rm T}$, and also
consistent with being independent of $y$, although there is a trend towards
smaller $A_{\rm P}$ values at more central rapidity. These measurements are
consistent with theoretical expectations [3, 4, *Norrbin:2000zc], provide
significant constraints on models of $D_{s}^{\pm}$ production, and can be used
as input for $C\\!P$ violation measurements.
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## Appendix A: Fitting functions for partial reconstruction
The signal probability density function (PDF) is given by:
$\displaystyle f_{\rm sig}(\Delta m_{\text{prt}})$ $\displaystyle=f_{\rm
eff}(\Delta m_{\text{prt}})\cdot BG(\Delta
m_{\text{prt}};\sigma_{l},\sigma_{r},\mu),~{}{\rm where}$ (4) $\displaystyle
BG(\Delta m_{\text{prt}})$
$\displaystyle=\begin{cases}\frac{\sigma_{l}}{\sigma_{l}+\sigma_{r}}G(\Delta
m_{\text{prt}};\mu,\sigma_{l})&\text{if $\Delta m_{\text{prt}}\leq\mu$,}\\\
\frac{\sigma_{r}}{\sigma_{l}+\sigma_{r}}G(\Delta
m_{\text{prt}};\mu,\sigma_{r})&\text{if $\Delta
m_{\text{prt}}>\mu$.}\end{cases}$
$G(\Delta m_{\text{prt}}$;$\mu$,$\sigma$) is a Gaussian function with mean
$\mu$ and width $\sigma$, and $BG$($\Delta m_{\text{prt}}$) is a bifurcated
Gaussian function. The efficiency function $f_{\text{eff}}(\Delta
m_{\text{prt}})$ is defined as:
$f_{\text{eff}}(\Delta m_{\text{prt}})=\begin{cases}\frac{|a(\Delta
m_{\text{prt}}-\Delta m_{0})|^{N}}{1+|a(\Delta m_{\text{prt}}-\Delta
m_{0})|^{N}}&\text{if $\Delta m_{\text{prt}}-\Delta m_{0}\geq 0$,}\\\
0&\text{if $\Delta m_{\text{prt}}-\Delta m_{0}<0$,}\end{cases}$ (5)
where $a$, $N$, and $\Delta m_{0}$ are fit parameters. The resolution function
(the bifurcated Gaussian function) is multiplied by the efficiency function
$f_{\text{eff}}(\Delta m_{\text{prt}})$ in order to account for the “turn-off”
behaviour of the quantity $\Delta m_{\text{prt}}$ near the threshold (pion
mass). There are in total six shape parameters in this signal PDF which are
left to vary in the fit.
The background PDF is taken as a threshold function with the inclusion of
extra components to obtain a good description of the WS combinations. It is
defined similarly:
$f_{\rm bkg}(\Delta m_{\text{prt}})=f^{*}(\Delta
m_{\text{prt}})\cdot(c_{2}\Delta m_{\text{prt}}^{2}+c_{1}\Delta
m_{\text{prt}}+1)-f_{1}\cdot BG(\Delta m_{\text{prt}})+f_{2}\cdot G(\Delta
m_{\text{prt}}),$ (6) $f^{*}(\Delta m_{\text{prt}})=\left[1-\exp(-(\Delta
m_{\text{prt}}-\Delta m^{p}_{0})/c_{p})\right]\cdot a_{p}^{\Delta
m_{\text{prt}}/\Delta m^{p}_{0}}+b_{p}(\Delta m_{\text{prt}}/\Delta
m^{p}_{0}-1).$ (7)
The parameters used in the background functions $BG$ and $G$ are different
than the ones used in the signal functions. There are in total 11 shape
parameters in the background PDF that are determined by the fit. We also fit
using $f^{*}(\Delta m_{\text{prt}})$ as the background PDF alone to estimate
the systematic uncertainty on the efficiency ratio.
## Appendix B: Fitting functions for full reconstruction
The signal PDF is defined as:
$\displaystyle f_{\rm sig}(\Delta m_{\text{full}})$
$\displaystyle=f_{1}G(\Delta m_{\text{full}};\mu_{1},\sigma_{1})+f_{2}G(\Delta
m_{\text{full}};\mu_{2},\sigma_{2})$
$\displaystyle+(1-f_{1}-f_{2})f_{\text{student}}(\Delta m_{\text{full}};\Delta
m_{0},\nu_{l},\nu_{h},\sigma_{\rm ave},\delta\sigma),$ (8)
where $G(\Delta m_{\text{full}})$ is a Gaussian function defined in Appendix
A, and $f_{\text{student}}(\Delta m_{\text{full}})$ is obtained from the
Student’s t-distribution
$f(t)=\frac{\Gamma(\nu/2+1/2)}{\Gamma(\nu/2)\sqrt{\nu\pi}}\cdot\left(1+\frac{t^{2}}{\nu}\right)^{(-\nu/2-1/2)},$
(9)
where $\Gamma$ is the Gamma function. We define $t=(\Delta
m_{\text{full}}-\Delta m_{0})/\sigma$ with $\Delta m_{0}$ and $\sigma$ the
mean and width. In order to obtain the asymmetric t-function, the width
parameter $\sigma$ and number of degrees of freedom $\nu$ are allowed to be
different for the high and low sides of $\Delta m_{\text{full}}$. Widths for
high and low sides of $\Delta m_{\text{full}}$ are then defined as:
$\sigma_{h}=\sigma_{\rm ave}+\delta\sigma$, and $\sigma_{l}=\sigma_{\rm
ave}-\delta\sigma$, and $\nu$ parameters for high and low sides are denoted as
$\nu_{h}$ and $\nu_{l}$, respectively. The bifurcated Student’s t-function can
then be defined as:
$f_{\text{student}}(\Delta
m_{\text{full}})=\begin{cases}\frac{r_{h}p_{h}}{\sqrt{\pi}}\cdot\left(1+\frac{(\frac{\Delta
m_{\text{full}}-\Delta
m_{0}}{\sigma_{h}})^{2}}{\nu_{h}}\right)^{(-\nu_{h}/2-1/2)}&\text{if $\Delta
m_{\text{full}}-\Delta m_{0}\geq 0$,}\\\
\frac{r_{l}p_{l}}{\sqrt{\pi}}\cdot\left(1+\frac{(\frac{\Delta
m_{\text{full}}-\Delta
m_{0}}{\sigma_{l}})^{2}}{\nu_{l}}\right)^{(-\nu_{l}/2-1/2)}&\text{if $\Delta
m_{\text{full}}-\Delta m_{0}<0$.}\end{cases}$ (10)
Auxiliary terms are defined as:
$p_{h}=\frac{\Gamma(\nu_{h}/2+1/2)}{\Gamma(\nu_{h}/2)\sqrt{\nu_{h}}|\sigma_{h}|},~{}p_{l}=\frac{\Gamma(\nu_{l}/2+1/2)}{\Gamma(\nu_{l}/2)\sqrt{\nu_{l}}|\sigma_{l}|},~{}r_{h}=\frac{2p_{l}}{p_{h}+p_{l}},~{}r_{l}=\frac{2p_{h}}{p_{h}+p_{l}}~{}.$
(11)
In total there are 11 shape parameters in the signal PDF, all of them are
allowed to vary in the fit. The background PDF is extracted from WS events,
and is defined as:
$f_{\rm bkg}(\Delta m_{\text{full}})=(1-f_{3})\cdot f^{*}(\Delta
m_{\text{full}})+f_{3}\cdot BG(\Delta m_{\text{full}}),$ (12)
where $f^{*}(\Delta m_{\text{full}})$ is defined in Eq. 7. We add a correction
function, a bifurcated Gaussian, in order to have a better fit; the shape and
the fraction of the bifurcated Gaussian is determined empirically from WS
events. (We also use a background shape without this correction term to
estimate the systematic uncertainty on the efficiency ratio.) We include a
“signal” term in the fit to WS events to account for doubly-Cabibbo suppressed
decays.
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|
arxiv-papers
| 2012-05-04T09:42:11 |
2024-09-04T02:49:30.514924
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, R. Bernet, M. -O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook,\n H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M.\n Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras,\n P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B.\n Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N. Y. David,\n I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda, L. De\n Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono,\n C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P.\n Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez,\n D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S.\n Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C.\n Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson,\n V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es,\n G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, T. Hampson, S. Hansmann-Menzemer, N. Harnew, J. Harrison, P. F.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A.\n Hernando Morata, E. van Herwijnen, E. Hicks, P. Hopchev, W. Hulsbergen, P.\n Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten,\n J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, I. Komarov,\n R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J. -P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y.\n Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.\n von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier,\n A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J.\n Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave,\n U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z.\n Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, S. Miglioranzi, D. A.\n Milanes, M. -N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva,\n M. Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.\n M. Otalora Goicochea, P. Owen, B. K. Pal, J. Palacios, A. Palano, M. Palutan,\n J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G.\n Passaleva, G. D. Patel, M. Patel, S. K. Paterson, G. N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo,\n D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch,\n A. Puig Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S.\n Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues,\n F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M.\n Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M. -H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco,\n A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A.\n Smith, E. Smith, M. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier,\n S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R. Waldi, R. Wallace,\n S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale,\n M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams,\n M. Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1205.0897"
}
|
1205.0918
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-113 LHCb-PAPER-2012-010 4 May 2012
Measurement of relative branching fractions of $B$ decays to $\psi(2S)$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons
The LHCb collaboration †††Authors are listed on the following pages.
The relative rates of $B$-meson decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ mesons are
measured for the three decay modes in $pp$ collisions recorded with the LHCb
detector. The ratios of branching fractions ($\cal B$) are measured to be
$\begin{array}[]{lll}\frac{{\cal B}(B^{+}\rightarrow\psi(2S)K^{+})}{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})}&=&0.594\pm 0.006\,(\mathrm{stat})\pm 0.016\,(\mathrm{syst})\pm
0.015\,(R_{\psi}),\\\ \vskip 3.0pt\cr\frac{{\cal
B}(B^{0}\rightarrow\psi(2S)K^{*0})}{{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0})}&=&0.476\pm 0.014\,(\mathrm{stat})\pm 0.010\,(\mathrm{syst})\pm
0.012\,(R_{\psi}),\\\ \vskip 3.0pt\cr\frac{{\cal
B}(B^{0}_{s}\rightarrow\psi(2S)\phi)}{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)}&=&0.489\pm 0.026\,(\mathrm{stat})\pm 0.021\,(\mathrm{syst})\pm
0.012\,(R_{\psi}),\end{array}$
where the third uncertainty is from the ratio of the $\psi(2S)$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ branching fractions to
$\mu^{+}\mu^{-}$.
Submitted to Eur. Phys. J. C
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G.
Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C.
Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-
Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35,
G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S.
Hansmann-Menzemer11, N. Harnew52, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M.
Kucharczyk20,23,35,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36,
D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42,
R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J.
Lefrançois7, O. Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9,
M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac
Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35,
J. Magnin1, S. Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev27, E.
Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R. McNulty12, M.
Meissner11, M. Merk38, J. Merkel9, S. Miglioranzi35, D.A. Milanes13, M.-N.
Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski23, R.
Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B.
Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I.
Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M.
Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A. Nomerotski52,35, A.
Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48,
O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P.
Owen50, B.K. Pal53, J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35,
A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N. Patrick46, C.
Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G.
Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, I. Polyakov28, E. Polycarpo2, D.
Popov10, B. Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V.
Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43, B.
Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M.
Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa
Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T.
Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31,
O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K. Sobczak5, F.J.P.
Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9, A.
Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S.
Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D.
Websdale50, M. Whitehead45, J. Wicht35, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y.
Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32,
M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Decays of $B$ mesons to two-body final states containing a charmonium
resonance such as a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or
$\psi(2S)$ offer a powerful way of studying electroweak transitions. Such
decays probe charmonium properties and play a role in the study of $C\\!P$
violation and mixing in the neutral $B$ system [1].
The relative branching fractions of $B^{+}$, $B^{0}$ and $B^{0}_{s}$ mesons
into ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ mesons have
previously been studied by both the CDF and D0 collaborations [2, 3, 4]. Since
the current experimental results for the study of $C\\!P$ violation in
$B^{0}_{s}$ mixing using the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decay
[5, 6, 7] are statistically limited, it is important to establish other
channels where this analysis can be done. One such channel is the
$B^{0}_{s}\rightarrow\psi(2S)\phi$ decay.
In this paper, measurements of the ratios of the branching fractions of $B$
mesons decaying to $\psi(2S)X$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X$ are reported, where $B$ denotes a $B^{+}$, $B^{0}$ or $B^{0}_{s}$
meson (charge conjugate decays are implicitly included) and $X$ denotes a
$K^{+}$, $K^{*0}$ or $\phi$ meson. The data were collected by the LHCb
experiment in $pp$ collisions at the centre-of-mass energy
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ during 2011 and correspond to an
integrated luminosity of 0.37$\mbox{\,fb}^{-1}$.
## 2 Detector description
The LHCb detector [8] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of $b$\- and
$c$-hadrons. The detector includes a high precision tracking system consisting
of a silicon-strip vertex detector surrounding the $pp$ interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet with a
bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip
detectors and straw drift-tubes placed downstream. The combined tracking
system has a momentum resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Data were taken
with both magnet polarities to reduce systematic effects due to detector
asymmetries. Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
and electromagnetic and hadronic calorimeters. Muons are identified by a muon
system composed of alternating layers of iron and multiwire proportional
chambers. The trigger consists of a hardware stage based on information from
the calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction.
Events with a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ final state are triggered using two hardware
trigger decisions: the single-muon decision, which requires one muon candidate
with a transverse momentum $p_{\mathrm{T}}$ larger than 1.5 $\mathrm{GeV}/c$,
and the di-muon decision, which requires two muon candidates with transverse
momenta $p_{\mathrm{T}_{1}}$ and $p_{\mathrm{T}_{2}}$ satisfying the relation
$\sqrt{p_{\mathrm{T}_{1}}\cdot p_{\mathrm{T}_{2}}}>1.3~{}\mathrm{GeV}/c$. The
di-muon trigger decision in the software trigger requires muon pairs of
opposite charge with $p_{\mathrm{T}}>500~{}\mathrm{MeV}/c$, forming a common
vertex and with an invariant mass in excess of $2.9~{}\mathrm{GeV}/c^{2}$.
## 3 Event selection
In this analysis, the decays $B^{+}\rightarrow\psi
K^{+}$($B^{0}\rightarrow\psi K^{*0}$, $B^{0}_{s}\rightarrow\psi\phi$) are
reconstructed, where $\psi$ represents $\psi(2S)$ or
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, reconstructed in the
$\psi\rightarrow\mu^{+}\mu^{-}$ decay modes. A $K^{+}$($K^{*0}$, $\phi$)
candidate is added to the di-muon pair to form a $B^{+}$($B^{0}$, $B^{0}_{s}$
) candidate.
The starting point of the analysis is the reconstruction of either a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi(2S)$ meson decaying
into a di-muon pair. Candidates are formed from pairs of opposite sign tracks
that both have a transverse momentum larger than
500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Good reconstruction quality is
assured by requiring the $\chi^{2}$ per degree of freedom of the track fit to
satisfy $\chi^{2}/\rm{ndf}<5$. Both tracks must be identified as muons. This
is achieved by requiring the muon identification variable, the difference in
logarithm of the likelihood of the muon and hadron hypotheses [9] provided by
the muon detection system, to satisfy $\Delta\log\mathcal{L}^{\mu-h}>-5$. The
muons are required to form a common vertex of good quality
($\mathrm{\chi^{2}_{vtx}}<20$). The resulting di-muon candidate is required to
have decay length significance from its associated primary vertex greater than
5 and have an invariant mass between 3020 and
3135${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in the case of a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate or between 3597 and
3730${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for a $\psi(2S)$ candidate.
These correspond to [$-5\sigma$; $3\sigma$] windows around the nominal mass.
The asymmetric window allows for the QED radiative tail.
The selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$
candidates are then combined with a $K^{+}$, $K^{*0}$ or $\phi$ to create $B$
meson candidates. Only the $K^{*0}\rightarrow K^{+}\pi^{-}$ and
$\phi\rightarrow K^{+}K^{-}$ decay modes are considered. Pion-kaon separation
is provided by the ring-imaging Cherenkov detectors. To identify kaons the
difference in logarithm of the likelihood of the kaon and pion hypotheses [9]
is required to satisfy $\Delta\log\mathcal{L}^{K-\pi}>-5$. In the case of
pions the difference in logarithm of the likelihood of the pion and kaon
hypotheses [9] is required to satisfy $\Delta\log\mathcal{L}^{\pi-K}>-5$. As
in the case of muons, a cut is applied on the track $\mathrm{\chi^{2}/ndf}$
provided by the track fit at 5. The kaons and pions are required to have a
transverse momentum larger than 250${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
to have an impact parameter significance with respect to any primary vertex
larger than 2. In the $B^{0}$ channel, the mass of the kaon and pion system is
required to be
$842<M_{K^{+}\pi^{-}}<942{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and in the
$B^{0}_{s}$ channel the mass of the kaon pair is required to be
$1010<M_{K^{+}K^{-}}<1030{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
In addition, we require the decay time of the $B$ candidate ($c\tau$) to be
larger than 100$\,\upmu\rm m$ to reduce the large combinatorial background
from particles produced in the primary $pp$ interaction. A global refit of the
three-prong (four-prong) combination is performed with a primary vertex
constraint and with the di-muon pair mass constrained to the nominal value
[10] using the Decay Tree Fit (DTF) procedure [11]. The reduced $\chi^{2}$ of
this fit ($\mathrm{\chi^{2}_{DTF}/ndf}$) is required to be less than 5, where
the DTF algorithm takes into account the number of decay products to determine
the number of degrees of freedom. The $B^{+}$ candidates, where a muon from
the $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ decay is reconstructed as both muon
and kaon, are removed by requiring the angle between the same sign muon and
kaon to be greater than 3 mrad.
LHCbLHCbLHCbLHCbLHCbLHCb(a)(b)(c)(d)(e)(f)$\mathrm{M}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}}~{}[{\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}]$$\mathrm{M}_{\psi(2S)K^{+}K^{-}}~{}[{\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}]$$\mathrm{M}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\pi^{-}}~{}[{\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}]$$\mathrm{M}_{\psi(2S)K^{+}\pi^{-}}~{}[{\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}]$$\mathrm{M}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}~{}[{\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}]$$\mathrm{M}_{\psi(2S)K^{+}}~{}[{\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}]$
Candidates/(4 ${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)Candidates/(4
${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)Candidates/(5
${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)Candidates/(5
${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)Candidates/(4
${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)Candidates/(4
${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)
Figure 1: Mass distributions of (a)
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, (b)
$B^{+}\rightarrow\psi(2S)K^{+}$, (c)
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-}$,
(d) $B^{0}\rightarrow\psi(2S)K^{+}\pi^{-}$, (e)
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
and (f) $B^{0}_{s}\rightarrow\psi(2S)K^{+}K^{-}$. The total fitted function
(solid) and the combinatorial background (dashed) are shown. The variation in
resolution of the different modes is fully consistent with the energy released
in the decays and in agreement with simulation.
## 4 Measurement of
$\boldsymbol{N_{\psi(2S)X}/N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X}}$
The mass distributions for selected candidates are shown in Fig. 1. The number
of the $B^{+}\rightarrow\psi K^{+}$ candidates is estimated by performing an
unbinned maximum likelihood fit. The same procedure is used to determine the
number of the $B^{0}\rightarrow\psi K^{+}\pi^{-}$ candidates in a
$842~{}<~{}M_{K^{+}\pi^{-}}~{}<~{}942{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
mass window and the number of the $B^{0}_{s}\rightarrow\psi K^{+}K^{-}$
candidates in a
$1010~{}<~{}M_{K^{+}K^{-}}~{}<~{}1030{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
mass window. The number of signal candidates is determined by fitting a
double-sided Crystal Ball function [12, 13] for signal together with an
exponential function to model the background. The tail parameters of the
Crystal Ball function are fixed to values determined from simulation.
The $B^{0}$ mass distributions include the contributions from resonant decays
($B^{0}\rightarrow\psi K^{*0}$), non-resonant decays ($B^{0}\rightarrow\psi
K^{+}\pi^{-}$) and combinatorial background. The contributions from resonant
and non-resonant modes are separated with the sPlot technique [14]. The
$K^{+}\pi^{-}$ invariant mass is used as a discriminating variable to unfold
the $B^{0}$ mass distribution of non-$K^{*0}$ $K^{+}\pi^{-}$ combinations. A
fit is then performed to the unfolded $B^{0}$ distribution, which contains
both non-resonant $B^{0}\rightarrow\psi K^{+}\pi^{-}$ decays and background,
to determine the number of non-resonant decays. The final number of resonant
decays is calculated by subtracting the number of non-resonant decays from the
total number of decays. For the $B^{0}_{s}$ modes the number of non-resonant
decays are obtained by a similar procedure using the $K^{+}K^{-}$ invariant
mass as the discriminating variable. The signal yields and their ratios are
summarized in Table 1.
Table 1: Summary of the signal yields for the six $B$ modes considered and the ratios of the number of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ decays: $N^{\rm{total}}$ is the summed signal yield for resonant and non-resonant modes, $N^{\rm{non-res}}$ is the signal yield for non-resonant modes only and $N^{\rm{res}}_{\psi X}$ is the signal yield for resonant decays (through $K^{*0}$ or $\phi$). The uncertainties are statistical only. $B$ decay modes | $N^{\rm{total}}$ | $N^{\rm{non-res}}$ | $N^{\rm{res}}$ | $N^{\rm{res}}_{\psi(2S)X}/N^{\rm{res}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X}$
---|---|---|---|---
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $141,769\pm 410$ | — | $141,769\pm 410$ | $0.0857\pm 0.0009$
$B^{+}\rightarrow\psi(2S)K^{+}$ | $12,154\pm 130$ | — | $12,154\pm 130$
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-}$ | $35,770\pm 207$ | $1,253\pm 30$ | $34,517\pm 209$ | $0.0612\pm 0.0018$
$B^{0}\rightarrow\psi(2S)K^{+}\pi^{-}$ | $2,223\pm 60$ | $112\pm 12$ | $2,111\pm 61$
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ | $7,654\pm 92$ | $66\pm 13$ | $7,588\pm 93$ | $0.0652\pm 0.0034$
$B^{0}_{s}\rightarrow\psi(2S)K^{+}K^{-}$ | $495\pm 25$ | $0^{+1}_{-0}$ | $495\pm 25$
## 5 Efficiencies and systematic uncertainties
The branching fraction ratio is calculated using
$\frac{{\cal B}(B\rightarrow\psi(2S)X)}{{\cal
B}(B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)}=\frac{N^{\rm{res}}_{\psi(2S)X}}{N^{\rm{res}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X}}\times\frac{\varepsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X}}{\varepsilon_{\psi(2S)X}}\times\frac{{\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})}{{\cal
B}(\psi(2S)\rightarrow\mu^{+}\mu^{-})},$ (1)
where $N^{\mathrm{res}}$ is the number of signal candidates and $\varepsilon$
is the overall efficiency.
The overall efficiency is the product of the geometrical acceptance of the
detector, the combined reconstruction and selection efficiency, and the
trigger efficiency. The efficiency ratio is estimated using simulation for all
six modes. The simulation samples used are based on the Pythia 6.4 generator
[15] configured with the parameters detailed in Ref. [16]. Final state QED
radiative corrections are included using the Photos package [17]. The EvtGen
[18] and Geant4 [19] packages are used to generate hadron decays and simulate
interactions in the detector, respectively. The digitized output is passed
through a full simulation of both the hardware and software trigger and then
reconstructed in the same way as the data.
The overall efficiency ratio is $0.901\pm 0.016$, $1.011\pm 0.014$ and
$0.994\pm 0.014$ for the $B^{+}$, the $B^{0}$ and the $B^{0}_{s}$ channels
respectively. Since the selection criteria for
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ and
$B\rightarrow\psi(2S)X$ decays are identical, the ratio of efficiencies is
expected to be close to unity. The deviation of the overall efficiency ratio
from unity in the case of the $B^{+}\rightarrow\psi K^{+}$ decays is due to
the difference between the $p_{\rm{T}}$ spectra of muons for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ decays. For the
$B^{0}$ and $B^{0}_{s}$ channels this difference is small. It has been checked
that the behaviour of the efficiencies of all selection criteria is consistent
in the data and simulation.
Since the decay products in each of the pairs of channels considered have
similar kinematics, most uncertainties cancel in the ratio. The different
contributions to the systematic uncertainties affecting this analysis are
discussed in the following and summarized in Table 2.
Table 2: Systematic uncertainties (in %) on the relative branching fractions. Source | $B^{+}$ channel | $B^{0}$ channel | $B^{0}_{s}$ channel
---|---|---|---
non-resonant decays | — | $1.5$ | $3.4$
data-simulation agreement | $1.7$ | $0.5$ | $2.0$
magnet polarity | $1.4$ | $0.6$ | $0.7$
finite simulation sample size | $0.3$ | $0.5$ | $0.6$
trigger | $1.1$ | $1.1$ | $1.1$
background shape | $0.6$ | $0.2$ | $0.2$
signal shape | $0.7$ | $0.8$ | $0.5$
angular distribution | — | $\\!\\!\\!\\!\\!\\!<0.1$ | $0.6$
particle misidentification | $0.4$ | $\\!\\!\\!\\!\\!\\!<0.1$ | $\\!\\!\\!\\!\\!\\!<0.1$
Sum in quadrature | $2.7$ | $2.2$ | $4.3$
The dominant source of systematic uncertainty arises from the subtraction of
the non-resonant components in the $B^{0}$ and the $B^{0}_{s}$ decays. The
non-resonant background is studied with two alternative methods. First,
determining the number of $B^{0}_{(s)}\rightarrow\psi K^{*0}(\phi)$ decays
directly using the sPlot technique by unfolding and fitting the $B^{0}_{(s)}$
mass distribution of candidates containing genuine $K^{*0}(\phi)$ resonances.
Second, using the $B^{0}_{(s)}$ mass distribution as the discriminating
variable to unfold the $K^{+}\pi^{-}(K^{+}K^{-})$ mass distribution of genuine
$B^{0}_{(s)}$ candidates and fitting this distribution to determine the number
of non-resonant decays. The corresponding uncertainties are found to be 1.5%
in the $B^{0}$ channel and 3.4% in the $B^{0}_{s}$ channel.
The other important source of uncertainty arises from the estimation of the
efficiencies due to the potential disagreement between data and simulation.
This is studied by varying the selection criteria in data and simulation. The
corresponding uncertainties are found to be 1.7% in the $B^{+}$ channel, 0.5%
in the $B^{0}$ channel and 2.0% in the $B^{0}_{s}$ channel. The observed
difference in the efficiency ratios for the two magnet polarities is
conservatively taken as an estimate of the systematic uncertainty. This is
1.4% in the $B^{+}$ channel, 0.6% in the $B^{0}$ channel and 0.7% in the
$B^{0}_{s}$ channel.
The trigger is highly efficient in selecting $B$ meson decays with two muons
in the final state. For this analysis the di-muon pair is required to trigger
the event. Differences in the trigger efficiency between data and simulation
are studied in the data using events which were triggered independently on the
di-muon pair [20]. Based on these studies, an uncertainty of 1.1% is assigned.
A further uncertainty arises from the imperfect knowledge of the shape of the
signal and background in the $B$ meson mass distribution. To estimate this
effect, a linear and a quadratic function are considered as alternative models
for the background mass distribution. In addition, a double Gaussian shape and
a sum of double-sided Crystal Ball and Gaussian shapes are used as alternative
models for the signal shape. The maximum observed change in the ratio of
yields in the $\psi(2S)$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
modes is taken as systematic uncertainty.
The central value of the relative efficiency is determined by assuming that
the angular distribution of the $B\rightarrow\psi(2S)X$ decay is the same as
that of the $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$. The
systematic uncertainty due to the unknown polarization of the $\psi(2S)$ in
the $B$ meson decays is estimated as follows. The simulation samples were re-
weighted to match the angular distributions found from the data and the
relative efficiency was recalculated. The difference between the baseline
analysis and the re-weighted simulation is taken as the systematic
uncertainty, as shown in Table 2.
Finally, the uncertainty due to potential contribution from the Cabibbo-
suppressed mode with a $\pi$ misidentified as $K$ is found to be 0.4% in the
$B^{+}$ channel and negligible in the $B^{0}$ and $B^{0}_{s}$ channels. The
uncertainty due to the cross-feed between $B^{0}$ and $B^{0}_{s}$ channels
with a $\pi$ misidentified as $K$ (or a $K$ misidentified as $\pi$) is
negligible.
## 6 Results
Since the di-electron branching fractions are measured more precisely than
those of the di-muon decay modes, we assume lepton universality and take
$R_{\psi}={\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})/{\cal
B}(\psi(2S)\rightarrow\mu^{+}\mu^{-})={\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow e^{+}e^{-})/{\cal
B}(\psi(2S)\rightarrow e^{+}e^{-})=7.69\pm 0.19$ [10]. The results are
combined using Eq. 1 to give
$\begin{array}[]{lll}\frac{{\cal B}(B^{+}\rightarrow\psi(2S)K^{+})}{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})}&=&0.594~{}\pm 0.006\,({\rm stat})\pm 0.016\,({\rm syst})\pm
0.015\,(R_{\psi}),\\\ \vskip 3.0pt\cr\frac{{\cal
B}(B^{0}\rightarrow\psi(2S)K^{*0})}{{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0})}&=&0.476~{}\pm 0.014\,({\rm stat})\pm 0.010\,({\rm syst})\pm
0.012\,(R_{\psi}),\\\ \vskip 3.0pt\cr\frac{{\cal
B}(B^{0}_{s}\rightarrow\psi(2S)\phi)}{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)}&=&0.489~{}\pm 0.026\,({\rm stat})\pm 0.021\,({\rm syst})\pm
0.012\,(R_{\psi}),\\\ \end{array}$
where the first uncertainty is statistical, the second is systematic and the
third is the uncertainty on the $R_{\psi}$ value [10].
The resulting branching fraction ratios are compatible with, but significantly
more precise than, the current world averages of ${\cal
B}(B^{+}\rightarrow\psi(2S)K^{+})/{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})=0.60\pm
0.07$ and ${\cal B}(B^{0}_{s}\rightarrow\psi(2S)\phi)/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=0.53\pm 0.10$ [10] and the CDF result of ${\cal
B}(B^{0}\rightarrow\psi(2S)K^{*0})/{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0})=0.515\pm 0.113\pm 0.052$ [2]. The
$B^{0}_{s}\rightarrow\psi(2S)\phi$ decay is particulary interesting since,
with more data, it can be used for the measurement of $C\\!P$ violation in
$B^{0}_{s}$ mixing.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
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* [2] CDF collaboration, F. Abe et al., Observation of $B^{+}\rightarrow\psi(2S)K^{+}$ and $B^{0}\rightarrow\psi(2S)K^{*}(892)^{0}$ decays and measurements of B-meson branching fractions into ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ final states, Phys. Rev. D58 (1998) 072001, arXiv:hep-ex/9803013
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|
arxiv-papers
| 2012-05-04T11:07:44 |
2024-09-04T02:49:30.524426
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1205.0918"
}
|
1205.0934
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-115 LHCb-PAPER-2011-041 31 May 2012
Measurement of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ branching fraction
The LHCb collaboration †††Authors are listed on the following pages.
The $B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ branching fraction is measured in a
data sample corresponding to $0.41\>\mbox{\,fb}^{-1}$ of integrated luminosity
collected with the LHCb detector at the LHC. This channel is sensitive to the
penguin contributions affecting the $\sin 2\beta$ measurement from $B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$. The time-integrated branching
fraction is measured to be ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})=(1.83\pm 0.28)\times 10^{-5}$. This is
the most precise measurement to date.
(Submitted to Phys. Lett. B)
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, L. Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6,
G. Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, F. De
Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31,
O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A.
Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9, A. Sparkes47, P.
Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S. Stoica26, S.
Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K. Subbiah35, S.
Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S. T’Jampens4, E.
Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van Tilburg11, V.
Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N. Torr52, E.
Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38,
M. Ubeda Garcia35, A. Ukleja25, P. Urquijo53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, H. Voss10, R. Waldi56, S. Wandernoth11, J. Wang53,
D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, K.
Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, O.
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Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated to 6
56Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
In the Standard Model (SM) $C\\!P$ violation arises through a single phase in
the quark mixing matrix [1, *Cabibbo:1963yz]. In decays of neutral $B$ mesons
to a final state which is accessible to both $B$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$, the interference between the
amplitude for the direct decay and the amplitude for decay via oscillation
leads to a time-dependent $C\\!P$-violating asymmetry between the decay time
distributions of the two mesons. The mode $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ allows for the measurement of such an asymmetry, which is parametrised by
the $B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing phase
$\phi_{d}$. In the SM this phase is equal to $2\beta$ [3], where $\beta$ is
one of the angles of the unitarity triangle of the mixing matrix. This phase
is already measured by the $B$ factories [4, *Babar:2009yr] but an improved
measurement is necessary to resolve conclusively the present tension in the
unitarity triangle fits [6] and determine possible small deviations from the
SM value. To achieve the required precision, knowledge of the doubly Cabibbo-
suppressed higher order perturbative corrections, known as _penguin diagrams_
, becomes mandatory. The contributions of these penguin diagrams are difficult
to calculate reliably and therefore need to be extracted directly from
experimentally accessible observables. Due to $SU(3)$ flavour symmetry, these
penguin diagrams can be studied in other decay modes where they are not
suppressed relative to the tree level diagram. The $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode is the most promising candidate from the theoretical perspective
since it is related to the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode through the interchange of all $d$ and $s$ quarks ($U$-spin symmetry,
a subgroup of $SU(3)$) [7] and there is a one-to-one correspondence between
all decay topologies in these modes, as illustrated in Fig. 1. A further
discussion regarding the theory of this decay and its potential at LHCb is
given in Ref. [8, *DeBruyn:2010ge].
Figure 1: Decay topologies contributing to the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ and $B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ channel: tree diagram to the left and
penguin diagram to the right.
To extract the parameters related to penguin contributions in these decays, a
time-dependent $C\\!P$ violation study of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode is required. The measurement of its branching fraction is an
important first step, allowing to test the $U$-spin symmetry assumption that
lies at the basis of the proposed approach. The CDF collaboration reported the
first observation of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ decay [10]. This letter presents a more precise measurement of this
branching fraction at the LHCb experiment.
The strategy of the analysis is to measure the ratio of $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ and $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ event yields, which is then converted into a $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ branching fraction. We make use of the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}$ branching fraction and
of the ratio of $B^{0}_{s}$ to $B^{0}$ meson production at the LHC, denoted
$f_{s}/f_{d}$ [11, *Aaij:2011jp].
We use an integrated luminosity of $0.41$$\mbox{\,fb}^{-1}$ of $pp$ collision
data recorded at a centre-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$
during 2010 and the first half of 2011. The detector [13] is a single-arm
spectrometer designed to study particles containing $b$ or $c$ quarks. It
includes a high precision tracking system consisting of a silicon-strip vertex
detector surrounding the $pp$ interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov (RICH) detectors. Muons are
identified by a muon system composed of alternating layers of iron and
multiwire proportional chambers.
The signal simulation sample used for this analysis was generated using the
Pythia $6.4$ generator [14] configured with the parameters detailed in Ref.
[15]. The EvtGen [16], Photos [17] and Geant4 [18] packages were used to decay
unstable particles, generate QED radiative corrections and simulate
interactions in the detector, respectively.
## 2 Data samples and selection
We search for $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ decays111$B$ stands for $B^{0}$ or
$B^{0}_{s}$. where ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ and $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$. Events are selected by a trigger system
consisting of a hardware trigger, which requires muon or hadron candidates
with high transverse momentum with respect to the beam direction, $p_{\rm T}$,
followed by a two stage software trigger [19]. In the first stage a simplified
event reconstruction is applied. Events are required to have either two
oppositely charged muons with combined mass above
$2.7\>\mathrm{\,Ge\kern-1.00006ptV}/c^{2}$, or at least one muon or one
high-$p_{\rm T}$ track ($\mbox{$p_{\rm
T}$}>1.8\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) with a large impact parameter
with respect to any primary vertex. In the second stage a full event
reconstruction is performed and only events containing
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$
$\mu^{+}$$\mu^{-}$ candidates are retained.
In order to reduce the data to a manageable level, very loose requirements are
applied to suppress background while keeping the signal efficiency high.
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are created from
pairs of oppositely charged muons that have a common vertex and a mass in the
range $3030$–$3150\>\mathrm{\,Me\kern-1.00006ptV}/c^{2}$. The latter
corresponds to about eight times the $\mu^{+}\mu^{-}$ mass resolution at the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass and covers part of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ radiative tail. The
$K^{0}_{\rm\scriptscriptstyle S}$ selection requires two oppositely charged
particles reconstructed in the tracking stations on either side of the magnet,
both with hits in the vertex detector (long $K^{0}_{\rm\scriptscriptstyle S}$
candidate) or not (downstream $K^{0}_{\rm\scriptscriptstyle S}$ candidate).
The $K^{0}_{\rm\scriptscriptstyle S}$ candidates must be made of tracks
forming a common vertex and have a mass within eight standard deviations of
the $K^{0}_{\rm\scriptscriptstyle S}$ mass and must not be compatible with the
$\mathchar 28931\relax$ mass under the mass hypothesis that one of the two
tracks is a proton and the other a pion.
We select $B$ candidates from combinations of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidates with mass
$m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ in the range $5200$–$5500~{}\mathrm{\,Me\kern-1.00006ptV}/c^{2}$. The
latter is computed with the masses of the $\mu^{+}\mu^{-}$ and
$\pi^{+}\pi^{-}$ pairs constrained to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ masses, respectively. The mass and decay
time of the $B$ are obtained from a decay chain fit [20] that in addition
constrains the $B$ candidate to originate from the primary vertex. The
$\chi^{2}$ of the fit, which has eight degrees of freedom, is required to be
less than $128$ and the estimated uncertainty on the $B$ mass must not exceed
$30\>\mathrm{\,Me\kern-1.00006ptV}/c^{2}$. $B$ candidates are required to have
a decay time larger than $0.2\>{\rm\,ps}$ and $K^{0}_{\rm\scriptscriptstyle
S}$ candidates to have a flight distance larger than five times its
uncertainty. The offline selected signal candidate is required to be that used
for the trigger decision at both software trigger stages. About 1% of the
selected events have several candidates sharing some final state particles. In
such cases one candidate per event is selected randomly.
## 3 Measurement of event yields
Following the selection described above, a neural network (NN) classifier [21]
is used to further discriminate between signal and background. The NN is
trained entirely on data, using samples that are independent of those used to
make the measurements. The training maximises the separation of signal and
background events using weights determined by the _sPlot_ technique [22]. We
use the $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ signal in the data as a proxy for the
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ decay. The background events are
taken from mass sidebands in the region 5390–5500
$\mathrm{\,Me\kern-1.00006ptV}/c^{2}$, thus avoiding the $B^{0}_{s}$ signal
region. A normalisation sample of one quarter of the candidates randomly
selected is left out in the NN training to allow an unbiased measurement of
the $B^{0}$ yield.
Figure 2: Mass distribution of the $B$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ candidates used to determine the PDF. The solid line is the total PDF
composed of the $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ signal shown in grey and the
combinatorial background represented by the dotted line.
We perform an unbinned maximum likelihood fit to the mass distribution of the
selected candidates, shown in Fig. 2, and use it to assign background and
signal weights to each candidate. The probability density function (PDF) is
defined as the sum of a $B^{0}$ signal component, a combinatorial background
and a small contribution from partially reconstructed $B$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$$X$ decays at masses below the $B^{0}$ mass. The mass lineshape of the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ signal in both data and simulation
exhibits non-Gaussian tails on both sides of the signal peak due to detector
resolutions depending on angular distributions in the decay. We model the
signal shape by an empirical model composed of two Crystal Ball (CB) functions
[23], one of which has the tail extending to high masses. The two CB
components are constrained to have the same peak and width, which are allowed
to vary in the fit. The parameters describing the CB tails are taken from
$B^{+}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{+}$
events which exhibit the same behaviour as $B$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$. The combinatorial background is described by a second order polynomial.
The $B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ signal is not included in this fit.
We extract $(14.4\pm 0.2)\times 10^{3}$ $B^{0}$ events from the fit.
The NN uses information about the candidate kinematics, vertex and track
quality, impact parameter, particle identification information from the RICH
and muon detectors, as well as global event properties like track and primary
vertex multiplicities. The variables that are used in the NN are chosen not to
induce a correlation with the mass distribution. This was verified using
simulated events.
To maximise the separation power, a first NN classifier using only the five
most discriminating variables is used to remove 80% of the background events
while keeping 95% of the $B^{0}$ signal. These variables are the $\chi^{2}$ of
the decay chain fit, the angle between the $B$ momentum and the vector from
the primary vertex to the decay vertex, the $p_{\rm T}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$, the estimated uncertainty on the $B$ mass
and the impact parameter $\chi^{2}$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$.
The weighting procedure is then repeated on the remaining candidates and a
second NN classifier containing 31 variables is trained. A cut is then made on
the second NN output in order to optimise the expected sensitivity to the
$B^{0}_{s}$ yield [24].
For the candidates passing the NN requirement, we determine the ratio of
$B^{0}_{s}$ and $B^{0}$ yields for candidates containing a downstream
$K^{0}_{\rm\scriptscriptstyle S}$ or a long $K^{0}_{\rm\scriptscriptstyle S}$
separately. The $B^{0}$ yield is measured in an unbinned likelihood fit to the
normalisation sample and scaled to the full sample. The $B^{0}_{s}$ yield is
fitted on the full sample. In both fits, the PDF is identical to that used to
determine the _sWeights_ with the addition of a PDF for the $B^{0}_{s}$
component, which is constrained to have the same shape as the $B^{0}$ PDF,
shifted by the measured $B^{0}_{s}-B^{0}$ mass difference [25]. The results of
the fits on the full samples are shown in Fig. 3 separately for candidates
with downstream and long $K^{0}_{\rm\scriptscriptstyle S}$.
Figure 3: Fit to full sample after the optimal NN cut has been applied with
downstream $K^{0}_{\rm\scriptscriptstyle S}$ to the left and long
$K^{0}_{\rm\scriptscriptstyle S}$ to the right.
The fitted yields are listed in Table 1. The long and downstream results are
compatible with each other and are combined using a weighted average.
Table 1: $B^{0}$ and $B^{0}_{s}$ yields. Only statistical errors are quoted. The $B^{0}$ yield is obtained in a fit to one quarter of the events which have not been used in the NN training (normalisation sample) and then scaled to the full sample. | downstream $K^{0}_{\rm\scriptscriptstyle S}$ | long $K^{0}_{\rm\scriptscriptstyle S}$
---|---|---
$B^{0}$ in normalisation sample | $1502\ \pm\ 39$ | $970\ \pm\ 31$
$B^{0}$ in normalisation sample (scaled to full) | $6007\ \pm\ 157$ | $3879\ \pm\ 124$
$B^{0}_{s}$ in full sample | $72\ \pm\ 11$ | $44\ \pm\ 8$
Ratio of $B^{0}_{s}$ to $B^{0}$ | $0.0120\ \pm\ 0.0018$ | $0.0112\ \pm\ 0.0020$
Ratio of $B^{0}_{s}$ to $B^{0}$ (weighted average, $r$) | $0.0117\pm 0.0014$
## 4 Corrections and systematic uncertainties
Differences in the total selection efficiencies between the $B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ and $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ arise because of the slight difference in momentum spectra of the $B$
mesons and/or the final state particles. We find, using simulated events, that
the geometrical acceptance of the LHCb detector is lower for the $B^{0}_{s}$
mode by $(1.3\pm 0.5)\%$ where the error is due to the limited sample of
simulated events. We correct for the ratio of acceptances and assign a
conservative systematic uncertainty of 1.8%, which is the sum of the measured
difference and its error.
The trigger, reconstruction and selection efficiencies also depend on the
transverse momentum of the final state particles. Applying the trigger
transverse momentum cuts on simulated $B^{0}$ and $B^{0}_{s}$ decays we find
differences of up to 1%, which is taken as systematic uncertainty.
Due to the selection cuts and the correlation of the neural network with the
decay time, a decay time acceptance function results in different selection
efficiencies for the $B^{0}_{s}$ and the $B^{0}$. We determine the lifetime
acceptance of the whole selection chain using simulated events, and find that
the ratio of the time-integrated decay time distributions for $B^{0}$ and
$B^{0}_{s}$ is ${0.975\pm 0.007}$. The uncertainties on the parametrisation of
the lifetime acceptance cancel almost perfectly in the ratio, while the ones
related to the $B^{0}$ and $B^{0}_{s}$ lifetimes and the $B^{0}_{s}$ decay
width difference $\Delta\Gamma_{s}$ do not.
The largest systematic uncertainty comes from the assumed mass PDF, in
particular the fraction of the positive tail of the $B^{0}$ extending below
the $B^{0}_{s}$ signal. We have studied the magnitude of this effect by
leaving both tails of the CB shapes free in the fit, or by allowing the two CB
shapes to have different widths. The maximal deviation we observe in the
ratios of downstream or long candidates is 5%, which we take as systematic
uncertainty. The effect of the uncertainty on the $B^{0}_{s}$–$B^{0}$ mass
difference is found to be $0.4\%$.
The corrections and systematic uncertainties affecting the branching fraction
ratio are listed in Table 2. The total uncertainty is obtained by adding all
the uncertainties in quadrature.
Table 2: Summary of corrections and systematic uncertainties on the ratio of branching fractions. Source | Correction factor
---|---
Geometrical acceptance ($\epsilon_{\text{geom}}$) | $0.987\pm 0.018$
Trigger and reconstruction | ${1.000\pm 0.010}$
Decay time acceptance ($\epsilon_{\text{time}}$) | ${0.975\pm 0.007}$
Mass shape | ${1.000\pm 0.050}$
$B^{0}_{s}$-$B^{0}$ mass difference | ${1.000\pm 0.004}$
Total | $0.962\pm 0.053$
We verify that the global event variable distributions, like the number of
primary vertices and the hit multiplicities, are the same for $B^{0}$ and
$B^{0}_{s}$ initial states using the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channel. We verify that the NN classifier is stable even when variables are
removed from the training. We search for peaking backgrounds in simulated $b$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$X$ events, and in
data by inverting the $\mathchar 28931\relax$ veto and the
$K^{0}_{\rm\scriptscriptstyle S}$ flight distance cut. No evidence of peaking
backgrounds is found. All these tests give results compatible with the
measured ratio though with a larger statistical uncertainty.
## 5 Determination of branching fraction
Using the measured ratio $r=0.0117\pm 0.0014$ of $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ and $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ yields, the geometrical
($\epsilon_{\text{geom}}$) and lifetime ($\epsilon_{\text{time}}$) acceptance
ratios, and assuming $f_{s}/f_{d}=0.267{\>}^{+{\>}0.021}_{-{\>}0.020}$ [11,
*Aaij:2011jp] we measure the ratio of branching fractions
$\displaystyle\frac{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})}{{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})}$
$\displaystyle=r\times\epsilon_{\text{geom}}\times\epsilon_{\text{time}}\times\frac{f_{d}}{f_{s}}$
(1) $\displaystyle=0.0420\pm 0.0049\text{\>(stat)}\pm 0.0023\text{\>(syst)}\pm
0.0033{\>(f_{s}/f_{d})}$
where the quoted uncertainties are statistical, systematic, and due to the
uncertainly in $f_{s}/f_{d}$, respectively. Using the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}$ branching fraction of
$(8.71\pm 0.32)\times 10^{-4}$ [26], we determine the time-integrated
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ branching fraction
$\displaystyle{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})=\left[1.83\right.$ $\displaystyle\pm
0.21\text{\>(stat)}\pm 0.10\text{\>(syst)}\pm 0.14{\>(f_{s}/f_{d})}$
$\displaystyle\pm\left.0.07\text{\>({$\cal B$}($B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}${}$K^{0}$))}\right]\times
10^{-5}$
where the last uncertainty comes from the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}$ branching fraction. This
result is compatible with, and more precise than, the previous measurement
[10].
## 6 Comparison with $SU(3)$ expectations
It was pointed out in Ref. [27] that because of the sizable decay width
difference between the heavy and light eigenstates of the $B^{0}_{s}$ system,
there is an ambiguity in the definition of the branching fractions of
$B^{0}_{s}$ decays. Due to $B^{0}_{s}$ mixing, a branching fraction defined as
the ratio of the time integrated number of $B^{0}_{s}$ decays to a final state
and the total number of $B^{0}_{s}$ mesons, is not equal to the
$C\\!P$-average of the decay rates in the flavour eigenstate basis
${\cal B}(B^{0}_{s}\rightarrow
f)_{\text{theo}}=\frac{\tau_{B^{0}_{s}}}{2}\left(\Gamma(B^{0}_{s}\rightarrow
f)+\Gamma(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
f)\right)\big{|}_{t=0},$ (2)
used in the theoretical predictions; the restriction to $t=0$ removes the
effects due to the non-zero $B_{s}$ decay width. To obtain the latter quantity
from the time-integrated decay rates the following correction factor
$\frac{1-y_{s}^{2}}{1+{\cal A}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}_{\Delta\Gamma}y_{s}}=0.936\pm 0.015,$
(3)
is applied, where $y_{s}=\Delta\Gamma_{s}/2\Gamma_{s}$ is the normalised decay
width difference between the light and heavy states and ${\cal
A}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}_{\Delta\Gamma}$ is the final-state dependent asymmetry of the $B^{0}_{s}$
decay rates to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ final state. In calculating this
correction factor we use $y_{s}=0.075\pm 0.010$ [28, *Asner:2010qj] and the SM
expectation
$\mathcal{A}_{\Delta\Gamma_{s}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}=0.84\pm 0.18$ [27].
With this correction, and assuming ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})_{\text{theo}}=\frac{1}{2}{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0})_{\text{theo}}$ we get the
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ branching fraction at $t=0$
$\displaystyle{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0})_{\text{theo}}=(3.42$
$\displaystyle\pm 0.40\text{\>(stat)}\pm 0.19\text{\>(syst)}\pm
0.27{\>(f_{s}/f_{d})}$ $\displaystyle\pm 0.13\text{\>({$\cal B$}($B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}${}$K^{0}$))}\pm
0.05\>(y_{s},\mathcal{A}_{\Delta\Gamma_{s}}))\cdot 10^{-5}.$
This branching fraction can be compared to theoretical expectations from
$SU(3)$ symmetry, which implies an equality of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ and $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$\pi^{0}$ decay widths [8]
$\Xi_{SU(3)}\equiv\frac{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0})_{\text{theo}}}{2{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}\pi^{0})}\frac{\tau_{B^{0}}}{\tau_{B^{0}_{s}}}\frac{\left[m_{B^{0}}\Phi(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}\pi^{0})\right]^{3}}{\left[m_{B^{0}_{s}}\Phi(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0})\right]^{3}}\,\stackrel{{\scriptstyle
SU(3)}}{{\longrightarrow}}1,$ (4)
where the factor two is associated with the wave function of the $\pi^{0}$,
$\tau_{B^{0}_{(s)}}$ is the mean $B^{0}_{(s)}$ lifetime and $\Phi$ refers to
the two-body phase-space factors; see e.g. Ref. [7].
Taking the measured ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0})_{\text{theo}}$ and using the world
average [26, 25] for all other quantities, this ratio becomes
$\Xi_{SU(3)}=0.98\pm 0.18$
and is consistent with theoretical expectation of unity under $SU(3)$
symmetry.
## 7 Conclusion
The branching fraction of the Cabibbo-suppressed decay $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ is measured in a
$0.41\>\mbox{\,fb}^{-1}$ data sample collected with the LHCb detector. We
determine the ratio of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ and $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ branching fractions to be
$\frac{{\cal B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})}{{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})}=0.0420\pm 0.0049\text{\>(stat)}\pm
0.0023\text{\>(syst)}\pm 0.0033{\>(f_{s}/f_{d})}.$ Using the world-average
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}$
branching fraction we get the time-integrated branching fraction ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})=[1.83\pm 0.21\text{\>(stat)}\pm
0.10\text{\>(syst)}\pm 0.14{\>(f_{s}/f_{d})}\pm 0.07\text{\>({$\cal
B$}($B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}${}$K^{0}$))}]\times 10^{-5}$. The total uncertainty of 16% is dominated
by the statistical uncertainty. This branching fraction is compatible with
expectations from $SU(3)$.
With larger data samples, a time dependent $C\\!P$-violation measurement of
this decay will be possible, allowing the experimental determination of the
penguin contributions to the $\sin 2\beta$ measurement from $B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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|
arxiv-papers
| 2012-05-04T12:27:55 |
2024-09-04T02:49:30.532011
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito,\n A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back,\n V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I.\n Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R.\n Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S.\n Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn, S. De\n Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De\n Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D.\n Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista,\n F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J.\n Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T.\n Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P. F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, R. F.\n Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M.\n Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.\n M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, S. Miglioranzi, D. A. Milanes,\n M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo,\n D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch,\n A. Puig Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S.\n Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues,\n F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M.\n Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S.\n Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza De Paula,\n B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S.\n Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S.\n Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P.\n Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, H. Voss, R. Waldi, S. Wandernoth, J. Wang,\n D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D.\n Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson,\n J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing,\n Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Patrick Koppenburg",
"url": "https://arxiv.org/abs/1205.0934"
}
|
1205.0975
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-109 LHCb-PAPER-2012-003 January 27, 2014
Observation of double charm production involving open charm in pp collisions
at $\sqrt{s}=7~{}\mathrm{TeV}$
The LHCb collaboration†††Authors are listed on the following pages.
The production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
mesons accompanied by open charm, and of pairs of open charm hadrons are
observed in pp collisions at a centre-of-mass energy of 7 TeV using an
integrated luminosity of $355~{}\mathrm{pb}^{-1}$ collected with the LHCb
detector. Model independent measurements of absolute cross-sections are given
together with ratios to the measured
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and open charm
cross-sections. The properties of these events are studied and compared to
theoretical predictions.
(Published in the Journal of High Energy Physics.)
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, K. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-Joergensen52, N. Torr52, E.
Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38,
M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J.
Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, H.
Voss10, R. Waldi55, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42,
A.D. Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Due to the high energy and luminosity of the LHC, charm production studies can
be carried out in a new kinematic domain with unprecedented precision. As the
cross-sections of open charm [1] and charmonium [2] production are large, the
question of multiple production of these states in a single proton-proton
collision naturally arises. Recently, studies of double charmonium and
charmonium with associated open charm production have been proposed as probes
of the quarkonium production mechanism [3]. In pp collisions, additional
contributions from other mechanisms, such as Double Parton Scattering (DPS)
[4, 5, 6, 7] or the intrinsic charm content of the proton [8] to the total
cross-section, are possible, though these constributions may not be mutually
exclusive.
In this paper, both the production of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons together with
an associated open charm hadron (either a $\mathrm{D}^{0}$, $\mathrm{D}^{+}$,
$\mathrm{D}^{+}_{\mathrm{s}}$ or $\Lambda_{\mathrm{c}}^{+}$)111The inclusion
of charge-conjugate modes is implied throughout this paper, unless explicitly
stated otherwise. and double open charm hadron production are studied in pp
collisions at a centre-of-mass energy of 7 TeV. We denote the former process
as ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and
the latter as $\mathrm{C}\mathrm{C}$. In addition, as a control channel,
$\mathrm{c}\mathrm{\bar{c}}$ events where two open charm hadrons are
reconstructed in the LHCb fiducial volume (denoted
$\mathrm{C}\overline{\mathrm{C}}$) are studied. While the production of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ events
have not been observed before in hadron interactions, evidence for the
production of four charmed particles in pion-nuclear interactions has been
reported by the WA75 collaboration [9].
Leading order (LO) calculations for the
$\mathrm{gg}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ process in
perturbative QCD exist and give consistent results [10, 11, 12]. In the LHCb
fiducial region ($2<y_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<4.5$,
$p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<10~{}\mathrm{GeV}/c$), where
$y_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ and
$p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$
stand for rapidity and transverse momentum respectively, the calculated
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production
cross-section is $4.1\pm 1.2~{}\mathrm{nb}$ [12] in agreement with the
measured value of $5.1\pm 1.0\pm 1.1~{}\mathrm{nb}$ [13]. Similar calculations
for the
$\mathrm{gg}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{c}\bar{\mathrm{c}}$ and
$\mathrm{gg}\rightarrow\mathrm{c}\bar{\mathrm{c}}\mathrm{c}\bar{\mathrm{c}}$
matrix elements exist [14, 15]. The calculated cross-sections for these
processes in the acceptance region considered here
($2<y_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}},y_{\mathrm{C}}<4$,
$p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<12~{}\mathrm{GeV}/c$,
$3<p^{\mathrm{T}}_{\mathrm{C}}<12~{}\mathrm{GeV}/c$) are
$\sigma\left({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}+{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\overline{\mathrm{C}}\right)\sim 18~{}\mathrm{nb}$ and
$\sigma\left(\mathrm{C}\mathrm{C}+{\overline{\mathrm{C}}}{\overline{\mathrm{C}}}\right)\sim
100~{}\mathrm{nb}$, where $\mathrm{C}$ stands for the open charm hadron. The
predictions are summarized in Table 1. These LO $\alpha_{s}^{4}$ perturbative
QCD results are affected by uncertainties originating from the selection of
the scale for the $\alpha_{s}$ calculation that can amount to a factor of two.
The DPS contribution can be estimated, neglecting partonic correlations in the
proton, as the product of the cross-sections of the sub-processes involved
divided by an effective cross-section [4, 5, 6, 7]
$\sigma^{\mathrm{DPS}}_{\mathrm{C}_{1}\mathrm{C}_{2}}=\upalpha\dfrac{\sigma_{\mathrm{C}_{1}}\times\sigma_{\mathrm{C}_{2}}}{\sigma^{\mathrm{DPS}}_{\mathrm{eff}}},$
(1)
where $\upalpha=\tfrac{1}{4}$ if $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are
identical and non-self-conjugate (e.g. $\mathrm{D}^{0}\mathrm{D}^{0}$),
$\upalpha=1$ if $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are different and either
$\mathrm{C}_{1}$ or $\mathrm{C}_{2}$ is self-conjugate (e.g.
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$),
and $\upalpha=\tfrac{1}{2}$ otherwise. Using this equation and the measured
single charm cross-sections given in [1, 2] together with the effective cross-
section measured in multi-jet events at the Tevatron
$\sigma^{\mathrm{DPS}}_{\mathrm{eff}}=14.5\pm 1.7^{+1.7}_{-2.3}~{}\mathrm{mb}$
[16], the size of this contribution is estimated (see Table 1). However, this
approach has been criticized as being too naive [17].
Extra charm particles in the event can originate from the sea charm quarks of
the interacting protons themselves. Estimates for the possible contribution in
the fiducial volume used here are given in the Appendix and summarized in
Table 1. It should be stressed that the charm parton density functions are not
well known, nor are the $p^{\mathrm{T}}$ distributions of the resulting charm
particles, so these calculations should be considered as upper estimates.
Table 1: Estimates for the production cross-sections of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and $\mathrm{C}\mathrm{C}$ modes in the LHCb fiducial range given by the leading order $\mathrm{gg}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{c}\bar{\mathrm{c}}$ matrix element, $\sigma^{\mathrm{gg}}$ [14, 15, 18], the double parton scattering approach, $\sigma^{\mathrm{DPS}}$ and the sea charm quarks from the interacting protons, $\sigma^{\mathrm{sea}}$. Mode | $\sigma^{\mathrm{gg}}$ | $\sigma^{\mathrm{DPS}}$ | $\sigma^{\mathrm{sea}}$
---|---|---|---
[14, 15] | [18]
| $\left[\mathrm{nb}\right]$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$ | $10\pm 6\phantom{0}$ | $7.4\pm 3.7$ | $146\pm 39\phantom{0}$ | $220$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ | $5\pm 3$ | $2.6\pm 1.3$ | $60\pm 17$ | $100$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $1.0\pm 0.8$ | $1.5\pm 0.7$ | $24\pm 7\phantom{0}$ | $30$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\Lambda^{+}_{\mathrm{c}}$ | $0.8\pm 0.5$ | $0.9\pm 0.5$ | $56\pm 22$ |
| $\left[\rm\,\upmu b\right]$
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ | | | $1.0\phantom{0}\pm 0.25$ | $1.5$
$\mathrm{D}^{0}{}\mathrm{D}^{+}$ | | | $0.85\pm 0.2\phantom{0}$ | $1.4$
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ | | | $0.33\pm 0.07$ | $0.4$
$\mathrm{D}^{0}{}\Lambda_{\mathrm{c}}^{+}$ | | | $0.75\pm 0.25$ |
$\mathrm{D}^{+}{}\mathrm{D}^{+}$ | | | $0.17\pm 0.05$ | $0.3$
$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$ | | | $0.14\pm 0.03$ | $0.2$
$\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$ | | | $0.32\pm 0.12$ |
## 2 The LHCb detector and dataset
The LHCb detector [19] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, and is designed for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the proton-proton interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov (RICH) detectors. Photon,
electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, and electromagnetic
and hadronic calorimeters. Muons are identified by a muon system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
consists of a hardware stage based on information from the calorimeter and
muon systems, followed by a software stage which applies a full event
reconstruction.
Events with a ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ final state are triggered using two
hardware trigger decisions: the single-muon decision, which requires one muon
candidate with a transverse momentum $p^{\mathrm{T}}$ larger than 1.5
$\mathrm{GeV}/c$, and the di-muon decision, which requires two muon candidates
with transverse momenta $p^{\mathrm{T}}_{1}$ and $p^{\mathrm{T}}_{2}$
satisfying the relation $\sqrt{p^{\mathrm{T}}_{1}\cdot
p^{\mathrm{T}}_{2}}>1.3~{}\mathrm{GeV}/c$. The di-muon trigger decision in the
software trigger requires muon pairs of opposite charge with
$p^{\mathrm{T}}>500~{}\mathrm{MeV}/c$, forming a common vertex and with an
invariant mass $2.97<m_{\upmu^{+}\upmu^{-}}<3.21~{}\mathrm{GeV}/c^{2}$. Events
with purely hadronic final states are accepted by the hardware trigger if
there is a calorimeter cluster with transverse energy
$E^{\mathrm{T}}>3.6~{}\mathrm{GeV}$. The software trigger decisions select
generic displaced vertices from tracks with large $\chi^{2}$ of impact
parameter with respect to all primary pp interaction vertices in the event,
providing high efficiency for purely hadronic decays [20].
To prevent a few events with high occupancy from dominating the CPU time in
the software trigger, a set of global event cuts is applied on the hit
multiplicities of each sub-detector used by the pattern recognition
algorithms. These cuts were chosen to reject events with a large number of
pile-up interactions with minimal loss of data.
The data used for this analysis comprises $355\pm 13~{}\mathrm{pb}^{-1}$ of
$\mathrm{pp}$ collisions at a centre-of-mass energy of
$\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$ collected by the LHCb experiment
in the first half of the 2011 data-taking period. Simulation samples used are
based on the Pythia 6.4 generator [21] configured with the parameters detailed
in Ref. [22]. The EvtGen [23] and Geant4 [24] packages are used to describe
hadron decays and for the detector simulation, respectively. The prompt
charmonium production is simulated in Pythia according to the leading-order
colour-singlet and colour-octet mechanisms.
## 3 Event selection
To select events containing multiple charm hadrons, first
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$, $\mathrm{D}^{0}$,
$\mathrm{D}^{+}$, $\mathrm{D}^{+}_{\mathrm{s}}$ and $\Lambda_{\mathrm{c}}^{+}$
candidates are formed from charged tracks reconstructed in the spectrometer.
Subsequently, these candidates are combined to form
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$,
$\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ candidates.
Well reconstructed tracks are selected for these studies by requiring that the
$\chi^{2}_{\rm{tr}}$ provided by the track fit satisfy
$\chi^{2}_{\mathrm{tr}}/{\mathrm{ndf}}<5$, where ndf represents the number of
degrees of freedom in the fit, and that the transverse momentum is greater
than $650~{}(250)~{}\mathrm{MeV}/c$ for each muon (hadron) candidate. For each
track, the global likelihoods of the muon and hadron hypotheses provided by
reconstruction of the muon system are evaluated, and well identified muons are
selected by a requirement on the difference in likelihoods
$\Delta\ln\mathcal{L}_{\upmu/\mathrm{h}}>0$.
Good quality particle identification by the ring-imaging Cherenkov detectors
is ensured by requiring the momentum of the hadron candidate to be between
$3.2~{}\mathrm{GeV}/c$ ($10~{}\mathrm{GeV}/c$ for protons) and
$100~{}\mathrm{GeV}/c$, and the pseudorapidity to be in the range $2<\eta<5$.
To select kaons (pions) the corresponding difference in logarithms of the
global likelihood of the kaon (pion) hypothesis provided by the RICH system
with respect to the pion (kaon) hypothesis,
$\Delta\ln\mathcal{L}_{\mathrm{K}/\uppi}$
($\Delta\ln\mathcal{L}_{\uppi/\mathrm{K}}$), is required to be greater than 2.
For protons, the differences in logarithms of the global likelihood of the
proton hypothesis provided by the RICH system with respect to the pion and
kaon hypotheses, are required to be
$\Delta\ln\mathcal{L}_{\mathrm{p}/\uppi}>10$ and
$\Delta\ln\mathcal{L}_{\mathrm{p}/\mathrm{K}}>10$, respectively.
Pions, kaons and protons, used for the reconstruction of long-lived charm
particles, are required to be inconsistent with being produced in a pp
interaction vertex. Only particles with a minimal value of impact parameter
$\chi^{2}$ with respect to any reconstructed proton-proton collision vertex
$\chi^{2}_{\mathrm{IP}}>9$, are considered for subsequent analysis. These
selection criteria are summarized in Table 2.
Table 2: Selection criteria for charged particles used for the reconstruction
of charm hadrons. Track selection
---
$\upmu^{\pm},\mathrm{h}^{\pm}$ | $\chi^{2}_{\mathrm{tr}}/\mathrm{ndf}<5~{}$
$\upmu^{\pm}$ | $p^{\mathrm{T}}>650~{}\mathrm{MeV}/c$
$\mathrm{h}^{\pm}$ | $p^{\mathrm{T}}>250~{}\mathrm{MeV}/c~{}\&~{}2<\eta<5~{}\&~{}\chi^{2}_{\mathrm{IP}}>9$
$\uppi^{\pm},\mathrm{K}^{\pm}$ | $3.2<p<100~{}\mathrm{GeV}/c$
$\mathrm{p}^{\pm}$ | $10<p<100~{}\mathrm{GeV}/c$
Particle identification
$\upmu^{\pm}$ | $\Delta\ln\mathcal{L}_{\upmu/\mathrm{h}}>0$
$\uppi^{\pm}$ | $\Delta\ln\mathcal{L}_{\uppi/\mathrm{K}}>2$
$\mathrm{K}^{\pm}$ | $\Delta\ln\mathcal{L}_{\mathrm{K}/\uppi}>2$
$\mathrm{p},\bar{\mathrm{p}}$ | $\Delta\ln\mathcal{L}_{\mathrm{p}/\mathrm{K}}>10~{}\&~{}\Delta\ln\mathcal{L}_{\mathrm{p}/\uppi}>10$
The selected charged particles are combined to form
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$,
$\mathrm{D}^{0}\rightarrow\mathrm{K}^{-}\uppi^{+}$,
$\mathrm{D}^{+}\rightarrow\mathrm{K}^{-}\uppi^{+}\uppi^{+}$,
$\mathrm{D}^{+}_{\mathrm{s}}\rightarrow\mathrm{K}^{-}\mathrm{K}^{+}\uppi^{+}$
and $\Lambda_{\mathrm{c}}^{+}\rightarrow\mathrm{p}\mathrm{K}^{-}\uppi^{+}$
candidates. A vertex fit is made to all combinations and a selection criterion
on the corresponding $\chi^{2}_{\mathrm{VX}}$ applied. The transverse
momentum, $p^{\mathrm{T}}$, for open charm hadron candidates is required to be
larger than $3~{}\mathrm{GeV}/c$. To ensure that the long-lived charm particle
originates from a primary vertex, the minimal value of the charm particle’s
$\chi^{2}_{\mathrm{IP}}$ with respect to any of the reconstructed proton-
proton collision vertices is required to be $<9$. In addition, the decay time
$c\tau$ of long-lived charm mesons is required to be in excess of
$100\,\,\upmu\rm m$, and in the range $100<c\tau<500\,\,\upmu\rm m$ for
$\Lambda_{\mathrm{c}}^{+}$ candidates. To suppress the higher combinatorial
background for $\Lambda_{\mathrm{c}}^{+}$ candidates, only pions, kaons and
protons with a transverse momentum in excess of $0.5~{}\mathrm{GeV}/c$ are
used in this case.
A global decay chain fit of the selected candidates is performed [25]. For
channels containing a ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ meson it is required that the muons be consistent with originating
from a common vertex and that this be compatible with one of the reconstructed
$\mathrm{pp}$ collision vertices. In the case of long-lived charm hadrons, the
momentum direction is required to be consistent with the flight direction
calculated from the locations of the primary and secondary vertices. To remove
background from $\mathrm{b}$-hadron decays the reduced $\chi^{2}$ of this fit,
$\chi^{2}_{\mathrm{fit}}/\mathrm{ndf}$, is required to be $<5$. To further
reduce the combinatorial background as well as cross-feed due to particle
misidentification, for the decay mode
$\mathrm{D}^{0}\rightarrow\mathrm{K}^{-}\uppi^{+}$ a selection criterion on
the cosine of the angle between the kaon momentum in the $\mathrm{D}^{0}$
centre-of-mass frame and the $\mathrm{D}^{0}$ flight direction in the
laboratory frame, $\theta^{*}$ is applied. For
$\mathrm{D}^{+}_{\mathrm{s}}\rightarrow\mathrm{K}^{+}\mathrm{K}^{-}\uppi^{+}$
candidates, the invariant mass of the $\mathrm{K}^{+}\mathrm{K}^{-}$ system is
required to be consistent with the $\upphi$ meson mass. These selection
criteria are summarized in Table 3.
Table 3: Criteria used for the selection of charm hadrons. | | ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ | $\mathrm{D}^{0}$ | $\mathrm{D}^{+}$ | $\mathrm{D}^{+}_{\mathrm{s}}$ | $\Lambda_{\mathrm{c}}^{+}$
---|---|---|---|---|---|---
| | $\upmu^{+}\upmu^{-}$ | $\mathrm{K}^{-}\uppi^{+}$ | $\mathrm{K}^{-}\uppi^{+}\uppi^{+}$ | $\left(\mathrm{K}^{+}\mathrm{K}^{-}\right)_{\upphi}\uppi^{+}$ | $\mathrm{p}\mathrm{K}^{-}\uppi^{+}$
$y$ | | $2<y<4$ | $2<y<4$ | $2<y<4$ | $2<y<4$ | $2<y<4$
$p^{\mathrm{T}}$ | $\left[\mathrm{GeV}/c\right]$ | $<12$ | $3<p^{\mathrm{T}}<12$ | $3<p^{\mathrm{T}}<12$ | $3<p^{\mathrm{T}}<12$ | $3<p^{\mathrm{T}}<12$
$\chi^{2}_{\mathrm{VX}}$ | | $<20$ | $<9$ | $<25$ | $<25$ | $<25$
$\chi^{2}_{\mathrm{IP}}$ | | — | $<9$ | $<9$ | $<9$ | $<9$
$\chi^{2}_{\mathrm{fit}}/\mathrm{ndf}$ | | $<5$ | $<5$ | $<5$ | $<5$ | $<5$
$c\tau$ | $\left[{}\,\upmu\rm m\right]$ | — | $c\tau>100$ | $c\tau>100$ | $c\tau>100$ | $\begin{array}[]{c}c\tau>100\\\ c\tau<500\end{array}$
$\left|\cos\theta^{*}\right|$ | | — | $<0.9$ | — | — | —
$\mathrm{m}_{\mathrm{K}^{+}\mathrm{K}^{-}}$ | $\left[\mathrm{GeV}/c^{2}\right]$ | — | — | — | $<1.04$ | —
$\min p^{\mathrm{T}}_{\mathrm{h}^{\pm}}$ | $\left[\mathrm{GeV}/c\right]$ | — | — | — | — | $>0.5$
The invariant mass distributions for selected
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$, $\mathrm{D}^{0}$,
$\mathrm{D}^{+}$, $\mathrm{D}^{+}_{\mathrm{s}}$ and $\Lambda_{\mathrm{c}}^{+}$
candidates are presented in Figs. 1 and 2 for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and open charm
mesons, respectively. The distributions are modelled by a double-sided Crystal
Ball function [13, 26] for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$, and a modified Novosibirsk function [27]
for the $\mathrm{D}^{0}\rightarrow\mathrm{K}^{-}\uppi^{+}$,
$\mathrm{D}^{+}\rightarrow\mathrm{K}^{-}\uppi^{+}\uppi^{+}$ and
$\mathrm{D}^{+}_{\mathrm{s}}\rightarrow\mathrm{K}^{+}\mathrm{K}^{-}\uppi^{+}$
and $\Lambda_{\mathrm{c}}^{+}\rightarrow\mathrm{p}\mathrm{K}^{-}\uppi^{+}$
signals. In each case the combinatorial background component is modelled with
an exponential function. The signal yields are summarized in Table 4 together
with an estimate of the contamination from the decays of $\mathrm{b}$ hadrons,
$f_{\mathrm{b}}^{\mathrm{MC}}$. The latter has been estimated using simulated
events, normalized to the corresponding measured cross-sections.
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$$m_{\upmu^{+}\upmu^{-}}$$\left[\mathrm{GeV}/c^{2}\right]$
$\dfrac{\mathrm{dN}}{\mathrm{d}m_{\upmu^{+}\upmu^{-}}}~{}~{}\left[\dfrac{1}{1~{}\mathrm{MeV}/c^{2}}\right]$
$\begin{array}[]{r}\mathrm{LHCb}{}\end{array}$ Figure 1: Invariant mass
distribution for selected ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ candidates. The results of a fit to the model described in the text is
superimposed on a logarithmic scale. The solid line corresponds to the total
fitted PDF whilst the dotted line corresponds to the background component.
$\mathrm{D}^{0}\rightarrow\mathrm{K}^{-}\uppi^{+}$$\mathrm{D}^{+}\rightarrow\mathrm{K}^{-}\uppi^{+}{}\uppi^{+}$$\mathrm{D}^{+}_{\mathrm{s}}\rightarrow\mathrm{K}^{+}\mathrm{K}^{-}\uppi^{+}$$\Lambda_{\mathrm{c}}^{+}\rightarrow\mathrm{p}\mathrm{K}^{-}\uppi^{+}$$m_{\left(\mathrm{K^{+}K^{-}}\right)_{\upphi}\uppi^{+}}$$m_{\mathrm{p}\mathrm{K}^{-}\uppi^{+}}$$m_{\mathrm{K}^{-}\uppi^{+}}$$m_{\mathrm{K}^{-}\uppi^{+}\uppi^{+}}$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$
$\tfrac{\mathrm{dN}}{\mathrm{d}m_{\mathrm{KK}\uppi}}~{}~{}\left[\tfrac{1}{1~{}\mathrm{MeV}/c^{2}}\right]$
$\tfrac{\mathrm{dN}}{\mathrm{d}m_{\mathrm{K}\uppi}}~{}~{}\left[\tfrac{1}{1~{}\mathrm{MeV}/c^{2}}\right]$
$\tfrac{\mathrm{dN}}{\mathrm{d}m_{\mathrm{pK}\uppi}}~{}~{}\left[\tfrac{1}{1~{}\mathrm{MeV}/c^{2}}\right]$
$\tfrac{\mathrm{dN}}{\mathrm{d}m_{\mathrm{K}\uppi\uppi}}~{}~{}\left[\tfrac{1}{1~{}\mathrm{MeV}/c^{2}}\right]$
$\begin{array}[]{r}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{r}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$a) b) c) d) Figure 2: Invariant mass distributions for selected a) $\mathrm{D}^{0}$, b) $\mathrm{D}^{+}$, c) $\mathrm{D}^{+}_{\mathrm{s}}$ and d) $\Lambda_{\mathrm{c}}^{+}$ candidates. The solid line corresponds to the total fitted PDF whilst the dotted line shows the background component. Table 4: Yields, $S$, and contamination from $b$-hadron decays, $f_{\mathrm{b}}^{\mathrm{MC}}$, for the prompt charm signal. | | ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ | $\mathrm{D}^{0}$ | $\mathrm{D}^{+}$ | $\mathrm{D}^{+}_{\mathrm{s}}$ | $\Lambda_{\mathrm{c}}^{+}$
---|---|---|---|---|---|---
| | $\upmu^{+}\upmu^{-}$ | $\mathrm{K}^{-}\uppi^{+}$ | $\mathrm{K}^{-}\uppi^{+}\uppi^{+}$ | $\left(\mathrm{K}^{+}\mathrm{K}^{-}\right)_{\upphi}\uppi^{+}$ | $\mathrm{p}\mathrm{K}^{-}\uppi^{+}$
$S$ | $\left[10^{6}\right]$ | 49.57 | 65.77 | 33.25 | 3.59 | 0.637
$f_{\mathrm{b}}^{\mathrm{MC}}$ | $\left[\%\right]$ | 1.6 | 1.7 | 1.3 | 2.6 | 4.5
The selected charm candidates are paired to form di-charm candidates:
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$,
$\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$. A global fit of
the di-charm candidates is performed [25], similar to that described above for
single charm hadrons, which requires both hadrons to be consistent with
originating from a common vertex. The reduced $\chi^{2}$ of this fit,
$\chi^{2}_{\mathrm{global}}/\mathrm{ndf}$, is required to be less than 5. This
reduces the background from the pile-up of two interactions each producing a
charm hadron to a negligible level. The remaining contamination from the pile-
up and decays from beauty hadrons is extracted directly from the data as
follows. The distributions of $\chi^{2}_{\mathrm{global}}/\mathrm{ndf}$ for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$,
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ and
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ events are shown in Fig. 3. For
the region $\chi^{2}_{\mathrm{global}}/\mathrm{ndf}>5$ the distributions are
well described by functions of the form222The functional form is inspired by
the $\chi^{2}$ distribution.
$f\left(x\right)\propto(\alpha x)^{\frac{n}{2}-1}\mathrm{e}^{-\frac{\alpha
x}{2}},$ (2)
where $\alpha$ and $n$ are free parameters. Fits with this functional form are
used to extrapolate the yield in the region
$\chi^{2}_{\mathrm{global}}/\mathrm{ndf}>5$ to the region
$\chi^{2}_{\mathrm{global}}/\mathrm{ndf}<5$. Based on these studies we
conclude that background from pile-up is negligible.
$\log_{10}\chi^{2}_{\mathrm{global}}/\mathrm{ndf}$$\log_{10}\chi^{2}_{\mathrm{global}}/\mathrm{ndf}$
Candidates
Candidates
$\begin{array}[]{r}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{r}\mathrm{LHCb}{}\end{array}$a)
b) $\begin{array}[]{cl}{\text{\char
108}}&\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}\\\
{\color[rgb]{0,0,1}\square}&\mathrm{D}^{0}{}\mathrm{D}^{0}\end{array}$ Figure
3: a) Background subtracted distribution of
$\log_{10}\chi^{2}_{\mathrm{global}}/\mathrm{ndf}$ for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$
events. The solid line corresponds to the fit result in the region
$\chi^{2}_{\mathrm{global}}/\mathrm{ndf}>5$ by the function described in the
text, the dashed line corresponds to the extrapolation of the fit results to
the $\chi^{2}_{\mathrm{global}}/\mathrm{ndf}<5$ region. b) Likewise for
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ (blue squares and red line) and
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ (black circles and green line).
The mass distributions for all pairs after these criteria are applied are
shown in Figs. 4 to 8 for channels with sufficiently large data samples.
$m_{\mathrm{K}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{KK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{pK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\upmu^{+}\upmu^{-}}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\upmu^{+}\upmu^{-}}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\upmu^{+}\upmu^{-}}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\upmu^{+}\upmu^{-}}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
Candidates
Candidates
Candidates
Candidates
a)b)c)d)$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$
Figure 4: Invariant mass distributions for a)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$, b)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$, c)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and d)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}$ candidates.
$m_{\mathrm{K}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{KK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{pK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
Candidates
Candidates
Candidates
Candidates
a)b)c)d)LHCb LHCb LHCb LHCb Figure 5: Invariant mass distributions for
$\mathrm{D}^{0}{}\mathrm{C}$ candidates: a) $\mathrm{D}^{0}{}\mathrm{D}^{0}$,
b) $\mathrm{D}^{0}{}\mathrm{D}^{+}$, c)
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ and d)
$\mathrm{D}^{0}{}\Lambda_{\mathrm{c}}^{+}$.
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{KK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{pK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
Candidates
Candidates
Candidates
a)b)c)LHCb LHCb LHCb Figure 6: Invariant mass distributions for
$\mathrm{D}^{+}{}\mathrm{C}$ candidates: a) $\mathrm{D}^{+}{}\mathrm{D}^{+}$,
b) $\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$, and c)
$\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$.
$m_{\mathrm{K}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{KK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{pK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi}~{}~{}~{}\left[\mathrm{GeV}/c^{2}\right]$
Candidates
Candidates
Candidates
Candidates
a)b)c)d)LHCb LHCb LHCb LHCb Figure 7: Invariant mass distributions for
$\mathrm{D}^{0}{}\bar{\mathrm{C}}$ candidates: a)
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, b)
$\mathrm{D}^{0}{}\mathrm{D}^{-}$, c)
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ and d)
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$.
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{KK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{pK}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
$m_{\mathrm{K}\uppi{}\uppi}~{}\left[\mathrm{GeV}/c^{2}\right]$
Candidates
Candidates
Candidates
a)b)c)LHCb LHCb LHCb Figure 8: Invariant mass distributions for
$\mathrm{C}\overline{\mathrm{C}}$ candidates: a)
$\mathrm{D}^{+}{}\mathrm{D}^{-}$, b)
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ and c)
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$.
## 4 Signal determination
The event yields are determined using unbinned extended maximum likelihood
fits to the mass distributions of the di-charm sample. The fit model is based
on the probability density functions (PDFs) for single open or hidden charm
production described in Section 3. These basic PDFs are used to build the
components of the two dimensional mass fit. Let $i$ and $j$ denote the two
resonance species. The reconstructed signal samples consist of the following
components:
* •
Di-charm signal. This is modelled by a product PDF of the individual signal
components for the first and the second particle.
* •
Combinatorial background. This is modelled by a product PDF of the individual
background components $i$ and $j$ denoted by $B_{i}(m_{i})$ and
$B_{j}(m_{j})$.
* •
Single production of component $i$ together with combinatorial background for
component $j$. This is modelled by a product PDF of the signal component $i$
denoted $S_{i}(m_{i})$ and the background component $j$ denoted
$B_{j}(m_{j})$.
* •
Single production of component $j$ together with combinatorial background for
component $i$. This is modelled by a product PDF of the signal component $j$
denoted $S_{j}(m_{j})$ and the background component $i$ denoted
$B_{i}(m_{i})$.
The total PDF is then
$\displaystyle F(m_{i},m_{j})$ $\displaystyle\propto$ $\displaystyle
N^{S_{i}\times S_{j}}\times S_{i}(m_{i})S_{j}(m_{j})+N^{S_{i}\times
B_{j}}\times S_{i}(m_{i})B_{j}(m_{j})$ (3) $\displaystyle+$ $\displaystyle
N^{B_{i}\times S_{j}}\times B_{i}(m_{i})S_{j}(m_{j})+N^{B_{i}\times
B_{j}}\times B_{\mathrm{i}}(m_{i})B_{j}(m_{j}),$
where $N^{S_{i}\times S_{j}}$, $N^{S_{i}\times B_{j}}$, $N^{B_{i}\times
S_{j}}$ and $N^{B_{i}\times B_{j}}$ are the yields of the four components
described above. The correctness of the fitting procedure is evaluated in
simulation studies. As discussed in Section 3 both the contribution of pile-up
background and $\mathrm{b}$-hadron decays is small and can be neglected. The
goodness of fit is found to be acceptable using the distance to the nearest
neighbour method described in Refs. [28, 29].
As a cross-check of the results, the signal yields have been determined from
the single charm hadron mass spectra using the technique described in Ref.
[13]. In this approach, for each pair of charm species the invariant mass
distributions of the first charm candidate are fitted to obtain the yield in
bins of the invariant mass of the second candidate and vice versa. This
technique gives signal yields consistent within 10% of the statistical
uncertainty and also allows the statistical significance of the result to be
easily evaluated. This exceeds five standard deviations for most of the modes
considered. The signal yields for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$,
$\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ events are
presented in Tables 5 and 6 together with the estimate of the goodness of fit.
Table 5: Yields of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ events, $S$, statistical significance of the signals, $S_{\sigma}$, determined from fits based on the technique described in Ref. [13], and goodness-of-fit characteristic ($\chi^{2}$ probability), $P$. When no significance is quoted, it is in excess of $8\sigma$. Mode | $S$ | $S_{\sigma}$ | $P~{}\left[\%\right]$
---|---|---|---
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$ | $4875\pm 86$ | | 59
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ | $3323\pm 71$ | | 26
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\phantom{0}328\pm 22$ | | 65
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\Lambda^{+}_{\mathrm{c}}$ | $\phantom{0}116\pm 14$ | $7.3\sigma$ | 98
Table 6: Yields of $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ events, $S$, statistical significance of the signals, $S_{\sigma}$, determined from fits based on the technique, described in Ref. [13], and goodness-of-fit characteristic, $P$. When no significance is quoted, it is in excess of $8\sigma$. Mode | $S$ | $S_{\sigma}$ | $P~{}\left[\%\right]$
---|---|---|---
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ | $\phantom{0}1087\pm 37\phantom{0}$ | | 4.5
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ | $10080\pm 105$ | | 33
$\mathrm{D}^{0}{}\mathrm{D}^{+}$ | $\phantom{0}1177\pm 39\phantom{0}$ | | 24
$\mathrm{D}^{0}{}\mathrm{D}^{-}$ | $11224\pm 112$ | | 36
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\phantom{00}111\pm 12\phantom{0}$ | $8\sigma$ | 10
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ | $\phantom{00}859\pm 31\phantom{0}$ | | 13
$\mathrm{D}^{0}{}\Lambda_{\mathrm{c}}^{+}$ | $\phantom{000}41\pm 8\phantom{00}$ | $5\sigma$ | 9
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$ | $\phantom{00}308\pm 19\phantom{0}$ | | 35
$\mathrm{D}^{+}{}\mathrm{D}^{+}$ | $\phantom{00}249\pm 19\phantom{0}$ | | 15
$\mathrm{D}^{+}{}\mathrm{D}^{-}$ | $\phantom{0}3236\pm 61\phantom{0}$ | | 67
$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\phantom{000}52\pm 9\phantom{00}$ | $5\sigma$ | 54
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ | $\phantom{00}419\pm 22\phantom{0}$ | | 59
$\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$ | $\phantom{000}21\pm 5\phantom{00}$ | $2.5\sigma$ | 36
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$ | $\phantom{00}137\pm 14\phantom{0}$ | $8\sigma$ | 7
## 5 Efficiency correction
The yields are corrected for the detection efficiency to obtain the measured
cross-sections. The efficiency for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$,
$\mathrm{C}\overline{\mathrm{C}}$ and $\mathrm{C}\mathrm{C}$ events
$\varepsilon^{\mathrm{tot}}$ is computed for each signal event and is
decomposed into three factors
$\varepsilon^{\mathrm{tot}}=\varepsilon^{\mathrm{reco}}\times\varepsilon^{\mathrm{ID}}\times\varepsilon^{\mathrm{trg}},$
(4)
where $\varepsilon^{\mathrm{reco}}$ is the efficiency for acceptance,
reconstruction and selection, $\varepsilon^{\mathrm{ID}}$ is the efficiency
for particle identification and $\varepsilon^{\mathrm{trg}}$ is the trigger
efficiency. The first term in Eq. (4), $\varepsilon^{\mathrm{reco}}$ is
factorized into the product of efficiencies for the first and second charm
particle and a correction factor
$\varepsilon^{\mathrm{reco}}=\varepsilon^{\mathrm{reco}}_{1}\times\varepsilon^{\mathrm{reco}}_{2}\times\xi^{\mathrm{trk}},$
(5)
where the efficiencies $\varepsilon^{\mathrm{reco}}_{(1,2)}$ are evaluated
using the simulation, and the correction factor333This is the product of the
individual corrections for each track. $\xi^{\mathrm{trk}}$ is determined from
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ data using a
tag-and-probe method and accounts for relative differences in the track
reconstruction efficiency between data and simulation.
The efficiency $\varepsilon^{\mathrm{reco}}_{i}$ is determined using the
simulation in bins of rapidity $y$ and transverse momentum $p^{\mathrm{T}}$ of
the charm hadron. In the case of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson, the effect of
the unknown polarization on the efficiency is accounted for by binning in
$|\cos\theta^{*}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}|$,
where $\theta^{*}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$
is the angle between the $\upmu^{+}$ momentum in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ centre-of-mass frame
and the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ flight
direction in the laboratory frame.
The efficiency for hadron identification as a function of momentum and
pseudorapidity is determined from the data using samples of
$\mathrm{D}^{*+}\rightarrow\left(\mathrm{D}^{0}\rightarrow\mathrm{K}^{-}\pi^{+}\right)\pi^{+}$,
and $\Lambda\rightarrow\mathrm{p}\pi^{-}$ [30, 31]. The efficiency for dimuon
identification,
$\varepsilon^{\mathrm{ID}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ is obtained from the analysis of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ sample as a function of transverse
momentum and rapidity of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$.
For the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$
sample the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ particle
is required to trigger the event whilst for the $\mathrm{C}\mathrm{C}$ and
$\mathrm{C}\overline{\mathrm{C}}$ case either of the two charm mesons could
trigger the event. The trigger efficiency for the di-charm system in the two
cases is thus
$\displaystyle\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}}$ $\displaystyle=$
$\displaystyle\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ (6a)
$\displaystyle\varepsilon^{\mathrm{trg}}_{\mathrm{C}\overline{\mathrm{C}},\mathrm{C}\mathrm{C}}$
$\displaystyle=$ $\displaystyle
1-(1-\varepsilon^{\mathrm{trg}}_{\mathrm{C}_{1}})\times(1-\varepsilon^{\mathrm{trg}}_{\mathrm{C}_{2}}).$
(6b)
In both cases the trigger efficiency for a single charm hadron
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ or $\varepsilon^{\mathrm{trg}}_{\mathrm{C}}$ is determined directly
from the data using the inclusive prompt charm sample as a function of $y$ and
$p^{\mathrm{T}}$. This is done using a method that exploits the fact that
events with prompt charm hadrons can be triggered either by the decay products
of the charm hadron, or by the rest of the event [13, 32]. The overlap between
the two cases allows the trigger efficiency to be estimated.
As discussed in Sect. 2, global event cuts are applied in the trigger on the
sub-detector hit multiplicites to reject complex events. The efficiency of
these cuts $\varepsilon^{\mathrm{GEC}}$ is studied using the distributions of
hit multiplicity after background subtraction. These distributions have been
extrapolated from the regions unaffected by the cuts into the potentially
affected regions and compared with the observed distributions in order to
determine $\varepsilon^{\mathrm{GEC}}$.
The efficiency-corrected signal yield $N^{\mathrm{corr}}$ is determined using
the ${}_{s}{\mathcal{P}}lot$ [33] technique. Each candidate is given a weight
for it to be signal, $\omega_{i}$, based on the result of the fit to the mass
distributions described before. The weight is then divided by the total event
efficiency and summed to give the efficiency-corrected yield
$N^{\mathrm{corr}}=\displaystyle\sum\limits_{i}\frac{\omega_{i}}{\varepsilon^{\mathrm{tot}}_{i}}.$
(7)
In the case of the $\mathrm{D}^{0}{}\mathrm{C}$ and
$\mathrm{D}^{0}{}\overline{\mathrm{C}}$ final states the corresponding yields
have been corrected to take into account the double Cabibbo-suppressed decay
(DCS) mode $\mathrm{D}^{0}{}\rightarrow\mathrm{K}^{+}\pi^{-}$, which mixes the
$\mathrm{D}^{0}{}\mathrm{C}$ and $\mathrm{D}^{0}{}\overline{\mathrm{C}}$
reconstructed final states
$\begin{pmatrix}N^{\prime}_{\mathrm{D}^{0}{}\mathrm{C}}\\\
N^{\prime}_{\mathrm{D}^{0}{}\bar{\mathrm{C}}}\end{pmatrix}=\frac{1}{\sqrt{1-r^{2}}}\begin{pmatrix}1&-r\\\
-r&1\end{pmatrix}\times\begin{pmatrix}N^{\mathrm{corr}}_{\mathrm{D}^{0}{}\mathrm{C}}\\\
N^{\mathrm{corr}}_{\mathrm{D}^{0}{}\bar{\mathrm{C}}}\end{pmatrix},$ (8)
where $r$ is
$r^{\mathrm{DCS}}=\dfrac{\Gamma\left(\mathrm{D}^{0}{}\rightarrow\mathrm{K}^{+}\pi^{-}\right)}{\Gamma\left(\mathrm{D}^{0}{}\rightarrow\mathrm{K}^{-}\pi^{+}\right)}=\left(3.80\pm
0.18\right)\times 10^{-3}$ [34]. This value of $r^{\mathrm{DCS}}$ accounts
also for the effect of $\mathrm{D}^{0}$-$\overline{\mathrm{D}}^{0}$ mixing.
For the $\mathrm{D}^{0}{}\mathrm{D}^{0}$ and
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ cases the value of
$r=2r^{\mathrm{DCS}}$ is used.
## 6 Systematic uncertainties
The sources of systematic uncertainty that enter into the cross-section
determination in addition to those related to the knowledge of branching
ratios and luminosity are discussed below. The dominant source of systematic
uncertainty arises from possible differences in the track reconstruction
efficiency between data and simulation which are not accounted for in the per-
event efficiency. This includes the knowledge of the hadronic interaction
length of the detector which results in an uncertainty of 2% per final state
hadron [32]. An additional uncertainty is due to the statistical uncertainty
on the determination of the per-event efficiency due to the finite size of the
simulation and calibration samples. This is estimated by varying the obtained
efficiencies within their corresponding uncertainties. The unknown
polarization of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
mesons affects the acceptance, reconstruction and selection efficiency
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ [2]. In this analysis the effect is reduced by explicitly taking into
account the dependence of
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ on $|\cos\theta^{*}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}|$ in the efficiency determination. The remaining dependence results in
a systematic uncertainty of 3% for channels containing a
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$.
Additional uncertainties are due to differences between data and simulation,
uncertainty on the global event cuts, knowledge of the branching fractions of
charm hadrons, ${\cal B}_{i}$. Uncertainties due to the parameterization of
the signal and background components are found to be negligible.
The absolute luminosity scale was measured at specific periods during the data
taking, using both van der Meer scans [35] where colliding beams are moved
transversely across each other to determine the beam profile, and a beam-gas
imaging method [36, 37]. For the latter, reconstructed beam-gas interaction
vertices near the beam crossing point determine the beam profile. The
knowledge of the absolute luminosity scale is used to calibrate the number of
tracks in the silicon-strip vertex detector, which is found to be stable
throughout the data-taking period and can therefore be used to monitor the
instantaneous luminosity of the entire data sample. The dataset for this
analysis corresponds to an integrated luminosity of $355\pm
13~{}\mathrm{pb}^{-1}$.
The sources of systematic uncertainty on the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$
production cross-section measurements are summarized in Table 7 and those for
open charm in Tables 8 and 9. The total systematic uncertainties have been
evaluated taking correlations into account where appropriate.
Table 7: Relative systematic uncertainties ($\%$) for the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ cross-sections. Source | ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$ | ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ | ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ | ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\Lambda^{+}_{\mathrm{c}}$
---|---|---|---|---
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ reconstruction | $\varepsilon^{\mathrm{reco}}_{1}$ | $1.3$ | $1.3$ | $1.3$ | $1.3$
$\mathrm{C}$ reconstruction | $\varepsilon^{\mathrm{reco}}_{2}$ | $0.7$ | $0.8$ | $1.7$ | $3.3$
Muon ID | $\varepsilon^{\mathrm{ID}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ | $1.1$ | $1.1$ | $1.1$ | $1.1$
Hadron ID | $\varepsilon^{\mathrm{ID}}_{\mathrm{had}}$ | $1.1$ | $1.9$ | $1.1$ | $1.5$
Tracking | $\xi^{\mathrm{trk}}$ | $4.9$ | $7.0$ | $7.0$ | $7.0$
Trigger | $\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}}$ | $3.0$ | $3.0$ | $3.0$ | $3.0$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ polarization | $\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ | $3.0$ | $3.0$ | $3.0$ | $3.0$
Global event cuts | $\varepsilon^{\mathrm{GEC}}$ | $0.7$ | $0.7$ | $0.7$ | $0.7$
Luminosity | $\mathcal{L}$ | $3.7$ | $3.7$ | $3.7$ | $3.7$
${\cal B}({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\rightarrow\upmu^{+}\upmu^{-})$ | ${\cal B}_{1}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$
$\mathrm{C}$ branching fractions | ${\cal B}_{2}$ | 1.3 | 4.3 | 6.0 | 26
Total | | 8 | 10 | 11 | 28
Table 8: Relative systematic uncertainties ($\%$) for the $\mathrm{D}^{0}\mathrm{C}$ cross-sections. The uncertainties for $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ are equal. Source | $\mathrm{D}^{0}{}\mathrm{D}^{0}$ | $\mathrm{D}^{0}{}\mathrm{D}^{+}$ | $\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\mathrm{D}^{0}{}\Lambda_{\mathrm{c}}^{+}$
---|---|---|---|---
$\mathrm{D}^{0}\mathrm{C}$ reconstruction | $\varepsilon^{\mathrm{reco}}_{1}\times\varepsilon^{\mathrm{reco}}_{2}$ | $1.4$ | $1.4$ | $2.3$ | $3.6$
Hadron ID | $\varepsilon^{\mathrm{ID}}_{\mathrm{had}}$ | $1.2$ | $1.8$ | $1.6$ | $2.4$
Tracking | $\xi^{\mathrm{trk}}$ | $8.5$ | $10.7\phantom{0}$ | $10.6\phantom{0}$ | $10.6\phantom{0}$
Trigger | $\varepsilon^{\mathrm{trg}}_{\mathrm{C}\mathrm{C},\mathrm{C}\overline{\mathrm{C}}}$ | 1.8 | 2.5 | 3.9 | 5.2
Global event cuts | $\varepsilon^{\mathrm{GEC}}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$
Luminosity | $\mathcal{L}$ | $3.7$ | $3.7$ | $3.7$ | $3.7$
${\cal B}(\mathrm{D}^{0}\rightarrow\mathrm{K}^{-}\pi^{+})$ | ${\cal B}_{1}$ | $1.3$ | $1.3$ | $1.3$ | $1.3$
$\mathrm{C}$ branching fractions | ${\cal B}_{2}$ | 1.3 | 4.3 | 6.0 | 26
Total | | 10 | 12 | 14 | 30
Table 9: Relative systematic uncertainties ($\%$) for the $\mathrm{D}^{+}\mathrm{C}$ cross-sections. The uncertainties for the $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ are equal. Source | $\mathrm{D}^{+}{}\mathrm{D}^{+}$ | $\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$
---|---|---|---
$\mathrm{D}^{+}\mathrm{C}$ reconstruction | $\varepsilon^{\mathrm{reco}}_{1}\times\varepsilon^{\mathrm{reco}}_{2}$ | $1.4$ | $2.2$ | $4.0$
Hadron ID | $\varepsilon^{\mathrm{ID}}_{\mathrm{had}}$ | $2.3$ | $2.4$ | $3.0$
Tracking | $\xi^{\mathrm{trk}}$ | $12.8\phantom{0}$ | $12.8\phantom{0}$ | $12.8\phantom{0}$
Trigger | $\varepsilon^{\mathrm{trg}}_{\mathrm{C}\mathrm{C},\mathrm{C}\overline{\mathrm{C}}}$ | 3.7 | 5.8 | 5.0
Global event cuts | $\varepsilon^{\mathrm{GEC}}$ | $1.0$ | $1.0$ | $1.0$
Luminosity | $\mathcal{L}$ | $3.7$ | $3.7$ | $3.7$
${\cal B}(\mathrm{D}^{+}\rightarrow\mathrm{K}^{-}\pi^{+}\pi^{+})$ | ${\cal B}_{1}$ | $4.3$ | $4.3$ | $4.3$
$\mathrm{C}$ branching fractions | ${\cal B}_{2}$ | 4.3 | 6.0 | 26
Total | | 17 | 17 | 31
## 7 Results
The model-independent cross-section for double charm production in the
fiducial range is computed as
$\sigma=\dfrac{N^{\mathrm{corr}}}{\mathcal{L}\times{\cal B}_{1}\times{\cal
B}_{2}\times\varepsilon^{\mathrm{GEC}}},$ (9)
where $\mathcal{L}$ is the integrated luminosity obtained as described in
Sect. 6, ${\cal B}_{(1,2)}$ stand for the corresponding branching ratios,
$\varepsilon^{\mathrm{GEC}}$ is the efficiency of the global event cuts, and
$N^{\mathrm{corr}}$ is the efficiency-corrected event yield, calculated
according to Eq. (7). The branching ratios used for these calculations are
taken from Ref. [34]. We reiterate that the inclusion of charge conjugate
processes is implied, so that e.g.,
$\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}}$
is the sum of production cross-sections for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{C}$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\overline{\mathrm{C}}$.
The cross-sections for the production of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and associated open
charm, $\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}}$, are measured in the fiducial volume
$2<y_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}},y_{\mathrm{C}}<4$,
$p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<12~{}\mathrm{GeV}/c$,
$3<p^{\mathrm{T}}_{\mathrm{C}}<12~{}\mathrm{GeV}/c$. The results are
summarized in Table 10 and Fig. 9.
The systematic uncertainties related to the reconstruction and trigger are
reduced if ratios to the cross-sections for prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$,
$\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$, and prompt
open charm production, $\sigma_{\mathrm{C}}$, with the same fiducial
requirements are considered (taking into account correlated uncertainties) [1,
2]. These ratios are presented in Table 11.
The cross-sections for $\mathrm{C}\mathrm{C}$ and
$\mathrm{C}\overline{\mathrm{C}}$ events in the fiducial volume
$2<y_{\mathrm{C}}<4$, $3<p^{\mathrm{T}}_{\mathrm{C}}<12~{}\mathrm{GeV}/c$ are
measured and listed in Table 12 and Fig. 9. The Table also includes the ratio
of $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ production
cross-sections,
$\sigma_{\mathrm{C}\mathrm{C}}/\sigma_{\mathrm{C}\overline{\mathrm{C}}}$, and
the ratios of the product of the prompt open charm cross-sections to the
$\mathrm{C}\mathrm{C}$ ($\mathrm{C}\overline{\mathrm{C}}$) cross-sections,
$\sigma_{\mathrm{C}_{1}}\sigma_{\mathrm{C}_{2}}/\sigma_{\mathrm{C}_{1}\mathrm{C}_{2}}$.
Several of the estimations given in Table 1 are also shown in Fig. 9 to
compare with our measurements. The expectations from gluon-gluon fusion
processes [14, 15, 18] are significantly below the measured cross-sections
while the DPS estimates qualitatively agree with them. The observed ratio of
$\mathrm{C}\mathrm{C}$/$\mathrm{C}\overline{\mathrm{C}}$ events is relatively
large, e.g. compared with
$\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}/\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=(5.1\pm 1.0\pm 1.1)\times 10^{-4}$ [13].
For the ratios $\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\sigma_{\mathrm{C}}/\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}}$ and
$\sigma_{\mathrm{C}_{1}}\sigma_{\mathrm{C}_{2}}/\sigma_{\mathrm{C}_{1}\mathrm{C}_{2}}$
listed in Tables 11 and 12, the systematic uncertainties largely cancel. In
addition, theoretical inputs such as the choice of the strong coupling
constant and the charm quark fragmentation fractions should cancel allowing a
more reliable comparison between theory and data. Figure 10 shows the ratios
$\mathcal{R}_{\mathrm{C}_{1}\mathrm{C}_{2}}$ defined as
$\mathcal{R}_{\mathrm{C}_{1}\mathrm{C}_{2}}\equiv\upalpha^{\prime}\dfrac{\sigma_{\mathrm{C}_{1}}\times\sigma_{\mathrm{C}_{2}}}{\sigma_{\mathrm{C}_{1}\mathrm{C}_{2}}},$
where $\upalpha^{\prime}$ is defined similarly to $\upalpha$ in Eq. (1) for
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and
$\mathrm{C}\mathrm{C}$ cases. When considering
$\mathrm{C}\overline{\mathrm{C}}$ production, $\upalpha^{\prime}=\tfrac{1}{4}$
is used for the $\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ and
$\mathrm{D}^{+}{}\mathrm{D}^{-}$ cases and $\upalpha^{\prime}=\tfrac{1}{2}$
for the other $\mathrm{C}\overline{\mathrm{C}}$ modes. For the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and
$\mathrm{C}\mathrm{C}$ cases these ratios have a clear interpretation in the
DPS approach [4, 5, 6] as the effective cross-section of Eq. (1) which should
be the same for all modes. For the $\mathrm{C}\overline{\mathrm{C}}$ case,
neglecting the contribution from
$\mathrm{c}\bar{\mathrm{c}}\mathrm{c}\bar{\mathrm{c}}$ production, this ratio
is related by a model-dependent kinematical factor to the total charm
production cross-section and should be independent of the final state under
consideration. The values for the effective DPS cross-section calculated from
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$
cross-section are in good agreement with the value measured in multi-jet
production at the Tevatron $\sigma^{\mathrm{DPS}}_{\mathrm{eff}}=14.5\pm
1.7^{+1.7}_{-2.3}~{}\mathrm{mb}$ [16].
Table 10: Production cross-sections for ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$. The first uncertainty is statistical, and the second is systematic. Mode | $\sigma~{}~{}\left[\mathrm{nb}\right]$
---|---
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$ | $161.0\pm 3.7\pm 12.2$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ | $\phantom{0}56.6\pm 1.7\pm\phantom{0}5.9$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\phantom{0}30.5\pm 2.6\pm\phantom{0}3.4$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\Lambda^{+}_{\mathrm{c}}$ | $\phantom{0}43.2\pm 7.0\pm 12.0$
Table 11: Ratios of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ production cross-section to prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ cross-section and prompt open charm cross-section, and ratios of the product of prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and open charm cross-sections to the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ cross-section. The first uncertainty is statistical, the second is systematic, and the third is due to the unknown polarization of the prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ [2]. Mode | $\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}}/\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}~{}\left[10^{-3}\right]$ | $\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}}/\sigma_{\mathrm{C}}~{}\left[10^{-4}\right]$ | $\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}\sigma_{\mathrm{C}}/\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}}~{}\left[\mathrm{mb}\right]$
---|---|---|---
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$ | $16.2\pm 0.4\pm 1.3^{+3.4}_{-2.5}$ | $6.7\pm 0.2\pm 0.5$ | $14.9\pm 0.4\pm 1.1^{+2.3}_{-3.1}$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ | $\phantom{0}5.7\pm 0.2\pm 0.6^{+1.2}_{-0.9}$ | $5.7\pm 0.2\pm 0.4$ | $17.6\pm 0.6\pm 1.3^{+2.8}_{-3.7}$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $\phantom{0}3.1\pm 0.3\pm 0.4^{+0.6}_{-0.5}$ | $7.8\pm 0.8\pm 0.6$ | $12.8\pm 1.3\pm 1.1^{+2.0}_{-2.7}$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\Lambda^{+}_{\mathrm{c}}$ | $\phantom{0}4.3\pm 0.7\pm 1.2^{+0.9}_{-0.7}$ | $5.5\pm 1.0\pm 0.6$ | $18.0\pm 3.3\pm 2.1^{+2.8}_{-3.8}$
Table 12: Production cross-sections for $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$, ratios of the $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ cross-sections and ratios of the product of prompt open charm cross-sections to the $\mathrm{C}\mathrm{C}$ ($\mathrm{C}\overline{\mathrm{C}}$) cross-sections. The first uncertainty is statistical and the second is systematic. The symmetry factor 2 is explicitly indicated for the $\mathrm{D}^{0}{}\mathrm{D}^{0}$, $\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, $\mathrm{D}^{+}{}\mathrm{D}^{+}$ and $\mathrm{D}^{+}{}\mathrm{D}^{-}$ ratios. Mode | $\sigma~{}~{}\left[\mathrm{nb}\right]$ | $\sigma_{\mathrm{C}\mathrm{C}}/\sigma_{\mathrm{C}\overline{\mathrm{C}}}~{}~{}\left[\%\right]$ | $\sigma_{\mathrm{C}_{1}}\sigma_{\mathrm{C}_{2}}/\sigma_{\mathrm{C}_{1}\mathrm{C}_{2}}~{}\left[\mathrm{mb}\right]$
---|---|---|---
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ | 690$\,\pm\,$ | 40 $\,\pm\,$ | 70 | 10.9 | $\,\pm\,$ | 0.8 | $2\times(42$$\,\pm\,$ | 3 $\,\pm\,$ | 4)
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ | 6230$\,\pm\,$ | 120 $\,\pm\,$ | 630 | $2\times(4.7$$\,\pm\,$ | 0.1 $\,\pm\,$ | 0.4)
$\mathrm{D}^{0}{}\mathrm{D}^{+}$ | 520$\,\pm\,$ | 80 $\,\pm\,$ | 70 | 12.8 | $\,\pm\,$ | 2.1 | 47$\,\pm\,$ | 7 $\,\pm\,$ | 4
$\mathrm{D}^{0}{}\mathrm{D}^{-}$ | 3990$\,\pm\,$ | 90 $\,\pm\,$ | 500 | 6.0$\,\pm\,$ | 0.2 $\,\pm\,$ | 0.5
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ | 270$\,\pm\,$ | 50 $\,\pm\,$ | 40 | 15.7 | $\,\pm\,$ | 3.4 | 36$\,\pm\,$ | 8 $\,\pm\,$ | 4
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ | 1680$\,\pm\,$ | 110 $\,\pm\,$ | 240 | 5.6$\,\pm\,$ | 0.5 $\,\pm\,$ | 0.6
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$ | 2010$\,\pm\,$ | 280 $\,\pm\,$ | 600 | — | 9$\,\pm\,$ | 2 $\,\pm\,$ | 1
$\mathrm{D}^{+}{}\mathrm{D}^{+}$ | 80$\,\pm\,$ | 10 $\,\pm\,$ | 10 | 9.6 | $\,\pm\,$ | 1.6 | $2\times(66$$\,\pm\,$ | 11 $\,\pm\,$ | 7)
$\mathrm{D}^{+}{}\mathrm{D}^{-}$ | 780$\,\pm\,$ | 40 $\,\pm\,$ | 130 | $2\times(6.4$$\,\pm\,$ | 0.4 $\,\pm\,$ | 0.7)
$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$ | 70$\,\pm\,$ | 15 $\,\pm\,$ | 10 | 12.1 | $\,\pm\,$ | 3.3 | 59$\,\pm\,$ | 15 $\,\pm\,$ | 6
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ | 550$\,\pm\,$ | 60 $\,\pm\,$ | 90 | 7$\,\pm\,$ | 1 $\,\pm\,$ | 1
$\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$ | 60$\,\pm\,$ | 30 $\,\pm\,$ | 20 | 10.7 | $\,\pm\,$ | 5.9 | 140$\,\pm\,$ | 70 $\,\pm\,$ | 20
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$ | 530$\,\pm\,$ | 130 $\,\pm\,$ | 170 | 15$\,\pm\,$ | 4 $\,\pm\,$ | 2
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$$\mathrm{D}^{0}{}\mathrm{D}^{-}$$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$$\mathrm{D}^{+}{}\mathrm{D}^{-}$$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$$\mathrm{D}^{0}{}\mathrm{D}^{0}$$\mathrm{D}^{0}{}\mathrm{D}^{+}$$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$$\mathrm{D}^{+}{}\mathrm{D}^{+}$$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$$\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}$$\sigma$$\left[\mathrm{nb}\right]$$\begin{array}[]{l}\mathrm{LHCb}{}\end{array}$
Figure 9: Measured cross-sections
$\sigma_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}}$, $\sigma_{\mathrm{C}\mathrm{C}}$ and
$\sigma_{\mathrm{C}\overline{\mathrm{C}}}$ (points with error bars) compared,
in ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$
channels, to the calculations in Refs. [14, 15] (hatched areas) and Ref. [18]
(shaded areas). The inner error bars indicate the statistical uncertainty
whilst the outer error bars indicate the sum of the statistical and systematic
uncertainties in quadrature. Charge-conjugate modes are included.
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$$\mathrm{D}^{0}{}\mathrm{D}^{-}$$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$$\mathrm{D}^{+}{}\mathrm{D}^{-}$$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$$\mathrm{D}^{0}{}\mathrm{D}^{0}$$\mathrm{D}^{0}{}\mathrm{D}^{+}$$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$$\mathrm{D}^{+}{}\mathrm{D}^{+}$$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$$\mathrm{D}^{+}{}\Lambda_{\mathrm{c}}^{+}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}$$\mathcal{R}_{\mathrm{C}_{1}\mathrm{C}_{2}}$$\left[\mathrm{mb}\right]$$\begin{array}[]{r}\mathrm{LHCb}{}\end{array}$
Figure 10: Measured ratios $\mathcal{R}_{\mathrm{C}_{1}\mathrm{C}_{2}}$
(points with error bars) in comparison with the expectations from DPS using
the cross-section measured at Tevatron for multi-jet events (light green
shaded area). The inner error bars indicate the statistical uncertainty whilst
the outer error bars indicate the sum of the statistical and systematic
uncertainties in quadrature. For the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ case the
outermost error bars correspond to the total uncertainties including the
uncertainties due to the unknown polarization of the prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons.
## 8 Properties of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}$, $\mathrm{C}\mathrm{C}$, and
$\mathrm{C}\overline{\mathrm{C}}$ events
The data samples available also allow the properties of the multiple charm
events to be studied. The transverse momentum spectra for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and open charm
mesons in ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}$ events are presented in Fig. 11.
$p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$$p^{\mathrm{T}}_{\mathrm{C}}$$\left[\mathrm{GeV}/c\right]$$\left[\mathrm{GeV}/c\right]$
$\tfrac{\mathrm{d}\ln\sigma}{{\mathrm{d}}p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}~{}\left[\tfrac{1}{500~{}\mathrm{MeV}/c}\right]$
$\tfrac{\mathrm{d}\ln\sigma}{{\mathrm{d}}p^{\mathrm{T}}_{\mathrm{C}}}~{}\left[\tfrac{1}{500~{}\mathrm{MeV}/c}\right]$
LHCb LHCb a) b) $\begin{array}[]{clcl}{\color[rgb]{1,0,0}\text{\char
108}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}&{\text{\char
109}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}\\\
{\color[rgb]{0,0,1}\text{\char
110}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}&&\\\ {\color[rgb]{1,0,1}\text{\char
115}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}&&\\\ {\color[rgb]{0,1,1}\text{\char
116}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}&&\\\
\end{array}$$\begin{array}[]{clcl}{\color[rgb]{1,0,0}\text{\char
108}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}&{\color[rgb]{1,0,0}\bigcirc}&\mathrm{D}^{0}\\\
{\color[rgb]{0,0,1}\text{\char
110}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}&{\color[rgb]{0,0,1}\square}&\mathrm{D}^{+}\\\
{\color[rgb]{1,0,1}\text{\char
115}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}&{\color[rgb]{1,0,1}\triangle}&\mathrm{D}^{+}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}&&\end{array}$ Figure 11: a) Transverse
momentum spectra of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
for ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and
prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ events. b)
Transverse momentum spectra for open charm hadrons for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and
prompt $\mathrm{D}^{0}$, $\mathrm{D}^{+}$ and $\mathrm{D}^{+}_{\mathrm{s}}$
events.
The transverse momentum spectra of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson in
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ events
are similar for all species of open charm hadrons. The shape of the transverse
momentum spectra of open charm hadrons also appears to be the same for all
species. The $p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ spectra are harder than the corresponding spectrum of prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$, while the
$p^{\mathrm{T}}$-spectra for open charm hadrons seem to be well compatible in
shape with the spectra for prompt charm production. To allow a more
quantitative comparison, each spectrum is fitted in the region
$3<p^{\mathrm{T}}<12~{}\mathrm{GeV}/c$ with an exponential function. The
results are summarized in Table 13 and Fig. 14. They agree reasonably well
within the uncertainties.
Table 13: Slope parameters of the transverse momentum spectra in the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ mode and for prompt charm particles. These parameters are determined from fits to the spectra in the region $3<p^{\mathrm{T}}<12~{}\mathrm{GeV}/c$. Mode | $p^{\mathrm{T}}$-slope $\left[\frac{1}{\mathrm{GeV}/c}\right]$
---|---
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ | $\mathrm{C}$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$ | $-0.49\pm 0.01$ | $-0.75\pm 0.02$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ | $-0.49\pm 0.02$ | $-0.65\pm 0.02$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $-0.60\pm 0.05$ | $-0.68\pm 0.05$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\Lambda^{+}_{\mathrm{c}}$ | $-0.46\pm 0.08$ | $-0.82\pm 0.08$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ | $-0.633\pm 0.003$ |
$\mathrm{D}^{0}$ | | $-0.77\pm 0.03$
$\mathrm{D}^{+}$ | | $-0.70\pm 0.03$
$\mathrm{D}^{+}_{\mathrm{s}}$ | | $-0.57\pm 0.13$
$\Lambda_{\mathrm{c}}^{+}$ | | $-0.79\pm 0.08$
The transverse momentum spectra of charm hadrons from $\mathrm{C}\mathrm{C}$
and $\mathrm{C}\overline{\mathrm{C}}$ events are presented in Figs. 12 and 13.
The fitted slope parameters of an exponential function are summarized in Table
14 and Fig. 14. The $p^{\mathrm{T}}$-slopes, though similar for
$\mathrm{C}\overline{\mathrm{C}}$ and $\mathrm{C}\mathrm{C}$ events, are
significantly different from those for both single prompt charm particles and
those found in ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}$ events.
$p^{\mathrm{T}}_{\mathrm{C}}$$p^{\mathrm{T}}_{\mathrm{C}}$$\left[\mathrm{GeV}/c\right]$$\left[\mathrm{GeV}/c\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{{\mathrm{d}}p^{\mathrm{T}}_{\mathrm{C}}}~{}\left[\tfrac{1}{250~{}\mathrm{MeV}/c}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{{\mathrm{d}}p^{\mathrm{T}}_{\mathrm{C}}}~{}\left[\tfrac{1}{250~{}\mathrm{MeV}/c}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{0}{}\mathrm{D}^{+}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}\end{array}$$\begin{array}[]{cl}{\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{+}{}\mathrm{D}^{+}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}\end{array}$ Figure 12:
Transverse momentum spectra of charm hadrons from $\mathrm{C}\mathrm{C}$: a)
$\mathrm{D}^{0}{}\mathrm{D}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{+}$,
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ and b)
$\mathrm{D}^{+}{}\mathrm{D}^{+}$ and
$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$ .
$p^{\mathrm{T}}_{\mathrm{C}}$$p^{\mathrm{T}}_{\mathrm{C}}$$\left[\mathrm{GeV}/c\right]$$\left[\mathrm{GeV}/c\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\mathrm{p}_{\mathrm{T}}^{\mathrm{C}}}~{}\left[\tfrac{1}{250~{}\mathrm{MeV}/c}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\mathrm{p}_{\mathrm{T}}^{\mathrm{C}}}~{}\left[\tfrac{1}{250~{}\mathrm{MeV}/c}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char 108}}&\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}\\\ {\color[rgb]{0,0,1}\text{\char 110}}&\mathrm{D}^{0}{}\mathrm{D}^{-}\\\ {\color[rgb]{1,0,1}\text{\char 115}}&\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}\\\ {\color[rgb]{0,1,1}\text{\char 116}}&\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$$\begin{array}[]{cl}{\color[rgb]{0,0,1}\text{\char 110}}&\mathrm{D}^{+}{}\mathrm{D}^{-}\\\ {\color[rgb]{1,0,1}\text{\char 115}}&\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}\\\ {\color[rgb]{0,1,1}\text{\char 116}}&\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$ Figure 13: Transverse momentum spectra of charm hadrons from $\mathrm{C}\overline{\mathrm{C}}$: a) $\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{-}$, $\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ and $\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$; b) $\mathrm{D}^{+}{}\mathrm{D}^{-}$, $\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ and $\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$. Table 14: Slope parameters of transverse momentum spectra for the $\mathrm{C}\mathrm{C}$ and $\mathrm{C}\overline{\mathrm{C}}$ modes. Mode | $p^{\mathrm{T}}$-slope $\left[\frac{1}{\mathrm{GeV}/c}\right]$
---|---
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ | $-0.51\pm 0.02$
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$ | $-0.48\pm 0.01$
$\mathrm{D}^{0}{}\mathrm{D}^{+}$ | $-0.40\pm 0.02$
$\mathrm{D}^{0}{}\mathrm{D}^{-}$ | $-0.46\pm 0.01$
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $-0.51\pm 0.05$
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ | $-0.44\pm 0.02$
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$ | $-0.41\pm 0.03$
$\mathrm{D}^{+}{}\mathrm{D}^{+}$ | $-0.48\pm 0.04$
$\mathrm{D}^{+}{}\mathrm{D}^{-}$ | $-0.46\pm 0.01$
$\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$ | $-0.39\pm 0.07$
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ | $-0.42\pm 0.02$
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$ | $-0.38\pm 0.05$
$\left.\begin{array}[]{l}\mathrm{D}^{0}\\\ \mathrm{D}^{+}\\\
\mathrm{D}^{+}_{\mathrm{s}}\\\ \Lambda_{\mathrm{c}}^{+}\\\
{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}\end{array}\right\\}\text{prompt}$$\left.\begin{array}[]{l}\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}\\\
\mathrm{D}^{0}{}\mathrm{D}^{-}\\\
\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}\\\
\mathrm{D}^{+}{}\mathrm{D}^{-}\\\
\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}\right\\}\mathrm{C}\overline{\mathrm{C}}$$\left.\begin{array}[]{l}\mathrm{D}^{0}{}\mathrm{D}^{0}\\\
\mathrm{D}^{0}{}\mathrm{D}^{+}\\\
\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}\\\
\mathrm{D}^{+}{}\mathrm{D}^{+}\\\
\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}\end{array}\right\\}\mathrm{C}\mathrm{C}$$\left.\begin{array}[]{l}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}\\\ {\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}\\\ {\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\\\
{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}\end{array}{}\right\\}p^{\mathrm{T}}_{\mathrm{C}}$$\left.\begin{array}[]{l}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}\\\ {\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}\\\ {\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\\\
{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}\end{array}\right\\}p^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$$p^{\mathrm{T}}$-slope$\left[\dfrac{1}{\mathrm{GeV}/c}\right]$LHCb
Figure 14: Slope parameters of the transverse momentum spectra for prompt
charm particles [1] and charm particles from
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$,
$\mathrm{C}\overline{\mathrm{C}}$ and $\mathrm{C}\mathrm{C}$ production.
The correlations in azimuthal angle and rapidity between the two charm hadrons
have also been studied by measuring the distributions of $\Delta\phi$ and
$\Delta y$, where $\Delta\phi$ and $\Delta y$ are the differences in azimuthal
angle and rapidity between the two hadrons. These distributions for the charm
hadrons in ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}$ events are shown in Fig. 15. No significant azimuthal
correlation is observed. The $\Delta y$ distribution is compared to the
triangular shape that is expected if the rapidity distribution for single
charm hadrons is flat and if there are no correlations.
$\left|\Delta\phi\right|/\pi$$\Delta y$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta\phi\right|}~{}\left[\tfrac{\pi}{0.1}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\Delta
y}~{}\left[\tfrac{1}{0.125}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\end{array}$$\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\end{array}$ Figure 15: Distributions of
the difference in azimuthal angle (a) and rapidity (b) for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{0}$,
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ events. The dashed line shows the
expected distribution for uncorrelated events.
The azimuthal and rapidity correlations for $\mathrm{C}\mathrm{C}$ and
$\mathrm{C}\overline{\mathrm{C}}$ events are shown in Figs. 16, 17, and 18. In
the $\mathrm{C}\mathrm{C}$ case the $\Delta\phi$ distribution is reasonably
consistent with a flat distribution. In contrast, for
$\mathrm{C}\overline{\mathrm{C}}$ events a clear enhancement is seen for
$\Delta\phi$ distributions at small $\left|\Delta\phi\right|$. This is
consistent with $\mathrm{c}\bar{\mathrm{c}}$ production via the gluon
splitting mechanism [38]. The $\mathrm{C}\overline{\mathrm{C}}$ events suggest
some enhancement at small $\left|\Delta y\right|$, while the
$\mathrm{C}\mathrm{C}$ sample shows no clear difference from the triangular
shape given the present statistics.
$\left|\Delta\phi\right|/\pi$$\left|\Delta y\right|$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta\phi\right|}~{}\left[\tfrac{\pi}{0.1}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta
y\right|}~{}\left[\tfrac{1}{0.1}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{0}{}\mathrm{D}^{+}\end{array}$$\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{0}{}\mathrm{D}^{+}\end{array}$ Figure 16: Distributions of
the difference in azimuthal angle (a) and rapidity (b) for
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ and $\mathrm{D}^{0}{}\mathrm{D}^{+}$ events.
The dashed line shows the expected distribution for uncorrelated events.
$\left|\Delta\phi\right|/\pi$$\left|\Delta\phi\right|/\pi$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta\phi\right|}~{}\left[\tfrac{\pi}{0.05}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta\phi\right|}~{}\left[\tfrac{\pi}{0.05}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}\\\
{\color[rgb]{0,0,1}\text{\char 110}}&\mathrm{D}^{0}{}\mathrm{D}^{-}\\\
{\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$$\begin{array}[]{cl}{\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{+}{}\mathrm{D}^{-}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$ Figure 17:
Distributions of the difference in azimuthal angle for
$\mathrm{C}\overline{\mathrm{C}}$ events: a)
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{-}$,
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$; b)
$\mathrm{D}^{+}{}\mathrm{D}^{-}$,
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$. $\left|\Delta
y\right|$$\left|\Delta y\right|$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta
y\right|}~{}\left[\tfrac{1}{0.2}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}\left|\Delta
y\right|}~{}\left[\tfrac{1}{0.2}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}\\\
{\color[rgb]{0,0,1}\text{\char 110}}&\mathrm{D}^{0}{}\mathrm{D}^{-}\\\
{\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$$\begin{array}[]{cl}{\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{+}{}\mathrm{D}^{-}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$ Figure 18:
Distributions of the difference in rapidity for
$\mathrm{C}\overline{\mathrm{C}}$ events: a)
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{-}$,
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$; b)
$\mathrm{D}^{+}{}\mathrm{D}^{-}$,
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$. The dashed line shows the
expected distribution for uncorrelated events.
Finally, the invariant mass distributions of the pairs of charm hadrons in
these events have been studied. The mass spectra for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ and
$\mathrm{C}\mathrm{C}$ events are shown in Fig. 19. The spectra appear to be
independent of the type of the open charm hadron.
$m_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}}$$m_{\mathrm{C}\mathrm{C}}$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}m_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}}}~{}\left[\tfrac{1}{500~{}\mathrm{MeV}/c^{2}}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}m_{\mathrm{C}\mathrm{C}}}~{}\left[\tfrac{1}{500~{}\mathrm{MeV}/c^{2}}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\end{array}$$\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\mathrm{D}^{0}\\\ {\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{0}{}\mathrm{D}^{+}\end{array}$ Figure 19: a) Invariant mass
spectra for ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}$, ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}$ and ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ events. b) Invariant mass spectra for
$\mathrm{D}^{0}{}\mathrm{D}^{0}$ and $\mathrm{D}^{0}{}\mathrm{D}^{+}$ events.
The invariant mass spectra for $\mathrm{C}\overline{\mathrm{C}}$ events are
shown in Fig. 20. Again, the spectra are similar and independent of the type
of the open charm meson. The enhancement at small invariant mass is most
likely due to the gluon splitting process [38]. For the region of invariant
masses above $6~{}\mathrm{GeV}/c^{2}$ the spectra are similar for
$\mathrm{C}\overline{\mathrm{C}}$ and $\mathrm{C}\mathrm{C}$ events.
$m_{\mathrm{C}\overline{\mathrm{C}}}$$m_{\mathrm{C}\overline{\mathrm{C}}}$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}m_{\mathrm{C}\overline{\mathrm{C}}}}~{}\left[\tfrac{1}{500~{}\mathrm{MeV}/c^{2}}\right]$
$\tfrac{\mathrm{d}\ln\sigma^{*}}{\mathrm{d}m_{\mathrm{C}\overline{\mathrm{C}}}}~{}\left[\tfrac{1}{500~{}\mathrm{MeV}/c^{2}}\right]$
LHCb LHCb a) b) $\begin{array}[]{cl}{\color[rgb]{1,0,0}\text{\char
108}}&\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}\\\
{\color[rgb]{0,0,1}\text{\char 110}}&\mathrm{D}^{0}{}\mathrm{D}^{-}\\\
{\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$$\begin{array}[]{cl}{\color[rgb]{0,0,1}\text{\char
110}}&\mathrm{D}^{+}{}\mathrm{D}^{-}\\\ {\color[rgb]{1,0,1}\text{\char
115}}&\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}\\\
{\color[rgb]{0,1,1}\text{\char
116}}&\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}\end{array}$ Figure 20:
Invariant mass spectra for $\mathrm{C}\overline{\mathrm{C}}$ events: a)
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{-}$,
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$; b)
$\mathrm{D}^{+}{}\mathrm{D}^{-}$,
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$.
## 9 Conclusion
The production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
mesons accompanied by open charm, and pairs of open charm hadrons has been
observed in pp collisions at $\sqrt{\mathrm{s}}=7~{}{\mathrm{TeV}}$. This is
the first observation of these phenomena in hadronic collisions. Signals with
a statistical significance in excess of five standard deviations have been
observed for four ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{C}$ modes: ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{0}$, ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}$, ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\Lambda^{+}_{\mathrm{c}}$, for six $\mathrm{C}\mathrm{C}$ modes:
$\mathrm{D}^{0}{}\mathrm{D}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{+}$,
$\mathrm{D}^{0}{}\mathrm{D}^{+}_{\mathrm{s}}$,
$\mathrm{D}^{0}{}\Lambda_{\mathrm{c}}^{+}$, $\mathrm{D}^{+}{}\mathrm{D}^{+}$,
and $\mathrm{D}^{+}{}\mathrm{D}^{+}_{\mathrm{s}}$, and for seven
$\mathrm{C}\overline{\mathrm{C}}$ channels:
$\mathrm{D}^{0}{}\overline{\mathrm{D}}^{0}$, $\mathrm{D}^{0}{}\mathrm{D}^{-}$,
$\mathrm{D}^{0}{}\mathrm{D}^{-}_{\mathrm{s}}$,
$\mathrm{D}^{0}{}\bar{}\Lambda_{\mathrm{c}}^{-}$,
$\mathrm{D}^{+}{}\mathrm{D}^{-}$,
$\mathrm{D}^{+}{}\mathrm{D}^{-}_{\mathrm{s}}$ and
$\mathrm{D}^{+}{}\bar{}\Lambda_{\mathrm{c}}^{-}$.
The cross-sections and the properties of these events have been studied. The
predictions from gluon-gluon fusion [14, 15, 18] are significantly smaller
than the observed cross-sections. Better agreement is found with the DPS model
[4, 5, 6, 7] if the effective cross-section inferred from the Tevatron data is
used. The absence of significant azimuthal or rapidity correlations provides
support for this hypothesis.
The transverse momentum spectra for these events have also been studied. The
transverse momentum spectra for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ from
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ events
are significantly harder than those observed in prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production. On the
other hand the spectra for open charm mesons in
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{C}$ events
appear to be similar to those observed for prompt charm hadrons. Similar
transverse momentum spectra for $\mathrm{C}\mathrm{C}$ and
$\mathrm{C}\overline{\mathrm{C}}$ events are observed. However, the
expectation of similar transverse momentum spectra for
$\mathrm{C}\overline{\mathrm{C}}$ events and prompt charm events appears to be
invalid.
For $\mathrm{C}\overline{\mathrm{C}}$ events significant rapidity and
azimuthal correlations are observed. These, as well as the invariant mass
spectra for $\mathrm{C}\overline{\mathrm{C}}$ events, suggest a sizeable
contribution from the gluon splitting process to charm quark production [38].
## Acknowledgements
We thank M. H. Seymour and A. Siódmok for the points raised in Ref. [39], and
have revised the paper to address these issues. We would like to thank J.-P.
Lansberg, A.K. Likhoded and A. Szczurek for many fruitful discussions. We
express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## Appendix: Contribution from sea charm quarks
Estimates for the expected cross-section in the LHCb fiducial region due to
the sea charm quarks from the interacting protons have been made as follows.
The LHCb rapidity window $2<y<4$ corresponds to a $x$ range for the additional
charm quarks of
$\frac{2m^{\mathrm{T}}_{\mathrm{c}}}{\sqrt{s}}\sinh{2}<x<\frac{2m^{\mathrm{T}}_{\mathrm{c}}}{\sqrt{s}}\sinh{4}$
(10)
where $m^{\mathrm{T}}_{\mathrm{c}}$ is the transverse mass of the charm quark.
Assuming the extra charm mesons are distributed over $p^{\mathrm{T}}$ in a
similar way to the inclusive charm mesons measured in [1, 2] one can take
$m^{\mathrm{T}}_{\mathrm{c}}\approx m_{\mathrm{c}}\oplus
2~{}\mathrm{GeV}/c^{2},$ (11)
where $2~{}\mathrm{GeV}/c$ is the mean transverse momentum of charm quarks
produced. This leads to the $x$ range of $0.0026<x<0.02$. Integration of
Alekhin’s LO parton distribution functions [40] over this $x$ range gives
$0.25$ additional charm quarks per event. In this calculation the parton
density functions are taken at the scale $\mu\approx
m^{\mathrm{T}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$. The
cross-sections of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
plus open charm mesons can then be estimated using the probabilities for the
$c$-quark transition to different mesons given in [1, 2]. Similarly cross-
sections for double open charm production can be estimated. Taking $\mu\approx
m^{\mathrm{T}}_{\mathrm{D}}$ and integrating Alekhin’s LO parton density
functions [40] on gets approximately 0.17 additional charm quarks per event.
This calculation assumes that all extra charm quarks from protons hadronize to
open charm states visible in the detector. The real cross-sections may be
smaller, but the ratio of different open charm states is expected to remain
the same. The integrated parton density functions provide no information about
the $p^{\mathrm{T}}$ distribution of charm quarks. Under the assumption that
the $p^{\mathrm{T}}$-spectrum coincides with the distribution of prompt charm
particles measured at LHCb [1], the cross-sections in the LHCb fiducial range
are calculated (see last column of Table 1).
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|
arxiv-papers
| 2012-05-04T15:15:36 |
2024-09-04T02:49:30.546000
|
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"license": "Public Domain",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, V. Balagura,\n W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C.\n Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, K. de Bruyn, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, S. De Capua, M. De\n Cian, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff,\n H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F.\n Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D.\n van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D.\n Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C.\n Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S.\n Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco,\n M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J.-C. Garnier, J. Garofoli, J. Garra Tico, L.\n Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon,\n Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.\n C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J.\n Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M.\n Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y. M. Kim, M. Knecht, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N.\n La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O.\n Leroy, T. Lesiak, L. Li, L. Li Gioi, M. 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Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr,\n E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez,\n P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, H. Voss, R. Waldi, S. Wandernoth, J. Wang,\n D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D.\n Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson,\n J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing,\n Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1205.0975"
}
|
1205.0995
|
# Natural flavor conservation in a three Higg-doublet Model
A. C. B. Machado ana@ift.unesp.br Instituto de Física Teórica–Universidade
Estadual Paulista
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
São Paulo - SP, 01140-070, Brazil Centro de Ciências Naturais e Humanas,
Universidade Federal do ABC, Santo André-SP, 09210-170
Brazil. V. Pleitez vicente@ift.unesp.br Instituto de Física
Teórica–Universidade Estadual Paulista
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
São Paulo - SP, 01140-070, Brazil
(03/08/13)
###### Abstract
We consider an extension of the electroweak standard model with three Higgs
doublets transforming under the $S_{3}$ symmetry as a doublet and a singlet.
Because of the $S_{3}$ symmetry and an appropriate vacuum alignment the mass
matrices in all the scalar sectors have the same form and are diagonalized by
the same matrix which is of the tribimaximal type. Moreover, independently of
the fermion representations under $S_{3}$, there is no flavor changing neutral
currents at tree level, although in order to obtain realistic mass matrices,
quarks and leptons must transform as singlets under $S_{3}$. Two of the scalar
doublets are fermiophobic and they are mass degenerate unless soft terms
violating the $S_{3}$ symmetry are added. The third Higgs doublet which is
singlet under $S_{3}$, correspond to the standard model one in the sense that
its mass and couplings to fermions and bosons are exactly the same of the SM
at tree level.
###### pacs:
12.60.Fr 12.15.-y
## I Introduction
The resonance with mass of approximately 125 GeV that was discover at LHC is
in agreement, within the experimental error, with the standard model (SM)
Higgs field cmsatlas . Hence, it implies that all possible new physics
scenarios have to include such scalar field. The SM symmetry breaking is the
simplest and economic but nothing prevent the model to have more scalars
multiplets transforming in several representations of $SU(2)$: more doublets,
singlets and even triplets. All of then have already been considered in
literature as extension of the SM. It means that the source of electroweak
symmetry breaking may be more complicated than in the SM. Moreover, if there
exist more scalar fields the SM Higgs couplings to fermion and gauge bosons
are usually reduced. In fact, this case is still possible with the current LHC
data, see for instance Ref. holdom .
Here we will show an extension of the SM with three Higgs doublets
transforming under $S_{3}$ as a doublet $(D)$and a singlet ($S$) and having an
appropriate vacuum alignment. An interesting feature of our model is that
unlike other multi-Higgs models, in which the SM-like Higgs boson mixes with
the others and this is why its coupling to fermion and gauge boson are
reduced, this does not happens in the present model because of the $S_{3}$
symmetry and the vacuum alignment: the combination of the three doublets
transforming as singlet under $S_{3}$ ($S$) is exactly the SM Higgs with all
its couplings the same as in the SM with one Higgs doublet and it does not mix
with the other $SU(2)$ doublets. The latter ones are in a doublet of $S_{3}$
there is no mixing between them too. The difference of the singlet $S$ and the
SM Higgs doublet is that the former has trilinear and quartic interactions
with the members of the doublet $D$.
On the other hand, it is well known, from experiments, that processes which
change flavor through the effects of neutral currents are quite suppressed
fcncexp . It is also well known, since the seminal paper of Glashow and
Weinberg gw , that the suppression of flavor changing neutral currents (FCNCs)
is natural if it depends only on the symmetry and the representation content
of the model. In general multi-Higgs models give rise to unwanted FCNCs.
Notwithstanding in these models it is possible to have natural flavor
conservation if the fermion masses, of a given charge, are generated by a
single source or, as in the standard electroweak model, a unique Higgs doublet
generates all fermion masses. In order to obtain natural flavor conservation
discrete Abelian symmetries like $Z_{2}$ gw , continuous Abelian u1 , and non-
Abelian continuos su3 or discrete a4 family symmetries are usually imposed
or, to assume the alignment in flavor space of the Yukawa matrices in the so
called A2HDM pich . In particular, the two-Higgs doublet models are the most
considered in literature, see for instance Ref. branco . Discrete symmetries
have also been used to gain some predictivity in the flavor problem, usually
to reproduce ansatze of mass matrix textures, and they origin may be related
to more fundamental (new) physics nilles . However, in all these cases the
goal was to explain the texture of the fermion mass matrices, for instance,
for getting the so called tribimaximal mixing matrix in the leptonic sector
tbm , in particular the $S_{3}$ symmetry was considered since many years ago
s3 . Things become more complicated when there are more than two Higgs
doublets. Thus, larger symmetries may be needed in order to simplify the
analysis of the scalar potential which may be or not related to the fermion
mass matrices. Discrete symmetries for three-Higgs doublet model, without
references to these matrices, were classified in Ref. ivanov .
Here we will consider a three-Higgs doublet model (model A) in which the mass
square matrices of all the scalar fields at the tree level have the same form
and for this reason, they are diagonalized with the same unitary matrix, $U$,
and there is mass degenerated states in the $S_{3}$ symmetry if it is not
softly broken. The matrix $U$ is, in some cases, of the tribimaximal type.
This is a consequence of the representation content of the discrete symmetry,
$S_{3}$, imposed to the model, one doublet $D$ and a singlet $S$ as well an
appropriate vacuum alignment, and it also implies that the model naturally
suppresses FCNC mediated by neutral scalars in the quark and lepton sectors,
if we extend the criterion for having natural flavor conservation to include
an appropriate vacuum alignment with the minimum of the scalar potential being
global and stable, at least at tree level. In a different version (model B)
the mass square are matrices are all diagonal at tree level if the $S_{3}$
symmetry is not softly broken.
The outline of this paper is as follows. In Sec. II we give the most general
scalar potential invariant under the gauge and $S_{3}$ symmetries. In Sec. III
we consider in details the scalar sector of the model A (see below) when all
VEVs are equal to each other, i.e., $v_{1}=v_{2}=v_{3}$; the same is doing in
Sec. IV for the model B (see below) but now with $v_{1}=v_{SM}$ and
$v_{2}=v_{3}=0$. The Yukawa interactions are the same in both models and are
briefly discussed in Sec. V. Our conclusions are in Sec. VII and in the
Apendices we show the constraint equations for arbitrary VEVs for model A in
the Appendix A and model B in the Appendix. B.
## II Three Higgs-scalar doublet models
The model is an extension of the electroweak standard model with three Higgs
scalars, all $SU(2)$ doublets having $Y=+1$ and the most general scalar
potential invariant under $SU(2)\otimes U(1)_{Y}\otimes S_{3}$ symmetry is
given by:
$\displaystyle V(D,S)$ $\displaystyle=$
$\displaystyle\mu^{2}_{s}S^{\dagger}S+\mu^{2}_{d}[D^{\dagger}\otimes
D]_{1}+\lambda_{1}([D^{\dagger}\otimes
D]_{1})^{2}+\lambda_{2}[(D^{\dagger}\otimes D)_{1^{\prime}}(D^{\dagger}\otimes
D)_{1^{\prime}}]_{1}$ (1) $\displaystyle+$
$\displaystyle\lambda_{3}[(D^{\dagger}\otimes D)_{2}(D^{\dagger}\otimes
D)_{2}]+\lambda_{4}(S^{\dagger}S)^{2}+\lambda_{5}[D^{\dagger}\otimes
D]_{1}S^{\dagger}S+\lambda_{6}[[S^{\dagger}D]_{2}[S^{\dagger}D]_{2}]_{1}$
$\displaystyle+$ $\displaystyle\\{\lambda_{7}S^{\dagger}[D\otimes
D^{\dagger}]_{1}S+\lambda_{8}[(S^{\dagger}\otimes D)_{2}(D^{\dagger}\otimes
D)_{2}]_{1}+H.c.\\}$
We recall that denoting $\textbf{2}=(x_{1},x_{2})$, we have
$\textbf{2}\otimes\textbf{2}=\textbf{1}\oplus\textbf{1}^{\prime}\oplus\textbf{2}^{\prime}$
where $\textbf{1}=x_{1}y_{1}+x_{2}y_{2}$,
$\textbf{1}^{\prime}=x_{1}y_{2}-x_{2}y_{1}$,
$\textbf{2}^{\prime}=(x_{1}y_{2}+x_{2}y_{1},x_{1}y_{1}-x_{2}y_{2}$), and
$\textbf{1}^{\prime}\otimes\textbf{1}^{\prime}=\textbf{1}$ ishimori .
We have two possibilities to build the doublet $D$ and the singlet $S$ if the
three scalar doublets are in the reducible triplet representation of $S_{3}$,
say, $\textbf{3}=(H_{1},H_{2},H_{3})$ where
$H_{i}=(H^{+}_{i}\,H^{0}_{i})^{T}$. This reducible representation is broken
down to the irreducible singlet and doublet ones, i.e.,
$\textbf{3}~{}=~{}\textbf{2}+\textbf{1}\equiv D+S$, where:
$\displaystyle S=\frac{1}{\sqrt{3}}(H_{1}+H_{2}+H_{3})\sim\textbf{1},$
$\displaystyle
D\equiv(D_{1},D_{2})=\left[\frac{1}{\sqrt{6}}(2H_{1}-H_{2}-H_{3}),\frac{1}{\sqrt{2}}(H_{2}-H_{3})\right]\sim\textbf{2}.$
(2)
We shall call this model A. On the other hand, the one of the $SU(2)$ doublet,
say $H_{1}$, is a singlet of $S_{3}$ and the other two, say $H_{2},H_{3}$
transform as the irreducible doublet of $S_{3}$:
$S=H_{1}\sim\textbf{1},\quad D=(H_{2},H_{3})\sim\textbf{2}.$ (3)
Here, we shall call this model B.
Both sort of models have been already consider in literature. For instance,
model A was considered in Ref.kubo . Model B was considered a long time ago
pakvasa and more recently in Refs. mondragon , and in a five-Higgs doublets
supersymmetric model in kaneko . All the previous articles have used the
$S_{3}$ symmetry to addressed the texture of the fermion mass matrices.
However, here we shall give up this issue and concentrate on the extension of
the standard model with three Higgs doublets, using the $S_{3}$ symmetry just
to simplifying the analysis of the scalar sector and suppress flavor changing
neutral currents at the tree level. Thus, fermions transform trivially under
$S_{3}$ and the mass matrices are of the general form but there is no flavor
violation mediated by neutral scalars. Models A and B seem very similar
specially when we choose two vacuum alignment which apparently lead one model
into another, since they could be related by a change of weak basis in the
representation of $S_{3}$ ishimori . However, they have different mass
matrices and trilinear and quartic interactions are also different.
## III Model A: $v_{1}=v_{2}=v_{3}$
We make as usual, the decomposition of the symmetry eigenstates as
$H^{0}_{i}=(1/\sqrt{2})(v_{i}+\eta^{0}_{i}+iA^{0}_{i}),\;i=1,2,3$, and assume,
for the sake of the simplicity, all VEVs being real.
The case when all VEVs are different from zero was considered in Ref. kubo
and the general constraint equations for this case are given in the Appendix
A. Here let us consider the case when $v_{1}=v_{2}=v_{3}=v$, and the
constraint equations in Eq. (35) become
$t_{1}=t_{2}=t_{3}=v(\mu^{2}_{s}+3\lambda_{4}v^{2}),$ (4)
and $t_{i}=0$ implies $\mu^{2}_{s}=-3\lambda_{4}v^{2}<0,\;\lambda_{4}>0$.
All scalar mass matrices has the form
$M^{2}_{n}=\left(\begin{array}[]{ccc}a_{n}&b_{n}&b_{n}\\\ b_{n}&a_{n}&b_{n}\\\
b_{n}&b_{n}&a_{n}\end{array}\right),$ (5)
where $a_{n},b_{n}>0$ (or $a_{n},b_{n}<0$ ) and $n$ denotes the scalar sector:
$n=h,a,c$ for the scalar, pseudoscalar and charged scalar fields.
This matrix is diagonalized as follows:
$U^{T}_{TBM}M^{2}_{n}U_{TBM}=\textrm{diag}(a_{n}+2b_{n},a_{n}-b_{n},a_{n}-b_{n})$,
hence $a_{n}-2b_{n}\geq 0$ and $a_{n}+b_{n}\geq 0$, with
$U_{TBM}=UP=(e_{1}\,e_{2}\,e_{3})=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&-\sqrt{\frac{2}{3}}&0\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{2}}\end{array}\right),$
(6)
where we have denoted the eigenvectors by $e_{1},e_{2},e_{3}$, and the
matrices $U$ and $P$ in (6) are give by
$U=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&0&0\\\
\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right),\quad
P=\left(\begin{array}[]{ccc}1&0&0\\\
0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right).$ (7)
In the case of CP-even neutral real scalars, we have
$3a_{h}=2\mu_{d}^{2}+(8\lambda_{4}+3\bar{\lambda}^{\prime})v_{SM}^{2}$, and
$6b_{h}=-2\mu_{d}^{2}+(16\lambda_{4}-3\bar{\lambda}^{\prime})v_{SM}^{2}$,
where $\bar{\lambda}^{\prime}=(\lambda_{5}+\lambda_{6}+2\lambda_{7})$, and the
eigenvalues are the following:
$\displaystyle m^{2}_{h_{1}}=\lambda_{4}v^{2}_{SM},\qquad
m^{2}_{h_{2}}=m^{2}_{h_{3}}\equiv
m^{2}_{h}=\mu^{2}_{d}+\frac{3}{2}\bar{\lambda}^{\prime}v^{2}_{SM},$ (8)
where we have used $v=v_{SM}/\sqrt{3}$. Denoting as $h^{0}_{i}$ the mass
eigenstates, we have $\eta^{0}_{i}=(U_{TBM})_{ij}h^{0}_{j}$, where $U_{TBM}$
is given in (6). The scalar $h^{0}_{1}$ can be identified with the standard
model Higgs scalar.
In the CP-odd neutral scalars sector, the mass matrix is given as in (5) but
now with
$3a_{a}=2\mu_{d}^{2}+(2\lambda_{4}+3\bar{\lambda}^{\prime\prime})v_{SM}^{2}$
and
$6b_{a}=-2\mu_{d}^{2}+(4\lambda_{4}-3\bar{\lambda}^{\prime\prime})v_{SM}^{2}$,
where $\bar{\lambda}^{\prime\prime}=(\lambda_{5}+\lambda_{6}-2\lambda_{7})$
and in this case we obtain the following masses:
$\displaystyle m^{2}_{a_{1}}=0,\qquad m^{2}_{a_{2}}=m^{2}_{a_{3}}\equiv
m^{2}_{a}=\mu^{2}_{d}+\frac{3}{2}\bar{\lambda}^{\prime\prime}v^{2}_{SM}$ (9)
Denoting $a^{0}_{i}$ the pseudo-scalar mass eigenstates, we have
$A^{0}_{i}=(U_{TBM})_{ij}a^{0}_{j}$.
Similarly in the charged scalars sector we use (5) with
$6a_{c}=2\mu_{d}^{2}+(2\lambda_{4}+3\lambda_{5})v_{SM}^{2}$ and
$12b_{c}~{}=~{}-2\mu_{d}^{2}+(4\lambda_{4}-3\lambda_{5})v_{SM}^{2}$ and in
this case we obtain the following masses:
$\displaystyle m^{2}_{c_{1}}=0,\qquad m^{2}_{c_{2}}=m^{2}_{c_{3}}\equiv
m^{2}_{c}=\frac{1}{4}(2\mu^{2}_{d}+\lambda_{5}v^{2}_{SM}).$ (10)
Finally, if $H^{+}_{i}$ denote the charged scalar symmetry eigenstates and
$h^{+}_{i}$ the respective mass eigenstates, we have
$H^{+}_{i}=(U_{TBM})_{ij}h^{+}_{j}$. The mass degeneracy is due to a residual
symmetry as we will see below.
The $\mu^{2}_{d}$ parameter appears in Eqs. (8), (9) and (10). This parameter
may be $>0$ or $<0$, and since it is not protected by any symmetry, it may be
larger than the electroweak scale. On one hand, if $\mu^{2}_{d}>v^{2}_{SM}$ or
$|\mu^{2}_{d}|<v^{2}_{SM}$ the masses of the scalar $h^{0}_{2,3}$,
pseudoscalar $a^{0}_{2,3}$ and the charged scalar $h^{\pm}_{2,3}$ are heavier
than $h^{0}_{1}$, independently of the values of the $\lambda$’s and $v_{SM}$.
On the other hand, if $\mu^{2}_{d}<0$ and
$\bar{\lambda}^{\prime},\bar{\lambda}^{\prime\prime}>0,\lambda_{5}>0$, all
these particles may be lighter than $h^{0}_{1}$.
Using the mixing matrix in Eq. (6), we can write the Higgs scalars doublet $D$
and the singlet $S$, in terms of the mass eigenstates, $h^{0}_{i},a^{0}_{i}$
and $h^{\pm}_{i}$, as
$\displaystyle S\equiv h_{1}=\left(\begin{array}[]{c}h^{+}_{1}\\\
\frac{1}{\sqrt{2}}(3v+h^{0}_{1}+ia^{0}_{1})\end{array}\right),\;D\equiv-(h_{2},h_{3}),\;h_{k}=\left(\begin{array}[]{c}h^{+}_{k}\\\
\frac{1}{\sqrt{2}}(h^{0}_{k}+ia^{0}_{k})\end{array}\right),$ (15)
where $k=2,3$ and we see that the would-be Goldstone bosons are all in the
singlet $S$. Notice that the doublet $S=h_{1}$ is the only source of fermion
masses, thus there is no flavor violation at tree level (see Sec. V). This is
independent of the way that fermions transform under $S_{3}$, however as we
will show below, in order to obtain realistic mass matrices at tree, fermions
must transform as singlet under $S_{3}$. Notice that in the unitary gauge
$h^{+}_{1}$ and $a_{1}$ are the two would be Goldstone bosons required by the
SM.
Compare the situation in Eq. (15) with the case of two doublets in which the
mass eigenstates neutral scalar are denoted usually as $h_{1}$ and $h_{2}$,
and the pseudoscalar $A$, while $H_{u},H_{d},$ and $A_{u},A_{d}$ denote the
symmetry eigenstates, we have
$h_{1}=\sqrt{2}(H_{d}\sin\alpha-
H_{u}\cos\alpha),\;\;h_{2}=-\sqrt{2}(H_{d}\cos\alpha+H_{u}\sin\alpha),\;\;A=\sqrt{2}(A_{d}\sin\beta-
A_{u}\cos\alpha)$ (16)
$\alpha$ is a mixing angle between the two neutral scalars and where
$\tan\beta=v_{u}/v_{d}$. In this case the SM Higgs is given by
$h_{SM}=h_{1}\sin(\alpha-\beta)-h_{2}\cos(\alpha-\beta)$, and only if
$\sin(\alpha-\beta)=1$ it is identical with the SM branco .
The potential in Eq. (1), which is written in terms of the $SU(2)$ scalar
doublets with their component being symmetry eigenstates in Eq. (2), can be
written with $SU(2)$ scalar doublet with their components being the mass
eigenstates given in Eq. (15):
$\displaystyle V(h_{i})$ $\displaystyle=$ $\displaystyle
3\lambda_{4}v^{2}h^{\dagger}_{1}h_{1}+\mu^{2}_{d}(h^{\dagger}_{2}h_{2}+h^{\dagger}_{3}h_{3})+\lambda_{1}(h^{\dagger}_{2}h_{2}+h^{\dagger}_{3}h_{3})^{2}+\lambda_{2}(h^{\dagger}_{2}h_{3}-h^{\dagger}_{3}h_{2})^{2}$
(17) $\displaystyle+$
$\displaystyle\lambda_{3}[(h^{\dagger}_{2}h_{3}+h^{\dagger}_{3}h_{2})^{2}+(h^{\dagger}_{2}h_{2}-h^{\dagger}_{3}h_{3})^{2}]+\lambda_{4}(h^{\dagger}_{1}h_{1})^{2}+\lambda_{5}h^{\dagger}_{1}h_{1}(h^{\dagger}_{2}h_{2}+h^{\dagger}_{3}h_{3})$
$\displaystyle+$
$\displaystyle\lambda_{6}[|h_{1}^{\dagger}h_{2}|^{2}+|h_{1}^{\dagger}h_{3}|^{2}]+\\{\lambda_{7}[(h^{\dagger}_{1}h_{2})^{2}+(h^{\dagger}_{3}h_{1})^{2}]+\lambda_{8}[h^{\dagger}_{1}h_{2}(h^{\dagger}_{2}h_{3}+h^{\dagger}_{3}h_{2})$
$\displaystyle+$ $\displaystyle
h^{\dagger}_{1}h_{3}(h^{\dagger}_{3}h_{3}-h^{\dagger}_{2}h_{2})]+H.c.\\}.$
Notice from the scalar potential in Eq. (17), that there is still a residual
$S_{2}\equiv Z_{2}$ symmetry: it is invariant under the exchange of the
doublets $h_{2}\leftrightarrow h_{3}$. The mass degeneracy is due to this
unbroken $S_{2}$ symmetry. That is described mathematically by the following
operator
$E=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&1&0\end{array}\right).$ (18)
If the eigenvectors of the mass square matrix are also eigenvectors of $E$ we
can distinguish the three eigenvectors $e_{1},\,e_{2},\,e_{3}$, with which the
tribimaximal matrix in Eq. (6) is obtained, even if $e_{2}$ and $e_{3}$ are
mass degenerated: $Ee_{2}=e_{2}$ and $Ee_{3}=-e_{3}$.
We will show later on under which conditions the potential in Eq. (1) is
bounded from below. For the moment, just notice that when $v_{1}=v_{2}=v_{3}$,
if $\lambda_{4}>0$. Under these conditions the minimum of the scalar potential
($V_{min}=-\lambda_{4}v^{4}_{SM}$) is global and stable as long as the masses
square, given in (8), (9), and (10), are all positive and, with the conditions
for the $\lambda$’s given in Sec. VI are satisfied. However, the stability of
the solution $v_{1}=v_{2}=v_{3}$ under radiative corrections will be studied
elsewhere.
The residual $S_{2}$ symmetry can be broken, if necessary, to avoid the mass
degeneracy and also the domain wall problem, and this can be done by quantum
corrections and/or by soft terms in the scalar potential. As an illustration,
here we break the $S_{2}$ symmetry by adding the following quadratic terms
$\mu^{2}_{nm}H^{\dagger}_{n}H_{m}$, $n,m=2,3$ to the scalar potential in (1).
The mass matrices in all the scalar sectors are now of the form
$M^{2}_{n}=\left(\begin{array}[]{ccc}a_{n}&b_{n}&b_{n}\\\
b_{n}&a_{n}+\mu^{2}_{22}&b_{n}+\mu^{2}_{23}\\\
b_{n}&b_{n}+\mu^{2}_{23}&a+\mu^{2}_{33}\end{array}\right),$ (19)
where $\mu^{2}_{nm}$ are naturally small and real for the sake of simplicity
(however see below). Although even when $\mu^{2}_{22}=\mu^{2}_{33}=\nu^{2}$
and $\mu^{2}_{23}=\mu^{2}$, the matrix above is diagonalized by the
tribimaximal matrix, as the neutrinos masses altarelli ), this is not possible
with scalars fields: in there is no Goldstone bosons. In order to have the
correct number of these bosons we have to impose that
$\mu^{2}_{22}=\mu^{2}_{33}=-\mu^{2}_{23}\equiv\mu^{2}$. In this case the
matrix in (19) is still diagonalized by tribimaximal matrix in Eq. (6), and
the eigenvalues are now $(2a_{n}+b_{n},a_{n}-b_{n},a_{n}-b_{n}+\mu^{2})$ and
we still have $S=h_{1}$ and $D=-(h_{2},h_{3})$, as in the previous case, and
there is no FCNC and two of the doublets are inert too.
After the discussion above, it seems reasonable to include the vacuum
alignment as part of the mechanism of natural flavor conservation.
Independently of the neutral current issue, the model also implement inert
scalar doublets: $h_{2}$ and $h_{3}$ do not couple to matter and do not
acquire vacuum expectation value, they only have electroweak, trilinear and
quartic interactions inert1 ; inert2 . In this case, is the lightest scalar is
the neutral ones and fermiophobic, they are stable and good candidates for
dark matter dark .
## IV Model B: $v_{1}\equiv v_{SM}$, $v_{2}=v_{3}=0$
As we discussed in Sec. II, another representation content of the three Higgs
doublets that has been considered in literature is that in Eq. (3) and the
scalar potential is the same as in model A, see Eq. (1). The constraint
equations are given in the Appendix B. Notice from Eq. (36) the $\lambda_{8}$
term is the one which breaks the symmetry between $v_{2}$ and $v_{3}$, hence
only when $\lambda_{8}=0$ we can have the solutions: (i) with $v_{1}=v_{SM}$
and $v_{2}=v_{3}=0$. This can be accomplished by using a $Z_{2}$ symmetry
under which $D\to-D$ and all the other fields are even; (ii): all VEVs equals
$v_{1}=v_{2}=v_{3}\equiv v$. However, the latter solution is not the most
interesting since it implies extra massless scalars. Here we shall consider
only the solution (i), that seems similar to the model A with
$v_{1}=v_{2}=v_{3}$ since in both cases the $S_{3}$ doublet $D=(D_{1},D_{2})$
has no VEVs. However, the models are phenomenological different since the
trilinear are different the decays of the fermiophobic scalars are different
in model A and B.
The constraint equations in Eq. (36) implies, with the vacuum aligmnet given
above, $\mu^{2}_{s}=-\lambda_{4}v^{2}_{SM}$ and the mass square matrices are
all diagonal: there is no mixing among the fields in each charge sector. At
tree level the masses are the same as in model A, see Eqs. (8) -(10), but due
to the difference in the interactions, quantum corrections must be different
in both models. The mass eigenstates doublets of $SU(2)$ are denoted, as
before, by $h_{1,2,3}$. In this case the scalar potential in terms of the mass
eigenstates is given also in Eq. (17), but with $\lambda_{8}=0$. Unlike the
model A, there is no mixing among the mass eigenstate scalar fields therefore
these fields are in the irreducible representations of $S_{3}$ too:
$S=h_{1}\equiv H_{1}$ and $D=(h_{2},h_{3})\equiv(H_{2},H_{3})$. The symmetry
$h_{2}\leftrightarrow h_{3}$ is now another $S_{3}$ symmetry. This symmetry,
again if it is necessary, can be softly broken by adding terms like
$\mu^{2}_{nm}H^{\dagger}_{n}H_{m}$, $n,m=2,3$ ($\mu^{2}$’s are also consider
real for simplicity). Assuming that $\mu^{2}_{22}=\mu^{2}_{33}=\nu^{2}$, and
$\mu^{2}_{23}=\mu^{2}$, the mass matrices are given by
$M^{2}_{n}=\left(\begin{array}[]{ccc}m^{2}_{n_{1}}&0&0\\\
0&m^{2}_{n_{2}}+\nu^{2}&\mu^{2}\\\
0&\mu^{2}&m_{n_{2}}+\nu^{2}\end{array}\right),$ (20)
where $n=h,a,c$, for scalar, pseudoscalar and charged scalar field,
respectively. The mixing now is only in the fermiophobic sector and the masses
square are
$\displaystyle\bar{m}^{2}_{h_{1}}=m^{2}_{h_{1}},\;\;\bar{m}^{2}_{h_{2}}=m^{2}_{h}+2\mu^{2}-\nu^{2},\;\;\bar{m}^{2}_{h_{3}}=m^{2}_{h}+2\mu^{2}+\nu^{2},$
$\displaystyle\bar{m}^{2}_{a_{1}}=0,\;\;\bar{m}^{2}_{a2}=m^{2}_{h}+2\mu^{2}-\nu^{2},\;\;\bar{m}^{2}_{a3}=m^{2}_{h}+2\mu^{2}+\nu^{2},$
$\displaystyle\bar{m}^{2}_{c_{1}}=0,\;\;\bar{m}^{2}_{c_{2}}=m^{2}_{c}+\mu^{2}-\frac{1}{2}\nu^{2},\;\;\bar{m}^{2}_{c_{3}}=m^{2}_{c}+\mu^{2}+\frac{1}{2}\nu^{2}$
(21)
The mass matrices of the form in (20) are diagonalized by the orthogonal
matrix
$U=\left(\begin{array}[]{ccc}1&0&0\\\
0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right),$ (22)
and the mixing between $h_{2}$ and $h_{3}$ sector is maximal. Notice that
there are still mass degeneracy between $CP$-even ad $CP$-odd scalars.
Thus, in terms of the mass eigenstate fields, the scalar doublets of $SU(2)$
are written as $S=h_{1}$ and
$D\equiv-(D_{1},D_{2})=-(-h_{2}+h_{3},h_{2}+h_{3})$, where $h_{i}$ ar the
$SU(2)$ doublets written in terms of the mass eigenstate fields. Explicitly
$\displaystyle S\equiv h_{1}=\left(\begin{array}[]{c}h^{+}_{1}\\\
\frac{1}{\sqrt{2}}(3v+h^{0}_{1}+ia^{0}_{1})\end{array}\right),\;$ (25)
$\displaystyle
D_{1}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}(-h^{+}_{2}+h^{+}_{3})\\\
\frac{1}{2}[-h^{0}_{2}+h^{0}_{3}+i(-a^{0}_{2}+a^{0}_{3})]\end{array}\right),D_{2}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}(h^{+}_{2}+h^{+}_{3})\\\
\frac{1}{2}[h^{0}_{2}+h^{0}_{3}+i(a^{0}_{2}+a^{0}_{3})]\end{array}\right).$
(30)
## V The Yukawa sector
If in the lepton and quark sectors all fields transform, in both models, as
singlet under $S_{3}$, and for this reason they only interact with the singlet
$S$ as following:
$-\mathcal{L}_{yukawa}=\bar{L}_{iL}(G^{l}_{ij}l_{jR}S+G^{\nu}_{ij}\nu_{jR}\tilde{S})+\bar{Q}_{iL}(G^{u}_{ij}u_{jR}\tilde{S}+G^{d}_{ij}d_{jR}S)+H.c.,$
(31)
$\tilde{S}=i\tau_{2}S^{*}$.
As we said before, the fermion masses arise only through the VEV of the
singlet $S$ which is the only linear combination with a non-zero VEV, see Eqs.
(15) or (30). Hence, there is no FCNC in the lepton and quark sectors because
there is just one source of the fermion masses which are given by
$M^{f}=(v_{SM}/\sqrt{2})G^{f}$, $f=l,\nu,u,d$ and where $v_{SM}=246$ GeV. The
neutral interactions are
$(\sqrt{2}/v_{SM})\bar{f}_{L}\hat{M}^{f}f_{R}h^{0}_{1}$, $\hat{M}^{f}$ is the
diagonal mass matrix in the $f$-sector. These mass matrices are general enough
to accommodate a realistic $V_{PMNS}$ and $V_{CKM}$ mixing matrices. However,
since the right-handed neutrinos may have a Majorana mass term we can have a
type-I seesaw mechanism. Fermions are mass degenerated at tree level, if their
left-handed and right-handed components transform as a triplet 3 under
$S_{3}$, for this reason the choice that all fermions transform as singlets
under $S_{3}$ is to guarantee realistic mass and mixing matrices. Moreover,
this choice implies that the triplet of the scalar reducible triplets has to
be broken into the doublet and singlet.
## VI Analysis of the scalar potential
In order to have a potential bounded from below we consider only the quartic
terms in Eq. (17) by defining as in gunion :
$\displaystyle\chi_{1}=h_{1}^{\dagger}h_{1}~{};~{}\chi_{2}=h_{2}^{\dagger}h_{2}~{};~{}\chi_{3}=h_{3}^{\dagger}h_{3}$
(32)
$\displaystyle\chi_{4}=\Re(h_{1}^{\dagger}h_{2})~{};~{}\chi_{5}=\Im(h_{1}^{\dagger}h_{2})~{};~{}$
$\displaystyle\chi_{6}=\Re(h_{1}^{\dagger}h_{3})~{};~{}\chi_{7}=\Im(h_{1}^{\dagger}h_{3})~{};~{}\chi_{8}=\Re(h_{2}^{\dagger}h_{3})~{};~{}\chi_{9}=\Im(h_{2}^{\dagger}h_{3}),$
it is possible now rewrite the quartic terms in the scalar potential given in
Eq. (17) as follows:
$\displaystyle V(D,S)_{4}$ $\displaystyle=$
$\displaystyle[\sqrt{\lambda_{4}}\,\chi_{1}-\sqrt{\lambda_{1}+\lambda_{3}}\,(\chi_{2}+\chi_{3})]^{2}$
$\displaystyle+$
$\displaystyle[\lambda_{5}+2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}][\chi_{1}(\chi_{2}+\chi_{3})-\chi_{4}^{2}-\chi_{5}^{2}-\chi_{6}^{2}-\chi_{7}^{2}]$
$\displaystyle+$ $\displaystyle
2[\lambda_{6}+\lambda_{5}+2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}](\chi_{4}^{2}+\chi_{6}^{2})$
$\displaystyle+$ $\displaystyle[2\Re
e\lambda_{7}-2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}-\lambda_{5}-\lambda_{6}][\chi^{2}_{4}+\chi^{2}_{6}-\chi^{2}_{5}-\chi^{2}_{7}]$
$\displaystyle-$ $\displaystyle
4\lambda_{3}\,\chi_{1}+4(\lambda_{3}-\lambda_{2})\chi^{2}_{8}+4(\lambda_{3}-\lambda_{2})\chi^{2}_{9}$
$\displaystyle+$ $\displaystyle 4\Im
m\lambda_{8}[\chi_{5}(\chi_{3}-\chi_{2})+2\chi_{7}\chi_{8}]$
The constrains on the parameters $\lambda_{i}$ such that the potential with
the assumed vacuum alignment is bounded from below, are given by:
$\displaystyle\lambda_{1}+\lambda_{3}>0,~{}\lambda_{4}>0,~{}\lambda_{3}<0,~{}\lambda_{5}>-2\sqrt{(\lambda_{1}+\lambda_{3})\lambda_{4}}$
(34) $\displaystyle
2\Re(\lambda_{7})-\lambda_{5}-\lambda_{6}>2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}.$
Notice that although this is a scalar potential with three Higgs scalars
doublets under $SU(2)$ it is as simple as the two doublet case in Ref. gunion
.
## VII Conclusions
In general, multi-Higgs doublet models have complicated scalar potentials. In
particular, if the Higgs sector also has, like the fermion sectors, three
generations, we should expect the existence of extra symmetries to make simple
the interactions and mass spectra in the scalar sectors. The symmetry, if any,
would be discrete for not to have extra Goldstone bosons. In literature, this
sort of models has been consider to get a given texture of the fermion mass
matrix. However, the main concern in this paper is to use the $S_{3}$ symmetry
to have three-Higgs doublet models in which the scalar mass spectra and the
mixing among scalar fields are simple, despite the three Higgs doublets, and
the models remain closer to the standard model than a three-Higgs doublets
without $S_{3}$ symmetry. Moreover, like multi-Higgs models with no flavor
changing neutral currents mediated by neutral scalars, the only mixing
parameters appearing in the Yukawa interactions are the CKM and PMNS angles
and phases.
We have presented two models in which the three scalar doublets of $SU(2)$
transform in different ways under the irreducible representations of $S_{3}$
which have two fermiophobic doublets. The main difference between both models
is that in model B all mass matrices are diagonal and there is no mixing in
these sectors. When the soft terms are added, in model B unlike the model A
there is now mixing between the inert doublets. Hence, in the former model
there is at least one extra mixing angle. The cases considered here is that of
maximal mixing. In both models, as in the two-doublet model pich , there is
also an alignment of the Yukawa matrices, but in model A this is achieved even
without using a global $SU(2)$ transformation in order to define a basis in
which only one of the doublet gain a nonzero VEV, but because the $\langle
D_{1,2}\rangle=0$ where $D_{i}$ are the components of the $S_{3}$ doublet in
Eq. (2).
The neutral component of singlet $S$ can be identified with the standard model
Higgs scalar. But the inert doublet shall be produced in weak photon and
vector fusion and it implies that the $\gamma\gamma$, $WW$ and $ZZ$ decay
modes have an enhancement in both models. See Ref. 2gamma for details. The
present model also implies a Higgs boson with SM couplings but additional
invisible decay modes if the inert doublets are lighter than the SM-like
scalar invisible . This depend on the value of $\mu^{2}_{d}$ in Eqs. (8)-(10)
in model A when the $S_{3}$ symmetry is not softly broken. The same in Model
B. The issue of $CP$ violation in the present context is as follows. If we
would like to maintain the dark scenario of the doublet $D$ even when the VEVs
are complex, we should expect that
$v_{1}e^{i\theta_{1}}=v_{2}e^{i\theta_{2}}=v_{3}e^{i\theta_{3}}=Ve^{i\Theta}$
was a stable minimum of the scalar potential. However assume that this is the
case the phase $\Theta$, which appears only in the singlet $S$, can be
transformed away with a global $U(1)$ transformation as it happens in the
standard model. Thus, in the situation there is no spontaneous $CP$ violation
through the VEVs. However, it is still possible to have soft $CP$ violation
through the quadratic non-diagonal term in the scalar potential
$\mu^{2}h^{\dagger}_{2}h_{3}$ assuming that $\mu^{2}$ is complex, as in Ref.
wu . Note that the $\lambda_{6}$ term has a form
$\lambda_{6}([S^{\dagger}D]_{2})^{2}+\lambda^{*}_{6}([D^{\dagger}S]_{2})^{2}$,
assuming a global phase rotation $S\rightarrow Se^{ia_{S}}$ and $D\rightarrow
De^{ia_{D}}$, and considering that the content of the theory cannot change by
this global phases we have that $\lambda_{6}=|\lambda_{6}|e^{i\alpha}$, is
possible we chose $a_{S}-a_{D}=\alpha/2+\pi/2$, so, is possible that
$\lambda_{6}=-|\lambda_{6}|$, and we have the same analyses for $\lambda_{8}$.
It is interesting that there are dark matter candidates in this model but this
deserve a detailed study zapata . Finally, we would like to mention that
depending on the masses of the dark scalars the SM-like neutral scalar may
have invisible decays invisible .
We would like to stress that the existence of two fermiophobic doublets, the
tribimaximal mixing in model A and no mixing at all in model B, and the flavor
conservation in the neutral currents mediated by scalars are consequences of
three ingredients: i) the $S_{3}$ symmetry, ii) with the representation
content of the fermion and scalar multiplets, and, iii) the vacuum alignment.
For instance, a three Higgs doublet model without any additional symmetry but
with the same vacuum alignment does not have the features above.
###### Acknowledgements.
The authors would like to thank to CAPES (ACBM) for fully support and to CNPq
and FAPESP (VP) for partial support.
## Appendix A Constraint equations in model A
From (1) we obtain the constraint equations are written explicitly as
$\displaystyle
18t_{1}=6\mu^{2}_{d}(2v_{1}-v_{2}-v_{3})+6\mu^{2}_{s}V+(2(4\lambda_{1}+4\lambda_{3}+\lambda_{4}+2(-2\sqrt{2}\lambda_{8}+\bar{\lambda}^{\prime}))v_{1}^{3}$
$\displaystyle-3(4\lambda_{1}+4\lambda_{3}-2\lambda_{4}-\sqrt{2}\lambda_{8}-\bar{\lambda}^{\prime})v_{1}^{2}(v_{2}+v_{3})+6v_{1}((\lambda_{4}+2(\lambda_{1}+\lambda_{3}+\sqrt{2}\lambda_{8}))v_{2}^{2}$
$\displaystyle+(2\lambda_{4}-\lambda_{5}-\lambda_{6}-2(\lambda_{7}+\sqrt{2}\lambda_{8}))v_{2}v_{3}+(\lambda_{4}+2(\lambda_{1}+\lambda_{3}+\sqrt{2}\lambda_{8}))v_{3}^{2})$
$\displaystyle+(v_{2}+v_{3})((-4\lambda_{1}-4\lambda_{3}+2\lambda_{4}+\sqrt{2}\lambda_{8}+\bar{\lambda}^{\prime})v_{2}^{2}$
$\displaystyle+(4\lambda_{1}+4\lambda_{3}+4\lambda_{4}-4\lambda_{5}-4\lambda_{6}-8\lambda_{7}-7\sqrt{2}\lambda_{8})v_{2}v_{3}$
$\displaystyle+(-4\lambda_{1}-4\lambda_{3}+2\lambda_{4}+\sqrt{2}\lambda_{8}+\bar{\lambda}^{\prime})v_{3}^{2}))$
(35)
where $V=v_{1}+v_{2}+v_{3}$, and
$\bar{\lambda}^{\prime}=\lambda_{5}+\lambda_{6}+2\lambda_{7}$. We have denoted
$\langle H^{0}_{i}\rangle=v_{i}/\sqrt{2}$. We have verified that the solution
$v_{1}=v_{2}=v_{3}$ produce a global and stable minimum for a width range of
the parameters $\lambda$s and $\mu^{2}_{d}$ and the potential is bounded from
below.
## Appendix B Constraint equations in model B
With the representation in Eq. (3), the constrain equation are
$\displaystyle 2t_{1}$ $\displaystyle=$ $\displaystyle
v_{1}\left[2\mu^{2}_{s}+2\lambda_{4}v^{2}_{1}+\bar{\lambda}^{\prime}(v^{2}_{2}+v^{2}_{3})-\frac{\lambda_{8}}{v_{1}}\left(v^{3}_{2}+v_{2}v^{2}_{3}\right)\right],$
$\displaystyle 2t_{2}$ $\displaystyle=$ $\displaystyle
v_{2}\left[2\mu^{2}_{d}+\bar{\lambda}^{\prime}v^{2}_{1}+(\lambda_{1}+\lambda_{3})(v^{2}_{2}+v^{2}_{3})-3\lambda_{8}\left(v_{1}v_{2}-\frac{v_{1}v^{2}_{3}}{v_{2}}\right)\right],$
$\displaystyle 2t_{3}$ $\displaystyle=$ $\displaystyle
v_{3}[2\mu^{2}_{d}+\bar{\lambda}^{\prime}v^{2}_{1}+2(\lambda_{1}+\lambda_{3})(v^{2}_{2}+v^{2}_{3})+6\lambda_{8}v_{1}v_{2}],$
(36)
and we see that even in the general case when $v_{1}\not=v_{2}\not=v_{3}$ they
are different from the respective equations in model A, see Eq. (35).
Notice that the $\lambda_{8}$ term avoid the zero solution for $v_{1}$ and
$v_{2}$. If this term is forbidden with a $Z_{2}$ symmetry under which
$D\to-D$ and all the other fields being even under this symmetry, we can have
the solution $v_{1}=v_{SM}$ and $v_{2}=v_{3}=0$. However, without the
$\lambda_{8}$ term it is easy to see from (1) that the scalar potential is
invariant under $S_{3}\otimes S^{\prime}_{3}$. With the soft terms there are
the following contributions, $4\mu^{2}_{22}v_{2}$ and $\mu^{2}_{23}v_{3}$ in
$t_{2}$, and $\mu^{2}_{33}v_{3}$ and $\mu^{2}_{23}v_{2}$ in $t_{3}$. The soft
terms, with $\mu^{2}_{22}=\mu^{2}_{33}=\nu^{2}$, and $\mu^{2}_{23}=\mu^{2}$,
break one of the $S_{3}$ while the other is trivially realized by the scalar
singlet that does not mix with the others two.
The neutral component of singlet $S$ can be identified with the standard model
Higgs scalar. But the inert doublet shall be produced in weak photon and
vector fusion and it implies that the $\gamma\gamma$, $WW$ and $ZZ$ decay
modes may have an enhancement in both models. See Ref. 2gamma for details.
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* (27) H. Cardenas, A. C. B. Machado, V. Pleitez and J.-A. Rodriguez, Phys. Rev. D 87 035028 (2013), arXiv:1212.1665.
|
arxiv-papers
| 2012-05-04T15:57:44 |
2024-09-04T02:49:30.559010
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "A. C. B. Machado and V. Pleitez",
"submitter": "Vicente Pleitez",
"url": "https://arxiv.org/abs/1205.0995"
}
|
1205.1043
|
# High-resolution monitoring of parsec-scale jets in the Fermi era
Eduardo Ros Departament d’Astronomia i Astrofísica, Universitat de València,
E-46100 Burjassot, Valencia, Spain
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn,
Germany
###### Abstract
I review here the present observational efforts to study parsec-scale radio
jets in active galactic nuclei with very-long-baseline interferometry (VLBI)
as related to the new window to the Universe opened by the LAT instrument on-
board the Fermi Gamma-Ray Space Telescope. I describe the goals and
achievements of those radio studies, which aim to probe the emission
properties, morphological changes and related kinematics, magnetic fields from
the linear and circular polarization, etc., and I put those in the context of
the radio–gamma-ray connection. Both statistical studies based on radio
surveys and individual studies on selected sources are reported. Those should
shed some light in the open questions about the nature of emission in blazars.
## I INTRODUCTION
Since the measurements from CGRO/EGRET we know that the $\gamma$-sky is
dominated by the Galactic plane diffuse emission, pulsars, and blazars. This
has been confirmed with the outstanding findings of Fermi and its LAT
detector, launched in June 2008 and operative since August 2008. The 2nd
Fermi/LAT catalog (2LAC) [1] includes all extragalactic sources with a
significant detection over the first two years of scientific operation. The
so-called ‘clean sample’ of the 2LAC contains 886 sources, from which 310 are
flat-spectrum radio quasars (FSRQ), 395 are BL Lac objects, 157 candidate
blazars of unknown type, 8 misaligned active galactic nuclei (AGN), 4 narrow-
line Seyfert 1 (NLS1), 10 AGN of other types, and 2 starburst galaxies. Notice
that sources with only sporadic activity were missing, since they don’t reach
the test statistic threshold of 25 ($\mathrm{TS}>25$)111Remarkably, sources
like 3C 120 or 3C 111, which were present in earlier catalogs, are not listed
at the 2LAC..
Fermi/LAT has shown that BL Lacs are the most common $\gamma$-emitters, more
frequently than FSRQ. One should be aware of the strong biases introduced by
Doppler boosting on the observed flux from AGN, which biases the brightest
extra-galactic $\gamma$-ray objects and high frequency radio sources towards
fast jets with a small viewing angle, that is, close to the line of sight
(e.g., [2, 3]). The picture is completed by observations of the AGILE
$\gamma$-mission [4] (2008-) the rapid AGN results provided by the X-ray and
$\gamma$-ray mission Swift [5] (2005-) and the ground-based very-high-energy
(VHE) $\gamma$-ray Cherenkov telescopes such as HESS (2003-), MAGIC (2004-),
CANGAROO-III (2004-), or VERITAS (2006-).
Among the big questions raised in earlier editions of this workshop, it is
open if $\gamma$-ray flares originate in relativistic shocks, what is the
distance of the main energy dissipation site from the central engine, what are
the emission mechanisms at stake, and what relates the brightness in the radio
with the $\gamma$-rays. The observational tools to address the Fermi era are
VLBI campaigns to be related with the continuous all-sky $\gamma$-ray
observations by Fermi/LAT, complemented with multi-band campaigns including as
well IR, optical, UV, and X-rays. The information achieved by intensive flux
density monitoring campaigns are being addressed by other authors at this
conference.
To review this topic, I will first introduce the observables measured directly
and indirectly by VLBI and the other parameters to be compared with from the
$\gamma$-ray monitoring and from the spectral energy distribution (SED)
studies of AGN. I will continue describing the main survey campaigns and some
of their highlights so far. To complement this, I will present a (necessarily
incomplete) selection of studies on individual sources combining VLBI and
$\gamma$-observations.
## II WHAT IS MEASURED BY VLBI?
### II.1 VLBI targets
Blazars display powerful jets oriented towards the observer, and show high
brightness temperatures in the radio regime, which allow them to be observed
by VLBI. VLBI is a technique that provides resolutions of the order of the
milliarcsecond (parsecs at cosmological distances), working regularly from 3
mm up to 1 m wavelengths. At the longest wavelengths the presence of the
ionosphere as a dispersive propagation medium distorts the waves and limits
VLBI performance. At the shortest wavelengths, where the highest resolution is
achieved, atmospheric turbulence, and especially water vapour, disturb the
observations, causing coherence loss and limiting the integration time and
therefore its detection threshold. Wavelengths from 7 mm up to 18 cm are the
most commonly used.
One of the main targets of VLBI are AGN, given their high brightness
temperatures $T_{\mathrm{b}}$, of up to $10^{12}$ K in the core, dropping to
$10^{10}$ K or lower values in the jet (see below for the definition of
$T_{\mathrm{b}}$).
Distinct features or ‘blobs’ in the jet can be identified with shocks or
instabilities in the jet. The magnetic field orientation can be estimated by
the linear and circular polarisation, if observing in this mode. The
structural changes observed by combining several epochs can be associated with
helical jets (e.g., [6]) or binary black holes (e.g., [7]). Identifying
‘moving’ features from different observing times, the kinematics of those
components are established, and even the ejection times of the features at the
base of the jet could be related to the outburst observed at the single-dish
light curves of the sources [8]. At the base of the jet, the emission is self-
absorbed by synchrotron, so that the peak of brightness (also labeled as jet
’core’) corresponds to different physical locations at different frequencies.
The absolute position of the source can be recovered by astrometric methods
[9]. If this is not possible, images at different frequencies can be
registered by aligning the optically thin regions either by model fitting with
Gaussian functions [10], by cross-correlation of the jet features [11, 12], or
by a combination of astrometry and jet alignment [13, 14]. After the core-
shift correction, the synchrotron turnover frequency and flux density can be
computed all over the jet, and from those, physical parameters in the jet are
estimated, such as the magnetic field, and pressure gradients [15].
### II.2 Extracting information from VLBI images
So, from one single VLBI image, or combining them in time (kinematics) or
frequency (spectral studies), we can measure directly several physical
quantities in AGN, most of them affected the observed flux by Doppler beaming,
caused by relativistic effects and the small viewing angle of the jet, pointed
almost towards the observer. Given a region of the jet moving downstream with
a speed $\beta=v/c$ in the rest frame ($c$ is the light speed), with an angle
$\theta$ between the jet, the Lorentz factor will be
$\Gamma=(1-\beta^{2})^{-1/2}$ and the Doppler factor will be
$\delta=(\Gamma(1-\beta\cos\theta))^{-1}$. Beaming will affect several
magnitudes, and the relativistic effects and the small viewing angle will
affect several parameters being measured by VLBI. I summarize those in Table
1, and describe them in the following paragraphs.
Table 1: Summary of observable VLBI, SED, and $\gamma$-ray parameters Radio
---
Parameter | Units
Radio detection | |
Apparent speed | $\beta_{\mathrm{app}}$ | $c$
Flux density | $S$ | Jy
Brightness temperature | $T_{\mathrm{b}}$ | K
Apparent opening angle | $\psi$ | deg
Luminosity | $L_{\mathrm{R}}$ | W Hz-1
Jet-to-counterj. ratio | R | –
P.A. misalignmenta | $\Delta\phi$ | deg
Spectral index | $\alpha$ | –
Polarisation angle | $\chi$ | deg
Polarisation level | $m$ | %
Faraday rotation | RM | rad m-2
Viewing angle | $\theta$ | deg
Lorentz factor | $\Gamma$ | –
Doppler factor | $\delta$ | –
Ejection epoch | $t_{0}$ | yr
SED
Sync. frequency peak | $\nu_{\mathrm{max,sync}}$ | Hz
Lum. at sync. peak | $L_{(}\nu=\nu_{\mathrm{max,sync}})$ | W Hz-1
Inv. Compton frequency peak | $\nu_{\mathrm{max,IC}}$ | Hz
Lum. at inv. Compton peak | $L_{(}\nu=\nu_{\mathrm{max,IC}}$ | W Hz-1
Gamma
$\gamma$-detection | |
Flare epoch | $t_{\gamma\mathrm{-flare}}$ | yr
Flux | $S_{\gamma}$ | Jy
Flux variability | $\Delta S_{\gamma}/S_{\gamma}$ | –
Luminosity | $L_{\gamma}$ | W Hz-1
Photon index | $\Gamma_{\gamma}$ | –
Gamma-ray to radio flux ratio | $G_{r}$ | –
a Kiloparsec- and parsec-scale misalignment
#### II.2.1 Directly measured parameters by VLBI
After identifying features between different observing epochs, we can
determine the sky motion of those and compute the apparent speed
$\beta_{\mathrm{app}}=\beta\sin\theta/(1-\beta\cos\theta)$. For a given
$\beta$, the maximum speed $\beta_{\mathrm{app,max}}=\beta\Gamma$ is reached
when $\cos\theta_{\mathrm{max}}=\beta$.
After hybrid mapping or Gaussian model fitting, we can measure for each
feature the value of the flux density $S$, usually expressed in Jy ($10^{-26}$
W m-2 Hz-1). The luminosity can be obtained directly from the flux density:
$L=4\pi D_{L}^{2}S$ where $D_{L}$ is the luminosity distance (to be computed
from the redshift $z$ measured in the optical). The intrinsic and observed
luminosity are also affected by the $K$-correction and Doppler boosting (and
the spectral index $\alpha$, from $S\propto\nu^{+\alpha}$) as follows
$L_{\mathrm{obs}}=L_{\mathrm{int}}\times\delta^{n-\alpha}\times(1+z)^{-(1-\alpha)}$
($n=2\,,3$).
Having the flux density $S$ and the interferometer resolution (beam), we
compute the brightness temperature $T_{\mathrm{b}}=1.222\times
10^{12}S(1+z)/\nu^{2}ab$ where $T_{\mathrm{b}}$ is given in K, $\nu$ is the
observing frequency in Hz, $S$ is the flux density in Jy, and $a$ and $b$ are
the major and minor beam axes, respectively, in milliarseconds. The intrinsic
and observed brightness temperatures are related with the beaming Doppler
factor $\delta$ by $T_{\mathrm{b,obs}}=T_{\mathrm{b,int}}\times\delta$.
We can also measure the difference in position angle of the jet in parsec- and
kiloparsec-scales (the latter from connected interferometers such as the VLA
or MERLIN), and get the jet misalignment angle $\Delta\phi$.
For images with a high dynamic range we can obtain the jet-to-counterjet
ratio, which is
$R=((1+\beta\cos\theta)/(1-\beta\cos\theta))^{2-\alpha}=(\beta_{\mathrm{app}}^{2}+\delta^{2})^{2-\alpha}$.
By measuring how the jet gets broader (see e.g., [16, 17]), we measure the
apparent opening angle $\psi$. Since the jets are not precisely in the plane
of the sky, the apparent and the intrinsic opening angle are related by
$\psi_{\mathrm{int}}=\psi_{\mathrm{obs}}\sin\theta$.
Finally, if we observe in polarisation mode, we can measure the polarisation
level $m$ (as the ratio of linearly polarised and total intensity) as well as
the polarisation angle $\chi$ (also known as electric vector position angle or
EVPA). If we have several frequencies, we can compute the change of the EVPA
as a function of the squared wavelength and determine the Faraday rotation
measurement RM.
#### II.2.2 Indirectly measured parameters
From the parameters measured above we can see that the intrinsic parameters
$\beta$, $\theta$, or $\delta$ are degenerate and several solutions are
possible for a given value of $R$, $\beta_{\mathrm{app}}$, etc.
We can measure $\delta$ independently of the VLBI kinematical analysis by
computing the flux density variations measured by a densely sampled single-
dish or VLBI monitoring campaign [19]: The variability time can be defined
from the variations of the flux density $S$ by decomposing a flux density
flare into exponential flares as $\Delta S(t)=\Delta
S_{\mathrm{max}}e^{(t-t_{\mathrm{max}})/\epsilon\tau}$, where $\Delta
S_{\mathrm{max}}$ is the maximum amplitude of the flare in Jy,
$t_{\mathrm{max}}$ is the epoch of the flare maximum and $\tau$ is the rise
time of the flare. $\epsilon=1$ for $t<t_{\mathrm{max}}$ and $\epsilon=1.3$
for $t>t_{\mathrm{max}}$. In this way, $T_{\mathrm{b,obs(var)}}=1.474\times
10^{13}S_{\mathrm{max}}D_{L}^{2}\nu^{-2}\tau^{-2}(1+z)^{-1}$ where $D_{L}$ is
the luminosity distance in Mpc and $\nu$ the frequency in Hz. From here the
variability Doppler factor is
$\delta_{\mathrm{var}}=(T_{\mathrm{b,obs(var)}}/T_{\mathrm{b,int}})^{1/3}$,
where $T_{\mathrm{b,int}}$ is assumed to be $5\times 10^{10}$ K.
With the apparent speed $\beta_{\mathrm{app}}$ and $\delta_{\mathrm{var}}$, we
also obtain the bulk Lorentz factor
$\Gamma=(\beta_{\mathrm{app}}^{2}+\delta_{\mathrm{var}}^{2}+1)/(2\delta_{\mathrm{var}})$
and the viewing angle
$\theta=\tan(2\beta_{\mathrm{app}}/\beta_{\mathrm{app}}^{2}+\delta_{\mathrm{var}}^{2}-1)$.
Alternatively, an upper bound for the viewing angle can be obtained from the
jet-to-counterjet ratio as
$\theta<\beta^{-1}\arccos(R^{1/2-\alpha}-1/R^{1/2-\alpha}+1)$.
Last but not least, knowing the apparent speed $\beta_{\mathrm{app}}$ we can
compute the ejection time $t_{0}$ simply by extrapolating the time for which
the distance of the feature to the core is zero. If jet features are related
to plasma injections at the base of the jet, this should be seen at high
frequencies, as it has been observed in X-rays for 3C 120 [20] or NGC 1052
[21].
### II.3 SED properties
AGN SED (representation of intensity as a function of frequency) show usually
two bumps, caused by non-thermal synchrotron emission at the low energies and
(most probably) by inverse Compton up-scattering of ambient optical-UV
photons, although the contribution from energetic hadrons cannot be ruled out
(e.g., [22]). Therefore, the positions and luminosities of SED peaks of both
bumps are also used for correlation studies:
$(\nu_{\mathrm{max,sync}},L_{(}\nu=\nu_{\mathrm{max,sync}})$ and
$(\nu_{\mathrm{max,IC}},L_{(}\nu=\nu_{\mathrm{max,IC}})$. From this,
additional parameters such as the gamma-radio loudness
$G_{r}=L_{\gamma}/L_{R}$ has been defined [23].
### II.4 Correlating VLBI measurements and $\gamma$-properties
Most of the statistical studies with radio surveys are based in relating the
above mentioned parameters with the $\gamma$-measurements. The first check to
be performed are cross-correlating catalogs (radio with the Fermi ones–three-
month list [24], one-year AGN catalog [25], or the two-year AGN catalog [1]),
and comparing $\gamma$-detection with radio properties. After that, the values
of the radio parameters can be plotted against the high-energy parameters, and
correlations are searched by using different statistical methods.
I have listed the parameters from $\gamma$-ray observations in Table 1 as
well. In principle, those high-energy parameters are the $\gamma$-flux
$S_{\gamma}$ and its $\gamma$-flux variability $\delta S_{\gamma}/S_{\gamma}$,
the $\gamma$-luminosity222Notice that computing the luminosity in the
$\gamma$-regime is more problematic than in the (almost monochromatic) radio
regime, given the fact that photons with frequencies different in several
orders of magnitude are being used. An expression for the $\gamma$-luminosity
is given in [23] as $L_{\gamma}=4\pi
D_{L}^{2}S_{0,\,1}/(1+z)^{2-\Gamma_{\gamma}}$, where
$S_{0,\,1}=C_{1}E_{1}F_{0,\,1}(\Gamma_{\gamma}-1/\Gamma_{\gamma}-2)(1-(E_{1}/E_{2})^{\Gamma_{\gamma}-2})$,
being $F_{0,\,1}$ the upper limit on photon flux above a given energy
$E_{1}=0.1$ GeV, the upper energy $E_{2}=100$ GeV, and $C_{1}=1.602\times
10^{-3}$ erg/GeV =1 J/J as conversion factor. The authors fixed
$\Gamma_{\gamma}=2.1$. $L_{\gamma}$, and the $\gamma$ photon index
$\Gamma_{\gamma}$ (since the number of photons per unit time per unit area in
a frequency bandwidth is $dN/dE=(F_{\nu}/h\nu)\nu_{0}$, where
$\nu_{0}=h/E_{0}$, where $E_{0}=1$ keV and $h$ is Planck’s constant, from the
radio astronomy convention, $S=F_{\nu}\propto\nu^{+\alpha}$,
$dN/dE\propto\nu^{\Gamma_{\gamma}}\propto\nu^{\alpha-1}$. So,
$\Gamma_{\gamma}=\alpha-1$ if we want to compare the radio spectral index and
the high energy photon index. We can also add the flaring activity, including
flare epochs $t_{\gamma\mathrm{-flare}}$.
A naive approach is to check the relationship between the different
observables both in VLBI and in $\gamma$-rays, and draw physical conclusions
of them. Having a sample of objects at one band, the properties at the other
band can be divided between detections and non-detections in histograms, and
usually the sources are then divided into their optical classification (FSRQ,
BL Lac, Radio Galaxy, etc.) or into their high-energy peak classification
(HSP, ISP, LSP). This was especially the approach on the first VLBI-related
publications of the Fermi era. When more data have been available, plots of
properties at one band versus the other band provide some hints of the nature
of radio-loud/quiet and $\gamma$-loud/quiet objects.
## III VLBI SURVEYS
VLBI has been performed since the early 1970s, and more intensively since the
construction of the Very Long Baseline Array (VLBA333The VLBA is operated by
the US National Radio Astronomy Observatory, a facility of the US National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc.) in the early 1990s. Regular observations with open calls
are being performed by the European VLBI Network (EVN444The European VLBI
Network is a joint facility of European, Chinese, South African and other
radio astronomy institutes funded by their national research councils.) and
the Long Baseline Array (LBA555The Long Baseline Array is part of the
Australia Telescope which is funded by the Commonwealth of Australia for
operation as a National Facility managed by CSIRO.; expanded with the addition
of telescopes outside Australia) operate regularly. Several big surveys have
monitored the brightest AGN for decades, every one with a different approach
(a review on VLBI imaging surveys is presented in [26]). Reaching both
hemispheres, the geodetic networks, at present under the umbrella of the
International VLBI Service, collected data of hundreds of sources for
calibration purposes, and for determining tectonic motions, Earth Orientation
Parameters, the length of the day, and other geophysical parameters.
Table 2: VLBI surveys complementing $\gamma$-observations
Program | $\lambda$ | $N_{\mathrm{sources}}$ | $N_{\mathrm{epochs}}$a | Time | Ref.
---|---|---|---|---|---
GMVA 3mm | 3 mm | 121 | 2 | 2004- | [27]
Boston Univ. | 7 mm | 35 | 50 | 2007- | [28]
TeV Sample | 7 mmb | 7 | 5 | 2006- | [29]
MOJAVE/2 cm Survey | 2 cm | 300 | 20 | 1994- | [30]
Bologna low-$z$ | 2/3.6 cm | 42 | 2 | 2010- | [31]
TANAMI | 1.3/3.6 cm | 80 | 5 | 2008- | [17]
VIPS | 6 cm | 1127 | 1 | 2007 | [32]
VIPS subsample | 6 cm | 100 | 2 | 2010- | [33]
CJF | 6 cm | 293 | 3 | 1990s | [34]
ICRF | 3.6/13 cm | 500 | 10 | 1990s | [35]
VCS | 3.6/13 cm | 3400 | 1 | 1990s | [36]
a Typical number of epochs per source
b Also including $\lambda$1.3 cm & $\lambda$3.6 cm
With astronomical goals, the Caltech-Jodrell Bank Survey, initiated in the
early 1990s, was the first big effort, and later it was followed by the 2 cm
VLBA Survey—now turned into the MOJAVE project—in the mid 1990s, and newer
projects with additional purposes such as the Boston University blazar
monitoring program, the 86 GHz Survey with the Global Millimetre VLBI Array,
joined the list. During the 2000s, several projects were designed to collect
information for the Fermi era. I will list in the next sections some of the
findings of the ongoing surveys (summarised in Table 2) reported in refereed
journals. A summary of the correlations found is presented in Table 3.
Table 3: Correlations reported in survey studies following the notation of
table 1
R/$\gamma$ | Det? | $t_{\gamma\mathrm{-f}}$ | $S_{\gamma}$ | $L_{\gamma}$ | $\Gamma_{\gamma}$ | $\delta S_{\gamma}/S_{\gamma}$ | $G_{r}$ | $\nu_{\mathrm{m,s}}$
---|---|---|---|---|---|---|---|---
Det? | [37]a | | | | | | |
$\beta_{\mathrm{app}}$ | [50],[56] | | | [56] | | [50] | |
$S_{\mathrm{R}}$ | [49] | [52]d | [49, 52],[37] | | | | | [23]e
$T_{\mathrm{b}}$ | [49, 37],[17] | | [37] | | | | | [23]f
$\psi$ | [16, 17, 37]b | | | | | | [23] |
$\Delta\phi$ | [50] | | | | | | |
$m$ | [37] | | | | | | | [23]g
$\theta$ | [51]c | | | | | | |
$\Gamma$ | [51] | | | | | | |
$\delta$ | [51, 37] | | | | | | |
$G_{r}$ | | | | | [23]e | | | [23]
Note: No correlation is shown in italics.
a No correlation for BL Lac, dependent on $\delta$ for FSRQ
b No correlation is found in [16] for the intrinsic opening angle
$\psi_{\mathrm{int}}$
c The $\theta_{\mathrm{int}}$ distribution is narrower for the
$\gamma$-detected sources in [51]
d The radio core flux is delayed w.r.t. $\gamma$-flares, following [52]
e Negative correlation
f HSP BL Lac objects tend to be less compact than other objects
g HSP BL Lac objects tend to lower polarisation levels.
### III.1 VIPS
The VLBA Imaging and Polarimetry
Survey666http://www.phys.unm.edu/$\sim$gbtaylor/VIPS/ is a one-epoch survey
including polarimetry at 5 GHz perfomed in the mid 2000s. The first stage of
the observations was described in [32]. Results on the radio properties of
sources detected by Fermi/LAT showed no correlation between $S$ and
$S_{\gamma}$ [37]. Furthermore, from this study radio-bright BL Lac objects
detected by Fermi/LAT were similar to the non-LAT ones, but for FSRQ there is
a difference on the emission related to Doppler boosting: not surprisingly,
only the FSRQ with higher $\delta$ are $\gamma$-loud. Polarisation at the base
of the jet is a signature as well for $\gamma$-ray loud AGN.
### III.2 Boston University Blazar Program
The Boston University (BU) blazar group has been performing a monitoring
campaign with VLBA images sampled monthly at 43 GHz since 2007 (continuing the
monitoring performed on some sources on the early 2000s), whose calibrated
data are publicly available777http://www.bu.edu/blazars/VLBAproject.html. Due
to the nature of the survey (reduced number of sources, intensively
monitored), the published results are focused on multi-band studies of
individual sources (see below). A description of the overall project is given
in [28]. In general it is interpreted that a high $\gamma$-state is related to
an outburst at the millimetre regime. The outburst is associated to the
passing of a traveling shock through a recollimation shock in the base of jet
[45]. Other explanations are possible, e.g., a pinch instability in a helical
jet.
### III.3 MOJAVE
The 2 cm Survey project [46] was started in 1994, short after VLBA completion,
with the aim of monitoring a sample of bright and representative AGN at sub-
parsec scales. It was continued from 2002 onwards and until now under the name
‘Monitoring Of Jets in Active galactic nuclei with VLBA Experiments’
(MOJAVE888http://www.physics.purdue.edu/astro/MOJAVE/). The project studies a
complete sample of 135 objects above $-30^{\circ}$ observed at 15 GHz
including dual polarisation, and its database contains images of up to 300
sources. It consists of continuous long-term monitoring including source-
specific observing cadences, yielding high-quality jet motions. The well-
defined sample enables solid statistics of the parent population (e.g., [47,
48]). The high-quality imaging and monitoring results have also made possible
numerous individual source studies performed by the MOJAVE group or by others,
since all calibrated data are made publicly available.
Studies based on the Fermi three-month bright source list [24] show that the
sources being more compact, brighter in radio, with higher radio activity, and
with higher $\delta$ values, are favorably detected by Fermi [49]. The
$\gamma$-ray bright sources tend to have faster jets, especially in the case
of quasars [50]. Concerning the apparent opening angles, jets detected in
$\gamma$-rays tend to be broader than the non-detected ones, but the intrinsic
opening angles are similar for detected and non-detected ones, connecting
beaming and $\gamma$-detection [16]. A later study shows that LAT-detected
blazars have higher $\delta$ values than the non-detected ones, and the
viewing angle distribution is different for the $\gamma$-ray bright and weak
sources; the comoving frame viewing angle distribution is narrower for
$\gamma$-bright sources [51]. Furthermore, a correlation analysis showed a
delay between the VLBI core brightness and the $\gamma$-ray emission ($\gamma$
leads 15-GHz radio) [52]. Results on a joint $\gamma$-ray and radio-selected
sample show that the $\gamma$-ray loudness $G_{r}$ increases with the SED
$\nu_{\mathrm{IC}}$, and that the high-synchrotron-peaked (HSP999HSP show a
peak in high-energy bump of the spectral energy distribution above $10^{15}$
Hz, whereas intermediate- (ISP) and low- (LSP) synchrotron-peaked have peaks
between $10^{14}$ Hz and $10^{15}$ Hz, and below $10^{14}$ Hz, respectively. )
BL Lac objects have lower radio core $T_{\mathrm{b}}$ values [23]. To finish
with the recently published results, positive correlation was found between
$L_{\mathrm{R}}$ and $L_{\gamma}$ [48]. A study on the relationship of RM in
the MOJAVE images with $\gamma$-ray is presented in [18] and in a future
publication. After including optical properties, a positive correlation is
present between $L_{\mathrm{R}}$ and the $\gamma$-ray-optical loudness for
quasars, and a negative correlation between $L_{\mathrm{opt}}$ and the
$\gamma$-ray-radio loudness [48]. A preliminary study on the relationship
between the SED properties of the MOJAVE sources and the radio properties has
been also first presented presented in [53], and will be published elsewhere.
### III.4 TANAMI
The TANAMI project (Tracking Active Galactic Nuclei with Austral
Milliarcsecond Interferometry101010http://pulsar.sternwarte.uni-
erlangen.de/tanami/ uses the Australian Long Baseline Array with additional
telescopes to study sources at declination below $-30^{\circ}$. With a
different approach, it complements the regions of the sky not covered by
MOJAVE. The project was started in November 2007 and observes at 8.4 GHz and
22 GHz. First images at 8.4 GHz have been published, and a preliminary
analysis shows that Fermi/LAT-detected sources have larger opening angles
$\psi$ [17]. 22 GHz and spectral index images and jet kinematics will be
published in a near future.
### III.5 Other studies and samples
Some of the TeV blazars have been studied systematically with VLBI. One of the
$\gamma$-related studies studies the properties of six blazars in [29], and
additional ones are included in [54]. A VLBI-$\gamma$ study based on the
Bologna Complete Sample aims to observe a sample of 94 nearby ($z<0.1$)
sources, from which 76 are being processed now [55]. A study comparing the CJF
sample with the 1LAC shows a tentative correlation between $L_{\gamma}$ and
$\beta_{\mathrm{app}}$, especially for BL Lac objects and $\gamma$-variable
sources [56]; the apparent speed distribution seems to be the same for
$\gamma$-detected and non-detected sources. Another interesting study is the
VLBI spectral analysis of 20 blazars at the 1.4–15.4 GHz range by [57, 58],
which data are used as well in connection to the $\gamma$-ray results, to test
if the high energy emission is located at the VLBI cores.
## IV INDIVIDUAL SOURCE STUDIES
Table 4 shows a selection of sources of special interest. I have tabulated
sources with with published combined VLBI-$\gamma$ results, and added some
sources with more than one flare reported in ATels (to keep the list short).
For a list of the sources detected at TeV, see http://tevcat.uchicago.edu/. A
very useful list compiled by the MOJAVE team listing all sources being
observed by the different surveys can be found at
http://www.physics.purdue.edu/astro/MOJAVE/blazarlist.html.
Table 4: Individual source studies ($\gamma$ & VLBI)
ID | Alt. ID | ATel | |
---|---|---|---|---
(B1950.0) | | No. | Programa | Ref
0219$+$428 | 3C 66A | 1753 | | [59]
0235$+$164 | | 1744, 1784 | M12+B | [60]
0313$+$411 | IC 310 | 2510 | | [61]
0316$+$413 | 3C 84 | 2737 | M12+B | [62]
0402$-$362 | | 2413, 2484, 3554, 3655, 3658 | T |
0454$-$234 | | 1898, 3703 | |
0528$+$134 | | 3412 | M12+B | [63]
0537$-$441 | | 2124, 2454, 2591 | T | [64]
0716$+$714 | | 1500, 3487, 3700 | M12+B |
0727$-$115 | | 1919, 2860 | M12 |
0805$-$077 | | 2048, 2136 | M12 |
0806$+$524 | | 1415, 3192 | M2 |
0836$+$710 | 4C +71.07 | 3233, 3831 | M12+B |
0851$+$202 | OJ 287 | 2256, 3680 | M12+B | [65]
0946$+$006 | PMN J0948$+$0022 | 2733, 3429, 3448 | M2 | [66, 67, 68]
1101$+$384 | Mrk 421 | … | M2+B | [69]
1222$+$216 | 4C +21.35 | 2021, 2348, 2349, 2584, 2641, 2684, 2687 | M12+B | [70, 86]
1226$+$023 | 3C 273 | 1707, 2009, 2168, 2200, 2376 | M12 | [86, 88]
1228$+$126 | M 87 | 2437b | M12 | [71, 72, 73]
1236$+$049 | | 1888, 3429 | M2 |
1253$-$055 | 3C 279 | 1864, 2154, 2886 | M12+B | [86]
1322$-$428 | Centaurus A | … | T | [38, 74]
1329$-$049 | OP $-$050 | 2728, 2829 | M2 |
1343$+$451 | | 2217, 3793 | M2 |
1424$-$418 | | 2104, 2583, 3329 | T |
1502$+$106 | OR 103 | 1905 | M12 | [75]
1510$-$089 | | 1743, 1897, 1968, 1976, 2033, 2385, 3470, 3473, 3694 | M12+B | [76, 77, 78]
1551$+$130 | OR +186 | … | M2 | [79, 80]
1622$-$253 | | 2231, 2531, 3424 | M2 |
1633$+$382 | 4C +38.41 | 2136, 2546, 3333 | M12+B | [81, 86]
1641$+$399 | 3C 345 | 2316 | M12+B | [82, 83]
1652$+$389 | Mrk 501 | … | M2 | [84]
1803$+$784 | | 2386, 3322 | M12 |
2200$+$420 | BL Lac | 2402, 3368, 3387, 3459, 3462 | M12+B | [87]
2251$+$158 | 3C 454.3 | 1628, 1634, 2009, 2200, 2322, 2326, 2328, 2534, 2995, 3034, 3041, 3043 | M12+B | [85, 86]
2345$-$167 | | 2408, 2972 | M12 |
Note: Table updated as of February 1st, 2012
Note: ATels on the gravitational lens PKS 1830$-$211 are not listed
a Key: M1/2 MOJAVE 1/2; T: TANAMI; B: Boston U.
b The ATel 2437 reports on post-VHE-flare eEVN observations.
Here we describe in detail a selection sources sorted by right ascension in
B1950.0 coordinates, where $\gamma$-VLBI data have been reported.
AO 0235+164 This source has been studied in the framework of a multi-band
campaign, and multi-band light curve correlations at different bands is
presented together with VLBI analysis from the BU program [60]. These results
show hints of a new feature in the jet associated to the $\gamma$-outburst
observed by Fermi, which is interpreted as the propagation of an extended
moving perturbation through a re-collimation structure at the end of the
region where the jet is collimated and accelerated.
IC 310 The galaxy 0313$+$411 in the Perseus Cluster has been recently detected
in $\gamma$-rays with an extremely hard $\gamma$-spectrum. Sub-parsec-scale
VLBA images at 8.4 GHz detect a one-sided core-jet structure with blazar-like
radio emission oriented at the same position angle than the kiloparsec radio
structure [61]. Those findings suggest this object to have of blazar nature
rather than being a head-tail radio galaxy as it was classified in the past.
3C 84 The Fermi-detection of 0316$+$413 in the Perseus Cluster was reported in
[41], including MOJAVE data, where a brightening of the central sub-parsec-
scale region is reported, especially by comparing images from August 2008 and
September 2007. Results from 14 epochs in the 2010s carried out with the
Japanese VLBI Network show an outburst associated with the central parsec near
the core. A jet component with $\beta_{\mathrm{app}}\sim 0.23c$ is getting
brighter during the $\gamma$-ray flare, which suggests a connection between
both events [62].
PKS 0528+134 Multi-band results during the 2nd half of 2009 show a quiescent
high-energy behavior, and the BU program images presented a stable state in
its parsec-scale radio jet at 43 GHz [63].
PKS 0537$-$441 First images from TANAMI are discussed by [64]. Those results
include monitoring and spectral information between 8.4 GHz and 22 GHz as
compared to Fermi/LAT light curves in $\gamma$-rays.
OJ 287 Multi-epoch, multi-waveband flux and linear polarization observations
of 0851$+$202 have been presented by [65] in the framework of the BU blazar
monitoring program. Those observational results suggest that the $\gamma$-ray
emission is caused by a prominent feature in the jet $>14$ pc away from the
central engine. The parsec-scale structure of this source has apparently
changed its direction since 2005, and the kinematic analysis of the VLBI
structure shows that two $\gamma$-flares happen while a feature passes through
a quasi-stationary shock in the jet. Detailed results of intensive monitoring
from 1995 to 2011 at 43 GHz are presented in [42], showing erratic wobbling in
its sub-parsec jet structure.
PMN J0948+0022 The narrow-line Seyfert 1 source 0946$+$006 has been studied
with a multi-band campaign tied to the $\gamma$-flare in spring 2009, and
MOJAVE and e-EVN data show a relationship between a compact parsec-scale image
with a bright core and the $\gamma$-ray observations [66]. MOJAVE results
showing the pc-scale structure and a swing in $\chi$ are presented in the
framework of the multi-band study presented in [67], but no robust kinematics
was possible with four epochs at the time of publication. The results from the
eEVN yield a value of $T_{\mathrm{b}}=3.4\times 10^{11}$ K, confirming that
this object is similar to FSRQ [68].
Mrk 421 A multi-band campaign on the TeV source 1101$+$384 [69] including VLBI
data shows a partially resolved pc-scale radio core; the radio source showed a
low activity at all wavebands during the campaign.
Centaurus A Detailed results from a multi-band campaign on 1322$-$428 is
presented in [38], including TANAMI observations where the VLBI core size is
used to calculate an upper limit on the size of the $\gamma$-ray emitting
central region ($<0.017$ pc), whereas the slow $\beta_{\mathrm{app}}$ measured
do not impose constrains in the value of $\Gamma$. More detailed VLBI imaging
is presented in [74] where some regions near the core have an inverted radio
spectrum, which are suggested as possible production sites for the high energy
photons, since they have high $T_{\mathrm{b}}$ and compact structure. Note
that this is the only source known so far with detectable $\gamma$-emission
beyond parsec-scales.
4C +21.35 The BU program study on 1222$+$236 combined with multi-waveband
observations has produced interesting results on the correlation of light
curves[70]. VLBI morphological results are interpreted in [86] as showing a
superluminal feature crossing at $\beta_{\mathrm{app}}\sim 14c$ a stationary
jet fieature simultaneously with a $\gamma$-ray high state [89].
M 87 The nearby galaxy 1228$+$126 was detected by Fermi/LAT and multi-band
results are presented in [72] including MOJAVE data. The source has been
intensively monitored in the mid 2000s to track the location and nature of the
high-energy and radio emission of the component HST-1 (see [44] and references
therein). The relationship between the 43 GHz VLBI brightening of the core in
2008 and a TeV emission was presented in [71]. In this context, further
monitoring has been performed including the eEVN [73], and the possibility
that the $\gamma$-ray emission observed by Fermi comes from HST1 is still
unclear.
3C 273 A relationship between a high state in the VLBI core and a $\gamma$-ray
flare has been reported, implying a location distance of 4–11 pc between both
emission regions [88]. Observations of 1226$+$023 by the BU blazar program
report that seven new features are present in the jet, four of them related to
$\gamma$-emission with $\beta_{\mathrm{app}}\sim 7-12c$, and a value of 0.7
knots ejected per year is reported [86].
3C 279 The BU blazar program reports on two knots appearing in the jet of
1253$-$055, and the time of their passage through the parsec-scale mm-wave
radio core coincide with two prominent $\gamma$-ray events in the light curve
[86]. This is so far the most distant source where TeV emission has been
detected.
PKS 1502+106 A multi-band campaign including Fermi/LAT and MOJAVE data has
shown the connection between $\gamma$ activity and a rotation in the EVPA at
this source [75].
PKS 1510$-$089 The source had a flare detected by AGILE in March 2009 [43],
also reported by Fermi/LAT [76]. The radio structure, spectra, and
polarisation of this source are presented in [76], including results from
MOJAVE and from [57, 58]. The BU blazar monitoring results are interpreted as
being caused by a bright knot of emission passing to the stationary VLBI
‘core’, which produces a long radio and X-ray outburst lasting months after
the flare [77]. Newer $\gamma$-flares are interpreted as well in this picture.
An analysis of earlier MOJAVE archival data reveals that emission at $\gamma$
and radio energies has origin in the same region [78].
4C +38.41 The source 1633$+$382 has been studied by [81, 70]. The BU
observations report that a high $\gamma$-ray state in September 2009 is
simultaneous with a high state in the VLBI core. The changes polarisation
vectors from VLBI and the optical support the idea of the $\gamma$-rays are
connected with processes near the mm-VLBI core.
3C 345 The source 1641$+$399 was identified as the source of $\gamma$-ray
emission reported from the 3C 345–NRAO 512 region [82]. The combination of
20-month data from Fermi/LAT and 43 GHz VLBA monitoring observations of 3C 345
shows that the quiescent and flaring components of $\gamma$-ray emission are
produced in a region of the jet of up to $\sim$23 pc (deprojected), favouring
the synchrotron self-Compton mechanism for $\gamma$-ray production [83].
Mrk 501 The source 1652$+$389 has emitted a mildly variable $\gamma$-flux over
the period 2008–2011. A multi-band campaign of this source shows multi-
frequency VLBA data from the projects BK150, BP143, and MOJAVE; its VLBI core
is used for the SED analysis of the source [84].
BL Lac The source 2200$+$420 has registered several major $\gamma$-ray flares
detected by Fermi and is observed intensively in VLBI since the 1980s. Results
on a multi-band campaign are presented in [87], including VLBI images from the
Fermi-related multi-wavelength campaign reported in [57, 58]. Multi-band and
spectral index VLBI images reveal a curved jet with a core region with
inhomogeneous structure and changing spectral properties rather than being a
single, uniform, self-absorbed feature. A turnover by synchrotron self-
absorption in the core takes place at $\sim 12$ GHz; and assuming $\delta=7.3$
from [19], a limit is set to the core magnetic field of $B<3$ G.
3C 454.3 The Fermi/LAT detection of 2251$+$158 is reported in [39]. The source
has registered several major $\gamma$-flares (see e.g., the AGILE observations
in [40]), as it seen in the amount of ATels in table 4. The source has been
intensively studied with VLBI. The results from the BU blazar program [85, 86]
report on the coincidence of $\gamma$-ray peaks with jet features crossing the
jet location at superluminal speeds. This phenomenon has occurred three times,
coinciding with the three major flares in December 2009, April and November
2010.
## V CONCLUSION
An enormous observational VLBI effort is being performed on AGN to address
their nature under the new light of $\gamma$-emission. Extensive studies on
AGN samples and intensive multi-band campaigns on individual sources are
underway. As of writing this review, new publications are in the process of
submission or revision, including newer correlations and observational data.
One of the next challenges is to establish a (statistically robust) connection
between ejection of features in VLBI jets and $\gamma$-flares and hardness
changes at the high spectrum, once enough data have been collected during the
Fermi era. The new windows opened to the Southern sky or to extreme regions in
the parameter space such as mm-VLBI will provide new results. We look forward
to the next $\gamma$-radio meeting and bigger symposia for new, exciting
findings.
###### Acknowledgements.
Thanks are due to K.I. Kellermann, M.L. Lister, T. Savolainen, and F. Schinzel
for careful reading of the manuscript and very useful suggestions. This work
has made use of data from the MOJAVE database that is maintained by the MOJAVE
team [30]. The author acknowledges partial support by the Spanish MICINN
through grant AYA2009-13036-C02-02, and by the COST action MP0905 ‘Black Holes
in a Violent Universe’.
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|
arxiv-papers
| 2012-05-04T19:20:39 |
2024-09-04T02:49:30.567281
|
{
"license": "Public Domain",
"authors": "Eduardo Ros (Dep. Astronomia, Univ. Valencia and Max-Planck-Institut\n fuer Radioastronomie)",
"submitter": "Eduardo Ros",
"url": "https://arxiv.org/abs/1205.1043"
}
|
1205.1073
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Models for Metal Hydride Particle Shape, Packing, and Heat Transfer
Kyle C. Smithkyle.c.smith@gmail.com
Timothy S. Fishercor1tsfisher@purdue.edu
Purdue University
School of Mechanical Engineering and Birck Nanotechnology Center
1205 W. State St.
West Lafafyette, IN 47907
[cor1]Corresponding author. Tel.: +1-765-494-5627; fax: +1-765-494-4731
A multiphysics modeling approach for heat conduction in metal hydride powders is presented, including particle shape distribution, size distribution, granular packing structure, and effective thermal conductivity. A statistical geometric model is presented that replicates features of particle size and shape distributions observed experimentally that result from cyclic hydride decreptitation. The quasi-static dense packing of a sample set of these particles is simulated via energy-based structural optimization methods. These particles jam (i.e., solidify) at a density (solid volume fraction) of $0.665 \pm 0.015$ – higher than prior experimental estimates. Effective thermal conductivity of the jammed system is simulated and found to follow the behavior predicted by granular effective medium theory. Finally, a theory is presented that links the properties of bi-porous cohesive powders to the present systems based on recent experimental observations of jammed packings of fine powder. This theory produces quantitative experimental agreement with metal hydride powders of various compositions.
hydride fragmentation jamming cohesion conduction effective medium theory
[2010] 70C20 80A20 62M40
§ INTRODUCTION
Metal hydrides offer high volumetric hydrogen storage density for on-board fuel cell vehicles <cit.>, and offer much potential for electrodes in electric vehicle batteries <cit.>. These metals undergo fragmentation induced by volumetric hydriding strain and embrittlement that results in the formation of irregular faceted particles <cit.>. Systematic reductions in average particle size have been observed as the number of hydriding cycles increases <cit.>. Also, hydrides can reach a state at which particles become mechanically stabilized, and the size distribution becomes invariant with further cycling <cit.>. This fragmentation process is essential to achieving fast hydriding kinetics, because the process exposes fresh chemically active surfaces <cit.>, but the fragmented nature of metal hydride particles inhibits hydriding heat dissipation <cit.>. Expanded graphite <cit.> and metal <cit.> additives composited with metal hydrides can enhance heat dissipation during the hydriding process, but these chemically inactive materials are parasitic to hydrogen storage density and permeability. In contrast, fragmented hydride packings exhibit high hydrogen storage density and permeability, but the dependence of effective thermal conductivity on the packed structure of hydride powders is not understood well. In the present work, models are developed to address fundamental aspects of these particles, their packings, and their associated thermal properties to enable materials engineering.
Ti$_{1.1}$CrMn particles with crack fissures produced from cyclic hydriding and dehydring. The particles produced after further cycling possess irregular faceted shapes.
Hydrogenation causes metal hydride particles to expand, with approximately 2 to 5 Å$^3$ per H atom <cit.>. Assuming a density of 6.0 g/cm$^3$ for Ti$_{1.1}$CrMn and hydride composition Ti$_{1.1}$CrMnH$_2$ <cit.>, volume expansion is expected to be approximately 9 to 23 %. Hydriding in metal particles requires diffusion of hydrogen through the lattice, with a hydride layer forming on the outer particle surface <cit.>. Thus, density mismatch at the metal-hydride interface induces large stresses on the material, inevitably leading to fracture. Hahne and Kallweit <cit.> documented the dependence of particle size distribution on the number of hydriding cycles. They observed a five-fold reduction in average particles size after 30 cycles. Particle morphology resulting from hydrogen-induced fracture can result in faceted particles having irregular shape. This behavior has been observed for a wide variety of intermetallic metal hydrides for which particle size distributions have been measured (e.g., in Ref. <cit.>), but no attempt has been made to predict the size distribution and shapes of irregular particles theoretically.
Particle shape resulting from hydrogen-induced fragmentation can result in faceted particles having irregular shapes (Fig. <ref>), in contrast to faceted particles produced from growth of single crystals. The particular metal hydride used as an example here, Ti$_{1.1}$CrMn, is prepared by water-cooled arc melting <cit.> that typically yields polycrystalline microstructures; disordered, polycrystalline alloy microstructures also have been shown to yield excellent hydriding characteristics <cit.>. Therefore, crystal defects and grain boundaries at which cracks initiate are expected to have random spatial and directional distribution. Materials such as CeH$_{2.84}$ <cit.> exhibit multiscale fractured structures as a result of phonon confinement energetics, but this phenomenon has not been observed for Ti$_{1.1}$CrMn.
Strain-induced fragmentation is not restricted to systems undergoing hydriding and has been suggested as a universal phenomenon <cit.>. For example, thin layers of dried mud crack readily as a result of contraction and substrate friction <cit.>. The patterns formed by cracked mud exhibit remarkable resemblance to Si thin film anodes subjected to lithiation <cit.>. More recently, investigations of lithiated Si nanopillars reveal that fracture planes have a highly anistropic directional distribution <cit.>.
Interparticle forces and macroscopic stresses result from this particle expansion and induce rearrangement and deformation of the powder. Also, after initial cycling, the decrepitated powder is typically packed into a reactor, but no models for effective thermal conductivity have incorporated packing of these materials. The influence of processing conditions on packed structure are important, because structures with enhanced heat conduction rates and active material density are highly desirable for compact, high-power fuel cell power systems. By considering that packed hydride powder is a granular material, understanding of the phases formed by granular media may be applied to hydrides. For instance, the jamming point represents the state at which a granular material develops rigidity and mechanical stability <cit.>, and the jamming threshold density $\phi_J$ [phJ]$\phi_J$jamming threshold density, - is the density $\phi$ [ph]$\phi$density, - (i.e., solid volume fraction) at which this transition occurs. Recently, we have introduced an energy-based formalism for repulsive contact that enables the rigorous simulation of jamming of arbitrarily-shaped faceted particles <cit.>. The structure of systems of tetrahedra prepared by such methods have shown excellent agreement with experiment (cf., <cit.>).
Particle shape and size distributions resulting from fracture cannot be replicated by a simple single-particle unit cell model, such as those of Asakuma and co-workers <cit.> and Zehner, Bauer, and Schlünder (see <cit.> for details). Reduced conduction through the gas phase as a result of boundary scattering and interfacial impedance mismatches of thermal energy carriers depends on the confining pore geometry of the packed powder. The gas-phase boundary scattering mechanism has often been interpreted as the Smoluchkowski effect in the packed bed literature <cit.>. Also, metal hydride particles inherently exhibit small contact areas in the packed state, with resultant constriction effects on solid-state thermal energy carriers (i.e., electrons and phonons). The degree of ballistic and diffusive conduction in the solid phase depends on the change in chemical composition of the metal hydride system due to hydrogenation and dehydrogenation. The extent to which each of these mechanisms limits heat transfer in metal hydride powders, and other types of porous media in general, is not well understood.
In this work a comprehensive approach is introduced for modeling particle shape and size distribution in Sec. <ref>, quasi-static packing through the energy-based simulation of granular jamming in Sec. <ref>, and heat transfer through direct solution of the heat diffusion equation in the resulting heterogeneous microstructures in Sec. <ref>. In Sec. <ref> a structure-transport theory inspired by recent experiments on fine powder is subsequently developed to relate non-cohesive jammed system properties to experimental packings with solid density below that of the non-cohesive jammed system.
§ PARTICLE SHAPE
§.§ Theory
We attempt to replicate the particle size and shape distribution of decrepitated metal hydride powder with an idealized statistical geometric model. The major underlying assumptions of the model follow: (1) infinitely extending planar surfaces are formed from instances of fracture and (2) planes of fracture have isotropic statistical orientation and position throughout the material. This model can be described as a 3D Poisson plane field, though no such statistical field has yet been proposed in the literature to our knowledge. The 3D Poisson plane field is closely related to the 2D Poisson line field that has been studied previously <cit.>. Both assumptions are well justified for polycrystalline metals, such as Ti$_{1.1}$CrMn. Simulation of this geometric field of planes is accomplished through sequential sectioning of a unit cube centered at the origin with randomly oriented and positioned planes. Random directions are determined through the generation of three random numbers $x_i$ [xi]$x_i$uniformly distributed random variables with $i=1,2,3$, - that are uniformly distributed on the interval [0,1] ($x_1$ for planar position; $x_2$ and $x_3$ for unit normal direction).
The random position $\boldsymbol{r}_{p}$ [rp]$\boldsymbol{r}_{p}$random position vector, m of a given plane can be expressed in terms of the parameter $x_1$, the plane unit normal vector $\hat{n}_p$ [np]$\hat{n}_p$plane unit normal vector, -, and the centroid $\boldsymbol{r}_{c}$ [rc]$\boldsymbol{r}_{c}$cube centroid position vector, m of the cubic-shaped domain undergoing subdivision:
\begin{equation}
\boldsymbol{r}_{p} = x_1 \hat{n}_p \sqrt{3}/2 + \boldsymbol{r}_{c}.
\end{equation}
The normal vector is expressed in terms of continuously distributed parameters, $x_1$, $x_2$, and $x_3$, assuming isotropic planar orientation, but the parameters could easily incorporate anisotropy by restricting them to a discrete set that reflects rotational symmetry of the underlying atomic structure (e.g., cubic or icosahedral). The present model may be applicable to the pulverization of Si anodes by incorporating anisotropic distributions consistent with those presented in Ref. <cit.>. The probability $p(d\omega)$ [p]$p$probability, - for fracture along planes having normal vectors subtending a differential solid angle $d\omega$ [om]$\omega$solid angle, steradians about the direction $\hat{n}_p$ can be expressed in terms of azimuthal and zenith angles in the spherical coordinate system, $\theta$ [th]$\theta$azimuthal angle, radians and $\psi$ [ps]$\psi$zenith angle, radians, as well as the probability density $c$ [c]$c$probability density, 1/steradians:
\begin{equation}
p(d\omega) = c d\omega = c d\theta d\psi \sin{\psi}.
\end{equation}
In the present study isotropic planar orientation is considered, and therefore $c$ is considered a constant independent of $\hat{n}_p$. Because uniformly distributed independent random variables ($x_1$, $x_2$, and $x_3$) are used to describe direction, the differential solid angle $d\omega$ must be recast with $\delta = \cos{\psi}$ [de]$\delta$zenith angle transformation variable, -; this substitution guarantees that $p(d\omega)$ is proportional to $d\theta$ and $d\delta$, each having uniform distributions. By defining $\delta$ and $\theta$ in terms of the parameters $x_2$ and $x_3$, the spherical coordinates specifying planar orientation can be expressed as:
\begin{equation}
(\theta,\psi) = (2 \pi x_2,\cos^{-1}(2x_3-1)).
\end{equation}
With these spherical coordinates, the planar normal vector can be expressed:
\begin{equation}
\hat{n}_p = \cos{\theta}\sin{\psi}\hat{i}+\sin{\theta}\sin{\psi}\hat{j}+\cos{\psi}\hat{k}.
\end{equation}
Particles generated by intersections of planes in the 3D Poisson field. Particles are colored according to their volume $V$ relative to the volume-weighted average particle volume $\bar{V}$ for the ensemble. Particles intersecting the domain boundary (black edges) are excluded to neglect edge effects.
[ijk]$\hat{i}$, $\hat{j}$, $\hat{k}$Cartesian unit vectors, -
This model was implemented numerically, and can be interactively accessed through nanohub.org <cit.>. Additional functionality exists in this tool that is outside the scope of this work; some of the extra tool functionality is described in our original work on this topic <cit.>. The resulting ensemble of particles formed by the intersection of 200 planes in the Poisson field is depicted in Fig. <ref>. The volume of individual particles in the ensemble spans nearly six orders of magnitude. Qualitatively, the field reflects many of the features of decrepitated metal hydride particles (cf., Fig. <ref>), including faceted particle shapes. Comparison of the resulting particle size distributions with experimental data in the subsequent section also confirms excellent quantitative agreement of this simple statistical geometric model.
§.§ Experiment
Details of metal hydrides considered.
Metal hydride History $D_{50}$ ($\mu$m) Reference
MmNi$_{3.5}$Co$_{0.7}$Al$_{0.8}$ 650 cycles $12$ <cit.>
LaNi$_{4.7}$Al$_{0.3}$ 30 cycles $7.2$ <cit.>
HWT 11 cycles $6.3$ <cit.>
Ti$_{1.1}$CrMn cycling/oxidation $3.6$ present
[D]$D$Volume effective sphere diameter, m
The particle size distribution of a high pressure metal hydride, Ti$_{1.1}$CrMn, was measured with laser diffraction by Malvern Instruments <cit.>. Prior to the measurements the material had undergone a process of cycled hydriding and dehydriding to chemically activate the material, as described in <cit.>. The pyrophoric tendency of Ti$_{1.1}$CrMn required controlled oxidation to passivate the exterior surfaces of metal hydride particles for exposure to air during particle size distribution measurements.
Particles were dispersed to obtain accurate measurements of particle size distributions by laser diffraction. Both wet and dry methods were utilized and are outlined in ISO13320-1 <cit.>. In the wet method particles were dispersed in water with and without ultrasonic excitation for subsequent analysis; this method is primarily intended to assess particle size in the absence of agglomeration <cit.>. For sonicated samples, the particle size distribution was measured as a function of sonicating time to determine when agglomerates were completely fragmented. Three minutes of ultra-sonication were required in order to achieve adequate dispersion of the passivated particles. In the dry method particles were accelerated in air by a pressure differential induced across a venturi in which shear stresses act to reduce agglomeration, but this process may have induced attrition of particles <cit.>. The resulting particle size distributions are displayed in Fig. <ref>.
Particle size distributions of passivated Ti$_{1.1}$CrMn by laser diffraction <cit.>. Air titration (dry) and aqueous (wet) methods were used to disperse particles.
The results indicate that the wet method yields larger particle sizes when sonication is not used; the median diameter was 3.8 and 5.7 $\mu$m with and without sonication, respectively. Despite the reduction in measured particle size upon sonication, agglomerates still appear to be present, as reflected by a small secondary peak near 20 $\mu$m. In contrast, the dry method results in much smaller median particle diameters of 1.4 and 2.0 $\mu$m for 4 and 1 bar applied pressure, respectively. Also, sparks were observed during acceleration of particles at the highest pressure <cit.>, indicating that particles had undergone attrition, resulting in fresh pyrophoric surfaces. These results support the notion that metal hydride particles possess an abundance of internal cracks even after the cyclic fragmentation process stabilizes. Such microstructure within particles must be accounted for in the modeling of hydrogen diffusion kinetics and heat flow in metal hydride powder.
The cumulative size distribution of aqueous sonicated Ti$_{1.1}$CrMn is displayed in Fig. <ref> in addition to those of several other chemically dissimilar metal hydrides from the literature. For the aqueous sonicated sample of Ti$_{1.1}$CrMn the weak secondary peak of the size distribution resulting from agglomeration was excluded. Despite composition contrast, each metal hydride exhibits similar particle size distribution. The most substantial deviation from similarity is for particles at the large end of the size range; this incongruence between measurements and model is likely due to the presence of agglomerates. In addition, the various metal hydride samples had undergone various degrees of cycling, as shown in Table <ref>. The lack of a physically consistent trend of particle size with the degree of cycling among the various hydrides in Table <ref> suggests that the median size of fully pulverized particles is primarily dependent on chemical composition and mechanical properties.
Metal hydride particles generated from a Poisson plane field formed by the intersection of 200 randomly oriented and positioned planes are compared to experimental size distributions of the four different metal hydrides in Fig. <ref>; 200 random planes were sufficient to converge the size distribution shape in Fig. <ref>. The size distribution of the Poisson plane field lies in the middle of the spread of experimental size distributions, indicating excellent agreement with the data. Such a finding is remarkable considering that a simple model lacking adjustable parameters yields a particle size distribution that closely matches experimental observations. This result suggests that the random nature of this process results in particle shapes that are relatively independent of material composition.
Modeled and experimental cumulative size distributions for cycled metal hydride powder from the present work and Refs. <cit.>. Volume effective sphere diameter $D$ is plotted for theoretical size distributions. $D$ is normalized by its median value $D_{50}$. Kernel density estimates of the simulated distributions are shown.
§ PACKING THEORY AND SIMULATION
400 particles from the Poisson plane field are randomly sampled to study athermal (i.e., low particle kinetic energy) jamming of frictionless metal hydride-like particles; the methods employed to find the jammed structure of these particles are briefly described here. The cumulative size distribution for this subset of particles closely matches that of the full Poisson plane field (Fig. <ref>). To simulate jamming, these particles are arranged in a dilute setting with random particle position and orientation in a periodic cubic supercell. Sequentially, isotropic affine compressive strain is applied to the assembly. Structural relaxation of particles is simulated at each density $\phi$ (i.e., solid volume fraction) via the minimization of elastic energy between contacting particles. A contact mechanics model is employed as in <cit.> for packings of Platonic solids and LiFePO$_4$ particles, where the contact energy $E_{\alpha\beta}$ between particles $\alpha$ and $\beta$ is:
[EAB]$E_{\alpha\beta}$energy between particles $\alpha$ and $\beta$, J
\begin{equation}
E_{\alpha\beta} = \frac{YV_{\alpha \cap \beta}^{2}}{2(V_{\alpha}+V_{\beta})},
\end{equation}
and $Y$ is Young's modulus [Y]$Y$Young's modulus, Pa, $V_{\alpha \cap \beta}$ [VAB]$V_{\alpha \cap \beta}$intersection volume between particles $\alpha$ and $\beta$, m$^3$ is the volume of intersection of the two undeformed polyhedral particles in contact, and $V_{\alpha}$ [VA]$V_{\alpha}$volume of particle $\alpha$, m$^3$ is the volume of particle $\alpha$. The present mechanics model permits computation of energy, forces, and moments for contact between two arbitrarily-shaped dissimilar particles; the generality of this model is especially important for the present study because of the random shape of particles in the Poisson plane field. Detailed expressions for forces and moments and the numerical methods employed for structural optimization are described in Ref. <cit.>.
Following the procedure outlined above, an initially dilute configuration of particles at $\phi = 0.005$ was consolidated sequentially until a stable state was reached at $\phi = 0.856$ with energy per particle of $1.5 \times 10^{-4}Y\bar{V}$, where $\bar{V}$ [Vb]$\bar{V}$volume-weighted average particle volume, m$^3$ is the volume-weighted average particle volume. The system was subsequently expanded to approach the jamming density, but the ill-conditioned nature of this system prohibited optimization to stable states near the jamming point. Subsequently, the system was expanded to a density of $\phi=0.643$ and optimized to nearly zero energy, confirming that the jamming threshold density $\phi_J$ exceeds 0.643. Further consolidation with fine strain stepping resulted in stagnant optimization progress near $\phi=0.65$, while estimates of the jamming threshold density based on limited contact depth data yield a jamming density of $0.68$ (see Ref. <cit.> for this approach). Based on these findings, we estimate the jamming density for this system to be $0.665\pm0.015$.
Metal $\phi_M$ and metal hydride $\phi_{MH}$ [phM]$\phi_{M(H)}$metal (hydride) packed density, - packed densities (i.e., solid volume fraction). Hydriding expansion was estimated at 2 to 5 Å$^3$ for each hydrogen atom present in the metal (see Ref. <cit.>). Uncertainty bounds of $\phi_{MH}$ are expressed in terms of the uncertainty in volumetric expansion during hydriding.
Metal hydride Expansion % $\phi_M$ $\phi_{MH}$ Reference
Ti$_{1.1}$CrMn $16 \pm 7$ 0.3 $0.35 \pm 0.02$ <cit.>
LaNi$_{4.7}$Al$_{0.3}$ $23 \pm 10$ 0.469 $0.577 \pm 0.047$ <cit.>
HWT $24 \pm 10$ 0.555 $0.688 \pm 0.056$ <cit.>
As displayed in Table <ref>, the jamming threshold density of this frictionless, non-cohesive system is higher than packed densities of experimentally prepared metal powder. The act of hydriding expands particles and, therefore, increases the density of powder confined by a container of fixed size. Estimates of the resultant hydride powder densities are included in Table <ref>. The results suggest that the HWT hydride was packed near jamming, while the other samples were very loosely packed relative to the simulated jamming point. Variation of these packed densities further suggests that cohesive van der Waals forces impede densification during packing (see <cit.> and references therein for example). Also, wet consolidation or mechanical vibration may be necessary to pack hydride near jamming and accordingly achieve jammed volumetric hydrogen storage densities. We expect that the present jammed systems without friction and cohesion represent the densest random packing of such particles <cit.>.
The jammed microstructure obtained through the aforementioned simulation is depicted in Fig. <ref> and reveals that small particles are well distributed and readily contact other small particles. Conversely, large particles may be isolated from other large particles and at most will only contact two other large particles. Visual inspection reveals that the full distribution of particles is engaged in the contact network in a non-trivial fashion. In addition to the system of 400 particles, a system of 100 particles was also studied. This ensemble was too small to reflect the structure in the large system limit, because unrealistic structural correlation was induced as a result of the similar size of the supercell relative to the largest particle in the system. Though only a single realization of each system size is studied here, convergence of structural and transport properties among multiple realizations and sizes is expected as long as the jammed cell size exceeds two length scales of the maximal particle.
Jammed packing of 400 particles randomly sampled from the Poisson plane field at $\phi=0.633$. Particle color indicates volume $V$ relative to the volume-weighted average particle volume $\bar{V}$. The cube-shaped periodic supercell is indicated by black edges.
§ HEAT TRANSFER
Transfer of heat by conduction occurs in materials via the flow of energy carriers, e.g., phonons, electrons, and molecules in semiconducting, metallic, and gas phases, respectively. For transport in large systems with long time scales, diffusive processes are characterized by frequent isotropic scattering of carriers via intrinsic mechanisms (e.g., through carrier-carrier interactions). Accordingly `large' systems with `long' times are rational only when compared to length and time scales of diffusive processes. In the present study of steady-state heat transfer, only the diffusive length scale, the mean free path between scattering events $\lambda_{mfp}$, [lamfp]$\lambda_{mfp}$mean free path, m must be considered. When the length scale through which heat is transported is smaller than $\lambda_{mfp}$, carriers are constricted only by boundaries, and the transport regime is referred to as ballistic. Multiscale features present in granular materials can result in ballistic and diffusive behavior throughout the heterogeneous medium. The reduction of heat flow through the gas phase occurs as a result of interfacial interactions of thermal energy carriers with the confining pore geometry of the granular medium.
§.§ Theory
The so-called Kapitza resistance and Smoluchkowski effects are known to occur in fluid-solid systems. The Kaptiza resistance refers to a temperature jump resulting from the finite density of fluid molecules transferring heat normal to the interface <cit.>. In contrast, the Smoluchowski effect involves the reduction of effective thermal conductivity of a gas as a result of molecular accommodation on the interface <cit.>; this process can also be understood according to thin film theory as a boundary scattering effect <cit.>. It is not clear, as yet, whether a simple Kaptiza resistance is sufficient to replicate the boundary scattering of gas molecules in porous media, i.e., the Smoluchowski effect. We therefore consider both effects to understand the connections and limitations of these phenomena in realistic porous media. Kaptiza resistance results from interactions between solid-gas interfaces oriented normal to the net heat flux $\vec{q}''$, [q]$\vec{q}''$heat flux vector, W/m$^{2}$ whereas interfaces aligned to the transport direction scatter molecules, as a result of the Smoluchowski effect.
To assess whether the Kaptiza resistance effect or the Smoluchowski effect dominates in the transport process, we introduce a Biot number $Bi$ [Bi]$Bi$Biot number, -, similar to <cit.>, as the ratio of bulk thermal resistance in the gas to interfacial resistance at the solid-gas interface:
\begin{equation}
Bi = \frac{\delta_p}{R''\kappa_g},
\label{eq:Bi}
\end{equation}
where $R''$ [RZ]$R''$Kaptiza resistance, m$^2$-K/W is the unit area Kaptiza resistance, $\kappa_g$ is the reduced gas thermal conductivity resulting from the Smoluchowski effect, and $\delta_p$ [dep]$\delta_p$characteristic pore size, m is the characteristic size of the pore. We express this quantity as a function of Knudsen number $Kn$ [KZn]$Kn$Knudsen number, - defined as $\lambda_{mfp}/\delta_p$ and consider diffuse interfacial models for both effects. For the Kaptiza resistance resulting from diffuse scattering of H$_2$ molecules at solid interfaces, we employ the approximation of the diffuse mismatch model <cit.>, where $R''$ is:
\begin{equation}
R''= \frac{4}{Cv}.
\label{eq:Gpp}
\end{equation}
$C$ [CZ]$C$volume-based specific heat, J/m$^3$ and $v$ [v]$v$mean molecular velocity, m/s are the specific heat (volume-based) and mean velocity of molecules, respectively. For the reduced gas thermal conductivity $\kappa_g$ [kag]$\kappa_g$gas conductivity, W/m-K we consider an analogous form of the result presented by Sondheimer for electron transport in thin films with diffuse interfaces <cit.>:
\begin{equation}
\kappa_g= \frac{\kappa_0}{(1+3Kn/8)},
\label{eq:kapg}
\end{equation}
where $Kn$ is Knudsen number and $\kappa_0$ [ka0]$\kappa_0$bulk gas conductivity, W/m-K is the bulk gas thermal conductivity. Eq. <ref> is valid for $Kn<<1$. Here the bulk gas conductivity $\kappa_0$ is expressed via kinetic theory as $\kappa_0=Cv\lambda_{mfp}/3$ <cit.>. Substitution of Eqs. <ref> and <ref> into Eq. <ref> and subsequent simplification yields:
\begin{equation}
Bi = \frac{3(1+3Kn/8)}{4Kn}.
\end{equation}
At low pressure (i.e., $Kn \sim 1$) interfacial and volumetric resistances are similar because $Bi$ is near unity, while in the opposite limit of high pressure ($Kn \rightarrow 0$) $Bi$ diverges to infinity. Thus, thermal resistance in the gas phase is large relative to that of the interface over much of the practical pressure range. The Smoluchowski effect is therefore primarily required to represent heat transfer in packed metal hydride beds; only at very low pressures are Kaptiza resistances comparable.
The above analysis is useful for understanding the relative importance of each boundary interaction mechanism. A directionally resolved solution of molecular convection and scattering within the gaseous region of the heterogeneous metal hydride powder is required to rigorously simulate the Smoluchowski effect in that medium. Methods for such analyses (e.g., Boltzmann transport equation or Monte Carlo) have yet to be performed in the metal hydride literature, mainly because of the high computational expense and unknown microscopic pore structure. In the present work, Fourier heat conduction in the gas and solid regions of the heterogeneous powder is simulated. Later, a reduced gas conductivity model is incorporated to replicate the Smoluchowski effect in a manner similar to prior work (see <cit.>).
§.§ Simulation
§.§.§ Methods
We consider a heterogeneous granular medium composed of gas and solid with thermal conductivities $\kappa_g$ and $\kappa_s$, [kas]$\kappa_s$solid conductivity, W/m-K respectively. The steady heat diffusion equation governs the microscopic temperature field $T$ [T]$T$temperature field, K in the medium:
\begin{equation}
\nabla \cdot (\kappa \nabla T) = 0,
\label{eq:heateq}
\end{equation}
where $\kappa$ [ka]$\kappa$local conductivity, W/m-K is local conductivity. The effective conductivity tensor $\boldsymbol{\kappa}$ [Kb]$\boldsymbol{\kappa}$effective conductivity tensor, W/m-K for the macroscopic medium relates the homogenized temperature gradient $(\nabla T)_h$ to the average heat flux vector $\langle \vec{q}'' \rangle$:
\begin{equation}
\langle \vec{q}'' \rangle = -\boldsymbol{\kappa}(\nabla T)_h.
\label{eq:kappatensor}
\end{equation}
Periodic temperature fall boundary conditions are applied to the granular medium to probe $\boldsymbol{\kappa}$:
\begin{equation}
T(\boldsymbol{r})=T(\boldsymbol{r}+\boldsymbol{r}_0)+ (\nabla T)_h \cdot \boldsymbol{r}_0.
\end{equation}
$\boldsymbol{r}$ [r]$\boldsymbol{r}$position in the heterogeneous medium, m is position in the heterogeneous medium, $\boldsymbol{r}_0$ [r0]$\boldsymbol{r}_0$integer multiple of lattice vectors, m is a linear combination of integer multiples of principal lattice vectors defining the periodic supercell, and $(\nabla T)_h$ is the homogenized temperature gradient. Three independent boundary values problems are solved with unit temperature drops in each of the principal Cartesian directions (i.e., $(\nabla T)_h=\hat{i}$, $\hat{j}$, and $\hat{k}$) to determine the components of $\boldsymbol{\kappa}$ based on its definition in Eq. <ref>. Mean effective conductivity $\bar{\kappa}$ [kab]$\bar{\kappa}$mean effective conductivity, W/m-K is computed as the mean of the principal conductivities of $\boldsymbol{\kappa}$.
To solve Eq. <ref> numerically on the heterogeneous domain, an approximate, robust meshing scheme is employed, wherein reconstructed interfaces are formed by that of a non-conformal multilevel Cartesian mesh. To utilize computer memory efficiently while achieving adequate solid-gas interface resolution, octree-based refinement is employed on Cartesian cells intersecting solid-gas interfaces. An example mesh employed to discretize the system of 400 three-dimensional irregular, polydisperse jammed particles is depicted in Fig. <ref>.
The two dimensional projection of a robust octree-based mesh for the simulation of transport within the granular microstructure. This system is composed of 400 irregular, polydisperse jammed particles, each of which exhibit a distinct color. The mesh shown exhibits three levels of refinement near particle boundaries that is necessary to resolve fine gaps between particles.
Refinement of the octree mesh was performed to verify adequacy of interfacial resolution. At densities approaching jamming ($\phi \rightarrow \phi_{J}^{-}$), a high degree of refinement is required to resolve gas-phase gaps between solid particles approaching face-face contact. For meshes having fine levels larger than these gaps, artificial continuity of solid phase can occur; to prevent this effect, cells exhibiting such artificial continuity are treated as gas phase. In the limit of infinite refinement, the number of such cells vanishes, and consequently the present scheme is topologically consistent with jammed structures.
In practice a coarse level mesh was used with cell size $\Delta x = 0.004L$ [Dx]$\Delta x$finite volume cell size, m for [L]$L$side length of primary supercell, m a system of 400 particles at $\phi=0.633$, where $L$ is the side length of the periodic supercell. Only 1% deviation in $\bar{\kappa}$ was observed among meshes having two and three levels of refinement with $\kappa_s/\kappa_g = 10^3$, and therefore a two-level mesh was used to obtain the present results. The two-level mesh contained $3.5\times10^7$ fine and $1.2\times10^7$ coarse cells. The aggregation-based algebraic multigrid method <cit.> was employed to iteratively solve the corresponding discrete set of equations with the implementation of Notay and co-workers.
§.§.§ Results
The mean effective conductivity $\bar{\kappa}$ of the jammed system at a density just below the jamming point ($\phi=0.633$) was computed for a range of solid-gas conductivity ratios $\kappa_s/\kappa_g$, as displayed in Fig. <ref>. $\bar{\kappa}/\kappa_g$ approaches an asymptote for high solid conductivities. This behavior implies that the solid phase does not form a high-conductivity path through which heat preferentially flows. The structure of the granular medium induced by the jamming of these highly irregular, faceted particles is responsible for this behavior. Heat is forced to flow through the gas phase in order to reach highly conductive solid islands. Such poor solid continuity is consistent with observations in the literature of very low effective conductivity of evacuated metal hydride powder (see <cit.>). In the evacuated state, no gas is present in the pores of the powder to transfer heat, and therefore only radiative transfer through vacuum between solid particles and conduction through the solid phase contact network are operable.
Other unit-cell micromechanical models not informed by the shape and packing of metal hydride particles, such as that of Zehner, Bauer, and Schlünder (see <cit.>), yield spurious divergence of $\bar{\kappa}/\kappa_g$ with $\kappa_s/\kappa_g$, as displayed in Fig. <ref>. In addition, the ZBS model significantly overestimates conductivity even at moderate $\kappa_s/\kappa_g$. In contrast, the granular effective medium approximation (GEMA) of Cohen and co-workers <cit.> exhibits consistency with this aspect of transport in metal hydride powder. GEMA involves the sequential embedding of inclusions within an effective medium formed by a lower density effective medium of inclusions and host material. A consequence of this sequential application of effective medium theory is the disjointed inclusion topology <cit.>, a feature that is consistent with the topology of the present jammed systems of faceted particles. Thus, GEMA provides an excellent estimate of metal hydride conductivity over much of the range of $\kappa_s/\kappa_g$.
Mean effective conductivity as a function of solid-fluid conductivity ratio $\kappa_s/\kappa_g$ for the jammed system at $\phi = 0.633$. Error bars represent bounds of the principal values of the effective conductivity tensor $\boldsymbol{\kappa}$. Theoretical data presented are based on the granular effective medium approximation (GEMA) in <cit.> and the Zehner-Bauer-Schlünder (ZBS) model (see <cit.>).
Shown in Fig. <ref> is the temperature field for the jammed system at a density just below the jamming point. Solid phase islands are readily apparent in the temperature profile, as they are nearly isothermal. Because of the lack of connectivity of these islands, $\bar{\kappa}/\kappa_g$ varies little in the high solid conductivity range ($\kappa_s/\kappa_g > 100$). Transition metals, such as Ni and Ti, are often employed in alloy form as hydrides for H$_2$ storage. The bulk conductivities among these pure metals span nearly an order of magnitude from 21.9 to 90.7 W/m-K at room temperature <cit.>. The conductivity of pressed solid pellets of H$_2$ storage alloys (e.g., LaNi$_{4.7}$Al$_{0.3}$ and HWT) are of the order of 10 W/m-K (see <cit.>). Bulk H$_2$ gas at 300 K exhibits a conductivity of 0.183 W/m-K <cit.>, and consequently $\kappa_s/\kappa_g$ for transition metal hydride powders is expected to be greater than $\sim$100. With a conductivity contrast ratio this high, enhancement of the intrinsic conductivity of metal hydride particles will yield only marginal enhancement of effective thermal conductivity because of the non-percolating solid-phase granular topology in the powder.
Temperature field of the jammed system with $\kappa_s/\kappa_g=10^3$ at $\phi=0.633$. The field is induced by a unitary periodic temperature fall applied to the primary simulation supercell along the indicated direction.
With this information the dependence of $\bar{\kappa}$ on $Kn$ can now be established by expressing $\kappa_g$ as a function of $Kn$. To do so, we adopt the semi-empirical relation of Kaganer (see <cit.>) employed in the recent metal hydride literature <cit.> that is similar to Eq. <ref>:
\begin{equation}
\kappa_g= \frac{\kappa_0}{1+2bKn}.
\label{eq:kapgKag}
\end{equation}
The gas-specific parameter $b$ [b]$b$gas-specific parameter, - depends on the accommodation coefficient and other parameters <cit.>; here we consider $b$ to have a value of 9.87, as in <cit.>. The bulk conductivity $\kappa_0$ of H$_2$ at 290 K is 0.178 W/m-K, as in <cit.>. As displayed in Fig. <ref>, moderately to highly conducting solids ($\kappa_s \gtrsim 20$W/m-K) yield very similar effective thermal conductivities. The Knudsen number $Kn$ scales inversely with gas pressure, and the present results therefore reflect the high-pressure limit of effective thermal conductivity in which the Smoluchowski effect is diminished.
Mean effective conductivity $\bar{\kappa}$ for the jammed system at $\phi=0.633$ as a function of Knudsen number $Kn$ for various solid conductivities $\kappa_s$ for H$_2$ gas at 290 K.
§ GENERALIZED STRUCTURE-TRANSPORT THEORY
As presented in Sec. <ref>, the simulated jammed density of particles from the Poisson plane field ($0.665 \pm 0.015$) exceeds that of experimental packings of metal hydride particles (cf., Table <ref>). The primary mechanisms expected to decrease the packing density from the present jamming density are attractive surface interactions between micron-sized metal hydride particles. Such interactions are known to decrease packing density dramatically <cit.>, but the non-cohesive jamming point remains a reference state after such systems have undergone compaction <cit.>. In cohesive packings, agglomerates form as jammed domains with cohesion <cit.> embedded in void space <cit.>; the agglomerates form a fractal-like structure with high dimensionality of 2.53 <cit.>. Based on this understanding, a low-density packing of metal hydride powder is hypothesized with a granular backbone having local density and structure close to that of the non-cohesive system. Consequently, the cohesive granular medium is modeled as bi-porous; pores reminiscent of the non-cohesive jammed state are formed with larger pores present due to the steric hindrance of cohesive clusters [Fig. <ref>].
Illustration of a hypothetical cohesive microstructure of metal hydride powder. Inspired by Refs. <cit.>, the microstructure is bi-porous with aggregate jammed regions having micro-pores with conductivity $\kappa_J$ [kaJ]$\kappa_J$jammed state conductivity, W/m-K and void regions forming macro-pores with gas phase conductivity $\kappa_g$.
Based on this hypothetical microstructure, the granular effective medium approximation (GEMA) is now employed to model the effective thermal conductivity of cohesive metal hydride powder. GEMA is based upon a sequential application of Maxwell-Garnett EMA to an effective inclusion-matrix medium (see <cit.> for a thorough development). [ka12]$\kappa_1,\kappa_2$phase 1 and 2 conductivities, W/m-K Firstly, EMA is applied to a pure phase $1$ with perfect continuity and a disjointed spherical inclusion of phase $2$. To the effective medium phase $e$ formed by these constituents another spherical inclusion of phase $2$ is introduced, and EMA is applied to yield the updated effective medium phase $e$. This process continues until phase $1$ is fully embedded in phase $2$ and results in a differential equation relating effective conductivity $\kappa_e$ [kae]$\kappa_e$effective medium conductivity, W/m-K to that of the respective phases and density (see <cit.>). The differential equation can be integrated for the process, yielding an implicit expression for the effective conductivity <cit.>:
\begin{equation}
\left(\frac{\kappa_2-\kappa_e}{\kappa_2-\kappa_1}\right)\left(\frac{\kappa_1}{\kappa_e}\right)^{1/3}= 1-\epsilon_1,
\label{eq:GEMA}
\end{equation}
where $\epsilon_1$ [ep1]$\epsilon_1$volume fraction of phase $1$, - is volume fraction of phase $1$.
The unique feature of this sequential embedding process is the preservation of the continuity of phase $1$, and of the disjointed topology of phase $2$ <cit.>. This feature is entirely consistent with our bi-porous hypothetical construction of dense, cohesive metal hydride powder. Locally jammed aggregates form a continuous phase $1$, while macro-pores form a disjointed phase $2$ in the medium. In the present cohesive metal hydride powder model, phase $1$ has conductivity $\kappa_1=\kappa_J$, where $\kappa_J$ is the effective conductivity of the non-cohesive jammed system, and phase $2$ has conductivity $\kappa_2=\kappa_g$, where $\kappa_g$ is the gas phase conductivity. The volume fraction of phase $1$ in the cohesive solid $\epsilon_1$ is expressed in terms of the non-cohesive jammed solid density $\phi_J$ and the cohesive solid density $\phi$ as $\epsilon_1=\phi/\phi_J$. Substituting these expressions into Eq. <ref>, the GEMA approximation for effective conductivity $\kappa_e$ is obtained:
\begin{equation}
\left(\frac{\kappa_g-\kappa_e}{\kappa_J-\kappa_g}\right)\left(\frac{\kappa_g}{\kappa_e}\right)^{1/3}= 1-\phi/\phi_J.
\label{eq:GEMAmh}
\end{equation}
Theoretical prediction of effective thermal conductivity for a biporous cohesive metal hydride powder in H$_2$ gas at high pressure (i.e., $Kn \rightarrow 0$). Triangles represent data for packed metal hydride powder in high pressure H$_2$ gas near 290 K (Ti$_{1.1}$CrMn at 230 bar ($Kn \approx 0.001$) <cit.>; LaNi$_{4.7}$Al$_{0.3}$ at 20 bar ($Kn \approx 0.008$) and HWT at 40 bar ($Kn \approx 0.004$) <cit.>).
Using this effective medium structure-transport theory, the variation of effective thermal conductivity $\kappa_e$ for cohesive metal hydride powder as a function of solid density $\phi$ is predicted in the high gas pressure limit. To maintain computational efficiency, the properties of the jammed state in the present theoretical model are estimated as that of the system at a density immediately below the jamming threshold at $\phi = 0.633$. Therefore, the non-cohesive jammed conductivity $\kappa_J$ is estimated by values presented in Fig. <ref> with $\phi_J=\phi=0.633$.
The validity of the present theory is supported by the experimental effective conductivity at high gas pressure (small $Kn$) of various hydride powders in Fig. <ref>. A solid-phase conductivity of 10 W/m-K yields good agreement with the present model. Realizing that $\kappa_s$ for solid pellets of LaNi$_{4.7}$Al$_{0.3}$ and HWT are approximately 10 W/m-K <cit.>, the results are especially encouraging. From the model results in Fig. <ref>, the high-pressure conductivity depends weakly on solid conductivity in comparison to the strong dependence on density. As stated in Sec. <ref>, little enhancement of metal hydride powder thermal conductivity can be expected by increasing the intrinsic thermal conductivity of solid metal particles. Instead, strategies that seek to modify the packing structure of the particles will achieve greater enhancement efficacy.
Normalized effective conductivity $\kappa_e/\kappa_0$ as a function of Knudsen number $Kn$ of the present theory. The dashed curves represent values for the density of $\phi = 0.3$ for cycled and packed Ti$_{1.1}$CrMn in Ref. <cit.>, while the solid curve represents values at higher density ($\phi = 0.325$) as a result of volumetric expansion upon hydriding (see Table <ref>). Curves for $\phi = 0.3$ are shown for various values of $\kappa_s$ as a reference. Solid squares represent the experimental data of Ref. <cit.> for cycled Ti$_{1.1}$CrMn, where Knudsen number $Kn$ was estimated with a characteristic pore size equal to the median particle diameter determined in the present work, $D_{50}=3.6 \mu$m.
Until this point only the high-pressure limit of gas-phase conductivity in which the Smoluchowski effect is inoperative has been considered. Employing the approximate results in Fig. <ref> that incorporate the Smoluchowski effect for the non-cohesive jammed system, the effects of reduced density with the present theory represented by Eq. <ref> are incorporated. In Fig. <ref> these results are compared to experimental data for the high-pressure metal hydride Ti$_{1.1}$CrMn from Ref. <cit.>. Theoretical results are displayed for two densities to reflect the effect of volumetric expansion of particles during hydriding. At low pressure (i.e., high $Kn$) the experimental data follow the theoretical results with a solid conductivity of 2 W/m-K; at high pressure the experimental data follow the theoretical results for solid conductivity of 10 W/m-K, which is closer to our expectations based on the solid conductivity of other intermetallic alloys, e.g., LaNi$_{4.7}$Al$_{0.3}$ and HWT in Ref. <cit.>.
Therefore, the `solid' phase appears to have a conductivity that depends strongly on H$_2$ gas pressure; physically, this scenario is rather unlikely because deviations with the theoretical results are predominantly within a range of $Kn$ for which hydriding has not initiated for Ti$_{1.1}$CrMn (see dotted line in Fig. <ref>). Composition changes to the solid phase occur only at very high pressure (low $Kn$). Instead, the high pressure air titration experiments suggest that the solid is composed of an abundance of internal cracks (cf., Fig. <ref>). At intermediate pressures, these cracks reduce the `effective solid' conductivity as a result of low gas density and conductivity within the cracks. Though this effect is not included herein, the present theoretical model and experimental observations support this conclusion.
An additional feature reflected by the theoretical model is the spike in conductivity at the initiation of hydriding. The difference in $\kappa_e$ between the theoretical curves for $\kappa_s=10$ W/m-K at the unhydrided density ($\phi=0.300$) and the approximate hydrided density ($\phi=0.325$) agrees well with the amplitude of the experimental spike. The elevated conductivity is not maintained after the immediate phase transition, suggesting that dynamic expansion of the powder's free surface occurred after this spike.
§ CONCLUSIONS
A numerical model for generating metal hydride particles possessing size and shape distributions similar to those observed experimentally has been developed and has been made available for public use on nanohub.org. Key features of decrepitated metal hydride powders are reflected by the geometric statistical model; this statistical geometric framework may be useful for modeling other strain-induced fragmented shapes (e.g., Si anodes and dried mud). Subsequent consolidation of sample particle sets generated from this procedure were simulated via energy-based structural optimization, reflecting a structure denser than that of cohesive metal hydride powders. Effective thermal conductivity of the modeled particle assemblies was simulated, and found to be in excellent agreement with the predictions of granular effective medium theory. In contrast, the commonly employed Zehner-Bauer-Schlünder semi-empirical model deviates strongly from the present results because of the granular nature of metal hydride packings. Finally, a generalized structure-transport theory is developed based on recent experimental results to model the effective conductivity of metal hydride powders with density below the jamming threshold. This theory reflects aspects of metal hydride powders of various compositions and packed densities. These findings suggest that engineering strategies to enhance hydride effective conductivity should seek to increase packing density. This model for particle shape can also be utilized for simulating metal hydride H$^+$ diffusion kinetics, H$_2$ permeability, and composite effective thermal and mechanical properties.
§ ACKNOWLEDGEMENTS
K.C.S. thanks the U.S. National Science Foundation and the Purdue Graduate School for financial support. Both authors thank the U.S. National Science Foundation's Office of International Science and Engineering for travel support that enabled illuminating foundational interactions on granular mechanics with Prof. Meheboob Alam of the J. Nehru Centre for Advanced Scientific Research. The authors thank Tyler Voskuilen of the Purdue Hydrogen Systems Laboratory for assistance in the preparation of oxidized hydride samples. The authors also thank Eric Maw, Claudia Mujat, and Gerald Sando of Malvern Instruments for particle size distribution measurements. Finally, the authors thank Jayathi Murthy and the PRISM center staff, as well as Phil Cheeseman of the Rosen Center for Advanced Computing, for access to and support of computing resources.
|
arxiv-papers
| 2012-05-04T21:14:42 |
2024-09-04T02:49:30.577694
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kyle C. Smith and Timothy S. Fisher",
"submitter": "Kyle Smith",
"url": "https://arxiv.org/abs/1205.1073"
}
|
1205.1094
|
# Quantum phase transition of Bose-Einstein condensates on a ring nonlinear
lattice
Zheng-Wei Zhou${}^{\text{1}}$ zwzhou@ustc.edu.cn Shao-Liang
Zhang${}^{\text{1}}$ Xiang-Fa Zhou${}^{\text{1}}$ Guang-Can Guo${}^{\text{1}}$
Xingxiang Zhou${}^{\text{1}}$ xizhou@ustc.edu.cn Han Pu${}^{\text{2}}$
hpu@rice.edu ${}^{\text{1}}$Key Laboratory of Quantum Information, University
of Science and
Technology of China, Hefei, Anhui 230026, P. R. China
${}^{\text{2}}$Department of Physics and Astronomy, and Rice Quantum
Institute, Rice University, Houston, Texas 77251-1892, USA
###### Abstract
We study the phase transitions in a one dimensional Bose-Einstein condensate
on a ring whose atomic scattering length is modulated periodically along the
ring. By using a modified Bogoliubov method to treat such a nonlinear lattice
in the mean field approximation, we find that the phase transitions are of
different orders when the modulation period is 2 and greater than 2. We
further perform a full quantum mechanical treatment based on the time-evolving
block decimation algorithm which confirms the mean field results and reveals
interesting quantum behavior of the system. Our studies yield important
knowledge of competing mechanisms behind the phase transitions and the quantum
nature of this system .
###### pacs:
03.75.Mn, 67.10.Fj, 05.45.Yv
## I Introduction
In the past few years, ultracold atoms confined in optical lattices have
generated a great amount of excitement in the physics community. They provide
the unique opportunity to realize various many-body models that are of
fundamental importance in physics review . More recently, nonlinear lattices
formed by periodically modulating atomic interaction strengths have also
received much attention. A comprehensive review of nonlinear wave phenomena
supported by nonlinear lattices can be found in Ref. malomed . There are two
basic physical systems that can potentially realize nonlinear lattices, both
of which can be described by nonlinear Schödinger equations. One uses
electromagnetic waves subject to inhomogeneous nonlinear optical media. The
other one is based on atomic Bose-Einstein condensates (BECs) with modulated
$s$-wave scattering length. In this work, we will focus on the latter system
although much of the physics are common to both.
We extend our previous work reported in Ref. Qian to study a BEC on a one-
dimensional (1D) nonlinear ring-shaped lattice. In Ref. Qian , we considered a
BEC on this ring lattice whose atomic scattering length is modulated according
to
$a(\theta)=a_{0}\sin(d\theta),$ (1)
where $\theta$ is the azimuthal angle along the ring and $d=2$ is the spatial
modulation frequency. We have shown that, as the modulation depth $a_{0}$ is
increased, the condensate can undergo a second-order symmtry-breaking quantum
phase transition from a soliton-like state to a spatially periodic condensate
that matches the scattering length modulation. In the present work, we
generalize our investigation to a larger spatial modulation frequency $d\geq
3$ and compare the results to the $d=2$ case. We developed a new mean-field
technique to study the semi-classical behavior of the system, and found that a
similar symmetry-breaking phase transition occurs for $d\geq 3$ as the
modulation depth is increased. However, the phase transition is now of first
order. We also carried out a numerical full quantum mechanical treatment of
the system based on the time-evolving block decimation (TEBD) algorithm. Both
static and dynamical properties of the system are investigated.
## II The Model Hamiltonian
The system considered here is similar to that studied in Ref. Qian ; Kanamoto
. $N$ bosons are confined in a toroid of radius $R$ and cross sectional area
$S$. By sufficiently tightening the radial confinement and freezing the atoms
in that direction, we can treat the atoms as a one dimension system on a ring.
The Hamiltonian of the system can be written in the following dimensionless
form:
$H=\int_{0}^{2\pi}d\theta\left[-\widehat{\psi}^{\dagger}\left(\theta\right)\frac{\partial^{2}}{\partial\theta^{2}}\widehat{\psi}\left(\theta\right)+\frac{U}{2}\widehat{\psi}^{\dagger}\left(\theta\right)\widehat{\psi}^{\dagger}\left(\theta\right)\widehat{\psi}\left(\theta\right)\widehat{\psi}\left(\theta\right)\right],$
(2)
where the first term in the integral represents the kinetic energy and the
second the interaction energy. For simplicity in notations, we measure energy
in units of $\hbar^{2}/(2mR^{2})$. The dimensionless interaction energy
$U\left(\theta\right)=8\pi a\left(\theta\right)R/S$, where
$a\left(\theta\right)$ is the periodically modulated $s$-wave scattering
length. In our work, we consider the situation where the scattering length
between atoms is modulated along the ring with $d$ periods:
$a\left(\theta\right)=a_{0}\sin\left(d\theta\right)$. As in Ref. Kanamoto , we
define the dimensionless interaction strength as:
$\gamma\left(\theta\right)\equiv-\frac{U\left(\theta\right)N}{2\pi}=\gamma_{0}\sin(d\theta),$
(3)
where $\gamma_{0}=-4a_{0}RN/S$ represents the modulation depth of the
interaction parameter $\gamma\left(\theta\right)$.
By taking the Fourier expansion of the field operator
$\widehat{\psi}\left(\theta\right)=\sum_{k}\frac{1}{\sqrt{2\pi}}e^{ik\theta}\widehat{a}_{k}$,
where $k$ takes integer values in order to satisfy the periodic boundary
condition and ${\widehat{a}_{k}}$ is the bosonic annihilation operator for
plane wave mode with wavenumber $k$, the Hamiltonian can be rewritten as:
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k}k^{2}\widehat{a}_{k}^{\dagger}\widehat{a}_{k}+\frac{i\gamma_{0}}{4N}\sum_{klmn}\widehat{a}_{k}^{\dagger}\widehat{a}_{l}^{\dagger}\widehat{a}_{m}\widehat{a}_{n}(\delta_{d+m+n-k-l}$
$\displaystyle-\delta_{-d+m+n-k-l}).$
Since the number of atoms is fixed, we have
$N=\sum_{l}\widehat{a}_{l}^{\dagger}\widehat{a}_{l}$. Because of this, the
kinetic energy term in the Hamiltonian can also be written as
$\frac{1}{N}\sum_{k,l}k^{2}\widehat{a}_{k}^{\dagger}\widehat{a}_{k}\widehat{a}_{l}^{\dagger}\widehat{a}_{l}$.
## III Mean-Field Treatment
In this section, we first consider the mean-field solution valid for $N\gg 1$.
In this case, the kinetic energy term can be approximated as
$\frac{1}{N}\sum_{k,l}k^{2}\widehat{a}_{k}^{\dagger}\widehat{a}_{k}\widehat{a}_{l}^{\dagger}\widehat{a}_{l}\approx\frac{1}{N}\sum_{k,l}k^{2}\widehat{a}_{k}^{\dagger}\widehat{a}_{l}^{\dagger}\widehat{a}_{k}\widehat{a}_{l}\,,$
and the total Hamiltonian can thus be cast into a biquadratic form as:
$H=\frac{1}{N}\sum_{i,j,k,l}{\alpha_{ijkl}}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\widehat{a}_{k}\widehat{a}_{l}\,.$
(4)
We will now describe a modified Bogoliubov method we use to find the
stationary solution and the excitations of the system.
### III.1 Modified Bogoliubov Approach
In the absence of atomic interactions, the ground state is quite trivial: all
the atoms occupy the zero-momentum mode $\widehat{a}_{0}$. In the presence of
interaction, this is no longer true. However, we may conjecture that the
system condenses into a different ground mode $\widehat{\chi}_{0}$. This mode
$\widehat{\chi}_{0}$, together with other orthogonal modes
$\\{\widehat{\chi}_{i}\\}$’s that form a complete set, are related to the
$\\{\widehat{a}_{i}\\}$ modes through a unitary transformation $U$:
$\left(\chi_{0},\chi_{1},\chi_{2},...\right)^{T}=U\,\left(a_{0},a_{1},a_{2},...\right)^{T}\,.$
(5)
In terms of $\\{\widehat{\chi}_{i}\\}$ and
$\\{\widehat{\chi}_{i}^{\dagger}\\}$, the biquadratic Hamiltonian in Eq. (4)
takes the following form:
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{N}\,\widehat{\chi}_{0}^{\dagger}\widehat{\chi}_{0}^{\dagger}\widehat{\chi}_{0}\widehat{\chi}_{0}+\left(\frac{\widehat{\chi}_{0}^{\dagger}\widehat{\chi}_{0}^{\dagger}}{N}\sum_{k,l\neq
0}c_{kl}\widehat{\chi}_{k}\widehat{\chi}_{l}+h.c.\right)+\frac{\widehat{\chi}_{0}^{\dagger}\widehat{\chi}_{0}}{N}\,\sum_{k,l\neq
0}d_{kl}\widehat{\chi}_{k}^{\dagger}\widehat{\chi}_{l}$ (6)
$\displaystyle+\left(\frac{\widehat{\chi}_{0}^{\dagger}}{N}\,\sum_{k,l,m\neq
0}p_{klm}\widehat{\chi}_{k}^{\dagger}\widehat{\chi}_{l}\widehat{\chi}_{m}+h.c.\right)+\frac{1}{N}\,\sum_{k,l,m,n\neq
0}q_{klmn}\widehat{\chi}_{k}^{\dagger}\widehat{\chi}_{l}^{\dagger}\widehat{\chi}_{m}\widehat{\chi}_{n}+\frac{1}{N}\,\sum_{k,l}r_{kl}\widehat{\chi}_{k}^{\dagger}\widehat{\chi}_{l}\,.$
This Hamiltonian can be simplified by a few considerations. First, it is
assumed that most atoms will be in the condensate mode $\widehat{\chi}_{0}$.
Under the mean field approximation, operators for the macroscopically occupied
condensate mode are replaced by $c$-numbers, i.e.,
$\widehat{\chi}_{0},\widehat{\chi}_{0}^{\dagger}\rightarrow\sqrt{N}$. Since
occupation numbers in other $\widehat{\chi}_{k}$ modes are very small, we can
drop terms involving 3 or more operators in $\widehat{\chi}_{k}$ and
$\widehat{\chi}_{k}^{\dagger}$ for $k\neq 0$. After this exercise, we obtain
the following effective Hamiltonian up to second order in
$\\{\widehat{\chi}_{k}\\}$ and $\\{\widehat{\chi}_{k}^{\dagger}\\}$:
$H_{\rm eff}=c_{0}N+\left(\sum_{k,l\neq
0}c_{kl}\widehat{\chi}_{k}\widehat{\chi}_{l}+h.c.\right)+\sum_{k,l\neq
0}d_{kl}\widehat{\chi}_{k}^{\dagger}\widehat{\chi}_{l}\,.$ (7)
This effective Hamiltonian can be diagonalized by the Bogoliubov
transformation and the system’s elementary excitaitons are quasiparticles in
nature. In order to investigate the stability of the system, we need to
analyze the energy spectrum of these quasiparticle excitations. For this
purpose, we should work with the following grand canonical operator to
acccount for the conservation of atom numbers:
$K=H_{\rm eff}-\mu N\,.$ (8)
Here, $\mu$ is the chemical potential and can be calculated from the
condensate energy $E=\left\langle H\right\rangle\approx c_{0}N$ as:
$\mu=\frac{\partial E}{\partial N}=c_{0}+\frac{\partial c_{0}}{\partial
N}N\,.$ (9)
Now, we may diagonalize the operator $K$ by using the Bogoliubov
transformation on $\\{\widehat{\chi}_{k}\\}$ and
$\\{\widehat{\chi}_{k}^{\dagger}\\}$ and obtain the excitation spectrum of
quasiparticles. (This will be elaborated on later.) If there is no imaginary
excitation frequencies, we claim that the condensate mode $\widehat{\chi}_{0}$
is dynamically stable.
In the above prescription, the key step is to search for the appropriate
unitary transformation defined in Eq. (5) that transforms Hamiltonian (4) into
(6). Although there are a great number of unknown parameters in the
undetermined unitary matrix $U$, further analysis shows that only the elements
in the first row of $U$ are necessary for determining the form of the
Hamiltonian (6).
To see this, we note that in order to transform Hamiltonian (4) into (6) via
the unitary matrix $U$, a fundamental requirement is to maintain the
biquadratic terms of the operators
$\left(\widehat{\chi}_{0}^{\dagger},\widehat{\chi}_{0}\right)$ and to
eliminate all the cubic terms. To this end, we may use the operators
$\left(\widehat{\chi}_{i}^{\dagger},\widehat{\chi}_{j}\right)$ to represent
the operators $\left(\widehat{a}_{l}^{\dagger},\widehat{a}_{m}\right)$ in
Hamiltonian (4) via:
$\widehat{a}_{i}=\sum_{j}u_{ji}^{*}\,\widehat{\chi}_{j}$
where $u_{ij}$ is the matrix element of the unitary matrix $U$, and we obtain
the following two equations :
$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\sum_{i,j,k,l}\,{\alpha_{ijkl}}\,u_{0i}u_{0j}u_{0k}^{*}u_{0l}^{*}\,,$
(10) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i,j,k,l}\,(\alpha_{ijkl}+\alpha_{ijlk})\left[u_{0i}u_{0j}u_{0k}^{*}(a_{l}-u_{0l}^{*}\chi_{0})\right]\,.$
(11)
Since $\chi_{0}=\sum_{i}u_{0i}a_{i}$, Eq. (11) can be recast into a set of
equations in operators $\left\\{a_{i}\right\\}$ (the number of this set of
equations depends on the cut-off of the Bose modes) and can be solved by
numerical method. Once the representation of the operator $\chi_{0}$ is
determined, the parameter $c_{0}$ and the chemical potential $\mu$ can be
obtained by solving Eqs. (10) and (9), respectively. Therefore, Eqs. (10) and
(11) together represent an algebraic form of the Gross-Pitaevskii (GP)
equation. In Appendix A, we will show that they are indeed equivalent to the
ordinary GP equation. The main advantage of Eqs. (10) and (11) is that, in
principle, all the stationary states (both dynamically stable and unstable
ones) of the system can be found. When more than one stable solutions are
found, the one that is dynamically stable and with the lowest energy will be
identified as the ground state of the system. In contrast, with ordinary GP
equation, using imaginary time evolution method one can only find the
dynamically stable states, and often just the ground state. Therefore, Eqs.
(10) and (11) are supeior for studying the phase transitions in our system,
where information beyond the ground state is needed.
Once the condensate state is determined, the Bogoliubov spectrum of
quasiparticle excitations above it can be found by diagonalizing $K$ defined
in Eq. (8) using the Bogoliubov transformation. The details are described in
Appendix B.
### III.2 Mean-field quantum phase transition
Using the modified Bogoliubov method outlined in the previous section, in
principle we can find all the stationary states (not just the ground state)
and their excitation spectrum for any modulated atomic interactions. This
provides a more thorough picture of the energy landscape of the system and
deeper insights into possbile quantum phase transitions induced when certain
parameters are varied (in our case, the modulation depth of the interaction
strength $\gamma_{0}$).
Our goal is to study the mean field quantum phase transition for different
modulation period $d$ as the modulation depth $\gamma_{0}$ is varied. We
concentrate on the low-energy states, meaning stationary states with energy
close to that of the ground state. We find that there are mainly two types of
stationary states in the low energy regime. One type has a density profile
matching the modulated period of the scattering length. We refer to such
states as symmetric states. The other type features a density profile that
spontaneously breaks the symmetry of the modulation. We refer to such states
as asymmetric states. The asymmetric states is always $d$-fold degenerate with
the peak density located at $\frac{(1+4i)\pi}{2d}$ ($i=0,1,...d-1$), where the
local interaction energy $U(\theta)$ reaches the minimum. Under the mean field
treatment (MFT), the symmetry-breaking quantum phase transition from the
symmetric type to the asymmetric type have been studied for uniform scattering
length Kanamoto and for $d=2$ periodic scattering length Qian . Here, by
taking advantage of the modified Bogoliubov method, we investigate the
critical point of the quantum phase transitions and the prime mechanism
driving such quantum phase transitions for arbitrary modulation period $d$. We
find that there is a fundamental difference between $d=2$ and $d>2$.
Figure 1: (Color online) Upper panel: Energy of the condensate for $d=2$. The
red solid line is for the dynamically stable symmetric state, the red dashed
line is for the dynamically unstable symmetric state and the blue solid line
is for the stable asymmetric state. Lower panel: Chemical potential of the
ground state. At the critical point $\gamma_{0}=0.528$ (indicated by arrows in
the plots) the ground state changes from the symmetric to the asymmetric type.
Shown in the insets are typical wave functions for symmetric and asymmetric
states.
Figure 1 shows the energies and chemcial potentials of low-energy Bose
condensate states by varying the parameter $\gamma_{0}$ for $d=2$. For small
$\gamma_{0}$, the ground state is a symmetric state. As $\gamma_{0}$ is
increased, a symmetry-breaking phase transition occurs at a critical value of
$\gamma_{0}=0.528$. At this point, the symmetric state becomes dynamically
unstable and the ground state changes to an asymmetric state. The ground state
chemcial potential shows a kink at this critical point, wheras the ground
state energy curve is smooth. This represents a second-order phase transition
Qian . A similar behavior is also found in attractive BEC with unmodulated
scattering length Kanamoto .
Figure 2: (Color online) Same plots as in Fig. 1 for $d=3$. The symmetry
breaking phase transition occurs at the critical point $\gamma_{0}=0.85$
(indicated by the arrows in the plots) and the dynamical instability sets in
for the symmetric state at $\gamma_{0}=1.04$.
Next, we turn our attention to the case of $d=3$. The energies and chemical
potentials as functions of $\gamma_{0}$ are plotted in Fig. 2. Similar to the
$d=2$ case, for small $\gamma_{0}$, the ground state is symmetric. As
$\gamma_{0}$ is increased, a symmetry-breaking phase transition occurs at a
critical value of $\gamma_{0}=0.85$. However, unlike in the $d=2$ case, at
this critical point, the symmetric state remains dynamically stable although
the energy of the asymmetric state drops below that of the symmetric state.
Furthermore, the ground state energy curve as a function of $\gamma_{0}$ shows
a kink at this critical value, while the ground state chemical potential
becomes discontinuous at this point. Therefore, the phase transition at this
point is of first order. The dynamical instability of the symmetric state does
not occur until $\gamma_{0}=1.04$.
To summarize our mean-field results, we have found that the nature of the
phase transitions changes for different modulation period $d$ due to the
presence of competing mechanisms in this system. When the modulation period
$d=2$, the symmetry-breaking phase transition is of second order and is
induced by dynamical instability of the associated states. For $d=3$, in
contrast, the symmetric-breaking phase transition is of first order and is
driven by the level crossing of different types of states. We have also
investigated the cases for $d=4$ and 5 and found similar behavior as in $d=3$.
## IV quantum mechanical treatment
So far, we have limited our discussion to the mean-field approximation. Now we
perform a fully quantum mechanical examination of the system using a numerical
method. In our previous work Qian , exact diagonalization is used for this
purpose. Here, we will use the TEBD algorithm Vidal . This has a two-fold
advantage compared to the exact diagonalization method: (1) It allows us to
treat larger systems; and (2) in addition to the static properties, we can
also use the TEBD method to study the dynamical behavior of the system.
We first discretize the space by introducing an equidistant grid
$\theta_{i}=i\Delta\theta$, $(i=0,1,...M-1)$. We then replace the field
operator $\widehat{\psi}\left(\theta_{i}\right)$ by
$\widehat{\psi}_{i}/\sqrt{\Delta\theta}$, where $\widehat{\psi}_{i}$ is a
bosonic annihilation operator. In doing so, integrals can be replaced by sums
and the second derivative in the kinetic energy term can be approximated by
the difference quotient
$\frac{\partial^{2}}{\partial\theta^{2}}\widehat{\psi}\left(\theta_{i}\right)\approx\left[\widehat{\psi}\left(\theta_{i+1}\right)+\widehat{\psi}\left(\theta_{i-1}\right)-2\widehat{\psi}\left(\theta_{i}\right)\right]/\Delta\theta^{2}$.
Finally, the discretized Hamiltonian is Michael1 :
$H=-\frac{1}{\Delta\theta^{2}}\sum_{i=0}^{M-1}\left(\widehat{\psi}_{i}^{\dagger}\widehat{\psi}_{i+1}+h.c.\right)+\frac{1}{\Delta\theta^{2}}\sum_{i=0}^{M-1}\widehat{\psi}_{i}^{\dagger}\widehat{\psi}_{i}+\sum_{i=0}^{M-1}\frac{U_{i}}{2\Delta\theta}\widehat{\psi}_{i}^{\dagger}\widehat{\psi}_{i}^{\dagger}\widehat{\psi}_{i}\widehat{\psi}_{i}.$
(12)
Here, the periodic boundary condition leads to the relation
$\widehat{\psi}_{0}=\widehat{\psi}_{M}$. For TEBD algorithm with periodic
boundary condition, we refer to Ref. Naidon . In our numerical treatment, we
typically divide the ring into $M=60$ equidistant grids. Our code is adapted
from the open source package maintained by the group of Lincoln Carr open .
### IV.1 many-body ground-state energy
In Fig. 3, we plot the ground-state energy per atom of the many-body system as
functions of $\gamma_{0}$ for the modulation period $d=2$ and $d=3$ . We see
that the ground-state energy curve approaches the MFT result as $N$ increases,
which is consistent with the usual quantum-semiclassical crossover behavior
for finite-size quantum systems.
Figure 3: (Color online) Many-body ground-state energy per atom for $N=6,12$
and modulation period $d=2$ (upper panel) and $d=3$ (lower panel). The black
solid lines are the mean-field results.
### IV.2 quantum correlation
In the quantum mechanical treatment, unlike the mean field results, the
spontaneous symmetry breaking of density distribution of ground state
wavefunction in real space dose not occur. The density profile of the quantum
mechanical ground state always matches the spatial modulation of the
scattering length Qian . However, we can still gain important insights into
the change in the characteristics of the wavefunctions by examining the
quantum correlation which is neglected in the mean-field study.
For further discussion, we define the partial number operator between the
interval $\theta\in[\varphi_{i},\varphi_{f}]$ as:
$\widehat{n}_{\left[\varphi_{i},\varphi_{f}\right]}=\frac{1}{2\pi}\int_{\varphi_{i}}^{\varphi_{f}}d\theta\,\widehat{\psi}^{\dagger}\left(\theta\right)\widehat{\psi}\left(\theta\right).$
(13)
Figure 4: (Color online) Bipartite correlation $g_{ij}^{\left(2\right)}$ and
tripartite correlation $g^{\left(3\right)}$ as functions of $\gamma_{0}$ for
$N=6$ and $d=3$.
For the particular case of $d=3$, we define three partial particle number
operators as follows:
$\widehat{n}_{i}=\widehat{n}_{\left[(i-1)\frac{2\pi}{3},i\frac{2\pi}{3}\right]},\quad
i=1,2,3.$ (14)
Using these number operators, we can define the bipartite and tripartite
correlation functions as:
$\displaystyle g_{ij}^{\left(2\right)}$ $\displaystyle=$
$\displaystyle\frac{\left\langle\widehat{n}_{i}\widehat{n}_{j}\right\rangle}{\left\langle\widehat{n}_{i}\right\rangle\left\langle\widehat{n}_{j}\right\rangle}\,,$
$\displaystyle g^{\left(3\right)}$ $\displaystyle=$
$\displaystyle\frac{\left\langle\widehat{n}_{1}\widehat{n}_{2}\widehat{n}_{3}\right\rangle}{\left\langle\widehat{n}_{1}\right\rangle\left\langle\widehat{n}_{2}\right\rangle\left\langle\widehat{n}_{3}\right\rangle}\,.$
We plot the bipartite and tripartite correlations as functions of $\gamma_{0}$
in Fig. 4. All these correlations are monotonically decreasing functions of
$\gamma_{0}$. In our calculation, the three two-body correlation functions
$g_{12}^{(2)}$, $g_{13}^{(2)}$ and $g_{23}^{(2)}$ are essentially identical,
which is also expected from the symmetry of the system. At $\gamma_{0}=0$, the
ground state is exactly known:
$\left|\Psi\right\rangle_{ground}=\frac{\left(a_{0}^{\dagger}\right)^{N}}{\sqrt{N!}}\left|{\rm
vac}\right\rangle$. The theoretical values of bipartite and tripartite
correlations can be obtained as:
$\displaystyle g_{ij}^{\left(2\right)}\left(\gamma_{0}=0\right)$
$\displaystyle=$ $\displaystyle
1-\frac{6}{\pi^{2}N}\sum_{l=1}^{\infty}\frac{1}{l^{2}}\,,$ $\displaystyle
g^{\left(3\right)}\left(\gamma_{0}=0\right)$ $\displaystyle=$ $\displaystyle
1-\frac{18}{\pi^{2}N}\sum_{l=1}^{\infty}\frac{1}{l^{2}}+O\left(\frac{1}{N^{2}}\right)\,,$
which are in good agreement with the numerical results. When $N$ goes to
infinity, all the bipartite and tripartite correlations approach unity at
$\gamma_{0}=0$.
Figure 4 shows that although both bipartite and tripartite correlations decay
as the interaction parameter $\gamma_{0}$ increases, the tirpartitle
correlation $g^{\left(3\right)}$ decays into zero much faster than the
bipartitle correlation $g_{ij}^{\left(2\right)}$. For the case of $N=6$ as
illustrated in Fig. 4, $g^{\left(3\right)}$ is essentially zero at
$\gamma_{0}=1.6$, while all the $g_{ij}^{\left(2\right)}$ are clearly non-zero
at the same $\gamma_{0}$. This is reminiscent of the three-body entangled
$W$-state, which can be written as
$|W\rangle=\frac{1}{\sqrt{3}}\,\left(|100\rangle+|010\rangle+|001\rangle\right)\,.$
For the $W$-state, the tripartite entanglement characterized by the 3-tangle
disappears and the bipartite entanglement characterized by concurrence remains
finite Coffman . For large $\gamma_{0}$, our mean-field treatment presented
earlier reveals that the ground state is characterized by asysmmetric state
with three-fold degeneracy. Each of these degenerate mean-field state features
a density peak at $\theta=(1+4i\pi)/6$ ($i=1,2,3$). In the quantum treatment,
this degeneracy is lifted by quantum fluctuations, and the non-degenerate
quantum ground state may be regarded as roughly a $W$-state formed by these
three mean-field states.
### IV.3 Single-particle density matrix
Another important quantity to characterize the many-body state is the single-
particle density matrix $\rho^{\left(1\right)}$ whose matrix element is
defined as Penrose ; Leggett :
$\rho_{ij}^{\left(1\right)}=\left\langle\widehat{\psi}_{i}^{\dagger}\widehat{\psi}_{j}\right\rangle\,,$
(15)
where the expectation value is calculated with respect to the ground state
obtained using the TEBD method. Roughly speaking, $\rho_{ij}^{\left(1\right)}$
represents the probability amplitude of finding one particle at site $i$ and
at the same time another particle at site $j$.
Figure 5: (Color online) The largest three eigenvalues of the single-particle
density matrix $\rho^{(1)}$ for $N=6$.
Because the matrix $\rho^{\left(1\right)}$ is Hermitian it can be diagonalized
as:
$\rho^{\left(1\right)}=N\sum_{i}p_{i}\,\varphi_{i}^{*}\varphi_{i}\,,$ (16)
where the eigenvalues $p_{i}$ are non-negative and satisfy the constraint
$\sum_{i}p_{i}=1$. For a bosonic system with $N\gg 1$, $p_{i}$ is closely
related to the condensate fraction of the system. It is easy to see that, the
system will be in a simple condensate state if and only if the largest
eigenvaule is of order unity and all the other $p_{i}$’s are of order
$O(N^{-1})$. If there are multiple $p_{i}$’s of order unity, the system is
said to be in a fragmented condensate because all the corresponding states
have appreciable occupation numbers. Finally, if all the $p_{i}$’s are of
order $O(N^{-1})$, then the system is not Bose condensed. It is reasonable to
speculate that, at small $N$, the condensate fraction versus the strength
parameter $\gamma_{0}$ exhibits quantum crossover behavior. For $N=6$ with the
modulation periods $d=2$ and $d=3$, we plot the three largest $p_{i}$’s versus
$\gamma_{0}$ in Fig. 5. The figure shows that at small $\gamma_{0}$, the
system may be characterized as a simple condensate. It becomes more and more
fragmented as $\gamma_{0}$ is increased. In the large $\gamma_{0}$ limit, the
eigenvalues approach some steady state values and there are $d$ eigenvalues
which are much larger than the rest. This is consistent with the earlier
argument that at large $\gamma_{0}$, the quantum many-body ground state can be
roughly regarded as superpositions of the $d$-fold degenerate mean-field
ground states.
To further quantify the crossover from a simple condensate to a fragmented
condensate, we adopt two methods as described below. In Fig. 6(a), we plot
$dp_{0}/d\gamma_{0}$ as a function of $\gamma_{0}$, where $p_{0}$ is the
largest single-particle density matrix eigenvalue. In Fig. 6(b), we plot the
overlap of the ground-state wave function Zanardi $|\langle
G(\gamma_{0})|G(\gamma_{0}+\delta\gamma)\rangle|$ for a small value of
$\delta\gamma=0.02$, where $|G(\gamma_{0})\rangle$ denotes the ground state at
$\gamma_{0}$. Both of these quantities measure how fast the characteristics of
the ground state change as $\gamma_{0}$ is varied. These two measures provide
consistent results: both exhibit a dip at some critical value
$\gamma_{0}=0.84$ for $d=2$ and $\gamma_{0}=1.08$ for $d=3$, which we may
define as the critical modulation depth where the system crosses from a simple
condensate to a fragmented condensate.
Figure 6: (Color online) (a) Derivative of the largest single-particle density
matrix eigenvalue with respect to the interaction strength modulation depth
and (b) overlap of the ground-state wave function versus $\gamma_{0}$ for the
particle number $N=6$.
### IV.4 Time evolution of the survival probability
Up to now, we have focused our attention on the static properties of the
ground state. The low-energy excitations of the system in the vicinity of the
crossover are also important because they expose fine characteristics valuable
for understanding the system dynamics in the crossover. A powerful tool to
study this is the time evolution of ground state survival probability Felker ;
Wang . It describes the dynamical behavior of the system’s ground state under
small perturbation in parameters in the system Hamiltonian. In our problem, we
parameterize the Hamiltonian using the interaction strength modulation depth
$\gamma_{0}$ such that $H=H(\gamma_{0})$. If $\gamma_{0}$ has a small
variation $\delta\gamma$ so that
$\gamma_{0}\rightarrow\gamma_{0}+\delta\gamma$, the ground state’s survival
probablity is defined as
$M(t)=\left|\left\langle
G\left(\gamma_{0}\right)\right|\exp\\{-iH\left(\gamma_{0}+\delta\gamma\right)t\\}\left|G\left(\gamma_{0}\right)\right\rangle\right|^{2}\,.$
(17)
The survival probability can be considered as a special case of the quantum
Loschmidt echo Peres . Incidentally, the numerical method we used to simulate
our system, the TEBD algorithm, is very convenient in calculating the system’s
time evolution and hence the survival probability.
We calculated the ground state survival probability $M(t)$ and plot the
results in Fig. 7 for $N=6$ and the perturbation in $\gamma_{0}$ is
$\delta\gamma=0.1$. $M(t)$ exhibits roughly sinusoidal oscillations in time.
The amplitude of the modulation reaches the maximum near critical points which
are consistent with those found in previous static study and shown in Fig. 6.
Indeed, the curve for the oscillation amplitude of the ground state’s survival
probability can be used to predict the overlap between the perturbed and
original ground state wave functions which is plotted in Fig. 6(b). The larger
the oscillation amplitude in $M(t)$, the more sensitive the ground state
wavefunction to the perturbation in the Hamiltonian.
Figure 7: (Color online) The amplitude of the oscillation of the survival
probability $M(t)$ for (a) $d=2$ and (b) $d=3$. The inset shows the dyanmics
of $M(t)$ for different values of $\gamma_{0}$.
## V Summary
In conclusion, we have made a systematic investigation of a condensate on a
nonlinear ring lattice. Our studies show that the properties of the system are
sensitive to the modulation period $d$ of the interaction strength. In
particular, the meanfield symmetry breaking phase transition is second order
when the modulation period $d$ is 2 but first order when $d>2$, due to
competing mechanisms present in the system driving these transitions. Our full
quantum mechanical treatment based on the TEBD method reveals the behavior of
many important quantities that are essential to the characterization of the
system physics.
That the mean-field symmetry breaking phase transition changes from second to
first order when $d$ changes from 2 to larger than 2 is somewhat surprising.
The change of the order of the phase transition may be related to the change
of the length scale associated with the modulation of the scattering length:
For $d>2$, this length scale is smaller compared with $d=2$. Recently,
Mayteevarunyoo et al. studied the symmetry breaking transition in a BEC
subject to a nonlinear double-well potential and found that the width of the
nonlinear potential plays an important role in controlling the transition
Mayteevarunyoo , a phenomenon that may be related to what we discovered in the
current work. This is certainly one of the peculiar properties of nonlinear
potentials that deserves further investigation.
## VI Acknowledgments
This work was funded by NFRP 2011CB921204 , and NNSF (Grant Nos. 60921091,
10874170, 10875110). Z. -W. Zhou gratefully acknowledges the support of the K.
C. Wong Education Foundation, Hong Kong. HP acknowledges support from U.S.
NSF. The authors thank Yong-Jian Han, Biao Wu, Shi-Liang Zhu, Hui Zhai, Wu-
Ming Liu and Wen-Ge Wang for helpful discussions and comments. Z. -W. Zhou
appreciates the hospitality of the Kavli Institute of Theoretical Physics in
Beijing, where part of this work was completed.
## Appendix A Proof of the equivalence between the algebraic type and time-
independent Gross-Pitaevskii equation
Following the argument in Sec. I, we obtain the Hamiltonian in the limit $N\gg
1$ as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{kl}\frac{k^{2}}{N}a_{k}^{\dagger}a_{l}^{\dagger}a_{k}a_{l}+\frac{i\gamma_{0}}{4N}\sum_{klmn}a_{k}^{\dagger}a_{l}^{\dagger}a_{m}a_{n}\left(\delta_{d+m+n-k-l}\right.$
(18) $\displaystyle\;\left.-\delta_{-d+m+n-k-l}\right)\,,$
which can be written in a simplified form:
$H=\frac{1}{N}\sum_{ijkl}{\alpha_{ijkl}}a_{i}^{\dagger}a_{j}^{\dagger}a_{k}a_{l}\,.$
(19)
In Sec. III, we obtain the algebraic type GP equations (10) and (11) which we
rewrite here as:
$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\sum_{ijkl}\,{\alpha_{ijkl}}\,u_{0i}u_{0j}u_{0k}^{*}u_{0l}^{*}\,,$
(20) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{ijkl}\,(\alpha_{ijkl}+\alpha_{ijlk})\left[u_{0i}u_{0j}u_{0k}^{*}(a_{l}-u_{0l}^{*}\chi_{0})\right]\,.$
(21)
Here, there is the unitary relation: $\sum_{l}u_{0l}^{*}u_{0l}=1$. Based on
Eqs. (18) and (20), we have:
$c_{0}=\sum_{i,j,k,l}\alpha_{ijkl}u_{0i}u_{0j}u_{0k}^{*}u_{0l}^{*}=A+B,$ (22)
where
$A=\sum_{kl}k^{2}u_{0k}u_{0l}u_{0k}^{*}u_{0l}^{*}=\sum_{k}k^{2}u_{0k}u_{0k}^{*}$
and
$B=\frac{i\gamma_{0}}{4}\sum_{klmn}u_{0k}u_{0l}u_{0m}^{*}u_{0n}^{*}\left(\delta_{d+m+n-k-l}-\delta_{-d+m+n-k-l}\right)$.
Furthermore, by considering Eq. (21), the following relation can be derived:
$\sum_{kl}k^{2}u_{0k}u_{0l}(u_{0k}^{*}a_{l}+u_{0l}^{*}a_{k})+\frac{i\gamma_{0}}{4}\sum_{klmn}u_{0k}u_{0l}(u_{0m}^{*}a_{n}+u_{0n}^{*}a_{m})\left(\delta_{d+m+n-k-l}-\delta_{-d+m+n-k-l}\right)=2c_{0}\chi_{0}\,.$
(23)
By taking advantage of the unitary relation: $\chi_{0}=\sum_{l}u_{0l}a_{l}$,
Eq. (23) can be decomposed into the following algebraic equations depending on
the boson operator $a_{l}$:
$l^{2}u_{0l}+\frac{i\gamma_{0}}{2}\sum_{kmn}u_{0k}u_{0n}u_{0m}^{*}\left(\delta_{d+m+l-k-n}-\delta_{-d+m+l-k-n}\right)=\mu
u_{0l}\,,$ (24)
where $\mu=2c_{0}-A=c_{0}+\frac{\partial c_{0}}{\partial N}N$.
The time-independent GP equation describing Bose gases on a ring with periodic
scattering length is:
$-\frac{\partial^{2}}{\partial\theta^{2}}\psi\left(\theta\right)-2\pi\gamma_{0}\sin(d\theta)\left|\psi\left(\theta\right)\right|^{2}\psi\left(\theta\right)=\mu\psi\left(\theta\right)\,,$
(25)
with the boundary condition:
$\psi\left(\theta\right)=\psi\left(2\pi+\theta\right)$. By replacing
$\psi\left(\theta\right)=\frac{1}{\sqrt{2\pi}}\sum_{l}u_{0l}e^{-il\theta}$
into Eq. (25), the algebraic equations same as Eq. (24) can be derived. This
demonstrates the equivalence between Eqs. (10) and (11) and the usual GP
equation.
## Appendix B Finding Excitation Spectrum
The Bogoliubov spectrum of quasiparticle excitations can be determined by
diagonalizing operator $K$ as defined in Eq. (8). However, here, we can not
obtain the representation of Eq. (8) using the modes
$\left\\{\chi_{l}^{\dagger},\chi_{m}\right\\}$ because the unitary
transformation $U$ is unknown except for its matrix elements in the first row.
This difficulty can be overcome by representing the last two terms at the
r.h.s. of Eq. (8) using the original mode operators
$\left\\{a_{l}^{\dagger},a_{m}\right\\}$:
$K=(c_{0}-\mu)N+\sum_{mn}A_{mn}a_{m}^{\dagger}a_{n}+\sum_{mn}(B_{mn}a_{m}a_{n}+h.c.)\,,$
(26)
where $\sum_{mn}B_{mn}a_{m}a_{n}=\sum_{i,j,k,l}\alpha_{ijkl}\left\langle
a_{i}^{\dagger}a_{j}^{\dagger}\right\rangle:a_{k}a_{l}:$ and
$\sum_{mn}A_{mn}a_{m}^{\dagger}a_{n}=\sum_{i,j,k,l}\alpha_{ijkl}\left\\{\left\langle
a_{i}^{\dagger}a_{k}\right\rangle:a_{j}^{\dagger}a_{l}:+\left\langle
a_{j}^{\dagger}a_{l}\right\rangle:a_{i}^{\dagger}a_{k}:+\left\langle
a_{i}^{\dagger}a_{l}\right\rangle:a_{j}^{\dagger}a_{k}:+\left\langle
a_{j}^{\dagger}a_{k}\right\rangle:a_{i}^{\dagger}a_{l}:\right\\}-\mu(\sum_{n}a_{n}^{\dagger}a_{n}-N)\,.$
Here $\left\langle{A}\right\rangle$ refers to the amplitude of the operator
$A$ projecting onto the operators $\chi_{0}^{\dagger}\chi_{0}$,
$\chi_{0}^{\dagger}\chi_{0}^{\dagger}$, or $\chi_{0}\chi_{0}$ and $:A:$ refers
to the corresponding component in the operator $A$ orthogonal to the operators
($\chi_{0}^{\dagger}\chi_{0}$, $\chi_{0}^{\dagger}\chi_{0}^{\dagger}$,
$\chi_{0}\chi_{0}$). For instance,
$\left\langle
a_{i}^{\dagger}a_{j}^{\dagger}\right\rangle:a_{k}a_{l}:=u_{0i}u_{0j}\left(a_{k}-u_{0k}^{*}\chi_{0}\right)\left(a_{l}-u_{0l}^{*}\chi_{0}\right).$
To find the energy spectrum for quasiparticle excitation, a generalized
Bogoliubov method ( see reference Milstein ) can be used to diagonalize Eq.
(26). Briefly, by introducing the row and column vectors
$\varsigma=\binom{a}{a^{\dagger}},\quad\varsigma^{\dagger}=\left(a^{\dagger}a\right)$
(27)
and defining the matrix
$M=\left(\begin{array}[]{ccc}A&2B\\\ 2B^{*}&A^{*}\end{array}\right)$ (28)
we may rewrite the operator $K$ as:
$K=(c_{0}-\mu)N+\frac{1}{2}\varsigma^{\dagger}M\varsigma-\frac{1}{2}{\rm
Tr}A\,.$ (29)
Here, the matrix $A=\left[A_{mn}\right]$ and the matrix
$B=\left[B_{mn}\right]$. Now $K$ can be diagonalized by introducing the
appropriate canonical transformation $T$: $\beta=T\varsigma$. Finally, the
operator $K$ has the diagonal form:
$K=(c_{0}-\mu)N+\frac{1}{2}\beta^{\dagger}\eta T\eta
MT^{-1}\beta-\frac{1}{2}{\rm Tr}A\,,$ (30)
where the matrix $\eta=I\oplus(-I)$ and $T\eta MT^{-1}$ is a diagonal matrix.
## References
* (1) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
* (2) Yaroslav V. Kartashov, Boris A. Malomed, and Lluis Torner, arXiv:1010.2254, to appear in Rev. Mod. Phys.
* (3) Lisa C. Qian, Michael L. Wall, Shaoliang Zhang, Zhengwei Zhou, and Han Pu, Phys. Rev. A 77, 013611 (2008).
* (4) R. Kanamoto, H. Saito, and M. Ueda, Phys. Rev. A 67, 013608 (2003).
* (5) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); ibid, 93, 040502 (2004).
* (6) B. Schmidt, L. I. Plimak, and M. Fleischhauer, Phys. Rev. A 71, 041601(R) (2005); B. Schmidt, and M. Fleischhauer, Phys. Rev. A 75, 021601(R) (2007).
* (7) I. Danshita and P. Naidon, Phys. Rev. A 79, 043601 (2009).
* (8) See http://physics.mines.edu/downloads/software/tebd/
* (9) V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000).
* (10) O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).
* (11) A. J. Leggett, Quantum Liquids, Oxford University Press (2006).
* (12) P. Zanardi and N. Paunkovi$\acute{c}$ Phys. Rev. E 74, 031123 (2006).
* (13) P. Felker and A. Zewail, Adv. Chem. Phys. 70, 265 (1988).
* (14) W. G. Wang, P. Qin, L. He, and P. Wang, Phys. Rev. E 81, 016214 (2010).
* (15) A. Peres, Phys. Rev. A 30, 1610 (1984).
* (16) T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
* (17) J. N. Milstein, The doctoral dissertation: ‘From Cooper Pairs to Molecules: Effective field theories for ultra-cold atomic gases near Feshbach resonances’ (University of Colorado), p118-120 (2004).
|
arxiv-papers
| 2012-05-05T03:03:05 |
2024-09-04T02:49:30.587478
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zheng-Wei Zhou, Shao-Liang Zhang, Xiang-Fa Zhou, Guang-Can Guo,\n Xingxiang Zhou, Han Pu",
"submitter": "Zheng-Wei Zhou",
"url": "https://arxiv.org/abs/1205.1094"
}
|
1205.1114
|
# A Durable Flash Memory Search Tree
James Clay III Kevin Wortman California State Univeristy, Fullerton
jaclay@fullerton.edu kwortman@fullerton.edu
###### Abstract
We consider the task of optimizing the _B-tree_ data structure, used
extensively in operating systems and databases, for sustainable usage on
multi-level flash memory. Empirical evidence shows that this new flash memory
tree, or _FM Tree,_ extends the operational lifespan of each block of flash
memory by a factor of roughly 27 to 70 times, while still supporting
logarithmic-time search tree operations.
## 1 Introduction
Flash memory has seen growing usage in recent years across all areas of
computing technology; devices utilizing flash include tablets, cell phones,
and USB drives. The lifespan of flash memories has been of serious concern
since their inception; flash memory degrades proportionally to the number of
times it is erased. While significant advances have been made to combat this
effect, flash usage continues to grow in areas where rewrite-intensive
operations are necessary. We consider the problem of extending the operational
lifetime of flash memory by avoiding erase operations, at the expense of some
additional time and space overhead.
## 2 Background
### 2.1 Flash Memory
_Multi-level flash memory_ is a form of NAND random access memory _(RAM)_
capable of storing one of $q>1$ discrete states in each flash cell. These $q$
values may interpreted as nonnegative integers in the range $[0,\,q-1].$ Cells
are aggregated into _blocks_ of fixed size. Individual multi-level flash cells
may be incremented, which is a fast and non-destructive operation. However,
idiosyncratically, bucket states can only be decreased by resetting an entire
block to state 0 _en masse._ These _resets_ or _erasures_ are costly both in
terms of time of the operation and lifetime of the device. Typical block
erases take between 1.5-2 milliseconds in comparison to seek or write times
which are in the tens or hundreds of microseconds. Each NAND flash device has
a set number of expected erasure cycles it can perform before failing. Wear
distribution can be done in multiple ways ranging from a purely round robin
approach to keeping the most actively used blocks in RAM. However recent
advances in multiple level flash memory data representation relaxes the
requirement that a block must be erased before it can be rewritten. Since
erasures are a limiting factor to both the durability and write throughput of
multi-level flash memory, we investigate approaches to avoid erasures, at the
cost of modest constant-factor expenditures of space and CPU time.
### 2.2 Tree Types
Our FM tree is an amalgam of ideas from established search tree data
structures. In this section we survey their properties. A _Bayer McCreight
B-Tree,_ henceforth denoted _B-tree_ , is a type of balanced search tree
developed for managing large blocks of data, particularly in file-systems and
databases. A B+ tree resembles a B-tree, with the addition of redundant node
links that facilitate tree traversal in common database operations. B- trees
are a relaxed version of the common B+ tree where the notoriously complex
post-deletion rebalancing operation is omitted. Perhaps surprisingly, Sen and
Tarjan showed that, despite postponing rebalance operations, B- trees still
boast asymptotically optimal update operations up to amortization, and may be
implemented simply with attractive constant factors.
### 2.3 A Durable B-tree
In creating a durable B-tree we have made a variety of changes to the
implementation of both operations on and storage of the keys within the
B-tree. These alterations reduce the maximum number of erasures per block and
the average number of erasures across all blocks.
* •
Block erasures are performed lazily, postponing them for as long as possible.
* •
The requirement that key/value pairs be sorted within nodes is relaxed. This
gives the insertion operation leeway to reuse key/value slots without erasing
the block, but makes searching a single node take $O(B)$ rather than $O(\log
B)$ time.
* •
As in the B- tree, the delete operation marks unused nodes _barren_ rather
than actually splicing the node out of the tree. Barren nodes are ignored
until the tree is eventually garbage collected and rebuilt. This allows a node
to be excised by toggling flash cell(s) representing a boolean barren flag,
without performing block erasures.
## 3 Analysis
We prove that an FM tree supports the search, insert, and delete operations
all in amortized $O(\log n)$ time, matching the lower bound for amortized
search tree data structures. We show that, for any sequence of operations, an
FM tree performs strictly fewer erasures than a conventional B-tree.
## 4 Experimental Results
We present a variety of experiments performed on a Python implementation of
the FM Tree. We emulate the flash memory, FM Tree, and B-tree to run a variety
of benchmarks. Every experimental trial consists of randomly generated data
sets that are inserted into both trees. Each tree is inserted with a baseline
of 1000 elements. Following these initial insertions, 10000 randomly chosen
insertions and deletions are performed. We repeat this process a total of 4
times independently and determine the average for each data point. We then
calculate the FM Tree performance by comparing the erasures, reads and writes
between it and the B-tree. This process indicates that the FM Tree performs 27
times to 72.2 times fewer erasures. While the total read count was higher, the
total writes and erasures performed were far lower. As these are the most
expensive operations in terms of time, realistically the FM Tree would also be
far faster than the B-tree.
## 5 Conclusion
We find that the FM-Tree is a more durable, faster variant of the B-tree with
properties that make it intrinsically better for operating on flash memory. We
also show that the erasure count for the FM Tree is drastically smaller (27 to
72 times) than that of a B-tree.
|
arxiv-papers
| 2012-05-05T08:41:24 |
2024-09-04T02:49:30.594711
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "James Clay III, Kevin Wortman",
"submitter": "James Clay III",
"url": "https://arxiv.org/abs/1205.1114"
}
|
1205.1171
|
# Divide-and-Conquer 3D Convex Hulls on the GPU
Jeffrey M. White Department of Computer Science, California State University,
Fullerton, jeffreymarkwhite@gmail.com Kevin A. Wortman Department of Computer
Science, California State University, Fullerton, kwortman@fullerton.edu
###### Abstract
We describe a pure divide-and-conquer parallel algorithm for computing 3D
convex hulls. We implement that algorithm on GPU hardware, and find a
significant speedup over comparable CPU implementations.
## 1 Introduction
The _3D convex hull problem_ is to identify, for a given set of $n$ points in
$\mathbb{R}^{3}$, the minimal set of input points such that the convex
envelope of those points contains all input points. The problem is fundamental
to computational geometry and has been studied extensively. Several $O(n\log
n)$ time algorithms are known, with various trade-offs in constant factors,
simplicity, numerical robustness, data structure dependencies, and
nondegeneracy requirements (see e.g. [2] [4] [7] [8] [10] [13] [15]). Chan’s
celebrated output-sensitive algorithm [5] runs in $O(n\log h)$ time, where $h$
denotes the number of faces in the output hull, which is asymptotically
optimal.
A _graphics processing unit (GPU)_ is a parallel coprocessor available in
commodity computers. An outgrowth of the computer gaming industry, GPUs
utilize a highly-parallel single instruction multiple data (SIMD)
architecture. At a high-level, GPUs work by applying a concise constant-space
function called a _kernel_ to all elements of an array simultaneously. Kernels
are written in _domain specific embedded languages (DSELs)_ such as NVIDIA’s
CUDA [12] or the OpenCL [11] open standard. Each kernel instance is passed an
integer _global identifier (id)_ which is customarily used to delineate the
ranges of input that each kernel invocation applies to. The potential
performance, measured in either gigaFLOPS or memory bandwidth, of GPUs is
substantially greater than that of multicore CPUs. However, realizing this
potential on practical problems, besides the embarrasingly-parallel graphics
applications for which GPUs were originally designed, has proven challenging.
By and large, existing parallel algorithms depend on facilities, such as
message passing and/or synchronization primitives, which are unavailable in
the GPU environment. Yet, GPUs are purpose-built for high performance
computation on low-dimensional geometric objects, and the opportunity to apply
them to computational geometry problems cannot be ignored.
While the 3D convex hull problem has been studied extensively in the standard
computational model, precious little past work is applicable to GPU
implementations. As stated above, GPU kernels cannot communicate with or
synchronize against each other. This limitation rendered unusable every PRAM-
model algorithm we surveyed (e.g. [3]). Further, running kernels have no
provision for dynamic memory; their collective input and output must be
allocated before the kernels execute _en masse_ and freed afterward.
Accordingly dynamic data structures are off limits. The absence of the doubly
connected edge list (DCEL) structure is a particularly formidable obstacle in
this context.
There are several results on computing 2D hulls on the GPU [9] [14] [16], but
results on the more general and complex 3D problem have been elusive. While
preparing this manuscript, we became aware of an independent result on the 3D
problem [17]. That algorithm uses heuristics to cull many, but not all,
interior points on the GPU, then feeds the remaining points to a black-box CPU
hull implementation (e.g. QuickHull [4]). The algorithm presented here
achieves competitive performance using a pure GPU divide-and-conquer approach,
whose worst case running time is not impacted by the presence of outlier
points, and which is conceptually simpler.
## 2 Algorithm
Figure 1: Algorithm Events.
Our algorithm is an adaptation of Chan’s _minimalist_ 3D convex hull algorithm
[6]. Note that this $O(n\log n)$-time algorithm is distinct from the $O(n\log
h)$-time algorithm mentioned earlier, also authored by Chan. The minimalist
algorithm is, by design, a straightforward top-down divide-and-conquer
algorithm for computing 3D convex hulls. It was originally motivated by
pedagogical needs for an algorithm that achieves a favorable $O(n\log n)$
running time, while being simple to explain and implement and avoiding
dependency on difficult data structures or algorithms. Serendipitously these
design constraints correspond to those imposed by the GPU.
The minimalist algorithm works by recasting the 3D problem as a 2D _kinetic_
problem. 3D $(x,y,z)$ points are mapped to $(x,y,\Delta y)$ points with an
initial $(x,y)$ starting point and $\Delta y$ vertical rate of speed. As time
$t$ advances, the points move at distinct velocities, which triggers
structural changes in the convex hull of the points (see Figure 1). Computing
the convex hull of the original 3D points may be visualized as computing a
_kinetic movie_ of these configurations for all values $-\infty<t<\infty$. The
algorithm represents this movie as a chronological sequence of _events_ when
input points are added to, or removed from, the hull. Input points are
presorted by $x$-coordinate; then event sequences for roughly equal-size
subsets are recursively generated, then combined by a Graham-scan-like $O(n)$
merging process. In the base case a single point nominates itself as the only
convex hull point.
While the minimalist algorithm boasts many of the features necessary for GPU
implementation, it cannot be ported to the GPU directly. GPU kernels cannot be
recursive, so the top-down divide-and-conquer approach is inappropriate.
Instead, the algorithm must be reoriented into one or more mapping steps where
an array of input data elements are mapped by a kernel to an array of output
data elements. We achieve this reorientation by rewriting the minimalist
algorithm to use _bottom-up_ divide and conquer. We define a _movie array_
data structure as a table of event logs. Our algorithm allocates a single
movie array, and initializes one trivial event log for each input point. Then,
our algorithm repeats a _merge step_ that combines each pair of event logs
with adjacent indices into a single event log. A merge step maps a movie array
with $n$ logs of length at most $l$ to a new array with at most $\lceil
n/2\rceil$ logs of length at most $2l$ each. Thus, after
$\lceil\log_{2}n\rceil$ merge steps, the movie array contains a single event
log for the entire point set. The key property of this algorithm with respect
to GPU computation is that each log merge may be performed entirely
independently of the others. Each kernel has a particular range of input movie
array indices to read from, and a corresponding range of output indices to
write to, and may perform its computation independently of other concurrent
kernel instances.
## 3 Implementation
Our implementation of the GPU algorithm follows the bottom-up divide-and-
conquer design as mentioned above. As shown in Figure 3, the point structure
in the CPU algorithm uses a doubly linked list connected by pointers. The idea
is to divide the sorted list down into trivial subsequences and build the list
back up to the desired set of faces on the convex hull. Memory pointers are
difficult (though not impossible) to move between the CPU and GPU since the
two devices have distinct memory spaces. Also, on the GPU each kernel instance
needs to seek to its assigned sub-input based on its global id, which could
take $O(n)$ time using a list structure. For these reasons, our GPU
implementation uses arrayed lists with integer indices rather than linked
lists with node addresses (Figure 3).
⬇
// CPU Algorithm Point
struct Point {
double x, y, z;
Point *prev, *next;
void act() {...}
};
// GPU Algorithm Point
struct Point {
cl_float x;
cl_float y;
cl_float z;
cl_int prev;
cl_int next;
};
Figure 2: Differences in the Point datatype.
⬇
// CPU Algorithm list of points
Point *P = new Point[n];
...
// Sorts points into a doubly
// linked list based x-coordinate.
Point *list = sort(P, n);
// event lists
Point **A = new Point *[2*n];
Point **B = new Point *[2*n];
// GPU Algorithm list of points
Point *P = (Point *)
malloc(n*sizeof(Point));
// event lists
cl_int *A = (cl_int *)
malloc(2*n*sizeof(cl_int));
cl_int *B = (cl_int *)
malloc(2*n*sizeof(cl_int));
Figure 3: Differences in list creation.
Modifying the way data is stored impacts the way data is accessed. Figure 4
shows the differences in act() function used for inserting and deleting points
from event logs. Figure 5 shows the differences in passing potential faces
into the event-time calculations.
⬇
// CPU Algorithm act() function call
point->act()
// CPU Algorithm act() function
struct Point {
...
void act() {
if (prev->next != this) {
// insert point
prev->next = next->prev = this;
}
else {
// delete point
prev->next = next;
next->prev = prev;
}
}
};
// GPU Algorithm act() function call
act(pointIndex);
// GPU Algorithm act() function
void act(int pointIndex) {
if (P[P[pointIndex].prev].next
!= pointIndex) {
// insert point
P[P[pointIndex].prev].next
= P[P[pointIndex].next].prev
= pointIndex;
}
else {
// delete point
P[P[pointIndex].prev].next
= P[pointIndex].next;
P[P[pointIndex].next].prev
= P[pointIndex].prev;
}
}
Figure 4: Differences in act() functions.
⬇
// CPU Algorithm time[0] calculation
t[0] = time(B[i]->prev,
B[i],
B[i]->next);
// GPU Algorithm time[0] calculation
t[0] = time(P[B[i]].prev,
B[i],
P[B[i]].next);
Figure 5: Differences in time calculations.
The implementation process began with converting the original CPU algorithm to
use arrays rather then pointers to represent the data. Point data is
implemented as its own data type with the $x,y,$ and $z$ values along with
indices to represent the next and previous pointers to reference other points
based on their array index. Also, instead of having two pointer lists, $A$ and
$B$, we have two arrays of indices that reference a master list $P$ of points.
⬇
dataOffsetValue = 2;
totalMergesLeft = numberOfPoints/2;
do {
numberOfThreads = totalMergesLeft;
runGPUkernels();
swap(A, B);
dataOffsetValue = dataOffsetValue*2;
totalMergesLeft = totalMergesLeft/2;
} while(totalMergesLeft > 1);
Figure 6: Main outer loop ran on the CPU to handle the execution of threads on
the GPU.
Another significant change we made to the design is the conversion from a top-
down design to a bottom-up design. Instead of using recursion, the heart of
the algorithm is placed within one while loop as shown in Figure 6. Before
implementing this routine as OpenCL kernel code, we wrote a simulation to run
on the serial CPU to ensure validity of the algorithm. The ultimate goal of
writing a simulation is to avoid the troublesome task of debugging GPU kernel
code. This simplified the task of converting the simulation code to GPU kernel
code and required only minimal modifications.
Figure 6 shows pseudocode for the main outer loop which runs on the CPU. The
main loop uses two movie array structures, both of which exist on the GPU. The
two structures alternate between serving as the input and output of a merge
step. This approach makes it possible to avoid transferring point data between
the GPU and CPU inside the loop, which is desirable as that is an expensive
operation. The dataOffsetValue is used to calculate the location of where the
head of the leftGroupIndex and rightGroupIndex exist on the globally accessed
master list of points $P$ as shown in Figure 7. To handle the way the CPU
algorithm swaps lists $A$ and $B$ in each divide routine, we swap the kernel
arguments of $A$ and $B$ in the swap(A, B) function after each iteration of
merges. Following the swap(A, B) function, dataOffsetValue is updated to tie
into the next set of group index calculations. Finally, totalMergesLeft is cut
in half to represent the number threads to take place in the next iteration of
merges. When totalMergesLeft reaches less than 2, the algorithm exits the main
while loop as there is no pair of hulls left to be merged together; only one
hull is left which represents the final solution.
⬇
// the index of where the head of the
// left group of the list can be found
// on the globally accessed array
leftGroupIndex
= global_ID*dataOffsetValue;
// the index of where the head of the
// right group of the list can be found
// on the globally accessed array
rightGroupIndex
= [leftGroupIndex+((global_ID+1)
*dataOffsetValue)]/2;
// the index of where the globally
// accessed event list begins for the
// group of merges based on the global_ID
eventListOffset = leftGroupIndex*2;
Figure 7: GPU kernel code: how the GPU knows which hulls should be merged and
which parts of the global data to access.
## 4 Experimental Results
The GPU algorithm shows significant improvements over the CPU algorithm. Peak
performance of the GPU algorithm reaches close to a 6x speedup over the CPU
algorithm. Figures 8 and 9 illustrate the runtime of both algorithms in
milliseconds.
The CPU algorithm runtime calculations are based on a Intel ® CoreTM i3-2330M
Processor with 2 cores capable of processing 2 threads each at a rate 2.2
gigahertz, and acheives about 25.0 gigaFLOPS according to the LINPACK
benchmark tool. The GPU algorithm runtime calculations are based on an ATI
Radeon HD 6470m graphics card with 32 stream cores each with 5 processing
elements capable of pipelining data at a rate of 700-750 megahertz. Peak
performance for this GPU can potentially reach 224-240 gigaFLOPS [1]. Both the
CPU and GPU, as described, rank low on the spectrum of hardware available
based on performance.
Points | GPU Alg. (ms) | CPU Alg. (ms)
---|---|---
4 | 0 | 0
8 | 0 | 0
16 | 0 | 0
32 | 0 | 0
64 | 0 | 0
128 | 0 | 1
256 | 0 | 3
512 | 0 | 7
1024 | 0 | 9
2048 | 1 | 10
4096 | 2 | 13
8192 | 4 | 25
16384 | 9 | 34
32768 | 23 | 59
65536 | 38 | 123
131072 | 56 | 233
262144 | 88 | 465
524288 | 173 | 941
1048576 | 354 | 1869
2097152 | 652 | 3732
4194304 | 1309 | 7494
8388608 | 2598 | 15047
Figure 8: Run time for $n$ data points. Figure 9: Run time graph for $n$ data
points.
Originally, a hybrid approach to the GPU algorithm seemed to be a more
attractive solution to solving the problem. The hybrid GPU algorithm would
perform nearly all of the merge steps on the GPU, then perform the last few
steps on the CPU after the totalMergesLeft variable reached a certain value.
The premise of this approach is that the last few iterations are poorly
parallelizable and could be more quickly performed by a serial CPU. To
accomplish this, the partially computed data would need to be copied from GPU
memory to memory that the CPU has access to. On the CPU side, there would be a
similar algorithm which would finish the rest of the computation using that
same bottom-up style algorithm.
Surprisingly, our experimental results showed that those last few merge
iterations take an insignificant amount of time – less than one millisecond.
So the hybrid approach is overly-complex, and implementing it would have been
an instance of _premature optimization._ The final design of the GPU algorithm
takes place entirely on the GPU rather then on both GPU and CPU hardware. The
GPU algorithm just requires the use of the CPU for the required OpenCL setup
routines and ultimately to read in the data and output the data; the GPU
completes all the extensive computations.
Something we found interesting is the ratio of speedup improvements over the
CPU algorithm as the data set increases. For a data set of four points, the
speed up is close to 6x. As the data set approaches 32,768 points, the speedup
decreases to about 2.5x. From 32,768 points and on, the speedup increases back
to about 6x.
Our roughly 6x speedup is notable since it approaches the maximum potential
improvement achievable on our hardware. According to the manufacturers, our
GPU is capable of roughly 9 times more gigaFLOPS than our CPU. So the greatest
conceivable speedup factor is roughly 9, which would correspond to an
embarrasingly-parallel problem with negligible overhead. Our implementation
comes close to realizing this full potential despite the obstacles inherent in
parallelizing the 3D convex hull problem.
## 5 Conclusion
We have shown that bottom-up adaptation of the minimalist divide-and-conquer
algorithm for 3D convex hulls is fast, practical, and reasonably
straightforward. The approach achieves run-time performance comparable to past
GPU implementations of convex hull algorithms.
In performing this exercise, we did make two counterintuitive conclusions.
First, while OpenCL and CUDA are intended to be high-level abstractions of GPU
hardware, we nonetheless faced many obstacles related to low-level concerns
such as memory management, memory hierarchies, and thread scheduling. Second,
our intuition was that the overhead of starting and scheduling kernel
applications would become a major bottleneck in the later steps of the
algorithm. However, empirical results demonstrated this to be a non-issue.
The following are potential areas for future work:
* •
Higher-level libraries or tools for implementing divide-and-conquer algorithms
on the GPU.
* •
A suite of compatible, parallel GPU implementations of fundamental
computational geometry algorithms.
* •
In particular, an arrangement data structure, e.g. doubly connected edge list,
is a prerequisite to implementing many well-motivated algorithms.
## References
* [1] AMD Accelerated Parallel Processing. Advanced Micro Devices, Inc., http://developer.amd.com/sdks/amdappsdk/assets/amd_accelerated_parallel%_processing_opencl_programming_guide.pdf.
* [2] S. G. Akl and G. T. Toussaint. A fast convex hull algorithm. Information Processing Letters, 7(5):219 – 222, 1978.
* [3] N. M. Amato and F. P. Preparata. A time-optimal parallel algorithm for 3D convex hulls. 1993\.
* [4] C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469–483, Dec. 1996.
* [5] T. Chan. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete & Computational Geometry, 16:361–368, 1996. 10.1007/BF02712873.
* [6] T. M. Chan. A minimalist’s implementation of the 3-d divide-and-conquer convex hull algorithm. 2003\.
* [7] B. Chazelle. An optimal convex hull algorithm in any fixed dimension. Discrete & Computational Geometry, 10:377–409, 1993. 10.1007/BF02573985.
* [8] W. F. Eddy. A new convex hull algorithm for planar sets. ACM Trans. Math. Softw., 3(4):398–403, Dec. 1977.
* [9] T. Jurkiewicz and P. Danilewski. Efficient quicksort and 2d convex hull for cuda, and msimd as a realistic model of massively parallel computations. November 2011.
* [10] D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287–299, 1986.
* [11] A. Munshi. The OpenCL specification version 1.0.
* [12] NVIDIA. NVIDIA CUDA Programming Guide 2.0. 2008\.
* [13] F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20(2):87–93, Feb. 1977.
* [14] A. Rueda and L. Ortega. Geometric Algorithms on CUDA. In Proceedings of the $3^{rd}$ International Conference on Computer Graphics Theory and Applications, 2008.
* [15] R. Seidel. A convex hull algorithm optimal for point sets in even dimension. Master’s thesis, Dept. of Computer Science, University of British Columbia, Vancouver, Canada, 1981.
* [16] S. Srungarapu, D. Reddy, K. Kothapalli, and P. Narayanan. Fast two dimensional convex hull on the gpu. In Advanced Information Networking and Applications (WAINA), 2011 IEEE Workshops of International Conference on, pages 7 –12, march 2011\.
* [17] M. Tang, J. yi Zhao, R. Tong, and D. Manocha. GPU accelerated convex hull computation. In Shape Modeling International (SMI) 2012, 2012.
|
arxiv-papers
| 2012-05-06T01:33:15 |
2024-09-04T02:49:30.602372
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeffrey M. White and Kevin A. Wortman",
"submitter": "Kevin Wortman",
"url": "https://arxiv.org/abs/1205.1171"
}
|
1205.1192
|
# A Note on Bounds of Scalar Operators in Perturbative SCFTs
Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Abstract
Bounds on anomalous dimensions of scalar operators in 4d superconformal field
theory are explored through perturbative viewpoint. Following the recent work
of Green and Shih, in which a conjecture involved this issue is verified at
the NLO, we consider the NNLO corrections to the bounds, which are important
in some situations and can be divided into two cases where
$\mathcal{O}(\lambda^{4})$ or $\mathcal{O}(y^{2})$ effects dominate
respectively. In the former case, we find that the conjecture is maintained at
NNLO, while in the later case the statement still holds due to null
correction.
May 2012
## 1 Introduction
Conformal field theory (CFT) ( see [1, 2, 3] for example ), which is tied to
important concepts in field theory and phenomenological application, has been
extensively explored. For example, the small hierarchy $\mu$ problem involved
in electroweak symmetry breaking in the minimal supersymmetric standard model
can be solved when this theory is coupled to a hidden superconformal field
theory (SCFT) [5, 6] (for other recipes, see [4] and reference therein for
example ). The reason for this viability is due to the different scaling
behaviors between chiral $\mu$ and real $B_{\mu}$ operator, which is expected
in SCFTs where the condition $\delta_{min}>0$ (see its definition in (1.4)) is
satisfied.
Given a CFT, the dimensions of operators and coefficients in the correlator
functions (or equivalently the OPE coefficients) of these operators exactly
determine or define the theory. Many efforts have been done by using arguments
of conformal symmetry, crossing symmetry and unitarity. Among these
developments, an interesting and well-known topic in unitary CFT is the
discovery of bounds on dimensions of operators. The full list of unitary
bounds, which includes fields with Lorentz spin $(j,\tilde{j})$ is presented
in [1] . Also, a-maximization [8] that follows from the arguments involved in
anomalies of global symmetries provides, in terms of unitary constraints, an
alternative method to determine the dimensions of chiral operators in SCFTs.
Very recently the bounds on anomalous dimension of primary scalar operators
are addressed [11, 12, 13, 14, 15, 16, 17] by applying conformal blocks [9]
and global symmetries to exploring the four-point correlators of scalar
primary operators. A conjecture is hinted by these works.
The 4d interacting SCFT $\mathcal{P}_{1}$ we are going to study contains a
chiral operator $\mathcal{O}$ of dimension $\Delta_{\mathcal{O}}=2-\epsilon$.
The OPE of $\mathcal{O}$ and its anti-chiral field $\mathcal{O}^{{\dagger}}$
is assumed to be
$\displaystyle{}\mathcal{O}(x)^{{\dagger}}\mathcal{O}(0)=\frac{1}{\mid
x\mid^{2\Delta_{\mathcal{O}}}}+\sum_{i}\frac{c_{i}}{\mid
x\mid^{2\Delta_{\mathcal{O}}-\Delta_{i}}}L_{i}+\cdots$ (1.1)
where $L_{i}$ are real scalar multiplets with dimension $\Delta_{i}=2+\nu_{i}$
( Here $\nu_{i}$ is a non-negative real number ). $c_{i}$ refer to the OPE
coefficients. The terms ignored in (1.1) denote descendants with higher spin.
We follow the convention in [10] where all primary scaling operators are
canonically normalized as in (1.1). We explore theories constructed through
deforming $\mathcal{P}_{1}$ by ,
$\displaystyle{}\mathcal{L}=\mathcal{L}_{\mathcal{P}_{1}}+\left(\frac{1}{4\pi^{2}}\int
d^{4}\theta X^{{\dagger}}X+\int
d^{2}\theta\frac{\lambda}{2\pi}X\mathcal{O}+h.c\right)$ (1.2)
with $X$ being a free chiral superfield. Our concern is to discuss the
anomalous dimensions of scalar primary operators $\mathcal{S}_{i}$ which
appear in the OPE of $X$ and $X^{{\dagger}}$,
$\displaystyle{}X^{{\dagger}}(x)X(0)=\frac{1}{\mid
x\mid^{2\Delta_{X}}}+\sum_{i}\frac{c_{i}}{\mid
x\mid^{2\Delta_{X}-\Delta_{i}}}\mathcal{S}_{i}+\cdots,$ (1.3)
When the anomalous dimension of $\Delta_{X}=1+\epsilon$ is small,
$0<\epsilon<<1$ as we assume throughout this paper, the deformed theory (1.2)
will renormalization group (RG) flow into a new interacting CFT
$\mathcal{P}_{2}$. As expected, the candidate operators $\mathcal{S}_{i}$ in
(1.3) include $X^{{\dagger}}X$, $L_{i}$ and their mixing. A variety of works
[11, 12, 13, 14, 15, 16, 17] tend to claim that the sign of $\delta_{min}$
defined as 111As mentioned in the previous discussion, the study of this
conjecture is of interest from point of view of phenomenology.
$\displaystyle{}\delta_{min}=\min{(\Delta_{i})}-2\Delta_{X}<0$ (1.4)
always holds in general.
The purpose in this article is to study the higher-order corrections on this
conjecture in the context of perturbative CFT, by following the method of
calculations proposed by Green and Shih [10]. The advantage of this method is
that the RG flow between the new and old fixed points is manifest. By using
this method, the conjecture is perturbatively verified at the next-to-leading
order ( NLO). We would like to address the question whether the the bound on
$\delta_{min}$ is robust as suggested. If not, then under which circumstances
it can be violated. As we will claim, despite smaller than NLO ones, the NNLO
corrections are important and even substantial in some circumstances. In
particular, the modifications to the vanishing matrix elements of anomalous
dimension of $\mathcal{S}_{i}$ at NLO can directly affect the sign of
$\delta_{min}$, even though they don’t substantially modify the values of
fixed points couplings $\lambda_{*}$ and $y_{i*}$.
In section 2, we divide the discussions into two cases. In the case where
$\mathcal{O}(\lambda^{4})$ dominates, we calculate the corrections to values
of couplings at the new fixed points in section 3, and estimate the
modification to the matrix of anomalous dimension and value of $\delta_{min}$,
which are found to be substantial, however, not enough to violate the
conjecture. In section 4, we consider the modification due to
$\mathcal{O}(y^{2})$ effects at NNLO, which is found to be actually null. We
claim that this observation exactly holds beyond NLO. Finally , we summarize
our results in section 5.
## 2 NNLO Corrections
Take the RG effects into account, the Lagrangian for $\mathcal{P}_{2}$ SCFT
can be written as,
$\displaystyle{}\mathcal{L}=\mathcal{L}_{\mathcal{P}_{1}}+\frac{1}{4\pi^{2}}\int
d^{4}\theta(1+\delta Z_{X})X^{{\dagger}}X+\int d^{4}\theta(y_{i}+\delta
y_{i})L_{i}+\left(\int
d^{2}\theta\frac{\lambda}{2\pi}\Lambda^{\epsilon}X\mathcal{O}+h.c\right)$
where we have introduced $\Lambda$ dependence so that $\lambda$ is a
dimensionless coupling. $y_{i}$ are the coupling constants appearing in
$L_{i}$ operators. $\delta Z_{X}$ and $\delta~{}y_{i}$ denote the effects of
wave-function renormalization. By using the holomorphic arguments, we find the
beta function for $\lambda$ is exactly given by,
$\displaystyle{}\beta_{\lambda}=-\epsilon\lambda+\lambda\gamma_{X}(\lambda,y_{i}),~{}~{}~{}~{}~{}~{}\gamma_{X}=-\frac{1}{2}\frac{\partial\delta
Z_{X}}{\partial\log\Lambda}$ (2.2)
Expanding the wave-function renormalization functionals $\delta Z_{X}$ and
$\delta y_{i}$ in power of $\lambda$ and $y_{i}$ which are both assumed to be
small as,
$\displaystyle{}\delta Z_{X}$ $\displaystyle=$ $\displaystyle
a_{1}\lambda^{2}+a_{1i}y_{i}+a_{2i}\lambda^{2}y_{i}+a_{2}\lambda^{4}+a_{2ij}y_{i}y_{i}+\mathcal{O}(\lambda^{6},y^{4},\lambda^{4}y^{2})$
$\displaystyle\delta y_{i}$ $\displaystyle=$ $\displaystyle
b_{1i}\lambda^{2}+b_{1ij}y_{j}+b_{2ij}\lambda^{2}y_{j}+b_{2i}\lambda^{4}+b_{2ijk}y_{j}y_{k}+\mathcal{O}(\lambda^{6},y^{4},\lambda^{4}y^{2})$
(2.3)
where $a_{i}$, $b_{i}$ are real coefficients, some of which have been
considered in [10] up to NLO,
$\displaystyle{}a_{1~{}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{\epsilon},$ $\displaystyle a_{1i}$
$\displaystyle=$ $\displaystyle 0,$ (2.4) $\displaystyle a_{2i}$
$\displaystyle=$
$\displaystyle\frac{8\pi^{4}c_{i}}{\nu_{i}-2\epsilon}\mathcal{I}(\nu_{i},\epsilon),$
and
$\displaystyle{}b_{1i~{}}$ $\displaystyle=$
$\displaystyle\frac{c_{i}}{2(2\epsilon+\nu_{i})},$ $\displaystyle b_{1ij}$
$\displaystyle=$ $\displaystyle 0,$ (2.5) $\displaystyle b_{2ij}$
$\displaystyle=$ $\displaystyle 0,$
In the following we take into account the NNLO corrections. In terms of the
assumption in (2) we can write the beta function of $\lambda$ and $y_{i}$ as,
$\displaystyle{}\beta_{\lambda}$ $\displaystyle=$
$\displaystyle-\epsilon\lambda+\lambda\left[\pi^{2}\lambda^{2}-4\pi^{4}\sum_{i}c_{i}y_{i}\mathcal{I}(\nu_{i},\epsilon)\lambda^{2}+2\epsilon
a_{2}\lambda^{4}-\sum_{i,j}a_{2ij}(\nu_{i}+\nu_{j})y_{i}y_{j}\right]$
$\displaystyle\beta_{y_{i}}$ $\displaystyle=$
$\displaystyle\nu_{i}y_{i}-\frac{1}{2}c_{i}\lambda^{2}-(4\epsilon+\nu_{i})b_{2i}\lambda^{4}+\sum_{j,k}b_{2ijk}(\nu_{j}+\nu_{k}-\nu_{i})y_{j}y_{k}$
(2.6)
which implies the values of couplings $\lambda_{*}$ and $y_{i*}$ at the fixed
point of $\mathcal{P}_{2}$,
$\displaystyle{}-\epsilon+\pi^{2}\lambda^{2}_{*}-4\pi^{4}\sum_{i}c_{i}y_{i*}\mathcal{I}(\nu_{i},\epsilon)\lambda^{2}_{*}+2\epsilon
a_{2}\lambda^{4}_{*}-\sum_{i,j}a_{2ij}(\nu_{i}+\nu_{j})y_{*i}y_{*j}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\nu_{i}y_{i*}-\frac{1}{2}c_{i}\lambda^{2}_{*}-(4\epsilon+\nu_{i})b_{2i}\lambda^{4}_{*}+\sum_{j,k}b_{2ijk}(\nu_{j}+\nu_{k}-\nu_{i})y_{*j}y_{*k}$
$\displaystyle=$ $\displaystyle 0$
A natural question we have not addressed is under which condition the
approximation up to NNLO is important and sufficient, especially in compared
with the NLO ones. For corrections to the second equation in (2),
$y_{i*}\simeq\frac{1}{2}\frac{c_{i}}{\nu_{i}}\lambda^{2}_{*}$ [10] is always
valid except that the new theory $\mathcal{P}_{2}$ is beyond the scope of
perturbation. This suggests $y_{i*}<<\lambda^{2}_{*}$ if $c_{i}<<\nu_{i}$, or
equivalently $c_{i}<<1$, which implies that the effect of $\mathcal{O}(y^{2})$
(even of $\mathcal{O}(\lambda^{2}y)$ ) is smaller in compared with that of
$\mathcal{O}(\lambda^{4})$. It is necessary to take the order of
$\mathcal{O}(\lambda^{4})$ into account and revise those discussions based on
orders up to $\mathcal{O}(\lambda^{2}y)$ but without
$\mathcal{O}(\lambda^{4})$, even though there exists no large hierarchy in the
OPE coefficients. Nevertheless, $y_{i*}>\lambda^{2}$ if
$c_{i}\sim\mathcal{O}(1)$. In this case the corrections arising from
$\mathcal{O}(y^{2})$ and $\mathcal{O}(\lambda^{2}y)$ dominate over
$\mathcal{O}(\lambda^{4})$.
## 3 SCFTs at $\mathcal{O}(\lambda^{4})$
We perform the perturbative calculations by using the OPEs in appendix A. The
rational is that correlation functions must be independent of $\Lambda$ scale,
which results in the requirement that the coefficients appearing in the same
operator that carries $\Lambda$ factor must cancel out. Doing so we obtain,
$\displaystyle{}a_{2~{}}$ $\displaystyle=$ $\displaystyle
16\pi^{4}\left(\frac{c^{2}_{i}}{\nu^{2}_{i}-4\epsilon^{2}}\right)\mathcal{I}(\nu_{i},\epsilon)-\frac{2\pi^{2}}{\epsilon^{2}}\mathcal{T}(\epsilon)$
$\displaystyle b_{2i}$ $\displaystyle=$
$\displaystyle-\frac{\pi^{2}c_{i}}{2\epsilon(\nu_{i}-2\epsilon)}\left[\mathcal{P}(\nu_{i},\epsilon)+\mathcal{Q}(\nu_{i},\epsilon)\right]$
(3.1)
where $\mathcal{I}(\nu_{i},\epsilon)$, $\mathcal{T}(\epsilon)$,
$\mathcal{P}(\nu_{i},\epsilon)$ and $\mathcal{Q}(\nu_{i},\epsilon)$ are all
dimensionless and smooth functionals as defined in appendix A.
Substituting (3) into (2)and (2) while neglecting the $\mathcal{O}(y^{2})$
effects results in,
$\displaystyle{}\beta_{\lambda}$ $\displaystyle=$
$\displaystyle-\epsilon\lambda+\lambda\left[\pi^{2}\lambda^{2}-4\pi^{4}\sum_{i}c_{i}y_{i}\mathcal{I}(\nu_{i},\epsilon)\lambda^{2}+2\epsilon
a_{2}\lambda^{4}+\cdots\right]$ $\displaystyle\beta_{y_{i}}$ $\displaystyle=$
$\displaystyle\nu_{i}y_{i}-\frac{1}{2}c_{i}\lambda^{2}-(4\epsilon+\nu_{i})b_{2i}\lambda^{4}+\cdots$
(3.2)
and consequently
$\displaystyle{}-\epsilon+\pi^{2}\lambda^{2}_{*}-4\pi^{4}\sum_{i}c_{i}y_{i*}\mathcal{I}(\nu_{i},\epsilon)\lambda^{2}_{*}+2\epsilon
a_{2}\lambda^{4}_{*}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\nu_{i}y_{i*}-\frac{1}{2}c_{i}\lambda^{2}_{*}-(4\epsilon+\nu_{i})b_{2i}\lambda^{4}_{*}$
$\displaystyle=$ $\displaystyle 0$ (3.3)
, respectively. The value of $y_{i*}$ is instead of,
$\displaystyle{}y_{i*}=\frac{c_{i}}{2\nu_{i}}\lambda^{2}_{*}\left[1+\epsilon^{-1}\lambda^{2}_{*}(\mathcal{O}(1)+\kappa\left(\mathcal{P}(\nu_{i},\epsilon)+\mathcal{Q}(\nu_{i},\epsilon)\right)\right]$
(3.4)
with the coefficient $\kappa$ is strictly of $\mathcal{O}(1)$ no matter how
$\nu_{i}$ is relative to $\epsilon$. So whether the higher-order corrections
to $y_{i*}$ in (3.4) are substantial depend on the finite quantities
$\mathcal{P}(\nu_{i},\epsilon)$ and $\mathcal{Q}(\nu_{i},\epsilon)$.
The $\mathcal{O}(\lambda^{4})$ corrections to $\gamma_{X}(\nu_{i},\epsilon)$
gives rise to,
$\displaystyle{}-\epsilon+\pi^{2}\lambda^{2}_{*}-2\pi^{2}\lambda^{4}_{*}\sum_{i}\frac{1}{{\nu_{i}}}\left[\pi^{2}c^{2}_{i}\left(1-16\frac{\epsilon\nu_{i}}{\nu^{2}_{i}-4\epsilon^{2}}\right)\mathcal{I}(\nu_{i},\epsilon)-\frac{\nu_{i}}{\epsilon}\left(\frac{3-\epsilon}{2}-2\mathcal{T}(\epsilon)\right)\right]=0$
Substitute the leading order approximation
$\lambda^{2}_{*}\simeq\frac{\epsilon}{\pi^{2}}$ into terms of order
$\mathcal{O}(\lambda^{4})$ in (3) gives rise to
$\displaystyle{}\lambda^{2}_{*}\simeq-\frac{\epsilon}{\pi^{2}}+\frac{1}{\pi^{2}}\mathcal{O}\left(\frac{\epsilon^{2}c_{i}^{2}}{\nu_{i}}\right)+\frac{\mathcal{T}(\epsilon)}{\pi^{2}}\mathcal{O}(\epsilon)$
(3.6)
it is clear to notice that the higher-order corrections can be substantial for
determining the fixed point coupling $\lambda_{*}$ when $c_{i}<\nu_{i}$ and
even dominate over the order of $\mathcal{O}(\lambda^{2}y_{i})$ when
$c_{i}<<\nu_{i}$. In the region of small $c_{i}$, $c_{i}<<\nu_{i}$, the
$\mathcal{O}(\lambda^{4})$ correction is substantial for determining the fixed
point coupling $\lambda_{*}$.
Now we calculate the anomalous dimensions of operators imposed of $L_{i}$,
$X^{{\dagger}}X$ and their mixing, which can be read from the $\tau$ matrix
defined as
$\tau\equiv\partial_{(y_{i},\lambda)}\beta_{(y_{i},\lambda)}\mid_{y_{i}*,\lambda_{*}}$.
By using (3) we obtain,
$\displaystyle{}\tau=\left(\begin{array}[]{cc}\nu_{i}\delta_{ij}&-c_{i}\lambda_{*}-4\sum_{i}(4\epsilon+\nu_{i})b_{2i}\lambda^{3}_{*}\\\
-4\pi^{4}\sum_{i}c_{i}\mathcal{I}(\nu_{i},\epsilon)\lambda^{3}_{*}&2\epsilon(1+\frac{5\epsilon^{2}}{\pi^{4}}a_{2})\end{array}\right)$
(3.9)
The deviation of the eigenvalues $\delta$ of this $\tau$ matrix to the case
without $\mathcal{O}(\lambda^{4})$ effects can be more clearly seen after we
make a $2\epsilon$ shift in $\tau$, which is a operation useful for us to
directly compare the value of $\delta_{min}$ with [10],
$\displaystyle{}\delta\tau=\left(\begin{array}[]{cc}(\nu_{i}-2\epsilon)\delta_{ij}&-c_{i}\lambda_{*}-4\sum_{i}(4\epsilon+\nu_{i})b_{2i}\lambda^{3}_{*}\\\
-4\pi^{4}\sum_{i}c_{i}\mathcal{I}(\nu_{i},\epsilon)\lambda^{3}_{*}&\frac{10\epsilon^{3}}{\pi^{4}}a_{2}\end{array}\right)$
(3.12)
The point is that all the diagonal elements aren’t zero, which remain after a
similarity transformation to $\tau$. So whether there exists such a negative
$\delta$ is not obvious anymore. In general it is quite difficult to obtain
the eigenvalues $\delta$ without given the information about relative values
of $\nu_{i}$ and $\epsilon$. We divide this task into a few cases. The first ,
also trivial case is $\nu_{i}<<\epsilon<<1$, in which there are already some
$L_{i}$ with dimension smaller than $2\Delta_{X}$. The other cases
$\epsilon<<\nu_{i}<<1$ and $\epsilon\sim\nu_{i}<<1$ are of more interest to
us.
### 3.1 $\epsilon<<\nu_{i}<<1$
Now we address the simplification for the functionals as defined in appendix A
in the region $\epsilon<<\nu_{i}<<1$. Each integral variable $X^{+}_{i}$ in
these functionals are evaluated in the region $|X^{+}_{i}|>\frac{1}{\Lambda}$,
with $\Lambda$ the cut-off scale introduced in (A), and integral over
Grassmann variables is equivalent to performing derivative over them. For
functional $\mathcal{I}(\nu_{i},\epsilon)$ (A), performing the integral gives
us,
$\displaystyle{}\mathcal{I}(\nu_{i},\epsilon)|_{\nu_{i}<<1,~{}\epsilon<<1}\simeq
1+\mathcal{O}(\epsilon,\nu_{i})$ (3.13)
Similar operation can be applied to $\mathcal{P}(\nu_{i},\epsilon)$
functional, which explicitly reads,
$\displaystyle{}\mathcal{P}(\nu_{i},\epsilon)|_{\nu_{i}<<1,~{}\epsilon<<1}$
$\displaystyle\simeq$
$\displaystyle\frac{2\epsilon-\nu_{i}}{2\epsilon+\nu_{i}}+\mathcal{O}(\epsilon,\nu_{i})\simeq-1+\mathcal{O}(\epsilon,\nu_{i})$
(3.14)
after setting $\nu_{i}=0$ and replacing $2\epsilon\rightarrow
2\epsilon+\nu_{i}$. The functionals $\mathcal{Q}(\nu_{i},\epsilon)$ and
$\mathcal{T}(\epsilon)$ are three-dimensional integrals, thus more involved
than $\mathcal{I}(\nu_{i},\epsilon)$ and $\mathcal{P}(\nu_{i},\epsilon)$. For
this case one can integrate over one variable, then follow the similar
operation for the two-dimensional integral. At leading order, we find
$\displaystyle{}\mathcal{Q}(\nu_{i},\epsilon)|_{\nu_{i}<<1,~{}\epsilon<<1}$
$\displaystyle\simeq$
$\displaystyle\left(\frac{2\epsilon-\nu_{i}}{2\epsilon+\nu_{i}}\right)\epsilon\Gamma(2\epsilon)+\cdots=-1+\mathcal{O}(\epsilon,\nu_{i})$
$\displaystyle\mathcal{T}(\nu_{i},\epsilon)|_{\nu_{i}<<1,~{}\epsilon<<1}$
$\displaystyle\simeq$
$\displaystyle+\epsilon^{2}\left[\Gamma(2\epsilon)+\cdots\right]=+\mathcal{O}(\epsilon)+\cdots$
(3.15)
where we have ignored the higher-order terms. The coefficients at the leading
order, related to the complicated Hypergeoemtric function
$~{}_{2}F_{1}(1,m-2\epsilon,1+2\epsilon,-1)$ (with integer $m$), are finite
and not shown explicitly. We will see the approximations (3.13)- (3.1) are
sufficient to illustrate the modification to anomalous dimensions of
$\mathcal{S}_{i}$.
Substitute (3.13)- (3.1) into (3), one can substantially simplify $a_{2}$ and
$b_{2i}$. Doing so, we obtain the leading-order approximation to the matrix
$\delta\tau$ in (3.12) under the limit $\epsilon<<\nu_{i}<<1$,
$\displaystyle{}\delta\tau$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\nu_{i}\delta_{ij}&-c_{i}\lambda_{*}\left[1+\frac{1}{\epsilon}(3+4\pi^{2})\right]\\\
-4\pi^{4}\sum_{i}c_{i}\lambda^{3}_{*}&-\frac{20\epsilon}{\pi^{2}}\mathcal{T}(\epsilon)\end{array}\right)$
(3.18)
Put the values of couplings at NLO back into (3.18), the characteristic
equation of $\delta$ is found to be,
$\displaystyle{}\left(\nu_{i}\delta_{ij}-\delta\right)\left(-\frac{20\epsilon}{\pi^{2}}\mathcal{T}(\epsilon)-\delta\right)-4\pi^{2}(3+4\pi^{2})c^{2}_{i}\epsilon=0$
(3.19)
Together with the small $c_{i}<<\nu_{i}$ condition assumed through out this
section, we notice that the last constant term in (3.19) is actually small
compared with $\nu_{i}\epsilon$ if $c_{i}$ is below the critical value
$c_{i*}\simeq\sqrt{\nu_{i}\epsilon}$, which implies that the minimal value of
$\delta$ is of order $\epsilon^{2}$,
$\displaystyle{}\delta_{min}\simeq-\frac{20\epsilon}{\pi^{2}}\mathcal{T}(\epsilon)+\mathcal{O}(\epsilon^{3})<0$
(3.20)
One thing happens when $c_{i}$ is above the critical value $c_{i*}$. The last
term dominate conversely, which modified the (3.19) as,
$\displaystyle{}\delta_{min}\simeq-\pi^{2}(3+4\pi^{2})\frac{c^{2}_{i}\epsilon}{\nu_{i}}\simeq-\pi^{2}(3+4\pi^{2})\left(\frac{c_{i}}{c_{i*}}\right)^{2}\epsilon^{2}<0$
(3.21)
### 3.2 $\nu_{i}\sim\epsilon<<1$
Since it is quite natural to expect that $\nu_{i}$ is of
$\mathcal{O}(\epsilon)$ or higher powers of $\epsilon$ in perturbative CFT, a
number of $\mathcal{P}_{2}$ theories can be covered in this limit. Now we
address the question that whether the statement in the previous discussion can
be generalized to this particular situation. At first, $a_{2}$ and $b_{2i}$
take the approximation 222Note that $a_{2}$ has a pole at $\nu_{i}=2\epsilon$.
Here, we assume $\nu_{i}$ is not equal to $2\epsilon$ for simplification.,
$\displaystyle{}a_{2}$ $\displaystyle=$
$\displaystyle-\frac{2\pi^{2}}{\epsilon^{2}}\mathcal{T}(\epsilon)$
$\displaystyle b_{2i}$ $\displaystyle=$
$\displaystyle-\frac{3\pi^{2}c_{i}}{4\epsilon(\nu_{i}+2\epsilon)}$ (3.22)
Substitute these values into (3.12), we obtain
$\displaystyle{}\delta\tau=\left(\begin{array}[]{cc}(\nu_{i}-2\epsilon)\delta_{ij}&-c_{i}\lambda_{*}\left[1+\mathcal{O}(\epsilon^{-1}\lambda^{2})\right]\\\
-4\pi^{4}\sum_{i}c_{i}\lambda^{3}_{*}&-\frac{20\epsilon}{\pi^{2}}\mathcal{T}(\epsilon)\end{array}\right)=\left(\begin{array}[]{cc}(\nu_{i}-2\epsilon)\delta_{ij}&\mathcal{O}(c_{i}^{\frac{1}{2}})\\\
\mathcal{O}(c_{i}\epsilon^{\frac{3}{2}})&-\frac{20\epsilon}{\pi^{2}}\mathcal{T}(\epsilon)\end{array}\right)$
(3.27)
Drop the off-diagonal elements in above matrix by using the relation
$c_{i}<<\nu_{i}\sim\epsilon$, we arrive at the conclusion that the statement
is also true in the region.
In summary, if $c_{i}<<\nu_{i}<<1$ is indeed produced given a
$\mathcal{P}_{2}$ theory, then we can conclude that the bound on the anomalous
dimension of $\mathcal{S}_{i}$ as conjectured in the literature is still valid
at NNLO , no matter the relative values of $\epsilon$ and $\nu_{i}$.Therefore,
the validity of this conjecture is directly transferred to examine these
conditions in $\mathcal{P}_{2}$ theory 333We want to remind the reader that
naively this statement can not be directly applied to BZ theory with large $N$
limit. However, in BZ theory $\nu_{i}\simeq\mathcal{O}(\epsilon^{2})$ [10],
which actually suggests some of anomalous dimension of $L_{i}$ is already
smaller than that of $X$. This statement is trivially satisfied in this
situation. .
## 4 SCFTs at $\mathcal{O}(y^{2})$
The $\mathcal{O}(y^{2})$ corrections dominate over $\mathcal{O}(\lambda^{4})$
when $c_{i}>>\nu_{i}$. The investigation of bounds on $c_{i}$ can be found in
[17, 18]. Instead of calculating the wave-function renormalization and beta
function as in appendix A, on must consider $L_{i}$ operators. But this task
can not be precisely achieved without knowing the explict form of $L_{i}$ (for
example $L_{i}$ are composite operators). The $\mathcal{O}(y^{2})$ effects can
only be analyzed either in a specific $\mathcal{P}_{1}$ theory or in certain
approximations.
### 4.1 BZ Theory As an illustration
One might wonder which $\mathcal{P}_{1}$ theory can provide such kind of
condition. Actually, given a special choice of the flavor number $N_{f}$ and
rank of gauge group $N_{c}$, the BZ theory [7] could be a simple realization.
It is classified in [10] that
$L=bTr(Q^{{\dagger}}Q+\tilde{Q}^{{\dagger}}\tilde{Q})$ in the BZ theory, with
$Q_{i}$ being the chiral matter superfields. Under the large $N$ limit with
$\frac{N_{f}}{3N_{c}}=1+\epsilon$ and normalizations taken in Ref [10] , it is
found that $c_{L}=\sqrt{\frac{2}{N_{f}N_{c}}}$ and $\nu_{L}\simeq
3\epsilon^{2}$. Impose the constraint $c_{i}<<\nu_{i}$, we find
$\epsilon^{2}<<\frac{1}{N_{c}}$. Take the perturbative condition
$y\simeq\frac{c_{i}}{\nu_{i}}\lambda^{2}\simeq\frac{c_{i}\epsilon}{\nu_{i}}$
into account , we obtain $\epsilon>>\frac{1}{N_{c}}$ for consistency. So if
$\epsilon$ which can be considered as an input parameter is left to be in the
narrow window
$\displaystyle{}\frac{1}{N_{c}}<<\epsilon<<\frac{1}{\sqrt{N_{c}}}$ (4.1)
then higher-order corrections in this BZ theory arising from
$\mathcal{O}(y^{2})$ indeed dominate over $\mathcal{O}(\lambda^{4})$.
To estimate the $\mathcal{O}(y^{2})$ corrections to the matrix of anomalous
dimensions at NLO [10],
$\displaystyle{}\tau=\left(\begin{array}[]{cc}\nu_{L}\simeq
3\epsilon^{2}&-\frac{3\epsilon^{2}}{N^{2}_{c}}\\\
-\frac{4}{3}\epsilon&2\epsilon\end{array}\right)$ (4.4)
one must consider the higher-order terms in the anomalous dimensions of $Q$
and $X$, especially those unsuppressed by $1/N$. From [19] (see also [10]) we
obtain,
$\displaystyle{}\delta\gamma_{Q}(\hat{g},\lambda)=\frac{2-\epsilon}{1+\epsilon}\hat{g}^{2}+\mathcal{O}(\hat{g}^{2}/N^{2}_{c}),~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta\gamma_{X}(\hat{g},\lambda)\simeq\frac{\hat{g}\hat{\lambda}}{N_{c}^{2}}++\mathcal{O}(\hat{g}\hat{\lambda}/N^{2}_{c})$
(4.5)
where $\hat{g}=\frac{N_{c}g^{2}}{16\pi^{2}}$. Substituting (4.5) into the
$\tau$ matrix leads to correction to (4.4),
$\displaystyle{}\delta\tau=\left(\begin{array}[]{cc}-12\hat{g}^{3}_{*}&0\\\
\frac{16}{3}\epsilon^{2}&\frac{4\epsilon^{2}}{N^{2}_{c}}\end{array}\right)=\left(\begin{array}[]{cc}\mathcal{O}(\epsilon^{3})&~{}0\\\
\mathcal{O}(\epsilon^{2})&~{}\mathcal{O}(\epsilon^{3})\sim\mathcal{O}(\epsilon^{4})\end{array}\right)<<\tau$
(4.10)
by using the constraint (4.1). Unlike the situation in the previous section,
each matrix element is smaller compared with those at NLO in this case. This
suggests that the ability to affect the sign of $\delta_{min}$ coming from
$\mathcal{O}(y^{2})$ is weaker than $\mathcal{O}(\lambda^{4})$.
### 4.2 Analysis of OPE
The simple example of BZ theory in the previous discussion provides us an
intuition that the $\mathcal{O}(y^{2})$ corrections are probably negligible
under the assumptions taken by us in the setup. Now we address this issue by
analyzing the OPEs in this case. The estimate of $\mathcal{O}(y^{2})$ effects
involved the calculations of coefficients $a_{2ij}$ and $b_{2ijk}$. The
possible combinations that contribute to coefficient $b_{2ijk}$ are null due
to the fact that all of the coefficients at $\mathcal{O}(y)$ vanish in (2) and
(2) . For $a_{2ij}$, by using the results in (2) and (2) all the combinations
of operators do not contribute, which gives us
$\displaystyle{}a_{2ij}=0,~{}~{}~{}~{}~{}and~{}~{}~{}~{}b_{2ijk}=0$ (4.11)
In summary, the NNLO corrections due to $\mathcal{O}(y^{2})$ are actually
null. The statement in the previous section holds also in the region of
$c_{i}>>\nu_{i}$ (but still on the realm of perturbative field theory).
What about the higher-order terms involved $y_{i}$ couplings. The vanishing
contributions both at NLO and NNLO indicates that the contributions arising
from $y_{i}$ beyond NLO do not exist, i.e, the coefficients in powers of
$y^{n}_{i}\lambda^{m}$ ($n=2,3,\cdots$, $m=0,1,\cdots$ ) are exactly zero. In
general, these operators are related to the following OPEs,
$\displaystyle
X^{{\dagger}}(z^{-}_{1})L_{i}(x_{2},\theta_{2},\bar{\theta}_{2})$
$\displaystyle X(z^{+}_{1})L_{i}(x_{2},\theta_{2},\bar{\theta}_{2})$
$\displaystyle
L_{i}(x_{1},\theta_{1},\bar{\theta}_{1})L_{j}(x_{2},\theta_{2},\bar{\theta}_{2})$
To determine the OPEs in (LABEL:D22), we use a crucial observation in our
setup. At first, the primary operators $\mathcal{S}_{i}$ are composed of
primary operators $L_{i}$ and $X^{{\dagger}}X$ because of the interaction
mediated by $\lambda$. This implies that $\mathcal{S}_{i}$ can be generally
expressed as444We understand this expression is not exact from the viewpoint
of superconformal symmetries, but it indeed captures the main property of
scaling dimension relevance, which is the central concern of this note. Also
note that operators composed of (super)derivative over operators on the RHS of
(4.13) are not permitted.,
$\displaystyle{}\mathcal{S}_{i}(x)=\frac{\cos\alpha_{i}}{\mid
x\mid^{\tilde{\Delta}_{i}-\Delta_{i}}}L_{i}-\frac{\sin\beta_{i}}{\mid
x\mid^{\tilde{\Delta}_{i}-\Delta_{X^{{\dagger}}X}}}X^{{\dagger}}X(x)+\cdots$
(4.13)
Angle $\alpha_{i}$ and $\beta_{i}$ are introduced to represent the mixings.
Here we refer $\tilde{\Delta}_{i}$ to the scaling dimension of
$\mathcal{S}_{i}$. What are ignored in (4.13) are irrelevant for our purpose.
Define the $d_{i}$ as the OPE coefficient in three-point correlator :
$\displaystyle{}<X^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})X(z^{+}_{1},\theta_{1})\mathcal{S}_{i}(x_{3},\theta_{3},\bar{\theta}_{3})>=\frac{d_{i}}{(X^{+}_{21})^{2\Delta_{X}-\tilde{\Delta}_{i}}(X^{+}_{23})^{\tilde{\Delta}_{i}}(X^{+}_{31})^{\tilde{\Delta}_{i}}}$
(4.14)
We can subtract the OPEs in (LABEL:D22) by the OPEs of $\mathcal{S}_{i}$s.
From (4.14) we obtain the two-point OPEs:
$\displaystyle{}\mathcal{S}(x_{3},\theta_{3},\bar{\theta}_{3})X^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})$
$\displaystyle\rightarrow$
$\displaystyle\frac{d_{i}}{(X^{+}_{23})^{\tilde{\Delta}_{i}}}X^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})+\cdots$
$\displaystyle\mathcal{S}(x_{3},\theta_{3},\bar{\theta}_{3})X(z^{+}_{1},\theta_{1})$
$\displaystyle\rightarrow$
$\displaystyle\frac{d_{i}}{(X^{+}_{31})^{\tilde{\Delta}_{i}}}X(z^{+}_{1},\theta_{1})+\cdots$
(4.15)
Now we derive the OPEs in (LABEL:D22). From (4.13) we obtain,
$\displaystyle{}L_{i}\simeq\frac{\cos\alpha_{i}}{\mid
x\mid^{\Delta_{i}-\tilde{\Delta}_{i}}}\mathcal{S}_{i}(x)+\frac{\sin\beta_{i}}{\mid
x\mid^{\Delta_{i}-2}}X^{{\dagger}}X(x)+\cdots$ (4.16)
Consequently, the OPEs (LABEL:D22) can be derived in terms of (4.16), (4.14)
and (4.2),
$\displaystyle{}L_{i}(x_{1},\theta_{1},\bar{\theta}_{1})L_{j}(x_{2},\theta_{2},\bar{\theta}_{2})\rightarrow\frac{\sin\beta_{i}\sin\beta_{j}}{\mid
x_{1}\mid^{\Delta_{i}-2}\mid
x_{2}\mid^{\Delta_{j}-2}(X_{21}^{+})^{2}}X^{{\dagger}}X(x_{1},\theta_{1},\bar{\theta}_{2})+\cdots$
(4.17)
and
$\displaystyle
X^{{\dagger}}X(x_{1},\theta_{1},\bar{\theta}_{1})L_{i}(x_{2},\theta_{2},\bar{\theta}_{2})L_{i}(x_{3},\theta_{3},\bar{\theta}_{3})\rightarrow$
$\displaystyle\frac{1}{(X^{+}_{12})^{\tilde{\Delta}_{i}}(X^{+}_{31})^{\tilde{\Delta}_{i}}\mid
x_{2}\mid^{\Delta_{i}-\tilde{\Delta}_{i}}\mid
x_{3}\mid^{\Delta_{j}-\tilde{\Delta}_{i}}}\left[\cos\alpha_{i}\cos\alpha_{j}d_{i}d_{j}X^{{\dagger}}X(x_{1},\theta_{1},\bar{\theta}_{2})+\cdots\right]$
where $....$ in the second line in (LABEL:D28) refer to similar structure of
$X^{{\dagger}}X$.
Consider the coefficient $a_{3ij}$ that appears in
$a_{3ij}y_{i}y_{j}\lambda^{2}$ as an example at the next-to-NNLO. The
combinations arising from multiple $X^{{\dagger}}X$ themselves do not
contribute, with only those possibilities in (LABEL:D22) left. Substitute
(LABEL:D28) and (4.17) into the operators that contribute to
$a_{3ij}y_{i}y_{j}\lambda^{2}$ , we find that both of them vanish due to the
residual Grassmann integrals. We conclude that the claim on null contribution
coming from $y_{i}$ coupling beyond NLO still holds.
## 5 Conclusions
In this note we study the effects of NNLO corrections on the conjecture that
$\delta_{min}<0$, in the context of perturbative CFT. As we have emphasized,
despite smaller than NLO ones, the NNLO corrections are important and even
substantial in some circumstances. In particular, the modifications to the
vanishing matrix elements of anomalous dimension at NLO can directly affect
the sign of $\delta_{min}$, although they don’t substantially modify the
values of fixed points couplings $\lambda_{*}$ and $y_{i*}$.
The main results include:
1. 1.
In the region of $c_{i}<<\nu_{i}<<1$ in a $\mathcal{P}_{2}$ theory as defined
in the introduction, the bound on the anomalous dimension of $\mathcal{S}_{i}$
as conjectured in the literature is still valid at NNLO, no matter the
relative values of $\epsilon$ and $\nu_{i}$.
2. 2.
In the region of $c_{i}>>\nu_{i}$ the NNLO corrections due to
$\mathcal{O}(y^{2})$ effects are actually null. the conjecture still holds.
3. 3.
The null contribution arising from $y_{i}$ couplings beyond NLO exactly
remains.
There are a few points that deserve further investigation. For instance, one
can examine the conjecture in background of strongly coupled SCFTs via method
of ADS/CFT. Throughout this note, we have not addressed the possibility that
there are residual global symmetries after imposing the deformation, it would
be also interesting to discuss this issue in the further.
$\bf{Acknowledgement}$
We would like to thank Tianjun Li, Jia-Hui Huang, Wei-Shui Xu for
communications, and the referee for valuable suggestions. This work is
supported in part by the Fundamental Research Funds for the Central
Universities with Grant No. CDJZR11300001.
## Appendix A OPEs and $\mathcal{O}(\lambda^{4})$ Effects
In superspace , the two-point functions for
$\mathcal{O}\mathcal{O}^{{\dagger}}$, $XX^{{\dagger}}$ and three-point
function for $L\mathcal{O}\mathcal{O}^{{\dagger}}$ are given by [3, 10],
$\displaystyle{}<\mathcal{O}(z^{+}_{1},\theta_{1})\mathcal{O}^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})>$
$\displaystyle=$ $\displaystyle\frac{1}{(X^{+}_{21})^{2(2-\epsilon)}}$
$\displaystyle<X(z^{+}_{1},\theta_{1})X^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})>$
$\displaystyle=$ $\displaystyle\frac{1}{(X^{+}_{21})^{2}}$ (A.1)
$\displaystyle<\mathcal{O}(z^{+}_{1},\theta_{1})\mathcal{O}^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})L(x_{3},\theta_{3},\bar{\theta}_{3})>$
$\displaystyle=$
$\displaystyle\frac{c_{i}}{(X^{+}_{21})^{2-2\epsilon-\nu_{i}}(X^{+}_{23})^{2+\nu_{i}}(X^{+}_{31})^{2+\nu_{i}}}$
where $X^{+}_{ij}=z^{-}_{i}-z^{+}_{j}+2i\theta_{j}\sigma\bar{\theta}_{i}$ is a
supertranslation invariant interval. Here $z^{\pm}=x\pm
i\theta\sigma\bar{\theta}$. We also need the following superspace OPEs that
can be derived from (A),
$\displaystyle{}\mathcal{O}^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})\mathcal{O}(z^{+}_{1},\theta_{1})$
$\displaystyle\rightarrow$
$\displaystyle\frac{1}{(X^{+}_{21})^{2(2-\epsilon)}}+\frac{c_{i}}{(X^{+}_{21})^{2-2\epsilon-\nu_{i}}}L_{i}+\cdots$
$\displaystyle
X^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})X(z^{+}_{1},\theta_{1})$
$\displaystyle=$
$\displaystyle\frac{1}{(X^{+}_{21})^{2}}+X^{{\dagger}}X(x_{1},\theta_{1},\bar{\theta}_{2})+\cdots$
(A.2)
and
$\displaystyle{}L(x_{3},\theta_{3},\bar{\theta}_{3})\mathcal{O}^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})$
$\displaystyle=$
$\displaystyle\frac{c_{i}}{(X^{+}_{23})^{2+\nu_{i}}}\mathcal{O}^{{\dagger}}(z^{-}_{2},\bar{\theta}_{2})+\cdots$
$\displaystyle
L(x_{3},\theta_{3},\bar{\theta}_{3})\mathcal{O}(z^{+}_{1},\theta_{1})$
$\displaystyle=$
$\displaystyle\frac{c_{i}}{(X^{+}_{31})^{2+\nu_{i}}}\mathcal{O}(z^{+}_{1},\theta_{1})+\cdots$
(A.3)
The terms ignored in (A) and (A) are superconformal descendant, which are
irrelevant for our calculations of beta function.
The terms involved in $\mathcal{O}(\lambda^{4})$ wave-function renormalization
can be read from (2) and (2),
$\displaystyle{}\frac{1}{4\pi^{2}}\int
d^{4}xd^{4}\theta(1+a_{1}\lambda^{2}+a_{2}\lambda^{4}+\cdots)X^{{\dagger}}X+\int
d^{4}xd^{4}\theta(y_{i}+b_{1i}\lambda^{2}+b_{2i}\lambda^{4}+\cdots)\lambda^{-\nu_{i}}L_{i}$
$\displaystyle+\frac{\lambda}{2\pi}\left(\int
d^{4}z^{+}_{2}d\theta^{2}_{2}\Lambda^{\epsilon}~{}\mathcal{O}X(z_{2}^{+},\theta_{2})+\int
d^{4}z^{-}_{1}d\bar{\theta}^{2}_{1}\Lambda^{\epsilon}~{}\mathcal{O}^{{\dagger}}X^{{\dagger}}(z_{1}^{-},\bar{\theta}_{1})\right)$
Evaluating (A) we obtain the counter terms of order
$\mathcal{O}(\lambda^{4})$,
$\displaystyle\lambda^{4}$ $\displaystyle\left[\frac{a_{2}}{4\pi^{2}}\int
d^{4}xd^{4}\theta
X^{{\dagger}}X(x,\theta,\bar{\theta})+b_{2i}\lambda^{-\nu_{i}}\int
d^{4}xd^{4}\theta L_{i}\right.$ (A.5) $\displaystyle+$
$\displaystyle\left.\frac{b_{2i}}{4\pi^{2}}\Lambda^{-\nu_{i}}\int
d^{4}x_{1}d^{4}x_{2}d^{4}\theta_{1}d^{4}\theta_{2}~{}X^{{\dagger}}X(x_{2},\theta_{2},\bar{\theta}_{2})L_{i}(x_{1},\theta_{1},\bar{\theta}_{1})\right.$
$\displaystyle+$ $\displaystyle\left.\frac{1}{(4\pi^{2})^{2}}\int
d^{4}x_{1}d^{4}x_{2}d^{4}\theta_{1}d^{4}\theta_{2}~{}X^{{\dagger}}X(x_{2},\theta_{2},\bar{\theta}_{2})~{}X^{{\dagger}}X(x_{1},\theta_{1},\bar{\theta}_{1})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{a_{1}}{(4\pi^{2})^{2}}\Lambda^{2\epsilon}\int
d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{4}x_{3}d^{2}\bar{\theta}_{1}d^{2}\theta_{2}d^{4}\theta_{3}~{}\mathcal{O}^{{\dagger}}X^{{\dagger}}(z_{1}^{-},\bar{\theta}_{1})~{}\mathcal{O}X(z_{2}^{+},\theta_{2})~{}X^{{\dagger}}X(x_{3},\theta_{3},\bar{\theta}_{3})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{b_{1i}}{4\pi^{2}}\Lambda^{2\epsilon-\nu_{i}}~{}\int
d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{4}x_{3}d^{2}\bar{\theta}^{2}_{1}d^{2}\theta_{2}d^{4}\theta_{3}~{}\mathcal{O}^{{\dagger}}X^{{\dagger}}(z_{1}^{-},\bar{\theta}_{1})~{}\mathcal{O}X(z_{2}^{+},\theta_{2})~{}L_{i}(x_{3},\theta_{3},\bar{\theta}_{3})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{(4\pi^{2})^{2}}\Lambda^{4\epsilon}~{}\int
d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{4}z^{-}_{3}d^{4}z^{+}_{4}d^{2}\bar{\theta}_{1}d^{2}\theta_{2}d^{2}\bar{\theta}_{3}d^{2}\theta_{4}~{}\mathcal{O}^{{\dagger}}X^{{\dagger}}(z_{1}^{-},\bar{\theta}_{1})~{}\mathcal{O}X(z_{2}^{+},\theta_{2})\right.$
$\displaystyle\times$
$\displaystyle\left.\mathcal{O}^{{\dagger}}X^{{\dagger}}(z_{3}^{-},\bar{\theta}_{3})~{}\mathcal{O}X(z_{4}^{+},\theta_{4})\right]$
which gives us,
$\displaystyle{}-b_{2i}$ $\displaystyle=$
$\displaystyle\left[\frac{a_{1}}{(4\pi^{2})^{2}}\Lambda^{2\epsilon+\nu_{i}}\int
d^{4}z^{-}_{1}d^{4}x_{3}d^{4}\theta_{3}~{}\frac{c_{i}}{(X^{+}_{32})^{2}(X^{+}_{13})^{2}(X^{+}_{12})^{2-2\epsilon-\nu_{i}}}\right.$
(A.6) $\displaystyle+$
$\displaystyle\left.\frac{b_{1i}}{4\pi^{2}}\Lambda^{2\epsilon}~{}\int
d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{2}\bar{\theta}_{1}d^{2}\theta_{2}~{}\frac{1}{(X^{+}_{12})^{6-2\epsilon}}\right.$
$\displaystyle+$ $\displaystyle
4\times\left.\frac{c_{i}}{(4\pi^{2})^{2}}\Lambda^{4\epsilon+\nu_{i}}\int\frac{d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{4}z^{-}_{3}d^{2}\theta_{2}d^{2}\bar{\theta}_{3}}{(X^{+}_{12})^{2}(X^{+}_{34})^{2}(X^{+}_{14})^{2-2\epsilon-\nu_{i}}(X^{+}_{32})^{2(2-\epsilon)}}\right]$
from the last three terms in (LABEL:A5) and OPEs given in (A). The factor 4 in
the last line in (A.6) counts the four symmetric permutations. Performing the
intergral of the first line in (A.6) gives,
$\displaystyle{}\frac{a_{1}c_{i}}{(4\pi^{2})^{2}}\Lambda^{2\epsilon+\nu_{i}}\int\frac{d^{4}X^{+}_{13}d^{4}X_{32}d^{4}\theta_{32}}{(X^{+}_{32})^{2}(X^{+}_{13})^{2}(X^{+}_{12})^{2-2\epsilon-\nu_{i}}}\equiv\frac{1}{2(\nu_{i}-2\epsilon)}a_{1}c_{i}\mathcal{P}(\nu_{i},\epsilon)$
(A.7)
with
$\displaystyle{}\mathcal{P}(\nu_{i},\epsilon)$ $\displaystyle=$
$\displaystyle\frac{(\nu_{i}-2\epsilon)}{8\pi^{4}}\Lambda^{2\epsilon+\nu_{i}}\int\frac{d^{4}X^{+}_{13}d^{4}X_{32}d^{4}\theta_{32}}{(X^{+}_{32})^{2}(X^{+}_{13})^{2}(X^{+}_{12})^{2-2\epsilon-\nu_{i}}}$
$\displaystyle=$
$\displaystyle\frac{(\nu_{i}-2\epsilon)(3-2\epsilon-\nu_{i})(2-2\epsilon-\nu_{i})}{8\pi^{4}}\Lambda^{2\epsilon+\nu_{i}}\int\frac{d^{4}X^{+}_{13}d^{4}X^{+}_{32}}{(X^{+}_{32})^{2}(X^{+}_{13})^{2}(X^{+}_{13}+X^{+}_{32})^{4-2\epsilon-\nu_{i}}}$
where we have changed the integration variables $z^{-}_{1}\rightarrow
X^{+}_{13}$, $x_{3}\rightarrow X^{+}_{32}$ ,
$\theta_{3}\rightarrow\theta_{32}$ and
$\bar{\theta}_{3}\rightarrow\bar{\theta}_{32}$, and use the equality
$X^{+}_{12}=X^{+}_{13}+X^{+}_{32}+2i\theta_{32}\sigma\bar{\theta}_{32}$ . The
second integral in (A.6) is equal to,
$\displaystyle{}\frac{b_{1i}}{4\pi^{2}}\Lambda^{2\epsilon}~{}\int\frac{d^{4}X^{+}_{12}d^{4}X^{+}_{23}d^{2}\bar{\theta}_{12}d^{2}\theta_{23}}{(X^{+}_{12})^{6-2\epsilon}}=0$
(A.9)
after we are free to change the integral variables $z^{-}_{1}\rightarrow
X^{+}_{12}$, $z^{+}_{2}\rightarrow X^{+}_{23}$,
$\bar{\theta}_{1}\rightarrow\bar{\theta}_{12}$ and
$\theta_{2}\rightarrow\theta_{23}$. The last integral in (A.6) can be
reexpressed as,
$\displaystyle{}4\times\frac{c_{i}}{(4\pi^{2})^{2}}\Lambda^{4\epsilon+\nu_{i}}\int\frac{d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{4}z^{-}_{3}d^{2}\theta_{2}d^{2}\bar{\theta}_{3}}{(X^{+}_{12})^{2}(X^{+}_{34})^{2}(X^{+}_{14})^{2-2\epsilon-\nu_{i}}(X^{+}_{32})^{2(2-\epsilon)}}\equiv\frac{\pi^{2}c_{i}}{2\epsilon(\nu_{i}-2\epsilon)}\mathcal{Q}(\nu_{i},\epsilon)$
(A.10)
with
$\displaystyle{}\mathcal{Q}(\nu_{i},\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\epsilon(\nu_{i}-2\epsilon)}{2\pi^{6}}\Lambda^{4\epsilon+\nu_{i}}\int\frac{d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{4}z^{-}_{3}d^{2}\theta_{2}d^{2}\bar{\theta}_{3}}{(X^{+}_{12})^{2}(X^{+}_{34})^{2}(X^{+}_{14})^{2-2\epsilon-\nu_{i}}(X^{+}_{32})^{2(2-\epsilon)}}$
(A.11) $\displaystyle=$
$\displaystyle\frac{\epsilon(\nu_{i}-2\epsilon)}{2\pi^{6}}\Lambda^{4\epsilon+\nu_{i}}\int\frac{d^{4}X^{+}_{12}d^{4}X^{+}_{32}d^{4}X^{+}_{34}d^{2}\theta_{42}d^{2}\bar{\theta}_{13}}{(X^{+}_{12})^{2}(X^{+}_{34})^{2}(X^{+}_{12}-X^{+}_{32}+X^{+}_{34}+2i\theta_{42}\sigma\bar{\theta}_{13})^{2-2\epsilon-\nu_{i}}(X^{+}_{32})^{2(2-\epsilon)}}$
$\displaystyle=$
$\displaystyle\frac{\epsilon(\nu_{i}-2\epsilon)(3-2\epsilon-\nu_{i})(2-2\epsilon-\nu_{i})}{2\pi^{6}}\Lambda^{4\epsilon+\nu_{i}}$
$\displaystyle\times$
$\displaystyle\int\frac{d^{4}X^{+}_{12}d^{4}X^{+}_{32}d^{4}X^{+}_{34}}{(X^{+}_{12})^{2}(X^{+}_{34})^{2}(X^{+}_{12}-X^{+}_{32}+X^{+}_{34})^{4-2\epsilon-\nu_{i}}(X^{+}_{32})^{2(2-\epsilon)}}$
after we change the integral variables $z^{-}_{1}\rightarrow X^{+}_{12}$,
$z^{+}_{2}\rightarrow-X^{+}_{32}$, $z^{-}_{3}\rightarrow X^{+}_{34}$,
$\theta_{2}\rightarrow-\theta_{42}$ and
$\bar{\theta}_{3}\rightarrow-\bar{\theta}_{13}$ and use the equality
$X^{+}_{14}=X^{+}_{12}-X^{+}_{32}+X^{+}_{34}+2i\theta_{42}\sigma\bar{\theta}_{13}$.
Collect the results in (A.10), (A.9)and (A.7), we have the final result about
$b_{2i}$,
$\displaystyle{}b_{2i}=-\frac{\pi^{2}c_{i}}{2\epsilon(\nu_{i}-2\epsilon)}\left[\mathcal{P}(\nu_{i},\epsilon)+\mathcal{Q}(\nu_{i},\epsilon)\right]$
(A.12)
Similarly, the methods can be applied to calculating $a_{2}$ in (LABEL:A5).
Doing so gives us the final result of $a_{2}$,
$\displaystyle{}-\frac{a_{2}}{4\pi^{2}}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{(4\pi^{2})^{2}}\Lambda^{2\epsilon}~{}\left[\left(\int\frac{d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{2}\theta_{2}d^{2}\bar{\theta}_{3}}{(X^{+}_{32})^{2}(X^{+}_{21})^{2(2-\epsilon)}}+permutations\right)+\int\frac{d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{2}\bar{\theta}_{1}d^{2}\theta_{2}}{(X^{+}_{21})^{6-2\epsilon}}\right]$
(A.13) $\displaystyle+$
$\displaystyle\frac{b_{1i}c_{i}}{4\pi^{2}}\Lambda^{2\epsilon-\nu_{i}}~{}\int\frac{d^{4}z^{-}_{1}d^{4}x_{3}d^{4}\theta_{3}}{(X^{+}_{12})^{2-2\epsilon-\nu_{i}}(X^{+}_{13})^{2+\nu_{i}}(X^{+}_{32})^{2+\nu_{i}}}$
$\displaystyle+$ $\displaystyle
4\times\frac{1}{(4\pi^{2})^{2}}\Lambda^{4\epsilon}~{}\int\frac{d^{4}z^{-}_{1}d^{4}z^{-}_{3}d^{4}z^{+}_{4}d^{2}\bar{\theta}_{3}d^{2}\theta_{4}}{(X^{+}_{32})^{2(2-\epsilon)}(X^{+}_{14})^{2(2-\epsilon)}(X^{+}_{34})^{2}}$
The first integral in the first line of (A.13)do not contributes, while the
second integral is the similar to (A.9),
$\displaystyle{}\frac{a_{1}}{(4\pi^{2})^{2}}\Lambda^{2\epsilon}~{}\int\frac{d^{4}z^{-}_{1}d^{4}z^{+}_{2}d^{2}\bar{\theta}_{1}d^{2}\theta_{2}}{(X^{+}_{12})^{6-2\epsilon}}=0$
(A.14)
The second one can be simplifed by introducing the
$\mathcal{I}(\nu_{i},\epsilon)$ function as in [10], which results in,
$\displaystyle{}b_{1i}c_{i}\Lambda^{2\epsilon-\nu_{i}}~{}\int\frac{d^{4}z^{-}_{1}d^{4}x_{3}d^{4}\theta_{3}}{(X^{+}_{12})^{2-2\epsilon-\nu_{i}}(X^{+}_{13})^{2+\nu_{i}}(X^{+}_{32})^{2+\nu_{i}}}\equiv-8\pi^{4}\frac{b_{1i}c_{i}}{\nu_{i}-2\epsilon}\mathcal{I}(\nu_{i},\epsilon)$
(A.15)
with
$\displaystyle{}\mathcal{I}(\nu_{i},\epsilon)=-\frac{(\nu_{i}-2\epsilon)(3-2\epsilon-\nu_{i})(2-2\epsilon-\nu_{i})}{8\pi^{4}}\Lambda^{2\epsilon-\nu_{i}}\int\frac{d^{4}X^{+}_{23}d^{4}X^{+}_{31}}{(X^{+}_{23}+X^{+}_{31})^{4-2\epsilon-2\nu_{i}}(X^{+}_{23})^{2+\nu_{i}}(X^{+}_{31})^{2+\nu_{i}}}$
The last integral in (A.13)
$\displaystyle{}4$ $\displaystyle\times$
$\displaystyle\frac{1}{(4\pi^{2})^{2}}\Lambda^{4\epsilon}~{}\int\frac{d^{4}z^{-}_{1}d^{4}z^{-}_{3}d^{4}z^{+}_{4}d^{2}\bar{\theta}_{3}d^{2}\theta_{4}}{(X^{+}_{32})^{2(2-\epsilon)}(X^{+}_{14})^{2(2-\epsilon)}(X^{+}_{34})^{2}}$
$\displaystyle=$ $\displaystyle
4\times\frac{1}{(4\pi^{2})^{2}}\Lambda^{4\epsilon}~{}\int\frac{d^{4}X^{+}_{12}d^{4}X^{+}_{34}d^{4}X^{+}_{14}d^{2}\bar{\theta}_{13}d^{2}\theta_{42}}{(X^{+}_{34}-X^{+}_{14}+X^{+}_{12}+2i\theta_{42}\sigma\bar{\theta}_{13})^{2(2-\epsilon)}(X^{+}_{14})^{4-2\epsilon}(X^{+}_{34})^{2}}\equiv\frac{8\pi^{6}}{(4\pi^{2})^{2}\epsilon^{2}}\mathcal{T}(\epsilon)$
with
$\displaystyle{}\mathcal{T}(\epsilon)=\frac{\epsilon^{2}(-2+2\epsilon)(-5+2\epsilon)}{2\pi^{6}}\Lambda^{4\epsilon}\int\frac{d^{4}X^{+}_{12}d^{4}X^{+}_{34}d^{4}X^{+}_{14}}{(X^{+}_{34}-X^{+}_{14}+X^{+}_{12})^{6-2\epsilon}(X^{+}_{34})^{2}(X^{+}_{14})^{2(2-\epsilon)}}$
(A.18)
after we change the integral variables $z^{-}_{1}\rightarrow X^{+}_{12}$,
$z^{-}_{3}\rightarrow X^{+}_{34}$, $z^{+}_{4}\rightarrow-X^{+}_{14}$,
$\bar{\theta}_{3}\rightarrow-\bar{\theta}_{13}$ ,
$\theta_{4}\rightarrow\theta_{42}$, and use the equality
$X^{+}_{32}=X^{+}_{34}-X^{+}_{14}+X^{+}_{12}+2i\theta_{42}\sigma\bar{\theta}_{13}$.
Consequently, we get the final expression of (A.13)
$\displaystyle{}a_{2}=16\pi^{4}\left(\frac{c^{2}_{i}}{\nu^{2}_{i}-4\epsilon^{2}}\right)\mathcal{I}(\nu_{i},\epsilon)-\frac{2\pi^{2}}{\epsilon^{2}}\mathcal{T}(\epsilon)$
(A.19)
With the help of Mathematica, the functionals defined above can be evaluated.
## References
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* [10] D. Green and D. Shih, Bounds on SCFTs from Conformal Perturbation Theory, arXiv:1203.5129 [hep-th].
* [11] A. Vichi, Improved bounds for CFTs with global symmetries, arXiv:1106.4037[hep-th].
* [12] R. Rattazzi, V. S. Rychkov, E. Tonni, A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 0812 (2008) 031, arXiv:0807.0004[hep-th].
* [13] R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev. D83 (2011) 046011, arXiv:1009.2725[hep-th].
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|
arxiv-papers
| 2012-05-06T08:22:25 |
2024-09-04T02:49:30.608680
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1205.1192"
}
|
1205.1202
|
# Variational problems related to some fractional kinetic equations
H. Hajaiej
###### Abstract
We establish the existence and symmetry of all minimizers of a constrained
variational problem involving the fractional gradient. This problem is closely
connected to some fractional kinetic equations.
## 1 Introduction
Fractional calculus has gained a lot of interest during the last decade due to
its numerous applications in many fields. It appears in wave propagation,
inhomogenous porous material, geology, hydrology, dynamics of earthquakes,
bioengineering, chemical engineering signal processing, medicine,
electrochemistry, thermodynamics, neural networks, statistical physics, [2],
[6], [8] and references therein.
Fractional equations involving the fractional laplacian have also played a
crucial role in some kinetic problems, [5], [10], [11], [12], [13] and [14],
in which particular solutions are obtained by solving the following
minimization problem :
$\displaystyle(P_{c}):I_{c}$ $\displaystyle=$
$\displaystyle\inf\Big{\\{}\int_{\mathbb{R}^{N}}|-\Delta^{s/2}(u)|^{2}-\int_{\mathbb{R}^{N}}F(|x|,u):u\in
S_{c}\Big{\\}}$ $\displaystyle E(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|-\Delta^{s/2}(u)|^{2}-\int_{\mathbb{R}^{N}}F(|x|,u),$
$\displaystyle F(r,t)$ $\displaystyle=$ $\displaystyle\int^{t}_{0}f(x,p)dp,$
$\displaystyle S_{c}$ $\displaystyle=$ $\displaystyle\Big{\\{}u\in
H^{s}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}u^{2}=c^{2}\Big{\\}},$
where $c$ is a prescribed number, $0<s<1$ and $H^{s}(\mathbb{R}^{N})$ is the
usual Beso’v space , [8].
Note also that the minimization problem $(P_{c})$ appears in disperative model
equations : The generalized Benjamin-On equation, the Benjamin-Bona-Mahong
equation and the fractional nonlinear Schrödinger equation.
In this paper, we address the question of existence, radiality and radial
decreasiness of all minimizers of $(P_{c})$ for integrands $F$ satisfying some
growth conditions. This result generalizes a recent one obtained by Frank and
Lenzmann, [7], in which the authors have considered the basic power
nonlinearity $F(r,t)=\frac{|t|^{\alpha+}}{\alpha+2}$.
Moreover, they have proved that the following minimization problems
$I_{M}=\inf\left\\{\frac{1}{2}\int_{\mathbb{R}}|-\Delta^{s/2}u|^{2}+\frac{1}{\alpha+2}\int_{\mathbb{R}}|u|^{\alpha+2}:u\in
S_{\sqrt{M}}\right\\}$ $None$
and
$J^{s,\alpha}(Q)=\inf_{u\in
H^{s}(\mathbb{R})\backslash\\{0\\}}\frac{(\int_{\mathbb{R}}|-\Delta^{s/2}u|^{2})^{\alpha/4s}(\int_{\mathbb{R}}u^{2})^{\frac{\alpha}{4s}(2s-1)+1}}{\int|u|^{\alpha+2}}$
$None$
are equivalent.
They have also added that : $u$ is a solution of (1.1) if and only if
$u=e^{i\theta}\lambda^{1/\alpha}Q(\lambda^{1/2s}(.+y))$, for some
$\theta\in\mathbb{R},y\in\mathbb{R},\lambda>0$ ; $Q$ is a solution of (1.2).
Finally they have stated that (1.2) (and therefore (1.1)) only has minimizers
when $0<\alpha<\alpha_{\max}$, where $\alpha_{\max}$ is defined as follows :
$\alpha_{\max}=\left\\{\begin{array}[]{ll}\frac{4s}{1-2s},0<s<\frac{1}{2}\\\
\infty,\frac{1}{2}\leq s<1\end{array}\right.$
This result seems to be erroneous and as we will show in section 2, (1.1)
admits minimizers if and only if $\alpha<4s$.
Moreover, in this paper, we will study $(P_{c})$ for general nonlinearities
$F$ such that : $|F(r,t)|\leq K(t^{2}+|t|^{\ell+2})$ where
$0<\ell<\frac{4s}{N}$.
Our main result, Theorem 2.1, states that :
1. 1.
If $0<\ell<\frac{4s}{N}$, $(P_{c})$ admits solutions and all minimizers are
radial and radially decreasing.
2. 2.
If $\ell=\frac{4s}{N}$, $(P_{c})$ admits solutions and all minimizers are
radial and radially decreasing if $c^{2}$ is small enough (some estimates will
be given below).
3. 3.
If $\displaystyle{\liminf_{t\rightarrow\infty}}F(r,t)/t^{\ell+2}\geq A>0$ for
some $\ell>\frac{4s}{N}$, then $I_{c}=-\infty$ for all $c$.
Now before stating our main result, let us first mention that definitions and
properties of the Schwarz symmetrization are detailed in [4].
If $u\in H^{s}_{+}(\mathbb{R}^{N})=\\{u\in H^{s}(\mathbb{R}^{N}):u\geq 0\\}$,
then the fractional Polya-Szegö inequality holds true :
$|\nabla_{s}u^{\ast}|^{2}_{2}=|-\Delta^{s/2}u^{\ast}|^{2}_{2}\leq|-\Delta^{s/2}u|^{2}_{2}=|\nabla_{s}u|^{2}_{2}$
$None$
which is a direct consequence of the generalized Riesz inequality, [4], since
as it was proven in [1] :
$|\Delta^{s/2}u|^{2}_{2}=C_{n,s}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\;\frac{|u(-x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\;.$
From now on $0<s<1,H^{s}(\mathbb{R}^{N})$ is the standard Besov space. The
norm of the Lebesgue space $L^{p}(\mathbb{R}^{N})$ is denoted by $|\,|_{p}$ .
$c$ is a prescribed number and $N\in\mathbb{N}^{\ast}$. In an integral where
no domain is given, it is to be understood that it extends on
$\mathbb{R}^{N}$.
## 2 Main Result
Theorem 2.1 Suppose that :
$(F_{0})F:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory
function :
$\bullet\quad F(.,t):[0,\infty)\rightarrow\mathbb{R}$ is measurable in
$\mathbb{R}_{+}\backslash\Gamma$ for all $t\in\mathbb{R}$, where $\Gamma$ is a
subset of $\mathbb{R}_{+}$ having one dimensional measure zero and ;
$\bullet\quad F(r,.):\mathbb{R}\rightarrow\mathbb{R}$ is continuous for every
$r\in[0,\infty)\backslash\Gamma$.
$(F_{1})\quad F(r,t)\leq F(r,|t|)$ for a.e $r\geq 0$ and every
$t\in\mathbb{R}$,
$(F_{2})$ For a.e $r\geq 0$ and every $t\geq 0$.
$0\leq F(r,t)\leq K(t^{2}+t^{\ell+2}),\mbox{ where }K>0\mbox{ and
}0<\ell<\frac{4s}{N}.$
$(F_{3})$ For every $\varepsilon>0$, there exist $R_{0}>0$ and $t_{0}>0$ such
that $F(r,t)\leq\varepsilon t^{2}$ for a.e $r\geq R_{0}$ and $0\leq t\leq
t_{0}$.
$(F_{4})(v,y)\rightarrow F(\frac{1}{v},y)$ is supermodular on
$\mathbb{R}_{+}\times\mathbb{R}_{+}$, i.e
$F(r,a)+F(R,A)\geq F(r,A)+F(R,a)$
for every $0\leq r<R$ and $0\leq a<A$.
Let $(\widetilde{P}_{c}):\inf\\{E(u):u\in H^{s}_{+}(\mathbb{R}^{N})$ and
$\displaystyle{\int_{\mathbb{R}^{N}}}u^{2}\leq c^{2}\\}=\widetilde{I}_{c}$
$(I)\quad\widetilde{I}_{c}<\widetilde{I}_{d}$ for $d<c$.
Then
1. 1.
$(P_{c})$ admits a Schwarz symmetric minimizer for any $c$. Moreover if
$(F_{4})$ holds with a strict sign, then for any $c$, all minimizers of
$(P_{c})$ are radial and radially decreasing, i.e, Schwarz symmetric.
2. 2.
If $\ell=\frac{4s}{N}$, then 1) holds true if and only if $c$ is small enough.
3. 3.
If $\displaystyle{\liminf_{t\rightarrow\infty}}F(r,t)/t^{\ell}=A>0$, with
$\ell>\frac{4s}{N}$ then $I_{c}=-\infty$.
Proof of 1 Fix $c$
Step 1 : $(P_{c})$ is well posed $(I_{c}>-\infty$ and all minimizing sequences
are bounded in $H^{s}(\mathbb{R}^{N}))$.
By $(F_{1})$ and $(F_{2})$, we can write :
$\begin{array}[]{ll}\int
F(|x|,u(x))dx&\leq\displaystyle{\int}F(|x|,|u(x)|)dx\\\ &\leq
Kc^{2}+K\displaystyle{\int}|u(u(x)|^{\ell+2}dx\end{array}$ $None$
Now using the fractional Gagliardo-Nirenberg inequality [8], [9], it follows
that :
$|u|_{\ell+2}\leq
K^{\prime}|u|^{1-\theta}_{2}|\nabla_{s}u|^{\theta}_{2}\;\quad|\nabla_{s}u|_{2}=\left(\int|-\Delta^{s/2}u|^{2}\right)^{1/2}\mbox{
and }\theta=\frac{N\ell}{2s(\ell+2)}.$
Thus
$\int|u(x)|^{\ell+2}dx\leq K^{\prime}\\{\int
u^{2}(x)dx\\}^{(1-\theta)(\ell+2)/2}|\nabla_{s}u|^{\theta(\ell+2)}_{2}.$
$None$
Therefore using Young inequality, we have
$\\{\displaystyle{\int}u^{2}(x)dx\\}^{(1-\theta)\frac{\ell+2}{2}}|\nabla_{s}u|_{2}^{\theta(\ell+2)}\leq$
$\frac{1}{p}\varepsilon^{p}\\{|\nabla_{s}u|^{2}_{2}\\}^{p\theta(\ell+2)/2}+\frac{1}{q\varepsilon^{q}}\left\\{\int
u^{2}(x)dx\right\\}^{q(1-\theta)\frac{(\ell+2)}{2}}$ $None$
for any $\varepsilon>0$ and $p>1$ where $\frac{1}{p}+\frac{1}{q}=1$.
Choosing $p=\displaystyle{\frac{2}{\theta(\ell+2)}=\frac{4s}{N\ell}}$, we get
:
$\displaystyle{\int}|u(x)|^{\ell+2}dx\leq\displaystyle{\frac{K^{\prime}}{p}}\varepsilon^{p}\\{|\nabla_{s}u|^{2}_{2}\\}+\frac{K^{\prime}}{q\varepsilon^{q}}\\{\displaystyle{\int}u^{2}(x)dx\\}^{\frac{q(1-\theta)(\ell+2)}{2}}$
$=\frac{K^{\prime}}{p}\varepsilon^{p}|\nabla_{s}u|^{2}_{2}+\frac{K^{\prime}}{q\varepsilon^{q}}c^{q(1-\theta)(\ell+2)}$
$None$
for any $u\in S_{c}$.
Therefore :
$\displaystyle E(u)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}|\nabla_{s}u|^{2}_{2}-Kc^{2}-K^{\prime}K\varepsilon^{p}|\nabla_{s}u|^{2}_{2}-\frac{KK^{\prime}}{q\varepsilon^{q}}c^{q(1-\theta)(\ell+2)}$
$\displaystyle=$
$\displaystyle(\frac{1}{2}-\frac{KK^{\prime}}{p}\varepsilon^{p})|\nabla_{s}u|^{2}_{2}-Kc^{2}-\frac{KK^{\prime}}{q\varepsilon^{q}}c^{q(1-\theta)(\ell+2)}.$
Thus $I_{c}>-\infty$ and all minimizing sequences are bounded in
$H^{s}(\mathbb{R}^{N})$.
Step 2 : Existence of a Schwarz symmetric minimizing sequence.
First note that if $u\in H^{s}(\mathbb{R}^{N})$ then $|u|\in
H^{s}(\mathbb{R}^{N})$.
Now by $(F_{1})$, we certainly have :
$E(|u|)\leq E(u)\quad\forall\;u\in H^{s}(\mathbb{R}^{N}).$
Now by (1.3), we know that
$|\nabla_{s}|u|^{\ast}|_{2}\leq|\nabla_{s}|u||_{2}$, and using Theorem 1 of
[4], we have :
$\int F(|x|,|u(x)|)dx\leq\int F(|x|,|u(x)|^{\ast})dx$
and
$\int u^{2}=\int(u^{\ast})^{2}.$
Thus without loss of generality, $(P_{c})$ always admits a Schwarz symmetric
minimizing sequence.
Let $(u_{n})$ be a Schwarz symmetric minimizing sequence of $(P_{c})$ for a
fixed $c$.
Step 3 : Let $(u_{n})=(u^{\ast}_{n})$ be a Schwarz symmetric minimizing
sequence then if $u_{n}$ converges weakly to $u$ $\Rightarrow E(u)\leq\lim\inf
E(u_{n})$.
Proof $|\nabla_{s}u|_{2}\leq\liminf|\nabla_{s}u_{n}|_{2}$ by the weak lower
semi-continuity of $\|\;\|_{2}$ of the fractional gradient in
$H^{s}(\mathbb{R}^{N})$.
Let us prove now that :
$\lim_{n\rightarrow\infty}\int F(|x|,u_{n}(x)dx=\int F(|x|,u(x)).$
Let $R>0$, let us first prove that :
$\lim_{n\rightarrow+\infty}\int_{|x|\leq R}F(|x|,u_{n}(x))dx=\int_{|x|\leq
R}F(|x|,u(x))dx.$
Since $u_{n}$ converges weakly to $u$ in $H^{s}(\mathbb{R}^{N})$, it converges
strongly to $u$ in $L^{\ell+2}(|x|\leq R)$. Thus there exists a subsequence
$(u_{n_{k}})$ of $(u_{n})$ such that $u_{n_{k}}\rightarrow u$ a.e in
$L^{2}(B(0,R))$ and $|u_{n_{k}}|\leq h$ with $h\in L^{\ell+2}(|x|\leq
R).\\{B(0,R)=\\{x\in\mathbb{R}^{N}:|x|\leq R\\}$
Now by $(F_{2}):F(|x|,u_{n_{k}}(x))\leq K(h^{2}(x)+h^{\ell+2}(x))$.
Noticing that $h^{2}+h^{\ell+2}\in L^{1}(|x|\leq R)$, we get thanks to the
dominated convergence theorem :
$\lim_{n\rightarrow+\infty}\int_{|x|\leq R}F(|x|,u_{n}(x))dx=\int_{|x|\leq
R}F(|x|,u(x))dx.$
Let us prove now that
$\displaystyle{\lim_{R\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{|x|>R}}F(|x|,u(x))dx=0$.
Let $n\in\mathbb{N}$, since $(u_{n})=(u^{\ast}_{n})$, we have that
$V_{N}|x|^{N}u^{2}_{n}(x)\leq\int_{|y|\leq|x|}u^{2}_{n}(y)dy\leq c^{2}.$
Thus
$u_{n}(x)\leq\frac{c}{V_{N}^{1/2}|x|^{N/2}}\leq\frac{c}{V_{N}^{1/2}R^{N/2}}\quad\forall\;|x|>R.$
Now let $\varepsilon>0$ and $R$ big enough, we obtain thanks to $(F_{3})$ that
:
$\int_{|x|>R}F(|x|,u_{n}(|x|)dx\leq\varepsilon\int_{|x|>R}u^{2}_{n}(x)dx<\varepsilon
c^{2},$
proving that
$\lim_{R\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{|x|>R}F(|x|,u_{n}(x))dx=0.$
But $u$ inherits all the properties of the sequence $(u_{n})$ used to get the
above limit, then it follows that :
$\lim_{R\rightarrow\infty}\int_{|x|>R}F(|x|,u(x))dx=0.$
Step 4 : $I_{c}$ is achieved.
Denoting $v$ the weak limit of a Schwarz minimizing sequence of
$(\widetilde{P}_{c})$. We certainly have, using previous steps, that
$E(v)\leq\liminf E(u_{n}),$
where
$\lim_{n\rightarrow\infty}E(u_{n})=\widetilde{I}_{c}.$
On the other hand $|v|^{2}_{2}=d^{2}\leq c^{2}$.
It follows then by hypothesis $(I)$, that :
$\widetilde{I}_{c}<\widetilde{I}_{d}\leq E(v)\leq\widetilde{I}_{c}$
which is impossible, then $|v|^{2}_{2}=c^{2}$. Suppose that
$|v|^{2}_{2}=d^{2}<c^{2}$.
Therefore $I_{c}\leq E(v)=\widetilde{I}_{c}\leq I_{c}$, proving that $(P_{c})$
is achieved by $v=v^{\ast}$ a.e.
Now to show that all minimizers of $(P_{c})$ are Schwarz symmetric, it is
sufficient to notice that if $(F_{4})$ holds with a strict sign then it
follows by Theorem 1 of [4] that
$E(u^{\ast})<E(u)\quad\mbox{ for any }u\in H^{s}_{t}(\mathbb{R}^{N})$
and the result follows.
Proof of 2) If $\ell=\displaystyle{\frac{4}{Ns}}$, (2.2) becomes :
$\int|u(x)|^{\ell+2}dx\leq
K^{\prime}c^{4/N}|\nabla_{s}u|^{2}_{2}\quad\forall\;u\in S_{c}.$
Hence
$\displaystyle E(u)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}|\nabla_{s}u|^{2}_{2}-Kc^{2}-KK^{\prime}c^{4/N}|\nabla_{s}u|^{2}_{2}$
$\displaystyle=$
$\displaystyle(\frac{1}{2}-KK^{\prime}c^{4/N})|\nabla_{s}u|^{2}_{2}-Kc^{2}.$
Thus $I_{c}>-\infty$ and all minimizing sequences are bounded in
$H^{s}(\mathbb{R}^{N})$ provided that $0<c<(\frac{1}{2KK^{\prime}})^{4/N}$.
Then previous steps (2,3 and 4) apply to $c$ such as in the latter interval.
Proof of 3) : It suffices to consider $u\in S_{c}$ and
$u_{\lambda}(x)=\lambda^{N/2}u(\lambda.)(\in S_{c})$, then the results follow
when $\lambda$ tends to infinity.
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A. Burchard, H Hajaiej: Rearrangement inequalities for functional with
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A. Mellet, S. Mischler, C. Mouhot : Fractional diffusion limit for collisional
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|
arxiv-papers
| 2012-05-06T11:09:41 |
2024-09-04T02:49:30.617940
|
{
"license": "Public Domain",
"authors": "H. Hajaiej",
"submitter": "Hichem Hajaiej",
"url": "https://arxiv.org/abs/1205.1202"
}
|
1205.1203
|
# Hierarchical Range Sectoring and Bidirectional Link Quality Estimation for
On-demand Collections in WSNs
Víctor Valls, José Luis Sánchez, Cristina Cano, Boris Bellalta, Miquel Oliver
Department of Information Technologies and Communications
Universitat Pompeu Fabra
Corresponding author: victor.valls@upf.edu
###### Abstract
The paper presents two mechanisms for designing an on-demand, reliable and
efficient collection protocol for Wireless Sensor Networks. The former is the
Bidirectional Link Quality Estimation, which allows nodes to easily and
quickly compute the quality of a link between a pair of nodes. The latter,
Hierarchical Range Sectoring, organizes sensors in different sectors based on
their location within the network. Based on this organization, nodes from each
sector are coordinated to transmit in specific periods of time to reduce the
hidden terminal problem. To evaluate these two mechanisms, a protocol called
HBCP (Hierarchical-Based Collection Protocol), that implements both
mechanisms, has been implemented in TinyOS 2.1, and evaluated in a testbed
using TelosB motes. The results show that the HBCP protocol is able to achieve
a very high reliability, especially in large networks and in scenarios with
bottlenecks.
## 1 Introduction
Wireless Sensor Networks (WSNs) are formed by small devices that can be
located in a wide range of scenarios, and usually in non-easily accessible
places: underground for pressure measurements, throughout the forest to detect
fires or inside buildings to take environmental measures, among many others.
Today’s trend to monitor everything, everywhere and at any time leads to a
wide range of WSNs applications, that have multiple and diverse requirements
in terms of reliability, scalability and network life-time [1].
Sensors are basically composed by a micro-controller (MCU), a transceiver to
communicate with other devices, a sensorboard to sense environmental data and
a battery to power the whole device. There are many different types of sensor
platforms, composed by different hardware components, but all of them have in
common the fact that they need to be powered by a battery of limited capacity.
Due to this factor, and the difficulty to access the sensors location to
replace the battery, it is essential to find mechanisms that maximize its
life-time while still achieving the required network performance.
There is an extensive literature regarding the maximization of the network
life-time, from MAC protocols like the B-MAC [2] or S-MAC [3] that try to
reduce the energy consumption by allowing sensor nodes to go to sleep and
wake-up periodically, to network protocols as LEACH [4], which has the aim to
balance the energy consumption during data collections to maximize the life-
time of the whole network. However, the performance of a given protocol varies
considerably depending on the type of topology and the application the network
has been deployed for. In general, there is no protocol stack that perfectly
fits all the scenarios and applications, and the election directly depends on
the network purpose and the environment characteristics.
Regarding data collection protocols for WSNs, they can be classified into two
groups based on the trigger point of view [5]. On the one hand, the ones where
nodes individually decide whether to send data based on an environmental
lecture, and on the other hand, on-demand protocols, like the Direct Diffusion
[6], where the sink triggers a query to collect data. The first set of
protocols are suitable for monitoring applications, where the others are
better, among others, for metering-like applications that have low collection
periodicity. In the latter case, the nodes only need to be active during a
short period of time, few minutes every month, and then they can remain in a
low-consumption state the time they do not need to be active. In this kind of
WSNs, it is possible to have large network life-times in the order of several
years with standard batteries.
This paper focuses on a scenario where the sink triggers a query to collect
one reading from each sensor in a network, i.e. on-demand collections. We
present two new mechanisms called Hierarchical Range Sectoring (HRS) and
Bidirectional Link Quality Estimation (BLQE). In addition, a new protocol
called Hierarchical-Based Collection Protocol (HBCP) is defined, which apart
from these two new mechanisms, also includes all the required functionalities
to perform efficient on-demand data collections in WSNs. The HBCP design
requirements are: $1$) collections must be carried out as fast as possible to
minimize the time that the network is awake and thus, save energy, $2$) it has
to be able to operate satisfactorily in different topologies and with
different workloads, independently of the network or application requirements
and $3$) a high next hop and end-to-end delivery rate is required (i.e., close
to $99$%). To assess the performance of the HBCP protocol, and of the HRS and
BLQE mechanisms in particular, the HBCP has been implemented in TinyOS 2.1 and
experimentally evaluated in a testbed.
The paper is divided as follows. Firstly, Section 2 describes the main
problems that WSNs suffer for the type of network we focus on. Secondly,
Section 3 presents a general description of the HRS and BLQE mechanisms, and
afterward, in Section 4, it is described in detail how the protocol works
throughout all the collection process. Section 5 presents how the HBCP has
been implemented in TinyOS 2.1, and in Section 6 the testbed evaluation and
the main results are described. Finally, Section 7 compares the HBCP with
other existing protocols, and Section 8 concludes the article.
## 2 Open Challenges for an On-demand Data Collection Protocol for WSNs
In this section the main challenges that an on-demand data collection protocol
for WSNs has to successfully handle are described. These are basically the
energy consumption and the scalability. In addition, for on-demand multihop
data collections that aim to perform the collection as fast as possible and
with minimum energy expenditure, the hidden terminal problem needs to be taken
into account as it may significantly harm the network reliability.
### 2.1 Energy Consumption
The sensor’s components that affect the most the battery consumption are the
MCU and the transceiver, and depending on the used platform, which defines
these components, the overall consumption may vary considerably. These two
components are key for the correct operation of the sensor, despite they do
not need to be active if they do not have a task to perform. Some platforms
have the possibility to turn off the radio and put the MCU into sleep mode in
order to save energy, so the nodes only wake up when they have some task to
perform. Moreover, the faster they do their task, the sooner they can go back
to sleep and save energy. However, determining whether a node has to be active
is not trivial, especially if they belong to a network where nodes have to
forward data from other nodes.
Furthermore, regarding the energy consumption of the radio, the behavior of
one node directly affects the life-time of the neighboring nodes. In some of
the current transceivers for WSNs, like the CC2420 [7], the energy consumed
when receiving a byte is equal or even a slightly higher than transmitting
one. Because of this, and the fact that for every transmitted packet the
neighboring nodes will overhear it, it is essential to reduce the number of
packets sent, especially when the density of the nodes is high.
### 2.2 Scalability
Scalability is another problem that multihop WSNs face, and it can appear due
to the number of hops to reach the sink or due to the density of nodes. Both
of them directly affect the network reliability, as the number of hops grows
the probability of an intermediary link break down increases, and as a result
the delivery probability drops to 0 ($\lim_{h\to\infty}(1-p)^{h}=0$ where
$(1-p)$ is the probability of a successful transmission and _h_ the number of
hops). Either way with the density of nodes, the larger number of nodes in a
certain area, the higher the probability of collisions. To check if the medium
is free to transmit is responsibility of the MAC layer, however, to base the
decision to transmit on carrier sensing does not necessarily mean that the
medium is idle at reception, due to the hidden terminal problem.
### 2.3 Multihop Communications and Hidden Terminals
The hidden terminal problem can be divided in two cases: 1) when nodes want to
send data to the same node and they cannot sense each other, and 2) when the
transmission of one node indirectly affects the on-going transmission between
two other nodes. With random medium access protocols, both cases have no
solution despite being mechanisms like the RTS-CTS [8] that try to minimize
the impact of collisions. The main difference between the two cases, is that
applying a time scheduling to alleviate this problem in the second case is
less complex than the one needed for the first case. In the first case each
node must be individually scheduled, while in the second case it is possible
to create a schedule for groups of nodes. In Section 3.1 is explained how
nodes can be isolated in _Sectors_ in order to reduce this kind of collisions.
## 3 HBCP Mechanisms
In the following sections, the mechanisms that the HBCP implements are
described. The two main mechanisms are the Hierarchical Range Sectoring and
the Bidirectional Link Quality Estimator which define the sectors and the
datapaths respectively. The other presented mechanisms, Randomized
Transmissions and Data Aggregation, are required to implement an efficient on-
demand data collection protocol for WSNs.
### 3.1 Hierarchical Range Sectoring (HRS)
HRS consists of grouping nodes in sectors according to their minimum number of
hops to reach the sink with good quality links (see Section 3.2), with the aim
to allow nodes to transmit simultaneously with a low collision probability. To
better understand the concept of _Sectoring_ see the following example: if a
node sends a broadcast packet at a fixed transmission power, all the nodes
that receive that packet with enough quality are at distance 1 from the
initial node. Then, if each of those nodes also send a broadcast packet, the
new nodes that receive those packets will be classified as distance 2; and so
on for the following sectors. This assures that generally, if the medium
conditions do not change, non-continuous sectors would not be able to sense
each other.
Figure 1: Network divided into different sectors, where two nodes belonging to
different sectors can transmit without interfering among them.
To have the network divided into sectors allows us to coordinate the
transmissions in order to reduce the hidden terminal problem explained in
Section 2.3. With this information, nodes can transmit for a given period of
time and do not interfere with the transmission of other sectors. In order to
allow multiple sectors to transmit at the same time, these must have at least
a distance of 3 between sectors [9][10]. Therefore, if the sector $n$ is
transmitting to the sectors $n\pm 1$ no other sector that interferes into
these sectors is allowed to transmit. However, meanwhile it would not affect
that nodes in the sector $n+3$ transmit, because they will just interfere to
the sectors $n+2$ and $n+4$. As an example refer to Figure 1, where it can be
observed that if nodes in sector 3 and 6 transmit at the same time no
collision can happen among nodes of these sectors. Whereas, if nodes located
in the sector 3 and 5 transmit simultaneously, the hidden terminal problem can
happen in the forth sector.
Notice that HRS is not able to completely eliminate all hidden terminal
problems in the network. There can be hidden terminals belonging to the same
sector $n$ and therefore, they can collide when sending data to the sector
$n-1$. Nevertheless, as it will be shown in Section 6.2, dividing the network
in sectors increases the end-to-end reliability as hidden terminal collisions
are reduced.
### 3.2 Bidirectional Link Quality Estimation (BLQE)
For the HRS is important to determine the quality of the links, and choose the
link with the greatest transmission success probability. In WSNs, most of the
routing protocols [9] [11] [12] prefer the LQI (Link Quality Indicator) over
the RSSI (Received Signal Strength Indicator) to estimate the quality of the
links because the LQI takes into account previous transmissions and computes
the successful transmission probability per link. Therefore, as the LQI
depends on the time and on the number of packets sent, and given that we focus
on a scenario where two collections are utterly independent, there is no worth
in using the LQI over the RSSI. Moreover, as it is stated in [13], the RSSI
has been under-appreciated even for values which it has shown to be very
reliable. For instance, for the CC2420 chip a RSSI of $-87$ dBm assures a
Packet Reception Rate (PRR) of $85\%$ [13]. The other region, from $-87$ dBm
to $-94$ dBm, is called the gray area where the PRR varies radically. However,
the RSSI is not a good indicator of link burstiness [14] and its degradation
depending on the antenna may not follow a polynomial function of distance
[15], which has to be taken into account when collecting the data.
Regarding the link symmetry, and although it is well known the asymmetry of
wireless links [16] [17], it has been shown in experimental evaluations that
in some radios like the CC2420, the links are very symmetric [13]. To extend
those results, we evaluated the link symmetry variation ($\gamma_{v}$) between
two TelosB [18] (Figure 2). This evaluation is done in an indoor environment
for 12 hours, and shows that the link has less than $1$ dBm of difference on
average between two nodes, and a total correlated fluctuation of nearly $3$
dBm.
Figure 2: Link symmetry evaluation between two TelosB
Based on this evidence, the link quality in BLQE is calculated by evaluating
if the RSSI belongs to a gray area taking into account both directions. This
is done with two parameters: the gray area threshold ($\gamma_{ga}$) which
defines if a link is in the gray area (useless to achieve a high packet
delivery rate), and the quality threshold ($\gamma_{q}$) which assures that
the link is not in the gray area for both directions. These thresholds
directly depend on the transceiver used. The ($\gamma_{q}$) is calculated as
follows:
$\gamma_{q}$ = $\gamma_{ga}$ $-\gamma_{v}$
The selection of the $\gamma_{v}$ value is not straightforward, so it depends
on several factors: the environmental conditions of the network, the
transceiver, the transmission power and the density of nodes. Additionally, in
dynamic wireless environments this parameter should be adapted depending on
the varying channel conditions. As a result, the estimation of this parameter
at each node would lead to an increased complexity of the protocol and energy
consumption. Therefore, in this work it has been considered to fix this value
for all nodes in the network to 5 dB.
In Section 3.1 we have fixed a distance of 3 between sectors to avoid
collisions with hidden nodes, however, this value is directly influenced by
the node platform and the transmission power. As it is shown in [15], where an
evaluation of the RSSI behavior of the CC2420 transceiver is provided, the
CC2420 is not omnidirectional, and the gray areas are affected by other
factors besides of the distance. When the quality of the links is evaluated,
the links in the gray areas cannot influence because they are filtered by the
$\gamma_{ga}$ threshold. However, when collecting the data, if the nodes in
the gray areas are not considered they can cause collisions. Hereby, the
sector distance when transmitting has to be adjusted to reduce as much as
possible the effect of the nodes in the gray areas.
### 3.3 Randomized Transmissions
To avoid collisions when two or more nodes want to transmit at the same
instant, the random MAC protocols usually wait a random backoff in order to
distribute the transmissions over a certain period of time. However, using a
backoff, there is a tradeoff between delay and collision probability. A larger
backoff time would provide less collision probability but will compromise the
transmission delay. The aim of the randomized transmissions mechanism is to
spread the transmissions of the nodes that want to transmit at the same
instant without modifying the MAC parameters. As an example, if there are 100
nodes that are ready to transmit, their transmissions are spread in an
interval of 10 seconds, and approximately, if uniform distribution is
considered, only 10 nodes will compete to access the medium during one second.
Also notice that if the time to transmit data is large compared to the number
of nodes, the randomized property of the random MAC protocol becomes less
important. However, as the aim of the network is to collect the data with the
minimum amount of time, the random MAC protocol is relevant. To set this
transmission time, different aspects have to be taken into account: the number
of nodes in the network, the MAC parameters, the transceiver characteristics
and the packet lengths.
### 3.4 Data Aggregation
Most of the times the packets sent in a WSN do not achieve the maximum packet
size, and usually payloads just represent a small part of the whole packet. In
order to achieve better packet efficiency and consume less energy, data
packets can be aggregated, which reduces transmissions and collisions. Packets
have larger sizes, but overall the total amount of bytes sent is less due to
the reduction of overhead.
## 4 Protocol Description
### 4.1 Headers and addresses
Before addressing how the protocol works, the format of the HBCP packets is
introduced. HBCP has two types of packets: data and discovery. The data
packets are the ones that carry the data sensed by the sensors, and the
discovery packets are the ones used to notify the nodes that a collection has
to be performed, create the datapaths and classify the nodes in sectors.
Each of these two packets have a common _HBCP header_ (Figure 3(a)), that is
followed by the _Data_ or _Discovery Frame_ (Figures 3(c) and 3(b)). The HBCP
header is composed by five fields: the _protocol id_ identifies the packet as
an HBCP packet, the _source address_ indicates the node id, the _packet size_
indicates the length of the whole network packet, and the last one is the
_Specific Frame_ which includes the _Data_ or the _Discovery Frames_.
Regarding the _Discovery Frame_ , it is divided in two parts: the collection
and the performance parameters. The collection parameters are three: _hop_ ,
_energy level_ and _coordination time_ , and these respectively mean the
sector the node belong to, the remaining energy of a node and the coordination
time that will be explained in Section 4.2.1. The performance parameters are
_max hops_ , which is the maximum number of sectors that a network can have,
the _drop threshold_ ($\gamma_{ga}$) which is the minimum quality of a
discovery packet in order to be evaluated, the _quality threshold_
($\gamma_{q}$) which is the indicator that bidirectional communications is
assured, and the _transmission power_ that the devices will use. The
_discovery time_ and the _collection time_ parameters will be explained in
detail in Section 4.2 and 4.3 respectively.
The _Data Frame_ is simpler than the _Discovery Frame_ , it only contains the
sector of the node that transmits the packet, the number of application data
(payloads) because of data aggregation (see Section 4.3.2), the total size of
the payload, and the sizes of each of the aggregated payloads and the total
payload.
(a) HBCP header
(b) Discovery frame
(c) Data frame
Figure 3: HBCP headers. The specific frame field of the _header_ is where the
_Discovery frame_ or _Data frame_ are placed.
### 4.2 Network Discovery
This section describes how the network sectors are created and how the
datapath to reach the sink is computed for each node in the network.
#### 4.2.1 Transmission Range Classification
The main goal of this function is to classify the nodes into different
sectors, where each sector, as it is explained in Section 3.1, corresponds to
the number of hops to reach the sink. Initially, a node does not have any kind
of information about the network topology, and every time that a data
collection ends, the routing information expires. The HBCP is designed for on-
demand collection applications with low frequency collections, hence, as the
time between two collections can be from several minutes to months, the
datapaths have to be built from scratch for every collection.
The mechanism to create the routes is to coordinately send the discovery
packets. The sink starts a collection sending a broadcast discovery packet
that is received by the nodes that belong to sector 1. Since we have assumed
that there is only one sink, just after the first transmission, every node
belonging to sector 1 knows it. To discover the next level each of the nodes
in sector 1 send also one discovery packet, and this process is repeated until
the maximum number of hops indicated in the discovery packet is reached.
However, there are two factors that affect the network formation: 1) the
collisions of the discovery packets, and 2) the capacity that a node in the
sector $n$ is able to receive all the packets from the $n-1$ sector during a
period of time. To alleviate the first problem nodes wait a random time,
independently of their backoff, and then transmit a discovery packet (see
Section 3.3). This random time is between 0 and the value indicated in the
discovery time field of the discovery frame. To address the second problem,
nodes include in their discovery packet the random time (coordination time)
they have used to transmit the discovery packet. Therefore, when a node in the
sector $n$ receives a discovery packet, it knows until when it will be
possible to receive more packets from the sector $n-1$. For instance, if a
node receives a discovery packet at $t_{0}$ with a coordination time $t_{c}$
and a discovery time $t_{d}$, the node will wait until $t_{0}+t_{d}-t_{c}$.
Regarding synchronization, it is worth to mention that for this type of
coordination the motes clock drift is not relevant. For example, the TelosB
mote at $20^{\circ}C$ has a large clock drift (40 ppm), what means a loss of
accuracy of 2.5 ms every minute.
Regarding the random time, it is worth noticing the tradeoff between delay and
collision probability. If this random time makes the network creation time to
substantially increase, the environmental conditions may change, especially in
volatile environments, and the path to reach the sink and sectors information
may not be valid anymore. Therefore, it is key to find the right balance
between collisions and reliability of paths.
#### 4.2.2 Datapath Selection
While the network nodes are sending the discovery packets, they use the
information included in them to select which is the best next hop to reach the
sink. The information used is the RSSI of the received discovery packet and
the remaining energy of the parent. In the discovery packet, the criteria that
the potential receivers have to follow when to select a parent (next hop) is
included. The two used parameters are the $\gamma_{ga}$ and the $\gamma_{q}$.
The first one is the minimum quality that a link must have, otherwise the
packet will be dropped and not even considered. The second one is the
threshold for a packet to be considered with good quality, hence the
bidirectional communication of the link is assured.
Figure 4: Discovery packet reception. The node receives the first discovery
packet (1), and from the information included in it knows until when it will
be possible to receive discovery packets from the previous sector. In this
example, the $4^{th}$ packet is the one with best quality, thus, the selected
parent of the node.
As it is explained in Section 4.2.1, when a node receives a discovery packet
it knows until when it would be possible to receive more discovery packets
from the same sector. During this period, each node processes all the received
discovery packets and picks up the one with the best quality (see Figure 4).
However, if the quality of all the discovery packets is in between of
$\gamma_{q}$ and $\gamma_{ga}$, the node will wait an additional discovery
period, and try to receive a discovery packet sent from a node in the same
sector it would belong to. If during this extra listening period the node
receives another discovery packet with enough quality, it will take the node
as a parent, otherwise, it will keep with the previous parent besides it does
not have a good quality link. As an example refer to Figure 5. Node 1 sends a
discovery message which assures bidirectional communications between nodes 2
and 3, but not with node 4. Therefore, node 4 waits another discovery period
to try to receive a discovery packet from nodes 2 and 3 with enough quality.
To ease the understanding of this part, the parent selection process is
algorithmically described in Algorithm 1.
Figure 5: In the first discovery period (straight line), node 4 receives a
discovery packet but cannot assure bidirectional communication with node 1.
Therefore, it waits for the second discovery period (dashed line) to obtain a
reliable link with node 3. Algorithm 1 Parent selection algorithm based on the
received RSSI. The $current\\_quality$ is the best RSSI of all the received
packets, and the $backup\\_parent$ the node used in case the second discovery
period does not find a better parent. Packets with a quality less than
$\gamma_{ga}$ are automatically discarded.
$current\\_quality=-\infty;$
function process discovery packet (packet)
if $isFirstDiscovery$ then
if $quality(packet)\geq current\\_quality$ then
$current\\_quality=quality(packet)$
if $quality(packet)\geq\gamma_{q}$ then
$parent=packet.source\\_id$
end if
$backup\\_parent=packet.source\\_id$
end if
end if
if $isSecondDiscovery$ then
if $quality(packet)\geq\gamma_{q}$ and $quality(packet)\geq current\\_quality$
then
$parent=packet.source\\_id$
$current\\_quality=quality(packet)$
end if
end if
end function
The idea of taking into account the remaining energy of the nodes is to do
traffic load balancing and try to avoid nodes with little remaining energy.
The low energy threshold ($\gamma_{e}$) has been fixed to 15%, but depending
on the platform or the type of application this value has to be readjusted to
meet the desired performance. When a node receives a discovery packet from a
parent in the previous sector, it analyzes the RSSI and the battery level, and
if the energy is below the fixed threshold, the link quality is readjusted to
prioritize other links. Using this mechanism, HBCP intends to exploit the
better datapaths until the node’s energy level is critic. The link priority is
done according to the quality ranges, if the RSSI is in the good quality range
but with little energy, the link quality will be fixed to its minimum within
the good quality range. For instance, a node has a $\gamma_{q}=K$ and receives
two discovery packets with a quality of $K+5$ and $K+10$ respectively. Without
taking into account the energy levels, the node will pick the second packet
because it has better quality. However, if the battery level of the second
node is critical, its quality will be downgraded to $K$. Hence, the node will
pick the first packet.
### 4.3 Collection
Once a node finishes the previously explained process, it has to wait for the
other nodes to start the collection. This waiting time can be calculated by
computing the difference between the sector it belongs to and the total number
of hops (which is included in the discovery packet), and then multiply it by
the discovery time.
#### 4.3.1 Data Collection Scheduling
Before to start the collection, a node knows beforehand how much time the data
collection will take. It will last the total collection time, which was
indicated in the discovery packet, times the number of hops. One of the first
options that was considered when designing the data collection protocol was to
collect the data from the outer nodes to the inner ones, but in this way the
nodes would have a high probability to run out of memory: the nodes in the
first sector would have to keep in memory all the network data before
transmitting it to the sink. Because of this, the data collection is carried
out allowing sectors that do not interfere between them to transmit at the
same time. The data arrives to the sink in different waves, and the
probability for the nodes to run out of memory is considerably reduced. Figure
6 is an example of how the protocol would behave in a network with ten hops.
Figure 6: Hop transmissions in a ten-hop network assuming a distance of 3
between sectors. In this case, in the first iteration (straight lines) sectors
3, 6 and 9 transmit at the same time to sectors 2, 5 and 8 respectively. In
the second iteration (dashed lines) these sectors will transmit to the next
sectors their own data and the data they received in the previous iteration.
After the third iteration (dotted lines), the first iteration will start
again.
#### 4.3.2 Data Queues and Forwarding
As it would happen in the network discovery, if multiple nodes try to access
the medium at the same time, collisions will likely occur. To reduce this
problem, nodes wait a random time up to the 80% of the data collection time.
This threshold has been set to allow a node with multiple packets in the queue
to have enough time to transmit them. After this initial random time a node
starts the forwarding process for all the packets in the queue. The data
aggregation is done checking how many sequential data payloads can fit into a
single packet. This aggregation mechanism is not optimum to reduce the total
number of transmissions, however, because its complexity is very low, it is
suitable for WSNs.
## 5 Implementation in TinyOS
This section presents how the HBCP architecture has been implemented in TinyOS
2.1111Code available at: http://code.google.com/p/hbcp/.
Figure 7: HBCP TinyOS components
Figure 7 shows the different HBCP components implemented in TinyOS. The
Routing Engine is the component responsible for analyzing the received
discovery packets, choose the best node as a parent and store the datapath
information. In the Forwarding Engine there are the queues that control the
incoming and the outgoing packets, and it is in charge of dispatching the
packets to the different components. Finally, the Network Manager is the
component in charge of controlling the behavior of the other components
depending on the node state. The role of the Network Manager is crucial due to
the number of concurrent timers in the code and because each component can
behave in a different way depending on the node state. For example, the
Routing Engine will not be allowed to analyze new received discovery packets
during the data collection.
Additionally, the Network Manager is the interface between the application and
the Network Layer. Therefore, it is in charge of notifying the Application
Layer that a collection is about to start, so it can obtain the value from its
sensor and forward it to the Network Layer.
## 6 Performance Evaluation
This section evaluates the HBCP performance, with special focus on the
behavior of the new proposed mechanisms in a real environment. The
implementation has been evaluated using the TelosB platform, with the CSMA
link layer without Low Power Listening (LPL) [19] active. LPL has not been
used because it is not suitable for networks that demand a great number of
transmissions during a short period of time, i.e, nodes are intended to work
intensively during few seconds, and afterward turn into sleep mode until the
next collection.
Parameter | Value
---|---
Max Hops | 10
Discovery time | $1000$ ms
Collection time | $2000$ ms
$\gamma_{ga}$ | $-87$ dBm
$\gamma_{q}$ | $-82$ dBm
Transmission power | $0$ dBm
Table 1: Parameters considered in the Testbed
To evaluate the two mechanisms, two testbeds are deployed. One for the
evaluation of the datapath creation and data collection, which consisted of
placing 30 nodes randomly throughout a three storey building, and checking
during 24 hours how the PRR behaves for the next hop and end-to-end. On the
second testbed, the performance of the data collection with different
collection times is evaluated, as well as how the protocol would behave if HRS
is not applied. One hundred collections are performed with and without HRS,
with the collection slot times set to 250, 500, 750, 1000, 1500 and 2000 ms.
Taking as a reference the previous work on the CC2420 chip presented in [13],
and assuming a $\gamma_{v}$ of $5$ dB, the $\gamma_{q}$ and the $\gamma_{ga}$
were fixed to $-82$ and $-87$ dBm respectively. For the two testbeds the
transmission power is fixed to $0$ dBm. Obviously, those values can vary
depending on the transceiver and the transmission power of each node in the
network.
The discovery time parameter is fixed to 1000 ms for the two testbeds, as this
value grants enough time for the nodes to receive at least one discovery
packet. The largest collection time in the first testbed (2000 ms) is
established by checking that the sink can receive all the packets in one
collection. Moreover, in the second testbed, where the collection time
parameter is evaluated, it is shown that 2000 ms is enough time to avoid
collision between nodes of the same sector.
Regarding the data link layer, the number of retransmissions is fixed taking
as reference the policy used in CTP [9], which is up to 32 retransmissions per
packet.
### 6.1 Testbed 1: Network formation
In all the collections, the tree that has been formed most of the times is the
one shown in Figure 8. In Figure 9(a) is depicted the average number of hops
per node for all the collections, where despite some changes, the nodes nearly
always (99.98%) tend to belong to the same sector. However, it does not happen
the same with the selected parent in each collection, as it is shown in Figure
9(b). For the majority of the collections, most of the nodes tend to chose the
same parent, but it is not as steady as the number of hops shown in Figure
9(a). There is a specific case that is interesting to highlight, and it is the
case of nodes 7 and 20, which drastically differentiate from the others
because of having a most used parent rate of 80% and 70% respectively.
Although this, the change of parent variation does not affect the node
performance in terms of end-to-end delivery (Figure 10(a)) and next-hop
acknowledgment rate (Figure 10(b)). The case of node number 20 is quite
particular, because as it shown in Figure 9(c), where the average RSSI per
node is depicted, the variance of this node is slightly over 1 dB and suggests
that the nodes that try to be its parent have somehow similar characteristics.
In the case of node 7, the cause of this variation is not straightforward and
might be due to environmental changes during the testbed, like doors openings
or people moving.
Figure 8: Most frequent tree created during the testbed.
Regarding the collection reliability, it is interesting to check the end-to-
end delivery and next-hop delivery rates, and how those vary depending on the
sector the nodes belong to. In the next hop delivery rate, the number of
packets acknowledged per node is checked, and as it is expected the next hop
link quality is independent of the sector the node belongs to (Figure 10(b)),
with an average over 99%. However, the number of hops matter in the end-to-end
delivery rate. If Figures 9(a) and 10(a) are compared, it can be observed how
the nodes that belong to further sectors have lower end-to-end delivery rates.
Nonetheless, the overall end-to-end delivery rate is considerably high, with
an average of 98.4%.
(a) Average number of hops per node. On average, in the 99.98% of the
collections, a node belonged to the same sector.
(b) Percentage of times the most used parent was elected for each node
(c) Average RSSI of the received discovery packet. The links are shown in
Figure 8.
Figure 9: Datapath metrics of the first testbed.
(a) End-to-end reliability per node. It only includes the application packets
generated by the node. (the dotted line is the average)
(b) Number of packets acknowledged per node. It includes the applications
packets generated by the node and the packet forwarded from other nodes (the
dotted line is the average).
(c) Number of packets transmitted per node per collection
Figure 10: Transmission metrics of the first testbed.
The last thing to check in this testbed is to see how data aggregation
benefits in terms of reducing the number of transmissions per node. In Figure
10(c) the average transmitted packets per collection and node are shown, and
as it can be presumably deduced from Figure 8, nodes 1 and 5 are the ones that
transmit a greater number of packets per collection, with an average of 8.66
and 11.79 respectively. In this scenario, due to the payload size, the maximum
packet aggregation was set to 3. The total average number of packets
transmitted per collection was 60.43, and compared to the 107 transmitted
packets that would have been expected without HRS, it is a reduction of over
40%. However, this results are only valid for this case, hence the transmitted
packet reduction highly depends on the number of hops and the location of the
nodes within the network. It must also be taken into account that if smaller
application packets were used, the number of transmissions could have been
reduced even more.
From Table 2 similar conclusions can be obtained, but instead of analyzing the
results for every node, the performance of each sector is analyzed. As it was
previously observed, the end-to-end delivery rate decreases with the number of
hops. But there is an exception, and it is in $5^{th}$ sector, where its
delivery rate is better than in the $4^{th}$ sector. Nonetheless, the
variation is quite small and it is due to the fact that the nodes in the
$5^{th}$ sector are not uniformly distributed along the nodes in the $4^{th}$
sector. Regarding the acknowledgment rate, in all sectors it is over 99% and
has some sort of correlation with the density of nodes in each sector.
| Sectors
---|---
Parameters | 1 | 2 | 3 | 4 | 5
Number of nodes | 2 | 4 | 4 | 15 | 5
Packets sent per coll. | 14.03 | 13.48 | 9.99 | 17.91 | 5
Exp. pkt. sent without HRS | 30 | 28 | 24 | 20 | 5
End-to-end delivery rate | 99.88 | 99.41 | 98.95 | 97.53 | 97.99
Acknowledgment rate | 99.8 | 99.44 | 99.12 | 99.18 | 99.82
Table 2: Sectors performance of the randomly deployed testbed
### 6.2 Testbed 2: Collection times
In the second testbed, the effect of different collection times is evaluated.
The aim is to check how the HBCP responds to this parameter, and to check
whether applying HRS adds any benefit to the end-to-end delivery rate.
Somehow, through varying the collection time per sector it is possible to
achieve an approximation of how the network would behave with a higher density
of nodes. As it was explained in Section 4.2.1, the collection slot time has
to be large enough to allow all the nodes in a sector to transmit, but with
the drawback that a large value may make the routes expire, especially in
volatile environments. In addition, the protocol must perform the collection
as fast as possible so the nodes can go back to sleep and save energy.
Two different topologies are deployed, the former with a bottle-neck: one node
in the first sector, another in the second sector, and twenty-three in the
third sector (1-1-23). In the latter nodes are uniformly distributed in each
sector (10-10-10). In the collection mechanism without HRS a node can transmit
at any instant within the whole collection time of all sectors.
The obtained results are plotted in Figure 11, and clearly show that nodes
that implement the HRS tend to converge more rapidly to its maximum end-to-end
delivery rate. Overall, using HRS is always better except in the case where
the nodes are uniformly distributed and the collection time is 250 ms. In this
case, with HRS the number of collisions between nodes in the same sector is
greater than the collisions without HRS. However, in the bottle-neck topology
it happens exactly the opposite, especially when the collection time ranges
from 500 to 1500 ms. During this 1000 ms span, as nodes in each sector have
enough time to transmit, the collisions between nodes of a same sector are
less than the ones of nodes from different sectors.
It is worth to notice that HRS reduces the number of collisions between nodes
in different sectors, but it increases the collisions of nodes in the same
sector. As the nodes in the same sector have to transmit during a limited
period of time, it is more likely that a node receives two packets from two
different nodes that cannot overhear each other. In contrast, this is less
likely without HRS, but there is no kind of control of the transmissions of
nodes from different sectors.
Figure 11: HBCP end-to-end delivery rate with and without HRS for different
collection times.
## 7 Related Work
Currently, there are several developed and implemented network protocols in
WSNs, however, most of which have been designed to send event-based data
rather than answering to explicit queries from the sink. In this section we
present three protocols: LEACH [4], CTP [9] and Direct Diffusion [6], which
have some similarities with the HBCP. Furthermore, it has been included some
mechanisms that, although very different in nature, aim to achieve the same
goal as the HRS and BLQE.
LEACH is a low-energy, self-organizing, adaptive clustering routing protocol
that uses randomization to distribute the energy consumption evenly among the
sensors in the network. It belongs to the hierarchical family of network
protocols, where the different sensors in the network are divided in clusters,
and its main purpose is to increase the whole sensor network life-time through
network traffic balancing. Sensors elect themselves as cluster-heads at a
given time with a certain probability, and the other nodes select from the
existing clusters which they want to belong to by choosing the cluster-heads
that has the minimum delivery cost (based on the RSSI). Once the network is
configured, a cluster-head creates a schedule for each node in the cluster,
similarly to TDMA, so the non-cluster-heads can turn off the radio except when
they have to transmit, which minimizes the energy dissipated per sensor. LEACH
does not define how data has to be collected, however, in contrast to HBCP,
nodes always have a route to reach the sink. In addition, one of the drawbacks
of LEACH is that the network cannot have more than two hops, therefore, it is
not suitable for large multihop networks.
The Collection Tree Protocol (CTP) is a collection protocol based on events
designed to improve the data collection reliability while sending a reduced
number of control packets (beacons). It has been implemented to be hardware
and MAC protocol independent, and the protocol evaluation in different
testbeds and platforms has shown a reliability results between 90-99.9%, while
sending up to 73% fewer control packets than existing approaches [11]. CTP is
basically characterized by the way routes are computed, and how beaconing is
adapted to do not overload the network. Routes are address-free and the nodes
just know its next hop and the cost to reach the sink. The parent selection
mechanism is done using the ETX (Expected Transmission) [20] value which is
computed according to the route cost and the parent reliability. Firstly, the
sink starts sending a beacon with an ETX equal to zero, the nodes that receive
that beacon announce their cost to reach the sink sending another beacon, and
that is repeated for the following levels on the network. In case that a node
receives more than one packet it will always choose the one with the lowest
ETX as it would be the best route to reach the sink. With this routing
algorithm a node knows if there is a loop in the network just comparing the
ETX values, because the parent ETX should always be lower than the one of its
son. The routes are periodically transmitted, however depending on the network
conditions, the routes maintenance frequency decreases, and only increases if
it is needed (link failure). Differently from the HBCP, CTP is thought to
collect data with high frequency. Moreover, as the data transmissions are
based on events, there is no kind of coordination, so nodes should always be
awake to be able to receive data to later forward it. Because of this, CTP is
not energy-efficient for low periodicity data collections.
Direct Diffusion is a data-centric routing protocol for on-demand collections,
where the sink broadcast a query to the whole network which contains
parameters related to the type of information it is looking for, such as name
of objects, interval, geographical area, interval, etc. When nodes receive a
query, they analyze it, and if their information matches with the sink
interests they start sending data to the sink for a given period of time,
which was fixed in the query. Despite the fact that HBCP and Direct Diffusion
are on-demand protocols, their purposes are very different. Direct Diffusion
is designed for event-based applications that require to receive data from a
portion or the entire network for a given interval of time, while HBCP is
designed to collect one reading from each sensor in the network. In addition,
Direct Diffusion does not implement any mechanism to reduce the collisions due
to hidden nodes, what makes it not suitable for collecting a large amount of
data during a short period of time.
There are some TDMA-like protocols (as the LMAC defined by Van Hoesel et. al)
[21] in which each sensor node picks a random slot from the free slots (slots
not used in a 2-hop distance) in order to avoid hidden terminal interference.
In LMAC sensor nodes exchange information about the slots they see as free. By
doing this, a sensor node can select a slot from the 2-hops away unused slots,
i.e., a slot considered free by all of its neighbors. The slot selection
starts by a message sent by the sink. After receiving it, sensor nodes select
the slot to use for transmitting data. Data transmissions are preceded by
control messages that serve to synchronize sensor nodes and to inform about
the intended destination of the data packet, then non interested nodes can go
to sleep. The LMAC concept of reusing slots is very close the HRS concept of
allowing nodes in different sectors to transmit during a given period of time.
Moreover, the reuse distance (slot/sector) used in the LMAC and the HRS is the
same. However, given that LMAC needs to synchronize every node in the network,
the configuration process is more complex and requires more transmissions of
control messages, and as a result would entail higher energy consumption.
The link quality estimation has gained attention in the last years because of
the impact that links have in the delivery rate and energy consumption
metrics. The basic metrics used for estimating the quality of a link are
three: RSSI, LQI and PRR. From these, the two most used are the RSSI and LQI
because they have shown that for certain thresholds they are capable of
providing 95% reliability [13]. However, differently from the RSSI, the LQI
quality estimation is more accurate but needs at least 120 packets for a good
estimation. Overall the different Link Quality Estimators (LQEs), the Fuzzy
Link Quality Estimator (F-LQE)[22], by Baccour et al., has shown to outperform
most of the LQE -based estimators [20][23][24][25]. F-LQE does not only take
into account one link metric but four: the smoothed packet reception rate, the
asymmetry level, the average signal to noise ratio and the link stability.
However, the penalty of obtaining such an accurate metric is time, which makes
the F-LQE not suitable for volatile environments where a fast link estimation
is required. In order to reduce the time required to estimate a link quality,
Boano et. al. presented the Triangle metric [26]. This LQE metric exploits the
correlation between the LQI, PRR and SNR metrics to obtain a link estimation
in a fast way. Results show that a good link estimation can be achieved with
just 10 packets, and that the accuracy of the link estimation increases with
the number of packets sent. Nevertheless, besides reducing the number of
packets to estimate the quality of a link, the total number of control packets
grows exponentially with the number of nodes. With BLQE, as we have shown in
Section 6, it is possible to obtain a good estimation of the quality of a link
using only a reduced number of control packets (linearly dependent on the
number of nodes). An approach that only uses the RSSI metric is the Kalman
filter based link quality estimator (KLE) [27], which estimates the quality of
a link with a single packet. With the RSSI of the received packet and the
ground noise, the KLE does an approximation of the SNR and the PRR of the
link. The main drawback of the KLE is that the function that maps the Received
Signal Strength (RSS) with the SNR does not adapt to dynamic environments.
BLQE also suffers in dynamic environments, however, the $\gamma_{v}$ parameter
is easier to compute and it obtains the link quality in both directions.
## 8 Conclusions
In this paper we have presented the Hierarchical-Based Collection Protocol for
on-demand data collections in WSNs, which main features are the HRS and BLQE
mechanisms. The first one estimates the link quality between two nodes based
on the RSSI of the received discovery packets, and the second one organizes
the nodes in different sectors, in a way that when performing the data
collection the hidden terminal problem is reduced. Both mechanisms, along with
the other mechanisms included in the HBCP, have been implemented in TinyOS
2.1, and have been experimentally evaluated using two testbeds. From the
results obtained from the testbeds, we have shown that the BLQE mechanism is
able to provide good quality links achieving a reliability over $99$%.
Regarding the HRS, its benefits are especially appreciated in networks with a
high number of hops and bottle-neck topologies. In addition to the main two
mechanisms, we have also observed that data aggregation can substantially
reduce the number of packets transmitted and therefore, reduce the number of
collisions. However, its benefits highly depend on the type of topology and
the maximum aggregation per packet which directly depends on the size of the
payloads.
One of the drawbacks of just using one packet per node to discover the network
is that a node may not receive any discovery packet due to collisions or
channel outages. In this case, the node will not send data to the sink because
it will not be aware that a data collection has been started. How the
discovery time affects the number of collisions, and which is the probability
that a node does not receive a discovery packet is an interesting point to be
addressed in future work. The discovery time is crucial to ensure that all
nodes participate in the collection and to reduce the amount of time the nodes
are awake. In addition, it would be worthwhile to reduce the impact of a link
break-down, especially in nodes in sectors near to the sink, which have to
forward multiple data packets. A node could store multiple parents addresses
or send broadcast data packets to nodes in the next sector.
Finally, if very large networks are considered, it would be worth to use
multiple sinks within the same network. This will decrease the number of hops
and also reduce the total collection time. The protocol presented in this work
can be easily adapted to work with multiple sinks if the queries sent are
synchronized. Therefore, and according to how datapaths are created in the
HBCP, only the nodes at the same distance (in hops) from two or more sinks
will receive queries from different sinks. The other nodes will only receive
queries from one sink, and will not even notice of the existence of other
sinks in the network. Furthermore, independently of the number of queries a
node receives, data will only be forwarded to one sink. One of the problems of
using multiple sinks is that the nodes at the same distance from different
sinks will receive a higher number of queries, which makes collisions more
likely to happen if the discovery time is static. Because of this, in future
work we would like to investigate how the discovery time can be dynamically
adapted depending on the network conditions.
## References
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* [21] L.F.W. van Hoesel and PJM Havinga. A lightweight medium access protocol (lmac) for wireless sensor networks: Reducing preamble transmissions and transceiver state switches. 2004\.
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* [23] A. Woo and D.E. Culler. Evaluation of efficient link reliability estimators for low-power wireless networks. Computer Science Division, University of California, 2003.
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* [25] A. Cerpa, J.L. Wong, M. Potkonjak, and D. Estrin. Temporal properties of low power wireless links: modeling and implications on multi-hop routing. In Proceedings of the 6th ACM international symposium on Mobile ad hoc networking and computing, pages 414–425. ACM, 2005.
* [26] C.A. Boano, M.A. Zúniga, T. Voigt, A. Willig, and K. Romer. The triangle metric: fast link quality estimation for mobile wireless sensor networks. In Computer Communications and Networks (ICCCN), 2010 Proceedings of 19th International Conference on, pages 1–7. IEEE, 2010.
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|
arxiv-papers
| 2012-05-06T11:14:30 |
2024-09-04T02:49:30.638709
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V\\'ictor Valls, Jos\\'e Luis S\\'anchez, Cristina Cano, Boris Bellalta,\n Miquel Oliver",
"submitter": "V\\'ictor Valls",
"url": "https://arxiv.org/abs/1205.1203"
}
|
1205.1322
|
THE RECEPTOR TOXIN ANTIBODY INTERACTION:
MATHEMATICAL MODEL AND NUMERICAL SIMULATION
P. Katauskis1, P. Skakauskas1, A. Skvortsov2
_1 Vilnius University, Lithuania_
_2 DSTO, VIC 3207, Melbourne, Australia_
E-mail: pranas.katauskis@mif.vu.lt, vladas.skakauskas@maf.vu.lt,
alex.skvortsov@dsto.defence.gov.au
††footnotetext: © P. Katauskis, V. Skakauskas, A. Skvortsov 2012
## 1 Introduction
An antibody, also known as an immunoglobulin, is a protein used by the immune
system to identify, neutralize, or kill foreign objects like bacteria,
viruses, or pollen which are termed as antigen. The production of antibodies
is the main function of the immune system. An antigen, when introduced into
the body, triggers the production of an antibody by immune system which will
then kill or neutralize the antigen that is recognized as a foreign invader.
The bio-medical application of antibodies against an effect of toxins
associated with various biological threats (epidemic outbreaks or bio-
terrorism) is well-documented (see, e.g., [1–3]).
For a long time the main target of antibody design has been the antibody
affinity. With progress in bio-engineering, many antibodies with different
affinity parameters have been generated. However, according to Skvortsov and
Gray [4] affinity is not a good predictor of protective or therapeutic
potential of an antibody. In fact, the treatment effect of an antibody can be
described by a parameter which includes the reaction rates of the receptor-
toxin-antibody (RTA) kinetics and relative concentration of reacting species.
As a result, any given value of this parameter determines a range of antibody
kinetic properties and its relative concentration in order to achieve a
desirable therapeutic effect.
The model considered by Skvortsov and Gray is a model of a well-mixed solution
of toxin, antibody, and cells and neglects diffusion fluxes of interacting
species. Diffusion fluxes are significant especially when the process of RTA
interaction is limited by diffusion. Skakauskas et al. [5] examined
numerically a RTA interaction model taking into account diffusion of all
species in the case where a spherical cell is embedded into an initially
uniformly distributed toxin–antibody solution which occupies a large volume
(compartment) lying between the cell and external surface. Initial values of
species and their values on the external surface were assumed to be the same
for all times. In this case fluxes of toxin, antibody and their complex across
the external surface are not zero. Some numerical results of the evaluation of
an antibody treatment efficiency parameter are given in this paper.
In the present paper by using the same model we study the influence of RTA
kinetic parameters and diffusivity of toxin, antibody, and their complex on
the behavior of the antibody protection parameter and concentrations of
species in more detail.
The paper is organized as follows. In Section 2 we introduce the
reaction–diffusion model for RTA interaction. Numerical results are presented
in Section 3. Summarizing remarks given in Section 4 conclude the paper.
## 2 The model
We study a case of a spherical cell embedded into a toxin–antibody solution
which occupies an extracellular domain $\Omega$ lying between the cell and an
external surface and use notations of paper [5]:
$\rho$ – spherical radius,
$S_{c}=\\{\rho:\rho=\rho_{c}\\}$ – the surface of the spherical cell,
$\rho_{c}$ is its radius,
$S_{e}=\\{\rho:\rho=\rho_{e}\\}$ – the surface of the external sphere
(external surface of $\Omega$), $\rho_{e}$ is its radius,
$\Omega=\\{\rho:\rho\in(\rho_{c},\rho_{e})\\}$ – the extracellular domain,
$r_{0}$ – the concentration of receptors on the cell surface,
$\theta(t,\rho)$ – the fraction of the toxin-bound receptors,
$r_{0}\theta$ – the concentration of the toxin-bound receptors (confined to
$S_{c}$),
$r_{0}(1-\theta)$ – the concentration of the free receptors,
$u_{T},$ $u_{A}$, and $u_{C}$ – the concentrations of toxin, antibody, and
toxin–antibody complex, respectively,
$u_{T}^{0},$ $u_{A}^{0},$ $u_{C}^{0}$ – the initial concentrations,
$\kappa_{T},$ $\kappa_{A}$, and $\kappa_{C}$ – the diffusivity of the toxin,
antibody, and toxin–antibody complex, respectively,
$k_{1}$, $k_{-1}$ – the forward and reverse constants of the toxin–antibody
reaction rate,
$k_{2}$ and $k_{-2}$ – the forward and reverse constants of the toxin and
receptor binding rate,
$k_{3}$ – the rate constant of the toxin internalization,
$\partial_{n}$ – the outward normal derivative on $S_{e}$ or $S_{c}$,
$\partial_{t}=\partial/\partial t$,
$\Delta=\rho^{-2}\dfrac{\partial}{\partial\rho}(\rho^{2}\dfrac{\partial}{\partial\rho})$
– the Laplace operator,
$\psi(t)$ – the antibody protection factor (a relative reduction of toxin
inside a cell due to application of antibody).
Dynamics of the concentrations $u_{T},$ $u_{A}$, $u_{C}$, and $\theta$ can be
described by the following equations:
$\begin{cases}\partial_{t}u_{T}=-k_{1}u_{T}u_{A}+k_{-1}u_{C}+\kappa_{T}\Delta
u_{T},\quad\rho\in\Omega,\ t>0,\\\ u_{T}=u_{T}^{0},\quad\rho=\rho_{e},\
t>0,\\\
\partial_{n}u_{T}=\frac{r_{0}}{\kappa_{T}}(-k_{2}(1-\theta)u_{T}+k_{-2}\theta),\quad\rho=\rho_{c},\
t>0,\\\ u_{T}|_{t=0}=u_{T}^{0},\quad\rho\in\Omega,\end{cases}$ (1)
$\begin{cases}\partial_{t}\theta=k_{2}(1-\theta)u_{T}-k_{-2}\theta-
k_{3}\theta,\quad\rho=\rho_{c},\ t>0,\\\
\theta|_{t=0}=0,\quad\rho=\rho_{c},\end{cases}$ (2)
$\begin{cases}\partial_{t}u_{A}=-k_{1}u_{T}u_{A}+k_{-1}u_{C}+\kappa_{A}\Delta
u_{A},\quad\rho\in\Omega,\ t>0,\\\ u_{A}=u_{A}^{0},\quad\rho=\rho_{e},\
t>0,\\\ \partial_{n}u_{A}=0,\quad\rho=\rho_{c},\ t>0,\\\
u_{A}|_{t=0}=u_{A}^{0},\quad\rho\in\Omega,\end{cases}$ (3)
$\begin{cases}\partial_{t}u_{C}=k_{1}u_{T}u_{A}-k_{-1}u_{C}+\kappa_{C}\Delta
u_{C},\quad\rho\in\Omega,\ t>0,\\\ u_{C}=0,\quad\rho=\rho_{e},\ t>0,\\\
\partial_{n}u_{C}=0,\quad\rho=\rho_{c},\ t>0,\\\
u_{C}|_{t=0}=0,\quad\rho\in\Omega.\end{cases}$ (4)
The initial and boundary conditions for the system above correspond to a case
where initially the toxin and antibody are distributed uniformly in the
extracellular domain $\Omega$. Values of all species on the outer boundary of
$\Omega$ for all times and their initial values are assumed to be the same. In
particular, zero value of the toxin–antibody complex is used for initial time
and for all times on the outer boundary of $\Omega$. We stress that in this
case the fluxes of all species are not zero on the outer boundary $S_{e}$ of
$\Omega$.
Eqs. (1)–(4) can be presented in non-dimensional form by using scales of
$\tau_{*}$ (time), $l$ (length), and $u_{*}$ (concentration). By substituting
variables $x=l\bar{x},$ $t=\tau_{*}\bar{t},$ $r_{0}=lu_{*}\bar{r}_{0}$,
$u_{T}=u_{*}\bar{u}_{T},$ $u_{A}=u_{*}\bar{u}_{A},$ $u_{C}=u_{*}\bar{u}_{C},$
$u_{T0}=u_{*}\bar{u}_{T}^{0},$ $u_{A0}=u_{*}\bar{u}_{A}^{0}$,
$\bar{k}_{1}=\tau_{*}u_{*}k_{1},$ $\bar{k}_{2}=\tau_{*}u_{*}k_{2},$
$\bar{k}_{-1}=\tau_{*}k_{-1},$ $\bar{k}_{-2}=\tau_{*}k_{-2},$
$\bar{k}_{3}=\tau_{*}k_{3}$, $\bar{\kappa}_{T}=\tau_{*}\kappa_{T}l^{-2},$
$\bar{\kappa}_{A}=\tau_{*}\kappa_{A}l^{-2},$
$\bar{\kappa}_{C}=\tau_{*}\kappa_{C}l^{-2}$ into (1)–(4) we can deduce the
same system, but only in the non-dimensional variables. Therefore, for
simplicity in what follows, we treat system (1)–(4) as non-dimensional.
The main antibody treatment efficiency parameter is the antibody protection
factor (a relative reduction of toxin attached to a cell due to application of
antibody) which can be defined by the following expression [4,5]:
$\psi(t)=\frac{\int_{S_{c}}\theta|_{u_{A}^{0}>0}\,\mathrm{d}S}{\int_{S_{c}}\theta|_{u_{A}^{0}=0}\,\mathrm{d}S}.$
(5)
By definition $0\leq\psi\leq 1$. The lower the value of $\psi$ the more
profound is therapeutic effect of antibody treatment.
## 3 Numerical results
We treated system (1)–(4) numerically for the spherically symmetric domain,
$\rho\in(\rho_{c},\rho_{e})$, and $t>0$ with an implicit finite-difference
scheme. Our selection of the values of parameters was motivated by the values
available in the literature [3,5–7] with the extended range to allow
exploration and illustration of the various transport and kinetics regimes
that are possible in the RTA system. We employ the following data that were
used in the most calculations in [5,8]: $u_{*}=6.02\cdot 10^{13}\
\mathrm{cm}^{-3},$ $\tau_{*}=1\ \mathrm{s},$ $r_{0}=1.6\cdot 10^{4}/S_{c}$,
where $1.6\cdot 10^{4}$ is the total number of receptors of the cell,
$l=10^{-2}\ \mathrm{cm},$ $S_{c}=4\pi\rho_{c}^{2}=4\pi\cdot 10^{-6}\
\mathrm{cm}^{2}$, $\bar{r}_{0}=2.115\cdot 10^{-3}$. The standard non-
dimensional values of the other parameters are the following:
$\begin{cases}k_{1}=1.3\cdot 10^{-2},\quad k_{-1}=1.4\cdot 10^{-4},\\\
k_{2}=1.25\cdot 10^{-2},\quad k_{-2}=5.2\cdot 10^{-4},\quad k_{3}=3.3\cdot
10^{-5},\\\
\kappa_{T}=10^{-2},\quad\kappa_{A}=10^{-2},\quad\kappa_{C}=10^{-2},\\\
\rho_{c}=10^{-1},\quad\rho_{e}=2,\\\ u_{A}^{0}=1,\quad
u_{T}^{0}=0.5.\end{cases}$ (6)
These values correspond to the ricin and 2B11 mono-clonal antibody
interaction. If values of $k_{1}$, $k_{2}$, $\kappa_{A}$, $\kappa_{C}$, and
$\kappa_{T}$ differ from those given in (6), they are specified in the legends
of plots.
As we indicated in the Introduction, the main purpose of our study was to
estimate the effect of diffusive and kinetic parameters of species on the
behavior of concentrations of species and protective properties of an antibody
against a toxin. Results of numerical solving of system (1)–(4) are presented
in Figs. 1–7.
Fig. 1. Influence of the external radius $\rho_{e}=2$ (solid line) and 5
(dashed line) and the toxin diffusivity $\kappa_{T}:$ $10^{-2}$ (1), $5\cdot
10^{-3}$ (2), $10^{-3}$ (3), $10^{-4}$ (4) on the cell protection
characteristic, $\psi$, in the case of $u_{T}^{0}=0.6.$
Fig. 2. Effect of the external radius $\rho_{e}=2$ (solid line) and 5 (dashed
line) and the antibody diffusivity $\kappa_{A}:$ $10^{-1}$ (1), $10^{-2}$ (2),
$10^{-3}$ (3), $5\cdot 10^{-4}$ (4), $10^{-4}$ (5) on the cell protection
factor, $\psi$, in the case of $\kappa_{T}=10^{-3}.$
Fig. 3. Effect of the toxin diffusivity $\kappa_{T}:$ $10^{-2}$ (solid line),
$5\cdot 10^{-3}$ (dashed line), $10^{-3}$ (dash-dotted line) and parameter
$k_{1}:$ $1.3\cdot 10^{-2}$ (1), $2\times 1.3\cdot 10^{-2}$ (2), $4\times
1.3\cdot 10^{-2}$ (3) on the cell protection function $\psi.$
Fig. 4. Effect of the toxin diffusivity $\kappa_{T}:$ $10^{-2}$ (solid line),
$5\cdot 10^{-3}$ (dashed line), $2.5\cdot 10^{-3}$ (dash-dotted line) and
parameter $k_{2}:$ $1.25\cdot 10^{-2}$ (1), $2\times 1.25\cdot 10^{-2}$ (2),
$4\times 1.25\cdot 10^{-2}$ (3) on the cell protection function $\psi.$
Fig. 5. Dynamics of toxin concentration $u_{T}$ for $\rho_{e}=2$ (solid line),
$\rho_{e}=5$ (dashed line), and $\kappa_{T}:$ $10^{-2}$ (1), $5\cdot 10^{-3}$
(2), $10^{-3}$ (3).
Fig. 6. Profiles of functions $\theta$ for $u_{A}^{0}=1$, $\rho_{e}=2$ (solid
line); $u_{A}^{0}=1$, $\rho_{e}=5$ (dashed line), and $\kappa_{T}:$ $10^{-2}$
(1), $5\cdot 10^{-3}$ (2), $10^{-3}$ (3). Line with bullets in the case of
$u_{A}^{0}=0.$
Fig. 7. Dynamics of functions $q_{A}=\partial u_{A}(t,\rho_{e})/\partial\rho$
and $q_{C}=\partial u_{C}(t,\rho_{e})/\partial\rho$ at $\rho_{e}=2$ for
$\kappa_{T}=10^{-3}$ and $\kappa_{A}=\kappa_{C}=10^{-2}.$
The plots of $\psi$ in Fig. 1 depict the dependence of the antibody protection
factor on the radius $\rho_{e}$ of the external surface $S_{e}$ and toxin
diffusivity $\kappa_{T}$. Parameter $\psi$ increases with $\kappa_{T}$
growing, but its behavior for large values of $\kappa_{T}$ is non-monotonic.
For large values of $\kappa_{T}$, parameter $\psi$ grows as $\rho_{e}$
decreases. But for small values of $\kappa_{T}$ its behavior is different. For
example, if $\kappa_{T}=10^{-3}$, then values of $\psi$ for $\rho_{e}=5$ are
larger than those for $\rho_{e}=2$ if $t<1400$ s approximately. But if
$\kappa_{T}\leq 10^{-4}$, then, for all $t$, values of $\psi$ for $\rho_{e}=5$
are larger than those for $\rho_{e}=2$ (see curves 3 and 4).
Fig. 2 illustrates the dependence of $\psi$ on the diffusivity $\kappa_{A}$ of
the antibody. The curves in this figure depict the increase of $\psi$ as
$\kappa_{A}$ decreases and non-monotonic time evolution of $\psi$ for small
values of $\kappa_{A}$. Moreover, in the case of small antibody diffusivity,
$\kappa_{A}=10^{-4}$, values of $\psi$ for $\rho_{e}=2$ are larger than those
for $\rho_{e}=5.$ But in the case of large antibody diffusivity,
$\kappa_{A}=10^{-1}$, values of $\psi$ for $\rho_{e}=2$ are smaller than those
for $\rho_{e}=5.$ only if $t\leq 1000$ s. For $t>1000$ s they behave vica
versa.
Figs. 3 and 4 exhibit the dependence of $\psi$ on diffusivity $\kappa_{T}$,
forward constant $k_{1}$ of the toxin and antibody reaction rate, and forward
constant $k_{2}$ of the toxin and receptor binding rate, respectively. Fig. 3
demonstrates the decrease of $\psi$ as $k_{1}$ increases. But different values
of $k_{1}$ do not change the monotonic behavior of all curves in time. From
Fig. 4 we see the non-monotonic behavior of $\psi$ as $k_{2}$ increases.
Moreover, $\psi$ increases with $k_{2}$ increasing. The bottom of the hollow
in Fig. 3 is located lower than that in Fig. 4. One can see in Fig. 4 that the
effect of toxin diffusivity variation on protection factor is sensitive to
changes of parameter $k_{2}$. Let us compare the minimal values of protection
factor. In the case of $k_{2}=1.25\cdot 10^{-2}$, the minimum of $\psi$ is
about 0.72 at $\kappa_{T}=10^{-2}$, 0.62 at $\kappa_{T}=5\cdot 10^{-3}$ and
0.5 at $\kappa_{T}=2.5\cdot 10^{-3}$, while the corresponding values of $\psi$
are about 0.86, 0.84 and 0.836 in the case of $k_{2}=5\cdot 10^{-2}$ (curves 1
and 3).
Numerical experiments show that diffusivity $\kappa_{C}$ practically does not
influence the time evaluation of $\psi$.
The plots of $u_{T}$ in Fig. 5 depict the dependence of the toxin
concentration at the cell surface on the diffusivity $\kappa_{T}$ and radius
$\rho_{e}$ of the external surface $S_{e}$. For any value of $\rho_{e}$,
$u_{T}$ decreases with $\kappa_{T}$ decreasing. For large values of
$\kappa_{T}$, function $u_{T}(t,\rho_{c})$ grows as $\rho_{e}$ decreases. But,
for small values of $\kappa_{T}$, its behavior is different. For example, for
$\kappa_{T}=10^{-3}$, values of $u_{T}(t,\rho_{c})$ for $\rho_{e}=5$ are
larger than those for $\rho_{e}=2$ only if $t<500$ s (see curves 3). Our
calculations show that influence of $\kappa_{C}$ on the behavior of
$u_{T}(t,\rho_{c})$ is insignificant. We observed the non-monotonic behavior
of $u_{T}(t,\rho_{c})$ for small $\kappa_{C}$, but difference between its
steady-state value and value at the bottom of the hollow is very small (of
order $10^{-3}$).
Calculations show that $u_{C}(t,\rho_{c})$ grows with $\kappa_{C}$ decreasing.
The behavior $u_{C}(t,\rho_{c})$ is monotonic for $\kappa_{C}\leq 5\cdot
10^{-2}$. Its values are smaller than initial ones of toxin for
$\kappa_{C}\in[5\cdot 10^{-3},5\cdot 10^{-2}]$. But $u_{C}(t,\rho_{c})$ can
reach a relatively large steady-state value for small $\kappa_{C}$ while
steady-state values of $u_{T}$ and $u_{A}$ are smaller than their initial
values. For example, the steady-state value of $u_{C}$ on $S_{c}$ is equal to
$2.2$ for $\kappa_{C}=10^{-3},\,\kappa_{A}=\kappa_{T}=10^{-2}$. Derivatives of
$u_{T}$ and $u_{A}$ with respect to $\rho$ on $S_{e}$ are of order $0.3$ while
derivative of $u_{C}$ on $S_{e}$ is of order $-3$. This means that $u_{C}$
increases faster towards the cell than $u_{T}$ and $u_{A}$ decays in the same
direction.
Curves in Fig. 6 depict the dependence of $\theta$ on $\kappa_{T}$ and
$\rho_{e}$ for $u_{A}^{0}=1$ and $u_{A}^{0}=0$. In the case where the antibody
is absent values of $\theta$ practically do not depend on diffusivity
$\kappa_{T}$ (see the bullets marked curve). $\theta$ decreases with
$\kappa_{T}$ decreasing. For any $\rho_{e}$, function $\theta$ grows as
$\kappa_{T}$ increases. If $\kappa_{T}\in 5\cdot[10^{-3},10^{-2}]$, then
values of $\theta$ for $\rho_{e}=2$ are larger than those for $\rho_{e}=5$.
But for small values of $\kappa_{T}$ its behavior is different. For example,
if $\kappa_{T}=10^{-3}$, values of $\theta$ for $\rho_{e}=5$ are larger than
those for $\rho_{e}=2$ only for about $t<1300$ s. This behavior is similar to
those of $u_{T}$ and $\psi$.
Two curves in Fig. 7 illustrate the non-monotonic behavior of derivatives
$\partial u_{A}(t,\rho_{e})/\partial\rho$ and $\partial
u_{C}(t,\rho_{e})/\partial\rho$ for small toxin diffusivity
($\kappa_{T}=10^{-3}$). For $\kappa_{T}=10^{-2}$ their behave is monotonic.
## 4 Concluding remarks
To conclude the paper we summarize results of study. The
receptor–toxin–antibody interaction is studied numerically by using a model
proposed in [5]. The model includes ”bulk” reaction of toxin and antibody,
surface binding of toxin and cell receptors, and diffusion of all species. The
main results of the numerical study are the following:
1\. The evolution of concentrations of some species (toxin and toxin-bound
receptors) and of the antibody protection factor for some cases (large toxin
diffusivity, small antibody diffusivity, and large forward constant of the
toxin–receptor binding rate) is non-monotonic
2\. The influence of small or large values of $\kappa_{T}$, $\kappa_{A}$, and
$k_{2}$ on the behavior of $u_{T}(t,\rho_{c}),$ $\theta(t)$ and $\psi(t)$ is
profoundly different in the cases of small or large $\rho_{e}$.
3\. The effect of $\kappa_{C}$ on the evolution of $u_{T},$ $\theta$, and
$\psi$ was found to be insignificant.
R e f e r e n c e s
1. 1.
_Oral H.B., Ozakin C., Akdis C.A._ Back to the future: antibody-based
strategies for the treatment of infectious diseases // Mol. Biotechnol. 2002.
T. 21. P. 225–239.
2. 2.
_Lobo E.D., Hansen R.J., Balthasar J.P._ Antibody pharmacokinetics and
pharmacodynamics // J. Pharm. Sci. 2004. T. 93. P. 2645–2668.
3. 3.
_Prigent J., Panigai L., Lamourette P., Sauvaire D., Devilliers K. et al._
Neutralising antibodies against ricin toxin // PloS ONE. 2011. T. 6. P.
e20166.
4. 4.
_Skvortsov A., Gray P._ Modeling and simulation of receptor–toxin–antibody
interaction // Proc. 18th World IMACS/ MODSIM Congress. Cairns, Australia,
2009. P. 185–191.
5. 5.
_Skakauskas V., Katauskis P., Skvortsov A._ A reaction–diffusion model of the
receptor–toxin–antibody interaction // Theor. Biol. Med. Model. 2001. T. 8:32.
P. 1–15.
6. 6.
_Sandvig K., Olsnes S., Pihl A._ Kinetics of binding of the toxic lectins
abrin and ricin to surface receptors of human cells // J. Biol. Chem. 1976. T.
251. P. 3077–3984.
7. 7.
Lectures Notes in Immunology: Antigen–antibody interactions, University of
Pavia.
$\mathrm{http://nfs.unipv.it/nfs/minf/dispense/immunology/lectures/files/antigens\\_antibodies.html}$
2011.
8. 8.
_Truskey G.A., Yuan F., Katz D.F._ Transport Phenomena in Biological Systems,
second ed. Prentice Hall, 2009. 888 p.
|
arxiv-papers
| 2012-05-07T08:49:47 |
2024-09-04T02:49:30.651916
|
{
"license": "Public Domain",
"authors": "P. Katauskis, P. Skakauskas, A. Skvortsov",
"submitter": "Alex Skvortsov",
"url": "https://arxiv.org/abs/1205.1322"
}
|
1205.1324
|
# Decomposition of Torsion Pairs on Module Categories
Fan Kong, Keyan Song, Pu Zhang
Abstract: In this article, we generalize the concept of torsion pairs and
study its structure. As a trial of obtaining all torsion pairs, we decompose
torsion pairs by projective modules and injective modules. Then we calculate
torsion pairs on the algebra $KA_{n}$ and tub categories. At last we try to
find all torsion pairs on the module categories of finite dimensional
hereditary algebras.
Key words: n-torsion pair, n-torsion pair seires, 1-type part partition,
2-type part partition, $\operatorname{Ext}$-projective,
$\operatorname{Ext}$-injective.
## 1 Introduction
The concept of torsion pair on abelian category was introduced by Dickson in
1966 [D]. From that time on, torsion pair has been always a useful tool for
studying the structure of module categories. However, it seems there is no
useful way to find all torsion pairs of a given algebra, although indeed there
are some ways to construct torsion pairs among which the most well known is
the tilting theory. As a trial, we try to give a way to obtain all torsion
pairs of hereditary algebras in this article. This topic is also discussed by
Assem and Kerner in [AK] where their most interest is to classify and
characterize the torsion pairs by partial tilting modules.
In section 2, we study the general theory where we introduce $n$-torsion pair
and $n$-torsion pair series as the generalization of classic torsion pair and
study its structure. We can see that these two generalizations are essentially
the same. In the rest of the paper we would know it is necessary and natural
to put forward this conception for studying the structure of torsion pairs.
The main skill in this section is from [R] and [TB] where they study HN-
filtration for some categories. There are really a lot of examples to
illustrate the necessity to study this finer structure of module categories.
For example perpendicular category is obtained by a 2-torsion pair series, and
the structure of partial tilting modules can be considered in this way. And
HN-filtration can be seen as a generalized $n-\text{torsion pair}$.
In [AK], Assem and Kerner show a relation between some particular partial
tilting moules and torsion pairs. In section 3, we adopt their ways by
restricting to projective modules and injective modules to try to decompose
all torsion pairs. And this is also an application of theories developed in
section 2. We give a method for how to decompose a classic torsion pair to
$n$-torsion pairs, and we give a one to one correspondence between all the
torsion pairs and some sepcial $n$-torsion pair on the module category of any
artin algebra.
In section 4, we apply the theory in section 3 to path algebras. As a
application, we give all the torsion pairs on path algebra $KA_{n}$ and tube
categories. Some of the results also have been shown in [BBM] and [BK]. But we
think our results will be much more clear in some aspects.
The section 5 is devoted to obtain all torsion pairs of hereditary algebras
which is our purpose. We define an operation called the translation of torsion
pairs. Combining this with the operation developed in section 3 and 4, the
issue of obtaining all torsion pairs comes down to find all torsion pairs on
regular component. For tame hereditary algebras, this problem is equivalent to
calculate all torsion pairs on the tube categories in section 4.
We should admit that our way of obtaining all torsion pairs is not very
satisfactory since it is mixed with $\operatorname{DTr}$-translation and the
extension between different parts of $n$-torsion pairs.
If there is no special instruction, all modules are left finitely generated
modules. For an artin algebra $\Lambda$, we denote by $\Lambda\text{-mod}$ the
category of all left finitely generated $\Lambda$-modules. Subcategories are
always assumed to be closed under isomorphism.
## 2 $n-\text{torsion pair}$ and $n-\text{torsion pair}$ series
In this section, we assume that $\Lambda$ is an artin algebra and
$\mathcal{C}$ is an extension-closed full subcategory of $\Lambda\text{-mod}$.
If ${\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n}$ are full
subcategories of $\Lambda\text{-mod}$, then we denote the minimal full
extension-closed subcategory containing
${\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n}$ by
$\langle{\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n}\rangle$.
If $\mathcal{D}$ is a subcategory of $\Lambda$-mod, then we denote the set
$\\{M\mid\operatorname{Hom}(M,N)=0,\forall N\in\mathcal{D}\\}$ by
$\leftidx{{}^{\bot}}\\!\mathcal{D}$, the
set$\\{N\mid\operatorname{Hom}(M,N)=0,\forall M\in\mathcal{D}\\}$ by
$\mathcal{D}^{\bot}$.
The following definition is well known but different from that in [ASS].
###### Definition 2.1.
A pair $(\mathcal{T},\mathcal{F})$ of full subcategories of $\mathcal{C}$ is
called a torsion pair on $\mathcal{C}$ if the following conditions are
satisfied:
$\left(1\right)$ $\operatorname{Hom}(X,Y)=0$ for all $X\in\mathcal{T}$,
$Y\in\mathcal{F}$.
$\left(2\right)$ $\forall X\in\mathcal{C}$, there exists an exact sequence on
$\Lambda\text{-mod}$:
$0\longrightarrow X_{\mathcal{T}}\longrightarrow X\longrightarrow
X_{\mathcal{F}}\longrightarrow 0$
such that $X_{\mathcal{T}}\in\mathcal{T}$ and
$X_{\mathcal{F}}\in{\mathcal{F}}.$
###### Remark 2.2.
Let $(\mathcal{T},\mathcal{F})$ be a torsion pair on $\mathcal{C}$. Then
$\mathcal{T}=\leftidx{{}^{\bot}}\\!\mathcal{F}\bigcap\mathcal{C}$;
$\mathcal{F}={\mathcal{T}}^{\bot}\bigcap\mathcal{C}$; $\mathcal{T}$ and
$\mathcal{F}$ are closed under extensions.
Now we give the following definition which is a generalization of the above.
###### Definition 2.3.
an $n$-tuple
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ of full
extension-closed subcategories of $\mathcal{C}$ is called an $n-\text{torsion
pair}$ if the following conditions are satisfied.
$\left(1\right)$
${\mathcal{C}}_{i}=\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
for $i=1,2,\cdots,n+1$.
$\left(2\right)$
$(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle)$
is a torsion pair on $\mathcal{C}$ for $i=1,2,\cdots,n+1$.
Moreover, if the first condition does not satisfy, we call
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ a defect
$n-\text{torsion pair}$ on $\mathcal{C}$.
The following lemma is obvious.
###### Lemma 2.4.
Let ${\mathcal{C}}_{1},{\mathcal{C}}_{2},{\mathcal{C}}_{3}$ be $3$ full
subcategories of $\Lambda\text{-mod}$. Then
$\left(1\right)$
$\langle{\mathcal{C}}_{1},{\mathcal{C}}_{2},{\mathcal{C}}_{3}\rangle=\langle\langle{\mathcal{C}}_{1},{\mathcal{C}}_{2}\rangle,{\mathcal{C}}_{3}\rangle=\langle{\mathcal{C}}_{1},\langle{\mathcal{C}}_{2},{\mathcal{C}}_{3}\rangle\rangle$.
$\left(2\right)$
$\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{1},{\mathcal{C}}_{2}\rangle}=\leftidx{{}^{\bot}}{{\mathcal{C}}_{1}}\bigcap\leftidx{{}^{\bot}}{{\mathcal{C}}_{2}}$,
${\langle{\mathcal{C}}_{1},{\mathcal{C}}_{2}\rangle}^{\bot}={{\mathcal{C}}_{1}}^{\bot}\bigcap{{\mathcal{C}}_{2}}^{\bot}$.
$\left(3\right)$
$\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{1}\rangle}=\leftidx{{}^{\bot}}{{\mathcal{C}}_{1}}$,${\langle{\mathcal{C}}_{1}\rangle}^{\bot}={{\mathcal{C}}_{1}}^{\bot}$.
###### Proposition 2.5.
Let $({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ be an
$n-\text{torsion pair}$ on $\mathcal{C}$. If
$(\tilde{\mathcal{C}}_{1},\tilde{\mathcal{C}}_{2},\cdots,\tilde{\mathcal{C}}_{k+1})$
be a $k-\text{torsion pair}$ on ${\mathcal{C}}_{i}$ for some $i$. Then
$({\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1},\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{k+1},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1})$
is a $(n+k)-\text{torsion pair}$ on $\mathcal{C}$.
Proof: Step 1. If
${\mathcal{C}}_{s}\in{\\{{\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{i-1}\\}}$,then
$\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{s-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{s+1},\cdots,{\mathcal{C}}_{i-1},\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{k+1},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
=
$\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{s-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{s+1},\cdots,{\mathcal{C}}_{i-1},{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
= ${\mathcal{C}}_{s}$.
similarly, if
${\mathcal{C}}_{s}\in{\\{{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\\}}$,then
$\mathcal{C}\bigcap\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1},\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{k+1},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{s-1}\rangle^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{s+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
= ${\mathcal{C}}_{s}$.
If
$\tilde{\mathcal{C}}_{s}\in{\\{\tilde{\mathcal{C}}_{1},\tilde{\mathcal{C}}_{2},\cdots,\tilde{\mathcal{C}}_{k+1}\\}}$,
then
$\mathcal{C}\bigcap\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1},\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s-1}\rangle^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle\tilde{\mathcal{C}}_{s+1},\cdots,\tilde{\mathcal{C}}_{k+1},\cdots,{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
=
$\mathcal{C}\bigcap\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle^{\bot}\bigcap\langle\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s-1}\rangle^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle\tilde{\mathcal{C}}_{s+1},\cdots,\tilde{\mathcal{C}}_{k+1}\rangle}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,}{\mathcal{C}}_{n+1}\rangle$
=
${\mathcal{C}}_{i}\bigcap\langle\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s-1}\rangle^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle\tilde{\mathcal{C}}_{s+1},\cdots,\tilde{\mathcal{C}}_{k+1}\rangle}$
= $\tilde{\mathcal{C}}_{s}$.
Thus, the checking of the first condition of definition 2.3 is finished.
Step 2. Without losing of generality, we may assume $1\leq s\leq k$, and we
want to check
$(\langle{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s}\rangle,\langle\tilde{\mathcal{C}}_{s+1},\cdots,{\mathcal{C}}_{n+1}\rangle)$
is a torsion pair on $\mathcal{C}$.
Given $X\in\mathcal{C}$, because
$(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle,\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{n+1}\rangle)$
is a torsion pair on $\mathcal{C}$, there is an exact sequence
$\begin{CD}0@>{}>{}>X_{1}@>{i_{1}}>{}>X@>{\pi_{1}}>{}>X_{2}@>{}>{}>0\end{CD}$
such that $X_{1}\in\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle$
and $X_{2}\in\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{n+1}\rangle$.
By torsion pair
$(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle)$,
there is an exact sequence
$\begin{CD}0@>{}>{}>X_{3}@>{i_{2}}>{}>X_{2}@>{\pi_{2}}>{}>X_{4}@>{}>{}>0\end{CD}$
such that $X_{3}\in\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle$
and $X_{4}\in\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
Because
$X_{3}\in{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}$
since
$X_{2}\in{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}$,
so
$X_{3}\in\mathcal{C}\bigcap\linebreak{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}={\mathcal{C}}_{i}$.
By $\text{torsion
pair}(\langle\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s}\rangle,\langle\tilde{\mathcal{C}}_{s+1},\cdots,\tilde{\mathcal{C}}_{k+1}\rangle)$
on $\mathcal{C}_{i}$, there is an exact sequence
$\begin{CD}0@>{}>{}>X_{5}@>{i_{3}}>{}>X_{3}@>{\pi_{3}}>{}>X_{6}@>{}>{}>0\end{CD}$
such that
$X_{5}\in\langle\tilde{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s}\rangle$
and
$X_{6}\in\langle\tilde{\mathcal{C}}_{s+1},\cdots,\tilde{\mathcal{C}}_{k+1}\rangle$.
By pushout of $i_{2}$ and $\pi_{3}$, we have the following commutative diagram
$\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\ X_{5}@>{}>{}>X^{{}^{\prime}}_{5}\\\
@V{i_{3}}V{}V@V{}V{i_{4}}V\\\
0@>{}>{}>X_{3}@>{i_{2}}>{}>X_{2}@>{\pi_{2}}>{}>X_{4}@>{}>{}>0\\\
@V{\pi_{3}}V{}V@V{}V{\pi_{4}}V\Big{\|}\\\
0@>{}>{}>X_{6}@>{}>{}>X_{\mathcal{F}}@>{}>{}>X_{4}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\ 00\end{CD}$
By snake lemma, $X_{5}=X^{{}^{\prime}}_{5}$, so we have an exact sequence
$\begin{CD}0@>{}>{}>X_{5}@>{i_{4}}>{}>X_{2}@>{\pi_{4}}>{}>X_{\mathcal{F}}@>{}>{}>0\end{CD}$
such that
$X_{\mathcal{F}}\in\langle\tilde{\mathcal{C}}_{s+1},\cdots,\tilde{\mathcal{C}}_{k+1},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
By pullback of $i_{4}$ and $\pi_{1}$, we have the following commutative
diagram:
$\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\
0@>{}>{}>X_{1}@>{}>{}>X_{\mathcal{T}}@>{}>{}>X_{5}@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{i_{4}}V\\\
0@>{}>{}>X_{1}@>{}>{i_{1}}>X@>{}>{\pi_{1}}>X_{2}@>{}>{}>0\\\
@V{}V{}V@V{}V{\pi_{4}}V\\\
X^{{}^{\prime}}_{\mathcal{F}}@>{}>{}>X_{\mathcal{F}}\\\ @V{}V{}V@V{}V{}V\\\
00\end{CD}$
By snake lemma, $X^{{}^{\prime}}_{\mathcal{F}}=X_{\mathcal{F}}$, so we have
the following exact sequence:
$\begin{CD}0@>{}>{}>X_{\mathcal{T}}@>{}>{}>X@>{}>{}>X_{\mathcal{F}}@>{}>{}>0\end{CD}$
such that
$X_{\mathcal{T}}\in\langle{\mathcal{C}}_{1},\cdots,\tilde{\mathcal{C}}_{s}\rangle$
and
$X_{\mathcal{F}}\in\langle\tilde{\mathcal{C}}_{s+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
Now we give the following definition which is very important to learn the
structure of $n-\text{torsion pair}$.
###### Definition 2.6.
Series
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
of torsion pairs on $\mathcal{C}$ is called an $n-$torsion pair series if
${\mathcal{T}}_{1}\subseteq{\mathcal{T}}_{2}\subseteq\cdots\subseteq{\mathcal{T}}_{n}$$(\text{equivalently,
}{\mathcal{F}}_{1}\supseteq{\mathcal{F}}_{2}\supseteq\cdots\supseteq{\mathcal{F}}_{n})$.
The following definition is an operation.
###### Definition 2.7.
Let $(\mathcal{T},\mathcal{F})$ be a torsion pair on $\mathcal{C}$, and
$\mathcal{D}$ be a subcategory of $\mathcal{C}$. We call
$(D^{1}_{(\mathcal{T},\mathcal{F})}(\mathcal{D}),D^{2}_{(\mathcal{T},\mathcal{F})}(\mathcal{D}))$
is a decomposition of $\mathcal{D}$ along $(\mathcal{T},\mathcal{F})$, where
$D^{1}_{(\mathcal{T},\mathcal{F})}(\mathcal{D})=\\{X\mid$ There exists an
exact sequence $0\rightarrow X\rightarrow M\rightarrow Y\rightarrow 0$ such
that $X\in\mathcal{T},Y\in\mathcal{F},M\in\mathcal{D}\\}$,
$D^{2}_{(\mathcal{T},\mathcal{F})}(\mathcal{D})=\\{Y\mid$ There exists an
exact sequence $0\rightarrow X\rightarrow M\rightarrow Y\rightarrow 0$ such
that $X\in\mathcal{T},Y\in\mathcal{F},M\in\mathcal{D}$}.
###### Lemma 2.8.
If
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}})\\}$
is a $2-\text{torsion pair}$ series on $\mathcal{C}$. Then
$\mathcal{F}_{1}\bigcap\mathcal{T}_{2}=D^{2}_{(\mathcal{T}_{1},\mathcal{F}_{1})}({\mathcal{T}}_{2})=D^{1}_{(\mathcal{T}_{2},\mathcal{F}_{2})}({\mathcal{F}}_{1}).$
Proof: $\mathcal{F}_{1}\bigcap\mathcal{T}_{2}\subseteq
D^{2}_{(\mathcal{T}_{1},\mathcal{F}_{1})}({\mathcal{T}}_{2})$ is clear.
Suppose $X\in{\mathcal{T}}_{2}$, by torsion pair
$({{\mathcal{T}}_{1},{\mathcal{F}}_{1}})$,there is an exact sequence
$0\longrightarrow X_{\mathcal{T}_{1}}\longrightarrow X\longrightarrow
X_{\mathcal{F}_{1}}\longrightarrow 0$
such that $X_{\mathcal{T}_{1}}\in\mathcal{T}_{1}$ and
$X_{\mathcal{F}_{1}}\in{\mathcal{F}_{1}}.$
However, $X_{\mathcal{F}_{1}}\in\leftidx{{}^{\bot}}\\!\mathcal{F}_{2}$ since
$X\in\leftidx{{}^{\bot}}\\!\mathcal{F}_{2}$. Thus,
$X_{\mathcal{F}_{1}}\in\leftidx{{}^{\bot}}\\!\mathcal{F}_{2}\bigcap\mathcal{C}={\mathcal{T}_{2}}$
and $X_{\mathcal{F}_{1}}\in{\mathcal{T}_{2}}\bigcap{\mathcal{F}_{1}}$. So
$\mathcal{F}_{1}\bigcap\mathcal{T}_{2}=D^{2}_{(\mathcal{T}_{1},\mathcal{F}_{1})}({\mathcal{T}}_{2})$.
The other half is similar.
$n-\text{torsion pair}$ series will give a filtration for every module which
is demonstrated below.
###### Proposition 2.9.
If
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
is an $n-\text{torsion pair}$ seires on $\mathcal{C}$. Then for every module
$X$ in $\mathcal{C}$, there is a filtration:
|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence for $i=1,2,\dots,n+1$, and
$S_{1}\in\mathcal{T}_{1},S_{i}\in\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}$ for
$1<i<n+1$, $S_{n+1}\in\mathcal{F}_{n}$ and $X_{j}\in\mathcal{T}_{j}$ for
$j<n+1$.
Proof. Using induction on $n$.
$n=1$, by the second condition of definition 2.1, there is a filtration
|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{2}}$
such that $0\rightarrow X_{1}\rightarrow X_{2}\rightarrow S_{2}\rightarrow 0$
is an exact sequence and $X_{1}\in\mathcal{T}_{1},S_{2}\in\mathcal{F}_{1}$.
Suppose that the proposition is true for $n=k$, let us consider $n=k+1$. By
torsion pair $({{\mathcal{T}}_{k+1},{\mathcal{F}}_{k+1}})$ on
$\mathcal{C}$,there is an exact sequence
$0\longrightarrow X_{k+1}\longrightarrow X\longrightarrow
S_{k+2}\longrightarrow 0$
such that $X_{k+1}\in{\mathcal{T}}_{k+1}$ and $S_{k+2}\in{\mathcal{F}}_{k+1}$.
Because
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{k},{\mathcal{F}}_{k}})\\}$
is a $k-\text{torsion pair}$ series on $\mathcal{C}$, by induction, there is a
filtration:
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---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k+1}}$$\textstyle{S_{1}}$$\textstyle{S_{k}}$$\textstyle{S_{k+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence for $i=1,2,\dots,k+1$, and
$S_{1}\in\mathcal{T}_{1},S_{i}\in\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}$ for
$1<i<k+1$, $S_{k+1}\in\mathcal{F}_{k},X_{i}\in\mathcal{T}_{i}$ for all $i$.
However $S_{k+1}\in\mathcal{T}_{k+1}$ since $X_{k+1}\in\mathcal{T}_{k+1}$. So
$S_{k+1}\in\mathcal{F}_{k}\bigcap\mathcal{T}_{k+1}$. The filtration is given.
###### Proposition 2.10.
If
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
is an $n-\text{torsion pair}$ series on $\mathcal{C}$. Then
$\mathcal{F}_{i}\bigcap\mathcal{T}_{i+k}=\langle\mathcal{F}_{i}\bigcap\mathcal{T}_{i+1},\mathcal{F}_{i+1}\bigcap\mathcal{T}_{i+2},\cdots,\mathcal{F}_{i+k-1}\bigcap\mathcal{T}_{i+k}\rangle$.
Proof.$"\supseteq"$ is obviously.
$"\subseteq":$For $X\in\mathcal{F}_{i}\bigcap\mathcal{T}_{i+k}$, by the above
lemma, there is a filtration of $X$:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{i+1}}$$\textstyle{S_{i+k}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence for $i=1,2,\dots,n+1$, and
$S_{1}\in\mathcal{T}_{1},S_{i}\in\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}$ for
$1<i<n+1$, $S_{n+1}\in\mathcal{F}_{n},X_{i}\in\mathcal{T}_{i}$ for $i<n+1$.
First, we claim that $X_{0}=X_{1}=\cdots=X_{i}=0$.
In fact, $\operatorname{Hom}(X_{i},X_{i+1})=0$ since $X_{i}\in\mathcal{T}_{i}$
and $X_{i+1}$ is submodule of $X$ belongs to $\mathcal{F}_{i}$. By the exact
sequence $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$, one gains $X_{i}=0$. Hence $X_{0}=X_{1}=\cdots=X_{i-1}=0$.
Second, we claim that $X_{i+k+1}=X_{i+k+2}=\cdots=X_{n+1}=X$.
In fact, $\operatorname{Hom}(X_{n+1},S_{n+1})=0$ since
$X_{n+1}=X\in\mathcal{F}_{i}\bigcap\mathcal{T}_{i+k}$ and
$S_{n+1}\in\mathcal{F}_{n}$. By exact sequence $0\rightarrow X_{n}\rightarrow
X_{n+1}\rightarrow S_{n+1}\rightarrow 0$, one gains $S_{n+1}=0$ and
$X_{n}=X_{n+1}=X$. Similarly, we have $X_{i+k+1}=X_{i+k+2}=\cdots=X_{n-1}=X$.
Now, we have the following filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{i+1}}$$\textstyle{S_{i+k}}$$\textstyle{S_{i+k+1}}$
Thus
$X\in\langle\mathcal{F}_{i}\bigcap\mathcal{T}_{i+1},\mathcal{F}_{i+1}\bigcap\mathcal{T}_{i+2},\cdots,\mathcal{F}_{i+k-1}\bigcap\mathcal{T}_{i+k}\rangle$.
The following is the relation between $n-\text{torsion pair}$ and
$n-\text{torsion pair}$ series.
###### Theorem 2.11.
There is a one to one correspondence between the set of $n-\text{torsion
pair}$ series on $\mathcal{C}$ and the set of $n-\text{torsion pair}$ on
$\mathcal{C}$:
$\begin{CD}\left\\{\begin{array}[]{c}({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}:\\\
n-\text{torsion pair series}\text{ on
}\mathcal{C}\end{array}\right\\}\autorightleftharpoons{$\alpha$}{$\beta$}\left\\{\begin{array}[]{c}({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1}):\\\
n-\text{torsion pair}\text{ on }\mathcal{C}\end{array}\right\\}\end{CD}$
such that
$\alpha(\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\})=(\mathcal{T}_{1},{\mathcal{F}}_{1}\bigcap{\mathcal{T}_{2}},\cdots,{\mathcal{F}}_{n-1}\bigcap{\mathcal{T}_{n}},{\mathcal{F}_{n}})$
and
$\beta(({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1}))=\\{(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{n+1}\rangle)\mid{i=1,2,\cdots,n}\\}$.
Proof: First, we check that
$(\mathcal{T}_{1},{\mathcal{F}}_{1}\bigcap{\mathcal{T}_{2}},\cdots,{\mathcal{F}}_{n-1}\bigcap{\mathcal{T}_{n}},{\mathcal{F}_{n}})$
is an $n-\text{torsion pair}$ on $\mathcal{C}$ .
$\left(1\right){\mathcal{F}}_{i-1}\bigcap{\mathcal{T}_{i}}=\mathcal{C}\bigcap{\mathcal{T}_{i-1}}^{\bot}\bigcap\mathcal{C}\bigcap\leftidx{{}^{\bot}}\\!\mathcal{F}_{i}=\mathcal{C}\bigcap{\mathcal{T}_{i-1}}^{\bot}\bigcap\leftidx{{}^{\bot}}\\!\mathcal{F}_{i}=\mathcal{C}\bigcap\langle\mathcal{T}_{1},{\mathcal{F}}_{1}\bigcap{\mathcal{T}_{2}},\cdots,\linebreak{\mathcal{F}}_{i-2}\bigcap{\mathcal{T}_{i-1}}\rangle^{\bot}\bigcap\leftidx{{}^{\bot}}\\!\langle{\mathcal{F}}_{i}\bigcap{\mathcal{T}_{i+1},\cdots,\mathcal{F}_{n}}\rangle$
by the above proposition.
$\left(2\right)$ Obviously,
$(\langle\mathcal{T}_{1},{\mathcal{F}}_{1}\bigcap{\mathcal{T}_{2}},\cdots,{\mathcal{F}}_{i-1}\bigcap{\mathcal{T}_{i}}\rangle,\langle{\mathcal{F}}_{i}\bigcap{\mathcal{T}_{i+1},\cdots,\mathcal{F}_{n}}\rangle)=(\mathcal{T}_{i},\mathcal{F}_{i})$.
Second, we check that
$\\{(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{n+1}\rangle)\\}_{i=1,2,\cdots,n}$
is an $n-\text{torsion pair}$ series on $\mathcal{C}$. But this is clear.
Third, we check that $\beta\alpha=1$.
$\beta\alpha(\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\})=\beta(\mathcal{T}_{1},{\mathcal{F}}_{1}\bigcap{\mathcal{T}_{2}},\cdots,{\mathcal{F}}_{n-1}\bigcap{\mathcal{T}_{n}},{\mathcal{F}_{n}})\\\
=\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
by the above proposition.
Last, we check that $\alpha\beta=1$.
$\alpha\beta(({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1}))=\alpha(\\{(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle)\\}\mid{i=1,2,\cdots,n})\\\
=\\{\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{n+1}\rangle\bigcap\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle\mid{i=1,2,\cdots,n+1}\\}\\\
=\\{\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}\mid{i=1,2,\cdots,n+1}\\}\\\
=({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$.
###### Proposition 2.12.
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ is an
$n-\text{torsion pair}$ on $\mathcal{C}$ if and only if
$\left(1\right)$ $\operatorname{Hom}(X,Y)=0$ for all $X\in\mathcal{C}_{i}$,
$Y\in\mathcal{C}_{j},i<j$.
$\left(2\right)$ For every $X\in\mathcal{C}$, there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence and $S_{i}\in\mathcal{C}_{i}$ for all $i$.
Proof: $"\Longrightarrow"$: Let
$\mathcal{T}_{i}=\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\mathcal{F}_{i}=\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{n+1}\rangle,i=1,2,\cdots,n.$
Then
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
is an $n-\text{torsion pair}$ series by proposition 2.12.
By the proof of the above proposition, we know
$\mathcal{C}_{i}=\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}$.
Hence, for every module $X$ in $\mathcal{C}$, there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence and
$S_{i}\in\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}=\mathcal{C}_{i}$.
$"\Longleftarrow"$: First, we show that
${\mathcal{C}}_{i}=\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
for $i=1,2,\cdots,n+1$.
$"\subseteq"$ is clear;
$"\supseteq"$: $\forall
X\in\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$,
there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{i-1}}$$\textstyle{S_{i}}$$\textstyle{S_{i+1}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence and $S_{i}\in\mathcal{C}_{i}$.
Just like the proof of proposition 2.10, we have
$X_{0}=X_{1}=\cdots=X_{i-1}=0$ and $X_{i}=X_{i+1}=\cdots=X_{n+1}=X$. So
$X_{i}=S_{i}\in\mathcal{C}_{i}$.
Second, we show that
$(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle,\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle)$
is a torsion pair on $\mathcal{C}$ by definition 2.1:
(1) Clear!
(2) $\forall X\in\mathcal{C}$, there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{i-1}}$$\textstyle{S_{i}}$$\textstyle{S_{i+1}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence and $S_{i}\in\mathcal{C}_{i}$.
It is clear that
$X_{i}\in\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle$, we claim
that $X/X_{i}\in\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
In fact, by snake lemma we have the following commutative diagram:
$\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\
0@>{}>{}>X_{i}@>{}>{}>X_{i+1}@>{}>{}>S_{i+1}@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\
0@>{}>{}>X_{i}@>{}>{}>X_{i+2}@>{}>{}>X_{i+2}/X_{i}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\ S_{i+2}@ =S_{i+2}\\\ @V{}V{}V@V{}V{}V\\\ 00\end{CD}$
Hence $X_{i+2}/X_{i}\in\langle{\mathcal{C}}_{i+1},{\mathcal{C}}_{i+2}\rangle$.
Use snake lemma again,we have the following commutative diagram:
$\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\
0@>{}>{}>X_{i}@>{}>{}>X_{i+2}@>{}>{}>X_{i+2}/X_{i}@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\
0@>{}>{}>X_{i}@>{}>{}>X_{i+3}@>{}>{}>X_{i+3}/X_{i}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\ S_{i+3}@ =S_{i+3}\\\ @V{}V{}V@V{}V{}V\\\ 00\end{CD}$
Hence
$X_{i+3}/X_{i}\in\langle{\mathcal{C}}_{i+1},{\mathcal{C}}_{i+2},{\mathcal{C}}_{i+3}\rangle$.
Similarly, we can obtain
$X_{n+1}/X_{i}\in\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
Now, $0\rightarrow X_{i}\rightarrow X_{n+1}\rightarrow
X_{n+1}/X_{i}\rightarrow 0$ is the desired exact sequence.
The following lemma is well known[D].
###### Lemma 2.13.
If $\mathcal{B}$ is a subcategory of $\Lambda-mod$, then
${}^{\bot}\\!((^{\bot}\\!\mathcal{B})^{\bot})=^{\bot}\\!\mathcal{B}$ and
$(^{\bot}\\!(\mathcal{B}^{\bot}))^{\bot}=\mathcal{B}^{\bot}$ and
$(^{\bot}\\!\mathcal{B},(^{\bot}\\!\mathcal{B})^{\bot})$ and
$(\mathcal{B}^{\bot},^{\bot}\\!(\mathcal{B}^{\bot}))$ are both torsion pairs.
The following means that the condition $(2)$ in Definition 2.3 will be
superfluous in some conditions.
###### Corollary 2.14.
Let ${\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n}$ be full
subcategories of $\Lambda\text{-mod}$, if
${\mathcal{C}}_{i}=\langle{\mathcal{C}}_{1},\cdots,$
${\mathcal{C}}_{i-1}\rangle^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
for $i=1,2,\cdots,n+1$. Then
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n})$ is an
$n-\text{torsion pair}$ on $\Lambda\text{-mod}$.
Proof: It is enough to show the second condition of the above proposition
since the first condition is clear.
By the above lemma, there is a fact:
$(\leftidx{{}^{\bot}}\\!\mathcal{C}_{n+1},\mathcal{C}_{n+1})$ is a torsion
pair since
${\mathcal{C}}_{n+1}={\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{n}\rangle}^{\bot}$.
Now, we use induction on $n$ to show.
If $n=1$, clear.
Suppose that the proposition is true for $n=k\geq 1$, we consider the case of
$n=k+1$.
Step 1,
claim:$\langle\mathcal{C}_{k+1},\mathcal{C}_{k+2}\rangle={\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{k}\rangle}^{\bot}$.
In fact, $"\subseteq"$ is clear.
$"\supseteq"$: $\forall
X\in{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{k}\rangle}^{\bot}$, by
torsion pair $(\leftidx{{}^{\bot}}\\!\mathcal{C}_{k+2},\mathcal{C}_{k+2})$,
$\exists$ an exact sequence $0\rightarrow X_{k+1}\rightarrow X\rightarrow
T_{k+2}\rightarrow 0$ such that
$X_{k+1}\in\leftidx{{}^{\bot}}\\!\mathcal{C}_{k+2}$ and
$T_{k+2}\in\mathcal{C}_{k+2}$.
$X_{k+1}\in{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{k}\rangle}^{\bot}$
since $X\in{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{k}\rangle}^{\bot}$.
Thus $X_{k+1}\in\mathcal{C}_{k+1}$ and
$X\in\langle\mathcal{C}_{k+1},\mathcal{C}_{k+2}\rangle$.
Step 2. By induction,
$(\mathcal{C}_{1},\cdots,{\mathcal{C}}_{k},\langle\mathcal{C}_{k+1},\mathcal{C}_{k+2}\rangle)$
is a $k-\text{torsion pair}$ on $\Lambda\text{-mod}$. So $\forall
X\in\Lambda\text{-mod}$, there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{k-1}}$$\textstyle{S_{k}}$$\textstyle{S}$
such that $S_{i}\in\mathcal{C}_{i}$ and
$S\in\langle\mathcal{C}_{k+1},\mathcal{C}_{k+2}\rangle$.
By torsion pair $(\leftidx{{}^{\bot}}\\!\mathcal{C}_{k+2},\mathcal{C}_{k+2})$,
there is an exact sequence $0\rightarrow S_{k+1}\rightarrow S\rightarrow
S_{k+2}\rightarrow 0$ such that
$S_{k+1}\in\leftidx{{}^{\bot}}\\!\mathcal{C}_{k+2}$ and
$S_{k+2}\in\mathcal{C}_{k+2}$.
Because $S\in\langle\mathcal{C}_{k+1},\mathcal{C}_{k+2}\rangle$, then
$S\in\langle\mathcal{C}_{1},\cdots,{\mathcal{C}}_{k}\rangle^{\bot}$, so
$S_{k+1}\in\langle\mathcal{C}_{1},\cdots,{\mathcal{C}}_{k}\rangle^{\bot}$,
hence $S_{k+1}\in\mathcal{C}_{k+1}$ since
$S_{k+1}\in\leftidx{{}^{\bot}}\\!\mathcal{C}_{k+2}$.
By pullback of $(X\rightarrow S,S_{k+1}\rightarrow S)$, we have the following
commutative diagram:
$\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\
0@>{}>{}>X_{k}@>{}>{}>X_{k+1}@>{}>{}>S_{k+1}@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\ 0@>{}>{}>X_{k}@>{}>{}>X@>{}>{}>S@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\ S_{k+2}@ =S_{k+2}\\\ @V{}V{}V@V{}V{}V\\\ 00\end{CD}$
Now, we find a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{k+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{k-1}}$$\textstyle{S_{k}}$$\textstyle{S_{k+1}}$$\textstyle{S_{k+2}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence and $S_{i}\in\mathcal{C}_{i}$.
###### Proposition 2.15.
Let $({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n})$ is an
$n-\text{torsion pair}$ on $\mathcal{C}$. Then
$\left(1\right)$
$\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{i+k}\rangle=\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+k+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$
$\left(2\right)$
$({\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{i+k})$ is a
$k-\text{torsion pair}$ on
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{i+k}\rangle$
$\left(3\right)$
$({\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1},\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{i+k}\rangle,{\mathcal{C}}_{i+k+1},\cdots,{\mathcal{C}}_{n+1})$
is an $(n-k)-\text{torsion pair}$.
Proof.(1) $"\subseteq"$: clear!
$"\supseteq"$: $\forall
X\in\mathcal{C}\bigcap{\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i}\rangle}^{\bot}\bigcap\leftidx{{}^{\bot}}{\langle{\mathcal{C}}_{i+k+1},\cdots,{\mathcal{C}}_{n+1}\rangle}$,
there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{i+1}}$$\textstyle{S_{i+k}}$$\textstyle{S_{n+1}}$
such that $X_{0}=X_{1}=\cdots=X_{i-1}=0$ and
$X_{i+k+1}=X_{i+k+2}=\cdots=X_{n+1}=X$.
(2) Checking by Definition $2.3$, the first condition holds by (1), and the
second condition holds by similar techniques in proof of proposition 2.10 and
(1).
(3) Checking by Definition $2.3$, the first condition obviously holds,
$\forall X\in\mathcal{C}$, there is a filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{i+1}}$$\textstyle{S_{i+k}}$$\textstyle{S_{n+1}}$
use the similar techniques in the last part of proof of proposition 2.12, we
have the following exact sequence:
$\begin{CD}0@>{}>{}>X_{i}@>{}>{}>X_{i+k}@>{}>{}>X_{i+k}/X_{i}@>{}>{}>0\end{CD}$
Let $\hat{S}=X_{i+k}/X_{i}$, then we have the desired filtration:
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{i+1}}$$\textstyle{\hat{S}}$$\textstyle{S_{n+1}}$
###### Corollary 2.16.
Suppose
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
is an $n-\text{torsion pair}$ series on $\mathcal{C}$. Let
$\mathcal{F}_{0}=\mathcal{T}_{n+1}=\mathcal{C},\mathcal{F}_{n+1}=\mathcal{T}_{0}=0$.
Then
$\\{(\mathcal{T}_{i+1}\bigcap\mathcal{F}_{i},\mathcal{F}_{i+1}\bigcap\mathcal{T}_{i+k+1}),\cdots,\linebreak(\mathcal{T}_{i+k}\bigcap\mathcal{F}_{i},\mathcal{F}_{i+k}\bigcap\mathcal{T}_{i+k+1})\\}$
is a $k-\text{torsion pair}$ seriies on
$\mathcal{T}_{i+k+1}\bigcap\mathcal{F}_{i}$ for $i=0,1,\cdots,\linebreak n-1$
and $k>0$.
Proof. Let
$\mathcal{C}_{1}=\mathcal{T}_{1},\mathcal{C}_{l}=\mathcal{F}_{l-1}\bigcap\mathcal{T}_{l}(2\leq
l\leq n),\mathcal{C}_{n+1}=\mathcal{F}_{n+1}$. So
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ is an
$n-\text{torsion pair}$ on $\mathcal{C}$, and
$({\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{i+k})$ is a
$k-\text{torsion pair}$ on
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,\linebreak{\mathcal{C}}_{i+k}\rangle$
. Thus
$\\{(\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{i+l}\rangle,\langle{\mathcal{C}}_{i+l+1},\cdots,{\mathcal{C}}_{i+k+l}\rangle)\mid
l=1,2,\cdots,k\\}$ is a $k-\text{torsion pair}$ series on
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{i+k}\rangle$.
But
$\langle{\mathcal{C}}_{i},\cdots,{\mathcal{C}}_{i+l}\rangle=\mathcal{F}_{i}\bigcap\mathcal{T}_{i+l},\
\langle{\mathcal{C}}_{i+l+1},\cdots,{\mathcal{C}}_{i+k+l}\rangle=\mathcal{F}_{i+1}\bigcap\mathcal{T}_{i+k+l}$.
The corollary is proved.
###### Corollary 2.17.
If $({\mathcal{D}}_{1},{\mathcal{D}}_{2},\cdots,{\mathcal{D}}_{n+1})$ is a
defect $n-\text{torsion pair}$ on $\mathcal{C}$. Then there is an unique
$n-\text{torsion pair}$
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ on
$\mathcal{C}$ such that $\mathcal{D}_{i}\subseteq\mathcal{C}_{i}$.
Proof. Let
$\mathcal{T}_{i}=\langle{\mathcal{D}}_{1},\cdots,{\mathcal{D}}_{i}\rangle,\mathcal{F}_{i}=\langle{\mathcal{D}}_{i+1},\cdots,{\mathcal{D}}_{n}\rangle$,
Then
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}}),\cdots,\linebreak({{\mathcal{T}}_{n},{\mathcal{F}}_{n}})\\}$
is an $n-\text{torsion pair}$ series on $\mathcal{C}$.
Let $\mathcal{C}_{i}=\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}$, then
$({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ is an
$n-\text{torsion pair}$ on $\mathcal{C}$ such that
$\mathcal{D}_{i}\subseteq\mathcal{C}_{i}$.
Suppose
$({\mathcal{C}}^{{}^{\prime}}_{1},{\mathcal{C}}^{{}^{\prime}}_{2},\cdots,{\mathcal{C}}^{{}^{\prime}}_{n+1})$
is an other $n-\text{torsion pair}$ on $\mathcal{C}$ such that
$\mathcal{D}_{i}\subseteq\mathcal{C}^{{}^{\prime}}_{i}$, then
$\mathcal{T}_{i}=\langle{\mathcal{D}}_{1},\cdots,{\mathcal{D}}_{i}\rangle\subseteq\langle{\mathcal{C}}^{{}^{\prime}}_{1},\cdots,{\mathcal{C}}^{{}^{\prime}}_{i}\rangle=\mathcal{T}^{{}^{\prime}}$,
Similarly, $\mathcal{F}\subseteq\mathcal{F}^{{}^{\prime}}$. Therefore,
$\mathcal{C}_{i}=\mathcal{F}_{i-1}\bigcap\mathcal{T}_{i}=\mathcal{F}^{{}^{\prime}}_{i-1}\bigcap\mathcal{T}^{{}^{\prime}}_{i}=\mathcal{C}^{{}^{\prime}}_{i}$.
The following proposition is very useful.
###### Proposition 2.18.
Suppose
$\\{({{\mathcal{T}}_{1},{\mathcal{F}}_{1}}),({{\mathcal{T}}_{2},{\mathcal{F}}_{2}})\\}$
is a $2-\text{torsion pair}$ series on $\mathcal{C}$. Then we have the
following 1 to 1 correspondence :
{$(\mathcal{T^{\prime}},\mathcal{F^{\prime}})$: $1-\text{torsion pair}$ on
${\mathcal{F}}_{1}\bigcap{\mathcal{T}}_{2}$ } $\autorightleftharpoons{F}{G}$
{$({{\mathcal{T}}_{3},{\mathcal{F}}_{3}})$: $1-\text{torsion pair}$ on
$\mathcal{C}$ such that
${\mathcal{T}}_{1}\subseteq{\mathcal{T}}_{3}\subseteq{\mathcal{T}}_{2}$} where
$F((\mathcal{T^{\prime}},\mathcal{F^{\prime}}))=(\langle{{\mathcal{T}}_{1}},\mathcal{T^{\prime}}\rangle,\langle\mathcal{F^{\prime}},{\mathcal{F}}_{2}\rangle),G(({\mathcal{T}}_{3},{\mathcal{F}}_{3}))=({\mathcal{T}}_{3}\bigcap{{\mathcal{F}}_{1},{\mathcal{F}}_{3}}\bigcap{\mathcal{T}}_{2})$.
Proof: By Proposition 2.5,Theorem $2.11$ and Proposition 2.15, it is clear.
###### Remark 2.19.
The above lemma has a lot of generalized forms since we have so many results.
And those forms can give a finer characterization for torsion pairs and module
categories. For example, Theorem $2.1$ in [AK].
The following is an example of $n-\text{torsion pair}$.
###### Example 2.20.
Let $T$ be a tilting module, ${T_{1},T_{2},\dots,T_{n}}$ be all non-isomorphic
indecomposable summands of $T$. Then
$(Gen(T_{1}),T_{1}^{\bot}\bigcap Gen(T_{1}\oplus
T_{2}),\dots,(T_{1}\oplus\dots\oplus T_{n-1})^{\bot}\bigcap Gen(T))$
is an $(n-1)-\text{torsion pair}$ on $Gen(T)$.
Proof. Let$X_{i}=T_{1}\oplus\cdots\oplus T_{i}$, then
$(GenX_{i},X_{i}^{\bot})$ is a torsion pair on $\Lambda\text{-mod}$. And
$\\{(GenX_{i},X_{i}^{\bot})\mid i=,1,2,\cdots,n\\}$ is an $n-\text{torsion
pair}$ series on $\Lambda\text{-mod}$. Therefore,
$(Gen(T_{1}),T_{1}^{\bot}\bigcap Gen(T_{1}\oplus
T_{2}),\dots,(T_{1}\oplus\dots\oplus T_{n-1})^{\bot}\bigcap
Gen(T_{n}),T^{\bot})$
is an $n-\text{torsion pair}$ on $\Lambda\text{-mod}$ byTheorem $2.11$. So
$(Gen(T_{1}),T_{1}^{\bot}\bigcap Gen(T_{1}\oplus
T_{2}),\dots,(T_{1}\oplus\dots\oplus T_{n-1})^{\bot}\bigcap Gen(T))$ is an
$(n-1)-\text{torsion pair}$ on $Gen(T)$ by Proposition $2.15$.
## 3 Decomposition by projective and injective modules
In this section, we always suppose $\Lambda$ is an artin algebra. For given
artin algebra $\Gamma$, we denote: $\mathcal{P}(\Gamma)$ is the category of
all projective modules in $\Gamma\text{-mod}$, $\mathcal{I}(\Gamma)$ is the
category of all injective modules in $\Gamma\text{-mod}$;
$\mathbf{E}(\Gamma)=\\{(\mathcal{T},\mathcal{F})$ is torsion pair on
$\Gamma\text{-mod}$
$\mid\mathcal{T}\bigcap\mathcal{P}(\Gamma)=\mathcal{F}\bigcap\mathcal{I}(\Gamma)=\phi\\}$.
For a set $\Psi$ we denote the number of the elements of $\Psi$ by $\\#\Psi$.
For a subcategory $\mathcal{D}$ of $\Lambda$-mod, let
$\operatorname{Ind}\mathcal{D}$ be the set of pairwise non-isomorphic
indecomposable modules in $\mathcal{D}$. For a module $M$, let
$\operatorname{Ind}M$ = $\operatorname{Ind}(\operatorname{add}M)$
###### Definition 3.1.
Suppose $\mathcal{C}$ is a full subcategory of $\Lambda$-mod. A
$\Lambda$-module $M$ is called $\operatorname{Ext}$-projective in
$\mathcal{C}$ if $\operatorname{Ext}^{1}_{\Lambda}(M,\mathcal{C})=0$. Dually,
it is called $\operatorname{Ext}$-injective in $\mathcal{C}$ if
$\operatorname{Ext}^{1}_{\Lambda}(\mathcal{C},M)=0$.
The following lemma is from [AK].
###### Lemma 3.2.
$\left(1\right)$ $(\Lambda
e)^{\bot}=\leftidx{{}^{\bot}}(D(e\Lambda))=\Lambda/\Lambda
e\Lambda\text{-mod}$
$\left(2\right)$ $(Gen(\Lambda e),\Lambda/\Lambda e\Lambda\text{-mod})$ and
$(\Lambda/\Lambda e\Lambda\text{-mod},Cogen(D(e\Lambda)))$ are both torsion
pairs on $\Lambda\text{-mod}$.
Proof: It is clear that $\Lambda/\Lambda
e\Lambda\text{-mod}=\\{M\in\Lambda\text{-mod}\mid\ eM=0\\}$.
We claim: $(\Lambda e)^{\bot}=\\{M\in\Lambda\text{-mod}\mid\
eM=0\\}=\leftidx{{}^{\bot}}(D(e\Lambda))$.
In fact, for any $M\in(\Lambda e)^{\bot},\operatorname{Hom}(\Lambda
e,M)=eM=0$; For any
$M\in\leftidx{{}^{\bot}}(D(e\Lambda)),\operatorname{Hom}\linebreak(M,D(e\Lambda))=\operatorname{Hom}(M,\operatorname{Hom}(e\Lambda,J))=\operatorname{Hom}(e\Lambda\otimes
M,J)=D(eM)=0$ $\Longleftrightarrow eM=0$.
By (1), (2) is clear.
###### Lemma 3.3.
Let $(\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3})$ be a $2-\text{torsion
pair}$ on $\Lambda\text{-mod}$:
$\left(1\right)$If $X\in\mathcal{C}_{2}$ is $\operatorname{Ext}$-projective in
$\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$, $P_{X}\twoheadrightarrow X$
is the projective cover of $X$. Then, there exists an exact sequence
$0\rightarrow K_{X}\rightarrow P_{X}\rightarrow X\rightarrow 0$ such that
$K_{X}\in\mathcal{C}_{1}$. Especially,
$P_{X}\in\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ and
$P_{X}\not\in\mathcal{C}_{1}$.
$\left(2\right)$ If $Y\in\mathcal{C}_{2}$ is $\operatorname{Ext}$-injective in
$\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$, $Y\hookrightarrow I_{Y}$ is
the injective envelope of $Y$. Then, there exists an exact sequence
$0\rightarrow Y\rightarrow I_{Y}\rightarrow C_{Y}\rightarrow 0$ such that
$C_{Y}\in\mathcal{C}_{3}$. Especially,
$I_{Y}\in\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$ and
$I_{Y}\not\in\mathcal{C}_{3}$.
Proof: We only proof (1); The proof of (2) is similar.
By $(\mathcal{C}_{1},\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle)$, there is
a exact sequence $0\rightarrow K_{X}\rightarrow P_{X}\rightarrow L\rightarrow
0$ such that $K_{X}\in\mathcal{C}_{1}$ and
$L\in\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$, obviously, there is an
epimorphism $\eta:L\rightarrow X$ if we apply $\operatorname{Hom}(-,X)$ to the
exact sequence. Since
$\operatorname{Ker}\eta\in\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$ and
$X$ is $\operatorname{Ext}$-projective in
$\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$, $\eta$ is split. Thus
$L=X\oplus\operatorname{Ker}\eta$, by the minimality of projective cover,
$L=X$.
###### Lemma 3.4.
Let $(\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3})$ be a $2-\text{torsion
pair}$ on $\Lambda\text{-mod}$, and $X\in\Lambda\text{-mod}$ has a filtration
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{S_{1}}$$\textstyle{S_{2}}$$\textstyle{S_{3}}$
$\left(1\right)$ If $X$ is projective and $S_{3}=0$, then $S_{2}$ is
$\operatorname{Ext}$-projective in
$\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$ or $S_{2}=0$
$\left(2\right)$If $X$ is injective and $S_{1}=0$, then $S_{2}$ is
$\operatorname{Ext}$-injective in
$\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ or $S_{2}=0$
Proof: We only proof (1); The proof of (2) is similar.
Since $S_{3}=0$, $X\cong X_{3}\cong X_{3}$. Then $0\rightarrow
X_{1}\rightarrow X\rightarrow S_{2}\rightarrow 0$ is an exact sequence such
that
$X_{1}\in\mathcal{C}_{1},S_{2}\in\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$.
By Proposition 1.11 in Chapter 6 of [ASS], $S_{2}$ is
$\operatorname{Ext}$-projective in
$\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$.
###### Proposition 3.5.
Let $({\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{n+1})$ be an
$n-\text{torsion pair}$ on $\Lambda$-mod. Then there exists bijections:
$\left(1\right)$ $F:\operatorname{Ind}\
\mathcal{P}(\Lambda)\rightarrow\\{X\in\operatorname{Ind}\ \mathcal{C}_{i}\mid$
$X$ is $\operatorname{Ext}$-projective in
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle\\}$;
$\left(2\right)$ $G:\operatorname{Ind}\
\mathcal{I}(\Lambda)\rightarrow\\{Y\in\operatorname{Ind}\ \mathcal{C}_{j}\mid$
$Y$ is $\operatorname{Ext}$-injective in
$\langle{\mathcal{C}}_{1},{\mathcal{C}}_{2},\cdots,{\mathcal{C}}_{j}\rangle\\}$.
Proof: We only proof (1); The proof of (2) is similar.
Step 1. For any indecomposable projective $\Lambda$-module $P$, there is a
filtration
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$$\textstyle{S_{i-1}}$$\textstyle{S_{i}}$$\textstyle{S_{i+1}}$$\textstyle{S_{n+1}}$
such that $0\rightarrow X_{i}\rightarrow X_{i+1}\rightarrow S_{i+1}\rightarrow
0$ is an exact sequence and $S_{i}\in\mathcal{C}_{i}$.
Assume that $S_{i}\in\\{S_{1},S_{2},\cdots,S_{n+1}\\}$ is the last non-zero
module, then $S_{i+1}=\cdots=S_{n+1}=0$ and $X_{i}=X_{i+1}=\cdots=X_{n+1}=P$.
Now, we consider the following filtration
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$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$$\textstyle{S^{\prime}_{i-1}}$$\textstyle{S_{i}}$$\textstyle{S_{i+1}}$
By lemma 3.4, $S_{i}$ is $\operatorname{Ext}$-projective in
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
We denote $F(P)=S_{i}$.
Step2. Suppose $X\in\operatorname{Ind}\mathcal{C}_{i}$ such that $X$ is
$\operatorname{Ext}$-projective in
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
Then we denote the projective cover of $X$ By $P_{X}$ and denote
$F^{-1}(X)=P_{X}$.
Step 3. It is clear that $F^{-1}F(P)=P$ for any indecomposable projective
module $P$. On the other hand, since
$(\langle{\mathcal{C}}_{1},\cdots,{\mathcal{C}}_{i-1}\rangle,{\mathcal{C}}_{i},\langle{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle)$
is a $3-\text{torsion pair}$ on $\Lambda$-mod, by Lemma 3.3, $FF^{-1}(X)=X$
for any $X\in\operatorname{Ind}\mathcal{C}_{i}$ which is
$\operatorname{Ext}$-projective in
$\langle{\mathcal{C}}_{i},{\mathcal{C}}_{i+1},\cdots,{\mathcal{C}}_{n+1}\rangle$.
###### Corollary 3.6.
Let $(\mathcal{T},\mathcal{F})$ be a torsion pair on $\Lambda\text{-mod}$.
Then
$\left(1\right)$ there is an idempotent $e$ such that
$\mathcal{T}\bigcap\mathcal{P}(\Lambda)=\operatorname{add}\Lambda e$, and
$\mathcal{T}\bigcap(\Lambda e)^{\bot}$ has no $\operatorname{Ext}$-projective
modules in $(\Lambda e)^{\bot}$;
$\left(2\right)$ there is an idempotent $e$ such that
$\mathcal{F}\bigcap\mathcal{I}(\Lambda)=\operatorname{add}D(e\Lambda)$, and
$\leftidx^{\bot}D(e\Lambda)\bigcap\mathcal{F}$ has no
$\operatorname{Ext}$-injective modules in $\leftidx^{\bot}D(e\Lambda)$.
Proof: We only proof (1); The proof of (2) is similar.
The first statement is clear, only the second one needs a proof:
$(Gen(\Lambda e),(\Lambda e)^{\bot})$ is a torsion pair since $\Lambda e$ is a
projective module. So we have a $2-\text{torsion pair}$ series
$\\{(Gen(\Lambda e),(\Lambda e)^{\bot}),(\mathcal{T},\mathcal{F})\\}$, and we
have a $2-\text{torsion pair}$ $(Gen(\Lambda e),(\Lambda
e)^{\bot}\bigcap\mathcal{T},\mathcal{F})$.
Suppose that $X\in\mathcal{T}\bigcap(\Lambda e)^{\bot}$ is
$\operatorname{Ext}$-projective in $(\Lambda e)^{\bot}$. Then obviously,
$X\not\in Gen(\Lambda e)$. Let $f:P_{X}\twoheadrightarrow X$ is the projective
cover of $X$. Then by proposition 3.4, $P_{X}\in\mathcal{T}$, and $X\in
Gen(\Lambda e)$, this is a contradiction!
###### Lemma 3.7.
Let $(\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3})$ be a $2-\text{torsion
pair}$ on $\Lambda\text{-mod}$:
$\left(1\right)$ If $\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ is closed
under kernel, $X\in\mathcal{C}_{1}$ is $\operatorname{Ext}$-projective in
$\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$, and
$f:P_{X}\twoheadrightarrow X$ is the projective cover of $X$, then $P_{X}=X$
or $P_{X}\not\in\leftidx^{\bot}\mathcal{C}_{3}$;
$\left(2\right)$ If $\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$ is closed
under cokernel, $X\in\mathcal{C}_{3}$ is $\operatorname{Ext}$-injective in
$\langle\mathcal{C}_{2},\mathcal{C}_{3}\rangle$, and $g:X\hookrightarrow
I_{X}$ is the injective envelope of $X$, then $I_{X}=X$ or
$I_{X}\not\in{\mathcal{C}_{1}}^{\bot}$.
Proof: We only proof (1); The proof of (2) is similar.
Suppose
$P_{X}\in\leftidx^{\bot}\mathcal{C}_{3}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$,
then exact sequence
$\begin{CD}0@>{}>{}>\operatorname{Ker}f@>{}>{}>P_{X}@>{f}>{}>X@>{}>{}>0\end{CD}$
is split in $\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ since
$X\in\mathcal{C}_{1}$ is $\operatorname{Ext}$-projective in
$\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$ and
$Kerf\in\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$.
###### Corollary 3.8.
Let $(\mathcal{T},\mathcal{F})$ be a torsion pair on $\Lambda\text{-mod}$.
$\left(1\right)$ If there are idempotents $e^{0},e^{1}$ such that
$\operatorname{add}\Lambda e^{0}\bigcap\operatorname{add}\Lambda e^{1}=0$,
$\mathcal{F}\bigcap\mathcal{I}(\Lambda)=\operatorname{add}D(e^{0}\Lambda)$,
$\mathcal{T}\bigcap\mathcal{P}(\Lambda/\Lambda
e^{0}\Lambda)=\operatorname{add}(\Lambda/\Lambda e^{0}\Lambda)e^{1}$. Then
$\mathcal{T}\bigcap\mathcal{P}(\Lambda)=\phi$ if and only if for any
$P\in\operatorname{add}\Lambda e^{1}$, $P\not\in\Lambda/\Lambda
e^{0}\Lambda\text{-mod}$;
$\left(2\right)$ If there are orthogonal idempotents
$\varepsilon^{0},\varepsilon^{1}$ such that
$\mathcal{T}\bigcap\mathcal{P}(\Lambda)=\operatorname{add}\Lambda e^{0}$,
$\mathcal{F}\bigcap\mathcal{I}(\Lambda/\Lambda\varepsilon^{0}\Lambda)=\operatorname{add}D(\varepsilon^{1}(\Lambda/\Lambda\varepsilon^{0}\Lambda))$.
Then $\mathcal{F}\bigcap\mathcal{I}(\Lambda)=\phi$ if and only if for any
$I\in\operatorname{add}\ D(\varepsilon^{1}\Lambda)$,
$I\not\in\Lambda/\Lambda\varepsilon^{0}\Lambda\text{-mod}$.
Proof: We only proof (1); The proof of (2) is similar.
$"\Rightarrow"$ Since $(\Lambda/\Lambda
e^{0}\Lambda\text{-mod},Cogen(D(e^{0}\Lambda)))$ is a torsion pair by lemma
3.2, $(\mathcal{T},\mathcal{F}\bigcap\Lambda/\Lambda
e^{0}\Lambda\text{-mod},\linebreak Cogen(D(e^{0}\Lambda))$ is a
$2-\text{torsion pair}$ on $\Lambda$-mod. Suppose $0\neq
P\in\operatorname{add}\Lambda e^{1}$. Then
$P/e^{0}P\in\operatorname{add}(\Lambda/\Lambda e^{0}\Lambda)e^{1}$ and
$P/e^{0}P\neq 0$. So by the above lemma, $P=P/e^{0}P\in\mathcal{T}$ or
$P\not\in\Lambda/\Lambda e^{0}\Lambda\text{-mod}$. Since
$\mathcal{T}\bigcap\mathcal{P}(\Lambda)=\phi$, $P\not\in\Lambda/\Lambda
e^{0}\Lambda\text{-mod}$.
$"\Leftarrow"$ Suppose $\mathcal{T}\bigcap\mathcal{P}(\Lambda)\neq\phi$. Then
there exists $0\neq P\in\mathcal{T}\bigcap\mathcal{P}(\Lambda)$. Then $P$ is
also projective in $\Lambda/\Lambda e^{0}\Lambda\text{-mod}$. So
$P\in\operatorname{add}\Lambda e^{1}$. This is a contradiction.
Now we start to show the structure of torsion pairs by decomposing them by
projective modules and injective modules. First we give some notations .
We always assume that $\Delta=\\{e_{1},e_{2},\cdots,e_{n}\\}$ is a fixed
complete set of primitive orthogonal idempotents of $\Lambda$. Given
$S=\\{\Delta_{0},\Delta_{1},\Delta_{2},\cdots,\Delta_{m}\mid\Delta_{i}\subseteq\Delta\\}$
such that $\Delta_{1},\Delta_{2},\cdots,\Delta_{m}\neq\phi$ and
$\Delta_{i}\bigcap\Delta_{j}=\phi$ for $i\neq j$, we have the following
notations : $e_{S}^{i}=\sum_{e\in\Delta_{i}}e$,
$\varepsilon_{S}^{i}=\sum^{i}_{j=0}e_{S}^{j}$;
$\Lambda_{S}^{0}=\Lambda,\Lambda_{S}^{1}=\frac{\Lambda_{S}^{0}}{\Lambda_{S}^{0}e_{S}^{0}\Lambda_{S}^{0}}=\frac{\Lambda}{\Lambda\varepsilon_{S}^{0}\Lambda},\cdots,\Lambda_{S}^{m+1}=\frac{\Lambda_{S}^{m}}{\Lambda_{S}^{m}e_{S}^{m}\Lambda_{S}^{m}}=\frac{\Lambda}{\Lambda\varepsilon_{S}^{m}\Lambda};\mathrm{P}_{i}(\Lambda_{S}^{i})=\oplus_{e\in\Delta_{i}}\Lambda_{S}^{i}e,\mathrm{I}_{i}(\Lambda_{S}^{i})=\oplus_{e\in\Delta_{i}}D(e\Lambda_{S}^{i})$.
###### Definition 3.9.
Suoppose $S$ is as the above. It is called a 2-type part partition if:
$\left(1\right)$ $\forall 0<2i\leq m$ and $e\in\Delta_{2i}$,
$e_{S}^{2i-1}\Lambda_{S}^{2i-1}e\neq 0$; $\left(2\right)$ $\forall 1<2i+1\leq
m$, and $e\in\Delta_{2i+1}$, $e\Lambda_{S}^{2i}e_{S}^{2i}\neq 0$.
Dually, $S$ is called a 2-type part partition if: $\left(1\right)$ $\forall
0<2i\leq m$ and $e\in\Delta_{2i}$, $e\Lambda_{S}^{2i-1}e_{S}^{2i-1}\neq 0$;
$\left(2\right)$ $\forall 1<2i+1\leq m$, and $e\in\Delta_{2i+1}$,
$e_{S}^{2i}\Lambda_{S}^{2i}e\neq 0$.
###### Lemma 3.10.
Let $I$ be an ideal of $\Lambda$, and $e,e^{\prime}$ be two idempotents. Then
$\operatorname{Hom}_{\Lambda/I}((\Lambda/I)\cdot
e,D(e^{\prime}\cdot\Lambda/I)=0$ if and only if $e^{\prime}\cdot\Lambda/I\cdot
e=0$.
Proof: Notice that $\operatorname{Hom}_{\Lambda/I}((\Lambda/I)\cdot
e,D(e^{\prime}\cdot\Lambda/I)=D(e^{\prime}\cdot\Lambda/I\cdot e)$.
We give the following notations for describing our theorem easily.
$\mathfrak{M}=\\{(\mathcal{T},\mathcal{F})\mid(\mathcal{T},\mathcal{F})$ is a
torsion pair on $\Lambda\text{-mod}\\}$;
$\mathfrak{N}=\\{(S=\\{\Delta^{\prime}_{0},\Delta^{\prime}_{1},\Delta^{\prime}_{2},\cdots,\Delta^{\prime}_{m}\\},(\mathcal{T}^{\prime},\mathcal{F}^{\prime}))\mid$
S is a 1-type part partition,
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in\mathbf{E}(\Lambda_{S}^{m+1})\\}$.
$\mathfrak{N}^{\prime}=\\{(S=\\{\Delta^{\prime}_{0},\Delta^{\prime}_{1},\Delta^{\prime}_{2},\cdots,\Delta^{\prime}_{m}\\},(\mathcal{T}^{\prime},\mathcal{F}^{\prime}))\mid$
S is a 2-type part partition,
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in\mathbf{E}(\Lambda_{S}^{m+1})\\}$.
Now we are in a position to give a demonstration of how to decompose a torsion
pair into $n-\text{torsion pair}$ by projective modules and injective modules.
Let$(\mathcal{T},\mathcal{F})$ be an torsion pair on $\Lambda\text{-mod}$:
a.Let
$\mathcal{T}^{0}=\mathcal{T},\mathcal{F}^{0}=\mathcal{F},\Lambda^{0}=\Lambda$,
there exists some $\Delta_{0}\subseteq\Delta$ such that
$\mathcal{T}^{0}\bigcap\mathcal{P}(\Lambda^{0})=\operatorname{add}\oplus_{e\in\Delta_{0}}\Lambda^{0}e=\operatorname{add}\mathrm{P}_{0}(\Lambda^{0})$.
Let
$\mathcal{T}^{1}=\mathcal{T}^{0}\bigcap(\mathrm{P_{0}}(\Lambda^{0}))^{\bot},\mathcal{F}^{1}=\mathcal{F}^{0}$,
$\Lambda^{1}=\Lambda/\Lambda e^{0}\Lambda$ where
$e^{0}=\sum_{e\in\Delta_{0}}e$. Then $(\mathcal{T}^{1},\mathcal{F}^{1})$ is a
torsion pair on $\Lambda^{1}\text{-mod}$ and
$\mathcal{T}^{1}\bigcap\mathcal{P}(\Lambda^{1})=\\{0\\}$ by corollary 3.6.
Hence we have a $2-\text{torsion pair}$
$(Gen\mathrm{P}_{0}(\Lambda^{0}),\mathcal{T}^{1},\mathcal{F}^{1})$ on
$\Lambda\text{-mod}$;
b.There exists some $\Delta_{1}\subseteq\Delta-\Delta_{0}$ such that
$\mathcal{F}^{1}\bigcap\mathcal{I}(\Lambda^{1})=\operatorname{add}\oplus_{e\in\Delta_{1}}D(e\Lambda^{1})=\operatorname{add}\mathrm{I}_{1}(\Lambda^{1})$.
Let
$\mathcal{T}^{2}=\mathcal{T}^{1},\mathcal{F}^{2}=\mathcal{F}^{1}\bigcap\leftidx{{}^{\bot}}\\!\mathrm{I}_{1}(\Lambda^{1})$,
$\Lambda^{2}=\Lambda/\Lambda\varepsilon^{1}\Lambda$ where
$\varepsilon^{1}=\sum_{e\in\Delta_{0}\bigcup\Delta_{1}}e$. Then
$(\mathcal{T}^{2},\mathcal{F}^{2})$ is a torsion pair on $\Lambda^{2}$-mod and
$\mathcal{F}^{2}\bigcap\mathcal{I}(\Lambda^{2})=\\{0\\}$ by corollary 3.6.
Hence we have a $3-\text{torsion pair}$
$(Gen\mathrm{P}_{0}(\Lambda^{0}),\mathcal{T}^{2},\mathcal{F}^{2},Cogen\mathrm{I}_{1}(\Lambda^{1}))$
on $\Lambda$-mod;
The above operation goes on alternatively, then it will eventually stop since
$\\#\Delta$ is finite.
Finally, we obtain:
(1)
$\\{\Delta_{0},\Delta_{1},\Delta_{2},\cdots,\Delta_{m}\mid\Delta_{i}\subseteq\Delta\\}$
such that $\Delta_{1},\Delta_{2},\cdots,\Delta_{m}\neq\phi$ and
$\Delta_{i}\bigcap\Delta_{j}=\phi$ for $i\neq j$;
(2) $(\mathcal{T}^{m+1},\mathcal{F}^{m+1})$ is a torsion pair on
$\Lambda^{m+1}-mod$ and
$(\mathcal{T}^{m+1},\mathcal{F}^{m+1})\in\mathbf{E}(\Lambda^{m+1})$;
(3)
$(Gen\mathrm{P}_{0}(\Lambda^{0}),Gen\mathrm{P}_{2}(\Lambda^{2}),\cdots,\mathcal{T}^{m+1},\mathcal{F}^{m+1},\cdots,Cogen\mathrm{I}_{3}(\Lambda^{3}),Cogen\mathrm{I}_{1}(\Lambda^{1}))$
is a $(m+2)-\text{torsion pair}$ on $\Lambda\text{-mod}$;
(4)
$\Lambda=\Lambda^{0}\rightarrow\Lambda^{1}\rightarrow\cdots\rightarrow\Lambda^{m+1}$
is a series of quotient algebras.
###### Theorem 3.11.
There is a one to one correspondence between $\mathfrak{M}$ and
$\mathfrak{N}$:
$\mathfrak{M}\autorightleftharpoons{F}{G}\mathfrak{N}$
Proof: Step 1. Suppose $(\mathcal{T},\mathcal{F})\in\mathfrak{M}$. we use the
above operation. Then we get
$S=\\{\Delta_{0},\Delta_{1},\Delta_{2},\cdots,\Delta_{m}\mid\Delta_{i}\subseteq\Delta\\}$
and $(\mathcal{T}^{m+1},\mathcal{F}^{m+1})\in\mathbf{E}(\Lambda^{m+1})$, so we
need to prove S is a 2-type part partition, but it follows from corollary 3.6
and lemma 3.8. Let
$F((\mathcal{T},\mathcal{F}))=(S,(\mathcal{T}^{m+1},\mathcal{F}^{m+1}))$.
Step 2. Suppose
$(S=\\{\Delta_{0},\Delta_{1},\Delta_{2},\cdots,\Delta_{m}\\},(\mathcal{T}^{\prime},\mathcal{F}^{\prime}))\in\mathfrak{N}$.
By induction on $m$. It is easy to see that
$(Gen\mathrm{P}_{0}(\Lambda_{S}^{0}),Gen\mathrm{P}_{2}(\Lambda_{S}^{2}),\cdots,\mathcal{T}^{\prime},\mathcal{F}^{\prime},\cdots,Cogen\mathrm{I}_{3}(\Lambda_{S}^{3}),Cogen\mathrm{I}_{1}(\Lambda_{S}^{1}))$
is a $(m+2)-\text{torsion pair}$ on $\Lambda-mod$. Let
$G((S,(\mathcal{T}^{\prime},\mathcal{F}^{\prime})))=(\mathcal{T},\mathcal{F})=(\langle
Gen\mathrm{P}_{0}(\Lambda_{S}^{0}),\linebreak
Gen\mathrm{P}_{2}(\Lambda_{S}^{2}),\cdots,\mathcal{T}^{\prime}\rangle,\langle\mathcal{F}^{\prime},\cdots,Cogen\mathrm{I}_{3}(\Lambda_{S}^{3}),Cogen\mathrm{I}_{1}(\Lambda_{S}^{1})\rangle)$.
Claim:$\mathcal{T}\cap\mathcal{P}(\Lambda)=\operatorname{add}\
\mathrm{P}_{0}(\Lambda^{0})$.
Otherwise, there exists some $e\in\Delta-\Delta_{0}$ such that $\Lambda
e\in\mathcal{T}$. By proposition 3.5 and the above $(m+2)-\text{torsion
pair}$, there exists $0\neq X\in
Gen\mathrm{P}_{2i}(\Lambda_{S}^{2i})(\text{or}\ \mathcal{T}^{\prime})$ for
some $i\neq 0$, such that $X$ is $\operatorname{Ext}$-projective in
$\Lambda_{S}^{2i}(\text{or}\ \Lambda_{S}^{m+1})$, and the projective cover of
$X$ is $\Lambda e$ since
$\mathcal{T}^{\prime}\cap\mathcal{P}(\Lambda_{S}^{m+1})=\phi$ and
$X\in\mathrm{P}_{2i}(\Lambda_{S}^{2i})$. However, since $S$ is a 2-type part
partition, $e_{S}^{2i-1}\Lambda_{S}^{2i-1}e\neq 0$. So
Hom${}_{\Lambda_{S}^{2i-1}}(X,D(e_{S}^{2i-1}\Lambda_{S}^{2i-1}))$ =
Hom${}_{\Lambda_{S}^{2i-1}}(\Lambda_{S}^{2i-1}e,D(e_{S}^{2i-1}\Lambda_{S}^{2i-1}))\neq
0$. Hence $X\not\in\leftidx^{\bot}\mathcal{F}$. So $\Lambda
e\not\in\leftidx^{\bot}\mathcal{F}$. A contradiction!
Step by step, we know
$F(\mathcal{T},\mathcal{F})=(S,(\mathcal{T}^{\prime},\mathcal{F}^{\prime}))$.
Step 3. Given $(\mathcal{T},\mathcal{F})\in\mathfrak{M}$, it is clear that
$GF(\mathcal{T},\mathcal{F})=(\mathcal{T},\mathcal{F})$.
Dually, if we start to decompose a torsion pair from the right hand (torsion-
free class), Then we have the following theorem :
###### Theorem 3.12.
There is a one to one correspondence between $\mathfrak{M}$ and
$\mathfrak{N}^{\prime}$:
$\mathfrak{M}\autorightleftharpoons{F^{\prime}}{G^{\prime}}\mathfrak{N}^{\prime}$
It’s natural to ask that what is the relation between the above two kinds of
decomposition. The following theorem indicates that the decomposition of a
torsion pair from left hand and right hand are the same.
###### Theorem 3.13.
Suppose $(\mathcal{T},\mathcal{F})\in\mathfrak{M}$,
$F((\mathcal{T},\mathcal{F}))=(S^{\prime}=\\{\Delta^{\prime}_{0},\Delta^{\prime}_{1},\Delta^{\prime}_{2},\cdots,\Delta^{\prime}_{u}\\},(\mathcal{T}^{\prime},\mathcal{F}^{\prime}))$
and
$F^{\prime}((\mathcal{T},\mathcal{F}))=(S^{\prime\prime}=\\{\Delta^{\prime\prime}_{0},\Delta^{\prime\prime}_{1},\Delta^{\prime\prime}_{2},\cdots,\Delta^{\prime\prime}_{v}\\},(\mathcal{T}^{\prime\prime},\mathcal{F}^{\prime\prime}))$.
Then
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})=(\mathcal{T}^{\prime\prime},\mathcal{F}^{\prime\prime}).$
Proof: It is clear that
$\mathcal{T}^{\prime}=\mathcal{T}\cap\Lambda^{u+1}_{S^{\prime}}\text{-mod},\mathcal{F}^{\prime}=\mathcal{F}\cap\Lambda^{u+1}_{S^{\prime}}\text{-mod}$.
And $(\mathcal{T}^{\prime\prime},\mathcal{F}^{\prime\prime})$ has the similar
property. So we only need to prove
$\Delta^{\prime}_{0}\cup\Delta^{\prime}_{1}\cup\cdots\cup\Delta^{\prime}_{u}=\Delta^{\prime\prime}_{0}\cup\Delta^{\prime\prime}_{1}\cup\cdots\cup\Delta^{\prime\prime}_{v}$.
For convenience, we give the following notations for any given $i\geq 0$:
$L^{i}_{S^{\prime}}=\langle Gen\mathrm{P}_{2j}(\Lambda^{2j}_{S^{\prime}})\mid
0\leq 2j\leq max\\{u,i\\}\rangle$;
$R^{i}_{S^{\prime}}=\langle
Cogen\mathrm{I}_{2j+1}(\Lambda^{2j+1}_{S^{\prime}})\mid 0\leq 2j+1\leq
max\\{u,i\\}\rangle$;
$L^{i}_{S^{\prime\prime}}=\langle
Gen\mathrm{P}_{2j+1}(\Lambda^{2j+1}_{S^{\prime\prime}})\mid 0\leq 2j+1\leq
max\\{v,i\\}\rangle$;
$R^{i}_{S^{\prime\prime}}=\langle
Cogen\mathrm{I}_{2j}(\Lambda^{2j}_{S^{\prime\prime}})\mid 0\leq 2j\leq
max\\{v,i\\}\rangle$.
We just prove
$\Delta^{\prime}_{0}\cup\Delta^{\prime}_{1}\cup\cdots\cup\Delta^{\prime}_{u}\subseteq\Delta^{\prime\prime}_{0}\cup\Delta^{\prime\prime}_{1}\cup\cdots\cup\Delta^{\prime\prime}_{v}$.
For this, we just need to prove: $\forall i\geq 0$,
$L^{2i+1}_{S^{\prime}}\subseteq
L^{2i+1}_{S^{\prime\prime}};R^{2i}_{S^{\prime}}\subseteq
R^{2i}_{S^{\prime\prime}}$.
For $i=0,R^{0}_{S^{\prime}}=\\{0\\}\subseteq
Cogen\mathrm{I}_{0}(\Lambda^{0}_{S^{\prime\prime}})=R^{0}_{S^{\prime\prime}}$,
$L^{1}_{S^{\prime}}=Gen\mathrm{P}_{0}(\Lambda^{0}_{S^{\prime}})\subseteq
Gen\mathrm{P}_{1}(\Lambda^{1}_{S^{\prime\prime}})=L^{1}_{S^{\prime\prime}}$.
Now we assume the theorem holds for $i\leq k-1$. Then
$\Lambda^{2k}_{S^{\prime\prime}}$ is a quotient algebra of
$\Lambda^{2k-1}_{S^{\prime}}$ since
$\Delta^{\prime}_{0}\cup\Delta^{\prime}_{1}\cup\cdots\cup\Delta^{\prime}_{2k-2}\subseteq\Delta^{\prime\prime}_{0}\cup\Delta^{\prime\prime}_{1}\cup\cdots\cup\Delta^{\prime\prime}_{2k-1}$.
So $\Lambda^{2k}_{S^{\prime\prime}}\text{-mod}$ is a full subcategory of
$\Lambda^{2k-1}_{S^{\prime}}\text{-mod}$.
Suppose $0\neq
X\in\operatorname{add}\mathrm{I}_{2k-1}(\Lambda^{2k-1}_{S^{\prime}})$. So $X$
is $\operatorname{Ext}$-injective in
$\Lambda^{2k}_{S^{\prime\prime}}\text{-mod}$. By torsion pair
$(\mathcal{F}\cap\Lambda^{2k}_{S^{\prime\prime}}\text{-mod},R^{2k-2}_{S^{\prime\prime}})$
on $\mathcal{F}$, there exists an exact sequence $0\rightarrow
X_{1}\rightarrow X\rightarrow X_{2}\rightarrow 0$ such that
$X_{1}\in\mathcal{F}\cap\Lambda^{2k}_{S^{\prime\prime}}\text{-mod}$ and
$X_{2}\in R^{2k-2}_{S^{\prime\prime}}$. For every
$Y\in\Lambda^{2k}_{S^{\prime\prime}}\text{-mod}$, applying
Hom${}_{\Lambda}(Y,-)$ to this exact sequence, we get an exact sequence:
$\operatorname{Hom}_{\Lambda}(Y,X_{2})\rightarrow\operatorname{Ext}^{1}_{\Lambda}(Y,X_{1})\rightarrow\operatorname{Ext}^{1}_{\Lambda}(Y,X)$.
Since Ext${}^{1}_{\Lambda}(Y,X)=0$ and
$Y\in\leftidx^{\bot}(R^{2k-2}_{S^{\prime\prime}})$,
Ext${}^{1}_{\Lambda}(Y,X_{1})=0$. So $X_{1}$ is $\operatorname{Ext}$-injective
in $\Lambda^{2k}_{S^{\prime\prime}}\text{-mod}$. Thus
$X_{1}\in\operatorname{add}\mathrm{I}_{2k}(\Lambda^{2k}_{S^{\prime\prime}})$.
So $X\in R^{2k}_{S^{\prime\prime}}$. Therefore, $R^{2k}_{S^{\prime}}\subseteq
R^{2k}_{S^{\prime\prime}}$, and similarly, we have
$L^{2k+1}_{S^{\prime}}\subseteq L^{2k+1}_{S^{\prime\prime}}$.
## 4 Examples
In this section, we will use the results developed in the previous two
sections to characterize torsion pairs on some particular module categories.
Those results will be related to [BBM], [BM], [HJR], [N], [HJ], [BK]. We
always assume $K$ is a filed. If Q is a quiver and $\Delta\in Q_{0}$ where
$Q_{0}$ is the set of vertices of Q, then we denote the full sub-quiver of Q
containing $\Delta$ by $Q(\Delta)$. We give the following definition.
###### Definition 4.1.
Let Q be a quiver , $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ a tuple
such that $\Delta_{i}\subseteq Q_{0},\ \Delta_{i}\bigcap\Delta_{j}=\phi\
\forall i\neq j,\Delta_{0}\neq\phi$. If $\forall i>0$ and $v\in\Delta_{2i+1}$
there is a path from some vertex in $\Delta_{2i}$ to $v$ in the sub-quiver
$Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-1})$, and $\forall i>0$ and
$v\in\Delta_{2i}$ there is a path from $v$ to some vertex in $\Delta_{2i-1}$in
the sub-quiver $Q((Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-2})$. Then we
call $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a 2-type part partition
of Q. The following diagram shows the relation:
---
$\textstyle{\Delta_{0}}$$\textstyle{\Delta_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots}$$\textstyle{\Delta_{1}}$$\textstyle{\Delta_{3}}$$\textstyle{\dots}$
Dually, we we call $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a 2-type
part partition of Q if $\forall i>0$ and $v\in\Delta_{2i+1}$ there is a path
from $v$ to some vertex in $\Delta_{2i}$ in the sub-quiver
$Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-1})$, and $\forall i>0$ and
$v\in\Delta_{2i}$ there is a path from some vertex in $\Delta_{2i-1}$ to $v$
in the sub-quiver $Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-2})$. The
following diagram shows the relation:
$\textstyle{\Delta_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots}$$\textstyle{\Delta_{0}}$$\textstyle{\Delta_{2}}$$\textstyle{\Delta_{4}}$$\textstyle{\dots}$
Especially, if $\forall i>0,\Delta_{2i-1}$ contains all sink points in
$Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-2})$, $\Delta_{2i}$ contains
all source points in $Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-1})$,
then we call $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a strong 1-type
part partition of Q. If $\forall i>0,\Delta_{2i-1}$ contains all source points
in $Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-2})$, $\Delta_{2i}$
contains all sink points in
$Q(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{2i-1})$, then we call
$\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a strong 2-type part
partition of Q.
If $\Delta_{0}\bigcup\Delta_{1}\bigcup\dots\bigcup\Delta_{m}=Q_{0}$ we call
$\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a complete partition of Q.
We have the following lemma.
###### Lemma 4.2.
Let Q be a acyclic quiver and $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$
is a strong 1-type part partition of Q. Then
$\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a 1-type part partition of
Q.
If $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a strong 2-type part
partition of Q. Then $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a
2-type part partition of Q.
For a quiver $Q$, we denote $\mathbf{E}(KQ)$ by $\mathbf{E}(Q)$. Now we have
the following theorem which is the path algebra’s version of Theorem 3.11.
###### Theorem 4.3.
Let Q be a acyclic quiver. Then we have a bijection between the set
$(\mathcal{T},\mathcal{F})$ which is a torsion pair on $KQ$-mod and the set of
the pair $(\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$;
$(\mathcal{T^{\prime}},\mathcal{F^{\prime}}))$, where
$\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a 1-type part partition of Q
and
$(\mathcal{T^{\prime}},\mathcal{F^{\prime}})\in\mathbf{E}(KQ(Q_{0}-\Delta_{0}-\Delta_{1}-\dots-\Delta_{m}))$.
The dual form of the theorem is similar, so we don’t demonstrate here. Now let
$A_{n}$ be the following quiver: $1\rightarrow 2\rightarrow
3\rightarrow\dots\rightarrow n$. Applying the above theorem to the quiver
$A_{n}$ , we have the following theorem.
###### Theorem 4.4.
There exists a bijection between torsion pairs on $KA_{n}$-mod and complete
strong 1-type part partition sets of $A_{n}$.
Proof: It is easy to see $\mathbf{E}(KA_{m})=\phi$ for every $m$. And a
complete partition of Q is a 2-type part partition if and only if it is strong
1-type part partition. The rest is clear by the above theorem.
If we observe the bijection above, then we obtain some simple corollaries.
###### Corollary 4.5.
Given a torsion pair $(\mathcal{T},\mathcal{F})$ on $KA_{n}$-mod, then there
exists a unique pair $(T,F)$ such that T, F are basic partial tilting modules,
$\\#\operatorname{Ind}(T\bigoplus F)=n$, and
$\mathcal{T}=Gen(T),\mathcal{F}=Cogen(F)$.
###### Corollary 4.6.
If $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a complete strong 1-type
part partition of $A_{n}$, then the corresponding torsion pair is induced by
tilting modules if and only if $v_{1}\in\Delta_{0}$.
If $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{m}\\}$ is a complete strong 2-type
part partition of $A_{n}$, then the corresponding torsion pair is induced by
cotilting modules if and only if $v_{n}\in\Delta_{0}$.
###### Proposition 4.7.
The number of torsion pairs on $KA_{n}$ is the $(n+1)-{th}$ Catalan number
$C_{n+1}=\frac{1}{n+2}{2n+2\choose n+1}$.
Proof: Adding one vertex to $A_{n}$, then we have the quiver
$A_{n+1}:1\rightarrow 2\rightarrow 3\rightarrow\dots\rightarrow n\rightarrow
n+1$. We have a torsion pair on $KA_{n+1}$-mod:
$(KA_{n}\text{-mod},\mathcal{P}(KA_{n+1}))$. So we have a bijection between
torsion pairs on $KA_{n}\text{-mod}$ and torsion pairs induced by cotilting
modules on $KA_{n+1}$-mod by proposition 2.18. The number of torsion pairs
induced by cotilting modules on $KA_{n+1}$-mod is well known which is the
$(n+1)-{th}$ Catalan number(Lemma $A.1$ in [BK]).
###### Definition 4.8.
Suppose $\Lambda$ is an artin algebra, $\mathcal{C}$ is a full subcategory of
$\Lambda$-mod. If there exists a set of full subcategories
$\\{\mathcal{C}_{i},i\in I\\}$ of $\mathcal{C}$ such that $\forall
M\in\mathcal{C}$, there uniquely exists a set of modules
$M_{i_{1}}\in\mathcal{C}_{1},M_{i_{2}}\in\mathcal{C}_{2},\dots,M_{i_{n}}\in\mathcal{C}_{n}$
where $i_{1},i_{2},\dots,i_{n}$ are mutually different such that $M\cong
M_{i_{1}}\bigoplus M_{i_{2}}\bigoplus\dots\bigoplus M_{i_{n}}$, then we call
$\mathcal{C}$ is the direct sum of $\\{\mathcal{C}_{i},i\in I\\}$, and we
denote $\mathcal{C}=\bigoplus_{i\in I}\mathcal{C}_{i}$.
We have the following correspondence.
###### Lemma 4.9.
Suppose $\Lambda$ is an artin algebra, $\mathcal{C}$ is a full subcategory of
$\Lambda$-mod, there exists a set of full subcategories
$\\{\mathcal{C}_{i},i\in I\\}$ of $\mathcal{C}$ such that
$\mathcal{C}=\bigoplus_{i\in I}\mathcal{C}_{i}$ and
$\operatorname{Hom}(X,Y)=0$ for every
$X\in\mathcal{C}_{i},Y\in\mathcal{C}_{j}$ and $i\neq j$. Then there exists a
bijection between torsion pairs on $\mathcal{C}$ and the tuple
$\\{(\mathcal{T}_{i},\mathcal{F}_{i})\\}_{i\in I}$ where
$(\mathcal{T}_{i},\mathcal{F}_{i})$ is a torsion pair on $\mathcal{C}_{i}$
Proof:Given ${(\mathcal{T},\mathcal{F})}$ a torsion pair on $\mathcal{C}$,
then
${(\mathcal{T}\bigcap\mathcal{C}_{i},\mathcal{F}\bigcap\mathcal{C}_{i})}_{i\in
I}$ is the corresponding tuple. Given the tuple
${(\mathcal{T}_{i},\mathcal{F}_{i})}_{i\in I}$ where
$(\mathcal{T}_{i},\mathcal{F}_{i})$, then $(\bigoplus_{i\in
I}\mathcal{T}_{i},\bigoplus_{i\in I}\mathcal{F}_{i})$ is the corresponding
torsion pair.
Let $\tilde{A}_{n}$ be the following quiver with vertices
$(\tilde{A}_{n})_{0}=\\{v_{1},v_{2},\dots v_{n}\\}$:
---
$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Let $J$ be the ideal of $K\tilde{A}_{n}$ generated by all arrows. We call a
finite-dimensional $K\tilde{A}_{n}$ module M is an ordinary module if there
exists N such that $J^{N}M=0$. In this condition M is a $K\tilde{A}_{n}/J^{N}$
module. So if M is indecomposable, then it is uniserial and determined by its
socle and length. Let $\mathcal{E}_{n}$ be the category of all ordinary
modules. Then $\mathcal{E}_{n}$ is closed under submodules, quotients and
extensions. We denote the simple module corresponding to the vertex $v_{i}$ by
$S_{i}$. We will give all torsion pairs on $\mathcal{E}_{n}$. For this we give
the following definition which is introduced in [BBM].
###### Definition 4.10.
Suppose $\Delta\in(\tilde{A}_{n})_{0}$. let $Ray(\Delta)$ be the category of
all modules with socle in $\operatorname{add}\bigoplus_{v_{i}\in\Delta}S_{i}$.
let $Coray(\Delta)$ be the category of all modules with top in
$\operatorname{add}\bigoplus_{v_{i}\in\Delta}S_{i}$.
For a subcategory $\mathcal{D}$ of $\mathcal{E}_{n}$. We denote
$L_{\mathcal{D}}$ be the set of all vertices $v_{i}$ such that there are
infinite indecomposable modules in $\mathcal{D}$ with $S_{i}$ as the top,
$R_{\mathcal{D}}$ be the set of all vertices $v_{j}$ such that there are
infinite indecomposable modules in $\mathcal{D}$ with $S_{i}$ as the socle .
By Definition 4.10 we have the following obvious lemma.
###### Lemma 4.11.
Suppose $\phi\neq\Delta\subseteq(\tilde{A}_{n})_{0}$. Then
$(Coray(\Delta),\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod})),(Q((\tilde{A}_{n})_{0}-\Delta)\text{-mod},Ray(\Delta))$
are two torsion pairs on $\mathcal{E}_{n}$.
Now we give the following proposition.
###### Proposition 4.12.
Suppose $\phi\neq\Delta\subseteq(\tilde{A}_{n})_{0}$. Then there is a
bijection:
$\left(1\right)$ $\\{(\mathcal{T^{\prime}},\mathcal{F^{\prime}}):\text{
\text{torsion pair} on }\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod
which is induced by cotilting modules}\\}\\\
\autorightleftharpoons{F}{$F^{\prime}$}\\{(\mathcal{T},\mathcal{F}):\text{
\text{torsion pair} on }\mathcal{E}_{n}\text{ such that
}L_{\mathcal{T}}=\Delta\\}$. In this condition
$F((\mathcal{T^{\prime}},\mathcal{F^{\prime}}))=(\langle
Coray(\Delta),\mathcal{T^{\prime}}\rangle,\mathcal{F^{\prime}}),F^{\prime}((\mathcal{T},\mathcal{F}))=(\mathcal{T}\bigcap\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod},\mathcal{F})$
$\left(2\right)$ $\\{(\mathcal{T^{\prime}},\mathcal{F^{\prime}}):\text{
\text{torsion pair} on }\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod
which is induced by tilting modules}\\}\\\
\autorightleftharpoons{G}{$G^{\prime}$}\\{(\mathcal{T},\mathcal{F}):\text{
\text{torsion pair} on }\mathcal{C}\text{ such that
}R_{\mathcal{F}}=\Delta\\}$. In this condition
$G((\mathcal{T^{\prime}},\mathcal{F^{\prime}}))=(\mathcal{T^{\prime},\langle\mathcal{F^{\prime}},\text{$Ray(\Delta)$}\rangle}),G^{\prime}((\mathcal{T},\mathcal{F}))=(\mathcal{T},\mathcal{F}\bigcap\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod})$.
Proof: We only proof (1) and (2) is similar.
$\left(1\right)$.
$\\{(Coray(\Delta),\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod}),(\mathcal{E}_{n},\\{0\\})\\}$
is a 2-torsion pair seires on $\mathcal{E}_{n}$. Then by Proposition 2.18, we
have a bijection between torsion pair $(\mathcal{T},\mathcal{F})$ on
$\mathcal{E}_{n}$ such that $Coray(\Delta)\subseteq\mathcal{T}$ and torsion
pairs on $\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod}$. It is obvious
in this condition $\Delta=L_{\mathcal{T}}$ if and only if in the corresponding
torsion pair $(\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ on
$\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod}$ $\mathcal{F^{\prime}}$
contains all projective modules which means it is induced by a cotilting
module.
The following lemma is from Corollary $4.5$ in [BBM].
###### Lemma 4.13.
Suppose $(\mathcal{T},\mathcal{F})\text{ is a \text{torsion pair}}$ on
$\mathcal{E}_{n}$. Then $L_{\mathcal{T}},R_{\mathcal{F}}$ are not both empty.
Now we have the following theorem which gives all torsion pairs on
$\mathcal{E}_{n}$.
###### Theorem 4.14.
The following are all mutually different torsion pairs on $\mathcal{E}_{n}$
which are classified as two kinds.
$\left(1\right)$
$(Coray(\Delta)\bigoplus\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ for some
$\phi\neq\Delta\subseteq(\tilde{A}_{n})_{0}$ and
$(\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ is a torsion pair on
$\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod}$ which is induced by
cotilting modules.
$\left(2\right)$ $(\mathcal{T^{\prime}},\mathcal{F^{\prime}}\bigoplus
Ray(\Delta))$ for some $\phi\neq\Delta\subseteq(\tilde{A}_{n})_{0}$ and
$(\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ is a torsion pair on
$\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod}$ which is induced by
tilting modules.
Proof: Suppose $(\mathcal{T},\mathcal{F})\text{ is a \text{torsion pair}}$ on
$\mathcal{E}_{n}$ and $L_{\mathcal{T}}\neq\phi$. Then we know that
$Coray({L_{\mathcal{T}})}\subseteq\mathcal{T}$ since $\mathcal{T}$ is closed
under quotients. And for the first kind it is obvious that $\langle
Coray(\Delta),\mathcal{T^{\prime}}\rangle=Coray(\Delta)\bigoplus\mathcal{T^{\prime}}$.
The other is similar.
Since $\phi\neq\Delta$, we know
$\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)\text{-mod}$ is a direct sum of
module categories of $A_{n}$-type algebras. so by Lemma 4.9 the torsion pair
is easily obtained. By the above theorem and the characterization of torsion
pairs induced by tilting or cotilting modules on $A_{n}$-type algebras, we
have the following bijection.
###### Theorem 4.15.
$\left(1\right)$ There is a bijection between the set of the torsion pairs
$(\mathcal{T},\mathcal{F})$ on $\mathcal{E}_{n}$ such that
$L_{\mathcal{T}}\neq\phi$ and the set of the complete sets of $\tilde{A}_{n}$
$\\{\Delta,\Delta_{1},\dots,\Delta_{m}\\}$ which is a strong 1-type part
partition and $\Delta$ is not empty.
$\left(2\right)$ There is a bijection between the set of the torsion pairs
$(\mathcal{T},\mathcal{F})$ on $\mathcal{E}_{n}$ such that
$R_{\mathcal{F}}\neq\phi$ and the set of the complete sest of $\tilde{A}_{n}$
$\\{\Delta,\Delta_{1},\dots,\Delta_{m}\\}$ which is a strong 2-type part
partition and $\Delta$ is not empty.
Proof: If $\\{\Delta,\Delta_{1},\dots,\Delta_{m}\\}$ is a strong 1-type part
partition, then $\\{\Delta_{1},\dots,\Delta_{m}\\}$ is strong 1-type part
partition in $\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)$. Then we get a
torsion pair $(\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ on
$\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)$-mod which is induced by a
cotilting module. Thus
$(Coray(\Delta)\bigoplus\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ is the
corresponding torsion pair on $\mathcal{E}_{n}$
If $\\{\Delta,\Delta_{1},\dots,\Delta_{m}\\}$ is a strong 2-type part
partition, then we get a torsion pair
$(\mathcal{T^{\prime}},\mathcal{F^{\prime}}\bigoplus Ray(\Delta))$ where
$(\mathcal{T^{\prime}},\mathcal{F^{\prime}})$ is a torsion pair on
$\tilde{A}_{n}((\tilde{A}_{n})_{0}-\Delta)$-mod which is induced by a tilting
module.
The rest is clear.
## 5 Torsion pairs on hereditary algebras
In this section we always assume $K$ is an algebraic closed field and $Q$ is a
acyclic quiver. We try to find a way to obtain all torsion pairs on $KQ$-mod.
This aim is also the motivation of the article. If $Q$ is not wild, we really
get a way. If it is wild, the issue comes down to the torsion pairs on regular
components of wild hereditary algebras. For this we denote the Auslander-
Reiten translation by $\tau$, its quasi-inverse by $\tau^{-}$, the finite-
dimensional projective $KQ$-module category by $\mathcal{P}(Q)$, the finite-
dimensional injective $KQ$-module category by $\mathcal{I}(Q)$. The following
two lemmas are well known.
###### Lemma 5.1.
Suppose $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is an exact
sequence on kQ-mod.Then
$\left(1\right)$ If $\operatorname{add}A\bigcap\mathcal{P}(Q)=\\{0\\}$, then
$0\rightarrow\tau A\rightarrow\tau B\rightarrow\tau C\rightarrow 0$ is an
exact sequence.
$\left(2\right)$ If $\operatorname{add}C\bigcap\mathcal{I}(Q)=\\{0\\}$, then
$0\rightarrow\tau^{-}A\rightarrow\tau^{-}B\rightarrow\tau^{-}C\rightarrow 0$
is an exact sequence.
###### Lemma 5.2.
Suppose $X,Y\in$ kQ-mod.
$\left(1\right)$ If $\operatorname{add}X\bigcap\mathcal{P}(Q)=\\{0\\}$, then
$\operatorname{Hom}(X,Y)\cong\operatorname{Hom}(\tau X,\tau Y)$
$\left(2\right)$ If $\operatorname{add}Y\bigcap\mathcal{I}(Q)=\\{0\\}$, then
$\operatorname{Hom}(X,Y)\cong\operatorname{Hom}(\tau^{-}X,\tau^{-}Y)$
We denote the set of torsion pairs on $KQ$-mod $(\mathcal{T},\mathcal{F})$
such that $\mathcal{I}(Q)\subseteq\mathcal{T}$ by $\mathbf{F}_{1}(Q)$ and the
set of torsion pairs on $KQ$-mod $(\mathcal{T},\mathcal{F})$ such that
$\mathcal{P}(Q)\subseteq\mathcal{F}$ by $\mathbf{F}_{2}(Q)$. And let
$\mathbf{F}(Q)=\mathbf{F}_{1}(Q)\bigcup\mathbf{F}_{2}(Q)$. It is obvious that
$\mathbf{E}(Q)=\mathbf{F}_{1}(Q)\bigcap\mathbf{F}_{2}(Q)$. As a consequence of
the above two lemmas, we have the following proposition.
###### Proposition 5.3.
Suppose there is no projective-injective $KQ$-module. Then there is a one to
one correspondence:
$\mathbf{F}_{1}(Q)\autorightleftharpoons{\sigma^{-}}{\sigma}\mathbf{F}_{2}(Q)$
such that
$\forall(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in\mathbf{F}_{1}(Q),\sigma^{-}(\mathcal{T}^{\prime},\mathcal{F}^{\prime})=(\tau^{-}\mathcal{T}^{\prime},\tau^{-}\mathcal{F}^{\prime}\bigoplus\mathcal{P}(Q));\forall(\mathcal{T}^{\prime\prime},\mathcal{F}^{\prime\prime})\in\mathbf{F}_{2}(Q),\linebreak\sigma(\mathcal{T}^{\prime\prime},\mathcal{F}^{\prime\prime})=(\mathcal{I}(Q)\bigoplus\tau\mathcal{T}^{\prime\prime},\tau\mathcal{F}^{\prime\prime})$.
Proof. We just prove that
$\forall(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in\mathbf{F}_{1},(\tau^{-}\mathcal{T}^{\prime},\tau^{-}\mathcal{F}^{\prime}\bigoplus\mathcal{P}(Q))$
is a torsion pair on $KQ$-mod.
By Lemma 5.2 (2), we know $\forall
X\in\mathcal{T}^{\prime},Y\in\mathcal{F}^{\prime}$,
$\operatorname{Hom}(\tau^{-}X,\tau^{-}Y)\cong\operatorname{Hom}(X,Y)=\\{0\\}$.
So the condition 1 in the Definition 2.1 is satisfied. By Lemma 5.1 (2), we
know except projective modules, every indecomposable module has a suitable
decomposition in
$(\tau^{-}\mathcal{T}^{\prime},\tau^{-}\mathcal{F}^{\prime}\bigoplus\mathcal{P}(Q))$.
But for projective modules, the suitable decomposition is obvious. So the
condition 2 in the Definition 2.1 is satisfied.
Just like the Auslander-Reiten translation, $\sigma^{-}$ and $\sigma$ also
gives a translation on $F(Q)$. For every
$(\mathcal{T},\mathcal{F})\in\mathcal{F}(Q),\text{if
}\mathcal{I}(Q)\subseteq\mathcal{T},$ then let
$\sigma^{-}(\mathcal{T},\mathcal{F})=(\tau^{-}\mathcal{T},\tau^{-}\mathcal{F}\bigoplus\mathcal{P}(Q))$;
if $\mathcal{P}(Q)\subseteq\mathcal{F}$, then let
$\sigma(\mathcal{T},\mathcal{F})=(\tau\mathcal{T}\bigoplus\mathcal{I}(Q),\tau\mathcal{F})$.
The above proposition tells us that this translation defines $\sigma$-obits
for elements in $\mathbf{F}(Q)$. We use $[\mathcal{T},\mathcal{F}]$ to denote
the $\sigma$-obit of $(\mathcal{T},\mathcal{F})$.
###### Definition 5.4.
Suppose $(\mathcal{T},\mathcal{F})\in\mathbf{F}(Q)$. We call the elements in
$[\mathcal{T},\mathcal{F}]\bigcap(\mathbf{F}_{2}(Q)-\mathbf{F}_{1}(Q))$ source
points of $[\mathcal{T},\mathcal{F}]$, the elements in
$[\mathcal{T},\mathcal{F}]\bigcap(\mathbf{F}_{1}(Q)-\mathbf{F}_{2}(Q))$ sink
points of $[\mathcal{T},\mathcal{F}]$, the elements in
$[\mathcal{T},\mathcal{F}]\bigcap\mathbf{F}_{1}(Q)\bigcap\mathbf{F}_{2}(Q)$
middle points of $[\mathcal{T},\mathcal{F}]$.
The following corollary is obvious.
###### Lemma 5.5.
Suppose $(\mathcal{T},\mathcal{F})\in\mathbf{F}(Q)$. Then
$[\mathcal{T},\mathcal{F}]$ has at most one source point and at most one sink
point. And
$[\mathcal{T},\mathcal{F}]\bigcap\mathbf{F}_{1}(Q)\bigcap\mathbf{F}_{2}(Q)=[\mathcal{T},\mathcal{F}]\bigcap\mathbf{E}(Q)$.
We denote the preprojective component of $KQ$-mod by
$\mathcal{P}_{\infty}(Q)$, the preinjective component of $KQ$-mod by
$\mathcal{I}_{\infty}(Q)$, the regular component of $KQ$-mod by
$\mathcal{R}(Q)$.
###### Theorem 5.6.
Suppose $(\mathcal{T},\mathcal{F})\in\mathbf{F}(Q)$. Then
$\left(1\right)$ $[\mathcal{T},\mathcal{F}]$ has a source point but no sink
point $\iff$ for every
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in[\mathcal{T},\mathcal{F}]$,
$\mathcal{I}_{\infty}(Q)\bigcap\mathcal{F}^{\prime}\neq\phi$ and
$\mathcal{P}_{\infty}(Q)\subseteq\mathcal{F}^{\prime}$.
$\left(2\right)$ $[\mathcal{T},\mathcal{F}]$ has a sink point but no source
point $\iff$ for every
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in[\mathcal{T},\mathcal{F}]$,
$\mathcal{P}_{\infty}(Q)\bigcap\mathcal{T}\neq\phi$ and
$\mathcal{I}_{\infty}(Q)\subseteq\mathcal{T}^{\prime}$.
$\left(3\right)$ $[\mathcal{T},\mathcal{F}]$ has a sink point and a source
point $\iff$ for every
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in[\mathcal{T},\mathcal{F}]$,
$\mathcal{I}_{\infty}(Q)\bigcap\mathcal{F}\neq\phi$ and
$\mathcal{P}_{\infty}(Q)\bigcap\mathcal{T}\neq\phi$.
$\left(4\right)$ $[\mathcal{T},\mathcal{F}]$ has no sink point and no source
point $\iff$ for every
$(\mathcal{T}^{\prime},\mathcal{F}^{\prime})\in[\mathcal{T},\mathcal{F}]$,
$\mathcal{I}_{\infty}(Q)\subseteq\mathcal{T}^{\prime}$, and
$\mathcal{P}_{\infty}(Q)\subseteq\mathcal{F}^{\prime}$.
We denote the set of torsion pairs $(\mathcal{T},\mathcal{F})$on $KQ$-mod such
that $\mathcal{I}_{\infty}(Q)\subseteq\mathcal{T}$, and
$\mathcal{P}_{\infty}(Q)\subseteq\mathcal{F}$ by $\mathbf{H}(Q)$. So it is
obvious that $\mathbf{H}(Q)\subseteq\mathbf{E}(Q)$. We denote the set of
torsion pairs on $\mathcal{R}(Q)$ by $\mathbf{R}(Q)$. We have the following
obvious lemma.
###### Lemma 5.7.
There is a one to one correspondence:
$\mathbf{H}(Q)\autorightleftharpoons{F}{F^{-}}\mathbf{R}(Q)$
such that
$\forall(\mathcal{T},\mathcal{F})\in\mathbf{H}(Q),F((\mathcal{T},\mathcal{F}))=(\mathcal{T}\bigcap\mathcal{R}(Q),\mathcal{F}\bigcap\mathcal{R}(Q))$;
$\forall(\mathcal{T}^{\prime},\mathcal{F})^{\prime}\in\mathbf{R}(Q),\linebreak
F^{-}((\mathcal{T}^{\prime},\mathcal{F}^{\prime}))=(\mathcal{T}^{\prime}\bigoplus\mathcal{I}_{\infty}(Q),\mathcal{F}^{\prime}\bigoplus\mathcal{P}_{\infty}(Q)).$
###### Remark 5.8.
Suppose $(\mathcal{T},\mathcal{F})\in\mathbf{F}(Q)$ and
$[\mathcal{T},\mathcal{F}]$ has at least one sink point or one source point.
We define the following operation $\Phi$:
Case $1$. If $[\mathcal{T},\mathcal{F}]$ has a sink point, then we denote the
sink point by $\Phi((\mathcal{T},\mathcal{F}))$.
Case $2$. If $[\mathcal{T},\mathcal{F}]$ has a source point but no sink point,
then we denote the source point by $\Phi((\mathcal{T},\mathcal{F}))$.
For any torsion pair on $KQ$-mod we apply the operation in Theorem 3.11 and
the operation $\Phi$ to it alternatively. At last we get a new torsion pair on
$KQ^{\prime}$-mod for some subquiver $Q^{\prime}$ of $Q$ such that the new
torsion pair belongs to $\mathbf{H}(Q^{\prime})$. This process is invertible
by Theorem 3.11 and Proposition 5.3. So by the above lemma if we know all
torsion pairs on regular components for all subquivers, then we can construct
all torsion pairs of $KQ$-mod.
From now on we suppose $Q$ is a acyclic quiver with a Euclid ground graph. We
start to find all the torsion pairs on $\mathbf{R}(Q).$ The following
definition and two lemmas are from [WB].
###### Definition 5.9.
Suppose $X\in KQ$-mod . Then $Q$ is regular uniserial if there are regular
submodules $0=X_{0}\subset X_{1}\subset\dots\subset X_{r}=X$ and these are the
only regular submodules of X.
###### Lemma 5.10.
If $\theta:X\rightarrow Y$ with $X,Y$ regular $KQ$-modules, then
$\operatorname{Im}(\theta),\operatorname{Ker}(\theta)$ and
$\operatorname{Coker}(\theta)$ are regular.
###### Lemma 5.11.
Every indecomposable regular $KQ$-module is regular universal.
As an consequence we have
###### Corollary 5.12.
If $KQ$ is an Euclid-type algebra, X is a regular module, then the quotient
modules of X forms a chain: $X=X^{r}\twoheadrightarrow\dots\twoheadrightarrow
X^{1}\twoheadrightarrow X^{0}$.
###### Corollary 5.13.
Let $KQ$ be an Euclid-type algebra, $f:X\rightarrow Y$ is an injective
morphism such that $X$ is a maximal regular submodule of the indecomposable
regular module of $Y$. Then $f$ is an irreducible morphism.
Proof. $X$ is indecomposable by Lemma $5.11$. Suppose $\exists g:X\rightarrow
Z,h:Z\rightarrow Z$ such that $f=hg$. Then by Lemma $5.11$, there is an
indecomposable direct summand $Z^{\prime}$ such that $\exists
g^{\prime}:X\rightarrow Z^{\prime},h:Z^{\prime}\rightarrow Y$ such that
$h^{\prime}g^{\prime}$ is an injective morphism. $Z^{\prime}$ is a regular
module. So by Lemma $5.11,h^{\prime}$ is an injective morphism. Since $X$ is a
maximal regular submodule, $h^{\prime}$ is anisomorphism or $g^{\prime}$ is an
isomorphism.
Now Let $\mathcal{R}(Q)=\bigoplus_{i\in I}\mathcal{R}_{i}(Q)$ where
$\\{\mathcal{R}_{i}(Q),i\in I\\}$ is the set of minimal additive categories
containing a connected component in AR-quiver of $KQ$. We denote the set of
torsion pairs on $\mathcal{R}_{i}(Q)$ by $\mathbf{R}_{i}(Q)$. By Lemma 4.9, we
have the following lemma.
###### Corollary 5.14.
There exists a bijection between $\mathbf{R}(Q)$ and the set of tuples
$\\{(\mathcal{T}_{i},\mathcal{F}_{i})\\}_{i\in I}$ with
$(\mathcal{T}_{i},\mathcal{F}_{i})\in\mathbf{R}_{i}(Q)$.
Proof: Let $X\in\mathcal{R}_{i}(Q)$. Then all regular submodules and all
regular quotient modules of $X$ are in $\mathcal{R}_{i}(Q)$ by the above
corollary. So we know if $i\neq j$, then $\operatorname{Hom}(X,Y)=0,\forall
X\in\mathcal{R}_{i}(Q)$ and $Y\in\mathcal{R}_{j}(Q)$. The rest is clear by
Lemma 4.9.
Now we start to demonstrate $\mathbf{R}_{i}(Q)$ . Suppose $\mathbf{R}_{i}(Q)$
has $n$ regular simple modules[WB]: $S_{1},S_{2},\dots,S_{n-1}$ where
$S_{i+1}=\tau S_{i}$. Let Let $\tilde{A}_{n}$ be the quiver in Section 4 and
$S_{1}^{\prime},S_{2}^{\prime},\dots,S_{n}^{\prime}$ are the correspondent
simple modules to the vertices. Then we construct a map:
$\overline{F}(S_{i}^{\prime})=S_{i}$. Then $\overline{F}$ induces a one to one
correspondence: $\mathcal{E}_{n}\rightarrow\mathcal{R}_{i}(Q)$ such that if
$X$ $\in\mathcal{E}_{n}$ and is indecomposable with the length $m$ and top
$S_{i}^{\prime}$, then $F(X)$ is the indecomposable regular module with the
regular length $m$ and top $S_{i}$. we have the following lemma.
###### Lemma 5.15.
$\left(1\right)$ $\forall
X,Y\in\mathcal{E}_{n},\operatorname{Hom}(X,Y)=0\iff\operatorname{Hom}(F(X),F(Y))=0$.
$\left(2\right)$ Suppose $Y\in\mathcal{E}_{n}$ and $X$ is a submodule of $Y$.
Then $F(Y/X)=F(Y)/F(X)$.
Proof: Clear by Lemma 5.11 and Corollary 5.12.
###### Theorem 5.16.
F induces a one to one correspondent between the set of torsion pairs on
$\mathcal{E}_{n}$ and $\mathbf{R}_{i}(Q)$.
Proof: Clear by the above lemma.
## References
* [AK] I. Assem, O. Kerner, Constructing torsion pairs, J. Algebra, 185 (1996) 19-41.
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* [TB] T. Bridgeland. Stability conditions on triangulated categories, Ann. of Math. (2), Vol. 166(2007), 317 C34.
* [WB] W. Crawley Boevey, Lectures on representations of quivers, http://www.amsta.leeds.ac.uk/ pmtwc/quivlecs.ps, 1992.
* [BBM] K. Baur, A. B. Buan, R. J. Marsh, torsion pairs and rigid objects in tubes, Preprint, rXiv:1112.6132v1 [math.RT], 2011.
* [BM] K. Baur, R. J. Marsh, A geometric model of tube categories, Preprint arXiv:1011.0743v2 [math.RT], 2010.
* [BK] A.B. Buan, H. Krause, Tilting and cotilting for quivers of type $\tilde{A}_{n}$, J. Pure Appl. Algebra 190(2004), no. 1-3, 1-21
* [D] S.E.Dickson, A torsion theory for abelian categories, Trans. AMS, 121(1966), 223-235.
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Fan Kong, Department of Mathematics, Shanghai Jiaotong University, 200240
Shanghai, People’s Republic of China.
Email: Kongfan08@yahoo.com.cn
Keyan Song, Department of Mathematics, Shanghai Jiaotong University, 200240
Shanghai, People’s Republic of China.
Email: sky19840806@163.com
Pu Zhang, Department of Mathematics, Shanghai Jiaotong University, 200240
Shanghai, People’s Republic of China.
Email: pzhang@sjtu.edu.cn
|
arxiv-papers
| 2012-05-07T08:55:32 |
2024-09-04T02:49:30.658450
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fan Kong, Keyan Song, Pu Zhang",
"submitter": "Fan Kong",
"url": "https://arxiv.org/abs/1205.1324"
}
|
1205.1437
|
# Spontaneous Bifurcation of Single Peaked Current Sheets by Chaotic Electron
Scattering
Kuang-Wu Lee lee@mps.mpg.de Jörg Büchner Max-Planck-Institut für
Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany
###### Abstract
It is shown that single-peaked collisionless current sheets in a Harris-type
equilibrium spontaneously bifurcate as a result of chaotic scattering of
electrons at fluctuating magnetic fields near the center of the sheet, as
demonstrated by a 2D kinetic particle-in-cell simulation. For this effect to
be simulated explicit particle advancing is necessary, since the details of
the electron motion have to be resolved. Unlike previous investigations of
triggering bifurcated current sheet (BCS) where initial perturbations or
external pressure was applied the bifurcation is spontaneous if thermal noise
is taken into account. A spontaneous current sheet bifurcation develops
quicker than a tearing mode or other plasma instabilities. It is shown that in
the course of the current sheet bifurcation the Helmholtz free energy
decreases while the entropy increases, i.e. the new, bifurcated current sheet
is in a more propable state than the single-peaked one.
PACS numbers: 05.70.-a, 05.70.Ce, 52.35.Ra
###### pacs:
Valid PACS appear here
††preprint: APS/123-QED
The stability and possible unstable decay of current sheets play a central
role in astrophyics as well as in the laboratory, e.g. for magnetic energy
dissipation and reconnection Büchner (2006). For the investigations of current
sheet stability often single-peaked current sheets are used, as the one
derived by Harris (1962).
The free energy of equilibria can cause a number plasma instabilities which
spontaneously arise from thermal noise like the dissipative tearing mode
instability in resistive Furth _et al._ (1963) and collisionless plasmas
Coppi _et al._ (1966). In fusion plasmas spontaneous magnetic reconnection is
found recently in the poloidal current sheet, which is an initial equilibria
in the Reverse Field Pinch (RFP)Zuin _et al._ (2009). Since the single-peaked
current sheet equilibria is unstable and magnetic reconnection frequently
takes place, the equilibria stability is an important issue for current sheet
evolution.
Magnetosphere is a natural plasma laboratory for the evolution of current
sheet equilibrium. Indeed, more recent detailed investigations of current
sheets, has shown that current sheets frequently are bifurcated (BCS) instead
of single-peaked (Sergeev _et al._ , 1993) (Hoshino _et al._ ,
1996)(Israelevich _et al._ , 2007a). It was also found that these BCS are
electron dominated, e.g. by statistical analyses of measurements onboard the
CLUSTER spaccraft mission (Israelevich _et al._ , 2007b).
BCS were also found in numerical simulations. In magnetic reconnection plan
BCS was interpreted, e.g., as a pair of slow mode shocks which develop in the
reconnection outflow region (Shiota _et al._ , 2005)(Thompson _et al._ ,
2006). However, BCS were also observed in minimum plasma inflow conditions,
for which magnetic reconnection is not expected (Tang _et al._ , 2006). In
the current direction, BCS formation without plasma inflow is discovered as a
result of anomalous momentum transport due to pressure-gradient driven lower
hybrid drift instability Daughton _et al._ (2004).
Figure 1: The $y$-integrated current profiles (left panel) and current density
at four different simulation times between $t=0$ and $t=8371,2\
\omega_{pe}^{-1}$. The upper right panels show the total current and lower
panels electron current densities.
In the direction perpendicular to current drift, current sheet splitting
similar to BCS is observed after the saturation of tearing mode instability
(Camporeale and Lapenta, 2005). Note that these authors used an implicit
numerical scheme for advancing the particles for their 2D particle-in-cell
(PIC) code simulations. In order to initialize the tearing mode instability
quickly they also imposed perturbations to trigger instability growth.
Schindler and Hesse (2008) performed a one-dimensional particle-in-cell (1D
PIC) simulation with an initial boundary pressing. They concluded that a
quasisteady boundary compression forces a single-peaked current sheet evolves
toward BCS, as an equilibrium relaxation process but not due to plasma
instability
We now have found that single-peaked collisionless current sheets might
spontaneously bifurcate, without boundary compression or imposing tearing mode
perturbation initially, as long as the electron thermal fluctuations are
considered properly. We found that magnetic field fluctuations can initially
start from thermal noise at the center of current. The magnetic field
fluctuations may lead to chaotic scattering of the electrons out of the
central (current peak) region of the sheet Büchner and Zelenyi (1989),
reducing the electron current flow there and adding current flows away from
the center of the sheet as already discussed for laminar current sheets
Zelenyi _et al._ (2003).
To prove this hypothesis quantitatively we carried out 2D electromagnetic
particle-in-cell code simulations to investigate the evolution of Harris
current sheet equilibrium. An explicit numerical scheme (XOOPIC) was
implemented since the details of the electron cyclotron trajectories have to
be calculated properly. The fastest processes up to electron plasma time
scales $\ \omega_{pe}^{-1}=\sqrt{\epsilon_{o}m_{e}/ne^{2}}$ is resolved. The
chosen current sheet has a width $2\lambda=1.15d_{i}$ which is slightly larger
than the ion inertial length $d_{i}=c/\ \omega_{pi}$ to cover the full width
of the ion dissipation region of the current sheet. The equilibrium magnetic
field of a Harris sheet equlibrium is
$B_{y}(x)=B_{0}\,tanh(x/\lambda)=\sqrt{4\mu k_{B}\ T\ N_{0}}tanh(x/\lambda)$,
where $k_{B}$, $T=T_{i}=T_{e}$, $B_{0}$ and $N_{0}$ are the Boltzmann
constant, ion/electron temperatures, asymptotic magnetic field and number
density at the center of the sheet, respectively. The ratio of electron plasma
frequency to electron cyclotron frequency is $\ \omega_{pe}/\
\omega_{ce}=2.87$. The ion to electron mass ratio used is $m_{i}/m_{e}=180$,
which is sufficient to separate the ion/electron motions. The grid size is of
the Debye length $dx=dy=\lambda_{De}=(\varepsilon_{0}T/Ne^{2})^{1/2}$, which
has sufficient spatial resolution of the detailed electron motion. The
simulation domain in the direction perpendicular to the current sheet was
$L_{x}=13.3d_{i}$ and in the current sheet direction $L_{y}=26.6d_{i}$. This
choice of $L_{y}$ allows a free development of the fastest growing tearing-
type and other eigenmodes of the sheet (see, e.g., Eq.(23) in (Brittnacher
_et al._ , 1995)). The boundary conditions for the particles and fields were
chosen to be periodic in the $y$ direction and conducting walls with particles
reflection in the $x$ direction.
The time step used was $dt=0.0872\ \omega_{pe}^{-1}=0.03\ \omega_{ce}^{-1}$ to
fulfill Courant condition and to resolve the detailed electron oscillation and
cyclotron motions. Note that this time step is much smaller than the one used
in implicit numerical schemes, where, e.g., in (Camporeale and Lapenta, 2005)
$dt=0.1\ \omega_{pi}^{-1}\approx 1.34\ \omega_{pe}^{-1}\approx 0.5\
\omega_{ce}^{-1}$ which does not track down the electron oscillation and
cyclotron motion. The later is significant for magnetic scattering, which we
will discuss soon. No initial perturbation was imposed as in (Camporeale and
Lapenta, 2005). Also, no boundary compression was imposed as in the 1D PIC
simulation of BCS formation (see Fig.1 in (Schindler and Hesse, 2008)).
The left panels of Fig.1 show the $y$ integrated total current density
$I_{z}(x)=I_{z,e}(x)+I_{z,i}(x)$ and the individual contributions of electron
$I_{z,e}$ and ion $I_{z,i}$: for the initial, single-peaked Harris-equilibrium
currents by a thin solid line, for $t=4883.2\ \omega_{pe}^{-1}$ by a dashed
line and for $t=8371.2\ \ \omega_{pe}^{-1}$ by a thick solid line. At the late
stage $t=8371.2\ \omega_{pe}^{-1}\approx 16.212\ \omega_{ci}^{-1}$ two current
peaks are formed which are well separated from each other. As one can see
already in the middle panel of the left Fig.1 the reduction of $I_{z}(x)$ near
the center of the current sheet is mainly due the spatial redistribution of
the electron current. Since the ion current is not changing much before
$t=8371.2\ \omega_{pe}^{-1}$ the right panels of Fig.1 depict the evolution of
the total current density distribution in the $x,y$ plane with time in the
upper row and the responsible for the current redistribution electron part in
the lower panels. While initially both the total and the electron currents are
concentrated near the center of the sheet (first column in the right part of
Fig.1), already at $t=4883.2\ \omega_{pe}^{-1}=9.5\
\omega_{ci}^{-1}=1.5\tau_{ci}$ a clear dip of the current density has
developed around the center of the sheet. Here $\tau_{ci}=2\pi\
\omega_{ci}^{-1}$ denotes a cyclotron period. Hence, the bifurcation takes
place at the time scale of an ion cyclotron period. This result indicates that
the ion-to-electron mass ratio $m_{i}/m_{e}=180$ used in the simulation the
electron and ion dynamics are sufficiently well separated.
Note further that the bifurcation due to boundary pressing takes place only
after $t\approx\ 400\ \omega_{ci}^{-1}$ (Schindler and Hesse, 2008). Their
mass ratio is $m_{i}/m_{e}=25$ and the bifurcation time in electron plasma
period is $400\ \omega_{ci}^{-1}\approx\ 4.4\times 10^{4}\ \omega_{pe}^{-1}$.
Figure 2: Absolute values of the fluctuating $|B_{x}|$ magnetic fields at the
center of the current sheet, normalized to the asymptotic magnetic field of
the Harris equilibrium (left panel) and the thermal electron $\kappa(y)$
values at center of the current sheet (right panel) for the same four
simulation times as in Figure 1, right panels.
In order to understand this phenomenon one has to realize that plasmas are
coarse-grained by their particles whose thermal motion lets currents and
magnetic fields fluctuate at small scales. Hence, while the self-consistent
magnetic field $B_{y}$ of a Harris-equilibrium vanishes at the center of the
sheet, fluctuating magnetic fields remain.
The left panel of Fig.2 shows $|B_{x}(x=0)|$, the absolute value of the normal
field $B_{x}$ component of the magnetic field at the center of the sheet,
divided by the asymptotic Harris sheet field $B_{yo}=B_{0}$ for the same
simulation times as chosen for the right panels in Fig. 1. As one can see,
$|B_{x}(x=0)|$ is finite most of the time unless the particles’ thermal
velocity is not taken into account as in the the uppermost column ($t=0$). It
is well known that particles transiting the center of current sheets can be
chaotically scattered of the curvature of the magnetic field is comparable to
the Larmor radii. According to Büchner and Zelenyi (1989) the scattering is
strongest, when the parameter $\kappa=\sqrt{R_{min}\rho_{max}}$ is unity,
where $R_{min}$ and $rho_{max}$ are the minimum curvature of the magnetic
field and and the maximum particle gyroradius at the current sheet center. The
right panel of Fig.2 depicts the $\kappa$ values for thermal electrons along
the y direction for the same moments of time for which the $B_{x}(y)$ fields
are shown in the left panel of this Figure and in the right panel of Fig.1. As
one can see in the right panel of Fig.2 before $t=4883\ \omega_{pe}^{-1}$
$\kappa$ reaches unity only at a few positions, i.e. only at a few places the
electrons are strongly scattered. After $t=4883\ \omega_{pe}^{-1}$ electrons
are strongly scattered chaotically at many positions.
The consequences of this strong scattering can be described best by means of
the action integral of the fast motion motion $I=\oint v_{z}dz$ Sonnerup
(1971). For $\kappa<1$ this action integral can be expressed in its normalized
form as $I^{\prime}=\left({\frac{\kappa^{2}y^{\prime}}{2k^{2}-1}}\right)\
f_{A/B}(k)$, where $k(y^{\prime})$ is a function of $y^{\prime}$, the
appropriately normalized $y$ coordinate of the slowly moving guiding center
and $f_{A/B}(k)$ are two different functions of complete elliptic integrals
for $k<1$ and $k>1$, respectively Büchner and Zelenyi (1989). For $\kappa<1$
the action integrals $I^{\prime}$ are adiabatically conserved, i.e. integrals
of motion, as long as they stay away from the separatrix which is reached when
$k\to 1$. For $k<1$ such quasi-adiabatic orbits cross the center of the
current sheets while for $k>1$ they will gyrate at some distance from the
sheet center. Particles with $1<I^{\prime}<I^{\prime}_{max}=1.16$ are trapped
on crossing the current sheet orbits $k<1$. But in thin current sheets most
particles will have $I^{\prime}<1$, which enables them to eventually reach a
separatrix in the velocity space, i.e. change from the region $k<1$ to $k>1$,
at $k=1$ changing from meandering across the sheet, drifting in the direction
of the original sheet current to a gyration away from the current sheet center
where they drift in the opposite direction, causing diamagnetic currents away
from the sheet center Büchner and Zelenyi (1989). While for $\kappa\ll 1$
particles the value of the quasi-integral of motion $I^{\prime}$ stays
practically unchanged during the sepratrix encounter, for $\kappa\to l$ it can
be essentially changed. If the obtained value of $I^{\prime}$ is very small
only a small amount of kinetic energy is left in the perpendicular to the
magnetic field velocity direction while most energy is in the slow drift
motion. These particles contribute significantly to the built up of
diamagnetic currents away from the sheet center. For small $I^{\prime}$ the
asymptotic expressions for the elliptic integrals reveal $I^{\prime}\approx
3/16\pi k_{tp}\kappa^{6}$, where $k_{tp}$ correspond to the turning point of
the drift motion in $y^{\prime}$ ($v_{y^{\prime}}=0$). From the condition for
turning the drift in the $y$ direction one obtains the position of the turning
point by solving the equation $\kappa^{2}y^{\prime}_{tp}=2k_{tp}^{2}-1$.
Solving this equation for $k_{tp}$s and calculating the mean $z$-position for
the turning point location one obtains a the distance $\Delta
z\sqrt{\lambda\rho_{the}}(16\pi/3)^{2}I^{\prime}_{the}$ from the current sheet
at which most of the particles drift in the dia-magnetic current direction.
Here $I^{\prime}_{the}$ means the quasi-adiabatic invariant for typical
thermal electrons, the bulk drift velocity of the electrons is much smaller
and can be neglected. With the time being $\Delta z$ will become the position
of the maximum of the dia-magnetic flow of the electrons where the initial
current profile is modified most by electron flows while the electron flow at
the center of the current sheet is reduced. For the parameters used in the
simulation one obtains $\Delta z\cong 27\lambda_{De}$. This theoretically
predicted distance corresponds to the one found in simulation.
The $Y$ integrated electron drift $V_{e,z}$ and density $N_{e}$ profiles are
show in Fig.3 for three moments of time. Note that such current profile was
obtained by Hoshino _et al._ (1996) from a statistical analysis of many
current sheet encounters in the Earth’s magnetotail. The electron number
density does not change much in the course of the bifurcation, which happens,
instead, in the electron drift velocity space. This is consistent with the
finding that the current depletion at the center is caused mainly by the
redistribution of the electrons by pitch-angle scattering in the velocity
space when crossing the sepratrix between meandering ($k<1$) and gyrating away
from the sheet center ($k>1$). Due to the mass ratio only the electrons
undergo strong chaotic scattering thin Harris-type current sheets.
Figure 3: Profiles of electron drift velocity $V_{e,z}$ (upper panel) and
number density $N_{e}$ (lower panel), both integrated along the $y$ axis.
The new, bifurcated current sheet is again in equilibrium. In fact there is an
infinite number of possible current sheet equilibria. A number of analytical
Neukirch _et al._ (2009) and non-analytical equilibrium solutions have been
found which are more realistic for space current sheets than Harris sheets
Cowley (1978). For a survey see, e.g., Schindler and Birn (2002). A current
sheet bifurcated out of a single peaked Harris current sheet by the chaotic
electron scattering is close to the equilibrium found by Camporeale and
Lapenta (2005), their case (b). Note that the BCS, naturally obtained via
chaotic electron scattering, is more probable than the single peaked Harris
sheet. This can be demonstrated by calculating the entropy of the system. For
a plasma with a continuous particle distribution, it is appropriate to
consider the relative Kullback-Leibler entropy which is always positive
(Kullback and Leibler, 1951). With respect to a reference distribution $q(v)$
at $t=0$ it can be written as
$\displaystyle
S_{KL}(t)=\int_{-\infty}^{\infty}dvf(v,t)ln(\frac{f(v,t)}{q(v)|_{t=0}})$
The relative entropy as a sum of the electron and ion contributions grows
increases in the course of current sheet bifurcation. Looking at the electrons
and ions separately one can see that the entropy of electrons is, indeed,
steadily increasing after about $t=3800\ \omega_{pe}^{-1}$ while the ion
entropy oscillates (right columns of Fig.4).
Figure 4: Evolution of the total relative entropy (left panel) and of the
electron (upper right panel) and ion entropies (lower right panel) separately.
The stability of a steady state equilibrium can be analyzed by calculating the
Helmholtz free energy $F=U-T\ S$ (Kan, 1954). Here the $U$ is the internal
energy, a sum of the particles kinetic energy and the field energy, $T$ is the
temperature energy and $S$ is the entropy. Less free energy $F$ corresponds to
a more stable equilibrium. In the course of the bifurcation, the internal
energy U is conserved while T and S are increasing. Hence the bifurcated
current sheet contains less free energy. This is why it is more favored in
nature than single-peaked equilibria like the Harris current sheet and it is
more stable.
2D PIC simulations confirmed that an initially single peaked Harris
equilibrium current sheet does spontaneously and quickly bifurcates a natural
consequence of chaotic electron scattering due to their fluctuations at the
center of the sheet. The bifurcation is faster than spontaneous plasma
instabilities as of the tearing mode. The single peaked sheet bifurcates
without initial perturbations or imposed external pressure. The bifurcated
state is more favorable because electrons gaining after chaotic scattering at
the sheet center toward small values of the quasi-adiabatic invariant of
motion $I^{\prime}$ spends longer time in the cucumber phase of their motion
away from the sheet center rather than meandering at the sheet center.
Therefore it can be simulated only by numerical schemes that explicitly
resolve the details of the electron motion. The spontaneous bifurcation of
single peaked current sheets explains the frequent observation of bifurcated
current sheets in quite situations without plasma inflows and reconnection as
they are more stable than single peaked sheets. Their influence on plasma
instabilities is stabilizing, delaying, lowering the growth rate, e.g., of the
tearing mode instability.
###### Acknowledgements.
The authors are grateful to the Max-Planck Society for funding Turbulent
transport and ion heating, reconnection and electron acceleration in solar and
fusion plasmas Project No. MIF-IF-A-AERO8047.
## References
* Büchner (2006) J. Büchner, Space Sci. Rev. 122, 149 (2006).
* Harris (1962) E. G. Harris, Nuovo Cimento 23, 115 (1962).
* Furth _et al._ (1963) H. P. Furth, J. Killeen, and M. N. Rosenbluth, Phys. Fluids 6, 459 (1963).
* Coppi _et al._ (1966) B. Coppi, G. Laval, and R. Pellat, Phys. Rev. Lett. 16, 1207 (1966).
* Zuin _et al._ (2009) M. Zuin, N. Vianello, M. Spolaore, V. Antoni, T. Bolzonella, R. Cavazzana, E. Martines, G. Serianni, and D. Terranova, Plasma Phys. Control Fusion 51, 035012 (2009).
* Sergeev _et al._ (1993) V. A. Sergeev, D. G. Mitchell, C. T. Russell, and D. J. Williams, J. Geophys. Res. 98, 17,345 (1993).
* Hoshino _et al._ (1996) M. Hoshino, A. Nishida, T. Mukai, Y. Saito, T. Yamamoto, and S. Kokubun, J. Geophys. Res. 101, 24775 (1996).
* Israelevich _et al._ (2007a) P. L. Israelevich, A. I. Ershkovich, and R. Oran, Planet. Space Sci. 55, 2261 (2007a).
* Israelevich _et al._ (2007b) P. L. Israelevich, A. I. Ershkovich, and R. Oran, J. Geophys. Res. 113, A04215 (2007b).
* Shiota _et al._ (2005) D. Shiota, H. Isobe, P. F. Chen, T. T. Yamamoto, T. Sakajiri, and K. Shibata, Astrophys. J. 634 (2005).
* Thompson _et al._ (2006) S. M. Thompson, M. G. Kivelson, M. El-Alaoui, A. Balogh, H. Réme, and L. M. Kistler, J. Geophys. Res. 111, A03212 (2006).
* Tang _et al._ (2006) C. L. Tang, L. Lu, Z. Y. Li, and Z. X. Liu, Chinese Phys. Lett. 23, 1054 (2006).
* Daughton _et al._ (2004) W. Daughton, G. Lapenta, and P. Ricci, Phys. Rev. Lett. 93, 105004 (2004).
* Camporeale and Lapenta (2005) E. Camporeale and G. Lapenta, J. Geophys. Res. 110, A07206 (2005).
* Schindler and Hesse (2008) K. Schindler and M. Hesse, Phys. Plasmas 15, 042902 (2008).
* Büchner and Zelenyi (1989) J. Büchner and L. M. Zelenyi, J. Geophys. Res. 94, 11821 (1989).
* Zelenyi _et al._ (2003) L. Zelenyi, H. Malova, and V. Popov, JETP Letters 78, 296 (2003), 10.1134/1.1625728.
* Brittnacher _et al._ (1995) M. Brittnacher, K. B. Quest, and H. Karimabadi, J. Geophys. Res. 100, 3551 (1995).
* Sonnerup (1971) B. U. Ö. Sonnerup, J. Geophys. Res. 76, 8211 (1971).
* Neukirch _et al._ (2009) T. Neukirch, F. Wilson, and M. G. Harrison, Phys. Plasmas 16, 122102 (2009).
* Cowley (1978) S. Cowley, Planet. Space Sci. 26, 1037 (1978).
* Schindler and Birn (2002) K. Schindler and J. Birn, J. Geophys. Res. 107, A8, 1193 (2002).
* Kullback and Leibler (1951) S. Kullback and R. A. Leibler, Ann. Math. Statist. 22, 79 (1951).
* Kan (1954) J. R. Kan, J. Plasma Phys. 7, 445 (1954).
|
arxiv-papers
| 2012-05-07T15:50:37 |
2024-09-04T02:49:30.671681
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kuang-Wu Lee and J\\\"org B\\\"uchner",
"submitter": "Kuang Wu Lee",
"url": "https://arxiv.org/abs/1205.1437"
}
|
1205.1488
|
# A Categorical Foundation for Bayesian Probability
Jared Culbertson and Kirk Sturtz
###### Abstract.
Given two measurable spaces $H$ and $D$ with countably generated
$\sigma$-algebras, a perfect prior probability measure $P_{H}$ on $H$ and a
sampling distribution $\mathcal{S}\colon H\rightarrow D$, there is a
corresponding inference map $\mathcal{I}\colon D\rightarrow H$ which is unique
up to a set of measure zero. Thus, given a data measurement $\mu\colon
1\rightarrow D$, a posterior probability $\widehat{P_{H}}=\mathcal{I}\circ\mu$
can be computed. This procedure is iterative: with each updated probability
$P_{H}$, we obtain a new joint distribution which in turn yields a new
inference map $\mathcal{I}$ and the process repeats with each additional
measurement. The main result uses an existence theorem for regular conditional
probabilities by Faden, which holds in more generality than the setting of
Polish spaces. This less stringent setting then allows for non-trivial
decision rules (Eilenberg–Moore algebras) on finite (as well as non finite)
spaces, and also provides for a common framework for decision theory and
Bayesian probability.
NB: This paper has been published in Applied Categorical Structures,
http://link.springer.com/article/10.1007/s10485-013-9324-9, please contact the
authors at jared.culbertson@us.af.mil for the correct reference.
## 1\. Introduction
Bayesian probability is a subject that has proven very successful in
prediction, inference and model selection [3, 11, 12]. Čencov [21] gives a
categorical foundation for non-Bayesian statistical inference, but as far as
the authors are aware, a categorical framework for Bayesian probability has
not been fully developed. Lawvere took the first steps in this direction by
defining the category of probabilistic mappings in the unpublished manuscript
[15]. Following this, Lawvere and Huber [16] gave a seminar in Zurich on
_Bayesian Sections_ , further developing this category as a basis for Bayesian
probability. The first appearance in the literature was an expansion of these
ideas by Giry [10], who showed that the endofunctor on the category of
measurable spaces $T\colon\mathcal{M}\to\mathcal{M}$ associated to the
probability adjunction given by Lawvere forms a monad, and that Lawvere’s
category of probabilistic mappings is the Kleisli category of that monad.
Subsequently, Meng [17], examined the category of convex sets and affine
linear maps, which can be shown to be equivalent to the category of
Eilenberg–Moore algebras of the Giry monad. This category can be thought of as
the category of “decision rules” since the objects of that category are
certain measurable functions $TX\to X$ whose fibers partition the space of
probability measures on a given space $X$ into positive convex measurable
sets.111Doberkat [6] refers to this process of making decision rules as
_derandomization_ , which in certain applications like probabilistic semantics
may be a more appropriate terminology. Based on the work by Giry restricting
the monad to Polish spaces, Doberkat[7, 5] has since characterized the
Eilenberg–Moore $T$-algebras for these topological spaces. That work, however,
was based upon giving the space of probability measures the $\sigma$-algebra
generated by the weak topology (as used for the Polish space monad) which
results in (nontrivial) finite spaces having no $T$-algebras. In the final
section, we show that this negative result can be circumvented by avoiding
topological conditions and using the initial $\sigma$-algebra generated by the
evaluation maps. Others, including Wendt [22], van Breugel [4] and
Abramsky–Blute–Panangaden [1] have also studied similar constructions.
Central to the study of Bayesian probability is the existence of regular
conditional probabilities. Many textbooks restrict to Polish spaces in order
to prove this existence (e.g., see [8]), though this is not a strictly
necessary condition. Several more general characterizations of conditions
which guarantee the existence of regular conditional probabilities have been
found, either restricting the spaces involved or the joint distributions which
are allowed. In [18], Pachl does not require even countably generated
$\sigma$-algebras, but relies instead on a certain notion of compactness. We
will prefer to follow [9], where Faden gives a necessary and sufficient
condition when we restrict to countably generated spaces. Namely, the
marginals of a joint distribution must give perfect measure spaces. The class
of perfect measures is broad and includes, for example, all Radon measures.
For this reason, we will only consider perfect measures and begin by showing
that the Giry monad restricted to perfect probability measures is still a
monad (this is straightforward based on Theorem 2.2). Note that everything
prior to Section 3.2 holds without restricting to perfect probability measures
(see [10]).
The main theorem (Theorem 4.1) states that inference maps are uniquely
determined by a prior probability and a sampling distribution. This result
follows from the existence of regular conditional probabilities, and we
restate an existence theorem (Theorem 3.1) of Faden [9]. Using our
characterization of Bayesian probability and the well-known fact that the
Kleisli category embeds into the category of $T$-algebras, we can then see
that the category of decision rules provides a common framework for both
decision theory and Bayesian probability. Some of these ideas are similar in
spirit to the general notion of distributions based on commutative monads
found in the recent paper of Kock [13].
## 2\. The Category of Perfect Probabilistic Mappings
We begin with an overview of the category of perfect probabilistic mappings,
which is a slight modification of Lawvere’s category (see [15]) of
probabilistic mappings. The restrictions to have countably generated spaces as
objects and restrict to perfect probability measures in the definition of the
morphisms are required to ensure the existence of regular conditional
probabilities, but are otherwise unnecessary. We include these restrictions
throughout the paper to avoid overcomplicating the statements of the main
theorems. See the introduction for a discussion of alternative definitions.
Most of the following fundamental results are widely known without the
restriction to perfect measures, and we indicate where the usual arguments
must be modified due to this additional constraint. We also review the
definition and foundational properties of perfect probability measures.
Let us fix the following notation. We will denote the $\sigma$-algebra of a
measurable space $X$ by $\Sigma_{X}$, and the category of countably generated
measurable spaces and measurable functions by $\mathcal{M}_{cg}$. For an
object $(X,\Sigma_{X})$ in $\mathcal{M}_{cg}$ we will often drop the
associated $\sigma$-algebra from the notation and denote it simply by $X$ when
the $\sigma$-algebra is obvious or inconsequential. We will use
$(1,\Sigma_{1})$ and $(2,\Sigma_{2})$ for the one-element and two-element
measurable sets with the discrete $\sigma$-algebras, but we will similarly
just write “$1$” or “$2$” when these are used as objects in some category.
###### Definition 2.1.
A measure space $(X,\Sigma,\mu)$ is called perfect if for any measurable
function $f\colon X\to\mathbb{R}$, there exists a Borel set $E\subset f(X)$
such that $\mu(f^{-1}(E))=\mu(X)$. A family $\\{\mu_{i}\\}_{i\in I}$ of
measures on $X$ is equiperfect if given $f$ as before, there exists a single
Borel set $E\subset f(X)$ with $\mu_{i}(f^{-1}(E))=\mu_{i}(X)$ for all $i\in
I$.
The following theorem collects many basic results about perfect probability
measures. We refer the reader to [19] and [9] and the references therein for
proofs and more details on perfect measures.
###### Theorem 2.2.
Let $(X,\Sigma_{X})$ and $(Y,\Sigma_{Y})$ be measurable spaces. Then
1. (a)
if $P$ is a {0,1}-valued probability measure, then $P$ is perfect
2. (b)
if $P$ and $Q$ are probability measures on $X$ such that $Q\ll P$, then $Q$ is
perfect,
3. (c)
if $\Sigma_{X}^{\prime}\subset\Sigma_{X}$ and a probability measure $P$ on $X$
is perfect, then $P$ restricted to $\Sigma_{X}^{\prime}$ is perfect,
4. (d)
if $f\colon X\to Y$ is a measurable function and a probability measure $P$ on
$X$ is perfect, then $f_{\ast}P$ is a perfect probability measure on $Y$,
5. (e)
if $J$ is a probability measure on $(X\times Y,\Sigma_{X}\otimes\Sigma_{Y})$
with marginals $P$ and $Q$, then $J$ is perfect if and only if $P$ and $Q$ are
perfect,
6. (f)
if $f\colon X\times\Sigma_{Y}\to[0,1]$ is measurable for each fixed
$B\in\Sigma_{Y}$ and $P$ is a perfect probability measure on $X$, then the
probability measure on $Y$ defined by $Q(B)=\int_{X}Q(x,B)\,dP$ is perfect if
and only if $\\{f(x,\cdot)\\}_{x\in X}$ is an equiperfect family of
probability measures on $Y$ except on a $P$-null set.
###### Definition 2.3.
The category of perfect probabilistic mappings $\mathcal{P}$ has countably
generated measurable spaces $(X,\Sigma_{X})$ as objects and an arrow between
two such objects $f\colon(X,\Sigma_{X})\to(Y,\Sigma_{Y})$ consists of a
function $f\colon X\times\Sigma_{Y}\to[0,1]$ such that
1. (i)
for all $B\in\Sigma_{Y}$, the function $f(\cdot,B)\colon X\rightarrow[0,1]$ is
measurable,
2. (ii)
the collection $\\{f(x,\cdot)\colon\Sigma_{Y}\rightarrow[0,1]\\}_{x\in X}$ is
an equiperfect family of probability measures on $Y$.
That is, morphisms in $\mathcal{P}$ can be thought of as parametrized families
of perfect probability measures that vary measurably. For an arrow
$f\colon(X,\Sigma_{X})\rightarrow(Y,\Sigma_{Y})$ we will often denote the
function $f(\cdot,B)\colon X\to[0,1]$ by $f_{B}$ and the function
$f(x,\cdot)\colon\Sigma_{Y}\to[0,1]$ by $f_{x}$.
Given two arrows
(1) $(X,\Sigma_{X})\xrightarrow{f}(Y,\Sigma_{Y})\xrightarrow{g}(Z,\Sigma_{Z})$
the composition $g\circ f\colon X\times\Sigma_{Z}\rightarrow[0,1]$ is defined
by
(2) $(g\circ f)(x,C)=\int_{y\in Y}g_{C}(y)df_{x}.$
This composition is well-defined due to Theorem 2.2 (f), and associativity
follows easily from the monotone convergence theorem. An important fact is
that every measurable function $f\colon X\to Y$ may be regarded as a
$\mathcal{P}$-morphism $\delta_{f}\colon X\to Y$, where the Dirac (or one
point) measure
(3) $\delta_{f}(x,B)=\begin{cases}1&\text{if }f(x)\in B\\\ 0&\text{if
}f(x)\notin B\end{cases}$
assigns to each $x\in X$ the Dirac measure on $Y$ which is concentrated at
$f(x)$. Taking the measurable function $f$ to be the identity map on a
particular measurable space $X$ gives the arrow
$\delta_{Id_{X}}\colon(X,\Sigma_{X})\to(X,\Sigma_{X})$, i.e., the identity
arrow for $X$ in $\mathcal{P}$. In fact, it is easy to check that the
association $f\mapsto\delta_{f}$ determines a functor
$\delta\colon\mathcal{M}_{cg}\to\mathcal{P}$ taking a measurable space to
itself. Note that this functor is not faithful, however, and so we do not get
an embedding of $\mathcal{M}_{cg}$ into $\mathcal{P}$. We will call a
$\mathcal{P}$ arrow $P\colon X\to Y$ deterministic if for every
$B\in\Sigma_{Y}$ the measurable functions $P_{B}\colon X\to[0,1]$ assume only
the values $0$ or $1$. In fact, every deterministic $\mathcal{P}$-arrow
$P\colon X\to Y$ is of the form $P=\delta_{f}$ for some measurable function
$f$, provided that $X$ has cardinality below the cardinality of the set of all
measurable sets.
The following lemma gives two useful properties which follow easily from
standard exercises in measure theory and the definition of composition in
$\mathcal{P}$.
###### Lemma 2.4.
If $p\colon X\to Y$ is a measurable function and $f\colon Y\to Z$ a
$\mathcal{P}$-morphism, then the composition
$X$$Y$$Z$$\delta_{p}$$f$
is given by $(f\circ\delta_{p})(x,C)=f_{p(x)}(C)$. On the other hand, if
$q\colon Y\to Z$ is a measurable function and $g\colon X\to Y$ a
$\mathcal{P}$-morphism, then the composition
$X$$Y$$Z$$\delta_{q}$$g$
is given by $(\delta_{q}\circ g)(x,C)=g_{x}(q^{-1}C)$.
There are several distinguished objects in $\mathcal{P}$ that play an
important role in many constructions. Any set $X$ with the indiscrete
$\sigma$-algebra $\Sigma_{X}=\\{X,\emptyset\\}$ is a terminal object since any
arrow $P\colon Y\rightarrow X$ is completely determined by the fact that
$P_{y}$ must be a probability measure on $X$. We denote the canonical terminal
object by $1$ since it is isomorphic to the one-element set. Notice that an
arrow $P\colon 1\to X$ is precisely a perfect probability measure on $X$ and
that $1$ is a separator for $\mathcal{P}$. In addition to having a separator,
the category $\mathcal{P}$ also has a coseparator, the two-element set
$2=\\{\top,\bot\\}$ with $\Sigma_{2}$ the discrete algebra on $2$. Moreover,
there is a set bijection
$\hom_{\mathcal{P}}(X,2)\simeq\hom_{\mathcal{M}_{cg}}(X,[0,1]).$
We briefly show how the Giry monad factors through $\mathcal{P}$. Let
$\mathscr{P}X$ denote the set of perfect probability measures on $X$, endowed
with the coarsest $\sigma$-algebra such that the evaluation maps
$ev_{B}\colon\mathscr{P}X\to[0,1]$ given by $ev_{B}(P)=P(B)$ are measurable.
Then we can define a functor $\mathscr{P}\colon\mathcal{P}\to\mathcal{M}_{cg}$
which sends a measurable space $X$ to the space $\mathscr{P}X$ of probability
measures on $X$. On arrows, $\mathscr{P}$ sends the $\mathcal{P}$-arrow
$f\colon X\to Y$ to the measurable function
$\mathscr{P}f\colon\mathscr{P}X\to\mathscr{P}Y$ defined pointwise on
$\Sigma_{Y}$ by
(4) $\mathscr{P}f(P)(B)=\int_{X}f_{B}\,dP.$
That is, $\mathscr{P}f(P)$ gives the probability measure on $Y$ defined by the
composition
(5) $1$$X$$Y$$P$$f$$f\circ P$
in $\mathcal{P}$. Since $\mathscr{P}X=\hom_{\mathcal{P}}(1,X)$ as sets,
another common notation for $\mathscr{P}X$ is $X^{1}$, but we will use the
functor notation for clarity.
In fact, $\mathscr{P}X$ is an important object in $\mathcal{P}$ as well, and
based on the definition of the $\sigma$-algebra on $\mathscr{P}X$, we can
define the evaluation morphism $\varepsilon_{X}\colon\mathscr{P}X\to X$ by
$\varepsilon_{X}(P,A)=P(A)$. With this, we are able to characterize the
relationship between $\mathcal{M}_{cg}$ and $\mathcal{P}$, first proved in
[10] for general measurable spaces and probability measures. To see that this
proof goes through when restricting to countably generated spaces and perfect
measures, we only need to observe that the $\sigma$-algebra defined above for
$\mathscr{P}X$ is countably generated when $X$ is countably generated and that
pushforwards take perfect measures to perfect measures.
###### Theorem 2.5.
The functors $\delta\colon\mathcal{M}_{cg}\to\mathcal{P}$ and
$\mathscr{P}\colon\mathcal{P}\to\mathcal{M}_{cg}$ form an adjunction
$\mathcal{M}_{cg}$$\mathcal{P}$$\delta$$\mathscr{P}$$\delta\dashv\mathscr{P}$
with the unit of the adjunction $\eta_{X}(x)=\delta_{\\{x\\}}$ and the counit
$\varepsilon_{X}\colon\mathscr{P}X\to X$.
Thus, we can realize the Giry monad as the composition
$T=\mathscr{P}\,\circ\,\delta$, and moreover, $\mathcal{P}$ is equivalent to
the smallest category through which $T$ factors—i.e., it is equivalent to the
Kleisli category $K(T)$ of the Giry monad. Hence every $\mathcal{P}$ arrow
$P\colon X\to Y$ corresponds uniquely to a measurable arrow $X\to TY$.
## 3\. Joint Distributions and Conditionals
Given a family of objects $\\{X_{i}\\}_{i\in I}$ we can form the cartesian
product $\prod_{i\in I}X_{i}$ and endow this set with the product
$\sigma$-algebra generated by all the projection maps $\prod_{i\in
I}X_{i}\stackrel{{\scriptstyle\pi_{j}}}{{\longrightarrow}}X_{j}$, one for each
index $j\in I$. It is easy to see that
(6) $\left(\left(\prod_{i\in I}X_{i},\bigotimes_{i\in
I}\Sigma_{X_{i}}\right),\\{\delta_{\pi_{i}}\\}_{i\in I}\right)$
does not give a categorical product. In fact, only weak products and
equalizers exist in $\mathcal{P}$, as the uniqueness condition fails for both
constructions. We use the terminology “product space” to denote the set
product of any family $\\{(X_{i},\Sigma_{X_{i}})\\}_{i\in I}$ of objects with
the product $\sigma$-algebra and not to imply that that object
$\left(\prod_{i\in I}X_{i},\bigotimes_{i\in I}\Sigma_{X_{i}}\right)$
with projections satisfies any universality condition. We will call a
probability measure $J\colon 1\to(\prod_{i\in I}X_{i},\otimes_{i\in
I}\Sigma_{X_{i}})$ on a product space a joint distribution. We do not mean to
imply that these are distributions of a random variable, but rather indicate a
measure on a product space which is not necessarily a product measure. These
joint distributions are the main objects of study in Bayesian probability.
Given any joint distribution $J\colon 1\to\prod_{i\in I}X_{i}$, for each $j\in
I$ we have the diagram
(7) $1$$X_{j}$$\prod_{i\in I}X_{i}$$J$$\delta_{\pi_{j}}$$\delta_{\pi_{j}}\circ
J$
where the composite $\delta_{\pi_{j}}\circ J$ is called the _marginal_
(distribution) of $J$ on component $X_{j}$ and is given by
$(\delta_{\pi_{j}}\circ J)(A_{j})=J(\pi_{j}^{-1}A_{j})$ by Lemma 2.4.
Given only the probability measures on the components, $\\{P_{i}\\}_{i\in I}$,
there are many joint distributions on the product space whose marginals are
the given family $\\{P_{i}\\}_{i\in I}$. By using _relationships_ in the form
of conditionals between the components, we bring into play additional
knowledge that permits the determination of the appropriate joint
distribution. If the uncertainty of component $X_{j}$, as expressed by a
probability measure $P_{j}$ on component $X_{j}$, depends conditionally on a
parameter which varies over component $X_{i}$ then we have the $\mathcal{P}$
arrow $h\colon X_{i}\to X_{j}$. These conditionals—which are the morphisms in
$\mathcal{P}$—are the key to determining a unique joint distribution. The
relationship between the components $X_{i}$ and $X_{j}$ is mediated by the
conditional $h$ and expresses the relationship $P_{j}=h\circ P_{i}$.
### 3.1. Constructing a Joint Distribution Given Conditionals
We now show how marginals and conditionals can be used to determine joint
distributions in $\mathcal{P}$. This development follows that of [1] where we
first learned of this approach (the category $\mathcal{S}toch$ of stochastic
kernels in that paper is nearly what we call $\mathcal{P}$, although our
restriction to countably generated spaces and perfect measures is replaced in
the paper by restricting to Polish spaces). Given a $\mathcal{P}$-arrow
$h\colon X\to Y$ and a perfect probability measure $P_{X}\colon 1\to X$ on
$X$, consider the diagram
(8) $1$$X$$Y$$X\times Y$$P_{X}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$J_{h}$$h$
where $J_{h}$ is the uniquely determined joint distribution on the product
space $X\times Y$ defined on the rectangles of the $\sigma$-algebra
$\Sigma_{X}\otimes\Sigma_{Y}$ by
(9) $J_{h}(A\times B)=\int_{A}h_{B}\,dP_{X}.$
Then it follows from Theorem 2.2 that $J_{h}$ is perfect. The marginal of
$J_{h}$ with respect to $Y$ then satisfies $\delta_{\pi_{Y}}\circ J_{h}=h\circ
P_{X}$ and the marginal of $J_{h}$ with respect to $X$ is $P_{X}$, both of
which are also perfect. By a symmetric argument, if we are given a probability
measure $P_{Y}$ and conditional probability $k\colon Y\to X$ then we obtain a
unique perfect joint distribution $J_{k}$ on the product space $X\times Y$
given on the rectangles by
(10) $J_{k}(A\times B)=\int_{B}k_{A}\,dQ.$
However, if we are given $P_{X},P_{Y},h,k$ as indicated in the diagram
(11) $1$$X$$Y$$X\times
Y$$P_{Y}$$P_{X}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$J_{k}$$J_{h}$$h$$k$
then we have that $J_{h}=J_{k}$ if and only if the compatibility conditions
(12) $\begin{array}[]{lcl}P_{X}&=&k\circ P_{Y}\\\ P_{Y}&=&h\circ
P_{X}\end{array}$
are satisfied. Thus if the compatibility conditions are satisfied, then we can
realize the product rule of probability in $\mathcal{P}$ as
(13) $\int_{A}h_{B}\,dP_{X}=J(A\times B)=\int_{B}k_{A}\,dP_{Y}.$
In the extreme case, suppose we have a $\mathcal{P}$-arrow $h\colon X\to Y$
which factors through the terminal object $1$ as
(14) $X$$Y$$1$$h$$!$$Q$
where $!$ represents the unique arrow from $X\to 1$. If we are also given a
perfect probability measure $P\colon 1\to X$, then we can calculate the joint
distribution determined by $P$ and $h=Q\circ!$ as
(15) $\begin{array}[]{lcl}J(A\times B)&=&\int_{A}(Q\circ!)_{B}\,dP\\\
&=&P(A)\cdot Q(B)\end{array}$
so that $J=P\otimes Q$. This is precisely the situation where we say that the
marginals $P$ and $Q$ are _independent_. Thus in $\mathcal{P}$ independence
corresponds to a special instance of a conditional—one that factors through
the terminal object.
### 3.2. Constructing Regular Conditionals given a Joint Distribution
The following result is the basis from which the inference maps in Bayesian
probability theory are subsequently constructed. The proof of the following
theorem can be found in [9], where several equivalent conditions are
identified.
###### Theorem 3.1.
If $J:1\to X\times Y$ is a $\mathcal{P}$-morphism with marginals $P_{X}$ on
$X$ and $P_{Y}$ on $Y$, then there exists a $\mathcal{P}$-arrow $f$ that makes
the diagram
(16) $1$$Y$$X$$P_{Y}$$P_{X}$$f$
commute and satisfies
(17) $J(A\times B)=\int_{B}f_{A}\,dP_{Y}.$
Moreover, the morphism $f$ is the unique $\mathcal{P}$-morphism with these
properties, up to a set of $P_{Y}$-measure zero.
Interestingly, we can use Theorem 3.1 to obtain a seemingly stronger
statement, i.e., that the regular conditional probability factors through the
product. Though this is not difficult to prove, we will prefer this stronger
statement in the sequel.
###### Theorem 3.2.
If $J:1\to X\times Y$ is a $\mathcal{P}$-morphism with marginal distributions
$P_{X}$ and $P_{Y}$ on $X$ and $Y$, then there exist $\mathcal{P}$ arrows $f$
and $g$ such that the diagram
(18) $1$$Y$$X$$X\times
Y$$P_{Y}$$P_{X}$$J$$\delta_{\pi_{Y}}$$\delta_{\pi_{X}}$$\delta_{\pi_{X}}\circ
f$$f$$g$$\delta_{\pi_{Y}}\circ g$
commutes and
(19) $\int_{A}(\delta_{\pi_{Y}}\circ g)_{B}\,dP_{X}=J(A\times
B)=\int_{B}(\delta_{\pi_{X}}\circ f)_{A}\,dP_{Y}.$
###### Proof.
We can apply Theorem 3.1 to see that there is a $\mathcal{P}$-arrow $f\colon
Y\to X\times Y$ satisfying $J=f\circ P_{Y}$ such that
(20) $\int_{C}f_{A\times B}\,dP_{Y}=J\left(A\times(B\cap C)\right).$
Then from Lemma 2.4, we know that $(\delta_{\pi_{X}}\circ f)(y,A)=f(y,A\times
Y)$ and so
(21) $\displaystyle\int_{B}(\delta_{\pi_{X}}\circ f)_{A}\,dP_{Y}$
$\displaystyle=\int_{B}f_{A\times Y}\,dP_{Y}$ (22)
$\displaystyle=J\left(A\times(Y\cap B)\right)$ (23) $\displaystyle=J(A\times
B)$
Similarly we obtain a $\mathcal{P}$-arrow $g\colon X\to X\times Y$ satisfying
$J=g\circ P_{X}$ and
(24) $\int_{A}(\delta_{\pi_{Y}}\circ g)_{B}\,dP_{X}=J(A\times B).$
With these facts, it is a simple exercise to check that the diagram commutes.
∎
Note that if the joint distribution $J$ is obtained by a probability measure
$P_{X}$ and a conditional $h\colon X\rightarrow Y$ using the method described
by Diagram 8, then using the above result and notation it follows $P_{X}$-a.s.
that $h=\delta_{\pi_{Y}}\circ g$.
###### Remark 3.3.
(Tonneli’s Theorem) Given a $\mathcal{P}$-morphism $J:1\to X\times Y$, with
marginals $P_{X}$ and $P_{Y}$ let $\gamma\colon X\rightarrow X\times Y$ and
$\varphi\colon Y\rightarrow X\times Y$ be the $\mathcal{P}$ arrows satisfying
$\gamma\circ P_{X}=J$ and $\varphi\circ P_{Y}=J$ whose existence is guaranteed
by Theorem 3.2. Given any measurable function $F\colon X\times
Y\rightarrow[0,1]$ we have the diagram
(25) $1$$Y$$X$$X\times
Y$$2$$P_{Y}$$P_{X}$$\delta_{\pi_{Y}}$$\delta_{\pi_{X}}$$\phi$$\gamma$$J$$\overline{F}$$\overline{f}=\overline{F}\circ\gamma$$\overline{g}=\overline{F}\circ\varphi$
where the top two triangles commute. Thus we can define
$\overline{f}=\overline{F}\circ\gamma$ and
$\overline{g}=\overline{F}\circ\varphi$ so that the entire diagram commutes.
From this, it follows that
(26) $\int_{X}f\,dP_{X}=\int_{X\times Y}F\,dJ=\int_{Y}g\,dP_{Y}$
and we can realize Tonneli’s Theorem as the special case with $J=P_{X}\otimes
P_{Y}$.
This formulation provides the context for the following optimal transportation
problem: given marginals $P_{X}$ and $P_{Y}$ that model the supply and demand
constraints, and a cost function $F$ (defined up to a scalar constant)
representing the unit cost to transport a product from $x\in X$ to $y\in Y$,
what joint distribution $J$ on $X\times Y$ with marginals $P_{X}$ and $P_{Y}$
minimizes the objective function $\int_{X\times Y}F\,dJ$? The optimal
assignment is then the conditional probability $X\rightarrow Y$ determined by
the optimal joint distribution $J$. For example, this problem is investigated
and a unique solution is given in certain cases in [14].
## 4\. Bayesian probability in $\mathcal{P}$
If we replace $X$ and $Y$ in Diagram 18 by $D$(ata) and $H$(ypotheses), and
the composites $\delta_{\pi_{Y}}\circ g$ by $\mathcal{S}$(ampling
distribution) and $\delta_{\pi_{X}}\circ f$ by $\mathcal{I}$(nference), then
we can define $P_{D}=\mathcal{S}\circ P_{H}$ to obtain
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In the context of Bayesian probability the probability measure $P_{H}$ is
often called a _prior probability_.
In this notation, the product rule in $\mathcal{P}$ given in Equation 13
becomes
(28)
$\int_{\mathcal{D}}\mathcal{I}_{\mathcal{H}}\,dP_{D}=J(\mathcal{H}\times\mathcal{D})=\int_{\mathcal{H}}\mathcal{S}_{\mathcal{D}}\,dP_{H}$
where $\mathcal{H}\in\Sigma_{H}$ and $\mathcal{D}\in\Sigma_{D}$. We will spend
the remainder of this sections showing how this interpretation of these spaces
in $\mathcal{P}$ provides a categorical foundation for Bayesian probability.
First, we briefly review the fundamental concepts in Bayesian probability
theory which can be found in [12] and then proceed to show how $\mathcal{P}$
is the appropriate category for this theory. Generally, a Bayesian model is
comprised of a number of items including
1. (i)
two measurable spaces $H$ and $D$ representing hypotheses and data,
respectively,
2. (ii)
a probability measure $P_{H}$ on the $H$ space called the prior probability,
3. (iii)
a $\mathcal{P}$ arrow $\mathcal{S}\colon H\rightarrow D$ called the sampling
distribution,
4. (iv)
a $\mathcal{P}$ arrow $\mathcal{I}\colon D\rightarrow H$ called the inference
map,
Note that the data space $D$ can also be thought of as the event space for
some experiment, and the $\sigma$-algebra on $D$ is determined by
distinguishable data. The prior probability measures are updated via the
inference map as one takes measurements, which correspond to probability
measures $\mu$ on $D$. These updated probability measures are then called
posterior probabilities and are given by
$\widehat{P}_{H}=\mathcal{I}\circ\mu$. The posterior $\widehat{P}_{H}$ then
becomes the prior probability for the next step and the process continues as
more measurements are taken. At each step in a Bayesian process, the posterior
probability is a representation of knowledge about the hypotheses based on all
of the data that has been accumulated up to that point.
Using the sampling distribution $\mathcal{S}$ and the prior probability
$P_{H}$ on $H$, we can define a joint distribution $J$ on the product space
$H\times D$ as in Section 3.1 by defining it on the rectangles as $J(A\times
B)=\int_{A}\mathcal{S}_{B}\,dP_{H}$. The $D$-marginal (prior probability on
data) is then $P_{D}=\mathcal{S}\circ P_{H}=\delta_{\pi_{D}}\circ J$. Using
Theorem 3.2, we have the following theorem.
###### Theorem 4.1.
Given $\mathcal{P}$ arrows representing a prior probability $P_{H}\colon 1\to
H$ and a sampling distribution $\mathcal{S}\colon H\to D$, the inference map
$\mathcal{I}\colon D\to H$ is determined uniquely up to a set of
$P_{D}$-measure zero.
###### Proof.
Let $\mathcal{I}\colon D\rightarrow H$ be the composition
$\delta_{\pi_{H}}\circ f$, where $\delta_{\pi_{H}}$ is the projection $H\times
D\to H$ and $f\colon D\rightarrow H\times D$ is the $\mathcal{P}$-arrow
satisfying
(29) $J(U\times V)=\int_{D}f_{U\times V}\,dP_{D}$
whose existence is given by Theorem 3.2. Thus
(30) $\int_{A}\mathcal{S}_{B}\,dP_{H}=J(A\times
B)=\int_{B}\mathcal{I}_{A}\,dP_{D}$
and this inference arrow $\mathcal{I}$ is unique in that if
$\mathcal{I}^{\prime}$ also satisfies equation 30 then the set $\\{y\in
Y\mid\mathcal{I}_{y}\neq\mathcal{I}^{\prime}_{y}\\}$ has $P_{D}$-measure zero.
∎
Thus the complete process works in the following way. A prior probability
$P_{H}$ and sampling distribution $\mathcal{S}$ are specified, from which one
determines the inference map $\mathcal{I}$. Once measurements $\mu\colon 1\to
D$ are taken, we then calculate the posterior probability by
$\mathcal{I}\circ\mu$. This updating procedure can be characterized by the
diagram
(31) $1$$H$$D$$P_{H}$$\mu$$\mathcal{S}$$\mathcal{I}$$\mathcal{I}\circ\mu$
where the solid lines indicate arrows given a priori, the dotted line
indicates the arrow determined using Theorem 3.2, and the dashed lines
indicate the updating after a measurement. Note that if there is no
uncertainty in the measurement, then $\mu=\delta_{\\{x\\}}$ for some $x\in D$,
but in practice there is usually some uncertainty in the measurements
themselves.
Following the calculation of the posterior probability, the sampling
distribution is then updated, if required. The process can then repeat: using
the posterior probability and the updated sampling distribution the updated
joint probability distribution on the product space is determined and the
corresponding (updated) inference map determined. We can then continue to
iterate as long as new measurements are received. For some problems (such as
with the standard urn problem with replacement of balls) the sampling
distribution does not change from iterate to iterate, but the inference map is
updated since the posterior probability on the hypothesis space changes with
each measurement. The model selection problem (either once at the beginning of
this process, or iteratively throughout) can also be modeled as a meta-
Bayesian process, where the hypothesis space is the space of potential models
and the data constitutes some information that would inform on the suitability
of a given model.
###### Remark 4.2.
We know from Theorem 4.1 that the inference map $\mathcal{I}$ is uniquely
determined by $P_{H}$ and $\mathcal{S}$ up to a set of $P_{D}$-measure zero.
However, there is no reason a priori that a measurement $\mu\colon 1\to D$ is
required to be absolutely continuous with respect to $P_{D}$. In $\mu$ is not
absolutely continuous with respect to $P_{D}$, then a different choice of
inference map $\mathcal{I}^{\prime}$ could yield a different posterior
probability—i.e., we could have
$\mathcal{I}\circ\mu\neq\mathcal{I}^{\prime}\circ\mu$. Thus we make the
assumption that measurement probabilities on $D$ are absolutely continuous
with respect to the prior probability $P_{D}$ on $D$. This is a reasonable
assumption, however, since if a data event is impossible (has $P_{D}$-measure
zero) under a certain model, then the model should not be expected to make an
meaningful inference when presented with that data. On the other hand, it is
easy to see that if a measurement $\mu\ll P_{D}$, then $\mathcal{I}\circ\mu\ll
P_{H}$, as expected.
We emphasize that this procedure can be employed for any perfect prior
probability and any regular conditional probability. For example, given a
perfect prior $P\colon 1\to\mathscr{P}X$ and the conditional
$\varepsilon_{X}\colon\mathscr{P}X\to X$ there corresponds a unique inference
map $\mathcal{I}\colon X\to\mathscr{P}X$ satisfying, for all $A\in\Sigma_{X}$
and for all $\mathcal{B}\in\Sigma_{\mathscr{P}X}$,
(32)
$\int_{\mathscr{P}X}{\varepsilon_{X}}_{A}\,dP=\int_{X}\mathcal{I}_{\mathcal{B}}\,d(\varepsilon_{X}\circ
P).$
In the case where $X=2=\\{\top,\bot\\}$ (the two element set), these “higher
order distributions” $P\colon 1\to\mathscr{P}2$ can be used to explicate the
concept of $A_{p}$ distributions as characterized by Jaynes [12, Chapter 18].
Using our notation, a proposition $A$, which is a morphism $A\colon 1\to 2$ in
the category $\mathcal{S}et$ of sets, has an associated probability of truth,
say $Pr(A)=p$. Hence $A$ determines a $\mathcal{P}$-morphism
$\overline{A}\colon 1\to 2$, with $\overline{A}(\\{\top\\})=p$. The
information supplied by the arrow $\overline{A}$ consists only of the single
value $p$ and fails to indicate how sensitive this proposition is to
additional data. The confidence that one has in the value $p$ can be supplied
by the higher order distributions which are probability measures on the space
$\mathscr{P}2$ of probability measures on $2$. Since $\mathscr{P}2$ consists
precisely of the Bernoulli distributions
$B_{\theta}=\theta\delta_{\top}+(1-\theta)\delta_{\bot}$, where
$\theta\in[0,1]$, it follows that $\mathscr{P}2\simeq[0,1]$. Consequently, any
distribution
(33) $1\xrightarrow{A_{p}}\mathscr{P}2$
has an expected value which can be calculated using the composition
(34) $1\xrightarrow{A_{p}}\mathscr{P}2\xrightarrow{\varepsilon_{2}}2.$
Thus $E(A_{p})=(\varepsilon_{2}\circ A_{p})(\\{\top\\})=p$ and any such
distribution provides a more informative measure. For example, the two
distributions on $\mathscr{P}2\simeq[0,1]$ specified by
$p=\delta_{\frac{1}{2}}$ and $p^{\prime}$ the uniform (Lebesque) measure both
have expected value $\frac{1}{2}$. Yet clearly, the first is deterministic,
expressing a (complete) confidence in the statement that the expected value of
the proposition
(35) $1\xrightarrow{~{}~{}\overline{A}~{}~{}}2$
where $\overline{A}(\top)=\frac{1}{2}$ is $\frac{1}{2}$. On the other hand,
the distribution $p^{\prime}$ also determines $\overline{A}$, but instead
expresses a maximal ignorance modeled by the uniform distribution.
## 5\. The Category of Decision Rules
Recall that the Giry monad $T$ factors through $\mathcal{P}$ via the
adjunction of Theorem 2.5. At the other end of the spectrum of categories
through which the Giry monad factors is the Eilenberg–Moore category
$\mathcal{M}_{cg}^{T}$, consisting of the Eilenberg–Moore algebras of the Giry
monad. From the theory of monads (see [2], for example), we know that
$\mathcal{P}$ then embeds into $\mathcal{M}_{cg}^{T}$, which has additional
structure that is useful for dealing with other aspects of probability theory
and decision making which $\mathcal{P}$ is not equipped for. Let us briefly
recall the definition of a $T$-algebra.
If $(T,\eta,\mu)$ is a monad in a category $\mathcal{C}$, a $T$-algebra
$(X,\alpha)$ is a pair consisting of an object $X$ in $\mathcal{C}$ and a
$\mathcal{C}$ arrow $\alpha\colon TX\to X$ such that the diagrams
(36)
$T^{2}X$$TX$$TX$$X$$T\alpha$$\mu_{X}$$\alpha$$\alpha$$X$$TX$$X$$\alpha$$\eta_{X}$$Id$
commute; the first diagram is called the associative law and the second
diagram the unit law. A morphism of $T$-algebras
$(X,\alpha)\stackrel{{\scriptstyle f}}{{\longrightarrow}}(Y,\beta)$ is an
arrow $f\colon X\to Y$ of $\mathcal{C}$ such that the diagram
(37) $TX$$TY$$X$$Y$$Tf$$\beta$$\alpha$$f$
commutes.
When $T$ is the Giry monad, an algebra
$TX\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}X$ consists of a
measurable space $X$ with a countably generated $\sigma$-algebra, the space
$TX=\mathscr{P}X$ of probability measures on $X$, and a measurable map
$\alpha$ satisfying the two defining properties of a $T$-algebra. The
measurable map $Tf$ in the definition of a morphism of $T$-algebras is the
pushforward map: given $P\in TX$ the pushforward by $f$ is
$Tf(P)=f_{\ast}P\in\mathscr{P}Y$.
The $T$-algebras $(X,\alpha)$ are often called decision rules since the
measurable map $\alpha$ assigns (decides) a value in $X$ to each probability
measure $P$ on $X$. Alternatively, we can think of a decision rule as
collapsing a probability distribution to a definite value, or derandomizing a
probability distribution as in [6]. For this reason, we often use the
descriptive characterization of Čencov [21] and call the category
$\mathcal{M}_{cg}^{T}$ the category of decision rules.222Čencov did not work
in $\mathcal{M}_{cg}^{T}$, but rather in $\mathcal{P}$, restricting to
measurable spaces $X$ such that $\mathscr{P}X$ has a $\sigma$-algebra
generated by finitely many atoms. The primary difference in the current
approach and that of Čencov is that we take a Bayesian viewpoint, while he is
attempting to describe the standard statistical inference perspective.
Embedding the main consequence of the existence of regular conditional
probabilities for Bayesian probability into $\mathcal{M}_{cg}^{T}$, we have
the following.
###### Theorem 5.1.
Given a measurable function $\mathcal{S}\colon H\rightarrow TD$, there exists
a measurable function $\mathcal{I}\colon D\to TH$ such that
$\hat{\mathcal{I}}=\mu_{H}\circ\mathcal{I}$ is a retraction of
$\hat{\mathcal{S}}=\mu_{D}\circ T\mathcal{S}$ in $\mathcal{M}_{cg}$.
###### Proof.
This follows immediately from Theorem 4.1 and the embedding of $\mathcal{P}$
into $\mathcal{M}_{cg}^{T}$ and can be summarized in the diagram
(38) $\leavevmode\hbox to193.46pt{\vbox
to100.66pt{\pgfpicture\makeatletter\hbox{\hskip
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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∎
One of the primary advantages to using the existence of regular conditional
probabilities guaranteed by Theorem 3.2 is that we only require that
measurable spaces have countably generated $\sigma$-algebras and that the
measures are perfect. In contrast, much of the previous work involving the
Giry monad and regular conditional probabilities requires resorting to
topological arguments and restricting to Polish spaces. For example, in [5],
Doberkat characterizes the $T$-algebras for the Giry monad under the Polish
space assumption, and proves the counter-intuitive result that there are no
non-trivial decision rules for finite spaces in $\mathcal{M}_{cg}^{T}$. In
contrast, we exhibit a finite space having a $T$-algebra when one does not
require topological restrictions.
###### Example 5.2.
An important such case is the decision rule $d\colon T2\rightarrow 2$ given by
(39) $d(P)=\begin{cases}\top&\text{if }P(\\{\top\\})=1\\\ \bot&\text{if
}P(\\{\top\\})<1.\end{cases}$
The function $d$ is measurable since
$d^{-1}(\\{\top\\})=\\{\delta_{\top}\\}\in\Sigma_{T2}$. The associativity
identity
(40)
$T^{2}2$$T2$$T2$$2$$Td$$d$$\mu_{2}$$d$$Q$$Qd^{-1}$$\mu_{2}(Q)$$d(\mu_{2}(Q))=d(Qd^{-1})$$Td$$d$$\mu_{2}$$d$
where $\mu$ is the monad multiplication defined by
(41) $\mu_{2}(Q)(A)=\int_{q\in T(2)}ev_{A}(q)\,dQ,$
is satisfied since both routes map the element $Q\neq\delta_{\delta_{\top}}\in
T^{2}(2)\mapsto\bot$ while $\delta_{\delta_{\top}}\mapsto\top$. The unit law
$Id_{2}=d\circ\eta_{2}$ is trivial to verify.
The decision rule $d\colon T2\rightarrow 2$ partitions the space $T2$ into
$\delta_{\top}$ and all measures on $2$ whose value on $\\{\top\\}$ is of
measure less than one. There are many other decision rules for $2$, and any
other finite or nonfinite space. Characterizing decision rules without the
requirement for continuity is an open problem.
## 6\. Acknowledgements
The authors would like to express gratitude to F.W. Lawvere, P.F. Stiller and
T. Nguyen for many fruitful conversations regarding these ideas. We also thank
the reviewer for the helpful comments provided that have led to a much clearer
exposition. This work was partially supported by the Air Force Office of
Scientific Research, for which the authors are extremely grateful.
## References
* [1] Abramsky, S., Blute, R., Panangaden, P.: Nuclear and trace ideals in tensored $\ast$-categories. J. Pure Appl. Algebra 143, 3–47 (1999)
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* [3] Berger, J.: Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer Series in Statistics. Springer-Verlag, New York (1985)
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* [6] Doberkat, E.E.: Derandomizing probabilistic semantics through Eilenberg–moore algebras for the Giry monad (2004). Internal Memorandum No. 149
* [7] Doberkat, E.E.: Kleisli morphisms and randomized congruences for the giry monad. J. Pure Appl. Algebra 211(3), 638–664 (2007)
* [8] Dudley, R.: Real Analysis and Probability. No. 74 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2002)
* [9] Faden, A.M.: The existence of regular conditional probabilities: necessary and sufficient conditions. Ann. Probab. 13(1), 288 – 298 (1985)
* [10] Giry, M.: A categorical approach to probability theory. Categorical Aspects of Topology and Analysis 915, 68–85 (1981)
* [11] Howson, C., Urbach, P.: Scientific Reasoning: The Bayesian Approach, 2nd edn. Open Court Publishing, Chicago (1993)
* [12] Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003)
* [13] Kock, A.: Commutative monads as a theory of distributions. Theory Appl. Categ. 26, 97–131 (2012)
* [14] Korman, J., McCann, R.: Optimal transportation with capacity constraints (2012). ArXiv:1201.6404v2
* [15] Lawvere, W.F.: The category of probabilistic mappings (1962). Unpublished
* [16] Lawvere, W.F.: Bayesian sections (2011). Personal communication
* [17] Meng, X.: Categories of convex sets and of metric spaces, with applications to stochastic programming and related areas. Ph.D. thesis, State University of New York at Buffalo (1988)
* [18] Pachl, J.K.: Disintegration and compact measures. Math. Scand. 43, 157 –168 (1978)
* [19] Rodine, R.H.: Perfect probability measures and regular conditional probabilities. Ann. Math. Statist. 37, 1273 – 1278 (1966)
* [20] Ryll-Nardzewski, C.: On quasi-compact measures. Fund. Math. 40, 125–130 (1953)
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|
arxiv-papers
| 2012-05-07T19:35:39 |
2024-09-04T02:49:30.679011
|
{
"license": "Public Domain",
"authors": "Jared Culbertson and Kirk Sturtz",
"submitter": "Jared Culbertson",
"url": "https://arxiv.org/abs/1205.1488"
}
|
1205.1574
|
# Broad-band spectral analysis of the Galactic Ridge X-ray Emission
Takayuki Yuasa**affiliation: Current affiliation and address: Japan Aerospace
Exploration Agency (JAXA), Institute of Space and Astronautical Science
(ISAS), 3-1-1 Yoshinodai, Chuo, Sagamihara, Kanagawa 252-5210, Japan
yuasa@astro.isas.jaxa.jp , Kazuo Makishima, and Kazuhiro Nakazawa Department
of Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-0033, Japan
(Received receipt date; Revised revision date; Accepted acceptance date)
###### Abstract
Detailed spectral analysis of the Galactic X-ray background emission, or the
Galactic Ridge X-ray Emission (GRXE), is presented. To study the origin of the
emission, broad-band and high-quality GRXE spectra were produced from 18
pointing observations with Suzaku in the Galactic bulge region, with the total
exposure of 1 Ms. The spectra were successfully fitted by a sum of two major
spectral components; a spectral model of magnetic accreting white dwarfs with
a mass of $0.66^{+0.09}_{-0.07}~{}M_{\odot}$, and a softer optically-thin
thermal emission with a plasma temperature of $1.2-1.5$ keV which is
attributable to coronal X-ray sources. When combined with previous studies
which employed high spatial resolution of the Chandra satellite (e.g.
Revnivtsev et al. 2009), the present spectroscopic result gives another strong
support to a scenario that the GRXE is essentially an assembly of numerous
discrete faint X-ray stars. The detected GRXE flux in the hard X-ray band was
used to estimate the number density of the unresolved hard X-ray sources. When
integrated over a luminosity range of $\sim 10^{30}-10^{34}~{}\mathrm{erg}\
\mathrm{s}^{-1}$, the result is consistent with a value which was reported
previously by directly resolving faint point sources.
###### Subject headings:
X-rays: diffuse background - Galaxy: bulge - novae, cataclysmic variables
## 1\. INTRODUCTION
Apparently extended X-ray emission has been observed along the Galactic plane
since early years of X-ray astrophysics (e.g. Cooke et al. 1969; Worrall et
al. 1982), and its origin has been one of the long-standing mysteries in the
research field. After its spatial structure, the emission has been called the
Galactic Ridge X-ray Emission (GRXE). Its total luminosity is estimated to be
$\sim 1-2\times 10^{38}~{}\mathrm{erg}\ \mathrm{s}^{-1}$ in the conventional
X-ray energy range of $2-10$ keV (e.g. Koyama et al. 1986; Valinia & Marshall
1998). The detection of intense emission lines from highly ionized Fe in the
GRXE spectrum indicated that optically-thin thermal X-ray emission from hot
plasmas must be the origin (Koyama et al., 1986; Yamauchi & Koyama, 1993), but
it was still unclear whether the plasma is gravitationally bound to discrete
sources, i.e. X-ray stars, or is of truly diffuse nature filling the
interstellar space.
Many scenarios which employ different origins have been proposed to explain
the GRXE. They can be divided into two major groups depending on the assumed
nature of the source, namely “Point Source” and “Diffuse” scenarios. The
former assumes that a collection of faint but numerous discrete X-ray point
sources in the Galaxy composes the GRXE (e.g. Revnivtsev et al. 2006), similar
to the case of the Cosmic X-ray Background (e.g. Giacconi et al. 1979; Ueda et
al. 1999; Mushotzky et al. 2000). In the latter scenario, in contrast,
literally-diffuse X-ray emitting materials are considered to fill the
interstellar space over the observed scales of the GRXE, i.e. is several tens
of degrees (longitudinal) by a few degrees (latitudinal) around the Galactic
center (e.g. Kaneda et al. 1997; Tanuma et al. 1999 and references therein).
Unlike the “Diffuse” scenario which confronted several difficulties of energy
supply and/or plasma confinement, almost only one uncertainty associated to
the “Point Source” scenario was whether there are enough number of faint
Galactic X-ray sources that can collectively explain the GRXE surface
brightness and spectrum, while individually satisfy the luminosity limit
($<10^{33}~{}\mathrm{erg}\ \mathrm{s}^{-1}$) not to be spatially resolved by
X-ray telescopes. In previous studies which used instruments with moderate
angular resolutions (e.g. $>1^{\prime}$), it was not possible to obtain a
definitive conclusion on this problem from direct source counting.
Chandra provided a great leap in the GRXE research thanks to its ever best
angular resolution in the X-ray wavelength ($\sim 0.5^{\prime\prime}$).
Revnivtsev et al. (2009) conducted a very deep Chandra observation on the
Galactic bulge region $(l,b)=(0.113,-1.424)$, and showed that more than 80% of
the $0.5-7$ keV GRXE surface brightness can be resolved into faint X-ray stars
with a limiting sensitivity of $\sim~{}10^{-16}~{}\mathrm{erg}\
\mathrm{cm}^{-2}\ \mathrm{s}^{-1}$. Based on X-ray spectral hardness of the
faint sources, as well as their infrared identifications, Ebisawa et al.
(2005) argued that most sources which show soft and hard spectral indices can
be regarded as X-ray emitting stellar coronae and accreting white dwarf
binaries like polars and intermediate polars (IPs), respectively. This has
been also supported by a study of low-luminosity X-ray source population near
the solar system (Sazonov et al., 2006).
The observation by Revnivtsev et al. (2009) has invested $\sim 1$ Ms of
Chandra exposure in a selected field. This is considered to be an ultimate
limit achievable with the current state of the art. However, questions still
remain: “how do individual types of X-ray sources contribute to the GRXE?” and
“is there any additional spectral component in its spectrum that cannot be
attributed to these types of sources?”.
To address these issues, we take an alternative approach; broad-band and high-
energy-resolution spectroscopy of the GRXE. For understanding competitive
contributions from multiple types of X-ray sources, it is essential to analyze
the GRXE spectra on a broad-band basis because some of the putative
constituents, namely magnetic accreting WDs, are known to emit strong signals
in the hard X-ray energy band above 10 keV. High energy resolution is also
necessary to understand the characteristic Fe emission lines which are emitted
presumably by several types of sources.
Some authors already examined broad-band spectral similarities between the
GRXE and several types of X-ray sources (Revnivtsev et al., 2006; Krivonos et
al., 2007) although spectral quality were not necessarily high. Compared to
these reports, our study improves data quality in terms of counting statistics
and energy resolution by using the Suzaku satellite (Mitsuda et al., 2007). In
the following sections, we try to spectroscopically decompose the GRXE along
the following approach:
1. (i)
Construct a broad-band GRXE spectrum using the data of Suzaku Galactic bulge
observations.
2. (ii)
Constrain contributions from magnetic CVs by analyzing the hard X-ray spectral
shape of the GRXE using the IP spectral model constructed in Yuasa et al.
(2010) and Yuasa (2011).
3. (iii)
Extrapolate the magnetic CV contribution down to a softer energy band below 10
keV, and quantify contribution from other types of X-ray sources to the soft-
band GRXE.
## 2\. OBSERVATIONS AND PREPARATION OF DATA SET
### 2.1. The Suzaku X-ray Observatory
Suzaku, which is the fifth Japanese X-ray astrophysical observatory launched
in 2005, carries the X-ray Imaging Spectrometer (XIS; Koyama et al. 2007) and
the Hard X-ray Detector (HXD; Takahashi et al. 2007).
The XIS consists of four X-ray imaging charge coupled device (CCD) cameras as
a focal plane detector of the X-ray Telescope (XRT; Serlemitsos et al. 2007).
The nominal energy covered by the XIS is $0.2-12$ keV (Koyama et al., 2007),
and the CCD provides a rectangular field of view (FOV) of $18^{\prime}\times
18^{\prime}$.
The HXD is a non-imaging collimated X-ray detector which consists of stacked
two main detection parts; silicon $p$-intrinsic-$n$ diodes (hereafter PIN) and
gadolinium silicate (GSO) scintillation crystals. They cover energy ranges of
$10-70$ keV (PIN) and $40-600$ keV (GSO). In the PIN energy band, a passive
collimator provides an FOV of $34^{\prime}\times 34^{\prime}$ at full-width at
half maximum, or a bottom-to-bottom aperture of $68^{\prime}\times
68^{\prime}$ (Kokubun et al., 2007). This tightly collimated FOV provides one
of strong points of the HXD in studies of apparently extended emission,
because contamination from bright discrete X-ray sources is considerably lower
than in previous hard X-ray instruments.
Thanks to the low-earth orbit (altitude of $\sim 570$ km) of Suzaku, the XIS
has lower non X-ray backgrounds (NXB) compared to XMM-Newton and Chandra, and
is suitable for observing extended emission with moderately low surface
brightness such as the GRXE. Simultaneous and broad-band spectral coverage
below and above 10 keV, achieved with the two detectors, allows us to
accurately measure spectral parameters of the GRXE, including in particular
the Fe-K lines and hard X-ray continua, respectively.
Figure 1.— (a) A composite three-color X-ray image of the Galactic center and
the Galactic bulge region created by adding data of 92 mapping observations
performed during $2005-2010$. It was constructed from the XIS0, 1, and 3 data.
The $0.5-2$, $2-5$, and $5-8$ keV data are represented by red, green, and
blue, respectively. The image is corrected for the vignetting effect of the
XRT and exposure differences, and the NXB was subtracted. Each rectangle
represents a single observation. Labeled green crosses indicate resolved point
sources of which the contributions were estimated in §2.4. (b) The same image
as panel a but with the fill FOVs and ObsIDs of the utilized observations
(Table 1) overlaid; dashed and solid rectangles for the XIS and HXD/PIN,
respectively.
### 2.2. Mapping of the Galactic Bulge Region
Since the launch, Suzaku has been actively performing observations of the
Galactic center and the Galactic bulge regions ($|l|\lesssim
5^{\circ}$,$|b|\lesssim 3^{\circ}$) to study the extreme environment near the
supermassive black hole, hot plasmas around supernovae, molecular clouds
reflecting X-rays, and a number of X-ray point-like sources in these regions
(e.g. Nobukawa et al. 2008; Uchiyama et al. 2011b). Figure 1 show a mosaic
image produced by collecting all these Suzaku data with a total exposure of
4.4 Ms. The data are taken from 92 mapping observations performed during
$2005-2010$ of which individual log is summarized for reference in Table 6.
### 2.3. Selecting Suitable Observations
To avoid signals from bright point sources which could contaminate GRXE
signals, we carefully selected observations that do not contain any bright
point source. The INTEGRAL General Reference Catalog (Ebisawa et al., 2003)
and the forth INTEGRAL/IBIS catalog (Bird et al., 2010) were used as
references to positions and typical intensities of known X-ray sources. We
discarded an observation when it is contaminated by X-ray point sources of
which flux is above 0.1 mCrab (equivalently $\sim 2\times
10^{-12}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ in the $2-8$ keV)
inside the HXD FOV. This criterion was set because a GRXE flux observed with
the HXD is typically 1-2 mCrab in this sky region, and the limiting HXD/PIN
flux of 0.1 mCrab corresponds to $5-10$% of the detected GRXE flux. Table 1
lists observations which passed the criterion, and used in the present
spectral analysis. In Figure 1 (b), the FOVs of the utilized observations are
overlaid on the same X-ray image as panel (a). To study possible spatial
variations of spectral properties of the GRXE, we divided the accepted
observations into two groups, and hereafter refer to them as Region 1 (on-
plane, $l<0$ regions) and Region 2 (off-plane, $b<0$ regions). Compared to
typical Suzaku observations, the total effective exposures of the selected
regions are extremely deep, 592.1 ks (XIS) and 524.0 ks (HXD) for Region 1,
and 418.8 ks (XIS) and 340.2 ks (HXD) for Region 2.
Table 1 Summary of the Galactic bulge observations. | Obs. IDaaObservation ID. | CoordinatebbAim point in the Galactic coordinate (degree). | Start time | ExposureccNet exposure in units of $10^{3}$ s. | Count rateddCount rates in units of counts s-1 calculated over $1-9$ and $17-50$ keV per one of the three XIS sensors and the HXD, respectively. NXB is subtracted. The XIS count rate includes CXB which accounts about 0.04 counts s-1 whereas that for the HXD ($\sim 0.01~{}\mathrm{counts}\ \mathrm{s}^{-1}$) is subtracted in the HXD rates.
---|---|---|---|---|---
| | $l$ | $b$ | UT | XIS | HXD | XIS | HXD
Region 1
1 | 501053010 | $-1.83$ | $-0.00$ | 2006-10-10 21:18:59 | 21.9 | 19.9 | 0.25 | 0.07
2 | 503014010 | $-2.10$ | $-0.05$ | 2008-09-18 04:46:49 | 55.4 | 51.2 | 0.18 | 0.05
3 | 503015010 | $-2.35$ | $-0.05$ | 2008-09-19 07:33:05 | 56.8 | 52.8 | 0.19 | 0.04
4 | 503016010 | $-2.60$ | $-0.05$ | 2008-09-22 06:47:49 | 52.2 | 49.3 | 0.18 | 0.03
5 | 503017010 | $-2.85$ | $-0.05$ | 2008-09-23 08:08:10 | 51.3 | 48.6 | 0.17 | 0.04
6 | 503021010 | $-1.62$ | $0.20$ | 2008-10-04 03:44:03 | 53.8 | 49.6 | 0.24 | 0.07
7 | 503076010 | $-1.50$ | $0.15$ | 2009-02-24 17:04:51 | 52.9 | 43.8 | 0.27 | 0.07
8 | 503077010 | $-1.70$ | $0.14$ | 2009-02-26 01:01:00 | 51.3 | 43.7 | 0.24 | 0.07
9 | 504001010 | $-1.47$ | $-0.26$ | 2010-02-26 09:15:00 | 51.2 | 42.2 | 0.20 | 0.05
10 | 504002010 | $-1.53$ | $-0.58$ | 2010-02-27 16:14:41 | 53.1 | 46.6 | 0.17 | 0.04
11 | 504003010 | $-1.45$ | $-0.87$ | 2010-02-25 04:33:17 | 50.9 | 41.3 | 0.19 | 0.02
12 | 504090010 | $-1.49$ | $-1.18$ | 2009-10-13 04:17:20 | 41.3 | 35.0 | 0.20 | 0.03
| | | | Total Exposure | 592.1 | 524.0 | |
Region 2
1 | 502004010 | $0.17$ | $-1.00$ | 2007-10-10 15:21:17 | 19.9 | 18.8 | 0.45 | 0.05
2 | 502059010 | $-0.00$ | $-2.00$ | 2007-09-29 01:40:51 | 136.8 | 110.5 | 0.35 | 0.02
3 | 503081010 | $0.03$ | $-1.66$ | 2009-03-09 15:41:50 | 59.2 | 57.6 | 0.49 | 0.01
4 | 504050010 | $0.10$ | $-1.42$ | 2010-03-06 03:55:37 | 100.4 | 80.5 | 0.60 | 0.02
5 | 504088010 | $-0.00$ | $-0.83$ | 2009-10-14 11:30:56 | 47.2 | 32.6 | 0.43 | 0.05
6 | 504089010 | $-0.05$ | $-1.20$ | 2009-10-09 04:05:59 | 55.3 | 40.2 | 0.54 | 0.02
| | | | Total Exposure | 418.8 | 340.2 | |
### 2.4. Contamination from Detected Point Sources
Although we selected observations with very low point-source contamination, we
still notice four faint X-ray point sources, as indicated with green crosses
in Figure 1, in the FOVs of Region1. Source a is a newly-found soft X-ray
source, and the remaining three are previously known sources; b=AX
J1742.6$-$3022 (Sakano et al., 2002), c=Suzaku J1740.5-3014 (Uchiyama et al.,
2011a), and d=IGR J17391$-$3021 (e.g. Bozzo et al. 2010). When extracting the
GRXE spectra from the XIS data, these sources are masked. On the other hand,
we cannot exclude them in HXD/PIN because it does not have imaging capability.
As a result, the HXD/PIN spectra contain signals from these point sources, in
addition to those from the GRXE. Therefore, before actually extracting GRXE
spectra, we study their effects on our HXD/PIN GRXE data.
We produced spectra of the four point sources from the XIS data of Obs.IDs
504001010, 503021010, and 402066010, as shown in Figure 2. To estimate their
fluxes in the HXD/PIN energy band, we fitted the spectra with phenomenological
models. Because there is no characteristic feature in the spectra of Sources
a,b, and d, we applied to them power-law models modified by the interstellar
absorption (i.e. `wabs`$\times$`powerlaw` in the XSPEC fitting package; Arnaud
1996). Source c shows clear emission-line features at $6-7$ keV, and is
identified as an intermediate polar by Uchiyama et al. (2011a) based on its
periodic time variation. Therefore, this source should be fitted with a multi-
temperature thermal plasma model. Because the counting statistics are not so
high, we restricted ourselves to a model consisting of two collisional
ionization equilibrium (CIE) plasma components plus a gaussian subject to
photo absorptions [i.e.
`wabs`$\times($`APEC`$+$`Gaussian`$)+$`wabs`$\times$`APEC`$)$]. The two plasma
components represent typical high and low temperatures of a multi-temperature
plasma in an accretion column of an intermediate polar (e.g. Yuasa et al.,
2010), and the Gaussian the fluorescence line from neutral Fe.
In all cases, the fits successfully reproduced the observed spectra of these
sources (Figure 2), and provided best-fit parameters as listed in Table 2.
Model-predicted intensities in the $2-8$ keV band were $<2.4\times
10^{-12}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ ($<0.11$ mCrab)
for all the sources. When HXD/PIN observes these sources in its FOV, the
angular transmission of the collimator (Takahashi et al., 2007) reduces these
fluxes, and therefore their effective contributions become much smaller.
Since typical NXB-subtracted HXD/PIN count rates in Regions 1 and 2 are $\sim
0.04$ and $\sim 0.02~{}\mathrm{counts}\ \mathrm{s}^{-1}$, respectively, a
$\lesssim 0.1$-mCrab point source, with $\lesssim 0.003~{}\mathrm{counts}\
\mathrm{s}^{-1}$, contaminates the data at most a few to ten percent of the
detected counts (before reduced by the collimator angular response). We
consider that these sources are negligible in the HXD band compared to the NXB
subtraction uncertainty ($\sim 0.003~{}\mathrm{counts}\ \mathrm{s}^{-1}$ at
1$\sigma$).
Other than these four sources, about ten point-source-like features are seen
in the XIS images. Since their intensities, if treated as point sources, are
all weak with fluxes on the order of $10^{-14}-10^{-13}~{}\mathrm{erg}\
\mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ in the $2-8$ keV band (equivalently
$0.5-5~{}\mu$Crab), their total flux amounts at most to a few percent of the
detected XIS count rates. Therefore, we neglected these possible weak point
sources when extracting the GRXE spectrum.
Summarizing these evaluations, we conclude that the total unwanted
contribution from the resolved point sources to the HXD/PIN data is
considerably less than $0.1~{}$mCrab, or equivalently $3\times
10^{-3}~{}\mathrm{counts}\ \mathrm{s}^{-1}$ over the $15-50$ keV band. This
means that the present analysis integrates X-ray signals emitted from
unresolved sources fainter than 0.1 mCrab, or $\sim 2\times
10^{-12}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ in the $2-8$ keV
band, as the “GRXE”. This limiting flux corresponds to an intrinsic X-ray
luminosity of a point source of $8\times 10^{33}~{}\mathrm{erg}\
\mathrm{s}^{-1}$ in the same energy band if a distance to the Galactic center
of 8 kpc is assumed. This limit is several to ten times lower than those of
the previous studies (Revnivtsev et al., 2006; Türler et al., 2010) in the
hard X-ray energy band, thanks to the tightly-collimated FOV of the HXD and
simultaneous imaging capability of the XIS.
A further question might be raised about possible presence of bright point
sources which are inside some of the HXD/PIN FOV but outside the regions
covered by the XIS. However, the INTEGRAL catalog already include dim point
sources which were found in the ASCA Galactic center survey (Sakano et al.,
2002), and we can conclude that there is no other bright point source which
could affect our analysis.
Figure 2.— Background-subtracted and response-inclusive XIS spectra of the four contaminating sources recognized in Region 1; (a) a newly-found soft X-ray point source, (b) AX J1742.6$-$3022, (c) Suzaku J1740.5$-$3014, and (d) IGR J17391$-$3021\. Crosses and solid curves in upper panels are data and the best-fit models, respectively. Lower panels present fitting residuals in terms of $\chi$. Inset labels represent model-predicted $2-8$ keV energy fluxes from the sources, followed by the same value scaled by that of the Crab nebula. Table 2 The best-fit parameters for the contaminating point sources. | Source | ModelaaPL = an absorbed power-law model. 2CIE = two collisional ionization equilibrium plasma plus a gaussian model. | $N_{\mathrm{H}}$ | $\Gamma$bbPhoton index of the power-law model. | $kT$ccPlasma temperature. Numbers in brackets are lower limits on the 90% confidence level. | $\chi^{2}/\nu$ | $F_{2-8~{}\mathrm{keV}}$ddModel predicted flux in the $2-8$ keV band (1Crab$=2.2\times 10^{-8}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$).
---|---|---|---|---|---|---|---
| | | cm-2 | | keV | | mCrab
a | New source | PL | $2.4^{+3.7}_{-1.9}$ | $3.1^{+1.5}_{-0.7}$ | – | 1.20(25) | 0.01
b | AX J1742.6-3022 | PL | $13^{+2}_{-2}$ | $2.7^{+0.3}_{-0.4}$ | – | 0.91(28) | 0.07
c | Suzaku J1740.5-3014 | 2CIE | $3.7^{+1.0}_{-1.2}$ | – | $10.8(>6.5)$ | 0.81(86) | 0.07
| | | $45^{+78}_{-35}$ | – | $31(>4)$ | |
d | IGR J17391-3021 | PL | $4.0^{+0.8}_{-0.6}$ | $1.9^{+0.2}_{-0.1}$ | – | 1.32(25) | 0.11
### 2.5. Time Variability
If any bright point source was inside the FOV of the selected observations,
and if it varied on a time scale less than $\sim$days, we should see count
rate variations within individual observations. This issue applies
particularly to the HXD, because it lacks imaging capability, and outer 75%
(or 50%, after considering angular transmission) of its FOV falls outside that
of the XIS where no simultaneous imaging coverage is available. Therefore it
is also important, before performing detailed spectral analysis, to confirm
that the background-subtracted signals of the HXD do not exhibit significant
time variations. This result also confirms that there was no bright fast
transient point source which could contaminate the HXD data, i.e. located in
the HXD FOV and outside the XIS FOV.
We extracted light curves of HXD count rates as exemplified in Figures 3, and
examined them carefully. The periodic variation of the total count rates (ALL
in the figure) is mostly due to time variation of the NXB caused by changes of
the cosmic-ray particle flux in orbit. After subtracting the NXB, residual
count rates in these observations (i.e. ALL-NXB) were $\sim
0.02-0.07~{}\mathrm{counts}\ \mathrm{s}^{-1}$, and showed little time
variation. Since the CXB amounts to $\sim 0.01~{}\mathrm{counts}\
\mathrm{s}^{-1}$in the HXD band, about $50-70\%$ of these residual counts can
be considered to originate from the GRXE.
Figure 3.— Examples of $15-50$ keV HXD/PIN light curves during the GRXE
observations; Obs.IDs 504002010 and 502059010 from Regions 1 and 2,
respectively. ALL (filled green rectangles), NXB (open blue triangles), and
ALL$-$NXB (filled black circles) are raw detector count rates, simulated count
rates of the NXB, and NXB-subtracted celestial signals including the CXB
($\sim 0.01~{}\mathrm{counts}\ \mathrm{s}^{-1}$), respectively. Each bin has a
width of 2000 s.
### 2.6. Extracting the GRXE Spectrum
Following the inspection of data quality, we extracted spectra of the
individual observations in Regions 1 and 2. The XIS data of the entire imaging
area of the CCD were used, except for two corners irradiated by calibration
radioactive isotopes and circularly-masked regions around the four point
sources. The XIS NXB was estimated by applying the standard tool `xisnxbgen`
to the same extracting region as the data. Although the XIS detects
significant signals below 2 keV as well as above that energy, we did not used
them because detailed spectral modeling in this energy band is hampered by
complex interstellar absorption. As the HXD NXB, we used the simulated NXB
file which is produced by the HXD team. Accuracies of these NXB subtractions
are $\sim 5\%$ and 1% ($1\sigma$) in the XIS and the HXD, respectively (Tawa
et al., 2008; Fukazawa et al., 2009). Since the HXD NXB amounts to $\sim
0.3~{}\mathrm{counts}\ \mathrm{s}^{-1}$ in the $15-50$ keV band, the
uncertainty corresponds to $\sim 0.003~{}\mathrm{counts}\ \mathrm{s}^{-1}$.
This is only $4-15\%$ of the CXB-subtracted GRXE count rates in the same
energy band (Table 1).
Figure 4 shows examples of the spectra, derived from the same observations as
were used in the previous light curve analysis. The raw spectra are shown as
“ALL”, together with “NXB” and “ALL$-$NXB” (NXB-subtracted celestial signals)
spectra. As expected from the light curves, the NXB subtraction has left
signals which are significant compared to the statistical and systematic
uncertainties of the NXB subtraction. We regard this as the GRXE signal, again
with $30-50\%$ contribution from the CXB. A difference of spectral slopes
below $\sim 3-4$ keV between the two regions is caused by the stronger
interstellar absorption on the Galactic plane (Region 1) than that in the off-
plane regions (Region 2). The intense Fe K$\alpha$ emission lines ($6-7$ keV)
are noticeable even in the individual XIS spectra. In addition, several
emission-line like features are recognizable in $2-4$ keV (e.g. at around 2.5
and 3.2 keV) which are thought to originate from lighter elements.
Within each Region (1 or 2), the spectra of individual observations did not
differ from one another within statistical errors. Therefore, we combined
their data for detailed broad-band spectral analyses, and created summed
spectra of the two regions. The derived spectra are shown in Figure 5. The
data summation has improved the data quality significantly, with little change
in the spectral shapes. Based on the NXB-subtraction accuracies, we consider
that the celestial signals are detected up to 10 keV and 50 keV in the XIS and
HXD/PIN, respectively. Figure 6 gives a close-up view of the Fe emission lines
in the $6-7$ keV band in Figure 5.
In Region 2, we limited our spectral analysis to narrower energy ranges ($2-9$
keV for the XIS and $15-40$ keV for HXD/PIN) due to lower counting statistics,
which, in turn, is because of the shorter exposure and the lower surface
brightness of the GRXE in off-plane regions. Compared to the individual
spectra, the emission lines from Fe and the other lighter elements are more
clearly seen in the XIS spectra.
Figure 4.— Examples of broad-band X-ray spectra obtained in individual
observations; Obs.IDs 503021010 and 502059010 representing Regions 1 and 2,
respectively. Black crosses are the celestial signal counts, which consist of
the GRXE and the CXB, derived by subtracting the NXB (blue curves) from the
raw counts (green crosses). Blue labels specify atomic emission lines clearly
seen in the XIS NXB (see Tawa et al. 2008). For clarity, spectra of XIS0 are
plotted, although the other two XIS cameras give fully consistent data.
Figure 5.— Spectra of the GRXE (including the CXB) summed over all
observations in Region 1 (top panel) and Region 2 (bottom panel), presented in
the same way as Figure 4. Figure 6.— A close-up view of the same GRXE
spectrum of Region 1 as Figure 5. Three K$\alpha$ lines from Fe in different
ionization states are clearly resolved; Fe I = neutral, Fe XXV = He-like, and
Fe XXVI = H-like.
### 2.7. Surface Brightness of the GRXE
To conduct spectral analyses, energy responses and effective areas of the XIS
and HXD/PIN should be calculated for individual observations, by taking into
account a global distribution of the GRXE surface brightness. This quantity is
known to correlate with that of near infrared diffuse emission which reflects
the stellar mass density (Revnivtsev et al., 2006). Among many plausible
functions that can model the distribution, we also used, after Revnivtsev et
al. (2006), so-called G3 model (Dwek et al., 1995) for the bulge/bar
structure. The model describes the stellar density $\rho(r)$ as
$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\rho_{0}r^{-1.8}\exp(-r^{3}),$ (1) $\displaystyle r$
$\displaystyle=$
$\displaystyle\left[\left(\frac{x}{x_{0}}\right)^{2}+\left(\frac{y}{y_{0}}\right)^{2}+\left(\frac{z}{z_{0}}\right)^{2}\right]^{1/2},$
(2)
Where $x$, $y$ and $z$ are three-dimensional coordinates centered on the
Galactic center, and $z$ axis corresponds to the pole of the Galactic
rotation. See Dwek et al. (1995) for detailed explanation of the assumed
coordinate system, and a rotation of the bar structure. For three scale-length
parameters, $x_{0}$, $y_{0}$, and $z_{0}$, we used values presented in Dwek et
al. (1995); $x_{0}=4.01$ kpc, $y_{0}=1.67$ kpc, and $z_{0}=1.12$ kpc.
The surface brightness $S$ at a certain sky position is assumed to follow a
line-of-sight integral,
$S(l,b)=S_{0}\int^{\infty}_{0}\rho(x,y,z)\mathrm{d}s,$ (3)
where $(l,b)$ is a sky position in the Galactic coordinate. $S_{0}$ and $s$
are a normalization parameter and a line-of-sight distance to $(x,y,z)$
measured from the Sun, respectively. In this calculation, a distance between
the Sun and the Galactic center was assumed to be 8.5 kpc.
Based on this, we constructed a surface brightness map, and used it as an
input in calculating the detector responses. To justify the procedure, we
compared the actually measured HXD/PIN count rates of individual observations
with the near-infrared surface brightness of Eqn. (3) convolved with the
HXD/PIN angular response. We chose HXD/PIN because the interstellar absorption
is negligible unlike below 10 keV, and the CXB count rate can be accurately
subtracted (whilst this is not possible in the XIS due to the absorption). A
nice correlation was obtained as plotted in Figure 7. Thus, we, after
Revnivtsev et al. (2006), reconfirmed that the usage of the model is
appropriate in the response calculation.
Figure 7.— Correlation of the HXD/PIN GRXE count rate ($15-50$ keV) and the
near IR surface brightness (Dwek et al., 1995) convolved with the HXD/PIN
angular transmission ($34^{\prime}\times 34^{\prime}$ at the full width at
half maximum). Count rates of individual observations of Regions 1 and 2 are
plotted in filled rectangles and open circles, respectively. Dashed line is
the best-fit linear function ($y=4.6x-0.005$). The estimated CXB count rate of
$0.016~{}\mathrm{counts}\ \mathrm{s}^{-1}$ was subtracted from the HXD/PIN
count rates.
## 3\. PARAMETERIZING THE HARD X-RAY SPECTRUM OF THE GRXE
### 3.1. Fit with the IP Spectral Model
Magnetic CVs, including IPs in particular, are thought to be a major
contributor to the GRXE in the hard X-ray band because of their relatively low
luminosities, high volume densities, and the hard spectral shapes (Revnivtsev
et al., 2006; Sazonov et al., 2006). This has been directly confirmed by
Chandra in the Galactic center region (Muno et al., 2004). X-ray spectral
shapes of IPs and Polars are quite similar, both having the multi-temperature
thermal nature; the only difference is that the plasma temperatures of Polars
are lower due to enhanced cyclotron cooling (e.g. Cropper et al. 1998), the
GRXE spectra integrated above 15 keV should approximately be reproduced by the
IP spectral model .
To test this idea, we fitted the HXD/PIN spectra of Regions 1 and 2 with the
IP spectral model which we numerically constructed and verified in our
previous study of nearby IPs (Yuasa et al., 2010). This spectral model
represents multi-temperature plasma emission from an accretion column on top
of the magnetic poles of an IP. The WD mass and Fe abundance, which are its
primary free parameters, determine the spectral shape. Compared to Yuasa et
al. (2010), we updated the model by taking into account a plasma cooling
function recently published by Schure et al. (2009) so as to accurately model
accreting gas with sub-solar abundances (Yuasa, 2011). Although this affected
little the spectral shape, it has increased the self-consistency of our model.
In the fitting, the models for Regions 1 and 2 were constrained to share the
same WD mass parameter, and allowed to have different normalizations. The Fe
abundance parameter was fixed at unity in the fitting because it cannot be
constrained without data of the emission lines. Even when we changed this
parameter, for example, to 0.5 solar, the result was not affected at all. The
CXB contribution is considered as a fixed model component using the
established models (e.g. Boldt 1987; Revnivtsev et al. 2003), rather than
subtracting from the data, because the CXB signals are affected by
interstellar absorption which is to be determined. The best-fit model
successfully reproduced the overall spectral shape in the two regions, and
gave $\chi^{2}_{\nu}=1.37(30)$ with a null hypothesis probability of 9%.
Therefore, we consider this fit as acceptable. Figure 8 shows the spectra,
with the best-fit model superposed. The best-fit WD mass parameter is
$M_{\mathrm{WD}}=0.66^{+0.09}_{-0.07}~{}M_{\odot}$.
Figure 8.— The HXD/PIN spectra of the GRXE of Region 1 (filled rectangles)
and Region 2 (open circles), simultaneously fitted with the IP spectral model.
Gray solid curves are the best-fit model spectra (CXB inclusive). Crosses in
the lower panel show fit residuals.
### 3.2. Fit with simpler models
To characterize the hard X-ray GRXE spectra using simpler alternatives than
the IP model, we also fitted the same HXD/PIN data with a CIE plasma emission
model (the `APEC` model in `XSPEC`) and a power-law function. Although these
fittings are rather empirical, the obtained representative spectral parameters
are considered to be useful for comparison with previous studies in similar
energy bands.
A CIE plasma emission model yielded an acceptable fit with
$\chi^{2}_{\nu}=1.37(31)$, and the best-fit plasma temperature of
$kT=15.7^{+2.7}_{-2.0}$ keV. If we simply take this value as a shock
temperature, and convert it to a WD mass using the relation between shock
temperature and the WD mass in an IP (Yuasa et al., 2010; Yuasa, 2011), a WD
mass of $0.48~{}M_{\odot}$ is derived. This WD mass is lower than that
obtained by the IP model fit, because the “color temperature” of the IP model
is generally lower than the actual shock temperature due to the multi-
temperature nature of the post-shock region.
A power-law fit to the HXD/PIN data gave a slightly worse but acceptable fit
with $\chi^{2}_{\nu}=1.43(31)$. The best-fit power-law index is $\Gamma=2.8\pm
0.2$ which is slightly softer than the value derived from the RXTE and
CGRO/OSSE data ($2.3\pm 0.2$; Valinia & Marshall 1998). If this power-law
function is extrapolated down to the XIS energy band, the model-predicted XIS
count rates exceed the actual data by more than a factor of 2 (i.e. too steep
to reproduce the observed XIS spectra). Since this discrepancy will not be
solved unless an artificial (and unrealistic) flattening of the power law in
$\lesssim 10$ keV is assumed, we consider that the power-law fit is not an
appropriate model to interpret the broad-band spectrum, although it roughly
reproduces the HXD/PIN spectra.
## 4\. SPECTRAL ANALYSIS OF THE GRXE IN THE SOFT X-RAY BAND
### 4.1. Identification of Atomic Emission Lines
As can be seen in Figure 5, the XIS spectra exhibit many emission lines from
astrophysically abundant heavy elements. To identify elements emitting these
lines, as well as ionization states, we fitted the XIS spectrum with a
phenomenological model consisting of a power-law continuum subject to an
interstellar absorption, and multiple Gaussians. As shown in Figure 9, the
model successfully reproduced the spectrum when we introduced 8 Gaussians for
prominent line features. The best fit center energies of these lines and their
equivalent widths are listed in Table 3. With these center energies,
identification is straightforward, and we found that He-like or H-like S, Ar,
Ca, and Fe ions are emitting these lines as labeled in the figure.
In the $2-3$ keV energy range, the foreground diffuse X-ray emission is known
to contribute to some extent to the emission lines (e.g. Kuntz & Snowden 2008;
Yoshino et al. 2009; Masui et al. 2009). However, the energy flux attributable
to this emission is only a few percent of the detected flux of the GRXE, and
therefore, is negligible at this stage. Instead, in a broad-band spectral
analysis presented below, we take into account this foreground emission.
Figure 9.— Emission lines seen in the GRXE spectrum of Region 1 taken with
XIS0 (black crosses). Plotted model (gray solid curve) is a phenomenological
one consisting of a power-law continuum (black dotted curve) and eight
Gaussians (black solid curves) for emission lines. Only prominent lines were
taken into account in the fit. Labels represent identified emission lines.
Table 3 Best-fit parameters for individual emission lines seen in the GRXE
spectrum.
Line | Energy | $\sigma$aaHydrogen column density of interstellar absorption. | E.W.bbPlasma temperatures of the CIE 1 and the CIE 2 components.
---|---|---|---
| (keV) | (eV) | (eV)
S XV K$\alpha$ | $2.45\pm 0.01$ | $<35$ | $25^{+2}_{-3}$
S XVI K$\alpha$ | $2.63\pm 0.01$ | $<25$ | $35^{+8}_{-9}$
Ar XVII K$\alpha$ | $3.13\pm 0.01$ | $<11$ | $41^{+6}_{-6}$
Ar XVIII K$\alpha$ | $3.33\pm 0.02$ | $<40$ | $14^{+7}_{-8}$
Ca XIX K$\alpha$ | $3.86\pm 0.07$ | $<89$ | $27^{+7}_{-9}$
Fe I K$\alpha$ | $6.41\pm 0.01$ | $<36$ | $76^{+9}_{-10}$
Fe XXV K$\alpha$ | $6.68\pm 0.01$ | $<17$ | $322^{+14}_{-15}$
Fe XXVI K$\alpha$ | $6.97\pm 0.01$ | $<34$ | $79^{+10}_{-11}$
11footnotetext: Upper limits (90% confidence level) of intrinsic line width in
terms of $1~{}\sigma$ of Gaussian.
22footnotetext: Equivalent width calculated against a continuum containing the
GRXE and the CXB signals.
### 4.2. Multi-temperature Nature of the GRXE
It is almost obvious from the above identification that at least two distinct
plasma components compose the GRXE. This is because, as long as collisional
ionization equilibrium is assumed, elements like S and Ar would be almost
fully ionized and would not emit emission lines if the plasma is hot enough
($10^{7.5-8}$ K) to highly ionize Fe up to Fe XXVI K$\alpha$ (6.9 keV;
H-like); see ionization balance calculated by, for example, Bryans et al.
(2006) and Bryans et al. (2009). This reconfirms the multi-temperature nature
of the GRXE first revealed by Kaneda et al. (1997) based on similar arguments
using ASCA/GIS data. In addition, the existence of the Fe I (neutral)
K$\alpha$ implies reprocessing of X-rays by a cold matter.
To quantify the multi-temperature nature of the GRXE found above, we first
performed a fit to the same spectra with a single-temperature CIE plasma
model. The CXB contribution was included as a fixed model, like in the HXD/PIN
analysis. As shown in Figure 10 (a), the best-fit model ($kT\sim 4$ keV)
significantly underpredicts the S XV K$\alpha$ (2.44 keV) and Ar XVII
K$\alpha$ (3.13 keV) line fluxes, and is unacceptable with
$\chi^{2}_{\nu}=2.91(245)$. These deficits cannot be resolved even when the
abundance parameter is increased, because the model would then over predict
He-like lines of these elements. This reconfirms that the GRXE spectrum cannot
be explained by the single-temperature CIE plasma model.
Figure 10.— (a) The GRXE spectrum of XIS0 (black crosses) fitted with a
single-temperature CIE plasma emission model and additional components, i.e.
the CXB (gray dashed curve) and a Gaussian for the Fe I K$\alpha$ line (gray
dotted curve). Red and magenta curves are the total model and the CIE plasma
component, respectively. Vertical dotted lines indicate those emission lines
which are not reproduced by the model. Fitting residuals are shown in the
lower panel. (b) An improved fit incorporating two single-temperature CIE
plasma emission components (magenta and blue curves). The CXB in this panel
has a spectral shape which is different from that of panel a below 3 keV
because the best-fit interstellar absorption column densities differ
($N_{\mathrm{H}}=2.8\times 10^{22}$ cm-2 in panel a, while
$N_{\mathrm{H}}=4.8\times 10^{22}$ cm-2 in b).
After Kaneda et al. (1997), we introduced an additional lower-temperature CIE
component to better reproduce the low-energy lines. The fit improved
considerably to $\chi^{2}_{\nu}=1.46(243)$, and yielded the best-fit
temperatures of $kT_{1}=0.79^{+0.46}_{-0.36}$ keV and
$kT_{2}=6.23^{+0.26}_{-0.26}$ keV. Although this fit, with a null hypothesis
probability less than 1%, is not yet acceptable in the strict sense, the
remaining issue can be ascribed to calibration inaccuracy rather than to model
inappropriateness, because the fit residuals are mainly observed in line-wing
regions. If we include a model systematic error of 2% to simulate the
insufficient response accuracy, the fit becomes formally acceptable with a
null hypothesis probability of 1%.
The derived two plasma temperatures are just representative ones, and may be
approximating a mixture of more than two distinct plasma temperatures. The
plasma may even have a continuous temperature distribution. However, the data
statistically do not require any additional component. Therefore, instead of
applying more complex models to the XIS data, we proceed to broad-band
spectral analysis combining the XIS and the HXD/PIN data.
## 5\. BROAD-BAND SPECTRAL DECOMPOSITION
As shown in §3.1, the hard X-ray spectrum of the GRXE is well reproduced with
spectral models which have convex spectral curvatures, or more physically, the
thermal nature (i.e. the IP model or a CIE plasma model). Besides, we
reconfirmed, in the soft X-ray band, the multi-temperature nature of the GRXE
(Kaneda et al., 1997). Based on these, we try, in this section, to reconstruct
the broad-band GRXE spectrum (Figure 5) using as small a number of physically
plausible spectral components as possible.
### 5.1. Model Construction
In analyzing the $2-50$ keV GRXE spectrum, let us choose, as our starting
point, the two CIE plasma components which gave a successful fit to the XIS
spectra. In addition to them, as noted above, we consider a contribution from
the foreground diffuse soft X-ray emission. Based on recent reports (Yoshino
et al., 2009; Masui et al., 2009; Kimura, 2010), we assumed the surface
brightness of this foreground emission to be $2.5\times
10^{-9}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}~{}\mathrm{sr}^{-1}$,
and included it in the spectral model as a CIE plasma component. The
temperature was fixed at $kT=0.7$ keV, because the foreground emission
contributes no more than 1%. Hereafter, we designate a sum of the two CIE
plasma emission plus the foreground emission as “Model 1” for simplicity.
Since the two plasma components in the GRXE can come through different
absorption column densities, they were subjected to independent absorption
factors. The abundances of major elements (i.e. Si, S, Ar, Ca, and Fe) were
allowed to vary individually, but were constrained to be the same between the
two CIE components. Thus, Model 1 can be expressed as,
$\displaystyle\mathrm{Model~{}1}$ $\displaystyle=$
$\displaystyle\mathrm{Abs.1}$
$\displaystyle\times(\mathrm{Foreground}+\mathrm{Abs.1}\times\mathrm{CXB}$
$\displaystyle+\mathrm{Abs.2}\times\mathrm{CIE1}+\mathrm{Abs.3}\times\mathrm{CIE2}).$
The 6.4-keV Fe K$\alpha$ emission line could provide a powerful tool for
understanding the origin of the GRXE because, near a WD, its line shape is
probably distorted by Compton scattering in the atmosphere or the
gravitational redshift. However, these studies would require a higher energy
resolution. In the present analysis, we therefore approximate it by a Gaussian
with its normalization left to freely vary.
Another model, Model 2, was constructed by replacing one of the two CIE plasma
components of Model 1 with our IP spectral model (`IP_PSR`; Yuasa et al.
2010). This can be supported by the successful application of this model to
the HXD/PIN GRXE spectrum (§3.1), and also by a suggestion that magnetic CVs
are a major contributor to the GRXE above 10 keV (e.g. Revnivtsev et al.
2006). The Fe abundance of the IP component was tied with that of the CIE
Plasma component. Other abundance parameters were again allowed to
individually vary (like Model 1). Since intrinsic partially-covering
absorption is a common feature of IPs (Suleimanov et al., 2005; Yuasa et al.,
2010), the IP model was subjected to an additional (partially-covering) dense
absorption, with a covering fraction $f$, of which absorption column density
was allowed to freely vary. Summarizing above, this Model 2 is constructed as
$\displaystyle\mathrm{Model~{}2}$ $\displaystyle=$
$\displaystyle\mathrm{Abs.1}\times(\mathrm{Foreground}~{}+~{}\mathrm{Abs.1}\times\mathrm{CXB}$
$\displaystyle+~{}\mathrm{Fe~{}I~{}K}\alpha~{}+~{}\mathrm{Abs.2}\times\mathrm{CIE}$
$\displaystyle+~{}[f\cdot\mathrm{Abs.3}+(1-f)\cdot\mathrm{Abs.4}]\times\mathrm{IP\\_PSR}).$
We further subdivided this model into Model 2a and 2b, by treating the
$M_{\mathrm{WD}}$ parameter of the IP component differently. In Model 2a,
$M_{\mathrm{WD}}$ was fixed at $0.66~{}M_{\odot}$ derived from the HXD/PIN
spectral fitting (§3.1). In contrast, the WD mass parameter of Model 2b was
allowed to freely vary.
### 5.2. Combined Spectral Fitting
We first fit the models to the stacked GRXE spectrum of Region 1, because it
has higher counting statistics and covers a wider energy range over $2-50$
keV. As drawn in Figure 11, all the three models well reproduced the broad-
band spectrum, and gave the best-fit parameters as listed in Tables 4 and 5.
Although the values of $\chi^{2}$ are not yet small enough to make the fits
formally acceptable, we consider again that this is caused by the inaccuracy
of the XIS response in the $2-4$ keV band as explained in §4.2. If we include
additional systematic errors (§4.2), Model 2 became acceptable with null
hypothesis probabilities larger than $1\%$, while Model 1 was still
unacceptable (with a probability of $0.2\%$).
The CIE1 component of Model 1 and CIE of Model 2 yielded a plasma temperature
of $kT=1.4-1.7$ keV, and account for the soft continua plus the Fe XXV
K$\alpha$ line with (almost) no contribution to the Fe XXVI K$\alpha$ photons.
Most of the hard X-ray flux detected in the HXD/PIN band is explained by the
higher temperature components, CIE2 with a temperature of $15.1^{+0.4}_{-0.7}$
keV in Model 1, or the IP component with WD masses of $0.66~{}M_{\odot}$ and
$0.48^{+0.05}_{-0.04}~{}M_{\odot}$ in Model 2a and 2b, respectively.
Similarly, the stacked Region 2 spectrum was fitted with the same model. Due
to poorer data quality, we fixed some parameters which mainly determines
spectral shapes of the individual components; the CIE plasma temperatures and
$M_{\mathrm{WD}}$ as summarized in Tables 4 and 5. The best-fit models are
plotted in Figure 12.
Figure 13 shows the best-fit Model 2b in Region 1 by removing the foreground
emission, the CXB, and the interstellar absorption. A hump-like structure seen
in the $10-30$ keV band is due to the partially-covering dense absorption
applied to the IP model.
Figure 11.— The GRXE spectrum of Region 1, fitted with Model 1 (top panel),
Model 2a (middle panel), and Model 2b (bottom panel). Black crosses are the
observed spectrum (NXB subtracted), and red curves are the sum of all model
components. In the Model 1 fit, the two CIE plasma components are shown in
magenta (lower temperature) and green (higher temperature). Magenta and orange
curves in the lower two panels are the CIE plasma and the IP model components,
respectively. Solid, dashed, and dotted gray curves are components
representing the foreground diffuse soft X-ray emission, the CXB, and
K$\alpha$ emission line from neutral Fe, respectively.
Figure 12.— The same as Figure 11, but for Region 2. The temperature of the lower-temperature CIE component was allowed to freely vary. Figure 13.— (a) The same best-fit GRXE model spectrum as Figure 11 but represented by removing the detector responses. Data points taken from previous studies are overlaid; green and blue crosses from Krivonos et al. (2007) and magenta crosses from Türler et al. (2010). Table 4 Result of the wide-band spectral fitting with Model 1**Blank parameters are fixed at the Region 1 values (see text).. | $n_{\mathrm{{H}}}$aaHydrogen column density of interstellar absorption. | $kT_{\mathrm{{CIE1}}}$bbPlasma temperatures of the CIE 1 and the CIE 2 components. | $n_{\mathrm{{H}}}$ccAdditional absorption column density applied to the CIE 2 component. | $kT_{\mathrm{{CIE2}}}$bbPlasma temperatures of the CIE 1 and the CIE 2 components. | $Z_{\mathrm{{Fe}}}$ddFe abundance of the CIE components. | $\chi^{2}_{\nu}$ | $\chi^{2}_{\nu,~{}3\%}$eeImproved fitting statistics achieved when the XIS response uncertainty was virtually took into account. | $F_{\mathrm{{CIE1}}}$ffModel-predicted fluxes of the CIE 1 and 2 components in $10^{-9}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ integrated over specified energy ranges and a $|l|<10^{\circ}$ and $|b|<10^{\circ}$ region. | $F_{\mathrm{{CIE2}}}$ffModel-predicted fluxes of the CIE 1 and 2 components in $10^{-9}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ integrated over specified energy ranges and a $|l|<10^{\circ}$ and $|b|<10^{\circ}$ region.
---|---|---|---|---|---|---|---|---|---
| $10^{22}$ cm-2 | keV | $10^{22}$ cm-2 | keV | $Z_{\odot}$ | | | $(2-10)$ | $(2-50)$
Region 1 | $4.0^{+0.2}_{-0.2}$ | $1.66^{+0.04}_{-0.04}$ | $15.9^{+1.5}_{-1.3}$ | $15.1^{+0.4}_{-0.7}$ | $0.97^{+0.06}_{-0.06}$ | $1.45(754)$ | $1.15$ | $4.9$ | $7.8$
Region 2 | $1.3^{+0.2}_{-0.2}$ | $1.31^{+0.03}_{-0.03}$ | $-$ | $-$ | $1.06^{+0.07}_{-0.08}$ | $1.85(353)$ | $1.25$ | $7.9$ | $11.8$
Table 5 Result of the wide-band spectral fitting with Models 2a and 2b**Blank parameters are fixed at the Region 1 values (see text).. | $n_{\mathrm{{H}}}$aaHydrogen column density of interstellar absorption | $kT$ | $Z_{\mathrm{{Fe}}}$bbFe abundance. | $n_{\mathrm{{H}}}$ccAdditional absorption column density applied to the IP component | $M_{\mathrm{{WD}}}$ddWD mass of the IP model. | $\chi^{2}_{\nu}$ | $\chi^{2}_{\nu,~{}3\%}$eeImproved fitting statistics achieved when the XIS response uncertainty was virtually took into account. | $F_{\mathrm{{CIE}}}$ffModel-predicted fluxes of the CIE and IP components in $10^{-9}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ integrated over specified energy ranges and a $|l|<10^{\circ}$ and $|b|<10^{\circ}$ region. | $F_{\mathrm{{IP}}}$ffModel-predicted fluxes of the CIE and IP components in $10^{-9}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ integrated over specified energy ranges and a $|l|<10^{\circ}$ and $|b|<10^{\circ}$ region. |
---|---|---|---|---|---|---|---|---|---|---
| $10^{22}$ cm-2 | keV | $Z_{\odot}$ | $10^{22}$ cm-2 | $M_{\odot}$ | | | $(2-10)$ | $(2-50)$ |
Model 2a | | | | | | | | | |
Region 1 | $3.6^{+0.2}_{-0.3}$ | $1.52^{+0.04}_{-0.04}$ | $0.86^{+0.05}_{-0.04}$ | $13.0^{+9.4}_{-1.3}$ | $0.66_{({\rm fixed})}$ | $1.30(753)$ | $1.04$ | $3.9$ | $8.6$ |
Region 2 | $1.2^{+0.2}_{-0.2}$ | $1.21^{+0.04}_{-0.03}$ | $0.82^{+0.05}_{-0.05}$ | $-$ | $-$ | $1.62(354)$ | $1.11$ | $7.2$ | $12.7$ |
Model 2b | | | | | | | | | |
Region 1 | $3.2^{+0.4}_{-0.6}$ | $1.44^{+0.06}_{-0.07}$ | $0.73^{+0.06}_{-0.05}$ | $11.8^{+8.6}_{-1.8}$ | $0.48^{+0.05}_{-0.04}$ | $1.28(752)$ | $1.02$ | $3.2$ | $8.6$ |
Region 2 | $1.1^{+0.2}_{-0.2}$ | $1.17^{+0.04}_{-0.04}$ | $0.67^{+0.04}_{-0.04}$ | $-$ | $-$ | $1.65(354)$ | $1.13$ | $6.5$ | $12.5$ |
## 6\. DISCUSSION
### 6.1. Interpreting the Obtained Spectral Parameters
As presented in the previous section, the GRXE spectrum up to 50 keV can be
well described only with thermal plasma models without any empirical power-law
continuum. We consider that this is a strong support for our working
hypothesis, the “Point Source” scenario of the GRXE origin because the derived
plasma temperatures and WD masses are well consistent with values expected
from this scenario. Contrarily, in the context of the “Diffuse” scenario,
problems of confinement and energy injection must be solved for sustain hot
interstellar plasma with a temperature higher than 10 keV in the Galactic
plane.
The lower-temperature CIE component (magenta in Figures 11 and 12) resulted in
ranges of $kT=1.4-1.7$ keV and $1.2-1.3$ keV in Region 1 and 2, respectively.
These temperatures are well consistent with those seen in X-ray spectra of
coronal X-ray sources (e.g. Favata et al. 1997; Covino et al. 2000) that are
binary stars harboring two late-type (or “normal”) stars like the Sun, and
exhibits X-ray emission via magnetic activities in their coronae. The small
temperature difference between the two regions ($\sim 0.3-0.4$ keV) can be
naturally understood if there is unresolved contribution from young supernova
remnants (SNRs) which tend to exhibit thermal X-ray spectra of plasma
(electron) temperatures of $kT\sim 2-4$ keV (e.g. Kinugasa & Tsunemi 1999 for
Kepler’s SNR; Tamagawa et al. 2009 for Tycho’s SNR). Recent discoveries or
possible discoveries of new SNRs in the Galactic center region (Sawada et al.,
2009; Nobukawa et al., 2008), especially in on-plane regions such as Region 1,
support this speculation on the contribution from unresolved SNRs.
The temperature of the CIE 2 component of Model 1, $15.1^{+0.4}_{-0.7}$ keV,
is consistent with a representative plasma temperature ($10-20$ keV) of
magnetic CVs (e.g. Yuasa et al. 2010). The Model 2a fit successfully
reproduced the broad-band spectrum although a WD mass parameter was fixed at
the value derived from the HXD/PIN analysis
($0.66^{+0.09}_{-0.07}~{}M_{\odot}$; §3.1). When it was unfixed in Model 2b, a
similar but slightly lower WD mass of $0.48^{+0.05}_{-0.04}~{}M_{\odot}$ was
obtained. This reduction in WD mass is also explained by unresolved
contributor from other types of CVs, especially dwarf novae in quiescent,
whose spectra can also be reproduced with a CIE plasma model with lower
representative temperatures ($kT\sim$a few$-10$ keV) than those of magnetic
CVs.
Is the reduction of the WD mass parameter by $\sim 0.10-0.15~{}M_{\odot}$
realistically possible with the contribution of dwarf novae? To test this, we
simulated a composite broad-band GRXE spectrum (excluding the low-temperature
CIE component) by adding spectral models of dwarf novae and IPs. A
representative plasma temperature of $kT=5$ keV and an average WD mass of
$0.6~{}M_{\odot}$ were assumed for the dwarf nova and the magnetic CV
components, respectively. Relative fluxes of individual components were
adjusted according to X-ray emissivities of these source types measured by
Sazonov et al. (2006). For better simulating the observed spectrum, we also
applied photo absorption models with hydrogen column densities of
$N_{\mathrm{H}}=3\times 10^{22}$ cm-2 (Galactic interstellar value for dwarf
novae) and $10\times 10^{22}$ cm-2 (dense intrinsic absorption for magnetic
CVs). Simulated count rates, or absolute intensities, were adjusted so as to
match those of the observed data. Figure 14 (a) presents a thus produced
spectrum. This spectrum mimics the GRXE spectrum (subtracted the low-
temperature CIE component).
We performed a similar fit to this simulated spectrum using only the IP model,
and obtained an acceptable fit which yields $\chi^{2}_{\nu}=1.13(287)$. The
best-fit model is plotted in Figure 14 (b), and it gave the best-fitting WD
mass of $0.51\pm 0.01~{}M_{\odot}$ and absorption column density of
$N_{\mathrm{H}}=7.6\pm 0.3\times 10^{22}$ cm-2. Thus, even when the dwarf nova
component is missing in the fit model, the composite simulated spectrum can be
well reproduced with the IP model with the WD mass parameter which is slightly
reduced from the assumed value of $0.6~{}M_{\odot}$ ($\Delta
M_{\mathrm{WD}}=0.09~{}M_{\odot}$). This supports the above discussion on the
reduced WD mass obtained in the broad-band spectral analysis.
Figure 14.— (a) A simulated GRXE spectrum (black crosses) composed of
spectral models of the dwarf nova (blue curve) and the magnetic CV (orange
curve; the IP model). (b) The same spectrum as panel a, but fitted with the IP
model (orange curve).
### 6.2. The Number Density of the Unresolved Hard X-ray Point Sources
The GRXE surface brightness can be used to constrain population of unresolved
point sources, mostly IPs (Revnivtsev et al., 2006). Using the GRXE flux
observed by HXD/PIN, we derive their population, or so-called X-ray luminosity
function. In the present calculation, we assume that all the point sources are
located at a distance of 8 kpc from the Sun (i.e. at the Galactic center
region) as done in many previous studies (e.g. Muno et al. 2009).
Generally, a luminosity function of X-ray stars $N$ can be expressed as a
power-law function, and therefore we write $N$ using the intrinsic luminosity
$L$
$N(>L)=N_{0}\left(\frac{L}{L_{0}}\right)^{-\alpha}.$ (5)
Here, $N(>L)$ means the surface number density of (unresolved) point sources
which have an intrinsic luminosity $L$, and has a dimension of [(number)
(solid angle)-2]. $N_{0}$ and $L_{0}$ are constant factors for scaling, and
the latter can be arbitrarily fixed. The parameter $\alpha$ denotes luminosity
dependence of $N$.
Now, we can calculate an X-ray surface brightness $S$ predicted from the
source population $N$ by integrating a product of the number of sources (i.e.
derivative of the cumulative density $N$) and the luminosity $L$ as
$S=\int^{L_{\mathrm{min}}}_{L_{\mathrm{max}}}\frac{\mathrm{d}N(>L)}{\mathrm{d}L}~{}L~{}\mathrm{d}L$
(6)
The integration is performed over a luminosity range
$L_{\mathrm{min}}-L_{\mathrm{max}}$ where $\mathrm{d}N(>L)/\mathrm{d}L$ is
non-zero.
For quantifying the luminosity function from the present GRXE flux, we set
$L_{\mathrm{min}}$ and $L_{\mathrm{max}}$ as follows. The present GRXE study
does not contain bright known X-ray sources which have energy fluxes higher
than $\sim 10^{-12}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ in the
$2-8$ keV band, or equivalently $\sim 0.1$ mCrab. The flux limit corresponds
to $1.7\times 10^{-12}~{}\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ when
expressed in the $15-50$ keV band (i.e. the HXD/PIN energy coverage). If we
consider that a point source located at 8 kpc from the Sun, the flux means an
intrinsic luminosity of $1.3\times 10^{34}~{}\mathrm{erg}\ \mathrm{s}^{-1}$ in
the same energy range. This should be regarded as $L_{\mathrm{max}}$. As for
$L_{\mathrm{min}}$, recent deep Chandra observations revealed that there exist
X-ray sources with luminosities as low as $10^{30}~{}\mathrm{erg}\
\mathrm{s}^{-1}$ in the $0.5-7$ keV band (Revnivtsev et al., 2009). Another
important fact is that Muno et al. (2009) securely measured the shape of the
luminosity function down to $3\times 10^{31}~{}\mathrm{erg}\ \mathrm{s}^{-1}$
in the $2-10$ keV band. This value provides stringent “upper limit” for
$L_{\mathrm{min}}$; i.e. its actual value should be much lower probably one
order of magnitude because there is no sign of break, or turn off, in their
luminosity function. In the present study,we tentatively considered two cases
with $L_{\mathrm{min}}$ being set these measured values when integrating of
Eqn. (6). If true $L_{\mathrm{min}}$ is lower, the normalization of the
calculated luminosity function will decrease.
Before the integration, these luminosity values should be converted to those
in the $15-50$ keV band which we concentrate on in the present calculation.
Since most of the unresolved point sources in this energy band are thought to
be accreting WDs (especially magnetic ones), they have hard spectra (i.e. high
plasma temperatures; Muno et al. 2004). We assumed, for the unresolved
sources, a typical X-ray spectrum consisting of a single-temperature CIE
plasma model with $kT=20$ keV suffered from the interstellar absorption of
$N_{\mathrm{H}}=6\times 10^{22}~{}\mathrm{cm}^{-2}$ (typical value for the
Galactic center region). Based on this spectral shape, we obtain converted
$L_{\mathrm{min}}$ values of $2\times 10^{30}~{}\mathrm{erg}\ \mathrm{s}^{-1}$
and $3.8\times 10^{31}~{}\mathrm{erg}\ \mathrm{s}^{-1}$ in the $15-50$ keV.
Figure 15.— X-ray luminosity functions of unresolved point sources, which
compose the GRXE in the higher energy band above 10 keV, calculated from the
HXD/PIN GRXE flux assuming $L_{\mathrm{min}}=2\times 10^{30}$ (green region)
and $3.8\times 10^{31}~{}\mathrm{erg}\ \mathrm{s}^{-1}$ (orange region) in the
$15-50$ keV band. Two solid lines which enclose each region have differently-
assumed power-law indices $\alpha=1.0$ and 1.5. Blue dashed line represents a
luminosity function actually measured by Chandra in similar sky region
($2^{\circ}\times 0.8^{\circ}$ around the Galactic center) in the $2-10$ keV
band (Muno et al., 2009).
The present HXD/PIN measurement gives the GRXE surface brightness, for
example, $S=2.34\times 10^{35}~{}\mathrm{erg}\
\mathrm{s}^{-1}~{}(\mathrm{PIN~{}FOV})^{-1}$ in Obs.ID 504001010 centered at
$(l,b)=(-1.47^{\circ},-0.26^{\circ})$ in the $15-50$ keV band. Since $S$, or
$N(>L)$ inside it, includes two unknown parameters $\alpha$ and $N_{0}$, an
additional constraint on one of the two parameters should be placed to
determine $N(>L)$. Previous studies (Ebisawa et al., 2005; Muno et al., 2009)
have revealed that the luminosity function of faint Galactic X-ray point
sources has $\alpha=1.0-1.5$. Based on this, we calculate $N(>L)$ using
assumed indices $\alpha=1.0$ and 1.5.
Figure 15 shows thus calculated luminosity functions of the unresolved
sources. Two filled regions represent two representative cases with lower
luminosity limits of $L_{\mathrm{min}}=2\times 10^{30}~{}\mathrm{erg}\
\mathrm{s}^{-1}$ (green region) and $3.8\times 10^{31}~{}\mathrm{erg}\
\mathrm{s}^{-1}$ (orange). The luminosity functions are well consistent with
that actually measured in similar sky regions by Chandra (Muno et al., 2009).
From this comparison, we consider that the number density of required faint
point sources is not unrealistically high unlike what proposed by some
previous studies (e.g. Ebisawa et al. 2005).
### 6.3. The Origin of the GRXE
The broad-band spectral decomposition has been anticipated for long time for
understanding the emission mechanism(s) and the energy supplier(s) of the
GRXE. This has been done in the present study, decomposing the GRXE into two
representative constituents. As was detailed in §5.2, the low-temperature and
the high-temperature CIE plasma emissions have plasma temperatures of $\sim 1$
keV and $>10$ keV according to our modeling. The latter component can be
successfully replaced by the IP spectral model as examined in the Model 2
fits. In addition, the luminosity function of unresolved hard X-ray sources
does not largely contradict with one directly measured with Chandra. Based on
these, we consider that the present results in $<10$ keV provide yet another
very strong support for the “Point Source” scenario of the GRXE alongside of
the imaging decomposition of Revnivtsev et al. (2009).
On the other hand, the results are against for several previous studies that
suggested the “Diffuse” scenario as the GRXE origin. The putative non-
equilibrium ionization (NEI) plasma proposed by Kaneda et al. (1997) is
rejected since we detected intense Fe XXVI K$\alpha$ line in the present sky
region which is not expected from their NEI model. This is a confirmation of a
similar denial argument by Ebisawa et al. (2008) in the different sky region.
Unlike Yamasaki et al. (1997) and Valinia & Marshall (1998), the observed
broad-band spectrum did not require a putative hard-tail component which was
suggested to smoothly connect to the gamma-ray background emission up to
hundreds of MeV.
### 6.4. The Mean WD Mass in the Galaxy
The WD mass of CVs is an important parameter when interpreting the hard X-ray
GRXE spectral shape. It was first estimated to be $\sim 0.5~{}M_{\odot}$ in
Krivonos et al. (2007) by roughly fitting their GRXE spectra with the IP model
by Suleimanov et al. (2005). Our hard X-ray analysis confirmed this, and gave
a slightly heavier WD mass of $0.66^{+0.09}_{-0.07}~{}M_{\odot}$. As Krivonos
et al. (2007) mentioned, the value could be interpreted as the mean WD mass of
cumulated magnetic CVs in the Galaxy.
If we accept this idea, it is intriguing to compare the derived WD mass with
that reported for isolated WDs based on optical spectroscopy. For example,
using large SDSS data, Kepler et al. (2007) reported a mean WD mass of
$0.593\pm 0.016~{}M_{\odot}$ for 1733 WDs which have helium and hydrogen outer
layers. Our value $0.66^{+0.09}_{-0.07}~{}M_{\odot}$ includes the average
within errors. The slight difference of the center values ($\sim
0.06~{}M_{\odot}$) could be a result of long-lasting mass accretion. However,
CVs are thought to evolve through the common-envelope phase of main-sequence
stars, in which two stars in a binary share their outer layers, possibly
resulting in WD masses different from those of isolated stars. Therefore, at
this moment, it is difficult to investigate the difference of the average WD
masses. More accurate WD mass determinations in CVs, and sophisticated
evolution model of them are necessary to accurately perform above comparison.
### 6.5. Connection to the Galactic Center X-ray Emission
A low-temperature CIE component ($kT\sim 1$ keV) similar to one seen in the
GRXE was also reported recently by Nobukawa et al. (2010) in the Galactic
center X-ray emission observed in $|l|<0.2^{\circ}$. Probably due to lack of
the hard X-ray spectral coverage in their analysis, the authors reported a
plasma temperature of the hotter CIE component of $7.0\pm 0.1$ keV, which is
considerably cooler than our result $kT=15.1^{+0.4}_{-0.7}$ keV. Note also
that their spectral analysis introduced, in addition to two CIE plasma
components, an intense power-law component which was not detected in our
broad-band GRXE data.
In the Galactic center region, the super-massive black hole ($\sim 3\times
10^{6}~{}M_{\odot}$) is suspected to have a close connection with the
unresolved (or literally diffuse) emission which is observed in its proximity
(e.g. predicted by Sunyaev et al. 1993 and observed e.g. by Inui et al. 2009).
This emission could have another origin which differ from that of the GRXE.
Besides, stellar density is higher by more than an order of magnitude in the
very central region of the Galaxy ($\sim 100$ pc from the Galactic center)
compared to those of the region we studied. Therefore, X-rays from faint point
sources (mostly coronal X-ray sources and IPs) might occupy a considerable
fraction of the detected signals, although Nobukawa et al. (2010) simply
neglected them. Since about a half of such faint point sources radiate its
energy in the hard X-ray band, we stress that a broad-band spectral analysis
including the hard X-ray energy range is essentially powerful to examine
possible existence of truly diffuse emission in the Galactic center.
## 7\. SUMMARY
We accumulated the data of Suzaku GRXE observations in the Galactic bulge
region achieving $\sim 1$ Ms exposure in total (§2.3), and produced the broad-
band GRXE spectra with high counting statistics covering the $2-50$ keV band
(§2.6). We showed that the hard X-ray GRXE spectrum taken with HXD/PIN is well
reproduced by the IP spectral model (§3.1) with a WD mass parameter of
$0.66^{+0.09}_{-0.07}~{}M_{\odot}$. From emission line analyses based on the
CIE modeling which is more appropriate than previous line studies (e.g. Kaneda
et al. 1997), we reconfirmed the multi-temperature nature of the GRXE (§4.2).
Based on these results, we constructed physical models of the GRXE (§5.2)
which consists of multi-temperature CIE plasma emissions. The models nicely
decomposed the broad-band GRXE spectral shape for the first time. Especially,
the model that includes the IP component, namely Model 2, gave better fits to
the data together with the low-temperature CIE plasma component. The derived
plasma temperature and the WD mass are quite consistent with those of coronal
X-ray sources and typical of magnetic accreting WDs (§6.1).
We also calculated the X-ray luminosity function and the number density of the
unresolved hard X-ray point sources, mostly IPs (§6.2). The luminosity
function is consistent with that obtained from a deep imaging observation near
our field, indicating that no unknown X-ray source type is additionally
required to explain the GRXE flux.
Combining the spectral decomposition result and the calculated source density,
we concluded that the present result supports the “Point Source” scenario of
the GRXE origin.
## Appendix A SUZAKU OBSERVATIONS IN THE GALACTIC CENTER REGION
For later reference, we summarize information of 92 Suzaku observations in the
Galactic center region in Table 6. XIS data of these observations are used to
produce the mosaic image shown in Figure 1. The total effective exposures of
the XIS and HXD/PIN are 4.4 and 3.9 Ms. The list contains observations with
various types of scientific aims; the Galactic Ridge X-ray Emission, supernova
remnants, transient X-ray binaries, molecular clouds reflecting X-rays, and
unidentified sources in TeV wavelength (H.E.S.S unID sources).
Table 6 Suzaku observations of the Galactic center. | Obs. IDaaObservation ID. | CoordinatebbAim point in Galactic coordinate (degree). | Start time | ExposureccNet exposure in units of $10^{3}$ s.
---|---|---|---|---
| | $l$ | $b$ | UT | XIS | PIN
1 | 100027010 | $0.06$ | $-0.08$ | 2005-09-23 07:18:25 | 44.8 | 37.9
2 | 100027020 | $-0.24$ | $-0.05$ | 2005-09-24 14:17:17 | 42.8 | 36.1
3 | 100027030 | $-0.44$ | $-0.39$ | 2005-09-24 11:07:08 | 2.1 | 1.9
4 | 100027040 | $-0.44$ | $-0.07$ | 2005-09-24 12:41:33 | 1.9 | 1.8
5 | 100027050 | $0.33$ | $0.01$ | 2005-09-25 17:29:12 | 2.0 | 1.8
6 | 100037010 | $-0.24$ | $-0.05$ | 2005-09-29 04:35:41 | 43.7 | 39.4
7 | 100037020 | $-0.44$ | $-0.39$ | 2005-09-30 04:30:44 | 3.3 | 3.1
8 | 100037030 | $-0.45$ | $-0.07$ | 2005-09-30 06:06:32 | 3.0 | 2.8
9 | 100037040 | $0.06$ | $-0.08$ | 2005-09-30 07:43:01 | 43.0 | 39.5
10 | 100037050 | $0.33$ | $0.01$ | 2005-10-01 06:22:41 | 2.4 | 2.2
11 | 100037060 | $0.64$ | $-0.10$ | 2005-10-10 12:28:01 | 76.6 | 70.8
12 | 100037070 | $1.00$ | $-0.10$ | 2005-10-12 07:10:24 | 9.2 | 9.5
13 | 100048010 | $0.06$ | $-0.08$ | 2006-09-08 02:23:24 | 63.0 | 60.3
14 | 102013010 | $0.06$ | $-0.08$ | 2007-09-03 19:01:10 | 51.4 | 44.5
15 | 402066010 | $-1.93$ | $0.45$ | 2008-02-22 11:52:49 | 36.5 | 31.3
16 | 403001010 | $1.36$ | $1.05$ | 2009-02-22 19:04:19 | 71.5 | 59.7
17 | 403009010 | $0.17$ | $0.03$ | 2009-03-21 02:03:28 | 110.8 | 91.7
18 | 500005010 | $0.43$ | $-0.11$ | 2006-03-27 23:00:22 | 88.4 | 64.6
19 | 500018010 | $-0.57$ | $-0.09$ | 2006-02-20 12:45:25 | 106.9 | 46.6
20 | 500019010 | $-1.09$ | $-0.04$ | 2006-02-23 10:51:11 | 13.3 | 12.2
21 | 501008010 | $-0.16$ | $-0.19$ | 2006-09-26 14:18:16 | 129.6 | 111.3
22 | 501009010 | $-0.07$ | $0.18$ | 2006-09-29 21:26:07 | 51.2 | 47.7
23 | 501010010 | $-1.29$ | $-0.64$ | 2006-10-07 02:16:52 | 50.7 | 45.7
24 | 501039010 | $0.78$ | $-0.16$ | 2007-03-03 12:20:20 | 96.4 | 91.1
25 | 501040010 | $0.61$ | $0.07$ | 2006-09-21 17:29:01 | 61.4 | 53.9
26 | 501040020 | $0.61$ | $0.07$ | 2006-09-24 05:03:12 | 44.8 | 40.0
27 | 501046010 | $-0.17$ | $0.34$ | 2007-03-10 15:03:10 | 25.2 | 25.0
28 | 501047010 | $-0.50$ | $0.34$ | 2007-03-11 03:55:59 | 25.6 | 19.1
29 | 501048010 | $-0.83$ | $0.34$ | 2007-03-11 19:04:59 | 27.5 | 24.1
30 | 501049010 | $-1.17$ | $0.33$ | 2006-10-08 10:22:40 | 19.6 | 17.6
31 | 501050010 | $-0.83$ | $-0.00$ | 2006-10-09 02:20:25 | 22.0 | 18.6
32 | 501051010 | $-1.17$ | $-0.00$ | 2006-10-09 13:40:09 | 21.9 | 21.1
33 | 501052010 | $-1.50$ | $-0.00$ | 2006-10-10 06:45:09 | 19.3 | 16.0
34 | 501053010 | $-1.83$ | $-0.00$ | 2006-10-10 21:18:59 | 21.9 | 19.9
35 | 501054010 | $-0.17$ | $-0.33$ | 2007-03-12 08:11:07 | 26.1 | 23.5
36 | 501055010 | $-0.50$ | $-0.33$ | 2007-03-12 23:59:09 | 27.2 | 21.2
37 | 501056010 | $-0.83$ | $-0.33$ | 2007-03-13 15:41:12 | 26.5 | 25.3
38 | 501057010 | $-1.17$ | $-0.34$ | 2006-10-11 10:07:27 | 20.5 | 19.1
39 | 501058010 | $1.30$ | $0.20$ | 2007-03-14 05:02:29 | 63.3 | 51.1
40 | 501059010 | $1.17$ | $0.00$ | 2007-03-15 18:55:51 | 62.2 | 54.4
41 | 501060010 | $1.50$ | $0.00$ | 2007-03-17 05:07:04 | 64.8 | 54.6
42 | 502002010 | $0.17$ | $-0.67$ | 2007-10-09 16:40:54 | 23.2 | 20.9
43 | 502003010 | $-0.17$ | $-0.67$ | 2007-10-10 03:41:13 | 21.5 | 18.9
44 | 502004010 | $0.17$ | $-1.00$ | 2007-10-10 15:21:17 | 19.9 | 18.8
45 | 502005010 | $-0.17$ | $-1.00$ | 2007-10-11 01:01:17 | 20.6 | 18.2
46 | 502006010 | $0.17$ | $0.33$ | 2007-10-11 11:34:01 | 22.6 | 21.7
47 | 502007010 | $0.17$ | $0.66$ | 2007-10-11 23:09:15 | 22.0 | 19.5
48 | 502008010 | $-0.17$ | $0.66$ | 2007-10-12 09:52:59 | 23.8 | 22.9
49 | 502009010 | $1.83$ | $-0.00$ | 2007-10-12 21:52:24 | 20.9 | 19.6
50 | 502010010 | $0.50$ | $0.33$ | 2007-10-13 07:32:00 | 21.6 | 21.2
Table 6 Continued. | Obs. IDaaObservation ID. | CoordinatebbAim point in Galactic coordinate (degree). | Start time | ExposureccNet exposure in units of $10^{3}$ s.
---|---|---|---|---
| | $l$ | $b$ | UT | XIS | PIN
51 | 502011010 | $0.83$ | $0.33$ | 2007-10-13 18:51:09 | 23.0 | 22.1
52 | 502016010 | $-1.08$ | $-0.48$ | 2008-03-02 18:08:00 | 70.5 | 61.8
53 | 502017010 | $-0.95$ | $-0.65$ | 2008-03-06 13:26:36 | 72.6 | 64.0
54 | 502018010 | $-1.27$ | $-0.42$ | 2008-03-08 16:02:17 | 79.0 | 70.2
55 | 502020010 | $1.05$ | $-0.17$ | 2007-09-06 00:26:47 | 139.1 | 124.5
56 | 502022010 | $0.23$ | $-0.27$ | 2007-08-31 12:33:33 | 134.8 | 116.8
57 | 502051010 | $0.92$ | $0.01$ | 2008-03-11 06:19:45 | 138.8 | 122.2
58 | 502059010 | $-0.00$ | $-2.00$ | 2007-09-29 01:40:51 | 136.8 | 110.5
59 | 503007010 | $0.33$ | $0.17$ | 2008-09-02 10:15:27 | 52.2 | 44.2
60 | 503008010 | $0.00$ | $-0.38$ | 2008-09-03 22:53:29 | 53.7 | 42.8
61 | 503009010 | $-0.32$ | $-0.24$ | 2008-09-05 06:57:08 | 52.4 | 40.3
62 | 503010010 | $-0.69$ | $-0.05$ | 2008-09-06 15:56:13 | 53.1 | 37.1
63 | 503011010 | $-0.97$ | $-0.13$ | 2008-09-08 09:08:09 | 57.6 | 40.2
64 | 503012010 | $-0.91$ | $-0.45$ | 2008-09-14 19:35:07 | 57.7 | 51.9
65 | 503013010 | $-1.30$ | $-0.05$ | 2008-09-16 00:51:19 | 104.8 | 93.9
66 | 503014010 | $-2.10$ | $-0.05$ | 2008-09-18 04:46:49 | 55.4 | 51.2
67 | 503015010 | $-2.35$ | $-0.05$ | 2008-09-19 07:33:05 | 56.8 | 52.8
68 | 503016010 | $-2.60$ | $-0.05$ | 2008-09-22 06:47:49 | 52.2 | 49.3
69 | 503017010 | $-2.85$ | $-0.05$ | 2008-09-23 08:08:10 | 51.3 | 48.6
70 | 503021010 | $-1.62$ | $0.20$ | 2008-10-04 03:44:03 | 53.8 | 49.6
71 | 503072010 | $-0.42$ | $0.17$ | 2009-03-06 02:39:12 | 140.6 | 135.5
72 | 503076010 | $-1.50$ | $0.15$ | 2009-02-24 17:04:51 | 52.9 | 43.8
73 | 503077010 | $-1.70$ | $0.14$ | 2009-02-26 01:01:00 | 51.3 | 43.7
74 | 503081010 | $0.03$ | $-1.66$ | 2009-03-09 15:41:50 | 59.2 | 57.6
75 | 503099010 | $-0.22$ | $1.13$ | 2009-03-10 19:39:08 | 29.7 | 30.6
76 | 503100010 | $-0.69$ | $1.13$ | 2009-03-15 06:41:41 | 25.7 | 24.1
77 | 503101010 | $-0.45$ | $0.89$ | 2009-03-16 14:43:17 | 33.9 | 30.8
78 | 503102010 | $-0.70$ | $0.66$ | 2009-03-17 07:49:09 | 33.7 | 30.1
79 | 503103010 | $-0.01$ | $1.20$ | 2009-03-11 10:56:59 | 18.3 | 16.4
80 | 504050010 | $0.10$ | $-1.42$ | 2010-03-06 03:55:37 | 100.4 | 80.5
81 | 504088010 | $-0.00$ | $-0.83$ | 2009-10-14 11:30:56 | 47.2 | 32.6
82 | 504089010 | $-0.05$ | $-1.20$ | 2009-10-09 04:05:59 | 55.3 | 40.2
83 | 504090010 | $-1.49$ | $-1.18$ | 2009-10-13 04:17:20 | 41.3 | 35.0
84 | 504091010 | $-1.50$ | $-1.60$ | 2009-09-14 19:37:36 | 51.3 | 47.8
85 | 504092010 | $-1.44$ | $-2.15$ | 2009-09-16 07:21:35 | 50.9 | 45.6
86 | 504093010 | $-1.50$ | $-2.80$ | 2009-09-17 13:54:31 | 53.2 | 46.9
87 | 903004010 | $-2.75$ | $-1.84$ | 2008-10-07 16:19:21 | 15.7 | 28.7
88 | 904002010 | $1.02$ | $2.53$ | 2009-08-28 12:20:31 | 23.1 | 21.9
89 | 904002020 | $1.02$ | $2.53$ | 2009-09-06 19:38:32 | 25.1 | 18.8
90 | 504003010 | $-1.45$ | $-0.87$ | 2010-02-25 04:33:17 | 50.9 | 41.3
91 | 504001010 | $-1.47$ | $-0.26$ | 2010-02-26 09:15:00 | 51.2 | 42.2
92 | 504002010 | $-1.53$ | $-0.58$ | 2010-02-27 16:14:41 | 53.1 | 46.6
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|
arxiv-papers
| 2012-05-08T01:59:26 |
2024-09-04T02:49:30.689884
|
{
"license": "Public Domain",
"authors": "Takayuki Yuasa, Kazuo Makishima, and Kazuhiro Nakazawa",
"submitter": "Takayuki Yuasa",
"url": "https://arxiv.org/abs/1205.1574"
}
|
1205.1670
|
# Rainbow Colouring of
Split and Threshold Graphs
L. Sunil Chandran Department of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{sunil, deepakr}@csa.iisc.ernet.in Deepak Rajendraprasad Department of
Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{sunil, deepakr}@csa.iisc.ernet.in
###### Abstract
A rainbow colouring of a connected graph is a colouring of the edges of the
graph, such that every pair of vertices is connected by at least one path in
which no two edges are coloured the same. Such a colouring using minimum
possible number of colours is called an optimal rainbow colouring, and the
minimum number of colours required is called the rainbow connection number of
the graph. A Chordal Graph is a graph in which every cycle of length more than
$3$ has a chord. A Split Graph is a chordal graph whose vertices can be
partitioned into a clique and an independent set. A threshold graph is a split
graph in which the neighbourhoods of the independent set vertices form a
linear order under set inclusion. In this article, we show the following:
1. 1.
The problem of deciding whether a graph can be rainbow coloured using $3$
colours remains NP-complete even when restricted to the class of split graphs.
However, any split graph can be rainbow coloured in linear time using at most
one more colour than the optimum.
2. 2.
For every integer $k\geq 3$, the problem of deciding whether a graph can be
rainbow coloured using $k$ colours remains NP-complete even when restricted to
the class of chordal graphs.
3. 3.
For every positive integer $k$, threshold graphs with rainbow connection
number $k$ can be characterised based on their degree sequence alone. Further,
we can optimally rainbow colour a threshold graph in linear time.
Keywords: rainbow connectivity, rainbow colouring, threshold graphs, split
graphs, chordal graphs, degree sequence, approximation, complexity.
## 1 Introduction
Connectivity is one of the basic concepts of graph theory. It plays a
fundamental role both in theoretical studies and in applications. When a
network (transport, communication, social, etc) is modelled as a graph,
connectivity gives a way of quantifying its robustness. This may be the reason
why connectivity is possibly the problem that has been studied on the largest
variety of computational models [25]. Due to the diverse application
requirements and manifold theoretical interests, many variants of the
connectivity problem have been studied. One typical case is when there are
different possible types of connections (edges) between nodes and additional
restrictions on connectivity based on the types of edges that can used in a
path. In this case we can model the network as an edge-coloured graph. One
natural restriction to impose on connectivity is that any two nodes should be
connected by a path in which no edge of the same type (colour) occurs more
than once. This is precisely the property called rainbow connectivity. Such a
restriction for the paths can arise, for instance, in routing packets in a
cellular network with transceivers that can operate in multiple frequency
bands or in routing secret messages between security agencies using different
handshaking passwords in different links [18] [5]. The problem was formalised
in graph theoretic terms by Chartrand et al. [7] in 2008.
An edge colouring of a graph is a function from its edge set to the set of
natural numbers. A path in an edge coloured graph with no two edges sharing
the same colour is called a rainbow path. An edge coloured graph is said to be
rainbow connected if every pair of vertices is connected by at least one
rainbow path. Such a colouring is called a rainbow colouring of the graph. A
rainbow colouring using minimum possible number of colours is called optimal.
The minimum number of colours required to rainbow colour a connected graph is
called its rainbow connection number, denoted by $rc(G)$. For example, the
rainbow connection number of a complete graph is $1$, that of a path is its
length, that of an even cycle is its diameter, that of an odd cycle of length
at least $5$ is one more than its diameter, and that of a tree is its number
of edges. Note that disconnected graphs cannot be rainbow coloured and hence
the rainbow connection number for them is left undefined. Any connected graph
can be rainbow coloured by giving distinct colours to the edges of a spanning
tree of the graph. Hence the rainbow connection number of any connected graph
is less than its number of vertices.
While formalising the concept of rainbow colouring, Chartrand et al. also
determined the precise values of rainbow connection number for some special
graphs [7]. Subsequently, there have been various investigations towards
finding good upper bounds for rainbow connection number in terms of other
graph parameters [4] [21] [15] [24] [2] and for many special graph classes
[19] [24] [2] [3]. Behaviour of rainbow connection number in random graphs is
also well studied [4] [11] [23] [9]. A basic introduction to the topic can be
found in Chapter $11$ of the book Chromatic Graph Theory by Chartrand and
Zhang [6] and a survey of most of the recent results in the area can be found
in the article by Li and Sun [18] and also in their forthcoming book Rainbow
Connection of Graphs [17].
On the computational side, the problem has received relatively less attention.
It was shown by Chakraborty et al. that computing the rainbow connection
number of an arbitrary graph is NP-Hard [5]. In particular, it was shown that
the problem of deciding whether a graph can be rainbow coloured using $2$
colours is NP-complete. Later, Ananth et al. [1] complemented the result of
Chakraborty et al., and now we know that for every integer $k\geq 2$, it is
NP-complete to decide whether a given graph can be rainbow coloured using $k$
colours. Chakraborty et al., in the same article, also showed that deciding
whether a given edge coloured graph is rainbow connected is NP-complete. It
was then shown by Li and Li that this problem remains NP-complete even when
restricted to the class of bipartite graphs [16].
On the positive side, Basavaraju et al. have demonstrated an $O(nm)$-time
$(r+3)$-factor approximation algorithm for rainbow colouring any graph with
radius $r$ [2]. Constant factor approximation algorithms for rainbow colouring
Cartesian, strong and lexicographic products of non-trivial graphs are
reported in [3]. Constant factor approximation algorithms for bridgeless
chordal graphs, and additive approximation algorithms for interval, AT-free,
threshold and circular arc graphs without pendant vertices will follow from
the proofs of their upper bounds [24]. To the best of our knowledge, no
efficient optimal rainbow colouring algorithm has been reported for any non-
trivial subclass of graphs.
### 1.1 Our Results
In this article we consider the problem of rainbow colouring split graphs and
a particular subclass of split graphs called threshold graphs (Definition 3).
We show the following results.
* 1.
The problem of deciding whether a graph can be rainbow coloured using $3$
colours remains NP-complete even when restricted to the class of split graphs
(Corollary 5). Any split graph can be rainbow coloured in linear time using at
most one more colour than the optimum (Algorithm 1).
This is similar to the problem of finding the chromatic index of a graph.
Though every graph with maximum degree $\Delta$ can be properly edge-coloured
in $O(nm)$ time using $\Delta+1$ colours using a constructive proof of
Vizing’s Theorem [20], it is NP-hard to decide whether the graph can be
coloured using $\Delta$ colours [12].
No two pendant edges (Definition 2) can share the same colour in any rainbow
colouring of a graph (Observation 2). The $+1$-approximation algorithm above
is obtained by carefully reusing the same colours on most of the remaining
edges of the graph. The hardness result is obtained by demonstrating a
reduction from the problem of $3$-colourability of $3$-uniform hypergraphs. In
fact, the technique in the reduction can be extended to show the following
result for chordal graphs.
* 2.
For every integer $k\geq 3$, the problem of deciding whether a graph can be
rainbow coloured using $k$ colours remains NP-complete even when restricted to
the class of chordal graphs (Theorem 6).
Though a similar hardness result is known for deciding the rainbow connection
number of general graphs, the above strengthening to chordal graphs is
interesting since, unlike for general graphs, a constant factor approximation
algorithm is already known for rainbow colouring chordal graphs. Chandran et
al. [24] have shown that any bridgeless chordal graph can be rainbow coloured
using at most $3r$ colours, where $r$ is the radius of the graph. The proof
given there is constructive and can be easily extended to a polynomial-time
algorithm which will colour any chordal graph $G$ with $b$ bridges and radius
$r$ using at most $3r+b$ colours. Since $\max\\{r,b\\}$ is easily seen to be a
lower bound for $rc(G)$, this immediately gives us a $4$-factor approximation
algorithm.
* 3.
For every positive integer $k$, threshold graphs with rainbow connection
number exactly $k$ can be characterised based on their degree sequence
(Definition 2) alone (Corollary 14). Further, we can optimally rainbow colour
a threshold graph in linear time (Algorithm 4).
In particular we show that if $d_{1}\geq\cdots\geq d_{n}$ is the degree
sequence of an $n$-vertex threshold graph $G$, then
$rc(G)=\begin{cases}1,&d_{n}=n-1\\\ 2,&d_{n}<n-1\textnormal{ and
}\sum_{i=k}^{n}2^{-d_{i}}\leq 1\\\
\max\\{3,p\\},&\textnormal{otherwise}\end{cases}$ (1)
where $k=\min\\{i:1\leq i\leq n,\,d_{i}\leq i-1\\}$ and $p=|\\{i:1\leq i\leq
n,\,d_{i}=1\\}|$.
Both the characterisation and the algorithm are obtained by connecting the
problem of rainbow colouring a threshold graph to that of generating a prefix-
free binary code.
### 1.2 Preliminaries
All graphs considered in this article are finite, simple and undirected. For a
graph $G$, we use $V(G)$ and $E(G)$ to denote its vertex set and edge set
respectively. Unless mentioned otherwise, $n$ and $m$ will respectively denote
the number of vertices and edges of the graph in consideration. The shorthand
$[n]$ denotes the set $\\{1,\ldots,n\\}$. The cardinality of a set $S$ is
denoted by $|S|$.
###### Definition 1.
Let $G$ be a connected graph. The length of a path is its number of edges. The
distance between two vertices $u$ and $v$ in $G$, denoted by $d(u,v)$ is the
length of a shortest path between them in $G$. The eccentricity of a vertex
$v$ is $ecc(v):=\max_{x\in V(G)}{d(v,x)}$. The diameter of $G$ is
$diam(G):=\max_{x\in V(G)}{ecc(x)}$ and radius of $G$ is
$radius(G):=\min_{x\in V(G)}{ecc(x)}$.
###### Definition 2.
The neighbourhood $N(v)$ of a vertex $v$ is the set of vertices adjacent to
$v$ but not including $v$. The degree of a vertex $v$ is $d_{v}:=|N(v)|$. The
degree sequence of a graph is the non-increasing sequence of its vertex
degrees. A vertex is called pendant if its degree is $1$. An edge incident on
a pendant vertex is called a pendant edge.
###### Definition 3.
A graph $G$ is called chordal, if there is no induced cycle of length greater
than $3$. A graph $G$ is a split graph, if $V(G)$ can be partitioned into a
clique and an independent set. A graph $G$ is a threshold graph, if there
exists a weight function $w:V(G)\rightarrow\mathbb{R}$ and a real constant $t$
such that two vertices $u,v\in V(G)$ are adjacent if and only if
$w(u)+w(v)\geq t$.
Before getting into the main results, we note two elementary and well known
observations on rainbow colouring whose proofs we omit.
###### Observation 1.
For every connected graph $G$, we have $rc(G)\geq diam(G)$.
###### Observation 2.
If $u$ and $v$ are two pendant vertices in a connected graph $G$, then their
incident edges get different colours in any rainbow colouring of $G$. In
particular, if $G$ has $p$ pendant vertices, then $rc(G)\geq p$.
## 2 Split Graphs: Hardness and Approximation Algorithm
We first show that determining the rainbow connection number of a split graph
is NP-hard, by demonstrating a reduction to it from the $3$-colouring problem
on $3$-uniform hypergraphs.
###### Definition 4.
A hypergraph $H$ is a tuple $(V,E)$, where $V$ is a finite set and $E\subseteq
2^{V}$. Elements of $V$ and $E$ are called vertices and (hyper-)edges
respectively. The hypergraph $H$ is called $r$-uniform if $|e|=r$ for every
$e\in E$. An $r$-uniform hypergraph is called complete if $E=\\{e\subset
V:|e|=r\\}$.
###### Definition 5.
Given a hypergraph $H(V,E)$ and a colouring $C_{H}:V\rightarrow\mathbb{N}$, an
edge is called $k$-coloured if the edge contains vertices of $k$ different
colours. An edge is called monochromatic if it is $1$-coloured. The colouring
$C_{H}$ is called proper if no edge in $E$ is monochromatic under $C_{H}$. The
minimum number of colours required to properly colour $H$ is called its
chromatic number and is denoted by $\chi(H)$.
We need a $3$-uniform hypergraph of chromatic number $3$ to avoid the
occurrence of a border case in the reduction. The following observation gives
us one.
###### Observation 3.
Let $K_{5}^{3}$ be the complete $3$-uniform hypergraph on $5$ vertices. Then
$\chi(K_{5}^{3})=3$.
###### Proof.
Assign colours $0,0,1,1,2$ to the $5$ vertices of $K_{5}^{3}$. This is a
proper colouring of $K_{5}^{3}$ since every edge contains $3$ vertices and
hence cannot be monochromatic. On the other hand, in any colouring of
$K_{5}^{3}$ using fewer than $3$ colours, some three vertices have to share
the same colour and hence the edge of $K_{5}^{3}$ constituted of those $3$
vertices will be monochromatic. Hence $\chi(K_{5}^{3})=3$. ∎
It follows from Theorem $1.1$ in [13] that it is NP-hard to decide whether an
$n$-vertex $3$-uniform hypergraph can be properly coloured using $3$ colours.
A reduction from this problem to a problem of computing the rainbow connection
number of a split graph is illustrated in the proofs of Theorem 4 and Theorem
6.
###### Theorem 4.
The first problem below (P1) is polynomial-time reducible to the second (P2).
1. P1.
Given a $3$-uniform hypergraph $H^{\prime}$, decide whether
$\chi(H^{\prime})\leq 3$.
2. P2.
Given a split graph $G$, decide whether $rc(G)\leq 3$.
$b$$a_{2}$$a_{0}$$a_{1}$$V_{H}$$E_{H}$pendant verticesclique$H$incidences
Figure 1: Split graph $G$ constructed from a 3-uniform hypergraph $H$. Note
that $V_{H}\cup\\{b\\}$ is a clique and $E_{H}\cup\\{a_{0},a_{1},a_{2}\\}$ is
an independent set in $G$.
###### Proof.
Let $H$ be the disjoint union of $H^{\prime}$ and a complete $3$-uniform
hypergraph on $5$ vertices ($K_{5}^{3}$). This ensures that $\chi(H)\geq 3$
(Observation 3) and that $\chi(H)=3$ iff $\chi(H^{\prime})\leq 3$. Let $V_{H}$
and $E_{H}$ be the vertex set and edge set, respectively, of $H$. We construct
a graph $G(V_{G},E_{G})$ from $H(V_{H},E_{H})$ as follows (See Figure 1).
$\displaystyle V_{G}$ $\displaystyle=$ $\displaystyle V_{H}\cup
E_{H}\cup\\{a_{0},a_{1},a_{2},b\\}$ (2) $\displaystyle E_{G}$ $\displaystyle=$
$\displaystyle\\{\\{v,e\\}:v\in V_{H},e\in E_{H},v\in e\textnormal{ in }H\\}$
(3) $\displaystyle\cup\,\\{\\{v,v^{\prime}\\}:v,v^{\prime}\in V_{H},v\neq
v^{\prime}\\}$ $\displaystyle\cup\,\\{\\{b,v\\}:v\in V_{H}\\}$
$\displaystyle\cup\,\\{\\{a_{i},b\\}:i=0,1,2\\}$
The graph $G$ thus constructed is a split graph with $V_{H}\cup\\{b\\}$ being
a clique and its complement with respect to $V_{G}$, which is
$E_{H}\cup\\{a_{0},a_{1},a_{2}\\}$, being an independent set. It is clear that
$G$ can be constructed from $H^{\prime}$ in polynomial-time. We complete the
proof by showing that $\chi(H)=3$ iff $rc(G)=3$.
Firstly, we show that if $rc(G)=3$, then $\chi(H)=3$. Since $\chi(H)\geq 3$,
it suffices to show that $H$ can be properly $3$-coloured. Let
$C_{G}:E_{G}\rightarrow\mathbb{Z}_{3}$ be a rainbow colouring of $G$. Define a
colouring $C_{H}:V_{H}\rightarrow\mathbb{Z}_{3}$ by
$C_{H}(v)=C_{G}(\\{b,v\\})$ for each $v\in V_{H}$. We claim that $C_{H}$ is a
proper colouring of $H$. For the sake of contradiction, suppose that one of
the hyper-edges $e_{H}$ of $H$ is monochromatic under $C_{H}$, i.e, all the
vertices in $e_{H}$ get the same colour $j$ for some $j\in\mathbb{Z}_{3}$.
This happens only when $C_{G}(\\{b,v\\})=j,\,\forall v\in e_{H}$. Hence all
the paths of length two from $b$ to $e_{H}$ in $G$ will use the colour $j$.
Since $\\{a_{0},a_{1},a_{2}\\}$ are pendant vertices, the edges from
$\\{a_{0},a_{1},a_{2}\\}$ to $b$ all have distinct colours in any rainbow
colouring of $G$ (Observation 2). Hence one of them, say $\\{a_{i},b\\}$, gets
the colour $j$. Then it is easy to see that there is no rainbow path from
$a_{i}$ to $e_{H}$ in $G$ under $C_{G}$ (Note that any rainbow path in a
$3$-coloured graph has length at most $3$). This contradicts the fact that
$C_{G}$ was a rainbow colouring of $G$.
Next, we show that if $\chi(H)=3$, then $rc(G)=3$. Since $G$ has $3$ pendant
vertices, $rc(G)\geq 3$ (Observation 2). So it suffices to show that $G$ can
be rainbow coloured using $3$ colours. Let
$C_{H}:V_{H}\rightarrow\mathbb{Z}_{3}$ be a proper colouring of $H$. Let
$V_{i}=\\{v\in V_{H}:C_{H}(v)=i\\}$, $i\in\mathbb{Z}_{3}$, be the colour
classes. Note that none of the colour classes is empty as $\chi(H)=3$. We
define a colouring $C_{G}:E_{G}\rightarrow\mathbb{Z}_{3}$ as follows (See
Figure 2). $C_{G}(\\{b,v\\})=C_{H}(v)$ for each $v\in V_{H}$. Consider a
hyper-edge $e_{H}=\\{v_{0},v_{1},v_{2}\\}$ of $H$. If $e_{H}$ is $3$-coloured
in $C_{H}$ then $C_{G}(\\{v_{i},e_{H}\\})=C_{H}(v_{i})+1$ (Note that the
colours are from $\mathbb{Z}_{3}$ and hence the addition is modulo $3$). If
$e_{H}$ is $2$-coloured in $H$, then without loss of generality, let
$C_{H}(v_{0})=C_{H}(v_{1})=i$ and $C_{H}(v_{2})=j$, $j\neq i$. Set
$C_{G}(\\{v_{0},e_{H}\\})=i+1$, $C_{G}(\\{v_{1},e_{H}\\})=i+2$, and
$C_{G}(\\{v_{2},e_{H}\\})\in\mathbb{Z}_{3}\setminus\\{i,j\\}$. This ensures
that for every hyper-edge $e\in E_{H}$, for each colour $i\in\mathbb{Z}_{3}$,
there exists a $2$-length rainbow path $P_{e,i}$ from $b$ to $e$ such that
colour $i$ does not appear in path $P_{e,i}$. The remaining edges of $G$ are
coloured as follows.
$\displaystyle C_{G}(\\{a_{i},b\\})$ $\displaystyle=$ $\displaystyle
i\quad\forall i\in\mathbb{Z}_{3}$ $\displaystyle C_{G}(\\{v,v^{\prime}\\})$
$\displaystyle=$ $\displaystyle i\quad\forall v,v^{\prime}\in V_{i},\,v\neq
v^{\prime},\,\forall i\in\mathbb{Z}_{3}$ $\displaystyle
C_{G}(\\{v,v^{\prime}\\})$ $\displaystyle=$ $\displaystyle 2\quad\forall v\in
V_{0},v^{\prime}\in V_{1}\cup V_{2}$ $\displaystyle C_{G}(\\{v,v^{\prime}\\})$
$\displaystyle=$ $\displaystyle 0\quad\forall v\in V_{1},v^{\prime}\in V_{2}.$
(4)
We show that $C_{G}$ is a rainbow colouring of $G$ by demonstrating a rainbow
path between every pair of non adjacent vertices in $G$. First we demonstrate
the paths from $\\{a_{0},a_{1},a_{2},b\\}$ to all their non-adjacent vertices.
(The numbers above an edge indicate the colour assigned to the edge under
$C_{G}$.)
$\displaystyle a_{i}\textnormal{ to }a_{j},i\neq j$ $\displaystyle:$
$\displaystyle a_{i}\stackrel{{\scriptstyle
i}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
j}}{{\mbox{---\negthinspace---}}}a_{j}$ $\displaystyle a_{i}\textnormal{ to
}v_{j}\in V_{j},i\neq j$ $\displaystyle:$ $\displaystyle
a_{i}\stackrel{{\scriptstyle
i}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
j}}{{\mbox{---\negthinspace---}}}v_{j}$ $\displaystyle a_{i}\textnormal{ to
}v_{i}\in V_{i}$ $\displaystyle:$ $\displaystyle a_{0}\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}V_{1}\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v_{0}$ $\displaystyle
a_{1}\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}V_{0}\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v_{1}$ $\displaystyle
a_{2}\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}V_{1}\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}v_{2}$ $\displaystyle a_{i}\textnormal{ to
}e\in E_{H}$ $\displaystyle:$ $\displaystyle a_{i}\stackrel{{\scriptstyle
i}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
P_{e,i}}}{{\mbox{---\negthinspace---}}}e$ $\displaystyle b\textnormal{ to
}e\in E_{H}$ $\displaystyle:$ $\displaystyle b\stackrel{{\scriptstyle
P_{e,0}}}{{\mbox{---\negthinspace---}}}e$ (5)
The rainbow path between any vertex $v\in V_{H}$ and a non-adjacent vertex
$e\in E_{H}$ is given by $v\stackrel{{\scriptstyle
i}}{{\mbox{---\negthinspace---}}}b\stackrel{{\scriptstyle
P_{e,i}}}{{\mbox{---\negthinspace---}}}e$, if $v\in V_{i}$. It remains to
demonstrate a rainbow path between any two vertices
$e=\\{v_{0},v_{1},v_{2}\\},e^{\prime}=\\{v_{0}^{\prime},v_{1}^{\prime},v_{2}^{\prime}\\}\in
E_{H}$. By $C_{H}(e)$ we denote the $3$-tuple
$(C_{H}(v_{0}),C_{H}(v_{1}),C_{H}(v_{2}))$. If $e$ is $3$-coloured, we relabel
$\\{v_{0},v_{1},v_{2}\\}$ so that $C_{H}(e)=(0,1,2)$ and hence
$C_{G}(\\{v_{i},e\\})=i+1$. If $e$ is $2$-coloured, we relabel
$\\{v_{0},v_{1},v_{2}\\}$ so that $C_{H}(e)=(i,i,j),j\neq i$ and such that
$C_{G}(\\{v_{0},e\\})=i+1$ and $C_{G}(\\{v_{1},e\\})=i+2$. We do the same for
$e^{\prime}$ too. Edges $e$ and $e^{\prime}$ may share some vertices, in which
case the same vertex will get different labels when considered under $e$ and
$e^{\prime}$. We consider the following cases separately: (i) both $e$ and
$e^{\prime}$ are $3$-coloured, (ii) $e$ is $3$-coloured and $e^{\prime}$ is
2-coloured and (iii) both $e$ and $e^{\prime}$ are $2$-coloured. The last case
is further split into $4$ sub-cases.
$\displaystyle C_{H}(e)=C_{H}(e^{\prime})=(0,1,2)$
$\displaystyle:e\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}v_{0}\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v_{2}^{\prime}\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}e^{\prime}$ (Case i) $\displaystyle
C_{H}(e)=(0,1,2),C_{H}(e^{\prime})=(i,i,j),i\neq j$
$\displaystyle:e\stackrel{{\scriptstyle
i+1}}{{\mbox{---\negthinspace---}}}v_{i}\stackrel{{\scriptstyle
i}}{{\mbox{---\negthinspace---}}}v_{1}^{\prime}\stackrel{{\scriptstyle
i+2}}{{\mbox{---\negthinspace---}}}e^{\prime}$ (Case ii) $\displaystyle
C_{H}(e)=(i,i,j),C_{H}(e^{\prime})=(i,i,k)$
$\displaystyle:e\stackrel{{\scriptstyle
i+1}}{{\mbox{---\negthinspace---}}}v_{0}\stackrel{{\scriptstyle
i}}{{\mbox{---\negthinspace---}}}v_{1}^{\prime}\stackrel{{\scriptstyle
i+2}}{{\mbox{---\negthinspace---}}}e^{\prime}$ (Case iii) $\displaystyle
C_{H}(e)=(0,0,j),C_{H}(e^{\prime})=(1,1,k)$
$\displaystyle:e\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}v_{0}\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v_{1}^{\prime}\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}e^{\prime}$ $\displaystyle
C_{H}(e)=(1,1,j),C_{H}(e^{\prime})=(2,2,k)$
$\displaystyle:e\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v_{0}\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}v_{1}^{\prime}\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}e^{\prime}$ $\displaystyle
C_{H}(e)=(2,2,j),C_{H}(e^{\prime})=(0,0,k)$
$\displaystyle:e\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}v_{0}\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v_{0}^{\prime}\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}e^{\prime}$ (6)
It is possible that $v_{i}$ may coincide with $v_{1}^{\prime}$ in Case (ii),
and $v_{0}$ may coincide with $v_{1}^{\prime}$ in the first sub-case of Case
(iii). In both those situations, we still get a $2$-length rainbow path
between the end points without using the middle edge indicated above. We have
exhausted all the cases and hence $C_{G}$ is a rainbow colouring of $G$. ∎
012220$b$$a_{1}$1$a_{0}$0$a_{2}$2$V_{0}$0$V_{1}$1$V_{2}$2$V_{H}$$E_{H}$$e_{3}$210$e_{2}$101
Figure 2: Rainbow colouring of split graph $G$ based on the $3$-colouring of
hypergraph $H$. In the figure, $e_{3}$ is a sample $3$-coloured edge and
$e_{2}$ is a sample $2$-coloured edge.
Since Problem P1 is known to be NP-hard, so is Problem P2. Further, it is easy
to see that the problem P2 is in NP. Hence the following corollary.
###### Corollary 5.
Deciding whether $rc(G)\leq 3$ remains NP-complete even when $G$ is restricted
to be in the class of split graphs.
The reduction used in the proof of Theorem 4 can be extended to show that for
every $k\geq 3$, it is NP-complete to decide whether a chordal graph can be
rainbow coloured using $k$ colours.
###### Theorem 6.
For any integer $k\geq 3$, the first problem below (P1) is polynomial-time
reducible to the second (P2).
1. P1.
Given a $3$-uniform hypergraph $H^{\prime}$, decide whether
$\chi(H^{\prime})\leq 3$.
2. P2.
Given a chordal graph $G$, decide whether $rc(G)\leq k$.
In particular, for every integer $k\geq 3$, the problem of deciding whether
$rc(G)\leq k$ remains NP-complete even when $G$ is restricted to be in the
class of chordal graphs.
$b_{0}$$b_{1}$$b_{k-3}$$a_{2}$$a_{0}$$a_{1}$$V_{H}$$E_{H}$clique$H$incidences
Figure 3: Chordal graph $G_{k}$ of diameter $k$ constructed from a 3-uniform
hypergraph $H$.
###### Proof.
Let $H$ be the disjoint union of $H^{\prime}$ and a complete $3$-uniform
hypergraph on $5$ vertices ($K_{5}^{3}$). This ensures that $\chi(H)\geq 3$
(Observation 3) and that $\chi(H)=3$ iff $\chi(H^{\prime})\leq 3$. Let $V_{H}$
and $E_{H}$ be the vertex set and edge set, respectively, of $H$. Let $k\geq
3$ be fixed. We construct a graph $G_{k}(V_{G},E_{G})$ from $H$ as follows
(See Figure 3).
$\displaystyle V_{G}$ $\displaystyle=$ $\displaystyle V_{H}\cup
E_{H}\cup\\{a_{0},a_{1},a_{2},b_{0},\ldots,b_{k-3}\\}$ (7) $\displaystyle
E_{G}$ $\displaystyle=$ $\displaystyle\\{\\{v,e\\}:v\in V_{H},e\in E_{H},v\in
e\textnormal{ in }H\\}$ (8)
$\displaystyle\cup\,\\{\\{v,v^{\prime}\\}:v,v^{\prime}\in V_{H},v\neq
v^{\prime}\\}$ $\displaystyle\cup\,\\{\\{b_{k-3},v\\}:v\in V_{H}\\}$
$\displaystyle\cup\,\\{\\{b_{i-1},b_{i}\\}:i=1,\ldots,k-3\\}$
$\displaystyle\cup\,\\{\\{a_{i},b_{0}\\}:i=0,1,2\\}$
The graph $G$ thus constructed is easily seen to be a chordal graph with
diameter $k$. It is clear that $G$ can be constructed from $H^{\prime}$ in
polynomial-time. We complete the proof by showing that $\chi(H)=3$ iff
$rc(G)=k$.
It is easy to see that when $k=3$, the graph $G_{3}$ constructed as above is
the same as the split graph constructed in the proof of Theorem 4. In that
proof we showed a rainbow colouring of $G_{3}$ using $3$ colours in the case
when $\chi(H)=3$. The same colouring can be extended to $G_{k}$ by giving
$k-3$ new colours exclusively to the edges
$\\{b_{i-1},b_{i}\\},\,i=1,\ldots,k-3$. Since the original $3$-colouring made
$G_{3}$ rainbow connected, it is easy to see that this colouring makes $G_{k}$
rainbow connected. Hence it is enough to show that if $rc(G_{k})=k$, then
$\chi(H)=3$.
Since $\chi(H)\geq 3$, it suffices to show that $H$ can be properly
$3$-coloured. Let $C_{G}:E_{G}\rightarrow\\{0,\ldots,k-1\\}$ be a rainbow
colouring of $G_{k}$. Since the subgraph $T_{k}$ of $G_{k}$ induced on
$\\{a_{0},a_{1},a_{2},b_{0},\ldots,b_{k-3}\\}$ is a tree with $k$ edges, it is
easy to see that in any rainbow colouring of $G_{k}$ the edges of $T_{k}$ get
$k$ distinct colours. Without loss of generality we rename the colours so that
$C_{G}(\\{a_{i},b_{0}\\})=i,\,i\in\\{0,1,2\\}$. Hence the edges in the path
from $b_{0}$ to $b_{k-3}$ get colours from $\\{3,\ldots,k-1\\}$. Define a
colouring $C_{H}:V_{H}\rightarrow\\{0,1,2\\}$ by
$C_{H}(v)=\min\\{C_{G}(\\{b_{k-3},v\\}),2\\}$ for each $v\in V_{H}$. We claim
that $C_{H}$ is a proper colouring of $H$. For the sake of contradiction,
suppose that one of the hyper-edges $e_{H}$ of $H$ is monochromatic under
$C_{H}$, i.e, all the vertices in $e_{H}$ get the same colour $j$ for some
$j\in\\{0,1,2\\}$. This happens only when
$\min\\{C_{G}(\\{b_{k-3},v\\}),2\\}=j,\,\forall v\in e_{H}$. If $j$ is $0$ or
$1$, all the paths of length two from $b_{k-3}$ to $e_{H}$ in $G$ will use the
colour $j$ and hence there is no ($k$-length) rainbow path from $a_{j}$ to
$e_{H}$. If $j$ is $2$, then too, all the paths of length two from $b_{k-3}$
to $e_{H}$ in $G$ will use one of the colours already used in the unique path
from $a_{2}$ to $b_{k-3}$ and hence there is no ($k$-length) rainbow path from
$a_{2}$ to $e_{H}$.
Since Problem P1 is known to be NP-hard, so is Problem P2. Further, it is easy
to see that the problem P2 is in NP. Hence the result. ∎
In the wake of Corollary 5, it is unlikely that there exists a polynomial-time
algorithm to optimally rainbow colour split graphs in general. In Section 3,
we show that the problem is efficiently solvable when restricted to threshold
graphs, which are a subclass of split graphs. Before that, we describe a
linear-time (approximation) algorithm which rainbow colours any split graph
using at most one colour more than the optimum (Theorem 7). First we note that
it is easy to find a maximum clique in a split graph, as follows.
The vertices of a graph can be sorted according to their degrees in $O(n)$
time using a counting sort [22]. If $G([n],E)$ is a split graph with the
vertices labelled so that $d_{1}\geq\cdots\geq d_{n}$, where $d_{i}$ is degree
of vertex $i$, then $\\{i\in V(G):d_{i}\geq i-1\\}$ is a maximum clique in $G$
and $\\{i\in V(G):d_{i}\leq i-1\\}$ is a maximum independent set in $G$ [10].
Hence we can assume, if needed, that a maximum clique or a maximum independent
set or an ordering of the vertices according to their degrees is given as
input to our algorithms.
Algorithm 1 ColourSplitGraph
0: $G([n],E)$, a connected split graph with a maximum clique $C$.
0: A rainbow colouring $C_{G}:E(G)\rightarrow\\{0,\ldots,\max\\{p,2\\}\\}$,
where $p$ is the number of pendant vertices in $V(G)\setminus C$.
1: $I\leftarrow V(G)\setminus C$ $/\negthickspace/$ $I$ is an independent set
in $G$
2: $P\leftarrow\\{i\in I:d_{i}=1\\}$, $p\leftarrow|P|$ $/\negthickspace/$ $P$
is the set of pendant vertices in $I$
3: $C_{G}(e)\leftarrow 0$, for all edges $e$ with both end points in $C$.
4: $C_{G}(e_{i})\leftarrow i$ for each pendant edge $e_{1},\ldots,e_{p}$
5: for $i\in I\setminus P$ do
6: Let $\\{e_{1},\ldots,e_{d_{i}}\\}$ be the edges incident on $i$
7: $C_{G}(e_{1})\leftarrow 1$
8: $C_{G}(e)\leftarrow 2$ for every other edge $e$ incident on $i$
9: end for$/\negthickspace/$ Now every vertex in $I\setminus P$ has a
$1$-coloured and a $2$-coloured edge to $C$
10: return $C_{G}$
###### Theorem 7.
For every connected split graph $G$, Algorithm 1 (ColourSplitGraph) rainbow
colours $G$ using at most $rc(G)+1$ colours. Further, the time-complexity of
Algorithm 1 is $O(m)$.
###### Proof.
If $G$ is a clique, then $C=V(G)$ and Algorithm 1 colours every edge of $G$
with colour $0$. This is an optimal rainbow colouring for $G$. Hence we can
assume that $G$ is not a clique in the following discussions. So
$d:=diam(G)\geq 2$. It is easy to check, by considering all pairs of non-
adjacent vertices, that Algorithm 1 indeed produces a rainbow colouring of
$G$. For example, between two vertices $v,v^{\prime}\in I\setminus P$, we get
a rainbow path $v\stackrel{{\scriptstyle
1}}{{\mbox{---\negthinspace---}}}C\stackrel{{\scriptstyle
0}}{{\mbox{---\negthinspace---}}}C\stackrel{{\scriptstyle
2}}{{\mbox{---\negthinspace---}}}v^{\prime}$. It is also evident that the
algorithm uses at most $k:=\max\\{p+1,3\\}$ colours. By Observation 1 and
Observation 2, $rc(G)\geq\max\\{p,d\\}\geq\max\\{p,2\\}=k-1$. Hence the
rainbow colouring produced by Algorithm 1 uses at most $rc(G)+1$ colours.
Further, the algorithm visits each edge exactly once and hence the time-
complexity is $O(m)$. ∎
The following bounds follow directly from Observation 1, Observation 2, and
Theorem 7.
###### Corollary 8.
For every connected split graph $G$ with $p$ pendant vertices and diameter
$d$,
$\max\\{p,d\\}\leq rc(G)\leq\max\\{p+1,3\\}.$
## 3 Threshold Graphs: Characterisation and Exact Algorithm
Threshold graphs form a subclass of split graphs (Observation 9b). The
neighbourhoods of vertices in a maximum independent set of a threshold graph
form a linear order under set inclusion (Observation 9c). We exploit this
structure to give a full characterisation of rainbow connection number of
threshold graphs based on degree sequences (Corollary 14). We use this
characterisation to design a linear-time algorithm to optimally rainbow colour
any threshold graph (Algorithm 4).
The following observations are easy to make from the definition of a threshold
graph (Definition 3).
###### Observation 9.
Let $G([n],E)$ be a threshold graph with a weight function
$w:V(G)\rightarrow\mathbb{R}$. Let the vertices be labelled so that
$w(1)\geq\cdots\geq w(n)$. Then
1. (a)
$d_{1}\geq\cdots\geq d_{n}$, where $d_{i}$ is the degree of vertex $i$.
2. (b)
$I=\\{i\in V(G):d_{i}\leq i-1\\}$ is a maximum independent set $G$ and
$V(G)\setminus I$ is a clique in $G$. In particular, every threshold graph is
a split graph.
3. (c)
$N(i)=\\{1,\ldots,d_{i}\\}$, for every $i\in I$. Thus the neighbourhoods of
vertices in $I$ form a linear order under set inclusion. Further, if $G$ is
connected, then every vertex in $G$ is adjacent to $1$.
###### Definition 6.
A binary codeword is a finite string over the alphabet $\\{0,1\\}$ (bits). The
length of a codeword $b$, denoted by $length(b)$, is the number of bits in the
string $b$. We denote the $i$-th bit of $b$ by $b(i)$. A codeword $b_{1}$ is
said to be a prefix of a codeword $b_{2}$ if $length(b_{1})\leq length(b_{2})$
and $b_{1}(i)=b_{2}(i)$ for all $i\in\\{1,\ldots,length(b_{1})\\}$. A binary
code is a set of binary codewords. A binary code $B$ is called prefix-free if
no codeword in $B$ is a prefix of another codeword in $B$.
The Kraft’s Inequality [14] gives a necessary and sufficient condition for the
existence of a prefix-free code for a given set of codeword lengths.
###### Theorem 10 (Kraft 1949 [14]).
For every prefix-free binary code $B=\\{b_{1},\ldots,b_{n}\\}$,
$\sum_{i=1}^{n}{2^{-l_{i}}}\leq 1$
where $l_{i}=length(b_{i})$, and conversely, for any sequence of lengths
$l_{1},\ldots,l_{n}$ satisfying the above inequality, there exists a prefix-
free binary code $B=\\{b_{1},\ldots,b_{n}\\}$, with
$length(b_{i})=l_{i},\,i=1,\ldots,n$.
###### Observation 11.
Given any sequence of lengths $l_{1}\leq\cdots\leq l_{n}$ satisfying the Kraft
Inequality, we can construct a prefix-free binary code
$B=\\{b_{1},\ldots,b_{n}\\}$, with $length(b_{i})=l_{i},\,i=1,\ldots,n$ in
time $O\big{(}\sum_{i=1}^{n}{l_{i}}\big{)}$. Further, we can ensure that every
bit in $b_{1}$ is $0$.
###### Proof.
A binary tree is a rooted tree in which every node has at most two child
nodes. A node with only one child node is said to be unsaturated. The level of
a node is its distance from the root. We assume that every edge from a parent
to its first (second) child, if it exists, is labelled $0$ ($1$). We can
represent a prefix-free binary code by a binary tree such that (i) every
codeword $b_{i}$ corresponds to a leaf $t_{i}$ of the binary tree at level
$length(b_{i})$ and (ii) the labels on the unique path from the root to a leaf
will be the codeword associated with that leaf [8]. We construct a prefix-free
binary code with the given length sequence by constructing the corresponding
binary tree as explained below.
Create the root, and for every new node created, create its first child till
we hit a node $t_{1}$ at depth $l_{1}$ for the first time. Declare $t_{1}$ as
a leaf. Once we have created a leaf $t_{i}$, $i<n$, we proceed to create the
next leaf as follows. Backtrack from $t_{i}$ along the tree created so far
towards the root till we hit the first unsaturated node. Create its second
child. If the second child is at level $l_{i+1}$, then declare it as the leaf
$t_{i+1}$. Else, recursively create first child till we create a node at level
$l_{i+1}$ and declare it as leaf $t_{i+1}$. Terminate this process once we
create the leaf $t_{n}$.
The process will continue till we create all the $n$ leaves. Otherwise, it has
to be the case that every internal node in the tree got saturated by the time
we created some leaf $t_{i}$, $i<n$. If we have a binary tree $T$ with every
internal node saturated, it is easy to see by an inductive argument that
$\sum_{t\in L}2^{-d_{t}}=1$, where $L$ is the set of leaves of $T$ and $d_{t}$
denotes the level of leaf $t$. Hence
$\sum_{j=1}^{n}2^{-l_{j}}>\sum_{j=1}^{i}2^{-l_{j}}=1$, contradicting the
hypothesis that the lengths $l_{1},\ldots,l_{n}$ satisfy the Kraft Inequality.
It follows from the construction that every bit of $b_{1}$ is $0$. Since every
edge in the tree constructed corresponds to a bit in at least one of the
codewords returned, the total number of edges in the tree constructed is at
most $\sum_{i=1}^{n}{l_{i}}$. Since each edge of the tree is traversed at most
twice, the construction will be completed in time
$O\big{(}\sum_{i=1}^{n}{l_{i}}\big{)}$. ∎
Now we give a necessary and sufficient condition for $2$-rainbow-colourability
of a threshold graph.
Algorithm 2 ColourThresholdGraph-Case1
0: $G([n],E)$, a connected threshold graph, with $d_{1}\geq\cdots\geq d_{n}$
and $\sum_{i=k}^{n}{2^{-d_{i}}}\leq 1$, where $d_{i}$ is the degree of vertex
$i$ and $k=\min\\{i:1\leq i\leq n,\,d_{i}\leq i-1\\}$.
0: A rainbow colouring $C_{G}:E(G)\rightarrow\\{0,1\\}$ of $G$.
1: $I=\\{k,\ldots,n\\}$ $/\negthickspace/$ $I$ is a maximal independent set in
$G$
2: Let $\mathcal{B}=\\{b_{k},\ldots,b_{n}\\}$ be a prefix-free code with
$length(b_{i})=d_{i}$ (constructed as mentioned in Observation 11)
3: for $i\in I$ do
4: $C_{G}(\\{i,j\\})=b_{i}(j),\,\forall j\in\\{1,\ldots,d_{i}\\}$
5: end for
6: for $i\in V(G)\setminus I$ do $/\negthickspace/$ $i<k$
7: $C_{G}(\\{i,j\\})=b_{k}(j),\,\forall j\in\\{1,\ldots,i-1\\}$
$/\negthickspace/$ Note that $length(b_{k})=d_{k}=k-1$
8: end for
9: return $C_{G}$
###### Theorem 12.
For every connected threshold graph $G([n],E)$ with $d_{1}\geq\cdots\geq
d_{n}$, $rc(G)\leq 2$ if and only if
$\sum_{i=k}^{n}2^{-d_{i}}\leq 1,$ (9)
where $d_{i}$ is the degree of vertex $i$ and $k=\min\\{i:1\leq i\leq
n,\,d_{i}\leq i-1\\}$. Further, if $G$ satisfies Inequality (9), then
Algorithm 2 (ColourThresholdGraph-Case1) gives an optimal rainbow colouring of
$G$ in $O(m)$ time.
###### Proof.
Note that $I:=\\{k,\ldots,n\\}$ is a maximal independent set in $G$
(Observation 9b) and that the summation on the left hand side of Inequality
(9) is over all the vertices in $I$. Hence $C:=\\{1,\ldots,k-1\\}$ is a clique
in $G$.
First we show that if $rc(G)\leq 2$, then the inequality is satisfied. Let
$C_{G}:E(G)\rightarrow\\{0,1\\}$ be a rainbow colouring of $G$. We can
associate a codeword with each vertex $i\in I$ by reading the colours assigned
by $C_{G}$ to edges $\\{i,c\\},c=1,\ldots,d_{i}$. Since every pair $i,j\in
I,d_{i}\leq d_{j}$ are non-adjacent, they need a $2$-length rainbow path
between them through a common neighbour $c\in\\{1,\ldots,d_{i}\\}$
(Observation 9c). This ensures that the codewords corresponding to $i$ and $j$
are complementary in at least one bit position. Hence the binary code formed
by codewords corresponding to all the vertices in $I$ form a prefix-free code.
Hence the inequality is satisfied (by Theorem 10).
Conversely, if the inequality is satisfied, then Algorithm 2 gives a colouring
$C_{G}$ of $E(G)$ using at most $2$ colours. We show that $C_{G}$ is indeed a
rainbow colouring of $G$. Consider any two non-adjacent vertices $i,j\in
V(G),\,i<j$. Since they are non-adjacent, either both of them are in $I$ or
otherwise $j$ is in $I$ and $i$ is from the clique $C$ such that $i>d_{j}$
(Since $N(j)=\\{1,\ldots,d_{j}\\}$). In the former case, $length(b_{i})\geq
length(b_{j})$ and there exists a $v\in\\{1,\ldots,d_{j}\leq d_{i}\\}$ such
that $b_{j}(v)\neq b_{i}(v)$ since $b_{j}$ is not a prefix of $b_{i}$ (They
both belong to a prefix-free code $B$). Hence $i\mbox{--}v\mbox{--}j$ is a
rainbow path. Similarly in the latter case, $length(b_{k})\geq length(b_{j})$
and there exists a $v\in\\{1,\ldots,d_{j}<i\\}$ such that $b_{j}(v)\neq
b_{k}(v)$ since $b_{j}$ is not a prefix of $b_{k}$. Hence
$C_{G}(\\{v,j\\})\neq C_{G}(\\{v,i\\})$ and $i\mbox{--}v\mbox{--}j$ is a
rainbow path. Hence $C_{G}$ is a rainbow colouring of $G$.
If $G$ is not a clique, then $rc(G)\geq 2$ (Observation 1), and hence the
above rainbow colouring is optimal. If $G$ is a clique then $k=n$ and
$|B|=|I|=1$. So the single codeword $b_{n}$ constructed as mentioned in
Observation 11 has all the bits $0$. So every edge of $G$ is coloured using
the single colour $0$, which is optimal for $G$.
Since $\sum_{i=1}^{n}{l_{i}}=\sum_{i=1}^{n}{d_{i}}=2m$, the prefix-free code
$B$ can be constructed in $O(m)$ time (Observation 11). Moreover, Algorithm 2
visits each edge only once. Hence the total time complexity is $O(m)$. ∎
Now we consider the case of threshold graphs which violate Inequality (9).
Algorithm 3 ColourThresholdGraph-Case2
0: $G([n],E)$, a connected threshold graph, with $d_{1}\geq\cdots\geq d_{n}$,
where $d_{i}$ is the degree of vertex $i$.
0: A rainbow colouring $C_{G}:E(G)\rightarrow\\{0,\ldots,\max\\{p,3\\}-1\\}$
of $G$, where $p$ is the number of pendant vertices in $G$.
1: $P\leftarrow\\{i\in V(G):d_{i}=1\\}$, $p\leftarrow|P|$ $/\negthickspace/$
$P$ is the set of pendant vertices in $G$
2: $C_{G}(\\{p_{i},1\\})\leftarrow i-1$ for each pendant vertex
$p_{1},\ldots,p_{p}$
3: if $p=n-1$ then
4: return $C_{G}$ $/\negthickspace/$ $G$ is a star
5: end if
6: $C_{G}(\\{1,2\\})=0$
7: for $i=3$ to $i=n-p$ do
8: $C_{G}(\\{i,1\\})=1$
9: $C_{G}(\\{i,2\\})=2$ $/\negthickspace/$ Every $v\in\\{3,\ldots,n-p\\}$ is
adjacent to vertices $1$ and $2$.
10: end for
11: $C_{G}(e)=0$ for each edge $e$ of $G$ not coloured so far.
12: return $C_{G}$
###### Theorem 13.
For every connected threshold graph $G$ which does not satisfy Inequality (9),
$rc(G)=\max\\{p,3\\},$
where $p$ is the number of pendant vertices in $G$.
Further, Algorithm 3 (ColourThresholdGraph-Case2) gives an optimal rainbow
colouring of $G$ in $O(m)$ time
###### Proof.
It is easy to check, by considering all pairs of non-adjacent vertices, that
Algorithm 3 indeed produces a rainbow colouring of $G$. It is also evident
that it uses at most $\max\\{p,3\\}$ colours. By Observation 2 and Theorem 12
, it follows hat $rc(G)\geq\max\\{p,3\\}$. Hence $rc(G)=\max\\{p,3\\}$ and
hence the rainbow colouring produced by Algorithm 3 is optimal. Further, since
Algorithm 3 visits each edge only once, its time complexity is $O(m)$. ∎
Algorithm 4 ColourThresholdGraph
0: $G([n],E)$, a connected threshold graph with $d_{1}\geq\cdots\geq d_{n}$,
where $d_{i}$ is the degree of vertex $i$.
0: An optimal rainbow colouring $C_{G}:E(G)\rightarrow\\{0,\ldots,rc(G)-1\\}$
of $G$.
1: $k=\min\\{i:1\leq i\leq n,\,d_{i}\leq i-1\\}$
2: if $\sum_{i=k}^{n}2^{-d_{i}}\leq 1$ then
3: $C_{G}=$ ColourThresholdGraph-Case1($G$)
4: else
5: $C_{G}=$ ColourThresholdGraph-Case2($G$)
6: end if
7: return $C_{G}$
Combining Theorem 12 and Theorem 13, we get a complete characterisation for
threshold graphs whose rainbow connection number is $k$, based on its degree
sequence alone. Further we can find the optimally rainbow colour every
threshold graph in linear-time.
###### Corollary 14.
Let $G([n],E)$, be a connected threshold graph with $d_{1}\geq\cdots\geq
d_{n}$, where $d_{i}$ is the degree of vertex $i$. Then,
$rc(G)=\begin{cases}1,&\textnormal{if $G$ is a clique}\\\ 2,&\textnormal{if
$G$ is not a clique and }\sum_{i=k}^{n}2^{-d_{i}}\leq 1\\\
\max\\{3,p\\},&\textnormal{otherwise,}\end{cases}$ (10)
where $k=\min\\{i:1\leq i\leq n,\,d_{i}\leq i-1\\}$ and $p=|\\{i:1\leq i\leq
n,\,d_{i}=1\\}|$.
Further, Algorithm 4 (ColourThresholdGraph) gives an optimal rainbow colouring
of $G$ in $O(m)$ time.
## References
* [1] Prabhanjan Ananth, Meghana Nasre, and Kanthi K. Sarpatwar. Rainbow Connectivity: Hardness and Tractability. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011), volume 13, pages 241–251, 2011\.
* [2] M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy. Rainbow connection number and radius. Arxiv preprint arXiv:1011.0620v1, 2010.
* [3] M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy. Rainbow connection number of graph power and graph products. Arxiv preprint arXiv:1104.4190, 2011.
* [4] Yair Caro, Arie Lev, Yehuda Roditty, Zsolt Tuza, and Raphael Yuster. On rainbow connection. Electron. J. Combin., 15(1):Research paper 57, 13, 2008.
* [5] Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Raphael Yuster. Hardness and algorithms for rainbow connection. J. Comb. Optim., 21(3):330–347, 2011.
* [6] G. Chartrand and P. Zhang. Chromatic Graph Theory. Chapman & Hall, 2008.
* [7] Gary Chartrand, Garry L. Johns, Kathleen A. McKeon, and Ping Zhang. Rainbow connection in graphs. Math. Bohem., 133(1):85–98, 2008.
* [8] Thomas M. Cover and Joy A. Thomas. Data Compression, pages 103–158. John Wiley & Sons, Inc., 2005.
* [9] A. Frieze and C.E. Tsourakakis. Rainbow connectivity of $g(n,p)$ at the connectivity threshold. Arxiv preprint arXiv:1201.4603, 2012.
* [10] Peter L. Hammer and Bruno Simeone. The splittance of a graph. Combinatorica, 1(3):275–284, 1981.
* [11] Jing He and Hongyu Liang. On rainbow k-connectivity of random graphs. Arxiv preprint arXiv:1012.1942v1 [math.CO], 2010.
* [12] Ian Holyer. The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4):718–720, 1981.
* [13] S. Khot. Hardness results for coloring 3-colorable 3-uniform hypergraphs. In Foundations of Computer Science, 2002. Proceedings. The 43rd Annual IEEE Symposium on, pages 23–32. IEEE, 2002.
* [14] L.G. Kraft. A device for quanitizing, grouping and coding amplitude modulated pulses. Master’s thesis, Electrical Engineering Department, Massachusetts Institute of Technology, 1949.
* [15] Michael Krivelevich and Raphael Yuster. The rainbow connection of a graph is (at most) reciprocal to its minimum degree. J. Graph Theory, 63(3):185–191, 2010.
* [16] S. Li and X. Li. Note on the complexity of determining the rainbow connectedness for bipartite graphs. Arxiv preprint arXiv:1109.5534, 2011.
* [17] X. Li and Y. Sun. Rainbow Connections of Graphs. Springerbriefs in Mathematics. Springer, 2012.
* [18] Xueliang Li and Yuefang Sun. Rainbow connections of graphs – a survey. Arxiv preprint arXiv:1101.5747v2 [math.CO], 2011.
* [19] Xueliang Li and Yuefang Sun. Upper bounds for the rainbow connection numbers of line graphs. Graphs and Combinatorics, pages 1–13, 2011. 10.1007/s00373-011-1034-1.
* [20] J. Misra and David Gries. A constructive proof of vizing’s theorem. Information Processing Letters, 41(3):131 – 133, 1992.
* [21] Ingo Schiermeyer. Rainbow connection in graphs with minimum degree three. In Combinatorial Algorithms, volume 5874 of Lecture Notes in Comput. Sci., pages 432–437. Springer, Berlin, 2009.
* [22] Harold. Seward, H. Information sorting in the application of electronic digital computers to business operations. Master’s thesis, Digital Computer Laboratory, Massachusetts Institute of Technology, 1954.
* [23] Yilun Shang. A sharp threshold for rainbow connection of random bipartite graphs. Int. J. Appl. Math., 24(1):149–153, 2011.
* [24] L. Sunil Chandran, Anita Das, Deepak Rajendraprasad, and Nithin M. Varma. Rainbow connection number and connected dominating sets. Journal of Graph Theory, 2011.
* [25] A. Wigderson. The complexity of graph connectivity. Mathematical Foundations of Computer Science 1992, pages 112–132, 1992.
|
arxiv-papers
| 2012-05-08T12:25:19 |
2024-09-04T02:49:30.704438
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Sunil Chandran and Deepak Rajendraprasad",
"submitter": "Deepak Rajendraprasad",
"url": "https://arxiv.org/abs/1205.1670"
}
|
1205.1695
|
# $N$-body simulations with a cosmic vector for dark energy
Edoardo Carlesi,1 Alexander Knebe,1 Gustavo Yepes,1 Stefan Gottlöber,3 Jose
Beltrán Jiménez,4,5 Antonio L. Maroto,2
1Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049,
Cantoblanco, Madrid, Spain
2Departamento de Física Teórica, Universidad Complutense de Madrid, 28040,
Madrid, Spain
3Leibniz Institut für Astrophysik, An der Sternwarte 16, 14482, Potsdam,
Germany
4Institute de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, 24 quai E. Ansermet, 1211 Genève, Switzerland
5Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway
E-mail: edoardo.carlesi@uam.es
(Accepted XXXX . Received XXXX; in original form XXXX)
###### Abstract
We present the results of a series of cosmological $N$-body simulations of a
Vector Dark Energy (VDE) model, performed using a suitably modified version of
the publicly available GADGET-2 code. The setups of our simulations were
calibrated pursuing a twofold aim: 1) to analyze the large scale distribution
of massive objects and 2) to determine the properties of halo structure in
this different framework. We observe that structure formation is enhanced in
VDE, since the mass function at high redshift is boosted up to a factor of ten
with respect to $\Lambda$CDM, possibly alleviating tensions with the
observations of massive clusters at high redshifts and early reionization
epoch. Significant differences can also be found for the value of the growth
factor, that in VDE shows a completely different behaviour, and in the
distribution of voids, which in this cosmology are on average smaller and less
abundant. We further studied the structure of dark matter haloes more massive
than $5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$, finding that no substantial
difference emerges when comparing spin parameter, shape, triaxiality and
profiles of structures evolved under different cosmological pictures.
Nevertheless, minor differences can be found in the concentration-mass
relation and the two point correlation function; both showing different
amplitudes and steeper slopes. Using an additional series of simulations of a
$\Lambda$CDM scenario with the same $\Omega_{M}$ and $\sigma_{8}$ used for the
VDE cosmology, we have been able to establish wether the modifications induced
in the new cosmological picture were due to the particular nature of the
dynamical dark energy used or a straightforward consequence of the
cosmological parameters used. On large scales, the dynamical effects of the
cosmic vector can be seen in the peculiar evolution of the cluster number
density function with redshift, in the distribution of voids and on the
characteristic form of the growth index $\gamma(z)$. On smaller scales,
internal properties of haloes are almost unaffected by the change of
cosmology, since no statistical difference can be observed in the
characteristics of halo profiles, spin parameters, shapes and triaxialities.
Only halo masses and concentrations show a substantial increase, which can
however be attributed to the change in the cosmological parameters.
###### keywords:
methods:$N$-body simulations – galaxies: haloes – cosmology: theory – dark
matter
††pagerange: $N$-body simulations with a cosmic vector for dark
energy–References††pubyear: 2011
## 1 Introduction
During the last twelve years, a large amount of cosmological high precision
data on Supernovae Ia (see Riess et al., 1998; Perlmutter et al., 1999; Guy et
al., 2010) cosmic microwave background anisotropies (Larson et al., 2011;
Sherwin et al., 2011), weak lensing (Huterer, 2010), baryon acoustic
oscillations (Beutler et al., 2011) and large scale structure surveys
(Abazajian et al., 2009) has provided evidence that the Universe we live in is
of a flat geometry and undergoing an accelerated expansion. These observations
motivate our belief in the existence of an ubiquitous fluid called dark energy
(DE) that by the exertion of a negative pressure, counters and eventually
overcomes the gravitational attraction that would otherwise dominate the
evolution of our Universe. The simplest explanation to the nature of this
fluid is found in the standard model of cosmology $\Lambda$CDM, where the role
of the DE is played by a cosmological constant $\Lambda$ obeying the equation
of state $p_{\Lambda}=-\rho_{\Lambda}$. Although perfectly consistent with all
the aforementioned observations, $\Lambda$CDM still lacks of appeal from a
purely theoretical point of view. In fact, if we belive the cosmological
constant to be the zero point energy of some fundamental quantum field, its
introduction in the Friedmann equations requires a fine-tuning of several tens
of orders of magnitude (depending on the energy scale we choose to be
fundamental in our theory) spoiling the naturalness of the whole $\Lambda$CDM
picture.
Another issue we encounter when dealing with the standard cosmological model
is the so called _coincidence problem_ , that is, the difficulty to explain in
a natural way the fact that today’s matter and dark energy densities have a
comparable value although they evolved in a completely different manner
throughout most of the history of the universe.
In an attempt to overcome these two difficulties of $\Lambda$CDM, Jimenez &
Maroto (2009) introduced the Vector Dark Energy (VDE) model, where a cosmic
vector field plays the role of a dynamical dark energy component, replacing
the cosmological constant $\Lambda$. Besides being compatible with supernovae
observations and CMB precision measurements, this scenario has the same number
of free parameters as $\Lambda$CDM. Moreover, the initial value of the vector
field (which is of the order of $10^{-4}M_{p}$111$M_{p}$ being the Planck
mass., a scale that could arise naturally in inflation) and its global
dynamics ensure the model to overcome the standard model’s naturarlness
problems. In the present work we study the impact of this VDE model on
structure formation and evolution by means of a series of cosmological
$N$-body simulations, analyzing the effects of this alternative cosmology in
the deeply non-linear regime and highlighting its imprints on cosmic
structures, in particular, emphasizing the differences emerging with respect
to the standard model $\Lambda$CDM.
The paper is organized as follows. In Section 2 we briefly introduce the VDE
model, discussing its most important mathematical and physical
characteristics. In Section 3 we describe the setup as well as the
modifications to the code and the initial conditions necessary to run the
$N$-body simulation. In the two Sections 4 and 5 we will present a detailed
analysis of the results, focusing on the main differences of the VDE models to
the standard $\Lambda$CDM cosmology, first analyzing the large-scale structure
and then (cross-)comparing properties of dark matter haloes. A short summary
of the results obtained and a discussion on their implications is then
presented in section Section 6.
## 2 The Model
$\begin{array}[]{cc}\includegraphics[width=227.62204pt]{energy_density}\includegraphics[width=233.3125pt]{a0}\end{array}$
Figure 1: Left plot: Evolution of the energy densities. Dashed (red) for
radiation, dotted (green) for matter and solid (blue) for vector dark energy.
We also show for comparison the cosmological constant energy density in
dashed-dotted line. We see the scaling behaviour of the cosmic vector in the
early universe and the rapid growth of its energy density contribution at late
times when approaching the final singularity. Right plot: Evolution of the
temporal component of the vector field where we see that it takes a constant
value at very high redshifts so that the cosmological evolutions is
insensitive to the precise redshift at which we set the initial value of the
cosmic vector.
In this section we will provide the basic mathematical and physical
description of the VDE model. For more details and an in-depth discussion on
the results obtained and their derivation we refer to Beltrán Jiménez & Maroto
(2008). The action of the proposed vector dark energy model can be written as:
$\displaystyle S=\int d^{4}x\sqrt{-g}\left[-\frac{R}{16\pi
G}-\frac{1}{4}\textbf{\emph{F}}_{\mu\nu}\textbf{\emph{F}}^{\mu\nu}\right.$
$\displaystyle\left.-\frac{1}{2}\left(\nabla_{\mu}\textbf{\emph{A}}^{\mu}\right)^{2}+\textbf{\emph{R}}_{\mu\nu}\textbf{\emph{A}}^{\mu}\textbf{\emph{A}}^{\nu}\right].$
(1)
where $\textbf{\emph{R}}_{\mu\nu}$ is the Ricci tensor,
$R=\textbf{\emph{g}}^{\mu\nu}\textbf{\emph{R}}_{\mu\nu}$ the scalar curvature
and
$\textbf{\emph{F}}_{\mu\nu}=\partial_{\mu}\textbf{\emph{A}}_{\nu}-\partial_{\nu}\textbf{\emph{A}}_{\mu}$.
This action can be interpreted as a Maxwell term for a vector field
supplemented with a gauge-fixing term and an effective mass provided by the
Ricci tensor. It is interesting to note that the vector sector has no free
parameters nor potential terms, being $G$ the only dimensional constant of the
theory. This is one of the main differences of this model with respect to
those based on scalar fields, which need the presence of potential terms to be
able to lead to late-time accelerated expansion.
The classical equations of motion derived from the action (1) are the Einstein
and vector field equations given by:
$\displaystyle\textbf{\emph{R}}_{\mu\nu}-\frac{1}{2}R\textbf{\emph{g}}_{\mu\nu}$
$\displaystyle=$ $\displaystyle 8\pi
G(\textbf{\emph{T}}_{\mu\nu}+\textbf{\emph{T}}_{\mu\nu}^{A}),$ (2)
$\displaystyle\Box\textbf{\emph{A}}_{\mu}+\textbf{\emph{R}}_{\mu\nu}\textbf{\emph{A}}^{\nu}$
$\displaystyle=$ $\displaystyle 0,$ (3)
where $\textbf{\emph{T}}_{\mu\nu}$ is the conserved energy-momentum tensor for
matter and radiation (and/or other possible components present in the
Universe) and $\textbf{\emph{T}}_{\mu\nu}^{\textbf{\emph{A}}}$ is the energy-
momentum tensor coming from the vector field sector (and that is also
covariantly conserved). In the following, we shall solve the equations of the
vector field during the radiation and matter eras, in which the contribution
of dark energy is supposed to be negligible. In those epochs, the geometry of
the universe is well-described by the flat Friedmann-Lemaître-Robertson-Walker
metric:
$ds^{2}=dt^{2}-a(t)^{2}d\textbf{\emph{x}}^{2}.$ (4)
For the homogeneous vector field we shall assume, without lack of generality,
the form $A_{\mu}=(A_{0}(t),0,0,A_{z}(t))$ so that the corresponding equations
read:
$\displaystyle\ddot{A}_{0}+3H\dot{A}_{0}-3\left(2H^{2}+\dot{H}\right)A_{0}=0,$
(5)
$\displaystyle\ddot{A}_{z}+H\dot{A}_{z}-2\left(\dot{H}+3H^{2}\right)A_{z}=0,$
(6)
where $H=\dot{a}/a$. These equations can be easily solved for a power law
expansion with $H=p/t$, in which case we obtain the following solutions:
$\displaystyle A_{0}(t)=A_{0}^{+}t^{\alpha_{+}}+A_{0}^{-}t^{\alpha_{-}},$ (7)
$\displaystyle A_{z}(t)=A_{z}^{+}t^{2p}+A_{z}^{-}t^{1-3p}.$ (8)
with $\alpha_{\pm}=\frac{1}{2}(1-3p\pm\sqrt{33p^{2}-18p+1})$ and $A_{0}^{\pm}$
and $A_{z}^{\pm}$ constants of integration. Thus, in the radiation dominated
epoch ($p=1/2$) we have the growing modes $A_{0}=$constant and $A_{z}\propto
t$, whereas for the matter dominated epoch ($p=2/3$) we have $A_{0}\propto
t^{(-3+\sqrt{33})/6}$ and $A_{z}\propto t^{4/3}$. Concerning the energy
densities, the corresponding expressions are given by:
$\displaystyle\rho_{A_{0}}=\frac{3}{2}H^{2}A_{0}^{2}+3HA_{0}\dot{A}_{0}-\frac{1}{2}\dot{A}_{0}^{2},$
(9)
$\displaystyle\rho_{A_{z}}=\frac{1}{2a^{2}}\left(4H^{2}A_{z}^{2}-4HA_{z}\dot{A}_{z}+\dot{A}_{z}^{2}\right).$
(10)
At this point, it is interesting to note that when we insert the full solution
for $A_{z}$ given in (8) in its corresponding energy density, we obtain:
$\displaystyle\rho_{A_{z}}=\frac{\left(A_{z}^{-}\right)^{2}}{2a^{8}}\left(25p^{2}-10p+1\right).$
(11)
so that the mode $A_{z}^{+}$ does not contribute to the energy density. That
way, even though $A_{z}$ grows with respect to $A_{0}$, the corresponding
physical quantity, i.e. its energy density, decays with respect to that of the
temporal component. It is easy to check that the ratio
$\rho_{A_{z}}/\rho_{A_{0}}$ decays as $a^{-4}$ in the radiation era and as
$a^{-6.37}$ in the matter era so that the energy density of the vector field
becomes dominated by the contribution of the temporal component. That
justifies to neglect the spatial components and deal uniquely with the
temporal one.
On the other hand, the potential large scale anisotropy generated by the
presence of spatial components of the vector field is determined by the
relative difference of pressures in different directions $p_{\parallel}$ and
$p_{\perp}$, that is given by:
$\displaystyle
p_{\parallel}-p_{\perp}=\frac{3}{a^{2}}\left(4H^{2}A_{z}^{2}-4HA_{z}\dot{A}_{z}+\dot{A}_{z}^{2}\right).$
(12)
This expression happens to be proportional to $\rho_{A_{z}}$ so that we have
that $(p_{\parallel}-p_{\perp})/\rho_{A}$ will decay as the universe expands
in the same manner as $\rho_{A_{z}}$ and the large scale isotropy of the
universe suggested by CMB observations is not spoiled. Hence, in the following
we shall neglect the spatial components of the vector field and we shall
uniquely consider the temporal one, since it gives the dominant contribution
to the energy-momentum tensor of the vector field. However, we should
emphasize here that this does not result in effectively having a scalar field.
As commented before, for a minimally-coupled scalar field, one needs to
introduce a certain potential (that will depend on some dimensional
parameters) to have accelerated expansion, whereas in the VDE model, we get
accelerated solutions with only kinetic terms and without introducing any new
dimensionful parameter.
The energy density of the vector field is given by:
$\displaystyle\rho_{A}=\rho_{A0}(1+z)^{\kappa},$ (13)
with $\kappa=4$ in the radiation era and $\kappa=(9-\sqrt{33})/2\simeq-1.63$
in the matter era. We can also calculate the effective equation of state for
dark energy as:
$\displaystyle
w_{DE}=\frac{p_{A}}{\rho_{A}}=\frac{-3\left(\frac{5}{2}H^{2}+\frac{4}{3}\dot{H}\right)A_{0}^{2}+HA_{0}\dot{A}_{0}-\frac{3}{2}\dot{A}_{0}^{2}}{\frac{3}{2}H^{2}A_{0}^{2}+3HA_{0}\dot{A}_{0}-\frac{1}{2}\dot{A}_{0}^{2}}.$
(14)
Again, using the approximate solutions in (7), we obtain;
$\displaystyle
w_{DE}=\left\\{\begin{array}[]{l}\frac{1}{3}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{radiation
era}\\\ \frac{3\sqrt{33}-13}{\sqrt{33}-15}\simeq-0.457\;\;\;\;\;\mbox{matter
era}\end{array}\right.$ (17)
From the evolution of the energy density of the vector field we see that it
scales as radiation at early times, so that $\rho_{A}/\rho_{R}=$ const.
However, when the Universe enters its matter era, $\rho_{A}$ starts growing
relative to $\rho_{M}$ eventually overcoming it at some point, in which the
dark energy vector field would become the dominant component. From that point
on, we cannot obtain analytic solutions to the field equations and we need to
numerically solve the corresponding equations. In Fig. 1 we show such a
numerical solution to the exact equations, which confirms our analytical
estimates in the radiation and matter eras. Notice that, since $A_{0}$ is
constant during the radiation era, the solutions do not depend on the precise
time at which we specify the initial conditions as long as we set them well
inside the radiation epoch. Thus, once the present value of the Hubble
parameter $H_{0}$ and the constant $A_{0}$ during radiation (which indirectly
fixes the total matter density $\Omega_{M}$) are specified, the model is
completely determined. In other words, this model contains the same number of
parameters as $\Lambda$CDM, i.e. the minimum number of parameters of any
cosmological model with dark energy.
The vector dark energy model does not only have the minimum required number of
parameters, but it also allows to alleviate the so-called naturalness or
coincidence problems that most dark energy models have. This is so because the
required value for the constant value that the vector field takes in the early
universe happens to be $\sim 10^{-4}M_{p}$. This value, in addition to be
relatively close to the Planck scale, could naturally arise from quantum
fluctuations during inflation, for instance. On the other hand, the fact that
the energy density of the vector field scales as radiation in the early
universe also goes in the right direction of alleviating the aforementioned
problems because the fraction of dark energy during that period remains
constant. Moreover, said fraction is
$\Omega_{A}^{early}\equiv\rho_{A}/\rho_{R}\simeq 10^{-6}$, which, again, is in
agreement with the usual magnitude of the quantum fluctuations produced during
inflation.
After dark energy starts dominating, the equation of state abruptly falls
towards $w_{DE}\rightarrow-\infty$ as the Universe approaches a finite time
$t_{end}$. As shown in Fig. 2, during the cosmological evolution the equation
of state crosses the so-called phantom divide line, so that we have
$w_{DE}(z=0)<-1$. The final stage of the universe in this model is a
singularity usually called Type III or Big Freeze, in which the scale factor
remains finite, but the Hubble expansion rate, the energy density and the
pressure diverge. This is a distinct feature of the VDE model as compared to
quintessence fields for which the equation of state is restricted to be $>-1$
so that no crossing of the phantom divide line is possible. In fact, for a
dark energy model based on scalar fields, one needs either non-standard
kinetic terms involving higher derivative terms in the action or the presence
of several interacting scalar fields to achieve a transition from $w>-1$ to a
phantom behaviour ($w<-1$). In either case, non-linear derivative interactions
or multiple scalar field scenarios, additional degrees of freedom are
introduced, whereas the VDE model is able to obtained the mentioned transition
with only the degree of freedom given by the temporal component of the vector
field.
Notice that in the VDE model the present value of the equation of state
parameter $w_{0}=-3.53$ is radically different from that of a cosmological
constant (cf. Fig. 1, where the redshift evolution of $w(z)$ is shown in the
range of our simulations). The values of other cosmological parameters also
differ importantly from those of $\Lambda$CDM (see Table 1). Despite this
fact, VDE is able to simultaneously fit supernovae and CMB data with
comparable goodness to $\Lambda$CDM (Beltrán Jiménez & Maroto (2008), Beltrán
Jiménez et al. (2009)). This might seem to be surprising if we notice that the
present equation of state for the VDE model is $w_{0}=-3.53$, which is far
from the usual constraints on this parameter obtained from cosmological
observations. Such constraints are usually obtained by assuming a certain
parametrization for the time variation of the dark energy equation of state.
However, the different used parameterizations are normally such that
$\Lambda$CDM is included in the parameter space. If we look at Fig. 2, we can
see that the evolution of the equation of state for the VDE model crucially
differs from those of $\Lambda$CDM or quintessence models and, indeed, it
cannot be properly described by the most popular parameterizations. This means
that we cannot directly apply the existing constraints to the VDE model, but a
direct comparison of its predictions to observations is required.
As a final remark, in the simulations we will not include inhomogeneous
perturbations of the vector field, but only the effects of having a different
background expansion will be considered.
In Fig. 3, we show the matter power spectrum for both $\Lambda$CDM and the VDE
model. The differences can be ascribed to the fact of having different
cosmological parameters that change the normalization and the matter-radiation
equality scale $k_{eq}$, which are the only two differences observed. Notice
that the transfer function is the same in both cases, since the slopes before
and after the $k_{eq}$ are the same, so that we do not expect strong effects
at early times which could affect the evolution of density parameters.
In particular, for CMB222We use the binned data of WMAP7, the $\chi^{2}$ for
the best fit parameters for $\Lambda$CDM is $48.3$, wheres for the VDE model
we obtain $\chi^{2}=51.8$ for the parameters used to run our simulations.
Thus, even though the equation of state evolution is as the one shown in Fig.
2, the VDE model provides good fits to observations.
Figure 2: Evolution of dark energy equation of state where we can see the
crossing of the phantom divide line and its evolution towards $-\infty$ as the
final singularity is approached.
## 3 The $N$-Body Simulations
In this section we will explain the (numerical) methods used in this work,
with a particular emphasis on the necessary modifications of the standard
$N$-body and halo finding algorithms, also describing the procedures followed
to test their accuracy and reliability.
### 3.1 Set-up
The $N$-body simulations presented in this work have been carried out using a
suitably modified version of the Tree-PM code GADGET-2 (Springel, 2005). It
has been also necessary to generate a particular set of initial conditions to
consistently account for the VDE-induced modifications to the standard
paradigm. In Table 1 we show the most relevant cosmological parameters used in
the different simulations. For the VDE model, we have used the value of
$\Omega_{M}$ provided by the best fit to SNIa data, whereas the remaining
cosmological parameters have been obtained by a fit of the model to the WMAP7
dataset. For $\Lambda$CDM we used the Multidark Simulation (Prada et al.,
2011) cosmological parameters with a WMAP7 $\sigma_{8}$ normalization (Larson
et al., 2011).
In addition, we also simulated a so-called $\Lambda$CDM-vde model, which
implements the VDE values for the total matter density $\Omega_{M}$ and
fluctuation amplitude $\sigma_{8}$ in an otherwise standard $\Lambda$CDM
picture. Although ruled out by current cosmological constraints, this model
provides nonetheless an interesting case study that allows us to shed light on
the effects of these two cosmological parameters on structure formation in the
VDE model. In particular, we want to be able to determine the impact of the
different parameters on cosmological scales, with a particular emphasis on the
very large structures and the most massive clusters, where observations are
starting to clash with the predictions of the current standard model (see Jee
et al. (2009), Baldi & Pettorino (2011), Hoyle et al. (2011), Carlesi et al.
(2011) and Enqvist et al. (2011)). Therefore, we need to determine whether the
results derived from our VDE simulations can be solely attributed to its
extremely different values for the cosmological parameters or actually by the
presence of the cosmic vector field. In other words, we want to separate the
signatures of the _dynamics_ -driven effects from the _parameter_ -driven
ones, with a focus on large scale structures, where the imprints are stronger
and more clearly connected to the cosmological model. We chose to run a total
of eight $512^{3}$ particle simulations summarized in Table 1 and explained
below:
* •
two VDE simulations, i.e. a 500 $h^{-1}\,{\rm Mpc}$ and a 1 $h^{-1}$Gpc box,
* •
two $\Lambda$CDM simulations with the same box sizes and initial seeds as the
VDE runs above,
* •
two more VDE simulations with a different random seed, again one in a 500
$h^{-1}\,{\rm Mpc}$ and one in a 1 $h^{-1}$Gpc box (both serving as a check
for the influence of cosmic variance), and
* •
two $\Lambda$CDM-vde simulations, one again in a 500 $h^{-1}\,{\rm Mpc}$ and
one in a 1 $h^{-1}$Gpc box.
All runs were performed on 64 CPUs using the MareNostrum cluster at the
Barcelona Supercomputing Center. Most of the results we will discuss and
analyze here are based on the 500 $h^{-1}\,{\rm Mpc}$ simulations as they have
the better mass resolution. The 1 $h^{-1}$Gpc runs primarily serve as a
confirmation of the results and have already been discussed in Carlesi et al.
(2011), respectively.
Table 1: $N$-body settings and cosmological parameters used for the GADGET-2 simulations, the two 500$h^{-1}\,{\rm Mpc}$ and the two 1$h^{-1}$Gpc have the same initial random seed (in order to allow for a direct comparison of the halo properties) and starting redshift $z_{\rm start}=60$. The number of particles in each run was fixed at $512^{3}$. The box size $B$ is given in units of $h^{-1}\,{\rm Mpc}$ and the particle mass in $10^{11}$ $h^{-1}{\rm{M_{\odot}}}$. Simulation | $\Omega_{m}$ | $\Omega_{de}$ | $\sigma_{8}$ | h | $B$ | $m_{p}$
---|---|---|---|---|---|---
$2\times$VDE-0.5 | 0.388 | 0.612 | 0.83 | 0.62 | 500 | $1.00$
$2\times$VDE-1 | 0.388 | 0.612 | 0.83 | 0.62 | 1000 | $8.02$
$\Lambda$CDM-0.5 | 0.27 | 0.73 | 0.8 | 0.7 | 500 | $0.69$
$\Lambda$CDM-1 | 0.27 | 0.73 | 0.8 | 0.7 | 1000 | $5.55$
$\Lambda$CDM-0.5vde | 0.388 | 0.612 | 0.83 | 0.7 | 500 | $1.00$
$\Lambda$CDM-1vde | 0.388 | 0.612 | 0.83 | 0.7 | 1000 | $8.02$
### 3.2 Code Modifications
In the following paragraph we are going to describe the procedures followed to
implement the modifications needed in order to run our $N$-body simulations
consistently and reliably. This is in principle a non-trivial issue, since, as
described in Section 2, we need to incorporate a large number of different
features that affect both the code used for the simulations and the initial
conditions.
Figure 3: Linear matter overdensity power spectra at $z=0$ and $z=60$ for VDE,
$\Lambda$CDM and $\Lambda$CDM-vde plotted versus wavenumber $k$. Vertical
solid thick black lines refer to the $k$-space interval covered by the 500
$h^{-1}\,{\rm Mpc}$ simulations whereas the thin ones refer to the 1
$h^{-1}$Gpc one. All matter power spectra at $z=0$ have been normalized to the
$\sigma_{8}$ values shown in Table 1 and then rescaled to $z=60$ via the
linear growth factor. We notice that for $k<0.05$ h Mpc-1, $\Lambda$CDM and
$\Lambda$CDM-vde have more power than VDE, whereas on smaller scales the
opposite is true. We also note that due to the different value of $\sigma_{8}$
normalization the $\Lambda$CDM-vde $P(k)$ is slightly larger than the
$\Lambda$CDM one at $z=0$ while the different growth factor, which is larger
in the $\Lambda$CDM-vde cosmology, affect the setting of the initial
conditions, where the latter power spectrum lies below the former. Figure 4:
The ratio of the Hubble function $H(a)h^{-1}$ for VDE and $\Lambda$CDM-vde to
the standard $\Lambda$CDM one. At earlier times VDE undergoes a relatively
faster expansion compared to $\Lambda$CDM, whereas the opposite is true at
smaller $z$s. On the other hand, $\Lambda$CDM-vde cosmology is characterized
by a slower relative expansion throughout the whole history of the universe.
In particular, we have to handle with care three features that distinguish it
from $\Lambda$CDM, i.e.:
* •
the matter power spectrum $P(k,z)$ (shown in Fig. 3) and its normalization
$\sigma_{8}$,
* •
the expansion history $H(z)$ (see Fig. 4), and
* •
the linear growth factor $D^{+}(z)$ (cf. Fig. 5).
Whereas the first and last point affect the system’s initial conditions, the
second one enters directly into the $N$-body time integration, and has to be
taken into account by a modification of the simulation code.
#### 3.2.1 Initial conditions
To consistently generate the initial conditions for our simulation, first we
normalized the perturbation power spectrum depicted in Fig. 3 to the chosen
value for $\sigma_{8}$ at $z=0$. Therefore, we normalized VDE and
$\Lambda$CDM-vde inital conditions to $\sigma_{8}=0.83$ while for $\Lambda$CDM
we used the WMAP7 value $\sigma_{8}=0.8$. Using the respective linear growth
factors, we rescaled the $P(k)$ to the initial redshift $z=60$ where then the
particles’ initial velocities and positions were computed using the Zel’Dovich
(Zel’Dovich, 1970) approximation.
We emphasize here that the main goal of our analysis is to find and highlight
the main differences of the VDE picture with respect to the standard one:
therefore, the choice of these different normalization parameters has to be
understood as unavoidable as long as we want the models under investigation to
be WMAP7 viable ones. Needless to say, in this regard the $\Lambda$CDM-vde
cosmology must be considered only as a tool to disentangle parameter-driven
effects from the dynamical ones, not being a concurrent cosmological paradigm
we want to compare VDE to.
#### 3.2.2 Hubble expansion
As pointed out by Li & Barrow (2011), the expansion history of the universe
has a very deep impact on structure formation and in particular the results of
an $N$-body simulation, as it affects directly every single particle through
the equations of motion written in comoving coordinates. In Fig. 4 the ratios
of the Hubble expansion factors for VDE and $\Lambda$CDM-vde to the standard
$\Lambda$CDM value are shown; we see that different models are characterized
by differences up to the $20\%$ in the expansion rate. To implement this
modification, we replaced the standard computation of $H(a)$ in GADGET-2 with
a routine that reads and interpolates from a pre-computed table.
### 3.3 Code Testing
To check the reliability of the modifications introduced into the simulation
code and during the generation of the initial conditions, we have confronted
the theoretical linear growth factor, computed using the Boltzmann-code CAMB
(Lewis et al., 2000) with the ones derived directly from the simulations.
Figure 5: Ratio of the growth function to the expansion factor $D(a)/a$ as
obtained from the 500 $h^{-1}\,{\rm Mpc}$ box simulations versus the
analytical one. The results shown an agreement between the theoretical
expectation and the numerically computed one within the $2\%$ level. The
results from the 1 $h^{-1}$Gpc box simulations are not shown since they
perfectly overlap with the ones presented here.
As shown in Fig. 5, our results yield an agreement within the $1\%$ level,
which proofs the correctness of our modifications as well as illustrating
(again) the differences in structure growth between the models.
We would like to note that for consistency reasons, both when calculating the
CAMB and the numerical value for the growth factor, we have used the
expression
$D^{+}(z)=\sqrt{\frac{P(z,k_{0})}{P(z_{0},k_{0})}}$ (18)
where $k_{0}$ is a fixed scale whithin the linear regime and $z_{0}$ is the
initial redshift of the simulation.
### 3.4 Halo Finding
In order to identify haloes in our simulation we have run the open source
MPI+OpenMP hybrid halo finder AHF333AHF stands for Amiga Halo Finder, to be
downloaded freely from http://www.popia.ft.uam.es/AMIGA described in detail in
Knollmann & Knebe (2009). AHF is an improvement of the MHF halo finder (Gill
et al., 2004) and has been extensively compared against practically all other
halo finding methods in Knebe et al. (2011). AHF locates local overdensities
in an adaptively smoothed density field as prospective halo centres. For each
of these density peaks the gravitationally bound particles are determined.
Only peaks with at least 20 bound particles are considered as haloes and
retained for further analysis.
But the determination of the mass requires a bit more elaboration as it is
computed via the equation
$M(R)=\Delta\times\rho_{c}(z)\times\frac{4\pi}{3}R^{3}$ (19)
where we applied $\Delta=200$ as the overdensity threshold and $\rho_{c}(z)$
refers to the critical density of the universe at redshift $z$. In this way
$M(R)$ is defined as the total mass contained within a radius $R$,
corresponding to the point where the halo matter density $\rho(r)$ is $\Delta$
times the critical value $\rho_{c}$. Using this relation, particular care has
to be taken when considering the definition of the critical density
$\rho_{c}(z)=\frac{3H^{2}(z)}{8\pi G}$ (20)
because it involves the Hubble parameter, that differs substantially at all
redshifts in the two models. This means that, identifying the halo masses, we
have to take into account the fact that the value of $\rho_{c}(z)$ changes
from $\Lambda$CDM to VDE. This has been incorporated into and taken care of in
the latest version of AHF where $H_{VDE}(z)$ is being read in from a
precomputed table, too.
We finally need to mention that we checked that the objects obtained by this
(virial) definition can be compared across different cosmological models and
using different mass definitions. To this extent we studied the ratio between
two times kinetic over potential energy $\eta=2T/|U|$ confirming that at each
redshift under investigation here the distributions of $\eta$ in $\Lambda$CDM
and VDE are actually comparable (not presented here though), meaning that the
degree of virialisation (which should be guaranteed by Eq. (19)) is in fact
similar. We therefore conclude that our adopted method to define halo mass
(and edge) in the VDE model leads to unbiased results and yields objects in
the same state of equilibrium as is the case for the $\Lambda$CDM haloes.
Please note that this test does not guarantee that all our objects are in fact
virialized; it merely assures us that the degree of virialisation is
equivalent. We will come back to this issue later when selecting only
equilibrated objects.
Figure 6: Projected density for $\Lambda$CDM,$\Lambda$CDM-vde and VDE showing
a $120\times 120$ $h^{-2}Mpc^{2}$ slice at the box center in the 500
$h^{-1}\,{\rm Mpc}$ box at $z=0$ projected on the $x$-$z$ plane. Bright areas
are associated with matter whereas underdense regions are denoted by darker,
black spots in the projected box. Results for the VDE-1, $\Lambda$CDM-1 and
$\Lambda$CDM-vde-1 simulations are not shown since the colour coding does not
provide useful insights on the different clustering patterns on smaller
scales.
## 4 Large Scale Structure and Global Properties
In the following section, we will discuss the global properties of large scale
structures identified in our simulations. Using all of our sets of simulations
for $\Lambda$CDM, $\Lambda$CDM-vde and VDE we will disentangle parameter-
driven effects from those due to the different dynamics of the background
expansion, which uniquely characterize VDE and therefore are worth pointing
out in the process of model selection.
Figure 7: Power Spectra at redshifts $z=0,1,2,4$ for $\Lambda$CDM-0.5,
$\Lambda$CDM-0.5-vde and VDE-0.5 The results from the 1 $h^{-1}$Gpc
simulations are not shown as they simply overlap to the present ones on the
smaller-$k$ end, without providing further insights on the small scales, where
we expect non linear effects to dominate. Figure 8: Mass function for
$\Lambda$CDM, VDE and $\Lambda$CDM-vde models at different redshifts, computed
for the 500 $h^{-1}\,{\rm Mpc}$ box simulations. We have also verified that
the corresponding values computed for the 1 $h^{-1}$Gpc simulations overlap to
the ones shown here for $M>10^{13}$$h^{-1}{\rm{M_{\odot}}}$, except for a
smoother high-mass end. In the lower panels of the plots, VDE and
$\Lambda$CDM-vde to $\Lambda$CDM ratios are represented by dotted lines while
VDE to $\Lambda$CDM-vde are shown using dash-dotted lines.
### 4.1 Density Distribution
In Fig. 6 we show the colour coded density field for the particle distribution
at redshift $z=0$, for a $120\times 120$ $h^{-1}\,{\rm Mpc}$2 slice at the box
center for the three 500 $h^{-1}\,{\rm Mpc}$ simulations projected on the
$x$-$z$ plane. As expected, we observe that the most massive structures’
spatial positions match in the three simulations, although in the
$\Lambda$CDM-vde and VDE a large overabundance of objects with respect to
$\Lambda$CDM, as we could expect due to the higher $\Omega_{M}$. This
observation will be confirmed on more quantitative grounds in the analysis
carried in the following sections, especially when refering to the study of
the cumulative mass function.
### 4.2 Matter Power Spectrum
In Fig. 7 we show the dark matter power spectrum $P(k)$ at redshifts
$z=0,1,3,4$ computed for the VDE-0.5, $\Lambda$CDM-0.5 and $\Lambda$CDM-
vde-0.5 simulations. For clarity, we do not show the 1 $h^{-1}$Gpc
simulations; however, we have checked their consistency with the 500
$h^{-1}\,{\rm Mpc}$ runs. We note that at all redshifts the differences
already seen in the input power spectra are preserved (cf. Fig. 3), meaning
that the VDE model has less power than $\Lambda$CDM on the large scales,
whereas the opposite is true for small scale. This particular shape of the
$P(k)$ is a peculiar feature of VDE cosmology, as other kinds of dynamical
quintessence (Alimi et al., 2010) and coupled DE (Baldi et al., 2010) show
completely different properties; with less power (in the former case) or a
$\Lambda$CDM-type of behaviour (in the latter) on small scales. At higher and
intermediate redshifts $\Lambda$CDM-vde shows almost no differences from
$\Lambda$CDM, as expected since the former is normalized to a lower initial
value with respect to the latter and therefore needs to equal it before
eventually overcoming it at smaller $z$’s, as imposed by the larger
$\sigma_{8}$ normalization. The effects of the different growth factor in this
model start to become evident only at $z<1$, where we see that the ratio of
the $P(k)$ starts to increase. Whereas the ratio of VDE to $\Lambda$CDM for
$k<0.05$ h $h^{-1}\,{\rm Mpc}$-1 is substantially unaltered at all redshifts,
small scales are affected by non-linear effects, eventually distorting its
shape.
### 4.3 Halo Abundance
In the following subsection we will study the abundance of massive objects at
different redshifts. Highlighting the differences arising among the three
models in the different mass ranges, we want to study VDE’s peculiar
predictions for the massive cluster distribution and highlight its distinction
from $\Lambda$CDM.
To this extent, we show in Fig. 8 the (cumulative) mass functions for the
three models at $z=0,1,2,4$, as computed from the VDE-0.5, $\Lambda$CDM-0.5
and $\Lambda$CDM-0.5-vde simulations; the corresponding VDE-1, $\Lambda$CDM-1
and $\Lambda$CDM-vde-1 results can be found in Carlesi et al. (2011); they are
not shown here again as they do not provide any new insights and rather
confirm (and extend) the results to be drawn from the $500$$h^{-1}\,{\rm Mpc}$
boxes, respectively: We note that the VDE cosmology is characterized by a
larger number of objects at all the mass scales and redshifts, outnumbering
$\Lambda$CDM by a factor constantly larger than 2. In particular, this
enhancement can be seen for the very large masses, where at low $z$ the
VDE/$\Lambda$CDM ratio reaches values of $\sim 10$. Although this value of the
ratio seems to be a mere result of the cosmic variance, due to the low number
of haloes found in this mass range, the computation of the mass function for
the second 500 $h^{-1}\,{\rm Mpc}$ VDE realization and the 1$h^{-1}$Gpc
simulations makes us believe that the expected enhancement in this region must
be at least a factor 5.
Interestingly enough, $\Lambda$CDM-vde has comparable characteristics to VDE,
which leads to the conclusion that the substantial enhancement in structure
formation is mainly parameter-driven, i.e. due to the overabundance of matter
and higher normalisation of matter density perturbations. Although this first
observation may seem in contrast with what we have found in Section 4.2, where
we have noticed that VDE has less power on large scales in comparison to
$\Lambda$CDM, we have to take into account that, in the hierarchical picture
of structure formation, objects on small scales form first to subsequently
give birth to larger ones. This means, in our case, that more power for large
$k$-values should be regarded as an important source of the overall
enhancement together with the overabundance of matter, as already pointed out
in the previous discussion. The evolution of the mass functions at different
redshift allows us to disentangle the effect of the modified expansion rate;
at higher redshift, in fact, both the $\Lambda$CDM and $\Lambda$CDM-vde mass
functions are suppressed with respect to the VDE model, mostly because of the
lack of power on small scales. These stronger initial fluctuations eventually
trigger the earlier start of structure formation, but – as time passes – the
effect of the increased expansion rate shown in Fig. 4 for the VDE cosmology
suppresses structure growth, leading to a mass function below the
$\Lambda$CDM-vde curve at around redshift one. At this point, the VDE
expansion rate starts decreasing with respect to the $\Lambda$CDM one,
comparatively enhancing very large structure growth and eventually causing the
two mass functions to be (nearly) indistinguishable at $z=0$.
Furthermore, if we look at Fig. 9, where we show the evolution with redshift
of the number density of objects above the $M=10^{14}$$h^{-1}{\rm{M_{\odot}}}$
threshold, we observe that the most massive structures in the two cosmologies
form at comparable rates. This seems to suggest that in the VDE picture there
is a subtle balance between the formation of new small haloes and their
merging into more massive structures. Such an effect comes as no surprise if
we again take into account that this model has two main opposite, different
features that affect the formation of structures: a strong suppression on all
scales induced by the faster expansion of the universe for a large redshift
interval and an enhancement due to a higher density of matter and a larger
power on the small scales.
An interesting consequence of this kind of behaviour is that the VDE
overabundance of massive objects may address some recent observational
tensions of $\Lambda$CDM; namely, the high redshift of reionization and the
presence of extremely massive clusters at $z>1$. Recent microwave background
observations seem to prefer a high reionization redshift, around $z\approx 10$
combined with a lower normalization of the matter perturbations,
$\sigma_{8}\approx 0.8$; whereas simulations have shown (see, for example,
Raičević et al., 2011) that early reionization can be achieved only for
$\sigma_{8}=0.9$ or larger. In VDE, the appearence of dark matter haloes with
masses larger than $10^{12}$$h^{-1}{\rm{M_{\odot}}}$ as early as $z=7$ (while
equivalent structures appear in $\Lambda$CDM only for $z>5$) might imply also
a larger $z_{\rm reion}$, provided the hierarchycal picture of structure
formation holds also in VDE at smaller mass scales. On the other hand, the
existence of $M>5\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$ clusters at $z>1$ (as
discussed in Jee et al., 2009; Brodwin et al., 2010; Foley et al., 2011) has
also been considered by many authors (e.g., Hoyle et al., 2011; Baldi &
Pettorino, 2011; Baldi, 2011; Enqvist et al., 2011) as a serious challenge to
the standard $\Lambda$CDM paradigm; for a more thorough discussion of this
issue in the context of VDE cosmology we refer to aforementioned articles as
well as Carlesi et al. (2011). However, the comparison to the $\Lambda$CDM-vde
paradigm also shown in Fig. 9 shows that indeed VDE acts as a source of
suppression of structure growth with respect to the enhancement triggered by
the increase in $\sigma_{8}$ and $\Omega_{M}$. This effect is indeed a general
result of uncoupled dynamical dark energy models (Grossi & Springel, 2009; Li
et al., 2011) as the presence of a larger fraction of dark energy at high $z$
enhances Hubble expansion (as shown in Fig. 4) preventing a stronger
clustering to take place.
In our case, it is also important to point out that the overprediction of
objects at $z=0$ may represent a shortcoming of the model, as observations on
the cluster number mass function seem to be in contrast with such a prediction
(see Vikhlinin et al. (2009), Wen et al. (2010) and Burenin & Vikhlinin
(2012)). Furthermore, we have to keep in mind that these results assume a
$\Lambda$CDM fiducial model, while the use of a different cosmology requires a
careful handling of the data and does not allow a straightforward comparison
to the observations, as they are affected by model-dependent quantities like
comoving volumes and mass-temperature relations.
Figure 9: Number density evolution for objects more massive than $10^{14}$
$h^{-1}{\rm{M_{\odot}}}$as a function of redshift. The larger amount of
massive clusters at higher redshift is a distinctive feature of VDE cosmology.
### 4.4 Void Function
In order to identify voids, our voidfinder starts with a selection of point
like objects in 3D. These objects can be haloes above a certain mass or a
certain circular velocity or galaxies above a certain luminosity. Thus the
detected voids are characterized by this threshold mass, circular velocity or
luminosity. Other voidfinders use different approaches (Colberg et al., 2008).
The void finding algorithm does not take into account periodic boundary
conditions used in numerical simulations. Therefore, we have periodically
extended the simulation box by 50 $h^{-1}\,{\rm Mpc}$. In this extended box we
represent all haloes with a mass above the threshold of $5\times
10^{12}{{h^{-1}{\rm{M_{\odot}}}}}$ as a point. In this point distribution we
search at first the largest empty sphere which is completely inside the box.
To find the other voids we repeat this procedure however taking into account
the previously found voids. We allow that newly detected voids intersect with
previously detected ones up to 25% of the radius of a previously detected
larger void.
In Fig. 10 we show the cumulative number of voids with radius larger than
$R_{\rm void}$ the center of which is in the original box. One can clearly see
that for a given void radius there exist more voids in the $\Lambda$CDM than
in th $\Lambda$CDM-vde and VDE models. The void distribution reflects the
behaviour of the mass function shown in Fig. 8. At redshift $z=0$ there exist
less haloes with $m_{h}>5\times 10^{12}{{h^{-1}{\rm{M_{\odot}}}}}$ in the
$\Lambda$CDM model than in the other two models. Thus on average larger voids
are expected.
Figure 10: Void function for VDE-0.5, $\Lambda$CDM-0.5 and $\Lambda$CDM-
vde-0.5 at z=0. For the $500$$h^{-1}\,{\rm Mpc}$ box we show the cumulative
number of empty spheres of radius $R$ which do not contain any object with
mass larger than $5\times 10^{12}$$h^{-1}{\rm{M_{\odot}}}$.
### 4.5 Growth Index
Figure 11: Growth index in the VDE and $\Lambda$CDM cosmologies from $z=5$ to
$z=0$. Whereas $\Lambda$CDM’s growth index has an almost constant behaviour
with a mild dependence on the redshift, VDE changes dramatically from a regime
where growth is relatively suppressed (until $z\approx 1.5$) to a relative
enhancement at earlier times, where $\gamma$ becomes larger.
The growth of the perturbations can be related to the evolution of the matter
density parameter by the general relation
$\Omega_{M}^{\gamma(a)}=\frac{d\ln(\delta(a))}{d\ln(a)}$ (21)
In the standard $\Lambda$CDM cosmology, the exponent $\gamma(a)$ can be
approximated by a constant value $\gamma\sim 0.55$, although a more detailed
calculation shows that this number is actually redshift dependend (see Bueno
Belloso et al., 2011). In Fig. 11 we show the evolution of this growth index
$\gamma(z)$ computed from our VDE-0.5, $\Lambda$CDM-vde and $\Lambda$CDM-0.5
simulations. As expected, we do observe that in VDE structure formation is
generally suppressed with respect to $\Lambda$CDM as an effect of the faster
expansion rate. This statement is true until $z\approx 1.5$, when the ratio
$H_{VDE}/H_{\Lambda CDM}$ starts decreasing causing the steep increase in the
growth index, eventually reducing again as soon as vector dark energy enters
into the phantom regime (see Section 2), undergoing an accelerated expansion
that strongly suppresses structure formation. This latter change, which takes
place at $z\approx 0.5$, is reflected by the peak of $\gamma(z)$, which is
reached for the same $z$. Actually, as stressed by different parametrizations
(Bueno Belloso et al., 2011), the growth index is extremely sensitive to the
value of the equation of state $\omega(z)$, although an explicit form in terms
of VDE cosmology still has to be found. Indeed, the extremely different
behaviour of this parameter at different redshifts is an interesting feature
that clearly distinguishes the two models in a unique way: In fact, parameter-
induced modification accounts for a $\approx 5\%$ change for the value of the
growth factor, as the comparison among $\Lambda$CDM and $\Lambda$CDM-vde
suggests. In this case we observe a slight increase of the value of
$\gamma(z)$ at all redshifts, due to the increased growth rate in
$\Lambda$CDM-vde also shown in Fig. 5. However, these changes have no impact
on the shape of this function, which keeps its mild dependence on $z$
unaltered. Therefore $\gamma(z)$ can be effectively used as a tool for model
selection, embodying effectively VDE’s peculiar equation of state $\omega(z)$
and expansion history. Current observational bounds on $\gamma$ constrain only
weakly its value at high $z$’s (see e.g. Nesseris & Perivolaropoulos, 2008) or
even favour a higher $\gamma(z=0)$ (Basilakos, 2012) in contrast to
theoretical calculations based on $\Lambda$CDM. In any case, it will surely be
something to be looked at in the near future, when deep surveys like Euclid
(Laureijs et al., 2011) will provide stringent constraints on this quantity
(Bueno Belloso et al., 2011).
## 5 Dark Matter Haloes
In this section we will discuss properties of (individual) haloes in VDE and
$\Lambda$CDM. In particular, we will compare the distributions of masses,
shape parameter, spin parameter, concentrations and formation redshifts as
well as the shape of dark matter density profiles. In this way, we will
determine the most important features that characterize on the average a
single cosmological model. But in addition we are also cross correlating
haloes in the two models studying differences on a one-to-one basis. By this
we will be able to determine how the properties of a single given structure
change when switching from one cosmological picture to the other.
### 5.1 General properties
To have a reliable description of the general halo properties we need to
properly select our sample from the catalogues, in order to include only those
objects composed of a number of particle sufficient to resolve its internal
structure without exceeding statistical uncertainty. Following Muñoz-Cuartas
et al. (2011) and Prada et al. (2011) we set this number to approximately 500,
even though other authors (see for example Macciò et al. (2007) and Bett et
al. (2007)) suggest that lower values can be used, too. However, since we are
dealing with different simulations run with particles of different mass, the
application of this criterion is not straightforward. In fact, since our aim
is to compare _equivalent_ structures (i.e. structures with the same
$M_{200}$) and not structures composed by an identical number of particles we
need to choose our sample imposing a mass threshold $M_{th}$. For the
simulations in the $500$$h^{-1}\,{\rm Mpc}$ box, we have chosen
$M_{th}=5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$, which corresponds to haloes
formed by at least 500 particles in VDE and $\Lambda$CDM-vde and 715 particles
in $\Lambda$CDM; while for the larger $1000$$h^{-1}\,{\rm Mpc}$ runs we
imposed a $M_{th}=3\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$ limit, i.e. 380 VDE
and $\Lambda$CDM-vde particles and 545 $\Lambda$CDM ones. In the latter set of
simulations, we see that we are including also haloes with a $\sim 20\%$ less
than 500 particles in the VDE and $\Lambda$CDM-vde cases; this has been done
since in the trade off between resolution and sample size we have felt more
comfortable using a larger number of haloes at the expense of a slight
reduction in accuracy, which will be nonetheless taken into account when
analyzing the results. The total number of haloes that comply with these
conditions in every simulations, as well as the number of haloes that satisfy
the relaxation criterion which will be discussed in Section 5.1.3, are shown
in Table 2. The state of virialisation of haloes will only be taken into
account below when investigating the density profiles; for the study of the
(distributions of the) two-point correlation functions, the spin, and even the
shape of haloes we prefer to include even un-relaxed objects as they should
clearly stick out in the distributions (if present in large quantities).
Table 2: Number of haloes above the mass (number) threshold $M_{th}$ ($N_{th}$) per simulation. It is also shown the number of relaxed haloes, defined as those complying with the criterion introduced in Section 5.1.3. Simulation | $M_{th}$ | $N_{th}$ | $N$ total | $N$ relaxed
---|---|---|---|---
$\Lambda$CDM-0.5 | $5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$ | 715 | 1704 | 1370
$\Lambda$CDM-vde-0.5 | $5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$ | 500 | 5898 | 5220
VDE-0.5 | $5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$ | 500 | 6274 | 5569
$\Lambda$CDM-1 | $3\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$ | 545 | 4045 | 3533
$\Lambda$CDM-vde-1 | $3\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$ | 380 | 9072 | 8117
VDE-1 | $3\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$ | 380 | 12174 | 11508
#### 5.1.1 Correlation function
Figure 12: The two point correlation function for objects in the
$500$$h^{-1}\,{\rm Mpc}$ simulations more massive than $5\times
10^{13}$$h^{-1}{\rm{M_{\odot}}}$.
To study the clustering properties of the haloes in VDE cosmology, we computed
the two point correlation function using the definition:
$\xi(r)=\frac{V}{N^{2}}\sum_{i=1}^{N}\frac{n_{i}(r;\Delta r)}{v(r;\Delta
r)}-1$ (22)
where $N$ is the total number of objects above the given mass threshold in the
simulation volume $V$, and $n_{i}$ is the total number of objects within a
shell of volume $v$ and thickness $\Delta r$ (of constant logarithmic spacing
in $r$) centered at the $i$th object. In this case, we have limited our
analysis to the 500$h^{-1}\,{\rm Mpc}$ boxes, ignoring the 1$h^{-1}$Gpc due to
their lack of small scale resolution. The results are plotted in Fig. 12,
where we can see that the $\xi(r)$ is slightly smaller at all scales in VDE.
Although in principle we would expect VDE cosmology to have an enhanced
clustering pattern due to the increased distribution of massive objects
observed in the mass function, the $N^{-2}$ dependence of the two point
correlation function drags the total value down, making the final distribution
function smaller than in $\Lambda$CDM. In fact, a similar behaviour can be
observed for $\Lambda$CDM-vde; with a two point correlation function below
$\Lambda$CDM at practically all scales. In Table 3 we show the results of
fitting $\xi(r)$ to a power law $(r_{0}/r)^{\gamma}$ from which we see that
VDE is characterized by a smaller correlation length $r_{0}$ and a steeper
slope $\gamma$.
#### 5.1.2 Spin parameter, shape and triaxiality
$\begin{array}[]{cc}\includegraphics[angle={270},width=227.62204pt]{lambda_distribution_best_fit_1}\end{array}$
Figure 13: Spin parameter versus the analytical log norm distribution
calculated with the best fit parameters. The fit has been performed using the
combined sample of haloes above $5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$
belonging to the three 500 $h^{-1}\,{\rm Mpc}$ boxes with those above the
$3\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$limit in the 1$h^{-1}$Gpc boxes.
Rotational properties of the haloes can be studied using the so called spin
parameter $\lambda$, a dimensionless number that measures the degree of
rotational support of the halo. Following Bullock et al. (2001), we define it
as
$\lambda=\frac{L_{200}}{\sqrt{2}M_{200}V_{200}R_{200}}$ (23)
where the quantities $L$ (the total angular momentum), $M$ (total mass), $V$
(circular velocity) and $R$ (radius) are all taken at the point where the
average halo density becomes 200 times the critical density. Different authors
have found (e.g. Barnes & Efstathiou, 1987; Warren et al., 1992; Cole & Lacey,
1996; Gardner, 2001; Bullock et al., 2001; Macciò et al., 2007, 2008; Muñoz-
Cuartas et al., 2011) that the distribution of this parameter is of lognormal
type
$P(\lambda)=\frac{1}{\lambda\sigma_{0}^{2}\sqrt{2\pi}}\exp\left[-\frac{\ln^{2}(\lambda/\lambda_{0})}{2\sigma_{0}^{2}}\right]\
,$ (24)
even though there are recent claims that this distribution has to be slightly
modified (Bett et al., 2007).
Fitting the above function to our numerical sample by a non-linear Levenberg-
Marquardt least square fit we find a remarkably good agreement, shown in Fig.
13 for the combined set of haloes of the 500$h^{-1}\,{\rm Mpc}$ and
1$h^{-1}$Gpc simulations. It is clear that the three models present no
substantial difference in the values of these distributions, meaning that the
change of cosmology has no impact on the rotational support of the dark matter
structures.
The shape of three dimensional haloes can be modelled as an ellipsoidal
distribution of particles (Jing & Suto, 2002; Allgood et al., 2006),
characterized by the three axis $a\geq b\geq c$ computed by AHF as the
eigenvalues of the inertia tensor
$I_{i,j}=\sum_{n}x_{i,n}x_{j,n}$ (25)
which is in turn obtained summing over all the coordinates of the particles
belonging to the halo.
We define the shape parameter $s$ and the triaxiality parameter $T$ as
$s=\frac{c}{a}\qquad T=\frac{a^{2}-b^{2}}{a^{2}-c^{2}}$ (26)
and we calculate the probability distributions $P(T)$ and $P(s)$ of the above
parameters for all the objects above the aforementioned mass thresholds in our
cosmological simulations, to see whether the VDE picture of structure
formation induces changes in the average shape and triaxiality. Similarly to
the previous case, we found again that halo shapes and triaxialities remain
practically unaltered by VDE cosmolgy. This result could be expected, keeping
in mind that VDE only affect background evolution: Once that structures start
to form, detaching from the background evolution, they become affected by
gravitational attraction only. Therefore, the internal structure of dark
matter haloes remains generally unaltered by the presence of an uninteracting
form of dark energy and cannot be used to discriminate between alternative
cosmological paradigms. We have also verified that these results hold also
when taking into account different halo samples separately, i.e. the massive
ones of the 1$h^{-1}$Gpc simulations and the smaller belonging to the
500$h^{-1}\,{\rm Mpc}$ boxes.
#### 5.1.3 Unrelaxed haloes
Before moving to the discussion of the properties of internal structure of the
haloes, and in particular the density profile, we need to introduce and
motivate a second criterion of selection for our halo sample, related to the
degree of _relaxation_ of the halo. An additional check is necessary since
only a fraction of the structures identified in our catalogues completely
satisfies the virial condition. In unvirialized structures, infalling matter
and merger phenomena may occour, heavily affecting the halo shape and thus
making the determination of radial density profiles and concentrations
unreliable. In fact, unrelaxed haloes are most likely to differ from an
idealized spherical or ellipsoidal shape since they have a highly asymmetric
matter distribution, which in turn makes the determination of the halo center
an ill-defined problem, as discussed by Macciò et al. (2007) and Muñoz-Cuartas
et al. (2011). Our halo finder AHF does not directly discriminate between
virialized and unvirialized structures giving catalogues containing both types
of objects; however, it provides kinetic $K$ and potential energy $U$ for
every halo identified, thus making the computation of the viral ratio $2K/|U|$
straightforward. Following one of the criteria used by Prada et al. (2011), we
will consider as relaxed all the haloes satisfying the condition
$\frac{2K}{|U|}-1<0.5$ (27)
without introducing additional parameters. Alternative ways of identifying
unrelaxed structures can be found throughout the literature (e.g. Macciò et
al., 2007; Bett et al., 2007; Neto et al., 2007; Knebe & Power, 2008; Prada et
al., 2011; Muñoz-Cuartas et al., 2011; Power et al., 2011); but since the
results they give are qualitatively similar for reasons of computational speed
and simplicity we will not make use of them. The total number of haloes
satisfying the relaxation condition is shown for every cosmology in Table 2.
#### 5.1.4 Density profiles
$N$-body simulations have shown that dark matter haloes can be described by a
Navarro Frenk White (NFW) profile (Navarro et al., 1996), which is given by
$\rho(r)=\frac{\rho_{0}}{\frac{r}{r_{s}}(1+\frac{r}{r_{s}})^{2}}$ (28)
where the $r_{s}$, the so called scale radius, and the $\rho_{0}$ are in
principle two free parameters that depend on the particular halo structure.
But $\rho_{0}$ can be written as a function of the critical density as
$\rho_{0}=\delta_{c}\rho_{c}$, where
$\delta_{c}=\frac{200}{3}\frac{c^{3}}{\log(1+c)-\frac{c}{1+c}}$
and $c=r_{\rm vir}/r_{s}$ is the _concentration_ of the halo relating the
virial radius $r_{v}$($=r_{200}$ in our case) to the scale radius $r_{s}$,
which will be discussed in detail in the following subsection. This
description is generally valid for $\Lambda$CDM, but simulations of ever
increased resolution have actually revealed that the very central regions are
not following the slope advocated by the NFW formula but rather follow a
Sersic or Einasto profile (cf. Navarro et al., 2004; Stadel et al., 2009).
Here we want to check to which degree the modified cosmological background
affects the distribution of matter inside dark matter haloes, i.e. its density
profile. All our (relaxed) objects in all simulations have been fitted to the
Eq. (28), and to estimate the goodness of this fit we compute for each halo
its corresponding $\chi^{2}$, defined in the usual way
$\chi^{2}=\sum_{i}\frac{(\rho_{i}^{\rm(th)}-\rho_{i}^{\rm(num)})^{2}}{\Delta\rho_{i}^{\rm(num)}}$
(29)
where the $\rho_{i}$’s are the numerical and theoretical overdensities in
units of the critical density $\rho_{c}$ at the $i$-th radial bin and
$\Delta\rho_{i}$ is the numerical Poissonian error on the numerical estimate.
Since different halo profiles will be in general described by a different
number of radial bins444Note that our halo finder AHF uses logarithmically
spaced radial bins whose number depends on the halo mass, i.e. more massive
haloes will be covered with more bins., to make our comparison between
different simulations and haloes consistent we need to use the reduced
$\chi^{2}$
$\chi^{2}_{\rm red}=\frac{\chi^{2}}{N_{pts}-N_{dof}-1}$ (30)
where $N_{pts}$ is the total number of points used (i.e. total number of
radial bins) and $N_{dof}$ is the number of degrees of freedom (free
parameters).
The comparison of the distributions of the reduced $\chi^{2}$ values for
$\Lambda$CDM-vde, $\Lambda$CDM and VDE haloes belonging to the two set of
500$h^{-1}\,{\rm Mpc}$ and 1$h^{-1}$Gpc simulations, shown in Fig. 14, allows
us to determine again that no substantial difference is induced by the VDE
picture, for the same reasons discussed in the case of spin, shape and
triaxiality distributions. The standard description of dark matter structures
is thus not affected by the presence of a VDE.
$\begin{array}[]{cc}\includegraphics[angle={270},width=227.62204pt]{nfw_chi2_distribution}\includegraphics[angle={270},width=227.62204pt]{nfw_chi2_distribution_1}\end{array}$
Figure 14: Reduced $\chi^{2}$ distribution for the best fit to a NFW profile,
on the vertical axis we plot the total fraction of haloes whose reduced
$\chi^{2}$ falls within the horizontal axis bin value. This is shown for
relaxed haloes above the $5\times 10^{13}$$h^{-1}{\rm{M_{\odot}}}$ threshold
belonging to the VDE-0.5, $\Lambda$CDM-vde-0.5 and the $\Lambda$CDM-0.5
simulations (left panel) as well as for those above the $3\times
10^{14}$$h^{-1}{\rm{M_{\odot}}}$ threshold belonging to VDE-1, $\Lambda$CDM-
vde-1 and $\Lambda$CDM-1 (right panel). The distributions show no particular
difference among the three cosmologies; however, in the three 1$h^{-1}$Gpc
simulations we notice how lower resolution affects the $\chi^{2}$
distribution, resulting in a ticker tail at higher values compared to the
500$h^{-1}\,{\rm Mpc}$ case, meaning that the fit to a NFW is on the average
worse.
#### 5.1.5 Halo concentrations
In the last step of the analysis of the general properties of haloes we will
turn to concentrations, which characterize the halo inner density compared to
the outer part. This parameter is usually defined as
$c=\frac{r_{\rm vir}}{r_{s}}$ (31)
where $r_{s}$ is the previously introduced scale radius, obtained through the
best fit procedure of the density distribution to a NFW profile. We would like
to remind that concentrations are correlated to the formation time of the
halo, since structures that collapsed earlier tend to have a more compact
center due to the fact that it has more time to accrete matter from the outer
parts. Dynamical dark energy cosmologies generically imply larger $c$ values
as a consequence of earlier structure formation, as found in works like those
by Dolag et al. (2004), Bartelmann et al. (2006) and Grossi & Springel (2009).
In fact, since the presence of early dark energy usually suppresses structure
growth, in order to reproduce current observations we need to trigger an
earlier start of the formation process, which on average yields a higher value
for the halo concentrations. However, this result does not hold in the case of
coupled dark energy, where the increased clustering strength induced by a
fifth force sets a later start of structure formation, as discussed in Baldi
et al. (2010).
$\begin{array}[]{cc}\includegraphics[angle={270},width=227.62204pt]{conc_log_best_fit}\end{array}$
Figure 15: Best fit of the mass-concentration relation for the combined sample
of relaxed haloes belonging to all the $\Lambda$CDM, $\Lambda$CDM-vde and VDE
simulations. The points represent the average concentration values for the
relaxed haloes in the corresponding mass bin; circles are for $\Lambda$CDM,
triangles for $\Lambda$CDM-vde and squares for VDE. Empty dots stand for bins
determined using haloes belonging to the 1 $h^{-1}$Gpc simulations; filled
ones refer to the 500 $h^{-1}\,{\rm Mpc}$ ones. The Poissonian error bars are
computed using the number of selected haloes within each mass bin.
In the hierarchical picture of structure formation, concentrations are usually
inversely correlated to the halo mass as more massive objects form later;
$N$-body simulations (Dolag et al., 2004; Prada et al., 2011; Muñoz-Cuartas et
al., 2011) and observations (Comerford & Natarajan, 2007; Okabe et al., 2010;
Sereno & Zitrin, 2011) have in fact shown that the relation between the two
quantities can be written as a power law of the form
$\log{c}=a(z)\log(\frac{M_{200}}{{{h^{-1}{\rm{M_{\odot}}}}}})+b(z)$ (32)
where $a(z)$ and $b(z)$ can have explicit parametrizations as functions of
redshift and cosmology (see e.g. Neto et al., 2007; Prada et al., 2011; Muñoz-
Cuartas et al., 2011). We can use our selected halo samples at $z=0$ from the
500 $h^{-1}\,{\rm Mpc}$ and 1 $h^{-1}$Gpc simulations to obtain the $a(z=0)$
and $b(z=0)$ values for the $\Lambda$CDM, $\Lambda$CDM-vde and VDE
cosmologies; the results of the best fit procedure to Eq. (32) are shown in
Table 3.
These values are in good agreement with the ones found by, for instance, Dolag
et al. (2004), Macciò et al. (2008) and Muñoz-Cuartas et al. (2011) (who quote
for $\Lambda$CDM values of $a(z=0)\approx-0.097$ and $b\approx 2.01$); the
$\sim 10\%$ discrepancy observed with their results is due to the fact that
our results are obtained over a smaller mass range, $5\times 10^{13}$–$2\times
10^{15}$ $h^{-1}{\rm{M_{\odot}}}$, whereas the previously cited works study it
over an interval larger by more than three orders of magnitude,
$10^{10}$–$10^{15}$$h^{-1}{\rm{M_{\odot}}}$. Still, according to our results,
the $c-M$ relation for both the VDE and $\Lambda$CDM-vde case is characterized
by a shallower $a$ exponent and a larger $b$. Although the magnitude of these
changes is different in the two models, we can safely conclude that also in
this case the results are mainly parameter-driven, i.e. due to the larger
value of $\Omega_{M}$. Furthermore, the large error bars for $M>10^{15}$
$h^{-1}{\rm{M_{\odot}}}$ scales, due to the low statistics of massive haloes
complying the relaxation requirements, makes it difficult to determine to
which extent the differences in the best fit relations among $\Lambda$CDM-vde
and VDE could be eventually reduced in the presence of a larger sample.
Table 3: Best fit values for the mass-concentration relation for $z=0$, obtained fitting the relation Eq. (32) to the relaxed haloes concentrations and the two point correlation function to a power law $(r_{0}/r)^{\gamma}$ for the $\Lambda$CDM, $\Lambda$CDM-vde and VDE cosmologies. $r_{0}$ values are given in $h^{-1}\,{\rm Mpc}$. Model | $a$ | $b$ | $r_{0}$ | $\gamma$
---|---|---|---|---
$\Lambda$CDM | -0.115 | 2.11 | 13.4 | -1.79
$\Lambda$CDM-vde | -0.112 | 2.21 | 12.1 | -1.91
VDE | -0.098 | 2.17 | 10.1 | -1.94
$\begin{array}[]{cc}\includegraphics[angle={270},width=227.62204pt]{concentration_best_fit}\end{array}$
Figure 16: Distribution of the ratio between the actual concentration and the
expected one (cf. Eq. (32)) and its fit to a lognormal distribution.
We also need to mention that in our simulations the actual halo concentrations
do not precisely follow equation Eq. (32) but rather scatter around it, as can
be seen in Fig. 15, where the average $c$ per mass bin is plotted against the
corresponding best fit relations. This is not really surprising, since
observations (Sereno & Zitrin, 2011) and $N$-body simulations (Dolag et al.,
2004) have shown that halo concentrations are lognormally distributed around
their theoretical value calculated using Eq. (32). In Fig. 16 we show that
this is indeed the case: the distribution of the $c(M)/c_{fit}(M)$, where
$c_{fit}(M)$ is the theoretical concentration value for a halo of mass $M$,
extremely close to a lognormal one with an almost model independent dispersion
$\sigma\approx 0.4$.
### 5.2 Cross Correlation
The next step in our analysis consists of studying the properties of the (most
massive) cross correlated objects found in the three models at $z=0$. Whereas
in the previous section our focus was on the distribution of halo properties,
this time we aim at understanding how they change switching from one model to
another.
The identification of ”sister haloes” among the different cosmologies can be
done using the AHF tool MergerTree, which determines correlated structures by
matching individual particles IDs in different simulation snapshots. For a
more elaborate discussion of its mode of operation we refer the reader to
Section 2.4 in Libeskind et al. (2010) where it has been described in greater
detail. This time we decided to restrict our halo sample further only picking
the first 1000 most massive ($\Lambda$CDM) haloes. The criterion of halo
relaxation has of course also been taken into account when dealing with
profiles and concentrations.
#### 5.2.1 Mass and spin parameter
Figure 17: Mass and spin parameter correlation ratios for the first 1000
(relaxed and possibly unrelaxed) haloes. Panels on the left show the
VDE/$\Lambda$CDM results, the ones in the center to $\Lambda$CDM-
vde/$\Lambda$CDMwhile on the right the ratio VDE/$\Lambda$CDM-vde is plotted.
Cross identified objects are characterized by larger masses in VDE and
$\Lambda$CDM-vde as a consequence of the higher $\Omega_{M}$ and $\sigma_{8}$
normalization value.
In the two upper panels of Fig. 17 we show the ratios of the masses $M$ and
spin parameter $\lambda$ for all the cross correlated sets of simulations; in
each panel we show the ratios for the 500$h^{-1}\,{\rm Mpc}$ simulation boxes
while on the right the 1$h^{-1}$Gpc ones. Both VDE and $\Lambda$CDM-vde show
average mass and spin values scattered around values larger than one when
compared to $\Lambda$CDM, whereas the cross comparison of VDE to $\Lambda$CDM-
vde shows average ratios close to unity at all mass scales. This substantial
increase in the ratios is due to the earlier beginning of structure formation,
triggered by the larger $\Omega_{M}$ and $\sigma_{8}$, as the comparison
VDE/$\Lambda$CDM-vde shows. As we already did in Section 5.1 when looking at
the halo properties in general, we conclude that also when observing the same
halo evolved under different cosmologies, the main effects are determined
exclusively by the set of cosmological parameters chosen, being the imprint of
the cosmological background evolution substantially negligible in this case.
This makes the identification of a cosmic vector through the determination of
halo properties impossible, since the background dynamics, which distinguishes
VDE from any other non interacting dynamical dark energy model, does not leave
any observable imprint on these scales.
#### 5.2.2 Halo concentrations and internal structure
As done in the previous section, in the determination of halo profiles and
concentrations properties we discard unrelaxed haloes, but this time in a way
so that our halo sample will still be composed of the first 1000 haloes
satisfying condition Eq. (27). This same halo sample has been used also in the
study of the $M_{vir}-z_{\rm form}$, in order to be able to compare these
results with the ones obtained from concentrations consistently – although in
principle formation redshifts are well defined even for unrelaxed haloes.
Again, our procedure consists in fitting all the selected structure to a NFW
profile, from which we will be able to derive the concentration parameter $c$
and a measure for the quality of the fit $\chi^{2}$; we will then compare
these results in each cross identified objects to see how a given halo
structure changes when evolved under a different cosmology. Although not shown
here, no particular trend in the differences among $\Lambda$CDM, $\Lambda$CDM-
vde and VDE pictures has been be found for either NFW $\chi^{2}$, shape and
triaxiality; since in all the cases the ratios of these properties among cross
correlated haloes are centered around unity. Not surprisingly, we also find
again a generally higher average value for the concentrations in VDE and
$\Lambda$CDM-vde with respect to $\Lambda$CDM, (see Fig. 17) a result which
again can be explained by the larger value of $\Omega_{M}$ and $\sigma_{8}$.
Similar concentrations for VDE and $\Lambda$CDM-vde haloes, shown in the upper
right panel of Fig. 17 can be also understood as a consequences of the similar
masses of the haloes examined and the similar $c-M$ relations found for the
two cosmologies. However, even if from $\Lambda$CDM-vde cosmology we conclude
that the different choice of $\Omega_{M}$, can explain in this case higher
halo concentration, we need to remind that such a results is also a general
feature of the dynamical nature of the dark energy fluid, as already found by
Dolag et al. (2004), Bartelmann et al. (2006) and Grossi & Springel (2009).
## 6 Conclusions
In this work we presented an in-depth analysis of the results of a series of
$N$-body dark matter only simulations of the Vector Dark Energy cosmology
proposed by Beltrán Jiménez & Maroto (2008). The main emphasis has been on the
comparison to the standard $\Lambda$CDM paradigm, using a mirror simulation
with identical number of particles, random seed for the initial conditions,
box size and starting redshift. An additional series of simulations for a
$\Lambda$CDM-vde cosmology have also been run using the VDE values for
$\Omega_{M}$ and $\sigma_{8}$ within a standard $\Lambda$CDM picture, to
disentangle the effects of the parameter induced modifications to the
dynamical ones coming directly from the VDE model.
The use of a modified version of the GADGET-2 code required us to check the
results with particular care. A consistency check of our simulations was
performed by comparing the numerical results for the evolution of the growth
factor to the analytical calculations, finding an excellent agreement between
the two. We further had to adapt the halo finding procedure, due to the fact
that the critical density as a function of redshift $\rho_{c}(z)$, entering
the definition of the halo edges, takes different values in VDE. Once halo
catalogues had been obtained, we carried our analysis at two different levels,
namely:
* •
we studied the very large scale clustering pattern through the computation of
matter power spectra, mass, void, and two-point correlation functions;
* •
we analyzed halo structure, comparing statistical distributions and averages
of spin parameters, concentrations, masses and shapes.
In the first point, making use of the full set of simulations, our analysis
covered the whole masse range $10^{12}$–$10^{15}$ $h^{-1}{\rm{M_{\odot}}}$ as
well as different redshifts, so that we could make specific VDE model
predictions for the number density evolution $n(>M,z)$ and growth index
$\gamma(z)$. A distinctive behaviour, very far from the standard $\Lambda$CDM
results, has been found for $\gamma(z)$, and, in particular, for the mass
function that in VDE cosmology can be up to 10 times larger than the standard
$\Lambda$CDM one. The latter result is due to the earlier onset of structure
formation and we have mentioned how it can be used to address current
$\Lambda$CDM observational tensions with large clusters at $z>1$ and possibly
with early reionization epoch (cf. also Carlesi et al., 2011).
Computing the cumulative mass function at different redshifts and making use
of the $\Lambda$CDM-vde simulations we have also observed how the condition
$H_{VDE}(z)>H_{\Lambda CDM}(z)$, holding up to $z\approx 1$, induces a
relative suppression of structure growth in this cosmological model, an effect
that clashes with the increased matter density and $\sigma_{8}$. In fact,
while on the one hand higher values of these parameters enhance the formation
of a larger number of objects, on the other hand, background dynamics
suppresses clustering and growth. The interplay and relative size of these
effects has been studied using the $\Lambda$CDM-vde simulations, showing that,
for example, faster expansion in the past determines for VDE an expectation of
clusters with $M>10^{14}$$h^{-1}{\rm{M_{\odot}}}$ up to $\approx 5$ times
smaller than what a simple increase in $\sigma_{8}$ and $\Omega_{M}$ would
determine. This effect has been also seen in the void distribution, where
suppression of clustering prevents small structures to merge into larger one
and to rather spread in the field, so that underdense regions happen to be
smaller and rarer than in $\Lambda$CDM and $\Lambda$CDM-vde. In these latter
cosmologies, in fact, a higher contrast between populated and less populated
regions is observed both in the power spectrum and in the colour coded matter
density.
In the second part of our work we have focused on the study of internal halo
structure. We found that VDE cosmology does not induce deviations in the
functional form of the dark matter halo density profiles, which are still well
described by a NFW (Navarro et al., 1996) profile, nor in the distributions
for the concentrations and spin parameters, which are of the lognormal type as
in $\Lambda$CDM. Shape and triaxiality are also unaffected: the distributions
for the relative parameters are identical and peaked in at the same values in
all the three cosmologies. The above results are a direct consequence of the
fact that dark matter haloes, once detached from the general background
evolution driven by the cosmic vector, evolve by means of gravitational
attraction only; which is unaffected by the specific nature of dark energy. A
net effect can be seen in masses, whose average values tend to be on the
larger than in the $\Lambda$CDM case by a factor of $\approx 2$, a
straightforward consequence of the larger $\Omega_{M}$ and $\sigma_{8}$, as
can be shown by a direct comparison of VDE to $\Lambda$CDM-vde results, that
turn out extremely close in these cases. On the other hand, the different
background evolution seems to affect $c-M$ relations only slightly, changing
the power law index $a(z)$ and normalization $b(z)$ by a $15\%$. In this case
we have also found that these values in general agree with previous results
from early dark energy studies such as those by Dolag et al. (2004), even
though in this case it would certainly be necessary to test the relation down
to smaller mass scales, where a better tuning of the parameter would be also
possible, and with a larger statistics on the higher scales. However, in
general, most of the halo-level effects which seem to characterize VDE can be
simply explained in terms of the different cosmological parameters, as we did
comparing these results to the outcomes of $\Lambda$CDM-vde simulations. For
the first time then, through the results of the series of $N$-body
simulations, we have shown that VDE cosmology provides a viable environment
for structure formation, also alleviating some observational tensions emerging
with $\Lambda$CDM. We have seen how the peculiar dynamics of this model leaves
its imprint on structure formation and growth, and in particular, how it
affects predictions for large scale clustering and halo properties. However, a
close comparison of the deep non-linear regime results with different sets of
observational data still needs to be performed, challenging us to improve the
accuracy of our simulations and at the same time devise new and reliable tests
which may shed some light not only on VDE but on the nature of dark energy in
general.
## Acknowledgements
We would like to thank Juan García-Bellido for his interesting suggestions and
discussions. EC is supported by the MareNostrum project funded by the Spanish
Ministerio de Ciencia e Innovacion (MICINN) under grant no.
AYA2009-13875-C03-02 and MultiDark Consolider project under grant
CSD2009-00064. EC also acknowledges partial support from the European Union
FP7 ITN INVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442). AK
acknowledges support by the MICINN’s Ramon y Cajal programme as well as the
grants AYA 2009-13875-C03-02, AYA2009-12792-C03-03, CSD2009-00064, and CAM
S2009/ESP-1496. G. Yepes would like to thank the MICINN for financial support
under grants AYA 2009-13875-C03, FPA 2009-08958, and the SyeC Consolider
project CSD2007-00050. JBJ is supported by the Ministerio de Educación under
the postdoctoral contract EX2009-0305 and also wishes to acknowledge support
from the Norwegian Research Council under the YGGDRASIL programme 2009-2010
and the NILS mobility project grant UCM-EEA-ABEL-03-2010. We also acknowledge
support from MICINN (Spain) project numbers FIS 2008-01323, FPA 2008-00592,
CAM/UCM 910309 and FIS2011-23000. The simulations used in this work were
performed in the Marenostrum supercomputer at Barcelona Supercomputing Center
(BSC).
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|
arxiv-papers
| 2012-05-08T13:37:43 |
2024-09-04T02:49:30.715424
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edoardo Carlesi, Alexander Knebe, Gustavo Yepes, Stefan Gottloeber,\n Jose Beltran Jimenez, Antonio L. Maroto",
"submitter": "Edoardo Carlesi Mr",
"url": "https://arxiv.org/abs/1205.1695"
}
|
1205.1835
|
# A Cusp in QED at $g=2$
Johann Rafelski and Lance Labun Department of Physics, University of Arizona,
Tucson, Arizona, 85721 USA
(April 2, 2013)
###### Abstract
We explore nonperturbative properties of the dimension-4 QED allowing a
gyromagnetic ratio $g\neq g_{\rm D}=2$. We determine the effective action
$V_{\mathrm{eff}}$ for an arbitrarily strong constant and homogeneous field.
Using the external field method, we find a cusp as a function of the
gyromagnetic factor $g$ in a) the QED $b_{0}$-renormalization group
coefficient, and in a b) subclass of light-light scattering coefficients
obtained in the long wavelength limit expansion. We discuss precision QED
results indicating an opportunity for resolution of known theory-experiment
disagreements. We show possibility of asymptotic freedom in an Abelian theory
for certain domains of $g$.
###### pacs:
12.20.-m, 11.15.Tk, 12.20.Ds, 13.40.-f
Motivation: No known particle has exactly the Dirac value $g_{\mathrm{D}}=2$
of the gyromagnetic ratio $g$. Determination of the higher order vacuum
fluctuation correction to $g$ provides the most precise test of Dirac-QED
(D-QED), and this test is facing current challenges: 1) There is a three
standard deviation discrepancy between theory and experiment involving the
directly measured muon gyromagnetic ratio Bennett:2006fi ; Venanzoni:2012yp ,
and theoretical result in 8th order $\alpha^{4}$ Kinoshita:2005sm . 2) In a
measurement of the muonic hydrogen Lamb shift, a discrepancy between D-QED and
experiment is reinterpreted as a revised size of the proton Antognini:1900ns ;
Pohl:2010zza , inconsistent by five standard deviations with other available
experimental information Mohr:2012tt .
In our opinion these recent experimental developments signal need to reexamine
the framework of D-QED focusing on the electron–magnetic field interaction.
Recall that the D-QED perturbative expansion is for the Dirac value
$g=g_{\mathrm{D}}=2$. However, the theory of charged particles interacting
with the photon field, which we refer to generically as QED, taken as a stand-
alone theory has a point-like electron or muon where in general $g\neq 2$ due
to modifications by other interactions. Perturbative expansion around
$g=g_{\mathrm{D}}$ is not appropriate should this value $g=g_{\mathrm{D}}$ be
a singular point. The aim of present work is to show that the there is a
signularity at $g=g_{\mathrm{D}}$, to study the nature of this singularity,
and to lay foundation for a framework allowing exploration of
$|g|>g_{\mathrm{D}}$.
To achieve these goals we consider the extension to $g\neq 2$ based on the
renormalizable dimension-4 action VaqueraAraujo:2012qa ;
AngelesMartinez:2011nt . We find a singularity at $g=g_{\mathrm{D}}$ employing
the external field method: this means that we study the vacuum properties in
presence of external constant and homogeneous electromagnetic fields,
integrating out fluctuations of spin-1/2 particles with $g\neq
g_{\mathrm{D}}$. The resulting effective potential $V_{\mathrm{eff}}$ is a
generalization of the Heisenberg-Euler-Schwinger (HES) effective action
Heisenberg:1935qt ; Weisskopf ; Schwinger:1951nm ; Reuter:1996zm ;
Dunne:2004nc to arbitrary value of $g$. Our result is regular for all
$|g|\leq g_{\mathrm{D}}$ Labun:2012jf .
For $|g|>g_{\mathrm{D}}$ the HES effective action derived in proper time
formulation Labun:2012jf based on dimension-4 renormalizable QED formulation
becomes singular. We propose a natural extension for all values
$|g|>g_{\mathrm{D}}$ which shows that $g=g_{\mathrm{D}}$ (and other periodic
recurrent values) is a cusp point as function of $g$. This extension resolves
the known difficulties in the theoretical framework of $g\neq g_{\mathrm{D}}$
theories Veltman:1997am . Considering the beta-function and light-light
scattering coefficients computed below, we discuss how the results we obtain
for $g\neq g_{\mathrm{D}}$ can be tested by experiment and may impact at the
required level the already described two challenges QED faces today.
Introducing magnetic moment $\mathbf{|g|\neq g_{\mathrm{D}}}$: One way to
account for $|g|\neq g_{\mathrm{D}}$ is to complement the Dirac action with an
incremental Pauli interaction term
$\delta\\!\mu\,(\vec{\sigma}\cdot\vec{B}+i\vec{\alpha}\cdot\vec{E})=\delta\\!\mu\,\sigma_{\alpha\beta}F^{\alpha\beta}/2$
where $\vec{E},\vec{B}$ are the electromagnetic fields, $F^{\alpha\beta}$ the
electromagnetic field strength tensor,
$\sigma_{\alpha\beta}=(i/2)[\gamma_{\alpha},\gamma_{\beta}]$ with
$\gamma_{\alpha}$ the usual Dirac matrices, and $\vec{\sigma}$, and
$\vec{\alpha}=\gamma_{5}\vec{\sigma}$ are the Pauli-Dirac matrices. This
incremental Pauli interaction is a dimension 5 operator,
$[\,\overline{\psi}\sigma_{\alpha\beta}F^{\alpha\beta}\psi]=L^{-5}$. The
coefficient $\delta\\!\mu$ consequently has dimension length, which in the
case of a composite particle such as the proton is naturally related to the
particle size. This Dirac-Pauli (DP) equation has been a popular and effective
tool to describe to lowest order the magnetic moment dynamics of a composite
particle of finite size, e.g. proton.
A distinct approach to introduce an effective action for $|g|\neq
g_{\mathrm{D}}$ is obtained by adding the full Pauli interaction term to the
Klein-Gordon action
${\cal
L}=\bar{\psi}\left[\Pi^{2}-m^{2}-\frac{g}{2}\frac{e\sigma_{\alpha\beta}F^{\alpha\beta}}{2}\right]\psi,$
(1)
where $\Pi_{\alpha}=i\partial_{\alpha}+eA_{\alpha}$. Note that the dimension
of the $\psi$ field is $[\psi]=L^{-1}$ and consequently the Pauli interaction
is dimension 4. We refer to the study of QED based on Eq. (1) as $g$-QED, and
the dynamical equation following from Eq. (1) as the Klein-Gordon-Pauli (KGP)
equation. $g$-QED is the $s=1/2$ case in the study of particles of all spins
in the Poincaré group framework initiated by Rarita and Schwinger
Rarita:1941mf . For recent developments see references in introduction to
Ref.VaqueraAraujo:2012qa .
Since there are at least two distinct paths to introduce $g\neq 2$ corrections
into relativistic particle dynamics, the question is in what sense these could
be equivalent and if not, which of the two forms is appropriate for study of
particle dynamics and/or vacuum structure and under what conditions:
1) The DP approach, involving a dimension-5 operator, requires new counter
terms in each order. This is limiting DP approach to situations in which the
physical particle properties are known and vacuum fluctuations need not be
considered. Even so, vacuum fluctuations and the effective action
$V_{\mathrm{eff}}$ have been considered in the DP i.e. modified D-QED approach
PauliTerm ; Diet78 ; Lav85 .
2) In $g$-QED the magnetic moment is point-like and consideration of $g\neq
g_{\rm D}$ does not require a higher dimensioned operator. Therefore the
quantum field theory is renormalizable VaqueraAraujo:2012qa ;
AngelesMartinez:2011nt , requires a finite number of counter terms, and vacuum
fluctuations can be considered in any higher order. It should be remembered
that $g$-QED constitutes an expansion around $g=0$ and not around $g=\pm 2$ as
is the case for DP approach. Properties of the KGP-originating effective
action were considered for general spin in Ref. Kruglov:2001dp , but this work
did not recognize the restricted validity domain of the perturbative approach,
for spin-1/2 $-2\leq g\leq 2$ .
3). A recent discussion of quantum field amplitudes with an anomalous moment
Larkoski:2010am also arrives at a second-order effective theory, but for a
reduced two-component spinor. In view of the derivation and properties of
their effective theory, a relation between KGP and DP can at best arise in an
infinite order resummation in some specific applications and not in general.
Considering that Eq. (1) is 2nd order in time and has four components, the
number of dynamical degrees of freedom present in Eq. (1) is 8. That is, there
are twice as many as in usual Dirac theory. For the case $g=2$ Eq. (1) can be
presented as the square of the operator $\gamma_{5}D,\
D=\gamma_{\alpha}(i\partial_{\alpha}+eA^{\alpha})-m$ and
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3},\gamma_{5}^{2}=1$ is the
5th Dirac matrix. This means that for $g=2$ Eq. (1) comprises exact
duplication of the Dirac degrees of freedom. For $g\neq 2$ one must search for
a projection restricting the full Hilbert space to the physical states.
Veltman has considered reduction of the number of dynamical components working
in a two-component formulation. However, there are unresolved challenges
Veltman:1997am in particular related to self-adjointness of the resulting
spectrum and thus conservation of probability in temporal evolution. By
individually characterizing states, we will present another resolution of this
problem that works in presence of externally applied fields and providing a
result within the external field method.
Eigenvalue-sum periodicity as a function of $g$: Our initial objective is to
identify the physics content of the 8 degrees of freedom and separate the
Hilbert space into two equal size parts that each individually comprise a
complete set of states at a fixed given value of $g$. To do so, we consider
the Landau-orbit spectrum of the operator in brackets in Eq. (1) in the
presence of a constant magnetic field $\vec{B}$
$E_{n}=\pm\sqrt{m^{2}+p_{z}^{2}+Q|e\vec{B}|\,[(2n\\!+\\!1)\mp g/2]},\quad
Q=\pm 1,$ (2)
where $p_{z}$ is the one dimensional continuous momentum eigenvalue and $n$ is
the Landau orbit quantum number. We have made explicit the presence of 8
eigenvalues for each value of $\vec{B}$, corresponding to three different
possible choices of the signs. There are the usual two roots in Eq. (2), a
known feature of relativistic dynamics also seen in the Landau spectrum of the
Dirac equation where the negative energy states become positive energy
antiparticle ‘hole’ states. The $\mp$ factor inside the root in Eq. (2) arises
from two possible particle spin projections onto magnetic field, corresponding
to the spin degeneracy.
Having recalled these usual features, we turn our attention to the new feature
and write the eigen energy Eq. (2) in the form
$K=\frac{E_{n}^{2}-m^{2}-p_{z}^{2}}{|e\vec{B}|}=Q\,[(2n\\!+\\!1)\mp g/2],\quad
Q=\pm 1.$ (3)
This exhibits a new spectrum duplication related to two possible values of
$Q$. The quantity $K$ is shown in the top portion of figure 1 as function of
$g$. We see that between $-2\leq g\leq 2$ there is an exact duplication of the
spectrum corresponding to $Q=1$ and $Q=-1$. The ‘squared’ Dirac operator
produces two eigenstate-space copies which can be separated in particular
applications. These are two sectors of the Hilbert space with the same
physical content, and the $Q=-1$ eigenvalues can be omitted. Thus for $-2\leq
g\leq 2$ the effective action is obtained by the usual procedure, and the
results have already been presented Labun:2012jf .
For $|g|>2$, new physics content arises for external fields of any strength,
including arbitrarily weak. First we note that taking Eq. (2) expression at
face-value, naively some eigenstates could have $E^{2}<m^{2}$, which implies
existence of bound localized states in the presence of a constant magnetic
field. Such solutions are not required for completeness and would violate
Lorentz symmetry; for these reasons, such states cannot be admitted in the
spectrum. This situation differs from the $m^{2}+p_{z}^{2}\to 0$ limit, in
which states having $K<0$ signal Nielsen-Olesen instabilities of the
conventional vacuum state Nielsen:1979xu . In the context of spin-1 charged
massless gluons in presence of color magnetic field, this instability has been
used to obtain the leading coefficient of QCD renormalization group
$\beta$-function Nielsen:1980sx . In $g$-QED for finite mass in the weak field
limit, there is no magnetic instability.
To compute the effective action we must define which states contribute to the
physical spectral sum. The first step is to accomplish (like for the case
$|g|\leq 2$) separation of the Hilbert space into two sectors. We divide the
states according to whether $K\geq 0$ or $K\leq 0$ and denote the respective
sectors ${\cal K}^{\pm}$. The limit $K=0$ where two states coincide occurs at
$g=2$ since the KGP operator can be written as exact square of the Dirac
operator. This situation recurs with the shift of $g$ by $4k,k\in\mathbb{Z}$.
There is no change in the number of states in each of the Hilbert space
sectors ${\cal K}^{\pm}$ as an equal number of single particle states is
exchanged between both sectors.
The principle we use to determine which states enter the spectral sum is that
there should be no localized bound states in a constant magnetic field. In the
notation just introduced, we require $K\geq 0$ and the ${\cal K}^{+}$ sector
is chosen as representing the physical spectrum. This is an extension from the
regular case $|g|\leq 2$, where the usual procedure sums over the $Q=+1$
states and is equivalent to summing over the ${\cal K}^{+}$ state space.
Seeing as $K\geq 0$ implies $E^{2}\geq m^{2}$, the physics is a continuous
extension of the case $g=2$, for which it is proved that $E^{2}\geq m^{2}$ for
arbitrary magnetic fields, i.e. there are no bound states Gornicki:1987hv .
Looking far outside the principal domain $-2\leq g\leq 2$, we see that
relativistic Landau eigenstates cross between ${\cal K}^{\pm}$ at each
$g_{k}=2+4k,k\in\mathbb{Z}$. As the graphic representation top frame of Fig. 1
shows, for each of the Hilbert space sectors ${\cal K}^{\pm}$ we have
periodicity of the Landau levels a function of $g$. Therefore, the sum
$\sum_{n}E_{n}$ over ${\cal K}^{+}$ leading to the real part of
$V_{\mathrm{eff}}(\vec{B}^{\,2})$, is a periodic function of $g$, a result we
will find explicitly. This periodicity does not apply to individual Landau
eigenvalues as is seen in Eq. (2). In computation of vacuum fluctuations the
truncation of the Landau eigenstate $n$-sum to any finite value breaks the
periodicity as well.
The choice of ${\cal K}^{+}$ as the physical state space has clear advantages
and resolves the challenges encountered by Veltman Veltman:1997am : In
addition to maintaining self-adjointness of the KGP system, it makes the
quantum field theories based on semi-spaces ${\cal K}^{\pm}$ each individually
unitary, because the number of states is conserved in transiting through the
singular points e.g at $|g|=2$, and for $|g|>2$ we omit the localized
solutions. Moreover, our proposal makes the spectrum and by extension the
quantum theory a continuous and analytic extension from the domain $|g|\leq
2$. Our approach preserves translation invariance of the vacuum, which would
be broken by any localized bound states in the constant-field-filled vacuum.
It is critical to note that had we separated the sectors along the sign of
$Q$, the contents of the theory would be different for $|g|>2$ and unitarity
would be violated since the ‘wrong’ levels would be included in the physical
half-space.
Figure 1: Top: Squared eigenvalues Eq. (2) of KGP in magnetic field; the solid
and (blue) dashed lines are for $Q=+1$, and respectively $(-)$ and $(+)$ spin
eigenvalue; the dotted and dash-dotted (red) lines are for $Q=-1$, and
respectively $(+)$ and $(-)$ spin eigenvalue. Bottom: coefficient functions:
$f_{1,0}(g)$ as defined in Eq. (12) and $f_{0,2}$ and $f_{2,0}$ as defined in
Eq. (13). Two full periods are shown. The values of $g$ where the sign of the
functions $f_{i,j}$ changes is indicated.
Effective action for $\mathbf{|g|\leq 2}$: We briefly summarize results for
${|g|\leq 2}$ Labun:2012jf , as these are needed to understand the present
case of ${|g|>2}$. For constant fields the effective action is manifestly
covariant, and can be written as a function of the Lorentz-invariant field-
like quantities $a,b$
$\displaystyle b^{2}-a^{2}$
$\displaystyle=\vec{B}^{2}-\vec{E}^{2}=\frac{1}{2}F_{\alpha\beta}F^{\alpha\beta}\equiv
2{\cal S},$ (4) $\displaystyle(ab)^{2}$
$\displaystyle=(\vec{E}\cdot\vec{B})^{2}=\left(\frac{1}{8}F^{\alpha\beta}\varepsilon_{\alpha\beta\kappa\lambda}F^{\kappa\lambda}\right)^{2}\equiv{\cal
P}^{2},$ (5)
where $\pm a$ are electric-field-like and $\pm ib$ are the magnetic-field-like
eigenvalues of $F^{\alpha\beta}$. $a$ is considered electric-like because
$a\to|\vec{E}|$ on taking the limit $b\to 0$, and similarly $b\to|\vec{B}|$ in
the limit $a\to 0$.
The Schwinger-Fock proper time method Schwinger:1951nm to evaluate the
effective action exploits properties of the ‘squared’ Dirac equation and thus
it can be used to study arbitrary value of $g$. The effective action can be
written in form
$V_{\mathrm{eff}}=\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{du}{u^{3}}\,e^{-i(m^{2}-i\epsilon)u}F(eau,ebu,\frac{g}{2}).$
(6)
For $g=0,2$ the proper time integrand $F(eau,ebu,g)$ was reviewed in Ref.
Dunne:2004nc . The generalization throughout the interval $|g|\leq 2$ is
accomplished by inserting into Schwinger’s Eq. (2.33) in the last term a co-
factor $g/2$ leading to Labun:2012jf .
$F(x,y,\frac{g}{2})=\frac{x\cosh(\frac{g}{2}x)}{\sinh
x}\frac{y\cos(\frac{g}{2}y)}{\sin y}-1,\ \ \left|\frac{g}{2}\right|\leq 1.$
(7)
The subtraction $-1$ in Eq. (7) removes the field-independent constant. The
logarithmically divergent charge renormalization term is isolated and
discussed below. Note that Eq. (6) would be divergent for $|g|>2$ if Eq. (7)
were to be used in this domain.
Effective action for $\mathbf{|g|>2}$: To extend Eq. (7) to $|g|>2$, we
consider in more detail the eigenvalue summation method we introduced above,
following the work of Heisenberg and Euler Heisenberg:1935qt and Weisskopf
Weisskopf . The mathematical tool used was the L. Euler summation formula,
leading to the Bernoulli functions $B_{2k}(x)$ and Bernoulli numbers ${\cal
B}_{2k}\equiv B_{2k}(0)$. The sum of the Landau energies Eq. (2) involves the
form $\sum_{n}f(x+n)$. L. Euler developed the technique for such sums, which
manifest an integer shift symmetry in the variable $x\to x+n^{\prime}$ Euler ;
Apostol . Due to this shift symmetry, the Bernoulli functions $B_{2k}(x)$ that
arise in the context of L. Euler summation of Landau energies $E_{n}$, Eq. (2)
are the periodic Bernoulli functions, given by the Fourier series Luo
$\tilde{B}_{2k}(t)=(-1)^{k-1}\frac{(2k)!}{2^{2k-1}}\sum_{n=1}^{\infty}\frac{\cos(2\pi
nt)}{(n\pi)^{2k}},$ (8)
(here only needed for an even value of index, $2k$). In the unit interval,
$0\leq t\leq 1$, the periodic Bernoulli functions are equal to the Bernoulli
polynomials, e.g. $\tilde{B}_{2}(t)=B_{2}(t)=t^{2}-t+1/6,\ 0\leq t\leq 1$.
Outside the unit interval, the periodic Bernoulli functions Eq. (8)
$\tilde{B}_{2k}(t)$ repeat the polynomials’ behavior on $0\leq t\leq 1$ in
each subsequent period.
Dividing the Landau energies by $2|e\vec{B}|$ to make the coefficient of $n$
unity, we see that $t\to g/4+1/2$ and hence we recognize that the periodic
Bernoulli functions with argument $t=g/4+1/2$ appears in the effective action,
arising from the summation of eigenvalues. The explicit representation of the
argument of Eq. (6) in terms of Bernoulli functions is arrived at employing
the analytic transformation of the integrand of Eq. (6) Muller:1977mm ;
Cho:2000ei .
$\displaystyle F(x,y,\frac{g}{2})$
$\displaystyle\>=(x^{2}\\!-\\!y^{2})\,2\\!\sum_{n=1}^{\infty}\frac{\cos
n\pi(\frac{g}{2}+1)}{(n\pi)^{2}}+\left(x^{2}\\!-\\!y^{2}\right)^{2}2\sum_{n=1}^{\infty}\frac{\cos
n\pi(\frac{g}{2}+1)}{(n\pi)^{4}}$ (9)
$\displaystyle+(xy)^{2}4\\!\left(\sum_{n=1}^{\infty}\frac{\cos
n\pi(\frac{g}{2}+1)}{(n\pi)^{4}}-3\left(\sum_{n=1}^{\infty}\frac{\cos
n\pi(\frac{g}{2}+1)}{(n\pi)^{2}}\right)^{2}\right)+F_{6}(x,y,\frac{g}{2}),$
where we separated the lowest powers of fields from an exact remainder
function
$\displaystyle
F_{6}(x,y,\frac{g}{2})=y^{2}f_{6}(y^{2},\frac{g}{2})-x^{2}f_{6}(-x^{2},\frac{g}{2})-f_{6}(-x^{2},\frac{g}{2})f_{6}(y^{2},\frac{g}{2})\>,\qquad
f_{6}(y^{2},\frac{g}{2})=2y^{4}\sum_{n=1}^{\infty}\frac{\cos
n\pi(\frac{g}{2}+1)}{(n\pi)^{4}(y^{2}+(n\pi)^{2})}.$ (10)
The new mathematical element on the RHS in Eq. (9) is that to assure the
necessary periodicity we introduced in accordance with Eq. (8) a series of
Bernoulli functions with $t=g/4+1/2$. We note that the right hand side of Eq.
(9) agrees exactly with the known expansion Cho:2000ei in the domain of Eq.
(7) $|g|\leq 2$. This expression provides an analytical continuation into the
domain $|g|>2$ having the periodicity property of the effective action
identified in study of the full set of eigenvalues.
Each term in Eq. (9) produces a well-defined result for all $g$ upon
performing the proper time integral Eq. (6). The form Eq. (9) is thus a unique
and convergent extension to $|g|>2$ determined by the Euler summation of the
eigenvalues Eq. (2). Even after the removal of the charge renormalization
subtraction term (first term on RHS of Eq. (9)) the remainder of the effective
action is manifestly periodic in $g$ but not in $e$.
Nonperturbative in $\mathbf{g}$ renormalization group $\mathbf{\beta}$
function: The first non constant term on the right hand side of Eq. (9)
proportional to $a^{2}-b^{2}$ isolates the logarithmically divergent one-loop
$\mathcal{O}(\alpha)$ $V_{\mathrm{eff}}$ subtraction required for charge
renormalization. The coefficient of this term is related to the
$\beta$-function coefficient $b_{0}$ as is discussed e.g. in section 5.1 in
Ref. Dunne:2004nc . The two next terms $(a^{2}-b^{2})^{2}$ and $(ab)^{2}$
correspond to lowest order effective field-field interaction potentials
describing light-light scattering. Setting in the remainder denominator on RHS
of Eq. (10) $y=0$ produces next term in the expansion, etc.
We now consider explicitly the running of the coupling constant $\alpha$
within the $g$-QED loop expansion of the $\mathbf{\beta}$-renormalization
function
$\beta\equiv\mu\frac{\partial\alpha}{\partial\mu},\quad\beta(\alpha)=-\frac{b_{0}}{2\pi}\alpha^{2}+\frac{b_{1}}{8\pi^{2}}\alpha^{3}+\ldots\,.$
(11)
The first sum in Eq. (9), for $g=2$, $\sum_{n=1}^{\infty}1/(\pi n)^{2}=1/6$
and implies the value of $b_{0}=-4/3$, where factor 4 indicates the 4
components of spin-1/2 particle. For arbitrary $g$, $b_{0}(g)$ is obtained
using Eq. (8) to identify this sum as $\tilde{B}_{2}(g/4+1/2)$. The character
of this function is manifest by reconnecting periodic domains of the familiar
Bernoulli polynomial $B_{2}(t)=t^{2}-t+1/6$ and the resulting $b_{0}(g)$
coefficient is given in each domain $g\in[g_{k-1},g_{k}]$
$b_{0}=\>-\frac{4}{3}f_{1,0}(g)=-\frac{4}{3}\left(\frac{3}{8}(g-4k)^{2}-\frac{1}{2}\right),$
(12)
where $f_{1,0}(g)$ is shown in bottom frame of Fig. 1. The subscripts of
$f_{i,j}$ indicate the powers of the Lorentz invariants in polynomial
expansion $f_{i,j}{\cal S}^{i}{\cal P}^{j}$ in Eq. (9). We see in Fig. 1 that
as a function of $g$, the Dirac value $g_{\rm D}=\pm 2$ is an upper cusp point
with $f_{1,0}(g)\leq f_{1,0}(2)=1$. For clarity, two periods are shown in Fig.
1.
Note that our result arises from the KGP equation applying a nonperturbative
method in $g$ to one loop expansion. This approach is necessary in order to
obtain the behavior of the $\beta$-function for $|g|>2$. At $g=\pm 2$ we find
the unexpected cusp. This feature is missing in perturbative consideration of
$\beta(g)$ at one loop level which produces the same functional dependence on
$g$ as seen in Eq. (12) setting $k=0$. As our study shows, a perturbative
expansion around $g=0$ has a finite convergence interval $|g|\leq 2$.
The following implications for $g$-QED of the properties of the
renormalization group coefficient $b_{0}(g)$ are noteworthy – we address in
the following discussion the range of values of $g$ shown in Fig. 1:
1.) We recognize $g$ is an independent ‘large’ coupling constant. The first-
order D-QED expands around $g=\pm 2$, which points are identified as being
non-analytic in the $g$-QED framework.
2.) For any value of $g$ not at the cusp the magnitude $|b_{0}|$ decreases
(and thus the speed of ‘running’ decreases) compared to its value at $g=2$.
Considering that the coefficient of the magnetic spin term in Eq. (1) is
dimensionless there is no new scale appearing in association with $g$.
3.) The presence of the cusp in $b_{0}$ implies that the running coupling of
$g$-QED, $\alpha(q)$ comprises the cusp as well.
4.) A cross check and confirmation of our result for $b_{0}(g)$ is obtained in
perturbative domain considering the limit $g\to 0$ where $b_{0}(g\to 0)$
differs only by a minus sign and the number of degrees of freedom from the
known behavior of scalar particle ‘QED’. The minus sign is due to the
commutation relation needed in closing the fermion trace in loops, whereas it
is absent in scalar boson loops.
5.) In the principal domain $|g|\leq 2$ the functional dependence on $g$ we
find agrees with the result Eqs. (53–57) seen in Ref. AngelesMartinez:2011nt .
Specifically, the leading term for large $q^{2}$ of the vacuum polarization
function, evaluated within the framework of $g$-QED is $-\alpha
b_{0}(g)/(2\pi)\ln(-q^{2}/m^{2})$, seen explicitly in Eq. (55) of Ref.
AngelesMartinez:2011nt .
6.) As the above limit shows, for a range of appropriate gyromagnetic moment
values $g$ (including $g=0$) $b_{0}(g)>0$ is possible. This produces
asymptotic freedom behavior for fermions interacting alone with an Abelian
charge. The switch between the infrared stable and the asymptotically free
behavior occurs in the principal $g$-domain twice, at $g=\pm 2/\sqrt{3}=\pm
1.155$ and continues periodically e.g. for $g=4-2/\sqrt{3}=2.845$. This
mechanism of asymptotic freedom generation by $g$-driven sign reversal is
implicit in Eq. (56) of Ref. AngelesMartinez:2011nt (valid in principal
domain $|g|\leq 2$), but the new mechanism allowing Abelian confinement has
not been recognized there. The values of $g$ where the sign of the functions
$f_{i,j}$ changes is indicated in Fig. 1, up to periodic recurrence.
Light-light scattering as function of $\mathbf{g}$: We find that the cusp at
$|g|=2$ reappears in a directly observable phenomenon inherent in the
Heisenberg-Euler action, the light by light scattering. For the general case
of both electric and magnetic fields present, using Eq. (9) we find up to
fourth order in the fields
$\displaystyle V_{\mathrm{eff}}\simeq$
$\displaystyle\frac{\alpha}{2\pi}\frac{e^{2}}{m^{4}}\\!\left(\frac{f_{2,0}}{45}\>{\cal
S}^{2}+\frac{7f_{0,2}}{45}\>{\cal P}^{2}\right)$ (13) $\displaystyle
f_{2,0}(g)$ $\displaystyle=-120\tilde{B}_{4}(g/4+1/2)$ (13a)
$\displaystyle=-\frac{15(g-4k)^{4}}{32}+\frac{15(g-4k)^{2}}{4}-\frac{7}{2}$
(13b) $\displaystyle f_{0,2}(g)$
$\displaystyle=-\frac{60}{7}\left[\tilde{B}_{4}\left(\frac{g}{4}+\frac{1}{2}\right)-3\tilde{B}_{2}^{2}\left(\frac{g}{4}+\frac{1}{2}\right)\right]$
(13c) $\displaystyle=\frac{15(g-4k)^{4}}{224}-\frac{1}{14}$ (13d)
where both $f_{2,0}$ and $f_{0,2}$ are normalized to $g=2$ values and
presented in Fig. 1. $f_{0,2}$ includes a product of two Bernoulli functions
with cusp and so has a steeper cusp. Importantly $f_{0,2}$ enters the ${\cal
P}^{2}$ term which one actually measures in laser light scattering off a
magnetic field Rikken:2001zz ; QAexperiment . In general, our finding is that
all $f_{i,j}(g)$ for $j>0$ have cusps at $g=2$ whereas all $f_{i,0}(g),i>1$
are continuous and differentiable at $g=2$, being proportional to higher order
$>2$ Bernoulli functions that have vanishing derivatives at $g=2$. Thus only
coefficients of terms involving powers of the pseudo scalar field invariant
${\cal P}^{2}=(\vec{E}\cdot\vec{B})^{2}$ display cusps at $g=2$.
Discussion: We found new physics arising for $|g|>2$ for arbitrarily weak
fields in $g$-QED. We proposed a new eigenstate sorting based on sign of $K$,
Eq. (3), leading to a self-adjoint theory that retains Poincaré symmetry and
contains a complete set of particle-antiparticle states, and thus preserves
probability in time evolution and analyticity as function of $g$, up to a
countable set of singular points.
While Eq. (7) is an analytic function of $g$, the integral of Eq. (7) with the
proper time weight Eq. (6) does not exist for $|g|>2$. Thus a naive extension
of HES effective action to $|g|>2$ is not possible. This parallels the
observation that the Klein-Gordon-Pauli operator Eq. (1) is not self-adjoint
for $|g|>2$. We have presented a careful study of how the eigenstate level
crossing can be recognized and states assigned to half-spaces of the full
Hilbert space, leading to a natural self-adjoint extension and a valid
theoretical $g$-QED framework for $|g|>2$. The cusp and related
nonperturbative in $g$ effects arise from implementation of the self-adjoint
extension described. The origin of the cusp is in the periodic crossing of
eigenenergies in the spectrum of Landau eigenstates seen in upper section of
Fig. 1 showing the quantity $K$, Eq. (3).
The top frame of Fig. 1 illustrates the case of a (weak) magnetic field only,
a similar result is obtained for the case of an electric field only, leading
to the unique form Eq. (9). Within this expression we have shown cusps at
$g=g_{\rm D}$ for two physical quantities computed for arbitrary $g$:
$\bullet$ The renormalization group coefficient $b_{0}$ proportional to
function $f_{1,0}$, see Fig. 1;
$\bullet$ The light-by-light scattering in the long-wavelength limit
comprising a smooth function $f_{2,0}$, and for the term
$(\vec{E}\cdot\vec{B})^{2}$ the cusp function $f_{0,2}$, see Fig. 1.
These nonperturbative in $g$ one-loop results require infinitely many
contributions from eigenvalues Eq. (2), and for this reason the effects
predicted here are only visible in phenomena arising from vacuum fluctuations,
such as the $\beta$-function and effective light-light interactions. We have
checked that these results can be arrived at directly by the method of
$\zeta$-function regularization following Weisskopf Weisskopf .
Our results agree in the fundamental domain $-2\leq g\leq 2$ with earlier
perturbative work: the functional dependence on $g$ is explicit and the same
for the vacuum polarization as had been obtained in Ref.
AngelesMartinez:2011nt in Eq. (56). We have shown by explicit computation
that an expansion around $g=0$ is valid for $|g|\leq 2$ only.
Conclusions and outlook: Difficulties of D-QED as a stand-alone theory have
been known for some time, beginning with the work of G. Källén Kallen:1957ib ,
and perturbative-D-QED is believed by many to be semi-convergent only.
Exploration of $g\neq 2$ in a renormalizable theory requires the dimension-4
$g$-QED based on KGP equation. However, $g$-QED has to begin with 8 degrees of
freedom and appropriate division into two half-Hilbert spaces is required.
Restriction to the usual Dirac-like 4 degrees of freedom is difficult, as a
theory with $g\neq 2$ is in general not unitary Veltman:1997am . We resolved
this problem, and from the solution we discovered that the Dirac value
$g=g_{\mathrm{D}}=2$ is a cusp point of the effective action
$V_{\mathrm{eff}}$, Eq. (6) evaluated in renormalizable $g$-QED approach.
This finding implies that the D-QED expansion around $g=g_{\mathrm{D}}$ could
be incomplete at sufficiently high order. To see the problem, imagine that we
partially resum $g-2$ diagrams with Dyson-Schwinger method finding an
effective electron with $g>2$. In the next step we want to compute the vacuum
polarization inserts in other $g-2$ diagrams. Attempts in D-QED framework will
encounter new divergences as the $g-2$ correction is dimension-5 operator. On
the other hand, we can accomplish this task in $g$-QED: we use the non-
perturbative in $g$ renormalization group coefficient $b_{0}$ to characterize
the vacuum polarization loop insert and there are no new divergences. However,
the result contains the cusp, and thus is different from the finite order
perturbative expansion of D-QED.
We believe that the higher order vacuum polarization modification we described
would be most visible for the Lamb shift of muonic hydrogen which is dominated
by the vacuum polarization Antognini:2012ofa ; Jentschura:2010ej ;
Czarnecki:2005sz . Similarly, we expect that our light-light scattering cusp
modifies the corresponding contribution to the muon $g-2$. We discussed in
seperate work how the top-quark loop modifies the two photon Labun:2012fg and
two gluon Labun:2012ra decay of the Higgs. Attempts to evaluate these results
using D-QED methods, that is a $g-2$ non-renormalizable extension, would fail
in evaluation of the loop integrals which in general are divergent in this
case.
Our study shows how a complete theory of a point-like fermion with $|g|>2$ can
be constructed within $g$-QED in order to allow dynamical description of real
world spin-1/2 particles. We have obtained the HES effective potential for an
elementary particle with gyromagnetic ratio $g\neq 2$ nonperturbatively in
$g$, see Eq. (6) and Eq. (9). We demonstrated a cusp as function of $g$ at the
Dirac value $g=g_{\mathrm{D}}=2$. We have shown how this cusp enters the
$\beta$-function and $(\vec{E}\cdot\vec{B})^{2n}$ terms of light-light
scattering. An interesting theoretical consequence is the possibility of
asymptotic freedom in an Abelian theory with anomalous magnetic moment
originating in the reversal in sign of the renormalization group coefficient
$b_{0}$ for $g$ in specific domains much different from $g=2$.
Acknowledgments: We thank I. Bialynicki-Birula, and R. Stora for careful
reading and valuable comments. This work was supported by a grant from the US
Department of Energy, DE-FG02-04ER41318.
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|
arxiv-papers
| 2012-05-08T21:40:28 |
2024-09-04T02:49:30.740145
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Johann Rafelski and Lance Labun",
"submitter": "Johann (Jan) Rafelski",
"url": "https://arxiv.org/abs/1205.1835"
}
|
1205.1861
|
# Carbon dioxide emissions trading and hierarchical structure in worldwide
finance and commodities markets
Zeyu Zheng Department of Environmental Sciences, Tokyo University of
Information Sciences, Chiba 265-8501, Japan Kazuko Yamasaki Department of
Environmental Sciences, Tokyo University of Information Sciences, Chiba
265-8501, Japan Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA 02215, USA Joel Tenenbaum Center for Polymer Studies
and Department of Physics, Boston University, Boston, MA 02215, USA H. Eugene
Stanley Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA 02215, USA
###### Abstract
In a highly interdependent economic world, the nature of relationships between
financial entities is becoming an increasingly important area of study.
Recently, many studies have shown the usefulness of minimal spanning trees
(MST) in extracting interactions between financial entities. Here, we propose
a modified MST network whose metric distance is defined in terms of cross-
correlation coefficient absolute values, enabling the connections between
anticorrelated entities to manifest properly. We investigate 69 daily time
series, comprising 3 types of financial assets: 28 stock market indicators, 21
currency futures, and 20 commodity futures. We show that though the resulting
MST network evolves over time, the financial assets of similar type tend to
have connections which are stable over time. In addition, we find a
characteristic time lag between the volatility time series of the stock market
indicators and those of the EU CO2 emission allowance (EUA) and crude oil
futures (WTI). This time lag is given by the peak of the cross-correlation
function of the volatility time series EUA (or WTI) with that of the stock
market indicators, and is markedly different ($>20$ days) from 0, showing that
the volatility of stock market indicators today can predict the volatility of
EU emissions allowances and of crude oil in the near future.
###### pacs:
PACS numbers:89.65.Gh, 89.20.-a, 02.50.Ey
## I Introduction and Method
In the study of complex systems, there has been much work demonstrating the
usefulness of extracting underlying structure from the correlations found in
statistical data Mantegna ; Bonanno2 ; Bonanno1 ; Onnela1 ; Onnela2 ; Song ;
Micciche ; Tumminello ; ManandStan ; Wang ; laloux ; Plerou1 ; Plerou2 ;
Utsugi . Any means of selection of statistically reliable information from
correlation matrices has been dubbed a “filtering procedure” Tumminello .
Useful examples of filtering procedures that use a correlation matrix from
return time series are: hierarchical clustering Mantegna ; Bonanno2 ; Bonanno1
; Onnela1 ; Onnela2 ; Song ; Micciche , procedures based on the random matrix
theory Wang ; laloux ; Plerou1 ; Plerou2 ; Utsugi , and networks from minimum
spanning trees Mantegna ; Bonanno2 ; Bonanno1 ; Onnela1 ; Onnela2 .
Correlation structure studies are not limited only to stock return time series
Mantegna ; Bonanno2 ; Bonanno1 , but also extend to quasi-synchronously
recorded time series of worldwide stock exchange market indices Bonanno1 and
stock return volatility increments Micciche .
Financial time series can include not only stock price, but also many other
types of data, such as commodities price, treasury yield, market index, etc….
Investigation of multi-type quasi-synchronous financial data may yield
insights into the interdependent relationships of markets and commodities.
Moreover, a relationship map of financial assets can highlight the movement of
speculative capital.
We investigate 69 daily financial series from the time period spanning January
2007 to September 2011. The data set includes 21 currency futures, 20
commodity futures which are taken from http://data.theice.com, and 28 stock
market indicators, which are taken from http://finance.yahoo.com (see
Appendix).
Recently, papers have shown the usefulness of the correlation structure
described by an ultrametric space and a corresponding hierarchical
organization for financial return time series Mantegna ; Bonanno2 ; Bonanno1 .
The approach requires the definition of a metric distance. Because correlation
does not fulfill the three metric axioms that define a metric, the
aforementioned papers use the Mantegna-Sornette distance defined by
$d_{ij}=\sqrt{2(1-\rho_{ij})}$ (1)
for each pair of elements $i$ and $j$, where $\rho_{ij}$ is the correlation
coefficient of the two time series given by feller
$\rho_{ij}\equiv\frac{\langle Y_{i}Y_{j}\rangle-\langle Y_{i}\rangle\langle
Y_{j}\rangle}{\sqrt{(\langle{Y_{i}}^{2}\rangle-\langle
Y_{i}\rangle^{2})(\langle{Y_{j}}^{2}\rangle-\langle
Y_{j}\rangle^{2})}}=\frac{\Big{\langle}\bigg{(}Y_{i}-\langle
Y_{i}\rangle\bigg{)}\bigg{(}Y_{j}-\langle
Y_{j}\rangle\bigg{)}\Big{\rangle}}{\sigma_{Y_{i}}\sigma_{Y_{j}}},$ (2)
where $\langle...\rangle$ denotes the mean.
This distance $d_{ij}$ fulfills the three axioms metric of a metric: i)
$d_{ij}=0$ if and only if $i=j$, ii) $d_{ij}=d_{ji}$, and iii) $d_{ij}\leq
d_{ik}+d_{kj}$ Mantegna ; Bonanno1 ; Bonanno2 .
In this work, we make a modification to the metric above, based on the
reasoning that if two time series have a very large negative correlation, they
should still be considered close to each other in ultrametric “correlation”
space, since this would indicate a strong connection, regardless of the sign.
Likewise, weak correlations of either sign indicate a weaker connection. On
this basis, strong correlations of either sign should be considered closer
than weak ones.
The value of this modification can be readily seen in situations where two
entities have strong anti-correlations, as has been observed in the
relationship between bond and stock markets (e.g. between UK gilts and the
FTSEMIB), between stocks and currency futures, and between industries and
their inputs (e.g. the price of oil and the value of airline stock) Norden ;
Kwan ; Hammoudeh ; businessweek . Here, use of the conventional Mantenga-
Sornette metric would likely result in any of these two entities manifesting
on opposite sides of a tree, even though we know them to be very closely
linked. Use of our generalized metric ensures that such entities will be
placed nearby when such appropriately strong relations exist.
For this reason, we replace the Pearson correlation coefficient in Eq. 1 with
the absolute value of the Pearson correlation coefficient, defining a new
distance as
$d_{ij}=\sqrt{2(1-|\rho_{ij}|)}$ (3)
with the $\rho_{ij}$ defined as before in Eq. 2 as the correlation coefficient
of assets $i$ and $j$.
This equation also fulfills three axioms of a metric distance. Because the set
of assets considered has no cases such that $\rho_{ij}=-1$, the first axiom is
satisfied on this set in that $d_{ij}=0$ if and only if the correlation is
total ($\rho=1$, meaning that the stocks perform the same stochastic process).
The second axiom (that of symmetry) is trivially satisfied because
$\rho_{ij}=\rho_{ji}$ by definition of the Pearson correlation coefficient.
For the validity of axiom (iii), consider three time series ${Y_{i}}$,
${Y_{j}}$ and ${Y_{k}}$, which have means equal to 0 and standard deviations
equal to 1. All times series have the same length. In order to prove
$d_{ij}\leq d_{ik}+d_{kj}$, firstly we define two new time series as
$Y^{\prime}_{i}\equiv\begin{cases}Y_{i},&\mbox{if }\rho_{Y_{i},Y_{k}}\geq 0\\\
-Y_{i},&\mbox{if }\rho_{Y_{i},Y_{k}}<0)\end{cases}$ (4)
$Y^{\prime}_{j}\equiv\begin{cases}Y_{j},&\mbox{if }\rho_{Y_{j},Y_{k}}\geq 0\\\
-Y_{j},&\mbox{if }\rho_{Y_{j},Y_{k}}<0.\end{cases}$ (5)
So we can rewrite $d_{ik}+d_{kj}$ of distance Eq. 3 by using $Y^{\prime}_{i}$
and $Y^{\prime}_{j}$ as
$\displaystyle d_{ik}+d_{kj}$
$\displaystyle=\sqrt{2(1-|\rho_{Y_{i},Y_{k}}|)}+\sqrt{2(1-|\rho_{Y_{j},Y_{k}}|)}$
(6)
$\displaystyle=\sqrt{2(1-\rho_{Y^{\prime}_{i},Y_{k}})}+\sqrt{2(1-\rho_{Y^{\prime}_{j},Y_{k}})}$
(7) $\displaystyle\geq\sqrt{2(1-\rho_{Y^{\prime}_{i},Y^{\prime}_{j}})}$ (8)
$\displaystyle\begin{cases}=\sqrt{2(1-|\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}|)},&\mbox{if
}\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}\geq 0\\\
\geq\sqrt{2(1-|\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}|)},&\mbox{if
}\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}<0\\\ \end{cases}$ (9)
$\displaystyle=\sqrt{2(1-|\rho_{Y_{i},Y_{j}}|)}\equiv d_{ij}.$ (10)
Thus $d_{ik}+d_{kj}\geq d_{ij}$ , and our metric satisfies the three axioms of
a metric.
For each of the 22 financial time series, we calculate the return time series,
defined as the change of logarithmic price of time series $i$
$R_{i}(t)\equiv\ln(Y_{i}(t+1))-\ln(Y_{i}(t)).$ (11)
Here $Y_{i}(t)$ is the daily price time series of financial asset $i$. For
each of the 22 time series, we also calculate the volatility time series which
is defined simply as the absolute value of the return $|R_{i}|$
$V_{i}(t)\equiv|R_{i}(t)|=|\ln(Y_{i}(t+1))-\ln(Y_{i}(t))|.$ (12)
Additionally, we define the cross-correlation for our analysis. Consider two
time series $\\{y_{t}\\}$ and $\\{y^{\prime}_{t}\\}$. The cross-correlation
between $\\{y_{t}\\}$ and $\\{y^{\prime}_{t}\\}$ is given by
$C_{y,y^{\prime}}(n)\equiv\overline{(y_{t}-\mu)(y^{\prime}_{t+n}-\mu^{\prime}))}/(\sigma\sigma^{\prime}),$
(13)
where $\mu$ and $\mu^{\prime}$ are the respective means and $\sigma$ and
$\sigma^{\prime}$ are the respective standard deviations of the series
$\\{y_{i}\\}$ and $\\{y_{i}^{\prime}\\}$.
The efficient market hypothesis, a basic tenet of modern economics, states
that markets are approximately efficient, meaning that one cannot consistently
achieve returns better than the market because all information about an asset
is already incorporated into that asset’s price Samuelson ; ManandStan . As a
result, it is believed that the long-range memory cannot exist in any return
time series. Suppose that long-range auto-correlations exist in a return time
series: investors may then obtain benefits by using information, which stands
in contradiction to the principle of an efficient market. Consider the cross-
correlation function (Eq. (13)) between return time series of asset $i$ and
asset $j$. Any significant cross-correlations $C_{R_{i},R_{j}}(n)$ in $n\neq
0$ of two return time series would also contradict the existence of an
efficient market. Therefore we can assume that significant cross-correlations
$C_{R_{i},R_{j}}(n)$ will only exist for the case $n=0$.
However, because trading occurs at different times in different cities, some
markets are open when others are closed. The effect of non-synchronous trading
in time series analysis has been well stated Lo ; Lin . In fact, the highest
degree of correlation between different markets may be detected at a one-day
time lag because of the time difference.
Therefore, significant cross-correlations $C_{R_{i},R_{j}}(n)$ may also exist
for $n=-1$ or $n=1$. Additionally, we only care about the magnitude and not
the sign of cross-correlation. Thus, we define the absolute correlation
coefficient as
$\rho_{ij}=\max\left(\left|C_{R_{i},R_{j}}(n)\right|\right)$ (14)
for $n=-1,0,1$.
In volatility time series, long-range correlations $C_{V_{i},V_{j}}(n)$ have
been shown to exist [10-13]. It follows that significant cross-correlations
$C_{V_{i},V_{j}}(n)$ may exist for $n\gg 0$ or $n\ll 0$. Additionally, the
existence of long-range negative correlations between past returns and future
volatility Bouchaud ; Perello , known as the leverage effect, has also been
reported. This correlation is moderate and decays exponentially over the long
term. However, while both of these types of correlations may help predict the
financial risks on a long-range time interval, we point out that neither the
negative correlation between returns and volatilities nor long-range
autocorrelation of volatility can be used to obtain benefits. This is because
the price volatility does not include the direction of price changes, and so
neither contradicts the efficient market hypothesis.
As a sample, the correlation functions of volatility $C_{V_{i},V_{j}}(n)$ and
return $C_{R_{i},R_{j}}(n)$ between the FTSE100 and DJIA are shown in Fig. 1.
The peaks (highest correlations) of both correlation functions are near n=0;
however, the correlation function for returns is fast decaying, quickly
approaching 0 for $n\neq 0$, while the volatility correlation function is slow
decaying with $C_{V_{i},V_{j}}(n)>0$ for $n>-50$ and $n<50$.
We now turn our attention to the stability and structure of MSTs, which are
made using the distance based defined in Eq. 3. Following our discussion on
MSTs, we will show the correlation function graphs of volatility that relate
the correlation $C(n)$ to the time-lag $n$, specifically focusing on the value
of $n$ that gives the LOWESS (the smoother which uses locally weighted
polynomial regression) equal to its maximum value. Here, the LOWESS is defined
by a complex algorithm, proposed by W. S Cleveland Cleveland in 1981. For
each value, we define 10th nearest neighbor values as the local region, which
is used to calculate the LOWESS value.
An MST is defined as the set of $n-1$ links that connect a set of $n$ elements
in the smallest possible total distance Graham . MSTs have been used in prior
papers Mantegna ; Bonanno2 ; Bonanno1 ; Onnela1 ; Onnela2 to connect
financial data, illustrating the MST’s usefulness in highlighting the
interactions between a number of financial time series.
In Fig. 2, we find that, although MSTs show significantly different structures
in different calendar years, the same type of financial assets tend to group
together consistently over time.
We also find that the stock market indicators (blue in Fig. 2) and currency
futures (green) groups show stronger interconnections than the commodities
(red) group. For stock market indicators and currency futures, financial
factors are the predominant reason for price changes. On the other hand,
commodity futures may be just as much affected by investment as they are by
actual supply and demand. Speculation in commodity futures may alter pricing
in a way that contradicts the law of supply and demand. The existence of such
contradictions depends on commodity type and the calendar year and therefore
serve to decrease the stability of the MST. If this reasoning is correct,
increasing the time span of the time series for cross-correlation should make
the observed connections more stable. In Fig. 3, we show the MST using the
time series from January 2007 to September 2011. We note that only two coal
futures are not connected to the commodity group and the stability is greater
than for single-year time series.
In Fig.4, we describe the cross-correlation functions of the volatility time
series. We show the cross-correlation function of main stock market indicators
with EUA in (a), and with WTI in (b). A systemic time shift between EUA or
WTI, and stock market indicators can clearly be seen. Since the maximum cross-
correlation coefficient in most functions is not much greater than 0.2, the
connections are not so strong, but certainly they are significant. Further
tests of Granger causality also show that the volatility of stock indices is
useful in forecasting the volatility of EUA and WTI approximately 20 to 120
days in advance.
As mentioned before, the correlation function of volatility is a slow-decaying
function. It is much more slowly decaying than the correlation function of
return time series (see Fig. 1), meaning that a long-range cross-correlation
relationship exists. If we consider significant cross-correlations between
different volatility time series to be an information transfer between
different assets, the time lag corresponding to the highest values of
correlation gives the time lag of that information transfer. It is worth
pointing out that the time lag between each pair of stock markets is
approximately 0, such as the time lag between DJIA and other 27 stocks
indicators are shown in Fig. 4 (d) . We show a simple summary of such time lag
in Fig. 4 (c) and (d) If there is information affecting stock markets at a
time of 0 days, the EUA and WTI crude oil futures will be affected by this
information roughly 30 and 90 days later respectively. Because both crude oil
and the ability to emit carbon are major inputs in the world economy, the
existence of this time lag can have strong implications in terms of potential
economic feedback loops.
## II Discussion
The drawn MSTs are reflective of a number of easily reasoned underlying
economic relationships, both through the stability and specificity of the
links.
As expected, we find a stable tendency for like financial assets to cluster.
Even during the otherwise anomalous 2007 subprime lending crisis and 2008
global financial crisis, this clustering tendency is preserved. This indicates
the existence of strong stable connections, which come out of the strong
cross-correlations, reflective of basic economic features and interactions.
These connections are stable over time and not affected by market conditions.
On the other hand, certain portions of the MSTs are consistently unstable,
like those relating to coal. This also may be reflective of economic
relationships. Unlike other commodities like oil, speculation in coal is
limited, so the movement of the coal futures may be simple supply and demand,
as opposed to driven by speculation. Coal’s lack of strong connections to
other commodities may be a result of investor’s low speculation in stock when
building their commodities portfolios.
Certain connections in our MSTs reveal underlying relationships that are
created by regulation. For example, we find that EUA futures mostly connect
with the base electricity and natural gas futures, which show stable
correlations among them. EU allowance permits, as a part of the European Union
Emission Trading Scheme, are either allocated or auctioned and allow a firm to
emit a designated amount of carbon dioxide Stavins . Since power generation
accounts for about one-quarter of total emissions of carbon dioxide, and
natural gas is the most resource of electricity generation in UK, the stable
connections of EUA to UK base electricity futures and UK natural gas futures
in our MST graph is reasonable.
Also intuitive is the location specific clustering of stock market indices.
AORD, an Australian index, consistently appears closely linked to those of
nearby countries like New Zealand, Japan, and China. Similarly, the HSI not
only keeps connections with most of the Asian stock indices like the JK11,
000001.SS, BSESN, and TWII, but also keeps connections with or otherwise stays
closely connected to indices from America and Australia. Thus, the MST created
reflects the common knowledge that Hong Kong is the financial center of Asia.
One benefit of the novelty of our approach is that it connects indices of
dissimilar type, yielding new insights. The index that connects most to coal
is OSEAX, that of Norway. Norway is rich in oil and natural gas, which
explains why the Norway stock index appears as the most “coal-like” of the
national indices. Similarly, the most “currency-like” of the national stock
indices seems to be AEX, the Dutch securities index.
We also note the relationship between the centrality in the network and real
world geographical knowledge. SOK/SEK is a currency future that is among the
furthest from the center. This is intuitive, since currency trading between
Norway and Sweden has little to do with financial activity in the rest of the
world. The same principle applies to the trading of Euros with the British
pound, shown as EUR/GBP. The Australian dollar, on the other hand, plays a
central role in its exchanges with far away currencies like the US dollar,
Euro, Japanese yen, and Canadian dollar.
## III Conclusion
In this paper, we have analyzed the correlation function of return and
volatility time series, constructed MSTs based on return time series, and
found consistent time lags in the correlation functions of the volatilities.
From these analysis, we have two main conclusions. (i) The stability of MST
structure clustering between like commodities reflects a basic rule of
economic activity, that the interaction between economic actors is not easily
affected by capital movement. The method of absolute cross-correlation
coefficient based MSTs has strong implications in the ongoing debate about the
relationships of different financial commodity time series. (ii) We find that
a time lag of correlation functions of volatility appears between stock
markets and EUA and WTI. From this finding, we hypothesize that there may be
systemic differences in the spread of financial risk, most often quantified as
volatility. In other words, as concerns risk, different types of markets may
have different sensitivities to economic information and other influencing
factors. It would be interesting, from theoretical point of view, to
generalize this time lag to predict the financial risks on a much longer time
interval. However, much more would need to be understood first, such as the
properties and mechanisms for this time lag. Hence, further empirical study is
needed first. We endeavor to address this question more in future work.
## IV Appendix: Set of Data
The data under investigation includes 28 stock market indicators, 21 currency
futures, and 20 commodity futures.
The stock market indicators investigated are:
Symbol | Meaning | Notes | Symbol | Meaning | Notes
---|---|---|---|---|---
000001.SS | SSE Composite Index | Shanghai stocks | ISEQ | Irish Stock Exchange Quotient |
AEX | Amsterdam Exchange Index | Dutch securities | JKII | Jakarta Islamic index |
AORD | All Ordinaries | Australian stocks | KLSE | Kuala Lumpur Stock Exchange |
ATX | Austrian Traded Index | | KS11 | Seoul Composite (South Korea) |
BSESN | SENSEX | Bombay stocks | MXX | Mexican Stock Exchange IPC |
BVSP | Bovespa Index | São Paulo stocks | N225 | NIKKEI 225 | Tokyo stocks
CAC | CAC 40 | French stocks | NZ50 | NZX 50 Index | New Zealand index
DJIA | Dow Jones Industrial Average | American index | OMX | OMX Stockholm 30 |
FTSE | FTSE 100 | London stocks | OMXC20 | OMX Copenhagen 20 |
FTSEMIB | FTSE MIB | Italian stocks | OSEAX | Oslo Børs All Share Index |
GDAXI | Deutscher Aktien Index | German blue chips | STI | Straits Times Index | Singapore stocks
GSPTSE | S&P/TSX Composite Index | Toronto stocks | SSMI | Swiss Market Index |
HSI | Hang Seng Index | Hong Kong stocks | TA100 | Tel Aviv 100 |
IBEX | IBEX 35 | Spanish stocks | TWII | TSEC weighted index | Taiwanese stocks
The commodity and currency futures investigated are all traded in the markets
of intercontinental exchange. The commodity futures are:
Symbol | Meaning | Notes | Symbol | Meaning | Notes
---|---|---|---|---|---
Barley | Western Barley Futures | | Gasoline | RBOB Gasoline Futures |
BrCrude | Brent Crude Futures | North Sea crude oil | HeatOil | Heating Oil Futures |
Canola | Canola Futures | | NatGas | UK Natural Gas Futures |
CCI | Consumer Confidence Index Futures | | PeakElec | UK Peak Electricity Futures |
Cocoa | Cocoa Futures | | RBCoal | Richards Bay Coal Futures |
Cotton | Cotton No. 2 Futures | | RCoal | Rotterdam Coal Futures |
FCOJA | FCOJ-A Futures | Florida orange juice | RJ/CBR | Thomson Reuters/Jefferies CRB Index | assorted commodities
Electric | UK Base Electricity Futures | | Sugar | Sugar No. 11 Futures |
Coffee | Coffee “C” Futures | | WTI | West Texas Intermediate | Texas crude oil
GasOil | Gas Oil Futures | | EUA | EU emission allowance |
Additionally, the currency futures in the form A/B refers to the value of
currency A in units of currency B. For example, USD/EUR would be the value of
US dollars in units of Euros. The currency futures investigated are:
Symbol | Currency | Traded in units of
---|---|---
AUD | Australian dollar | CAD, JPY, NZD, USD
CAD | Canadian dollar | JPY
EUR | Euro | AUD, CAD, GBP, JPY
GDP | British pound | AUD, CAD, JPY, NOK, NZD, ZAR, SEK
JPY | Japanese yen |
NOK | Norwegian krone | JPY, SEK
NZD | New Zealand dollar | JPY, USD
SEK | Swedish krona | JPY
ZAR | South African rand |
## References
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* (3) G. Bonanno, F. Lillo, and R.N. Mantegna, Quantitative Finance 1, 96-104 (2001).
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* (8) M. Tumminello, F. Lillo, and R. N. Mantegna, J. Econ. Behav. Org. 75 40 (2010); Phys. Rev. E, 76 031123 (2007).
* (9) R. Mantegna and H. E. Stanley, An Introduction to Econophysics Correlations and Complexity in Finance, Cambridge University Press (2000).
* (10) L. Laloux, P. Cizeau, J.-P. Bouchaud and M. Potters Phys. Rev. Lett. 83 1467 (1999).
* (11) V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, and H. E. Stanley, Phys. Rev. Lett. 83 1471 (1999).
* (12) V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr, and H. E. Stanley, Phys. Rev. E 65 066126 (2002).
* (13) A. Utsugi, K. Ino, and M. Oshikawa Phys. Rev. E 70 026110 (2004).
* (14) D. Wang, B. Podobnik, D. Horvatic, I. Grosse, and H. E. Stanley, Phys. Rev. E 83 046121 (2011).
* (15) The strict mathematical definition of a metric actually requires that the metric satisfy a fourth axiom, that of positive definiteness (non-negativity), which both the Mantegna-Sornette metric and ours satisfy trivially. See: A. V. Arkhangel’skii, and L. S. Pontryagin, General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences, (Springer, 1990).
* (16) L.Norden and M.Weber European Financial Management 15 529 (2009).
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* (19) http://www.businessweek.com/ap/2012-05/D9V3TCDG0.htm
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Figure 1: The cross-correlation function $C(n)$ of volatility (a) and return
(b) time series between the DJIA and FTSE100. Both show statistically
significant correlation coefficients at their maxima near time lag=0 (dotted
curve). Solid lines show the LOWESS (locally weighted scatter plot smoothing)
values of $C(n)$, smoothed over a span of 30 days.
Figure 2: The MST obtained from the absolute correlation coefficients
$|\rho_{ij}|$ of the set of 68 return time series during in individual
calendar years (a) 2007 (b) 2008 (c) 2009 (d) 2010. Red indicates commodity
futures, blue indicates currency futures, and green represents stock market
indicators. See Appendix for a listing of symbols and their meanings.
Figure 3: The minimal-spanning tree (MST), similar to Fig. 2, but for the
longer time span January 2007 to September 2011. See Appendix for symbol
definitions.
Figure 4: Cross-correlation function $C(n)$ of volatility daily time series
between 5 main stock market indices and (a) EUA (European carbon emissions
permits), or (b) WTI (light sweet crude oil). Red lines indicate the locally
weighted scatter plot smoothing values (LOWESS) of $C(n)$. Graphs show
systemic time shift for the highest cross-correlation value. This time shift
is observed in most stock markets. (c) indicates the average time-lag between
EUA, WTI, and 28 stock market indicators, with error bars showing the standard
deviations. (d) indicates the average time-lag between the DJIA and other 27
stock market indicators, with error bars showing the standard deviations. The
time lags (in days) are calculated from the time lag $n$ of highest LOWESS
values.
|
arxiv-papers
| 2012-05-09T03:19:44 |
2024-09-04T02:49:30.752377
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zeyu Zheng, Kazuko Yamasaki, Joel N. Tenenbaum, and H. Eugene Stanley",
"submitter": "Zeyu Zheng",
"url": "https://arxiv.org/abs/1205.1861"
}
|
1205.1997
|
Aaron F. McDaid, Thomas Brendan Murphy, Nial Friel and Neil HurleyModel-based
clustering in networks with Stochastic Community FindingStochastic Community
Finding
Aaron F. McDaidUniversity College Dublin, Irelandaaronmcdaid@gmail.com Thomas
Brendan MurphyUniversity College Dublin, Irelandbrendan.murphy@ucd.ie Nial
FrielUniversity College Dublin, Irelandnial.friel@ucd.ie Neil J.
HurleyUniversity College Dublin, Irelandneil.hurley@ucd.ie
In the model-based clustering of networks, _blockmodelling_ may be used to
identify roles in the network. We identify a special case of the Stochastic
Block Model (SBM) where we constrain the cluster-cluster interactions such
that the density _inside_ the clusters of nodes is expected to be greater than
the density _between_ clusters. This corresponds to the intuition behind
_community-finding_ methods, where nodes tend to clustered together if they
link to each other. We call this model Stochastic Community Finding (SCF) and
present an efficient MCMC algorithm which can cluster the nodes, given the
network. The algorithm is evaluated on synthetic data and is applied to a
social network of interactions at a karate club and at a monastery,
demonstrating how the SCF finds the ‘ground truth’ clustering where sometimes
the SBM does not. The SCF is only one possible form of constraint or
specialization that may be applied to the SBM. In a more supervised context,
it may be appropriate to use other specializations to guide the SBM.
Model-based clustering, MCMC, Social networks, Community finding,
Blockmodelling
## 1 Introduction
Clustering typically involves dividing objects into clusters where objects are
in some sense ‘close to’ the other objects in the same cluster. Much research
has been done into clustering points in Euclidean space where points are put
into the same cluster based on a distance metric between pairs points. But the
data we have is of a different form, we have a network as input data.
In network analysis, clustering is usually based on the idea that two nodes in
the network are ‘close to’ each other if they are linked to each other. This
is called _community-finding_ and is the main topic of this paper. There are a
large number of methods using heuristic algorithms and non-statistical
objective functions [9, 12, 1]. The complexity issues around some such
algorithms are also discussed in the literature [2, 3]. For a thorough review
of the broad area of research into clustering the nodes of a network, see [5].
In the rest of this paper we focus on statistical models and algorithms, as
they are relevant for our approach. We base our model in the Stochastic Block
Model (SBM) of [11]. That model is not, by default, a community-finding model.
For example, with the famous social network known as Zachary’s Karate Club the
SBM will, if asked for two clusters, divide the nodes into one small cluster
of high-degree nodes and another cluster containing a large number of smaller-
degree nodes. In community-finding, this would be seen as an ‘incorrect’
result; the members of the karate club went on to divide themselves into two
factions, where most of the friendship edges are, unsurprisingly, inside the
factions. Community-finding methods are expected to find this type of
clustering, where the edges tend to be inside clusters.
Many of the probabilistic models of networks are based on the SBM [4, 16] and
therefore they do not explicitly tackle community-finding. In this paper, we
make a change to the standard SBM to require that the blocks corresponding to
within-cluster connectivity will be expected to be denser than the blocks
corresponding to between-cluster connectivity. This will lead to an algorithm
which, unlike the SBM, will cluster the nodes according to the two factions in
the karate club, as would be expected in a community-finding algorithm.
Given a generative model and an observed network, we can check the posterior
distribution and obtain a clustering, or set of clusterings, which are a good
fit for the data. It is typically trivial to write MCMC algorithms to sample
from the relevant distribution. However, it can be challenging to create
suitably fast algorithms. We use _collapsing_ along with algorithmic
techniques such as the _allocation sampler_ [10]; a scalable application of
these ideas to the standard SBM is in [8].
In applying these concepts to the SCF we run into a problem though. It does
not appear to be possible to directly integrate out the relevant parameters to
give us a fully collapsed model. However, we will show in this paper how we
can work around this and still develop a suitable Metropolis-Hastings
algorithm with the correct transition probabilities without having to resort
to trans-dimensional RJMCMC[6]. This technique is not a typical application of
Metropolis-Hastings and it may have broader applicability, allowing faster
algorithms with the simplicity of collapsing, in models where full explicit
collapsing is not possible.
### 1.1 Structure of this paper
In Section 2 we will review the standard SBM of [11] \- defining the basic
notation and models which will be used throughout. In Section 3 we will define
our modification to the SBM which we call Stochastic Community Finding (SCF).
In Section 4 we will consider the issue of collapsing; this is straightforward
for the SBM, but not for the SCF. In Section 5 we discuss the algorithm used
in our software111C++ implementation, and datasets used, at
https://sites.google.com/site/aaronmcdaid/sbm which enables us to use
Metropolis-Hastings even though we cannot write down the collapsed posterior
mass in closed form. We then proceed to evaluations, first considering a
synthetic network in Section 6 and finally an analysis of Zachary’s Karate
Club and Sampson’s Monks in Section 7. We close with a discussion of possible
future directions in Section 8.
## 2 Stochastic Block Model
In this section, we define the Stochastic Block Model (SBM) of [11] before
discussing our modification in the next section. We restrict our attention in
this section to directed unweighted networks, where edges are simply present
or absent. There are many extensions222 directed _or_ undirected, unweighted
_or_ integer-weights and other more complex ‘alphabets’ to describe an edge,
self-loops modelled _or_ ignored., for example allowing weighted networks with
integer- or real-valued weights [8, 14].
We model a network of $N$ nodes, and the network is represented as an
adjacency matrix $x$. If there is a directed edge from node $i$ to $j$, we
have $x_{ij}=1$. If they are not connected, we have $x_{ij}=0$. By default, we
ignore self loops ($x_{ii}$) and they are simply left out of the formulae.
Given a network $x$, our goal is to identify a clustering $z$. We use a vector
$z$ of length $N$, where $z_{i}$ is the cluster to which node $i$ is assigned.
There are $K$ clusters, $1\leq z_{i}\leq K$.
Given $K$ clusters, there are $K\times K$ _blocks_ , one block for each pair
of clusters. There is a $K\times K$ matrix $\pi$ which records, for each
block, the expected density of edge-formation in that block. In other words,
given node $i$ which is in cluster $k=z_{i}$ and node $j$ which is in cluster
$l=z_{j}$, the probability of a connection is $\pi_{kl}$,
$x_{ij}\sim\text{Bernoulli}(\pi_{kl})$
In the undirected variant we would have $x_{ij}=x_{ji}$, and only a single
draw from the relevant Bernoulli would be used to assign to these. The
probability of two nodes connecting depends on the clusters to which the nodes
are assigned, but is otherwise independent of the particular nodes; this is
the definition of blockmodelling. The elements of $\pi$ have a prior;
$\pi_{kl}\sim\text{Beta}(\beta_{1},\beta_{2})$. Our default is to set
$\beta_{1}=\beta_{2}=1$ which means this prior is a Uniform distribution over
(0,1).
$z$ is itself a random variable. There is a vector $\theta$ of length $K$
which represents the probability, for each cluster, of any node being assigned
to that cluster.
$z_{i}\overset{iid}{\sim}\text{Multinomial}(1;\theta_{1},\theta_{2},\cdots\theta_{K})$
$\theta$ is also a random variable and we place a Dirichlet prior on it.
$\theta\sim\text{Dirichlet}(\alpha_{1},\alpha_{2},\cdots,\alpha_{K})$ (1)
The parameters to the Dirichlet prior are a choice to be made by the user, and
it is conventional to set each of the $\alpha_{k}$ to the same value,
$\alpha_{k}=\alpha$, and we set $\alpha$ to 1 by default in our experiments.
Given $N$ and $K$, this is a fully specified generative model to generate many
variables including the clustering $z$ and the network $x$. We investigated
this model in [8]. An important extension we introduced there is to place a
prior on $K\sim\text{Poisson}(1)$, thus allowing us to deal directly with the
number of clusters as a random variable and avoids the need for any separate
model selection criterion. See that paper for a more extended discussion of
model selection and validation of the accuracy of the method in estimating the
number of clusters.
$\mathrm{P}(x,\pi,z,\theta,K)=\mathrm{P}(K)\times\mathrm{p}(z,\theta|K)\times\mathrm{p}(x,\pi|z,K)$
where we use $\mathrm{P}(\dots)$ for probability mass functions, i.e. of
discrete quantities such as $z$ or $K$, and $\mathrm{p}(\dots)$ for
probability density functions.
## 3 Stochastic Community Finding
Now that we have defined the SBM, as introduced by [11], we define the
modification we are introducing in the Stochastic Community Finding (SCF)
model. In community-finding, as opposed to block-modelling, we expect that if
a pair of nodes are connected then the nodes are more likely to be clustered
together than if they were not connected.
$\mathrm{P}(z_{i}=z_{j}|x_{ij}=1)>\mathrm{P}(z_{i}=z_{j}|x_{ij}=0)$
Blockmodelling doesn’t have such a constraint. This is not a hard rule in
community-finding, it is a useful guide to help define the different goals in
community-finding and block-modelling. An equivalent statement is
$\mathrm{P}(x_{ij}=1|z_{i}=z_{j})>\mathrm{P}(x_{ij}=1|z_{i}\neq z_{j})$
This is the formulation we use to define the SCF. We require that all the
diagonal entries in $\pi$ be larger than the off-diagonal entries of $\pi$.
$\min(\pi_{mm})>\max(\pi_{kl})$ for all $m,k,l$ where $k\neq l$. Define a
function $v(\pi)$ which returns 1 if $\pi$ satisfies the constraint, and
returns 0 if it does not.
$v(\pi)=\left\\{\begin{array}[]{ccl}1&&\text{if}\;\min(\pi_{mm})>\max(\pi_{kl})\qquad\text{for}\;k\neq
l\\\ 0&&\text{otherwise}\par\end{array}\right.$ (2)
Under this constraint, the probability density of the SCF model is
proportional to $f(x,\pi,z,\theta,K)$ where
$f(x,\pi,z,\theta,K)=\mathrm{P}_{\text{SBM}}(x,\pi,z,\theta,K)\times v(\pi)$
and $\mathrm{P}_{\text{SBM}}$ is the probability density as defined by the
SBM. This probability mass function is essentially identical to the SBM except
that we have set the density to zero where the constraint on $\pi$ is not
satisfied. A simpler form of the SBM has been investigated [16] where all the
diagonal entries in the blockmodel are taken to be equal to $\lambda$ and the
all the off-diagonal entries are equal to $\epsilon$. Their model does not
explicitly require that $\lambda>\epsilon$, and hence it is not quite a
community-finding model.
## 4 Collapsing
Given a network $x$, our goal is to estimate the number of clusters and to
find the clustering $(K,z)$. In the SBM as investigated by [8], it is
straightforward to use _collapsing_ and integrate out the other variables that
we are not directly interested in such as $\pi$ and $\theta$,
$\displaystyle\mathrm{P}_{\text{SBM}}(x,z,K)$
$\displaystyle=\mathrm{P}_{\text{SBM}}(K)\times\mathrm{P}_{\text{SBM}}(z|K)\times\mathrm{P}_{\text{SBM}}(x|z,K)$
$\displaystyle=\mathrm{P}_{\text{SBM}}(K)\times\int\mathrm{P}_{\text{SBM}}(z,\theta|K)\;\mathrm{d}\theta\times\int\mathrm{P}_{\text{SBM}}(x,\pi|z,K)\;\mathrm{d}\pi$
allowing one to create an algorithm which, given $x$, samples $(z,K)$.
But this collapsing does not work in such a straightforward way with the SCF;
we cannot, to our knowledge, write down a closed form expression for
$f(x,z,K)$ where $\pi$ and $\theta$ have been integrated out. The problem is
that it is difficult to integrate out $\pi$ in the SCF due to the dependence
structure between the blocks which is introduced by the constraint in Equation
2. In the SBM, the elements of $\pi$ are independent of each other. Also,
given $z$, the various blocks within $x$ which correspond to the elements of
$\pi$ are independent of each other and dependent only on a single element of
$\pi$.
The model for $K$ and $\theta$ and $z$ are the same in the SCF as in the SBM,
therefore we will simply use $\mathrm{P}(\dots)$ and $\mathrm{p}(\dots)$ for
these. But for expressions involving $\pi$ it will make sense to use
$\mathrm{p}_{\text{SBM}}(\dots)$ and $f(\dots)$ to distinguish between the
(normalized) probability distribution of the SBM and the (non-normalized)
function for the SCF. We attempt to collapse as much as possible in order to
get an expression for $f(x,z,K)$, our desired stationary distribution:
$\begin{split}f(x,z,K)&=\mathrm{P}(K)\times\mathrm{P}(z|K)\times\int\mathrm{p}_{\text{SBM}}(x,\pi|z,K)\times
v(\pi)\;\mathrm{d}\pi\\\
&=\mathrm{P}(K)\times\mathrm{P}(z|K)\times\int\mathrm{P}_{\text{SBM}}(x|z,K)\times\mathrm{p}_{\text{SBM}}(\pi|x,z,K)\times
v(\pi)\;\mathrm{d}\pi\\\
&=\mathrm{P}(K)\times\mathrm{P}(z|K)\times\mathrm{P}_{\text{SBM}}(x|z,K)\times\int\mathrm{p}_{\text{SBM}}(\pi|x,z,K)\times
v(\pi)\;\mathrm{d}\pi\\\
&=\mathrm{P}_{\text{SBM}}(x,z,K)\times\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,z,K)\end{split}$
(3)
The final factor in the final expression,
$\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,z,K)$, can be interpreted as the
probability (under the SBM), given $(x,z,K$), that a draw of $\pi$ will
satisfy the constraint; it is this factor that, to our knowledge, cannot be
solved in closed form. The first factor in the final expression,
$\mathrm{P}_{\text{SBM}}(x,z,K)$, can be directly taken from [8] as the
relevant integration has been solved as described in the Appendices of that
paper. In the following expression, we define $n_{k}$ to be the number of
nodes in cluster $k$, i.e. $n_{k}$ is a function of $z$. Also, $p_{kl}$ is the
number of pairs of nodes in the block between clusters $k$ and $l$, i.e.
$p_{kl}=n_{k}n_{l}$, and $y_{kl}$ is the number of directed edges from nodes
in cluster $k$ to nodes in cluster $l$. We also use the Beta function
$\text{B}(a,b)=\frac{\Gamma(a)\Gamma(b))}{\Gamma(a+b)}$.
$\begin{split}\mathrm{P}_{\text{SBM}}(x,z,K)&=\mathrm{P}_{\text{SBM}}(K)\times\mathrm{P}_{\text{SBM}}(z|K)\times\mathrm{P}_{\text{SBM}}(x|z,K)\\\
&=\frac{1}{K!}\frac{1}{e}\times\frac{\Gamma(K\alpha)}{\Gamma(N+K\alpha)}\prod_{k=1}^{K}\frac{\Gamma(n_{k}+\alpha)}{\Gamma(\alpha)}\times\prod_{k=1}^{K}\prod_{l=1}^{K}\frac{\text{B}(y_{kl}+\beta_{1},p_{kl}-y_{kl}+\beta_{2})}{\text{B}(\beta_{1},\beta_{2})}\end{split}$
(4)
where $\alpha$ is the user-specified parameter to the Dirichlet prior (eq. 1).
In a conventional Metropolis-Hasting algorithm (as in [8]), it is convenient
to have closed form expressions of the posterior mass at each state in the
chain. However, it is not necessary to have such expressions and we will see
in the next section how we can work around this and develop a Markov Chain
with the correct transition probabilities for the SCF even though we do not
have a fully closed-form expression for $f(x,z,K)$.
## 5 MCMC algorithm
In this section, we will describe the algorithm we have used to sample from
the space of $(z,K)$, with probability proportional to $f(x,z,K)$ (Equation
3). We have extended the software we developed in [8] and we direct the reader
to that paper for detailed definition of all the moves.
### 5.1 Algorithm for the SBM
We will first summarize the procedure used in our SBM algorithm, and then
describe the change necessary to turn it into an SCF algorithm. This means our
initial goal is to describe an algorithm whose stationary distribution is
proportional to $\mathrm{P}_{\text{SBM}}(x,z,K)$. We define a _proposal
distribution_ which, given a current state $s=(z,K)$, will propose a new state
$t=(z^{\prime},K^{\prime})$.
The proposals are defined by $p$, where $p_{st}$ is the probability that,
given the chain is in state $s=(z,K)$, that it will propose to move to state
$t=(z^{\prime},K^{\prime})$. Clearly, $\sum_{t}p_{st}\;=\;1$ for all $s$.
Given that a proposal has been made to move from $s$ to $t$, where $s\neq t$,
we define an _acceptance probabality_ $a_{st}$. When the proposal is made, we
will decide whether to accept or reject the proposal using a Bernoulli
variable with probability $a_{st}$.
In the SBM, where the desired stationary distribution is proportional to
$\mathrm{P}_{\text{SBM}}(x,z,K)$, we were able to use a standard Metropolis-
Hasting [7] algorithm with acceptance probability
$a_{st}=\min\left(1,\frac{p_{ts}}{p_{st}}\frac{\mathrm{P}_{\text{SBM}}(x,t)}{\mathrm{P}_{\text{SBM}}(x,s)}\right)$
(5)
where $\mathrm{P}_{\text{SBM}}(x,s)$ is defined as
$\mathrm{P}_{\text{SBM}}(x,z,K)$ and $\mathrm{P}_{\text{SBM}}(x,t)$ is defined
as $\mathrm{P}_{\text{SBM}}(x,z^{\prime},K^{\prime})$. For the SBM, the
transition probabilities satisfy _detailed balance_ :
$\frac{t^{\text{SBM}}_{st}}{t^{\text{SBM}}_{ts}}=\frac{p_{st}a_{st}}{p_{ts}a_{ts}}=\frac{\mathrm{P}_{\text{SBM}}(K,(z^{\prime},K^{\prime})=t)}{\mathrm{P}_{\text{SBM}}(K,(z,K)=s)}$
(6)
One of the moves is a simple Gibbs update on the position of one node,
$z_{i}$. Node $i$ is considered for inclusion in each of the $K$ clusters.
Another move is called M3, which involves proposing a reassignment of all the
nodes in two randomly-selected clusters. AE is a move which proposes to split
a cluster into two, increasing $K$, or merging two clusters into one,
decreasing $K$. Together, these moves can visit all states $(z,K)$. For full
details see our earlier work [8], which was based on existing algorithms [10,
14].
### 5.2 Algorithm for the SCF
But our goal is to develop an algorithm for the SCF. We use the following
scheme: First, make a proposal such as those used in the collapsed SBM
algorithm [8]. Second, calculate the ‘SBM-acceptance probability’ according to
Equation 5. Third, make a draw from a Bernoulli with this probability to
decide whether to Reject or to (provisionally) Accept. If the proposal was
rejected, then there is no further work to be done, the proposal has been
rejected. _But_ , if the SBM-acceptance probability led to a (provisional)
‘acceptance’, then there is one final step required to decide on rejection or
acceptance of the move; we draw from the posterior of
$\pi|x,z^{\prime},K^{\prime}$, drawing a new $\pi$ conditioning on the
(proposed) new values of $z^{\prime}$ and $K^{\prime}$ in state $t$; we fully
accept the new state if and only if the $\pi$ satisfies the SCF validity
constraint in Equation 2. This procedure is giving in pseudocode in Table 1.
Given current state, $s=(z,K)$
---
Propose new state, $t=(z^{\prime},K^{\prime})$
Calculate SBM-acceptance propability, $a_{st}$
Draw a Bernoulli with probability $a_{st}$.
If Failure:
REJECT
Else:
Draw $\pi|x,z^{\prime},K^{\prime}$ from posterior
Test if $\pi$ satisfies $v(\pi$)
If Satisfactory:
ACCEPT
Else:
REJECT
Table 1: Pseudocode describing the acceptance and rejection rules in the SCF
algorithm
In this algorithm, a proposal $s\rightarrow t$ (with $s\neq t$) will only be
accepted if the SBM-acceptance succeeds _and_ if the $\pi|x,z,K$ satisfies the
constraint. Given that the current state is $s$, the probability of
transitioning to another state $t$ is
$t^{\text{SCF}}_{st}=p_{st}\times
a_{st}\times\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,z^{\prime},K^{\prime})$
We will shortly show that this algorithm is correct for drawing from the
desired stationary distribution, but first we describe how to draw $\pi$ from
the its posterior given $(x,z,K)$. $\pi$ is a $K\times K$ matrix, one element
for each block. In this posterior, as in the prior, these elements are
independent of each other and therefore we proceed by estimating each element
of $\pi$ separately. The prior on each element of $\pi$ is, as described
earlier, a Beta($\beta_{1},\beta_{2}$). The data for that block is the number
of edges which appears, $y_{kl}$, and the number of non-edges that are in that
block, $p_{kl}-y_{kl}$. In this case, the posterior is
Beta($\beta_{1}+y_{kl},\beta_{2}+p_{kl}-y_{kl}$). For each element in $\pi$,
this posterior Beta is prepared and one draw is made from each. If the
elements on the diagonal,
$\pi_{mm}\sim\text{Beta}(\beta_{1}+y_{mm},\beta_{2}+p_{mm}-y_{mm})$, are
greater than those off the diagonal,
$\pi_{kl}\sim\text{Beta}(\beta_{1}+y_{kl},\beta_{2}+p_{kl}-y_{kl})$, then the
move is accepted.
Now, we show that this satisfies detailed balance and that the stationary
distribution is proportional to $f(x,z,K)$. We reuse Equation 6 in this proof:
$\displaystyle\frac{t^{\text{SCF}}_{st}}{t^{\text{SCF}}_{ts}}=$
$\displaystyle\frac{p_{st}\times
a_{st}\times\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,(z^{\prime},K^{\prime})=t)}{p_{ts}\times
a_{ts}\times\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,(z,K)=s)}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{P}_{\text{SBM}}(x,(z^{\prime},K^{\prime})=t)}{\mathrm{P}_{\text{SBM}}(x,(z,K)=s)}\times\frac{\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,(z^{\prime},K^{\prime})=t)}{\mathrm{P}_{\text{SBM}}(v(\pi)=1|x,(z,K)=s)}$
$\displaystyle=$
$\displaystyle\frac{f(x,(z^{\prime},K^{\prime})=t)}{f(x,(z,K)=s)}$
We also use a method of label-switching which was introduced in [10] and which
we used in [8]. The chain will often visit states which are essentially
equivalent to earlier states, but where the cluster labels have merely been
permuted. The procedure involves permuting the labels of the clusters with the
goal of maximizing the similarity of the latest state to all the previous
states. This leads to more easily interpretable results from the chain.
If it is possible to solve Equation 3 exactly, this would probably allow us to
have larger acceptance probabilities and to increase the speed of the
algorithm accordingly. Currently, the algorithm can, in theory, get trapped
for some time in a state where the constraint typically fails for that state
and for neighbouring states, making it difficult for the algorithm to climb
towards better states. This is worth some further consideration, and perhaps
an algorithm based on an uncollapsed representation might be best. A naive
uncollapsed algorithm, where just one of $z$ or $\pi$ or $\theta$ is updated
in a move, would mix very slowly. It may be possible to use moves such as
those in the allocation sampler to propose changes simultaneously to the
clustering $z$ and to the density matrix $\pi$ and to the cluster-membership-
probability vector $\theta$; such an algorithm may mix as well as the
allocation sampler; such a method would also make it easier to efficiently
handle the constraint. However, this method would be complex to implement; it
may be worthwhile to investigate this further.
## 6 Evaluation with synthetic data
In this section, we evaluate the SCF on a simple synthetic network. We compare
the results with those found by the basic SBM algorithm. If we generate data
strictly according to the generative SCF model, then both algorithms tend to
be quite accurate, see our earlier work [8] for a detailed analysis of the
accuracy of the collapsed SBM MCMC algorithm. Therefore, in order to challenge
the algorithms, instead we construct a network where the SBM and SCF get
different results in order to demonstrate the preference of the SCF for
‘community-like’ structure. We consider the undirected network in Figure 1,
which has two star-like communities. Each of these communities has ten nodes,
made up of two central nodes and eight peripheral nodes. Every central node is
connected to every periphery node.
This network has a more heterogenous degree distribution; this very loosely
approximates the heavy-tailed degree distribution seen in many real-world
networks. If we generate data strictly according to the SBM or SCF the degree
distribution is more homogenous, especially the distribution of the degrees
within a single cluster.
Figure 1: The ‘$2\times 2$’ network. Two ‘roles’, peripheral and central. And
two communities also, left and right. The SCF finds the two communities, and
the SBM finds the roles.
In all the experiments in this section and the following section, we ran the
algorithm for 10,000,000 iterations. By default, we allow the algorithm to
select the number of clusters itself as the allocation sampler algorithm
naturally searches the entire search space. With this network, the SCF selects
$K=2$ and it clusters the nodes into the two star-like communities. The Markov
Chain spends 97.5% of its iterations in that ‘ground truth’ state.
On the other hand, the SBM select 4 clusters. It subdivides each of the two
true communities into two further communities - one containing the central
nodes and the other containing the peripheral nodes. We see this in Figure 2,
where very few edges are inside the found clusters. Even if we restrict the
SBM to consider only $K=2$, then it again divides the nodes into central and
periphery nodes. Regardless of the number of clusters, the SBM finds clusters
which do not contain any of the edges; this is the opposite of what we expect
in community finding.
Figure 2: The adjacency matrices showing the clusterings found by the SCF
(left) and SBM (middle) on the ‘$2\times 2$’ network (Figure 1). The SCF has
found the communities, with all edges inside the clusters, as expected. The
SBM has divided the nodes according to degree and community, but there are no
edges within any of the four clusters found by the SBM. On the right, we see
how the SBM finds only the roles if the number of clusters is fixed at $K=2$
in advance. Only the SCF has placed all the edges inside the clusters and
correctly estimated the number of communities.
In networks there may be multiple types of structure that can be detected; the
SCF focuses on finding the ‘community-like’ structure, where the clusters are
expected to be internally dense. In synthetic and empirical networks with a
heavy-tailed degree distribution the SBM may have a tendency to cluster nodes
according to their degree, or other structural roles, and not according to
community structure.
## 7 Empirical networks
In this section, we apply the SCF to two well-known social networks.
### 7.1 Sampson’s Monks
Sampson [13] gathered data on novices at a monastery333Sampson’s monk data as
an R package: http://rss.acs.unt.edu/Rdoc/library/LLN/html/Monks.html. There
are 18 novices in the network and a pair are linked if they reported a
positive friendship between them, giving us an undirected network. There were
factions within the group, which Sampson labelled _Loyal Opposition_ , _Young
Turks_ and _Outcasts_.
We ran the SCF method on this dataset for 10,000,000 iterations. It estimated
the number of clusters at 3, with 88.5% of the iterations. For 69% of the
iterations, the clustering was exactly equal to the factions reported by
Sampson. The network and adjacency matrix are shown in Figure 3. We also ran
the data through the SBM. It found very similar results. This suggests that if
the community structure is strong, then either algorithm can detect it.
However, the SBM is slightly less accurate and only 55% of the iterations
involve $K=3$. This suggests that there is other structure, perhaps the high-
degree versus low-degree structure, that is trying to assert itself.
Figure 3: Sampson’s monks, and the 3-way split found by the SCF which matches
with the factions found by Sampson.
### 7.2 Zachary’s Karate club
Now, we apply the SCF to a network of interactions at a karate club [15],
again demonstrating the ability of the SCF to detect community structure where
the SBM focusses on other types of structure.
The members of the karate club were asked about their social interactions with
other members, focusing on interactions outside of the lessons and
tournaments. This gives us a network of 34 members and 78 interactions. The
interaction data is weighted, according to the number of distinct social
interaction types reported by the members; a larger number is taken to
indicate a stronger friendship444Weighted karate club network:
http://vlado.fmf.uni-lj.si/pub/networks/data/Ucinet/zachary.dat. After the
survey was taken, the club split into two factions over a dispute of the cost
of the lessons. The network is visualized in Figure 4.
Figure 4: The karate club network of [15]. The width indicates the strength of
the relationship by counting the number of distinct interaction types recorded
between the two members. The club split in two after the survey and the colour
of the node records the split. On the right is the adjacency matrix of this
network. The rows and columns have been ordered according to which faction the
node is in; most of the edge weight is on the top-left and bottom-right, as
would be expected in good community structure. The SCF algorithm finds this
clustering when $K=2$.
This network has weighted edges and hence we apply our SCF constraint (eq. 2)
to the weighted variant of the SBM. The edges have a Poisson weight, and the
rate of the Poisson, $\pi_{kl}$, is different from each block and comes from a
Gamma prior; full details of this edge model are in the Appendix of our
earlier work[8].
If we fix the number of clusters at $K=2$, then the SCF will correctly cluster
the nodes according the split that occured in the club; the chain will spend
85.5% of its iterations in that state. This contrasts with the SBM, which
instead clusters the nodes into 9 high-degree and 25 low-degree nodes, a
clustering which is quite different from the factional split; this SBM
clustering is in Figure 5. The high-degree nodes include the leaders of each
faction.
Unfortunately, unlike our earlier networks, the SCF does not correctly
estimate the number of clusters within the karate club. We had to specify that
$K=2$ in order to find the correct clustering, whereas our MCMC algorithm
estimates $K=5$. The issue of model selection within this model may be worth
considering further.
Figure 5: The SBM, when told to find two clusters, divides the nodes according
to degree.
## 8 Conclusion
Community finding is popular in the social science literature, but many
statistical models are defined for block-modelling, not explicitly for
community-finding. In order to investigate community-finding, we have
introduced a constraint that the density inside clusters be larger than the
density between pairs of clusters. We have extended an existing block-
modelling method, which was based on the Stochastic Block Model (SBM), to take
account of this constraint. We evaluated the method and shown it can detect
community structure where the SBM cannot.
### Acknowledgement
This research was supported by Science Foundation Ireland (SFI) Grant No.
08/SRC/I1407.
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* Zanghi et al. [2008] Zanghi, H., Ambroise, C. & Miele, V. (2008). Fast online graph clustering via Erdős–Rényi mixture. Pattern Recognition, 41, 3592–3599.
|
arxiv-papers
| 2012-05-09T14:33:29 |
2024-09-04T02:49:30.762159
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aaron F. McDaid, Brendan Thomas Murphy, Nial Friel, Neil J. Hurley",
"submitter": "Aaron Francis McDaid",
"url": "https://arxiv.org/abs/1205.1997"
}
|
1205.2047
|
# Statistical multi-moment bifurcations in random delay coupled swarms
Luis Mier-y-Teran-Romero(1), Brandon Lindley(2), and Ira B. Schwartz(3) (1) US
Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section,
Plasma Physics Division, Washington, DC 20375
###### Abstract
We study the effects of discrete, randomly distributed time delays on the
dynamics of a coupled system of self-propelling particles. Bifurcation
analysis on a mean field approximation of the system reveals that the system
possesses patterns with certain universal characteristics that depend on
distinguished moments of the time delay distribution. Specifically, we show
both theoretically and numerically that although bifurcations of simple
patterns, such as translations, change stability only as a function of the
first moment of the time delay distribution, more complex patterns arising
from Hopf bifurcations depend on all of the moments.
Recently, much attention has been given to the study of interacting multi-
agent, particle or swarming systems in various natural and engineering fields.
Interestingly, these multi-agent swarms can self-organize and form complex
spatio-temporal patterns even when the coupling between agents is weak. Many
of these investigations have been motivated by a multitude of biological
systems such as schooling fish, swarming locusts, flocking birds, bacterial
colonies, ant movement, etc. Budrene and Berg (1995); Toner and Tu (1995);
Parrish (1999); Topaz and Bertozzi (2004), and have also been applied to the
design of systems of autonomous, communicating robots or agents Leonard and
Fiorelli (2002); Morgan and Schwartz (2005); Chuang et al. (2007) and mobile
sensor networks Lynch et al. (2008).
Many studies describe the swarm system at the individual, or particle, level
via models constructed with ordinary differential equations (ODEs) or delay
differential equations (DDEs) to describe the trajectories Vicsek et al.
(1995); Flierl et al. (1999). When there are a large number of densely-
distributed particles, authors have employed partial differential equations
(PDEs) to describe the average agent density and velocity Toner and Tu (1995,
1998); Edelstein-Keshet et al. (1998); Topaz and Bertozzi (2004). Recently,
the inclusion of noise in such particle-based studies has revealed
interesting, noise-induced transitions between different coherent patterns
Erdmann and Ebeling (2005); Forgoston and Schwartz (2008). Such noise driven
systems have led to the discovery of first and second order phase transitions
in swarm models Aldana et al. (2007).
A topic of intense ongoing research in interacting particle systems, and in
particular in the dynamics of swarms, is the effect of time delays. It is well
known that time delays can have profound dynamical consequences, such as
destabilization and synchronization Englert et al. (2011); Zuo et al. (2010),
and delays have been effectively used for purposes of control Konishi et al.
(2010). Initially, such studies focused on the case of one or a few discrete
time delays. More recently, however, the complex situation of several and
random time delays has been researched Ahlborn and Parlitz (2007); Wu et al.
(2009); Marti et al. (2006). An additional important case is that of
distributed time delays, when the dynamics of the system depends on a
continuous interval in its past instead of on a discrete instant Omi and
Shinomoto (2008); Dykman and Schwartz (2012).
There exists a complex interplay between the attractive coupling, time delay,
and noise intensity that produces transitions between different spatio-
temporal patterns Forgoston and Schwartz (2008); Mier-y Teran-Romero et al.
(2011) in the case of a single, discrete delay. Here, we consider a more
general swarming model where coupling information between particles occurs
with randomly distributed time delays. We perform a bifurcation analysis of a
mean field approximation and reveal the patterns that are possible at
different values of the coupling strength and parameters of the time delay
distribution.
We model the dynamics of a 2D system of $N$ identical self-propelling agents
that are attracted to each other in a symmetric manner. We consider the
effects of finite communication speeds and information-processing times so
that the attraction between agents occurs in a time delayed fashion. The time
delays are non-uniform but they are symmetric among agents
$\tau_{ij}(=\tau_{ji})$, for particles $i$ and $j$, as well as constant in
time. The dynamics of the particles is described by the following governing
equations:
$\displaystyle\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(1-|\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{r}}_{i}-\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau_{ij})),$ (1)
for $i=1,2\ldots,N$. The vector $\mathbf{r}_{i}$ denotes the position of the
$i$th agent at time $t$. The term
$\left(1-|\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{r}}_{i}$ represents
self-propulsion and frictional drag forces that act on each agent. The
coupling constant $a$ measures the strength of the attraction between agents
and the time delay between particles $i$ and $j$ is given by $\tau_{ij}$. When
$a=0$ the agents tend to move in a straight line with unit speed as time tends
to infinity. The $N(N-1)/2$ different time delays $0<\tau_{ij}(=\tau_{ji})$
are drawn from a distribution $\rho(\tau)$ whose mean and standard deviation
are denoted by $\mu_{\tau}$ and $\sigma_{\tau}$, respectively.
We obtain a mean field approximation of the swarming system by measuring the
particle’s coordinates relative to the center of mass
$\mathbf{r}_{i}=\mathbf{R}+\delta\mathbf{r}_{i}$, for $i=1,2\ldots,N$, where
$\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t)$. Following the
approximations from Lindley et al. (2012), we obtain a mean field description
of the swarm:
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{\
R}(t)-\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho(\tau)d\tau\right).$ (2)
The approximations necessary to obtain Eq. (2) require that $N$ be
sufficiently large so that
$\frac{1}{N(N-1)}\sum_{i=1}^{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\mathbf{R}(t-\tau_{ij})\approx\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho(\tau)d\tau$
and that the swarm particles remain close together.
Figure 1: Bifurcation structure of the translating state of the mean field
Eqs. (2) for the exponential distribution described in the text
($\sigma_{\tau}=0.2$). The translating state merges with: (_i_) the stationary
state along the continuous red curve $a\mu_{\tau}=1$; and (_ii_) the
circularly rotating state along the dashed black curve
$a\langle\tau^{2}\rangle=2$. The component of the translating state parallel
to the motion undergoes a Hopf bifurcation along the green, dotted curve.
(Color online.)
Equations (2) admit a uniformly translating solution
$\mathbf{R}(t)=\mathbf{R}_{0}+\mathbf{V}_{0}\cdot t$ ($\mathbf{R}_{0}$ and
$\mathbf{V}_{0}$ are any constant 2D vectors). The speed $|\mathbf{V}_{0}|$
must satisfy
$\displaystyle|\mathbf{V}_{0}|^{2}=1-a\int_{0}^{\infty}\tau\rho(\tau)d\tau=1-a\mu_{\tau},$
(3)
which shows that this solution is possible as long as the system parameters
lie below the hyperbola $a\mu_{\tau}=1$ in the $(a,\ \mu_{\tau})$ plane.
Remarkably, the speed of the of the translating state depends exclusively on
the mean of the distribution $\rho(\tau)$ and not on any of the higher
moments.
The linear stability of the translating state is examined by taking
$X(t)=\sqrt{1-a\mu_{\tau}}\cdot t+\delta X(t)$ and $Y(t)=\delta Y(t)$. The two
linearized equations decouple and the stability of motions parallel and
perpendicular to the translating direction are determined by the
characteristic equations $\mathcal{D}_{\parallel}(\lambda)$ and
$\mathcal{D}_{\perp}(\lambda)$, respectively:
$\displaystyle\mathcal{D}_{\parallel}(\lambda)=\mathcal{F}(\lambda)-(3a\mu_{\tau}-2)\lambda,\quad\mathcal{D}_{\perp}(\lambda)=\mathcal{F}(\lambda)-a\mu_{\tau}\lambda,$
(4)
where $\mathcal{F}(\lambda)=a\left(1-\langle
e^{-\lambda\tau}\rangle\right)+\lambda^{2}$. The function $\langle
e^{-\lambda\tau}\rangle$ is the moment generating function of $\rho(\tau)$
since the $n$-th moment is equal to
$\langle\tau^{n}\rangle=(-1)^{n}\frac{d^{n}}{d\lambda^{n}}\langle
e^{-\lambda\tau}\rangle|_{\lambda=0}$. Regardless of the choice of $a$ and
$\rho(\tau)$, the characteristic functions $\mathcal{D}_{\parallel}$ and
$\mathcal{D}_{\perp}$ have a zero eigenvalue arising from the translation
invariance of Eq. (2) not . There is a fold bifurcation as an eigenvalue of
$\mathcal{D}_{\parallel}$ crosses the origin when $a\mu_{\tau}=1$, which marks
the disappearance of the translating state as seen from Eq. (3). Numerical
analysis Engelborghs (2000) reveals an additional curve on the $(a,\
\mu_{\tau})$ plane (below the curve $a\mu_{\tau}=1$) along which perturbations
parallel to the translation undergo a Hopf bifurcation as a complex pair of
eigenvalues of $\mathcal{D}_{\parallel}$ cross the imaginary axis.
As for perturbations perpendicular to the translational motion, there is
another fold bifurcation as an eigenvalue of $\mathcal{D}_{\perp}$ crosses the
origin along the curve $a\langle\tau^{2}\rangle=2$, which represents a
bifurcation in which the translating state merges with a circularly rotating
state of infinite radius, as discussed below.
Considering a fixed $\sigma_{\tau}$, the overall stability picture of the
translating state of Eqs. (2) is as follows (see Fig. 1). For values of $(a,\
\mu_{\tau})$ below the curves $a\langle\tau\rangle=1$ and
$a\langle\tau^{2}\rangle=2$ (region A) the translating state is linearly
stable . These two curves may cross at a point that we call the ‘zero
frequency Hopf point’ (ZFH). The transverse direction of the translating state
becomes unstable along the curve $a\langle\tau^{2}\rangle=2$ where this state
merges with the circularly rotating state (along the mentioned curve the
rotating state has an infinite radius); transverse perturbations of the
translating state will thus produce a transition to the rotating state in
regions B and C. From the ZFH point, there emanates a Hopf bifurcation curve
where the parallel component of the translating state becomes unstable so that
in region C there is a transition from the translating state to oscillations
along a straight line. Finally, the translating state ceases to exist along
the curve $a\langle\tau\rangle=1$ where there is a pitchfork type bifurcation
with the stationary steady state solution. The possible behaviors in region D
are discussed below.
Figure 2: Speed of the translating state of Eqs. (1) and (2) as a function of
$\mu_{\tau}$; here $N=150$, $a=0.2$ (a) and $a=1.2$ (b). The red line (color
online) represents the mean field result from Eq. (3); the continuous segment
marks where the translating state is linearly stable and the dashed segment
where it is unstable. The symbols represent numerical simulations of Eqs. (1)
for different time delay distributions: sliding exponential and sliding
uniform $\sigma_{\tau}$ = 0.5 (a), $\sigma_{\tau}$ = 0.05 (b); widening
exponential $\mu_{\tau}=\sigma_{\tau}$ (a) and (b); widening uniform
$\mu_{\tau}=\sqrt{3}\sigma_{\tau}$ (a) and (b).
We compare the mean field bifurcation results with the full swarm system via
numerical simulations. Here, we make use of two different time delay
distributions with mean $\mu_{\tau}$ and standard deviation $\sigma_{\tau}$ to
test our findings. The first is an exponential distribution
$\rho(\tau)=e^{\frac{\tau-\mu_{\tau}+\sigma_{\tau}}{\sigma_{\tau}}}/\sigma_{\tau}$
for $\tau\geq\mu_{\tau}-\sigma_{\tau}$ and zero otherwise; we require
$\sigma_{\tau}\leq\mu_{\tau}$ for proper normalization. The second
distribution is a uniform $\rho(\tau)=\frac{1}{2\sqrt{3}\cdot\sigma_{\tau}}$
for
$\mu_{\tau}-\sqrt{3}\sigma_{\tau}\leq\tau\leq\mu_{\tau}+\sqrt{3}\sigma_{\tau}$
and zero otherwise; here, we require $\sqrt{3}\sigma_{\tau}\leq\mu_{\tau}$.
Moreover, we employ two versions of the mentioned distributions: a ‘sliding’
one ($\sigma_{\tau}$ = const.) and a ‘widening’ version
($\mu_{\tau}\propto\sigma_{\tau}$).
Fig. 2 compares the speed of the swarm obtained from Eq. (3) with the time-
averaged speed of the center of mass obtained from simulations (after the
decay of transients). The swarm particles are all located at the origin at
time zero and move with the speed obtained from Eq. (3) along the $x$ axis. In
these simulations, we use both ‘sliding’ ($\sigma_{\tau}$ fixed) and
‘widening’ (both $\mu_{\tau}$ and $\sigma_{\tau}$ vary) versions of the
exponential and uniform distributions described above. Fig. 2 shows that the
swarm converges to the translating state up to a value of $\mu_{\tau}$ beyond
which the swarm converges to a state in which it oscillates back and forth
along a line with a near zero time-averaged speed (the average is taken over
an interval much longer than the period). The transition to the oscillatory
regime occurs earlier than the mean field prediction. The full simulation
results show that in this deviation from the mean field, the swarm particles
become spread out too far apart and render the approximations leading to Eq.
(2) invalid.
Figure 3: Bifurcation curves of the mean field Eqs. (2) at fixed
$\sigma_{\tau}$ for the two time delay distributions $\rho(\tau)$ described in
the text: exponential (top) and uniform (bottom). The translational state
disappearance curve $a\mu_{\tau}=1$ (red), bifurcation of the translational
state with circularly rotating state curve $a\langle\tau^{2}\rangle=2$
(black). The first four members of the stationary state Hopf bifurcation
curves are also shown (blue, dashed green, dotted-dashed cyan and dotted
magenta). In each panel, $\sigma_{\tau}$ has the value (a) 0.2, (b) 0.95, (c)
0.2 and (d) 0.3, respectively. (Color online.)
In addition to the translating state, Eqs. (2) always possess a stationary
state solution $\mathbf{R}(t)=\mathbf{R}_{0}=$ const. In the full system, Eq.
(1), the stationary state for the center of mass manifests itself in a swarm
‘ring state’, where some particles rotate clockwise and others counter-
clockwise on a circle around a static center of mass. The characteristic
equation that governs the linear stability of the stationary state has the
form $\big{(}\mathcal{D}(\lambda)\big{)}^{2}=0$, where
$\mathcal{D}(\lambda)=\mathcal{F}(\lambda)-\lambda.$ Once more there is a zero
eigenvalue for all choices of $a$ and $\rho(\tau)$ that arises from the
translation invariance of Eqs. (2). Also, since $\mathcal{D}(0)=0$,
$\mathcal{D}^{\prime}(0)=a\mu_{\tau}-1$ and
$\lim_{\lambda\rightarrow\infty}\mathcal{D}(\lambda)=\infty$, the condition
$a\mu_{\tau}-1<0$ guarantees the existence of at least one real and positive
eigenvalue which renders the stationary state linearly unstable. Thus,
$a\mu_{\tau}=1$ is a bifurcation curve on the $(a,\mu_{\tau})$ plane along
which the uniformly translating state bifurcates with the stationary state.
The stationary state undergoes Hopf bifurcations when the equation
$\mathcal{D}(i\omega)=a\left(1-\langle
e^{-i\omega\tau}\rangle\right)-\omega^{2}-i\omega=0$ for $\omega\neq 0$ is
satisfied. The function $\langle e^{-i\omega\tau}\rangle$ is called the
characteristic function of $\rho(\tau)$ and is related to the moment
generating function of the distribution; its Taylor series contains all of the
moments of $\rho(\tau)$. This shows that the location of the Hopf bifurcations
depends on the values of all moments of the time delay distribution. This is
in contrast to the region where the translating state exists $a\mu_{\tau}<1$,
which involves the first moment only.
At the location of the Hopf bifurcations, circular orbits bifurcate from the
stationary state. This may be seen by changing Eq. (2) from the Cartesian
$(X,\ Y)$ to polar coordinates $(R,\ \phi)$, noticing that circular orbits
$R=R_{0}$, $\phi=\omega t$ are possible as long as
$\displaystyle\omega^{2}=a\left(1-\langle\cos\omega\tau\rangle\right),\qquad
R_{0}=\frac{1}{\omega}\sqrt{1-\frac{a}{\omega}\langle\sin\omega\tau\rangle},$
(5)
and then realizing that the first of (5) and the condition $R_{0}=0$ are
precisely the real and imaginary parts of the Hopf bifurcation conditions
$\mathcal{D}(i\omega)=0$.
Figure 4: Center of mass speed as a function of $\mu_{\tau}$ for the ‘sliding’
exponential time delay distribution (left) and the ‘sliding’ uniform
distribution (right). Here $N=150$, $a=2$. The continuous red curve represents
the mean field result from Eq. (5), while the symbols represent the results
from numerical simulations of Eqs. (1). (color online)
Generically, the Hopf conditions for the stationary state
$\mathcal{D}(i\omega)=0$ yield a family of curves in the $(a,\ \mu_{\tau})$
plane (Fig. 3). The first member of the Hopf family emanates from the crossing
of the curves $a\mu_{\tau}=1$ and $a\langle\tau^{2}\rangle=2$; the former
curve is where the translating state bifurcates from the stationary state in a
pitchfork-like bifurcation. Hence the name ‘Zero Frequency Hopf’ for the Hopf-
fold point (Fig. 3a). The first Hopf curve is supercritical and gives rise to
a circularly rotating orbit with radius and frequency given by the first
solution of Eqs. (5). Below this first Hopf curve and $a\mu_{\tau}=1$, in
region A the stationary state is stable. From Eqs. (5) it follows that this
circularly rotating orbit collides with the translating state along the curve
$a\langle\tau^{2}\rangle=2$ where its radius tends to infinity and its speed
to that of the translating state, $\sqrt{1-a\mu_{\tau}}$. Thus, in regions B
and C the system converges to the circularly rotating orbit. The different
regions change shape for the other panels of Fig. 3, but the dynamics remain
as described above.
We compare the mean field prediction for the location of the birth of the
circularly rotating state (first Hopf curve) with the full system. Fig. 4
shows the results from numerical simulations of Eqs. (1) at a fixed value of
$\sigma_{\tau}$ for increasing $\mu_{\tau}$. We plot the speed of the center
of mass averaged over a long time interval, after the decay of transients. In
that plot, the near-zero values of the mean speed (in the interval
$0\lesssim\mu_{\tau}\lesssim 1.4$) indicate that the particles have converged
to the ring state, while for all higher values of $\mu_{\tau}$ the swarm
converges to the rotating state. Remarkably, for the highest values of
$\mu_{\tau}$, the center of mass of the swarm moves faster than unit speed,
the asymptotic speed of uncoupled particles. The reason is that while in its
rotating orbit, the mean time-delayed position of the swarm is actually
“ahead” of the center of mass at the current time, causing the particles to
accelerate forward along the circular orbit.
In summary, we have considered a randomly delay distributed coupled swarm
model, and analyzed the bifurcations of various patterns as a function of
delay characteristics and coupling strength. In particular, we have shown that
the location and shape of the Hopf bifurcation curves is strongly-dependent on
all the moments of $\rho(\tau)$. This dependence, in addition to the fact that
the succeeding Hopf curves in Fig. 3 exhibit higher frequencies of rotation,
makes the higher-order patterns equally sensitive to all the moments of the
delay distribution. In the single delay case with distribution
$\rho=\delta(\tau-\tau_{0})$, where $\delta(\tau)$ is a Dirac delta function,
all of the succeeding Hopf bifurcations are all subcritical and continuous. In
contrast, when all moments are present, the bifurcations may not even be
continuous, presenting their structure as isolated closed curves bounded by
fold bifurcations, as seen in Fig. 3d for a uniform distribution. Finally, we
expect that in other globally delay-coupled systems Choi et al. (2000); Marti
and Masoller (2003); Kozyreff et al. (2001); Masoller et al. (2009), generic
types of behavior involving bifurcations including all moments of the
distribution should be present.
The authors gratefully acknowledge the Office of Naval Research for their
support. LMR is an NIH post doctoral fellow and BL is a post doctoral fellow
of the NRC.
## References
* Budrene and Berg (1995) E. Budrene and H. Berg, Nature 376, 49 (1995).
* Toner and Tu (1995) J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995).
* Parrish (1999) J. K. Parrish, Science 284, 99 (1999).
* Topaz and Bertozzi (2004) C. Topaz and A. Bertozzi, SIAM Journal on Applied Mathematics 65, 152 (2004).
* Leonard and Fiorelli (2002) N. Leonard and E. Fiorelli, in _Proc. of the 40th IEEE Conference on Decision and Control._ (2002), vol. 3, pp. 2968–2973.
* Morgan and Schwartz (2005) D. Morgan and I. B. Schwartz, Phys. Lett. A 340, 121 (2005).
* Chuang et al. (2007) Y.-L. Chuang, Y. R. Huang, M. R. D’Orsogna, and A. L. Bertozzi, in _Proc. of the 2007 IEEE International Conference on Robotics and Automation._ (2007), pp. 2292–2299.
* Lynch et al. (2008) K. M. Lynch, P. Schwartz, I. B. Yang, and R. A. Freeman, IEEE Trans. Robotics 24, 710 (2008).
* Vicsek et al. (1995) T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
* Flierl et al. (1999) G. Flierl, D. Grünbaum, S. Levins, and D. Olson, J. Theor. Biol. 196, 397 (1999).
* Toner and Tu (1998) J. Toner and Y. Tu, Phys. Rev. E 58, 4828 (1998).
* Edelstein-Keshet et al. (1998) L. Edelstein-Keshet, J. Watmough, and D. Grunbaum, J. Math. Biol. 36, 515 (1998).
* Erdmann and Ebeling (2005) U. Erdmann and W. Ebeling, Phys. Rev. E 71 (2005).
* Forgoston and Schwartz (2008) E. Forgoston and I. B. Schwartz, Phy. Rev. E 77 (2008), arXiv:0712.2950.
* Aldana et al. (2007) M. Aldana, V. Dossetti, C. Huepe, V. Kenkre, and H. Larralde, Phys. Rev. Lett. 98 (2007).
* Englert et al. (2011) A. Englert, S. Heiligenthal, W. Kinzel, and I. Kanter, Phys. Rev. E 83 (2011).
* Zuo et al. (2010) Z. Zuo, C. Yang, and Y. Wang, Phys. Lett. A 374, 1989 (2010).
* Konishi et al. (2010) K. Konishi, H. Kokame, and N. Hara, Phys. Rev. E 81 (2010).
* Ahlborn and Parlitz (2007) A. Ahlborn and U. Parlitz, Phys. Rev. E 75 (2007).
* Wu et al. (2009) D. Wu, S. Zhu, and X. Luo, EPL 86 (2009).
* Marti et al. (2006) A. C. Marti, M. Ponce C, and C. Masoller, Physica A 371, 104 (2006).
* Omi and Shinomoto (2008) T. Omi and S. Shinomoto, Phys. Rev. E 77 (2008).
* Dykman and Schwartz (2012) M. I. Dykman and I. B. Schwartz (2012), arXiv:1204.6519.
* Mier-y Teran-Romero et al. (2011) L. Mier-y Teran-Romero, E. Forgoston, and I. B. Schwartz (2011), IEEE TRO, accepted. arXiv:1205.0195.
* Lindley et al. (2012) B. Lindley, L. Mier-y Teran, and I. B. Schwartz, ICRA 2012 conference papers, accepted. arXiv:1204.4606. (2012).
* (26) In fact, $\mathcal{D}_{\perp}$ has a second zero eigenvalue.
* Engelborghs (2000) K. Engelborghs, Tech. Rep. TW-305, Department of Computer Science, K. U. Leuven, Belgium (2000), URL http://www.cs.kuleuven.ac.be/$∼$twr/research/software/de%lay/ddebiftool.shtml.
* Choi et al. (2000) M. Choi, H. Kim, D. Kim, and H. Hong, Phys. Rev. E 61, 371 (2000).
* Marti and Masoller (2003) A. Marti and C. Masoller, Phys. Rev. E 67 (2003).
* Kozyreff et al. (2001) G. Kozyreff, A. Vladimirov, and P. Mandel, Phys. Rev. E 64 (2001).
* Masoller et al. (2009) C. Masoller, M. C. Torrent, and J. Garcia-Ojalvo, Philos. T. Roy. Soc. A 367, 3255 (2009).
|
arxiv-papers
| 2012-05-09T17:46:46 |
2024-09-04T02:49:30.770047
|
{
"license": "Public Domain",
"authors": "Luis Mier-y-Teran-Romero, Brandon Lindley, and Ira B. Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1205.2047"
}
|
1205.2052
|
# Local equivalence of reversible
and general Markov kinetics
Alexander N. Gorban ag153@le.ac.uk Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, UK
###### Abstract
We consider continuous–time Markov kinetics with a finite number of states and
a positive equilibrium $P^{*}$. This class of systems is significantly wider
than the systems with detailed balance. Nevertheless, we demonstrate that for
an arbitrary probability distribution $P$ and a general system there exists a
system with detailed balance and the same equilibrium that has the same
velocity ${\mathrm{d}}P/{\mathrm{d}}t$ at point $P$. The results are extended
to nonlinear systems with the generalized mass action law.
###### keywords:
detailed balance , Lyapunov function , decomposition , entropy , uncertainty
###### PACS:
02.50.Ga , 05.20.Dd
## 1 Introduction
### 1.1 Detailed balance and beyond
The principle of detailed balance is one of the most celebrated results in
kinetics. A kinetic system is represented as a mixture of independent
elementary processes (collisions or elementary reactions, for example). Due to
the principle of detailed balance, at equilibrium, each elementary process
should be equilibrated by its reverse process. Kinetics is decomposed into
pairs of mutually inverse processes and in many problems we can consider these
pairs separately.
We study relations between the systems with and without detailed balance. In
this Section, we briefly overview the main results of the work. Then, in Sec.
1.2 we review the history of the problem. We prove the local equivalence
theorem for the Markov processes in Sec. 2 and give there the simple examples.
The nonlinear generalizations are presented in Sec. 3.
In Sec. 2 we start from the first order kinetics without the detailed balance
assumption. The general first order kinetic equation has the form:
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(q_{ij}p_{j}-q_{ji}p_{i})\,,$ (1)
where $q_{ij}$ ($i,j=1,\ldots,n$, $i\neq j$) are non-negative. This system of
equations (master equations or Kolmogorov’s equations) describes dynamics of
non-negative variables $p_{i}$ ($i=1,\ldots,n$). These variables may be
considered as probabilities (then $\sum_{i}p_{i}=1$) or concentrations. For
the corresponding states or components we use the notation $A_{i}$. In this
notation, $q_{ij}$ is the rate constant for transitions $A_{j}\to A_{i}$.
Let us assume that system (1) has a positive equilibrium $P^{*}=(p_{i}^{*})$,
$p_{i}^{*}>0$:
$\sum_{j,\,j\neq i}q_{ij}p^{*}_{j}=\left(\sum_{j,\,j\neq
i}q_{ji}\right)p^{*}_{i}\;\mbox{ for all }i=1,\ldots n\,.$ (2)
This is the so-called balance equation.
The detailed balance condition is much stronger. It assumes that the sums in
the left and right hand sides of Eq. (2) are equal term by term:
$q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}\;\mbox{ for all }i,j=1,\ldots n,\,i\neq j\,.$
(3)
For the number of states $n>2$, a simple comparison of dimensions demonstrates
that there are much more general systems with the given positive equilibrium
$P^{*}$ (2) (dimension is $(n-1)^{2}$) than the systems with detailed balance
with equilibrium $P^{*}$ (3) (dimension is $\frac{n(n-1)}{2}$). Surprisingly,
for every given distribution $P$, the set of possible velocities
${\mathrm{d}}P/{\mathrm{d}}t$ for general Markov kinetics with equilibrium
$P^{*}$ is the same that for Markov kinetics with detailed balance and the
same equilibrium. This is the central result of the paper (Theorem 1 in Sec.
2).
We demonstrate this in two steps. First, we use the representation of a
general Markov chain with a given positive equilibrium as a combination with
non-negative coefficients of several simple cycles with the same equilibrium.
Secondly, we demonstrate this equivalence for a simple cycle of transitions
with positive constants
$A_{1}\to A_{2}\to\ldots\to A_{n}\to A_{1}\,.$ (4)
For the equilibrium $P^{*}$ the constants of the cycle are
$q_{i+1i}=\kappa/p_{i}^{*}$ (we use here the standard convention about the
cyclic numeration, $n+1=1$).
Thus, if we observe the Markov kinetics at one point then we can not
distinguish general systems from systems with detailed balance because the
sets of possible velocities coincide. In particular, they have the same set of
Lyapunov functions.
Our main results allow us to decompose any Markov kinetics (or generalized
mass action law kinetics with semi-detailed balance) into pairs of mutually
inverse elementary processes with the same equilibrium. If the system does not
satisfy the principle of detailed balance then this decomposition depends on
the state. Nevertheless, in some problems it is still convenient to consider
these pairs separately.
In this paper, we give two examples of the application of Theorem 1: the
evaluation of logarithmic decrement for general Markov chains and a simple
proof of the Morimoto $H$-theorem for all the Csiszár–Morimoto divergencies.
We give also the nonlinear generalization of Theorem 1 for the systems which
obey the generalized Mass Action Law (MAL).
Figure 1: Elementary reaction with intermediate compounds.
Master equation is a source for many other kinetic equations. In particular,
in Sec. 3 we consider complex reactions with intermediate compounds (Fig. 1)
under two asymptotic conditions
* 1.
The compounds $B_{j}$ are in fast equilibrium with the corresponding input or
output reagents;
* 2.
They exist in very small concentrations compared to other components.
For compounds transitions the first order kinetics is assumed because of small
concentrations of compounds (only first order terms survive). These
assumptions allow us to produce the reaction rates for the overall reaction
from Fig. 1 in the form of the generalized MAL:
$r_{\rho}=\varphi_{\rho}\exp\left(\frac{\sum_{i}\alpha_{\rho
i}\mu_{i}}{RT}\right)\,,\;\varphi_{\rho}\geq 0\,.$ (5)
where $\mu_{i}$ is the chemical potential of the component $A_{i}$, $\rho$ is
the reaction number, $\alpha_{\rho i}$ are the input stoichiometric
coefficients (Fig. 1). Both $\alpha_{\rho i}$ and $\beta_{\rho i}$ are non-
negative integers. We use notations $\boldsymbol{\alpha}_{\rho}$ and
$\boldsymbol{\beta}_{\rho}$ for vectors wit coordinates $\alpha_{\rho i}$ and
$\beta_{\rho i}$. The positive functions $\varphi_{\rho}$ are called the
kinetic factors whereas $\exp\left({\sum_{i}\alpha_{\rho i}\mu_{i}}/RT\right)$
are the Boltzmann factors.
The balance condition of the first order kinetics of compounds (2) transforms
in the semi-detailed balance condition (that is known also as the complex or
the cyclic balance condition):
$\sum_{\rho,\,\boldsymbol{\alpha}_{\rho}=\boldsymbol{\nu}}\varphi_{\rho}\equiv\sum_{\rho,\,\boldsymbol{\beta}_{\rho}=\boldsymbol{\nu}}\varphi_{\rho}$
for any vector $\boldsymbol{\nu}$ from the set of all vectors
$\\{\boldsymbol{\alpha}_{\rho},\boldsymbol{\beta}_{\rho}\\}$. Kinetics with
the generalized MAL and the semi-detailed balance conditions give the natural
nonlinear generalizations of Markov processes. In particular, the entropy
production for these systems at any nonequilibrium state is positive.
If we assume for the Markov kinetics of compounds that the positive
equilibrium is the point of detailed balance (3) then the kinetic factors
$\varphi_{\rho}$ satisfy the stronger condition of detailed balance:
$\varphi_{\rho}^{+}\equiv\varphi_{\rho}^{-}\;\mbox{ for all }\rho\,,$
where $\varphi_{\rho}^{+}$ is the kinetic factor for the direct reaction and
$\varphi_{\rho}^{-}$ is the kinetic factor for the reverse reaction.
The class of systems with semi-detailed balance is much wider than the class
of systems with detailed balance. Nevertheless, locally they coincide: the set
of possible velocities for systems with semi-detailed balance coincide with
the set of possible velocities for the systems with detailed balance for given
thermodynamic functions and any given state (Sec. 3).
The systems with generalized MAL and semi-detailed balance are the nonlinear
analogs of the Markov processes and the local equivalence of the generalized
MAL systems with detailed and semi-detailed balance is the analog and a
consequence of Theorem 1.
### 1.2 A bit of history
In 1872, Boltzmann introduced the principle of detailed balance for collisions
and used it to prove his $H$-theorem [1]. Boltzmann’s proof of the positivity
of entropy production for systems with detailed balance is very transparent
because it is sufficient to prove this positivity just for a couple of
mutually inverse elementary processes.
In 1887, Lorentz [2] objected Boltzmann: he insisted that some collisions of
polyatomic molecules do not have reverse collisions and cannot satisfy the
principle of detailed balance. Immediately, Boltzmann realized that there
exists a much weaker condition sufficient for the $H$-theorem [3]. In 1981, it
was proven that the Lorentz objections are wrong and the principle of detailed
balance is valid for polyatomic molecules [4]. This is not very surprising
because the detailed balance follows from microreversibility (or
$T$-invariance of the fundamental equations of mechanics or quantum
mechanics). Nevertheless, the Boltzmann discovery is valuable by itself and is
used for many kinetic equations.
This condition was rediscovered several times. It is known as the semi-
detailed balance condition, the cyclic balance condition or the complex
balance condition. In 1952, Stueckelberg proposed a proof of the semi-detailed
balance condition for the Boltzmann equation [5]. His proof is based on the
Markov model of elementary events. Recently, the Stueckelberg approach was
extended to prove the semi-detailed balance condition for the generalized MAL
kinetics [6].
The complex balance condition for chemical kinetics was introduced by Horn and
Jackson in 1972 [7] independently of Boltzmann’s work. Now it is used for
mathematical modeling in chemical kinetics and engineering [8]. Boltzmann’s
idea about cyclic balance developed in physical kinetics was independently
rediscovered in the theory of Markov processes and it is proved that any
recurrent Markov process can be decomposed into directed cycles [10].
The principle of detailed balance was crucially important in the development
of the Metropolis–Hastings and other Markov chain Monte Carlo algorithms from
the very beginning [11]. Technically, it is much easier to use the detailed
balance conditions (3) than to follow more intricate balance conditions.
Detailed balance was considered as a necessary condition for construction of
Monte Carlo algorithms as a “systematic design principle” [12]. It was
demonstrated that it helps to reduce the uncertainty of some observables in
stochastic numerics [13].
Nevertheless, there are many examples of efficient Monte Carlo computations
without detailed balance. Sometimes computational models without detailed
balance are constructed because of the physical nature of the systems. For
example, the models of inelastic processes in particle physics [14] or in
granular media [15] may violate the principle of detailed balance. The general
theories of stochastic cellular automata with Gibbsian equilibria but without
compulsory detailed balance were developed [16]. It is widely recognized that
the balance equation (not the detailed balance) is necessary and sufficient
condition for invariance (stationarity) of the desired equilibrium
distribution. Under some more technical irreducibility conditions, the Monte
Carlo simulations will converge to this equilibrium [17, 18].
The interplay between reversible (with detailed balance) and irreversible
Markov chains is non-trivial and important for many applications. Recently, it
was demonstrated that the local deformation of the reversible Markov processes
into irreversible ones helps to create efficient computational Monte Carlo
algorithms [19].
Much efforts were applied to verification of microscopic reversibility and its
consequences, the detailed balance conditions and the Onsager reciprocal
relations, in many experimental systems [20, 21, 22, 23, 24]. To check
experimentally the detailed balance conditions it is necessary to deal with a
complex reaction that can be formally equilibrated without detailed balance.
If not, then one tests not the detailed balance but just the equilibrium
condition as it was mentioned in [25].
The detailed balance conditions are very natural and appealing. They simplify
many computations and proofs. Thus, they are used in many applications and
models. For some physical and chemical systems, detailed balance has a solid
background, the $T$-invariance of the fundamental equations, but it is also
used beyond the proven microreversibility. When modelling with detailed
balance meets some difficulty then the problem about relations between
reversible and irreversible systems arises again and detailed balance is
substituted by more general conditions. Nevertheless, the convenience, beauty
and some intrinsic benefits of the phenomenon, have forced researchers to
return to detailed balance if it is possible without a conflict with reality.
There are many examples of this “pendulum” in scientific publications: accept
detailed balance – criticize detailed balance – go to more general conditions
– realize the benefits of detailed balance – return to detailed balance – …
At the same time, the consequences of the principle of detailed balance are
extended to the situations where it was not used before. Thus, recently the
multiscale limit of the systems of reversible reactions was studied when some
of the equilibrium concentrations tend to zero. The extended principle of
detailed balance was proved for the systems with some irreversible reaction
[26, 27].
In Theorem 1, we compare the sets of possible velocities,
${\mathrm{d}}P/{\mathrm{d}}t$, for two classes of systems: (i) general first
order kinetics (1) with the given positive equilibrium $P^{*}$ and (ii) first
order kinetics with detailed balance (3) and the same positive equilibrium.
Understanding the structure of the sets of possible velocities can provide
additional information about attainable states of the system which is helpful
in the modelling context. There are many other reasons too. It is known that
different types of kinetics data bear different degrees of reliability. It
would therefore be very attractive to study the consequences of the
information of each level of reliability separately [28]. For example,
uncertainty about equilibria in the system is usually significantly lower than
that of the reaction rate constants. The value of the equilibrium gives us
some information about dynamics: the set of possible velocities
${\mathrm{d}}P/{\mathrm{d}}t$ is not arbitrarily wide at a given state and for
given equilibrium. For the systems with detailed balance, this set is a
polyhedral cone which allows a simple description (Sec. 2.2). Due to Theorem
1, however, this cone is also the set of possible velocities for the general
master equation. Therefore, to distinguish the detailed balance systems from
the general ones we have to involve data about ${\mathrm{d}}P/{\mathrm{d}}t$
for several significantly distant distributions $P$.
Another example is to employ the knowledge of the sets of possible velocities
for estimating attainable regions for kinetics. The idea to use the
equilibrium information to estimate the attainable sets in kinetics was
proposed in 1964 by Horn [29] and developed further in chemical kinetics and
chemical engineering [30, 31, 32, 33, 34] (for more detailed review see [28]).
If the sets of possible velocities coincide then the estimated attainable
regions coincide too. The knowledge of the sets of possible velocities is also
important for the analysis of observability, identifiability and
controllability of the systems.
## 2 Local equivalence of general Markov systems and systems with detailed
balance
### 2.1 Global decomposition into cycles, local decomposition into steps, and
the equivalence theorem
Let us start from master equation (1). The coefficient $q_{ij}$ is the rate
constant for transitions $A_{j}\to A_{i}$. Any set of non-negative
coefficients $q_{ij}$ ($i\neq j$) corresponds to a master equation. Therefore,
the set of all master equations (1) may be considered as the non-negative
orthant in $\mathbb{R}_{+}^{n(n-1)}\subset\mathbb{R}^{n(n-1)}$. (The non-
negative orthant is the set of all vectors with only non-negative components.)
We assume that system (1) has a positive equilibrium $P^{*}=(p_{i}^{*})$,
$p_{i}^{*}>0$ and the balance condition (2) holds. The sum of these $n$
balance conditions is a trivial identity. Let us delete any single equation
from (2), for example, the last one (for $i=n$). Each of the remaining
equations includes the variable $q_{in}$ which is not present in other
equations ($i=1,\ldots,n-1$). Therefore, for given positive $P^{*}$, there are
$n-1$ independent conditions on $q_{ij}$ ($i,j=1,\ldots,n$, $i\neq j$) in (2).
Thus, the balance conditions (2) define for a given positive equilibrium a
$(n-1)^{2}$-dimensional linear subspace $L^{*}$ in the $n(n-1)$ dimensional
space of $q_{ij}$ ($i\neq j$). A vector of positive coefficients
$q_{ij}^{*}=1/p_{j}^{*}$ satisfies (2) and belongs to $L^{*}$. This vector
belongs to the interior of the non-negative orthant $\mathbb{R}_{+}^{n(n-1)}$.
Therefore, the intersection $\mathbb{R}_{+}^{n(n-1)}\cap L^{*}$ includes a
vicinity of $q_{ij}^{*}$ in $L^{*}$ and is a $(n-1)^{2}$-dimensional cone.
Thus, the non-negative solutions of (2) form a $(n-1)^{2}$-dimensional closed
cone in $\mathbb{R}_{+}^{n(n-1)}$.
The systems with detailed balance (3) for a given positive equilibrium form a
smaller cone. Under these conditions, there are only $\frac{n(n-1)}{2}$
independent coefficients among $n(n-1)$ numbers $q_{ij}$. For example, we can
arbitrarily select $q_{ij}\geq 0$ for $i>j$ and then take
$q_{ij}=q_{ji}\frac{p_{i}^{*}}{p^{*}_{j}}$ for $i<j$. So, for given $P^{*}$,
the cone of the detailed balance systems (3) can be considered as a non-
negative orthant in $\mathbb{R}^{\frac{n(n-1)}{2}}$ embedded in
$\mathbb{R}_{+}^{n(n-1)}$.
If the balance condition (2) holds then system (1) may be rewritten in a
convenient equivalent form:
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\,.$
(6)
With this form of master equation, it is straightforward to calculate the time
derivative of the quadratic divergence, a weighted $l_{2}$ distance between
$P$ and $P^{*}$,
$H_{2}(P\|P^{*})=\sum_{i}\frac{(p_{i}-p^{*}_{i})^{2}}{p^{*}_{i}}$:
$\frac{{\mathrm{d}}H_{2}(P\|P^{*})}{{\mathrm{d}}t}=-\sum_{i,j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{i}}{p^{*}_{i}}-\frac{p_{j}}{p^{*}_{j}}\right)^{2}\leq
0\,.$ (7)
This time derivative is strictly negative if for a transition $A_{j}\to A_{i}$
the rate constant is positive, $q_{ij}>0$, and
$\frac{p_{i}}{p^{*}_{i}}\neq\frac{p_{j}}{p^{*}_{j}}$. Hence, if the state $P$
is not an equilibrium (i.e., the right hand side in (6) is not zero) then
$\frac{{\mathrm{d}}H_{2}(P\|P^{*})}{{\mathrm{d}}t}<0$.
Let us introduce the following notation for a given number of states $n$:
* 1.
$\mathcal{Q}^{n}_{\rm B}(P^{*})$ is the cone of the vectors of non-negative
coefficients $q_{ij}$ ($i\neq j$) which satisfy the balance conditions (2),
that is, the set of all Markov processes with the equilibrium distribution
$P^{*}$;
* 2.
$\mathcal{Q}^{n}_{\rm DB}(P^{*})$ is the cone of the vectors of non-negative
coefficients $q_{ij}$ ($i\neq j$) which satisfy the detailed balance
conditions (3), that is, the set of all Markov processes with detailed balance
and the equilibrium distribution $P^{*}$.
If a system satisfies the detailed balance condition (3) then the balance
condition (2) holds too: it holds term by term, even without summation.
Therefore, $\mathcal{Q}^{n}_{\rm DB}(P^{*})\subset\mathcal{Q}^{n}_{\rm
B}(P^{*})$. Moreover, comparing dimension we find that this inclusion is
strong. Indeed, $\dim\mathcal{Q}^{n}_{\rm B}(P^{*})=(n-1)^{2}$,
$\dim\mathcal{Q}^{n}_{\rm DB}(P^{*})=\frac{n(n-1)}{2}$. If $n>2$ then
$\frac{n(n-1)}{2}<(n-1)^{2}$, hence,
$\mathcal{Q}^{n}_{\rm DB}(P^{*})\subsetneqq\mathcal{Q}^{n}_{\rm
B}(P^{*})\subsetneqq\mathbb{R}_{+}^{n(n-1)}\,.$ (8)
The cone of the systems with detailed balance is, in some sense, much smaller
than the cone of the systems with the balance condition: the difference
between their dimensions is $\frac{(n-1)(n-2)}{2}$.
Now, let us consider the right hand side vector fields of the systems (1) at
the point $P\neq P^{*}$. For each cone of the coefficients $q_{ij}$ the
vectors of the possible velocities, ${\mathrm{d}}P/{\mathrm{d}}t$, also form a
cone. Let us introduce the following notation:
* 1.
$\mathbf{Q}^{n}_{\rm B}(P,P^{*})$ is the cone of the possible velocities,
${\mathrm{d}}P/{\mathrm{d}}t$, at the point $P$ for
$(q_{ij})\in\mathcal{Q}^{n}_{\rm B}(P^{*})$;
* 2.
$\mathbf{Q}^{n}_{\rm DB}(P,P^{*})$ is the cone of of the possible velocities,
${\mathrm{d}}P/{\mathrm{d}}t$, at point $P$ for
$(q_{ij})\in\mathcal{Q}^{n}_{\rm DB}(P^{*})$.
$\mathbf{Q}^{n}_{\rm B}(P,P^{*})$ is the cone of all possible velocities for
Markov kinetics at the point $P$ if the equilibrium is $P^{*}$.
$\mathbf{Q}^{n}_{\rm DB}(P,P^{*})$ is the cone of these velocities for Markov
kinetics with detailed balance.
Surprisingly, for every given distribution $P$, the set of possible velocities
${\mathrm{d}}P/{\mathrm{d}}t$ for general Markov kinetics with equilibrium
$P^{*}$ is the same that for Markov kinetics with detailed balance and the
same equilibrium. The following theorem is the central result of this work.
###### Theorem 1.
$\mathbf{Q}^{n}_{\rm B}(P,P^{*})=\mathbf{Q}^{n}_{\rm DB}(P,P^{*})$
This means that for every first order kinetic equation (1) with a given
positive equilibrium $P^{*}$ and for every point $P\neq P^{*}$ there exists a
first order kinetic equation with detailed balance and equilibrium $P^{*}$
that has the same velocity at $P$. At this point the right hand sides of the
kinetic equations coincide.
Therefore, if we observe the Markov kinetics at one point then we can never
distinguish general systems from systems with detailed balance. In particular,
they have the same set of Lyapunov functions:
###### Corollary 1.
If for a function $H(P,P^{*})$, ${\mathrm{d}}H/{\mathrm{d}}t\leq 0$ for any
system (1) with equilibrium $P^{*}$ and detailed balance then
${\mathrm{d}}H/{\mathrm{d}}t\leq 0$ for any system (1) with equilibrium
$P^{*}$.
The system with detailed balance has $n(n-1)/2$ dimensions available to match
a single $n$-dimensional velocity vector. Therefore, it is not surprising that
the cones $\mathbf{Q}^{n}_{\rm DB}(P,P^{*})$ has non-empty interior (solid
cones). But this is not enough to cover any $n$-dimensional velocity vector
using non-negative coefficients $q_{ij}$. This non-negativity condition
defines the borders of the cones and we can a priori just state the inclusion
$\mathbf{Q}^{n}_{\rm DB}(P,P^{*})\subseteq\mathbf{Q}^{n}_{\rm B}(P,P^{*})$.
The calculation of dimension does not give a hint about coincidence of these
cones.
The proof of Theorem 1 is constructed in two steps. First, we prove that for
every $P^{*}$ and $P$ the cone of possible velocities $\mathbf{Q}^{n}_{\rm
B}(P,P^{*})$ is the convex hull of the velocities at point $P$ of the simple
cyclic schemes, $A_{i_{1}}\to\ldots\to A_{i_{k}}\to A_{i_{1}}$ ($k\leq n$ and
all the numbers $i_{1},\ldots,i_{k}$ are different), with the same equilibrium
$P^{*}$. Secondly, we prove that it is sufficient to take $k=2$.
We will characterize $\mathbf{Q}^{n}_{\rm B}(P,P^{*})$ by its extreme rays. A
ray with direction vector $x\neq 0$ is a set $\\{\lambda x\\}$ ($\lambda\geq
0$). $l$ is an extreme ray of a cone $\mathbf{Q}$ if for any $u\in l$ and any
$x,y\in\mathbf{Q}$, whenever $u=(x+y)/2$, we must have $x,y\in l$. If a closed
convex cone does not include a whole straight line then it is the convex hull
of its extreme rays [35].
###### Lemma 1.
The cone $\mathbf{Q}^{n}_{\rm B}(P,P^{*})$ does not include a whole straight
line.
###### Proof.
If $v\neq 0$ is a possible value of the right hand side of (6) then the
derivative of $H_{2}(P\|P^{*})$ in direction $v$ is strictly negative (7).
Therefore, it is impossible that both $v$ and $-v$ belong to
$\mathbf{Q}^{n}_{\rm B}(P,P^{*})$. ∎
Let us consider a simple cyclic scheme, $A_{i_{1}}\to\ldots\to A_{i_{k}}\to
A_{i_{1}}$ ($k\leq n$ and all the numbers $i_{1},\ldots,i_{k}$ are different).
For a given positive equilibrium, $P^{*}$ the coefficients for this scheme
belong to a ray:
$q_{i_{j+1}\,i_{j}}=\frac{\kappa}{p_{i_{j}}^{*}}\;(j=1,\ldots,k)\,,$ (9)
where $\kappa\geq 0$ is a constant and we use the standard convention that for
a cycle $q_{i_{k+1}\,i_{k}}=q_{i_{1}\,i_{k}}$.
###### Lemma 2.
If system (1) has a positive equilibrium $P^{*}$ then for every $A_{i}$ either
all $q_{ji}=q_{ij}=0$ or the state $A_{i}$ belongs to a cycle with strictly
positive rate constants.
###### Proof.
Let $A_{i}$ not belong to a cycle with positive constants. We say that a state
$A_{j}$ is reachable from a state $A_{k}$ if there exists a non-empty chain of
transitions with non-zero coefficients which starts at $A_{k}$ and ends at
$A_{j}$: $A_{k}\to\ldots\to A_{j}$. Let $\mathcal{A}_{i\downarrow}$ be the set
of states reachable from $A_{i}$ and $\mathcal{A}_{i\uparrow}$ be the set of
states $A_{i}$ is reachable from.
$\mathcal{A}_{i\downarrow}\cap\mathcal{A}_{i\uparrow}=\emptyset$ because
$A_{i}$ does not belong to a cycle. If $\mathcal{A}_{i\uparrow}$ is not empty
then in equilibrium all the corresponding $p_{j}^{*}=0$
($j\in\mathcal{A}_{i\downarrow}$ because there is flow from
$\mathcal{A}_{i\uparrow}$ to $A_{i}$ and no flow back). If
$\mathcal{A}_{i\downarrow}$ is not empty then in equilibrium $p^{*}_{i}=0$
because there is a flow from $A_{i}$ to $\mathcal{A}_{i\downarrow}$ and no
flow back. Therefore, if the equilibrium is strictly positive and $A_{i}$ does
not belong to a cycle then
$\mathcal{A}_{i\uparrow}=\mathcal{A}_{i\downarrow}=\emptyset$, hence, all
$q_{ji}=q_{ij}=0$. ∎
We will use the following simple general statement: Let $\mathcal{Q}$ be a
cone in $\mathbb{R}^{m}$ without straight lines, $L$ be a linear map,
$L:\mathbb{R}^{m}\to\mathbb{R}^{k}$, and $\mathbf{Q}=L(\mathcal{Q})$ be a cone
in $\mathbb{R}^{k}$ without straight lines. Then for every extreme ray
$V\subset\mathbf{Q}$ there exists an extreme ray $W\subset\mathcal{Q}$ such
that $L(W)=V$. (In other words, there always exists an extreme ray in the
preimage of an extreme ray.) We will apply this statement to
$\mathcal{Q}=\mathcal{Q}^{n}_{\rm B}(P^{*})$ (the cone of all Markov processes
with the given equilibrium $P^{*}$) and
$\mathbf{Q}=\mathbf{Q}^{n}_{B}{(P,P^{*})}$ (the cone of the possible
velocities at point $P$ for all Markov processes with the given equilibrium).
The map $L$ transforms the right hand side of the Kolmogorov equation (1) into
its value at point $P$. This transformation “vector field $\mapsto$ its value
at point $P$” is, obviously, a linear map.
###### Lemma 3.
Any extreme ray of the cone $\mathcal{Q}^{n}_{\rm B}(P^{*})$ is a simple cycle
with constants (9).
###### Proof.
Let a non-zero Markov chain $Q$ with coefficients $q_{ij}$ belong to an
extreme ray of $\mathcal{Q}^{n}_{\rm B}(P^{*})$. Due to Lemma 2 this chain
includes a simple cycle with non-zero coefficients, $A_{i_{1}}\to\ldots\to
A_{i_{k}}\to A_{i_{1}}$ ($k\leq n$, all the numbers $i_{1},\ldots,i_{k}$ are
different, $q_{i_{j+1}\,i_{j}}>0$ for $j=1,\ldots,k$, and $i_{k+1}=i_{1}$).
For sufficiently small $\kappa$ ($0<\kappa<\kappa_{0}$),
$q_{i_{j+1}\,i_{j}}-\frac{\kappa}{p^{*}_{i_{j}}}>0$ ($j=1,\ldots,k$). Let
$Q_{\kappa}$ be the same simple cycle with the coefficients (9). Then for
$0<\kappa<\kappa_{0}$ vectors $Q\pm Q_{\kappa}$ also represent Markov chains
with the equilibrium $P^{*}$. Obviously,
$Q=\frac{(Q+Q_{\kappa})+(Q-Q_{\kappa})}{2}$, hence, $Q$ should be proportional
to $Q_{\kappa}$. ∎
Due to this Lemma, every Markov chain with positive equilibrium is a convex
combination of several simple cycles with the same equilibrium. This is the
global decomposition of a Markov chain into simple cycles. “Global” here means
that the same decomposition is valid for all distributions.
Now, we are in position to prove Theorem 1.
###### Proof.
We will prove that any extreme ray of the cone $\mathbf{Q}^{n}_{\rm
B}(P,P^{*})$ corresponds to a simple cycle of length 2 (a step):
$A_{i}\rightleftharpoons A_{j}$ with the rate constants (9)
$q_{ij}=\frac{\kappa}{p^{*}_{j}}\,,\;q_{ji}=\frac{\kappa}{p^{*}_{i}}$.
According to Lemma 3, it is sufficient to prove that for any simple cycle with
equilibrium $P^{*}$ and rate constants (9) and for any distribution $P$ the
right hand side of the Kolmogorov equation (1) is a conic combination (a
combination with non-negative real coefficients) of the right hand sides of
this equation for simple cycles of length 2 at the same point $P$.
Let us prove this by induction on the cycle length $k$. For $k=2$ it is true
(trivially). For a cycle of length $k>2$, $A_{1}\to A_{2}\to\ldots A_{k}\to
A_{1}$, with the rate constants given by (9), the right hand side of equation
(1) is the vector $\mathbf{v}_{k}$ with coordinates
$(\mathbf{v}_{k})_{j}=\frac{p_{j-1}}{p^{*}_{j-1}}-\frac{p_{j}}{p^{*}_{j}}$
(10)
Here, without loss of generality, we take $\kappa=1$, use index $j$ instead of
$i_{j}$ and apply the standard convention regarding cyclic order. Other
coordinates of $\mathbf{v}_{k}$ are zeros.
Let us decompose this $\mathbf{v}_{k}$ into a conic combination of a vector
$\mathbf{v}_{k-1}$ for a cycle of length $k-1$ and a vector $\mathbf{v}_{2}$
for a cycle of length 2. The flux $A_{j}\to A_{j+1}$ is ${p_{j}}/{p^{*}_{j}}$.
Let us find the minimum value of this flux and, for convenience, let us put
this minimal flux in the first position by a cyclic permutation. The target
cycle of length $k-1$ is $A_{2}\to\ldots A_{k}\to A_{2}$ with rate constants
given by formula (9) ($\kappa=1$). We just delete the vertex with the smallest
flux from the initial cycle of length $k$. The target cycle of length 2 is
$A_{1}\rightleftharpoons A_{2}$ with the rate constants (9)
$q_{21}=\frac{\kappa}{p^{*}_{1}}\,,\;q_{12}=\frac{\kappa}{p^{*}_{2}}$. We find
the constant $\kappa$ from the conditions:
$\mathbf{v}_{k}=\mathbf{v}_{k-1}+\mathbf{v}_{2}$ at the point $P$, hence, two
following reaction schemes, (a) and (b), should have the same velocities,
${\mathrm{d}}P/{\mathrm{d}}t$:
$\mbox{(a)
}A_{k}{\overset{1/p_{k}^{*}}{\rightarrow}}A_{1}{\overset{1/p_{1}^{*}}{\rightarrow}}A_{2}\mbox{
and (b)
}A_{k}{\overset{1/p_{k}^{*}}{\rightarrow}}A_{2}\,;A_{1}\underset{\kappa/p_{2}^{*}}{\overset{\kappa/p_{1}^{*}}{\rightleftharpoons}}A_{2}\,.$
From this condition,
$\kappa=\left(\frac{p_{k}}{p_{k}^{*}}-\frac{p_{1}}{p_{1}^{*}}\right)\left(\frac{p_{2}}{p_{2}^{*}}-\frac{p_{1}}{p_{1}^{*}}\right)^{-1}$
$\kappa\geq 0$ because ${p_{1}}/{p^{*}_{1}}$ is the minimal value of
${p_{j}}/{p^{*}_{j}}$. Finally,
$\mathbf{v}_{k}=\mathbf{v}_{k-1}+\mathbf{v}_{2}$. ∎
Further, we omit the index B or DB at the cone: $\mathbf{Q}^{n}_{\rm
B}(P,P^{*})=\mathbf{Q}^{n}_{\rm DB}(P,P^{*})=\mathbf{Q}^{n}(P,P^{*})$.
It is necessary to stress that the decomposition of the right hand side of the
Kolmogorov equation (1) into a conic combination of cycles of length 2 depends
on the ordering of the ratios $p_{i}/p_{i}^{*}$ and cannot be performed for
all values of $P$ simultaneously. Thus, this decomposition is local.
### 2.2 Quasichemical representation and the cones of possible velocities
For systems with detailed balance, the cone of possible velocities,
$\mathbf{Q}^{n}_{\rm DB}(P,P^{*})$, is a polyhedral cone. For a given $P^{*}$,
it is a piecewise constant function of $P$. The hyperplanes of the equilibria
$A_{i}\leftrightarrow A_{j}$ divide the standard simplex of distributions into
a finite number of polyhedra (compartments). In each compartment the dominant
direction of every transition $A_{i}\leftrightarrow A_{j}$ is fixed and the
cone of possible velocities is constant. Now we find that this construction
provides the cones of possible velocities for general Markov kinetics and not
only for systems with detailed balance. Let us describe these cones in detail.
The construction of cones of possible velocities was described in 1979 [30]
for systems with detailed balance in the general setting for generalized MAL,
for nonlinear chemical kinetics. These systems are represented by
stoichiometric equations of the elementary reaction coupled with the reverse
reactions:
$\alpha_{\rho 1}A_{1}+\ldots+\alpha_{\rho n}A_{n}\rightleftharpoons\beta_{\rho
1}A_{1}+\ldots+\beta_{\rho n}A_{n}\,,$ (11)
where $\alpha_{\rho i},\,\beta_{\rho i}\geq 0$ are the stoichiometric
coefficient, $\rho$ is the reaction number ($\rho=1,\ldots,m$). The
stoichiometric vector of the $\rho$th reaction is an $n$ dimensional vector
$\gamma_{\rho}$ with coordinates $\gamma_{\rho i}=\beta_{\rho i}-\alpha_{\rho
i}$.
The equilibria of the $\rho$th pair of reactions (11) form a surface in the
space of concentrations. The intersection of these surfaces for all $\rho$ is
the equilibrium (with detailed balance). These surfaces of the equilibria of
the pairs of elementary reactions (11) divide the space of concentrations into
several compartments. In each compartment the dominant direction of each
reaction (11) is fixed and, hence, the cone of possible velocities is also
constant. It is a piecewise constant function of concentrations (for a given
temperature):
$\mathbf{Q}={\rm cone}\\{\gamma_{\rho}{\rm
sign}(w_{\rho})\,|\,\rho=1,\ldots,m\\}\,.$
For example, let us join the transitions $A_{i}\rightleftharpoons A_{j}$ in
pairs (say, $i>j$) and introduce the stoichiometric vectors $\gamma^{ij}$ with
coordinates:
$\gamma^{ij}_{k}=\left\\{\begin{array}[]{ll}-1&\mbox{ if }k=i,\\\ 1&\mbox{ if
}k=j,\\\ 0&\mbox{ otherwise}.\end{array}\right.$ (12)
Let us rewrite the Kolmogorov equation for the Markov process with detailed
balance (3) in the quasichemical form:
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{i>j}w_{ij}^{*}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\gamma^{ji}\,.$
(13)
Here, $w_{ij}^{*}=q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}$ is the equilibrium flux
from $A_{i}$ to $A_{j}$ and reverse.
The cone of possible velocities for (13) is
$\mathbf{Q}^{n}(P,P^{*})={\rm cone}\left\\{\gamma^{ji}{\rm
sign}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\,\Big{|}\,i>j\right\\}\,.$
(14)
Here, we use the three-valued sign function (with values $\pm 1$ and 0). In
Fig. 2, the partition of the standard distribution simplex into compartments,
and the cones (angles) of possible velocities are presented for the Markov
chains with three states.
Figure 2: Partition of the distribution triangle for the Markov chains with
three states by the lines $\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}=0$
into twelve compartments. The corresponding cones (angles) of possible
velocities are presented. The clockwise and anticlockwise borders of the
trajectories are represented by bold lines.
A set of distributions $U$ is positively invariant with respect to system (1)
if for any initial distribution $P(0)\in U$, the solution of (1) $P(t)$
remains in $U$ for $t>0$. The bold broken lines in Fig. 2 follow along the
extreme rays of the angles of possible velocities (clockwise or
anticlockwise). They form the borders of a positively-invariant area for all
the Markov chains with the given equilibrium $P^{*}$.
These borders give, for example, a simple estimate of the logarithmic
decrement for Markov chains. For decaying oscillations, the logarithmic
decrement is the natural logarithm of the ratio of any two successive
amplitudes: $\delta\triangleq\ln\frac{x_{1}}{x_{2}}$. For a complex eigenvalue
$\lambda$, the period between two amplitudes $T=2\pi/|\Im\lambda|$ and
$\delta=2\pi\frac{|\Re\lambda|}{|\Im\lambda|}$. For systems with detailed
balance, eigenvalues are always real but for the general Markov chains they
may be complex. For example, for the simple cycle $A_{1}\to A_{2}\to A_{3}\to
A_{1}$ with the equilibrium equidistribution $p_{1,2,3}^{*}=1/3$, and the rate
coefficients $\kappa$, the nonzero eigenvalues of the linear system (1) are
$\lambda=\kappa(-\frac{3}{2}\pm i\frac{\sqrt{3}}{2})$ and
$\delta=2\pi\sqrt{3}$.
Let us follow the clockwise border trajectory (Fig. 2) starting from the state
$A_{1}$ (the corresponding distribution is $P=(1,0,0)$). This state belongs to
the line of equilibria of the transition $A_{2}\rightleftharpoons A_{3}$. The
first step is the equilibration of the transition $A_{1}\rightleftharpoons
A_{2}$ ($A_{3}$ does not change). After that, the equilibration of the
transition $A_{1}\rightleftharpoons A_{3}$ follows ($A_{2}$ does not change):
$\begin{split}&(1,0,0)\mapsto\left(\frac{p^{*}_{1}}{1-p^{*}_{3}},\frac{p^{*}_{2}}{1-p^{*}_{3}},0\right)\\\
&\mapsto\left(\frac{(p^{*}_{1})^{2}}{(1-p^{*}_{3})(1-p^{*}_{2})},\frac{p^{*}_{2}}{1-p^{*}_{3}},\frac{p^{*}_{1}p^{*}_{3}}{(1-p^{*}_{3})(1-p^{*}_{2})}\right)\,.\end{split}$
(15)
As the result of this sequence of equilibrations, when the clockwise border
line again approaches the equilibrium line of the transition
$A_{2}\rightleftharpoons A_{3}$, the value of $p_{1}$ is
$\frac{(p^{*}_{1})^{2}}{(1-p^{*}_{3})(1-p^{*}_{2})}$. After this turn in angle
$\pi$ every trajectory becomes closer to $P^{*}$. The contraction coefficient
is
$\frac{p^{*}_{1}p^{*}_{2}p^{*}_{3}}{(1-p^{*}_{1})(1-p^{*}_{2})(1-p^{*}_{3})}$
or less. The anticlockwise trajectory gives the same contraction. We estimated
the logarithmic decrement from below:
$\delta\left(=2\pi\frac{|\Re\lambda|}{|\Im\lambda|}\right)\geq
2\ln\left(\frac{(1-p^{*}_{1})(1-p^{*}_{2})(1-p^{*}_{3})}{p^{*}_{1}p^{*}_{2}p^{*}_{3}}\right)\,.$
(16)
### 2.3 Two $H$-theorems
The most general form of the $H$-theorem for Markov processes was proposed by
Morimoto [36]. He used the following $H$-functions: for each convex function
of the positive convex variable $h(x)$ the $h$-divergence between
distributions $P$ and $P^{*}$ is
$\begin{split}&H_{h}(P\|P^{*})=\sum_{i}p^{*}_{i}h\left(\frac{p_{i}}{p_{i}^{*}}\right)\,.\end{split}$
(17)
At the same time these divergencies were studied by Csiszár [37] and sometimes
they are called the Csiszár–Morimoto divergences. These functions were
introduced two years earlier by Rényi on the last page of his famous work [38]
together with the hint about the $H$-theorem. For more details see [39].
The time derivative of the Csiszár–Morimoto function $H_{h}(P\|P^{*})$ (17)
with respect to master equation (6) for a general Markov process is
$\begin{split}&\frac{{\mathrm{d}}H_{h}(P\|P^{*})}{{\mathrm{d}}t}=\sum_{i,j,\,j\neq
i}q_{ij}p^{*}_{j}\\\
&\times\left[h\left(\frac{p_{i}}{p_{i}^{*}}\right)-h\left(\frac{p_{j}}{p_{j}^{*}}\right)+h^{\prime}\left(\frac{p_{i}}{p_{i}^{*}}\right)\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\right]\leq
0\end{split}$ (18)
For a Markov process with detailed balance we use the quasichemical form of
master equation (13) and find immediately
$\begin{split}&\frac{{\mathrm{d}}H_{h}(P\|P^{*})}{{\mathrm{d}}t}=-\sum_{i,j,\,i>j}q_{ij}p^{*}_{j}\\\
&\times\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\left(h^{\prime}\left(\frac{p_{j}}{p_{j}^{*}}\right)-h^{\prime}\left(\frac{p_{i}}{p_{i}^{*}}\right)\right)\leq
0\,.\end{split}$ (19)
The inequality for the general Markov processes (18) follows from Jensen’s
inequality in the differential form, $h^{\prime}(x)(y-x)\leq h(y)-h(x)$. It is
valid for left and right limits of $h^{\prime}$ at any point $x>0$. The
inequality for systems with detailed balance (19) follows from the
monotonicity of $h^{\prime}$. In full agreement with Corollary 1, the
divergences $H_{h}(P\|P^{*})$ (17) are Lyapunov functions for systems with
detailed balance and for all the Markov processes as well. Theorem 1 has an
even stronger corollary.
###### Corollary 2.
For every Markov process $Q$ with positive equilibrium $P^{*}$ and for a
distribution $P\neq P^{*}$ there exists a Markov process $Q_{\rm DB}$ with the
same equilibrium that obeys the detailed balance condition and has the
following property: For every convex function $h$ the time derivative
${{\mathrm{d}}H_{h}(P\|P^{*})}/{{\mathrm{d}}t}$ for $Q$ coincides at point $P$
with the time derivative of $H_{h}(P\|P^{*})$ at this point for $Q_{\rm DB}$.
## 3 Nonlinear kinetics: detailed balance versus semi-detailed balance
The general equations of MAL without any restriction on the reaction rate
constants demonstrate all types of non-trivial dynamic behavior, from multiple
steady states to strange attractors [40, 41]. It is not a surprise because the
MAL systems can approximate with arbitrary accuracy any smooth vector field
which preserves the linear conservation laws and positivity of concentrations
[42, 43].
The systems with semi-detailed balance give the direct nonlinear
generalization of the general Markov kinetics. They were introduced by
Boltzmann for gas kinetics [3] and generalized later for MAL systems [7, 6,
8]. The systems with semi-detailed balance are the generalized MAL systems
with additional relations between rate constants.
To produce these relations, let us follow the classical work [5] and assume
that behind the reaction mechanism (11) there is the reaction mechanism with
intermediate compounds $B_{\rho}^{\pm}$ illustrated by Fig. 1. Each compound
is associated with a formal input or output complex $\sum_{i}\alpha_{\rho
i}A_{i}$ or $\sum_{i}\beta_{\rho i}A_{i}$. Such a complex may participate in
several reactions. Let there be $k$ different vectors among
$\\{\boldsymbol{\alpha}_{\rho},\boldsymbol{\beta}_{\rho}\\}$
($\boldsymbol{\alpha}_{\rho}=(\alpha_{\rho i})$ and
$\boldsymbol{\beta}_{\rho}=(\beta_{\rho i})$). We denote these different
vectors by $\boldsymbol{\nu}_{j}$ ($j=1,\ldots,k$). The correspondent
complexes are $\Theta_{j}=\sum_{i}\nu_{ji}A_{i}$. The reaction mechanism (11)
takes the form of the list of transitions $\Theta_{j}\to\Theta_{l}$ and the
extended reaction mechanism is the list of transitions
$\Theta_{j}\rightleftharpoons B_{j}\to B_{l}\rightleftharpoons\Theta_{l}\,.$
(20)
Stueckelberg introduced this representation for the collisions in Boltzmann’s
equations and used two asymptotic assumptions:
* 1.
The compounds $B_{j}$ are in fast equilibrium with the corresponding input or
output reagents and the reactions $\Theta_{j}\rightleftharpoons B_{j}$ in (20)
are always close to equilibrium (this is the quasiequilibrium assumption, QE);
* 2.
They exist in very small concentrations compared to other components (this
leads to the quasi steady state approximation, QSS).
We call the intermediates $B_{j}$ compounds following the classical work of
Michaelis and Menten [44]. In 1913, they introduced the same asymptotic
assumptions and representation for an enzyme reaction and demonstrated that in
this case the overall catalytic reaction obeys the MAL.
In more general settings, these two assumption, QE and QSS, allow us to
produce the reaction rates for the rates of the overall reactions in the form
of the generalized MAL. The rate of the reaction
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}$
is the product of two factors, a standard Boltzmann factor $W_{\rho}$ and a
kinetic factor $\varphi_{\rho}\geq 0$:
$r_{\rho}=\varphi_{\rho}W_{\rho}=\varphi_{\rho}\exp\left(\frac{\sum_{i}\alpha_{\rho
i}\mu_{i}}{RT}\right)\,,$ (21)
where $\mu_{i}$ is the chemical potential of the component $A_{i}$. The
corresponding kinetic equation is
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}r_{\rho}\boldsymbol{\gamma}_{\rho}\,,\;\;(\gamma_{\rho
i}=\beta_{\rho i}-\alpha_{\rho i})\,.$ (22)
Here $N$ is the vector of composition ($N_{i}$ is the amount of $A_{i}$), and
$V$ is the volume.
We use the notation $c_{i}$ for the concentration of $A_{i}$, $c$ is the
vector of concentrations, $\varsigma_{j}$ is the concentration of $B_{j}$. The
chemical potentials $\mu_{i}$ of the components $A_{i}$ are the partial
derivatives of the free energy density, $\mu_{i}=\partial f(c,T)/\partial
c_{i}$. The standard thermodynamic assumption about strong convexity of the
function $f(c,T)$ for all $T$ is accepted.
Let us demonstrate how the generalized MAL follows from the QE and QSS
approximations (for more details see Ref. [6]). The thermodynamic equilibria
for the extended mixture are defined as the conditional minima of the free
energy $F$. The free energy of a mixture of $A_{i}$ with small admixtures of
the compounds $B_{j}$ is:
$F=Vf(c,T)+VRT\sum_{j=1}^{q}\varsigma_{j}\left(\frac{u_{j}(c,T)}{RT}+\ln\varsigma_{j}-1\right)\,.$
(23)
The entropic terms $VRT\varsigma_{j}\ln\varsigma_{j}$ in this expression
corresponds to the ideal gas equations $p_{j}=\varsigma_{j}RT$ for the partial
pressure of the small admixtures of the compounds $B_{j}$. This ideal gas low
may be valid not only in gases but for osmotic pressure of small admixtures in
solutions (the Morse equation).
The thermodynamic equilibria of $B_{j}$ are: $\varsigma_{j}^{\rm
eq}=\varsigma_{j}^{*}/Z$, where $Z$ is a positive number and
$\varsigma_{j}^{*}$ is the standard equilibrium:
$\varsigma_{j}^{*}(c,T)=\exp\left(-\frac{u_{j}(c,T)}{RT}\right)\,.$ (24)
The thermodynamic equilibrium condition of the reactions
$\Theta_{j}\rightleftharpoons B_{j}$ under the condition of smallness of
$\varsigma_{j}$ (QE+QSS) can be solved explicitly:
$\varsigma_{j}^{\rm
qe}=\varsigma^{*}_{j}(c,T)\exp\left(\frac{\sum_{i}\nu_{ji}\mu_{i}(c,T)}{RT}\right)\,.$
(25)
The smallness of the concentration of the compounds implies that the rates of
the reactions $B_{i}\to B_{j}$ in the extended mechanism (20) are linear
functions of their concentrations. Let the rate constants for this first order
kinetics be $q_{ji}$.
In the selected approximations the extended reaction mechanism (20) returns to
the form $\Theta_{j}\to\Theta_{l}$. The reaction rate of the transition
$\Theta_{j}\to\Theta_{l}$ in the quasiequilibrium approximation is
$r_{lj}=q_{lj}\varsigma_{j}^{\rm qe}$. This is exactly the generalized MAL
(21) with
$\varphi_{lj}=q_{lj}\varsigma_{j}^{*},\,\boldsymbol{\alpha}_{\rho}=\boldsymbol{\nu}_{j},\,\boldsymbol{\beta}_{\rho}=\boldsymbol{\nu}_{l}\,.$
At the equilibrium $\varsigma^{*}/Z$, the first order kinetics of compounds
should satisfy the general balance condition (2):
$\sum_{j}q_{lj}\varsigma_{j}^{*}=\sum_{j}q_{jl}\varsigma_{l}^{*}\,.$
Therefore, the kinetic factors $\varphi_{\rho}$ satisfy the identity of semi-
detailed balance:
$\sum_{\rho,\,\boldsymbol{\alpha}_{\rho}=\boldsymbol{\nu}}\varphi_{\rho}\equiv\sum_{\rho,\,\boldsymbol{\beta}_{\rho}=\boldsymbol{\nu}}\varphi_{\rho}$
(26)
for any vector $\boldsymbol{\nu}$ from the set of all vectors
$\\{\boldsymbol{\alpha}_{\rho},\boldsymbol{\beta}_{\rho}\\}$. This identity is
exactly the Markov balance condition (2) for kinetics of compounds with
equilibrium $\varsigma^{*}/Z$. It has a very transparent sense: the
thermodynamic equilibrium is, at the same time, the equilibrium for the first
order kinetics of compounds, i.e. the it satisfies the balance condition (2)
for master equation.
Let us assume that the Markov kinetics of compounds satisfies the detailed
balance condition (3) at the thermodynamic equilibrium:
$q_{lj}\varsigma_{j}^{*}=q_{jl}\varsigma^{*}_{l}\,.$
Then the kinetic factors $\varphi_{\rho}$ satisfy the condition of detailed
balance:
$\varphi_{\rho}^{+}\equiv\varphi_{\rho}^{-}\,,$ (27)
where $\varphi_{\rho}^{+}$ is the kinetic factor for the direct reaction and
$\varphi_{\rho}^{-}$ is the kinetic factor for the reverse reaction.
This detailed balance condition assumes that the sums in the left and right
hand sides of Eq. (26) are equal term by term. Therefore, it is stronger than
the semi-detailed balance condition.
For linear systems, the semi-detailed balance condition turns into the
standard balance condition (2) and the detailed balance condition (27) turns
into (3). Of course, the class of systems with semi-detailed balance is much
wider than the class of systems with detailed balance. Nevertheless, locally
they coincide: for given thermodynamic functions (23) and any given
concentrations and temperature, the cone of possible velocities for systems
(22) with semi-detailed balance coincides with the cone of the possible
velocities for the systems with detailed balance.
Indeed, for the given values of concentrations we can perform the following
three operations, (i) return from the generalized MAL to the first order
kinetic equations of compounds, (ii) use Theorem 1 and find the system of
compounds with detailed balance, which has the same velocity at the same
point, and (iii) return back, to the generalized MAL. As a result, we get a
kinetic system for the components $A_{i}$ with detailed balance, the same free
energy $Vf(c,T)$ (23), and the same velocity at the selected values of
concentrations.
## 4 Conclusion
The definition of detailed balance includes the rates of all transitions at
equilibrium but observability of all these rates together is a very special
situation. Typically, one can observe the overall system velocity,
${\mathrm{d}}P/{\mathrm{d}}t$, or just some components of this velocity but
not the rates of individual transitions. According to our results, if we know
the equilibrium distribution $P^{*}$ and observe the system velocity at one
nonequilibrium point $P$ then we can never distinguish a general system from
the systems with detailed balance. This is true for Markov kinetics as well as
for the systems with the generalized MAL; detailed balance can never be
distinguished from the semi-detailed balance if we know the equilibrium and
observe the velocity at one nonequilibrium point. The difference between
velocities of the general kinetic systems and the systems with detailed
balance is hidden in the correlations between different nonequilibrium states
(or, for example, in the continuous pieces of trajectories). The cone of
possible velocities at a nonequilibrium state $P$ is a piece-wise constant
function of $P$, which can be constructed explicitly for the systems with
detailed balance (Fig. 2), and the same construction is valid for the general
kinetics. These results seem to be rather surprising.
For the nonlinear mass action systems, the systems with semi-detailed balance
give the proper analogue of the general Markov kinetics. The conditions of
semi-detailed balance were invented for the Boltzmann equation by Boltzmann
[3], studied by Stueckelberg [5] and rediscovered for the mass action kinetics
by Horn and Jackson [7]. Recently [6], it was proved that the generalized MAL
with the semi-detailed balance condition always follows from the Markov
kinetics of compounds in the Michaelis–Menten–Stueckelberg asymptotic. The
class of the systems with semi-detailed balance is wider than the class of
systems with detailed balance. Nevertheless, for a given equilibrium and for
any given value of concentration these two classes have the same sets of
possible velocities in the distribution space.
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|
arxiv-papers
| 2012-05-09T18:09:41 |
2024-09-04T02:49:30.776918
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1205.2052"
}
|
1205.2143
|
# Surfaces of Revolution with Constant Gaussian Curvature in Four-Space
Dang Van Cuong
###### Abstract
In this paper, we show that the constant property of the Gaussian curvature of
surfaces of revolution in both $\mathbb{R}^{4}$ and $\mathbb{R}_{1}^{4}$
depend only on the radius of rotation. We then give necessary and sufficient
conditions for the Gaussian curvature of the general rotational surfaces whose
meridians lie in two dimensional planes in $\mathbb{R}^{4}$ to be constant,
and define the parametrization of the meridians when both the Gaussian
curvature is constant and the rates of rotation are equal.
Mathematics Subject Classification. 53A05, 53C50, 53A35.
Key words and phrases. Surface of revolution, Surfaces with constant Gaussian
curvature.
## 1 Introduction
It is well known that a regular surface in $\mathbb{R}^{3}$ is zero Gaussian
curvature if and only if it is a part of a developable surface. A regular
surface with constant Gaussian curvature, $K,$ is locally isometric to $H^{2}$
provided $K=-1,$ $\mathbb{R}^{2}$ provided $K=0,$ and $S^{2}$ provided $K=1.$
For the surfaces of revolution in $\mathbb{R}^{3},$ it is easy to define the
parametrization of the surfaces with constant Gaussian curvature.
In recent years some mathematicians have taken an interest in the surfaces of
revolution in $\mathbb{R}^{4},$ for example V. Milosheva ([6]), U. Dursun and
N. C. Turgay ([3]), K. Arslan ([1]), …. In [4], V. Milosheva applied
invariance theory of surfaces in the four dimensional Euclidean space to the
class of general rotational surfaces whose meridians lie in two-dimensional
planes in order to find all minimal super-conformal surfaces. These surfaces
were further studied by U. Dursun and N. C. Turgay in [3], which found all
minimal surfaces by solving the differential equation that characterizes
minimal surfaces. They then determined all pseudo-umbilical general rotational
surfaces in $\mathbb{R}^{4}$. K. Arslan et.al in [1] gave the necessary and
sufficient conditions for generalized rotation surfaces to become pseudo-
umbilical, they also shown that each general rotational surface is a Chen
surface in $\mathbb{E}^{4}$ and gave some special classes of generalized
rotational surfaces as examples.
Let $M$ be a spacelike or timelike surface in Lorentz-Minkowski three-space
$\mathbb{R}_{1}^{3}$ generated by a one-parameter family of circular arcs, R.
L${\rm\acute{o}}$pez in [5] shown that if its Gaussian curvature $K$ is a
nonzero constant then $M$ is a surface of revolution, he also described the
parametrizations for $M$ when $K=0.$ In [2], by applying the
$\mathfrak{l}_{r}^{\pm}$-Gauss maps, Cuong defined the parameterizations of
minimal and totally umbilical spacelike surfaces of revolution in
$\mathbb{R}_{1}^{4}$.
In this paper, we introduce the notions of surfaces of revolution in
$\mathbb{R}^{4}$ and in Lorentz-Minkowski $\mathbb{R}_{1}^{4},$ we then give
necessary and sufficient conditions for the Gaussian curvature of these
surfaces to be constant. We then give a differential equation that
characterizes the general rotational surfaces whose meridians lie in two
dimensional planes in $\mathbb{R}^{4}$ with constant Gaussian curvature. In
the case that rates of rotation are equal, we can define the parametrization
of the meridians of these surfaces when its Gaussian curvature is constant.
## 2 Preliminaries
Let $M$ be a semi-Riemannian surface, that is, a semi-Riemannian manifold of
dimension two. For a coordinate system $u,v$ in $M$ the components of the
metric tensor (the coefficients of the first fundamental form) are
traditionally denoted by
$E=g_{11}=\langle\partial_{u},\partial_{u}\rangle,F=g_{12}=\langle\partial_{u},\partial_{v}\rangle,G=g_{22}=\langle\partial_{v},\partial_{v}\rangle.$
Since $M$ is two-dimensional, $T_{p}M$ is the only tangent plane at $p$. Thus
the sectional curvature $K$ becomes a real-valued function on $M$, called
Gaussian curvature of $M.$
Let $u,v$ be an orthogonal coordinate system in a semi-Riemannian surface,
that means $F=\langle\partial_{u},\partial_{v}\rangle=0.$ Then (see
Proposition 4.4, pp. 81, [8])
$K=-\frac{1}{eg}\left[\varepsilon_{1}\left(\frac{g_{u}}{e}\right)_{u}+\varepsilon_{2}\left(\frac{e_{v}}{g}\right)_{v}\right],$
where $e=|E|^{1/2},g=|G|^{1/2}$ and $\varepsilon_{1},\varepsilon_{2}$ are the
sign of $E,G$ , respectively.
If $M$ is a surface immersed in a manifold of constant curvature $C,$ the
normal bundle of $M$ has a orthonormal frame $\\{\nu_{i}\\}_{i=1,\dots,n}$
then by using the equation of Gauss we have
$K=C+\sum_{i=1}^{n}K_{\nu_{i}},$
where $K_{\nu_{i}}$ is $\nu_{i}$-curvature of $M$ associated with $\nu_{i}.$
For more detail, let see [7].
Therefore, if $M$ is a surface immersed in a manifold with constant curvature
we then can define the Gaussian curvature by two ways. In this paper, we only
use the formula of Gaussian curvature in term of the coefficients of the first
fundamental form of $M,$ and apply this formula to define the surfaces of
revolution whose Gaussian curvature are constant.
We now introduce the notion surfaces of revolution in $\mathbb{R}^{4}.$ Let
$C$ be a curve in ${\rm span}\\{e_{1},e_{2},e_{3}\\}$ parametrized by arc-
length
$z(u)=\left(f(u),g(u),\rho(u),0\right),\ u\in I,$ (1)
where $\rho(u)>0$. The orbit of $C$ under the action of the orthogonal
transformations of $\mathbb{R}^{4}$ leaving the plane $Oxy,$
$A=\left[\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\cos v&-\sin v\\\ 0&0&\sin
v&\cos v\end{matrix}\right],\ v\in\mathbb{R},$
is a surface given by
${[SR1]}\qquad{{\rm X}}(u,v)=\left(f(u),g(u),\rho(u)\cos v,\rho(u)\sin
v\right),u\in I,v\in[0,2\pi).$ (2)
Then $[SR_{1}]$ is called surface of revolution in $\mathbb{R}^{4}.$ That
means, $[RS_{1}]$ is orbit of a curve by rotating it around a plane.
We also have the another kind of surface of revolution in $\mathbb{R}^{4},$ it
is the obit of a plane curve rotated around both two planes. This surface is
defined as following. Let $C$ be a regular curve in
$\text{span}\\{e_{1},e_{3}\\}$ parametrized by arc-length
$r(u)=\left(f(u),0,g(u),0\right),\ u\in I,$
and
$B=\left[\begin{matrix}\cos\alpha v&-\sin\alpha v&0&0\\\ \sin\alpha
v&\cos\alpha v&0&0\\\ 0&0&\cos\beta v&-\sin\beta v\\\ 0&0&\sin\beta
v&\cos\beta v\end{matrix}\right],\ v\in\mathbb{R},$
be a subgroup of the orthogonal transformations group on $\mathbb{R}^{4}$,
where $\alpha,\beta$ are positive constants and $(f(u))^{2}+(g(u))^{2}\neq 0$.
The orbit of $C$ under the action of the subgroup $B$ is a surface in
$\mathbb{R}^{4}$ given by
$[SR_{2}]\qquad{\rm X}(u,v)=\left(f(u)\cos\alpha v,f(u)\sin\alpha
v,g(u)\cos\beta v,g(u)\sin\beta v\right),$ (3)
which is called General rotational surface whose meridians lie in two-
dimensional planes. Then $r(u)$ is called meridian and $\alpha,\beta$ are
called the rates of rotation.
Modifying this method, we can introduce the notion surfaces of revolution in
Lorentz-Minkowski space. The Lorentz-Minkowski space $\mathbb{R}^{4}_{1}$ is
the $4$-dimensional vector space
$\mathbb{R}^{4}=\\{(x_{1},\ldots,x_{4}):x_{i}\in\mathbb{R},i=1,\ldots,4\\}$
endowed the pseudo scalar product defined by
$\langle\textbf{x},\textbf{y}\rangle=\sum_{i=1}^{3}x_{i}y_{i}-x_{4}y_{4},$
where $\textbf{x}=(x_{1},\ldots,x_{4}),\textbf{y}=(y_{1},\ldots
y_{4})\in\mathbb{R}^{4}.$
Let $C$ be a spacelike (timelike) curve in
$\text{span}\\{e_{1},e_{2},e_{4}\\}$ parametrized by arc-length,
$z(u)=\left(f(u),g(u),0,\rho(u)\right),\ \rho(u)>0,\ \ u\in I.$
The orbit of $C$ under the action of the orthogonal transformations of
$\mathbb{R}_{1}^{4}$ leaving the spacelike plane $Oxy,$
$A_{S}=\left[\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\cosh v&\sinh v\\\
0&0&\sinh v&\cosh v\end{matrix}\right],\ v\in\mathbb{R},$
is a surface given by
$[SR_{3}]\qquad{\rm X}(u,v)=\left(f(u),g(u),\rho(u)\sinh v,\rho(u)\cosh
v\right),\ u\in I,\ v\in\mathbb{R}.$ (4)
The surface $[SR_{3}]$ is called the surface of revolution of hyperbolic type
in $\mathbb{R}_{1}^{4}$.
Let $C$ be a spacelike (timelike) curve in
$\text{span}\\{e_{1},e_{3},e_{4}\\}$ parametrized by arc-length,
$z(u)=\left(\rho(u),0,f(u),g(u)\right),\ \rho(u)>0,\ u\in I.$
The orbit of $C$ under the action of the orthogonal transformations of
$\mathbb{R}_{1}^{4}$ leaving the timelike plane $Ozt,$
$A_{T}=\left[\begin{matrix}\cos v&-\sin v&0&0\\\ \sin v&\cos v&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{matrix}\right],\ v\in\mathbb{R},$
is a surface given by
$[SR_{4}]\qquad{\rm X}(u,v)=\left(\rho(u)\cos v,\rho(u)\sin
v,f(u),g(u)\right),\ v\in\mathbb{R}.$ (5)
The surface $[SR_{4}]$ then is called the surface of revolution of elliptic
type in $\mathbb{R}_{1}^{4}.$
## 3 Main Results
The following theorems show that the constant property of Gaussian curvatures
of surfaces of revolution in four-space depends only on the radius of
rotation.
###### Theorem 3.1.
The Gaussian curvature of surface $[SR_{1}]$ is constant if and only if
1. 1.
$\rho(u)=C_{1}e^{Cu}+C_{2}e^{-Cu},$ provided $K=-C^{2},$
2. 2.
$\rho(u)=C_{1}\sin(Cu)+C_{2}\cos(Cu),$ provided $K=C^{2},$
3. 3.
$\rho=C_{1}u+C_{2},$ provided $K=0,$
where $C_{1},C_{2}$ are constant such that for each $u\in I,$ $\rho(u)>0.$
###### Proof..
The coefficients of the first fundamental form of $[SR_{1}]$ are defined
$E=\langle{\rm X}_{u},{\rm X}_{u}\rangle=1,\ F=\langle{\rm X}_{u},{\rm
X}_{v}\rangle=0,\ G=\langle{\rm X}_{v},{\rm X}_{v}\rangle=\rho^{2}.$
Therefore, the Gaussian curvature is defined
$K=-\frac{\rho^{\prime\prime}}{\rho}.$
Solving the equation $K=-C^{2},$ ($C^{2},0$) we have the conclusion of Theorem
3.1. ∎
We have the similar result for surfaces of revolution in $\mathbb{R}_{1}^{4}.$
###### Theorem 3.2.
The Gaussian curvature of Surface $[SR_{3}]$ (or $[SR_{4}]$) is constant,
$K=C,$ if and only if
1. 1.
$\rho(u)=C_{1}e^{\lambda u}+C_{2}e^{-\lambda u},$ when $\varepsilon C=-2\\\
lambda^{2}<0,$
2. 2.
$\rho(u)=C_{1}\sin(\lambda u)+C_{2}\cos(\lambda u),$ when $\varepsilon
C=\lambda^{2}>0,$
3. 3.
$\rho=C_{1}u+C_{2},$ when $C=0,$
where $C_{1},C_{2}$ are constant such that $\rho(u)>0,u\in I$ and
$\varepsilon$ is the sign of $E.$
###### Proof..
The coefficients of the first fundamental form of $[SR_{3}]$ (or $[SR_{4}]$)
are defined
$E=(f^{\prime}(u))^{2}+(g^{\prime}(u))^{2}-(\rho^{\prime}(u))^{2}=\varepsilon,\
F=0,\ G=\left(\rho(u)\right)^{2}>0,$
where $\varepsilon=\pm 1.$ It is similar to the proof of Theorem 3.1, we have
the result of this Theorem.∎
For the general rotational surfaces whose meridians lie in two-dimensional
planes in $\mathbb{R}^{4},$ the following Theorem gives us a differential
equation that characterizes the constant Gaussian curvature surfaces.
###### Theorem 3.3.
The Gaussian curvature of $[SR_{2}]$ is constant if and only if
$f(u)=\frac{\sqrt{G}}{\alpha}\cos\phi(u),\
g(u)=\frac{\sqrt{G}}{\beta}\sin\phi(u),$
where $\phi(u)$ are solutions of the following equation
$G\left(\frac{\sin^{2}\phi}{\alpha^{2}}+\frac{\cos^{2}\phi}{\beta^{2}}\right)(\phi^{\prime})^{2}+2G\sin\phi\cos\phi\left(\frac{1}{\beta^{2}}-\frac{1}{\alpha^{2}}\right)\phi^{\prime}+\frac{(G^{\prime})^{2}}{4G}\left(\frac{\cos^{2}\phi}{\alpha^{2}}+\frac{\sin^{2}\phi}{\beta^{2}}\right)-1=0,$
(6)
and
$G=\left(C_{1}e^{Cu}+C_{2}e^{-Cu}\right)^{2},\ \text{if}\ K=-C^{2},$ (7)
$G=\left(C_{1}\sin(Cu)+C_{2}\cos(Cu)\right)^{2},\ \text{if}\ K=C^{2},$ (8)
$G=\left(C_{1}u+C_{2}\right)^{2},\ \text{if}\ K=0,$ (9)
where $C,C_{1},C_{2}$ are constant such that $G\neq 0$.
###### Proof..
The coefficients of the first fundamental form of $[SR_{2}]$ are defined
$E=(f^{\prime})^{2}+(g^{\prime})^{2}=1,\ F=0,\
G=\alpha^{2}f^{2}+\beta^{2}g^{2}.$ (10)
We have
$K=-\frac{\left(\sqrt{G}\right)_{uu}}{\sqrt{G}}.$
Solving the equation $K=-C^{2}(C^{2},0),$ we have $G.$ Setting
$\alpha f(u)=\sqrt{G}\cos(\phi(u)),\beta g(u)=\sqrt{G}\sin(\phi(u))$
and substituting for (10) we then have (6). ∎
In the case two rates of rotation are equal, $\alpha=\beta,$ we can solve the
equation (6) and find the parametrization of the meridians.
###### Corollary 3.4.
If $\alpha=\beta$ then the Gaussian curvature of $[SR_{2}]$ is constant if and
only if
1. 1.
In the case $K=C^{2},$
$f(u)=\frac{C_{1}\sin(Cu)+C_{2}\cos(Cu)}{\alpha}\cos\phi(u),g(u)=\frac{C_{1}\sin(Cu)+C_{2}\cos(Cu)}{\alpha}\sin\phi(u)$
where
$\phi(u)=\int\frac{\sqrt{1-C^{2}[C_{1}\cos(Cu)-C_{2}\sin(Cu)]^{2}}}{C_{1}\sin(Cu)+C_{2}\cos(Cu)}du,$
$C_{1},C_{2}$ are constants such that the formula under the integral sign is
defined.
2. 2.
In the case $K=-C^{2},$
$f(u)=\frac{C_{1}e^{Cu}+C_{2}e^{-Cu}}{\alpha}\cos\phi(u),\
g(u)=\frac{C_{1}e^{Cu}+C_{2}e^{-Cu}}{\alpha}\sin\phi(u),$
where
$\phi(u)=\int\frac{\sqrt{1-C^{2}[C_{1}e^{Cu}-C_{2}e^{-Cu}]^{2}}}{C_{1}e^{Cu}+C_{2}e^{-Cu}}du,$
$C_{1},C_{2}$ are constants such that the formula under the integral sign is
defined.
1. 3.
In the case $K=0,$
$f(u)=\frac{C_{1}u+C_{2}}{\alpha}\cos\phi(u),g(u)=\frac{C_{1}u+C_{2}}{\alpha}\sin\phi(u),$
where
$\phi(u)=\int{\frac{\sqrt{1-C_{1}^{2}}}{C_{1}u+C_{2}}}du,$
$C_{1},C_{2}$ are constants such that $|C_{1}|\leq 1$ and $C_{1}u+C_{2}\neq
0,\forall u\in I.$
Acknowledgements The author would like to thank Prof. Frank Morgan for his
comment and appended.
## References
* [1] K. Arslan, B. Bayram, B. Bulca and G. ${\rm\ddot{O}}$zt${\rm\ddot{u}}$rk, General rotation surfaces in $\mathbb{E}^{4}$, Results. Math., (2012), DOI 10.1007/s00025-011-0103-3.
* [2] D. V. Cuong, $LS_{r}$-valued Gauss maps and spacelike surfaces of revolution in $\mathbb{R}_{1}^{4}$, App. Math. Sci., Vol. 6, (2012), no. 77, 3845 - 3860.
* [3] U. Dursun, N. C. Turgay, Minimal and Pseudo-Umbilical Rotational Surfaces in Euclidean Space $\mathbb{E}^{4}$, Mediterr. J. Math, (2012), DOI 10.1007/s00009-011-0167-z.
* [4] G. Ganchev, V. Milousheva, On the Theory of Surfaces in the Four-Dimensional Euclidean Space. Kodai Math. J., 31 (2008), 183-198.
* [5] R. L${\rm\dot{o}}$pez, Surfaces of constant Gauss curvature in Lorentz-Minkowski three-space, Rocky Mountain Journal of Mathematics, 33 (2003), Number 3, 971-993.
* [6] V. Milosheva, General rotational surfaces in $\mathbb{R}^{4}$ with meridians lying in two-dimension planes,C. R. Acad. Bulg. Sci., 63, 3, (2010) 339-348.
* [7] M. Navarro, F. S${\rm\acute{a}}$chez, A theorem of Gauss-Bonnet type in codimension 2 for Riemannian manifolds of even dimension, Abstraction and Application, 1 (2009), 4-17.
* [8] B. O’Neill, Semi-Riemannian Geometry, Academic Press, Orland 1983.
Dang Van Cuong
Department of Natural Sciences
Duy Tan University
Danang
Vietnam
E-mail address: dvcuong@duytan.edu.vn
|
arxiv-papers
| 2012-05-10T02:39:51 |
2024-09-04T02:49:30.790117
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dang Van Cuong",
"submitter": "Dang Van Cuong",
"url": "https://arxiv.org/abs/1205.2143"
}
|
1205.2285
|
2
# Complex-Demand Knapsack Problems and
Incentives in AC Power Systems
Lan Yu
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University, Singapore
Chi-Kin Chau
Computing and Information Science
Masdar Institute of Science and Technology
Abu Dhabi,UAE
yula0001@ntu.edu.sg ckchau@masdar.ac.ae
###### Abstract
We consider AC electrical systems where each electrical device has a power
demand expressed as a complex number, and there is a limit on the magnitude of
total power supply. Motivated by this scenario, we introduce the complex-
demand knapsack problem (C-KP), a new variation of the traditional knapsack
problem, where each item is associated with a demand as a complex number,
rather than a real number often interpreted as weight or size of the item.
While keeping the same goal as to maximize the sum of values of the selected
items, we put the capacity limit on the magnitude of the sum of satisfied
demands.
For C-KP, we prove its inapproximability by FPTAS (unless P = NP), as well as
presenting a $(1/2-\epsilon)$-approximation algorithm. Furthermore, we
investigate the selfish multi-agent setting where each agent is in charge of
one item, and an agent may misreport the demand and value of his item for his
own interest. We show a simple way to adapt our approximation algorithm to be
monotone, which is sufficient for the existence of incentive compatible
payments such that no agent has an incentive to misreport. Our results shed
insight on the design of multi-agent systems for smart grid.
###### Key Words.:
knapsack problem, approximation algorithm, FPTAS, incentive compatibility,
truthfulness, AC electrical system, smart grid
I.2.11Distributed Artificial IntelligenceMultiagent Systems F.2Theory of
ComputationAnalysis of Algorithms and Problem Complexity Algorithms, Theory
## 1 Introduction
Most studies of power allocation only consider devices without minimum power
requirements; we focus on those with such requirements, such as electric
vehicles (EVs) charging, which will not produce any value unless it is charged
enough to travel a threshold distance. Gerding et al. gerding2011online
studies online electric vehicle charging by expressing power demands as real
numbers. However, in alternating current (AC) electrical systems, alternating
power is provided. In this paper, we study electrical devices with a power
demand expressed as a complex number $d=d^{\rm R}+{\bf i}d^{\rm I}$. Although
$d^{\rm I}=0$ for purely resistive appliances; devices with capacitive or
inductive components have non-zero imaginary part $d^{\rm I}$ GS94power .
In power allocation, due to the constraint of power generation, there is a
limit $C$ on the magnitude of the total power supply, i.e., the magnitude of
the sum of satisfied demands should not exceed $C$. Since only a limited
number of devices can be served and different devices produce different values
when they receive enough power to work, there arises a natural allocation
problem: we want to select a subset of devices to provide power subject to the
power limit constraint such that the total value produced is maximized.
Moreover, in a multi-agent setting (e.g., in future smart grid, where
intelligent devices are automatically controlled by agents), the demand and
value of each device are private knowledge of an individual agent. The power
allocation algorithm collects the input information from each agent, and based
on that, computes which subset of demands to satisfy. Depending on the
(publicly known) algorithm, each selfish agent may misreport his demand or
value to the algorithm in order to get selected.
Naturally, to guarantee a good realization of our optimization goal (here,
maximizing social welfare, the total value of selected items), we would like
to design the algorithm in a way that incentivizes all agents to report their
true information. This falls into the study of Algorithmic mechanism design
N07book ; NR01alg , a burgeoning research area that deals with designing
algorithms (called mechanisms here) for settings where inputs are controlled
by selfish agents. Each agent is modeled to strategize so as to maximize his
utility, a quantity that indicates his overall benefit. A mechanism is
incentive compatible, or simply truthful, if no agent has an incentive to
misreport. A general approach in mechanism design is to enforce payment on
each agent to adjust his utility so that truth-telling always maximizes his
utility.
Now formally, we have the following mechanism design problem: we have a set
$K$ of distinct agents where each agent $k\in K$ owns an item with a positive
value $v_{k}$ and a complex-valued demand $d_{k}=d^{\rm R}_{k}+{\bf i}d^{\rm
I}_{k}$. Given capacity $C>0$, our task is to choose a subset $S\subseteq K$
of agents to satisfy their demands and assign each agent $k$ a nonnegative
payment $p_{k}$. The goal is to elicit true inputs and maximize the total
value of selected items subject to the constraint that $|\sum_{k\in
S}d_{k}|\leq C$. Here we limit our attention to the case where $d^{\rm
R}_{k},d^{\rm I}_{k}\geq 0$ for all $k$. This assumption is reasonable, since,
although demands do not necessarily lie in the first quadrant of the complex
plane, they are recommended111NEC NFPA 70-2005 (a standard for electrical
systems and appliances) suggests that high-consumption appliances should
conform to restricted power factor, which implies $d^{\rm R}_{k}\geq|d^{\rm
I}_{k}|$. to stay within the region $d^{\rm R}\geq|d^{\rm I}|$, which can be
obtained by rotating the first quadrant by $\pi/4$.
Usually, the design of a truthful mechanism is composed of two steps: first,
we solve the pure algorithmic problem; second, we identify certain condition
that guarantees the existence of incentive compatible payments and make it
satisfied by our algorithm.
We follow this path for our problem. Our algorithmic problem is a winner
determination optimization problem; we call it the complex-demand knapsack
problem (C-KP), as it turns out to be an interesting new variation of the
traditional knapsack problem KPP10book . In the original one-dimensional
knapsack problem (1-KP), the demand of an item is simply a nonnegative real
number, often interpreted as the weight or size of the item. The ”knapsack”,
with fixed real-valued capacity to hold the items, represents the limited
resource. The multi-dimensional generalization, the $m$-dimensional knapsack
problem ($m$-KP), captures the settings where there are $m$ independent
resource constraints on the $m$ dimensions (independent features) of the
demands. 1-KP can model the power allocation in direct current (DC) electrical
systems, where power demands can be expressed as real numbers, but fails for
AC systems, where each demand is a two-dimensional vector. Our problem is also
different from 2-KP since our capacity constraint is a quadratic one (on the
magnitude of the total satisfiable demand), rather than two independent linear
constraints in 2-KP. Moreover, it is natural to modify our problem by
including these two linear constraints (on the real and imaginary part of the
total satisfiable demand respectively), and thus introduce the generalized
complex-demand knapsack problem (GC-KP). In fact, many power generators do
have all three constraints of GC-KP.
It is well-known that 1-KP is NP-hard, and our complex-demand variations
include it as a special case when we set all $d^{\rm I}_{k}=0$. Hence we are
interested in good polynomial-time approximation algorithms. In this work, we
present an algorithm with constant $1/2-\epsilon$ approximation ratio for both
C-KP and GC-KP, and show the inapproximability of C-KP by FPTAS (unless P =
NP), based on its connections to well-studied 1-KP, 2-KP and 3-KP. There is
still a gap to close, and we conjecture that C-KP admits a PTAS.
As to the incentive part, the difficulty lies in the following: VCG mechanisms
N07book are both social welfare maximizing and truthful; however, they become
computationally infeasible when computing optimal social welfare is
computationally hard, as in our setting. Worse still, using algorithms
approximating maximum social welfare may not preserve truthfulness. To obtain
truthful and efficient mechanisms with a good approximation ratio, a leading
approach is through ”monotonization”: First prove that a certain notion of
monotonicity suffices for the existence of incentive compatible payments and
then design or adapt an existing algorithm to be monotone. This has been
successfully applied to problem settings with single-parameter
archer2001truthful and single-minded agents LOS99mono , with efficiently
computable payments specified; in fact, for the former, monotonicity is
necessary as well, which justifies the necessity of monotonization. An
additional nice property of the specified payments is that they guarantee
nonnegative utilities for all agents, which, in mechanism design, is an
important desired property called individual rationality ensuring voluntary
participation of the agents.
For the knapsack problem, if both demand and value of an item are private
information, which is the case we investigate here, we do not have single-
parameter agents. However, all variations we consider are special cases of
single-minded agents, each has a single object $d_{k}$ in mind, gets value
$v_{k}$ if he is assigned an object no worse than $d_{k}$ and 0 otherwise. For
example, in our power system setting, the power demand $d_{k}$ is the single
object the $k$th agent desires, and the value $v_{k}$ is produced as long as
the power he receives is $\geq d_{k}$ (according to comparisons between
complex numbers). The monotonicity property for single-minded agents looks
natural and reasonable: If an agent is selected with certain demand and value,
he should remain selected with a lower demand and a higher value, while the
inputs of other agents are fixed. Although this property easily holds for
exact optimization, it may not hold for approximation algorithms. For C-KP, we
succeed in monotonizing our constant approximation algorithm, based on an
existing monotone FPTAS for 1-KP in BKV05KS , and thus achieving incentive
compatibility.
Related Work The knapsack problem has many variations with respect to
divisibility of items, copies of items, dimensions of constraints, etc
KPP10book . In this work, we restrict our attention to the NP-hard one-
dimensional knapsack problem (1-KP) where each indivisible item has only one
single copy, and its multi-dimensional generalization, the $m$-dimensional
knapsack problem ($m$-KP).
For 1-KP, there is a pseudo-polynomial time algorithm using dynamic
programming achieving exact optimization when all item values are integers.
There is a simple fully polynomial-time approximation scheme (FPTAS), which
scales and rounds the item values and then applies the pseudo-polynomial time
algorithm on small integer values KPP10book . However, this FPTAS is not
monotone, since the scale factor involves the maximum item value. Briest et
al. BKV05KS monotonized it, by performing the same procedure with a series of
different scaling factors irrelevant to item values and taking the best
solution out of them. Hence 1-KP admits an incentive compatible FPTAS.
As to $m$-KP with $m\geq 2$, there is a polynomial-time approximation scheme
(PTAS) by Frieze and Clarke FC84alg based on the integer programming
formulation, but it is not evident to see whether it is monotone. On the other
hand, 2-KP is already inapproximable by FPTAS unless P = NP, by a reduction
from equipartition KPP10book . In fact, there is no efficient polynomial-time
approximation scheme (EPTAS) for 2-KP unless W[1] = FPT (See kulik2010there ).
Our Results We initiate the study of the complex-demand knapsack problem
(C-KP) and its hybrid with 2-KP, the generalized complex-demand knapsack
problem (GC-KP).
In Section 3, we present an approximation algorithm for C-KP, which projects
all demand vectors onto the $\pi/4$ line and uses an approximation algorithm
for 1-KP as a subroutine. Since 1-KP admits an FPTAS, we achieve approximation
ratio $1/2-\epsilon$ for any $\epsilon>0$, with running time polynomial in
$1/\epsilon$ and the size of the input. Moreover, the algorithm can be
monotonized, as shown in Section 4, due to the existence of the monotone FPTAS
for 1-KP.
On the other hand, in Section 5, we complete our study of C-KP by providing an
inapproximability result. We prove that there is no FPTAS for C-KP unless P =
NP, through a modification of the reduction from equipartition for 2-KP.
Finally, for GC-KP, the inapproximability result is inherited since it
includes C-KP as a special case. We also come up with an approximation
algorithm by applying the same idea as for C-KP, but we have to use a PTAS for
3-KP as a subroutine (Section 6). Again we achieve approximation ratio
$1/2-\epsilon$ for any $\epsilon>0$, but the running time is only guaranteed
to be polynomial in the size of the input. Regarding monotonization, a similar
trick as in Section 4 would work for GC-KP, if we could find a good monotone
approximation algorithm for 3-KP.
## 2 Preliminaries
### 2.1 The Knapsack Problems
Here we give the integer programming formulation of the knapsack problems
discussed in this paper. The decision of an allocation algorithm is specified
by indicator variables $x_{k}\in\\{0,1\\}$ for item $k\in K$, which has a
simple correspondence to the selected subset of items: $S=\\{k\in
K:x_{k}=1\\}$. We will switch back to the subset representation in later
sections for convenience of illustration.
The one-dimensional knapsack problem (1-KP) is defined as:
$\begin{array}[]{@{}lc}\mbox{({\sc 1-KP})}&\displaystyle\qquad\max\sum_{k\in
K}x_{k}v_{k}\end{array}$
subject to
$\sum_{k\in K}x_{k}d_{k}\leq C$
where
* •
$K$ is a set of items;
* •
$v_{k}$ is the positive value of item $k$ if its demand is satisfied;
* •
$d_{k}$ is the nonnegative real-valued demand of item $k$;
* •
$C$ is the positive real-valued capacity on the total satisfiable demand;
* •
$x_{k}$ indicates whether item $k$ is selected: $x_{k}=1$ means that the
demand of item $k$ is satisfied, and 0 otherwise.
1-KP can be generalized to multi-dimensions. The $m$-dimensional knapsack
problem ($m$-KP) is defined as:
$\begin{array}[]{@{}lc}\mbox{({\sc$m$-KP})}&\displaystyle\qquad\max\sum_{k\in
K}x_{k}v_{k}\end{array}$
subject to $m$ independent inequalities
$\sum_{k\in K}x_{k}d^{j}_{k}\leq C^{j}$
for $j=1,\ldots,m$, where
* •
$d^{j}_{k}$ is the nonnegative real-valued demand of item $k$ in dimension
$j$;
* •
$C^{j}$ is the positive real-valued capacity on the total satisfiable demand
in dimension $j$.
Each $m$-KP is a linear integer program, and $m$-KP is a special case of
$(m+1)$-KP for all $m$. We are especially interested in 1-KP, 2-KP and 3-KP,
whose previous results will be used to achieve ours. In particular, the two-
dimensional knapsack problem (2-KP) can also be formulated in terms of
complex-valued demands:
$\begin{array}[]{@{}lc}\mbox{({\sc 2-KP})}&\displaystyle\qquad\max\sum_{k\in
K}x_{k}v_{k}\end{array}$
subject to
$\sum_{k\in K}x_{k}d_{k}^{\rm R}\leq C^{\rm R}\mbox{\ and\ }\sum_{k\in
K}x_{k}d_{k}^{\rm I}\leq C^{\rm I}$
where
* •
$d^{\rm R}_{k},d^{\rm I}_{k}$ are the nonnegative real part and imaginary part
respectively of the complex-valued demand $d_{k}$ of item $k$;
* •
$C^{\rm R},C^{\rm I}$ are the positive real-valued capacities on the real part
and imaginary part respectively of the total satisfiable demand.
Our study concerns the capacity constraint on the magnitude of the total
satisfiable demand, which is no longer linear. We formulate the complex-demand
knapsack problem (C-KP) as follows:
$\begin{array}[]{@{}lc}\mbox{({\sc C-KP})}&\displaystyle\qquad\max\sum_{k\in
K}x_{k}v_{k}\end{array}$
subject to
$\Big{|}\sum_{k\in K}x_{k}d_{k}\Big{|}\leq C$
where
* •
$d_{k}=d_{k}^{\rm R}+{\bf i}d_{k}^{\rm I}$ is the complex-valued demand of
item $k$ where $d^{\rm R}_{k},d^{\rm I}_{k}$ are both nonnegative;
* •
$C$ is the positive real-valued capacity on the magnitude of the total
satisfiable demand.
Combining the constraints of C-KP and 2-KP results in the following
generalized complex-valued knapsack problem (GC-KP):
$\begin{array}[]{@{}lc}\mbox{({\sc GC-KP})}&\displaystyle\qquad\max\sum_{k\in
K}x_{k}v_{k}\end{array}$
subject to
$\Big{|}\sum_{k\in K}x_{k}d_{k}\Big{|}\leq C\mbox{\ and\ }\sum_{k\in
K}x_{k}d_{k}^{\rm R}\leq C^{\rm R}\mbox{\ and\ }\sum_{k\in K}x_{k}d_{k}^{\rm
I}\leq C^{\rm I}.$
### 2.2 Approximation Algorithm
For knapsack problems, given a solution represented by the selected subset of
items $S\subseteq K$, we denote the total value of selected items by
$v(S)=\sum_{k\in S}v_{k}$. Let $S^{\ast}$ denote an optimal solution.
For our value maximization objective, an algorithm is called a
$\rho$-approximation, if on each input, the output $S$ of the algorithm
satisfies $v(S)\geq\rho\cdot v(S^{\ast})$. Since the knapsack problems
considered in this paper are NP-hard, one looks for polynomial-time algorithms
with good approximation ratio $\rho$.
It is desirable to find constant approximation algorithms with $\rho$ as close
to 1 as possible; stronger than that are algorithms whose approximation ratio
can be arbitrarily close to 1:
One such candidate is a polynomial-time approximation scheme (PTAS), which is
a $(1-\epsilon)$-approximation algorithm for any $\epsilon>0$. The running
time of a PTAS is polynomial in the input size for every fixed $\epsilon$, but
the exponent of the polynomial might depend on $1/\epsilon$. One way of
addressing this is to define the efficient polynomial-time approximation
scheme (EPTAS), whose running time is the multiplication of a function in
$1/\epsilon$ and a polynomial in the input size independent of $\epsilon$. An
even stronger notion is a fully polynomial-time approximation scheme (FPTAS),
which requires the running time to be polynomial in both the input size and
$1/\epsilon$.
In this work, we design constant $1/2-\epsilon$ approximation algorithms for
C-KP and GC-KP based on the FPTAS for 1-KP and PTAS for 3-KP respectively.
### 2.3 Incentive Compatibility
In this subsection, we give a formal model of mechanism design with single-
minded agents based on our C-KP problem setting, state the monotonicity
condition, and specify the incentive compatible payments under it. Single-
minded agents are first introduced by Lehmann et al. LOS99mono , and here we
essentially present the model described in BKV05KS . Readers can refer to
N07book ; NR01alg for a formal definition of the general setting of mechanism
design.
We are given a set $K$ of agents, where agent $k$ controls item $k$. The
demand and value of item $k$ is agent $k$’s private information, which is
called his type, denoted by $t_{k}=(d_{k},v_{k})$. Each agent $k$ is single-
minded: he has the single demand $d_{k}$ in mind, and enjoys value $v_{k}$ if
and only if his demand is satisfied.
Here, with selfish behaviors, satisfying the demand of an agent is no longer
the same as selecting an agent, since an agent may get selected by reporting a
lower demand, but the assignment he receives is only guaranteed to cover his
reported demand, which may not be enough for his true demand. Therefore, we
need to modify an outcome $o$ of an allocation algorithm from the indicator
variable $x_{k}\in\\{0,1\\}$ for each agent $k$ to a specific assignment
$o_{k}$ agent $k$ receives (clearly $o_{k}=0$ when $x_{k}=0$). Let
$\mathbb{C}_{+}$ denote all complex numbers in the first quadrant of the
complex plane, we have $o_{k}\in\mathbb{C}_{+}$ and $o\in\mathbb{C}_{+}^{K}$.
Now we are able to represent the value agent $k$ derives from an outcome $o$
by his valuation function: $t_{k}(o)=v_{k}$ if $d_{k}\leq o_{k}$ and 0
otherwise. Conventionally we abuse the notation and use
$t_{k}:\mathbb{C}_{+}^{K}\rightarrow\mathbb{R}$ to denote the valuation
function associated with type $t_{k}$. The comparison $d_{k}\leq o_{k}$
interprets the condition that the assignment meets the demand. For C-KP, it
conforms to the partial order between complex numbers: $z_{1}\leq z_{2}$ iff
$z_{1}^{\rm R}\leq z_{2}^{\rm R}$ and $z_{1}^{\rm I}\leq z_{2}^{\rm I}$. It
can also be generalized to settings where the outcome set admits a partial
order and a minimum element. As required in the general model of mechanism
design, our valuation function only depends on the outcomes, which also
justifies the necessity to change our representation of outcomes.
For ease of notation, we let $t$ denote an input, a list of all agents’ types
$((d_{k},v_{k}):k\in K)$ and denote the input except that of agent $k$ by
$t_{-k}$. Clearly $t=(t_{k},t_{-k})$.222Unless specified as the true type,
$t_{k}$ may denote any reported type.
A mechanism $M=({\cal A},p)$ consists of an allocation algorithm $\cal A$
computing an allocation solution ${\cal A}(t)\in\mathbb{C}_{+}^{K}$ for each
input $t$ and a $|K|$-tuple $p(t)$ for each $t$ where $p_{k}(t)\in\mathbb{R}$
is the payment enforced on agent $k$. If $d_{k}\leq{\cal A}(t)_{k}$, we say
that agent $k$ is selected, i.e., he receives an assignment that meets his
input demand. We represent the set of selected agents as $S(A(t))$. Given the
mechanism, the utility, the overall benefit of agent $k$, when his true type
is $t_{k}$, equals his valuation minus the payment: $u_{k}(t)=t_{k}({\cal
A}(t))-p_{k}(t)$.
As mentioned in Section 1, given the mechanism, each agent may not report his
true type for his own benefit. Suppose agent $k$ has true type
$t_{k}=(d_{k},v_{k})$ and reports
$t_{k}^{\prime}=(d_{k}^{\prime},v_{k}^{\prime})$. Here the outcome of the
algorithm $\cal A$ is ${\cal A}(t_{k}^{\prime},t_{-k})$, but his valuation
function remains $t_{k}$, so he obtains valuation $t_{k}({\cal
A}(t_{k}^{\prime},t_{-k}))$, and his utility is
$u_{k}(t_{k}^{\prime},t_{-k})=t_{k}({\cal
A}(t_{k}^{\prime},t_{-k}))-p_{k}(t_{k}^{\prime},t_{-k})$. On the other hand,
if he reports his true type, his utility is $u_{k}(t_{k},t_{-k})=t_{k}({\cal
A}(t_{k},t_{-k}))-p_{k}(t_{k},t_{-k})$. Each selfish agent intends to maximize
his utility, so he will choose to misreport $t_{k}^{\prime}$ if it results in
higher utility, assuming other agents do not change their input, i.e.,
$u_{k}(t_{k}^{\prime},t_{-k})>u_{k}(t_{k},t_{-k})$. Therefore, a mechanism is
incentive compatible, or truthful, if and only if this can not happen, which
is equivalent to saying that, for any agent $k$, any $t_{-k}$ and any true
type $t_{k}$, truth-telling maximizes agent $k$’s utility, i.e.,
$u_{k}(t_{k},t_{-k})\geq u_{k}(t_{k}^{\prime},t_{-k})$ for any possible
$t_{k}^{\prime}$.
A sufficient condition to ensure truthfulness for single-minded agents is
monotonicity, specified as follows in our setting:
###### Definition 2.1
An allocation algorithm ${\cal A}$ is monotone if $k\in S({\cal
A}(t_{k},t_{-k}))$ implies $k\in S({\cal A}(t_{k}^{\prime},t_{-k}))$ for any
$t_{k}=(d_{k},v_{k})$ and $t_{k}^{\prime}=(d_{k}^{\prime},v_{k}^{\prime})$
with $v^{\prime}_{k}\geq v_{k}$, ${d}_{k}^{\prime}\leq d_{k}$.
Intuitively, in a monotone algorithm, if agent $k$ is selected with demand
$d_{k}$ and value $v_{k}$, he should be also selected when he has smaller
demand ${d}_{k}^{\prime}$ and larger value $v^{\prime}_{k}$.333Note that in
this definition, the specific assignments $o_{k}$ are irrelevant, so in
Section 4, we can stay with our original problem formulation when we argue
about the monotonicity of our algorithm. The following theorem states the
sufficiency of monotonicity BKV05KS ; LOS99mono :
###### Theorem 2.2
Let ${\cal A}$ be a monotone and exact algorithm for single-minded agents.
Then there exists payment $p^{\cal A}$ such that ${\cal M}_{\cal A}=({\cal
A},p^{\cal A})$ is incentive compatible.
We call a mechanism exact if for all inputs $t=((d_{k},v_{k}):k\in K)$ and all
agents $k$, $A(t)_{k}$ is either $d_{k}$ or $0$, i.e., either the exact demand
is satisfied or nothing is assigned. Without exactness, an agent may benefit
from underreporting his demand. It is not difficult to see that we can always
modify a truthful mechanism to be exact. After all, exactness is a reasonable
assumption since it is undesirable to waste resource in our allocation.
The incentive compatible payment $p^{\cal A}$ is specified as follows: Given a
monotone algorithm ${\cal A}$, if we fix $d_{k}$ and $t_{-k}$ for agent $k$,
then ${\cal A}$ defines a critical value $\theta_{k}(d_{k},t_{-k})$, such that
when $v_{k}$ is above the critical value, $k$ is selected; and when $v_{k}$ is
below the critical value, $k$ is not selected. Then we can define a payment
function $p^{\cal A}(t)$, where each selected agent pays the critical value:
$p_{k}^{\cal
A}(t)=\left\\{\begin{array}[]{ll}\theta_{k}(d_{k},t_{-k})&\mbox{if agent $k$
is selected\ }\\\ 0&\mbox{otherwise\ }\\\ \end{array}\right.$
By Theorem 2.2, if we are able to design a monotone algorithm, we can
transform it into a truthful mechanism. Moreover, the critical value for a
given input can be computed in polynomial time by a binary search on interval
$[0,v_{k}]$ for each agent $k$ during which we repeatedly test if $k$ is
satisfied by running algorithm ${\cal A}$. Therefore, a monotone polynomial
time allocation algorithm ${\cal A}$ implies a polynomial time truthful
mechanism.
In addition, the payment function $p^{\cal A}(t)$ guarantees that all agents
receive nonnegative utilities. This property, called individual rationality,
ensures voluntary participation of the agents, thus is also an important
desired property in mechanism design.
Therefore, the monotone polynomial time algorithm for C-KP we will present in
Section 4 implies a polynomial time mechanism that is both individually
rational and incentive compatible.
We need to point out that the mechanism requires the item values $\\{v_{k}\\}$
to be integers, because of the binary search needed in the payment
computation. This is a reasonable assumption, since values are usually rounded
up to the nearest cent or dollar. The approximation algorithm in Section 3
does not need this assumption, since the FPTAS for 1-KP rounds the item
values.
## 3 Approximation Algorithm for
C-KP
We present a polynomial-time $(\frac{1}{2}-\epsilon)$-approximation algorithm
for C-KP, which relies on a polynomial-time approximation algorithm for 1-KP
as a subroutine.
### 3.1 Basic Idea
Graphically, each demand $d_{k}=d^{\rm R}_{k}+{\bf i}d^{\rm I}_{k}$ of item
$k$ is a vector in the first quadrant. A feasible solution of our problem is a
subset of items whose sum of demands lies in region ${\cal D}$, the $1/4$ disk
of radius $C$ in the first quadrant. As shown in Fig. 1, ${\cal D}$ is divided
by chord $PQ$ into a closed triangle ${\cal D}_{1}$ and a circular segment
${\cal D}_{2}={\cal D}-{\cal D}_{1}$. The $\frac{\pi}{4}$ line intersects
chord $PQ$ at point $R$. Since we may preprocess the demands and eliminate
those whose magnitude exceeds capacity $C$, without loss of generality, we
assume all $|d_{k}|\leq C$.
Figure 1: Graphical picture for C-KP.
If we project all demands onto the $\frac{\pi}{4}$ line, i.e.,
$\tilde{d}_{k}\triangleq(d_{k}^{\rm R}+d_{k}^{\rm I})/\sqrt{2},$
we make all demands one-dimensional. Now a subset of demands has sum
$\sum\tilde{d}_{k}\leq C/\sqrt{2}$ (i.e., the sum vector does not go beyond
point $R$ on the $\frac{\pi}{4}$ line) if and only if its original sum vector
$\sum d_{k}$ lies inside the triangle ${\cal D}_{1}$. This is because that,
the sum of projections, $\sum\tilde{d}_{k}$, is the projection of $\sum d_{k}$
on the $\frac{\pi}{4}$ line. Therefore, the subproblem on feasible region
${\cal D}_{1}$ can be solved by an approximation algorithm for 1-KP with
demands changed to $\tilde{d}_{k}$ and capacity to $C/\sqrt{2}$.
On the other hand, the subproblem on feasible region ${\cal D}_{1}$ is almost
the whole story: First, evidently an optimal solution in ${\cal D}$ can
contain at most one demand in ${\cal D}_{2}$; second, if an optimal solution
consists of more than one demand, its sum can be broken into either two
separate subsums lying in ${\cal D}_{1}$, or, the sum of a vector in ${\cal
D}_{2}$ and a subsum in ${\cal D}_{1}$. Our algorithm takes the maximum
between an approximate solution for the subproblem on feasible region ${\cal
D}_{1}$ and an optimal solution on input demands lying in ${\cal D}_{2}$. This
only reduces the approximation ratio by at most a factor of 2.
### 3.2 Approximation Algorithm
We let ${\sf Alg}^{\rm a}[(d_{k},v_{k}:k\in K),C]$ be our algorithm for C-KP,
where $(d_{k},v_{k}:k\in K)$ are the complex-valued demands and values of
items and $C$ is the capacity. Moreover, we let ${\sf Alg}^{\rm
1d}[(d_{k},v_{k}:k\in K),C]$ be a polynomial-time approximation algorithm for
1-KP, where each demand is real-valued. We describe our algorithm as follows:
Algorithm 1 ${\sf Alg}^{\rm a}[(d_{k},v_{k}:k\in K),C]$
1: for $k\in K$ do
2: Set $\tilde{d}_{k}=\frac{d_{k}^{\rm R}+d_{k}^{\rm I}}{\sqrt{2}}$
3: end for
4: Set $S_{1}={\sf Alg}^{\rm 1d}[(\tilde{d}_{k},v_{k}:k\in
K),\frac{C}{\sqrt{2}}]$
5: Set $S_{2}=\\{\displaystyle{\arg\max}_{k\in K:d_{k}\in{\cal
D}_{2}}v_{k}\\}$
6: Set $S=\arg\max_{S_{1},S_{2}}\\{v(S_{1}),v(S_{2})\\}$
7: Output $S$
In ${\sf Alg}^{\rm a}$, we first project all demands onto the $\frac{\pi}{4}$
line, and use an approximation algorithm ${\sf Alg}^{\rm 1d}$ for 1-KP to
compute an allocation (denoted by $S_{1}$) considering the projected demands
and capacity $C/\sqrt{2}$. Then we look at all demands lying in region ${\cal
D}_{2}$ and choose one with maximum value as solution $S_{2}$. Note that
$S_{2}$ only consists of a single item. Finally, we compare the total value of
solutions $S_{1}$ and $S_{2}$ and pick the larger one. All ties are broken
arbitrarily.
### 3.3 Analysis
It is evident that our algorithm outputs a feasible solution in polynomial
time. For the approximation ratio, our main result is:
###### Theorem 3.1
If ${\sf Alg}^{\rm 1d}$ is a $\rho$-approximation algorithm for 1-KP, then
${\sf Alg}^{\rm a}$ is a $\frac{\rho}{2}$-approximation algorithm for C-KP.
###### Corollary 3.2
Since 1-KP has an FPTAS BKV05KS ; KPP10book , there is a
$(\frac{1}{2}-\epsilon)$-approximation algorithm for C-KP that runs in
polynomial-time in the size of input and $1/\epsilon$, for any $\epsilon>0$.
Now we prove Theorem 3.1.
###### Proof
Let $S^{*}$ be an optimal solution to C-KP, for which the feasible region is
$\cal D$. Let $S_{1}^{*}$, $S_{2}^{*}$ be an optimal solution for the
subproblem on feasible region ${\cal D}_{1}$ and ${\cal D}_{2}$ respectively.
By our observation in Subsection 4.1, $S_{1}^{*}$ is an optimal solution to
1-KP on projected demands and capacity $C/\sqrt{2}$. Since ${\sf Alg}^{\rm
1d}$ is a $\rho$-approximation algorithm to 1-KP, we have
$v(S_{1})\geq\rho\cdot v(S_{1}^{*})$. It is also evident that
$v(S_{2}^{*})=v(S_{2})$.
Next, we analyze the approximation ratio of ${\sf Alg}^{\rm a}$ in three
cases. Here for a subset $S\subseteq K$, we define
$d(S)\triangleq\sum_{k\in S}d_{k}=\sum_{k\in S}d_{k}^{\rm R}+{\bf i}\sum_{k\in
S}d_{k}^{\rm I}$
Case (1): ($\rho$-approximation) We consider an optimal solution $S^{\ast}$,
such that its sum of demands $d({S^{\ast}})\in{\cal D}_{1}$.
This is an easy case where $v({S^{\ast}})=v({S_{1}^{\ast}})$. We have
$v(S)\geq v(S_{1})\geq\rho\cdot v({S_{1}^{\ast}})=\rho\cdot v({S^{\ast}})$.
Case (2): ($\frac{\rho}{1+\rho}$-approximation) We consider an optimal
solution $S^{\ast}$, such that $d({S^{\ast}})\in{\cal D}_{2}$, and there
exists an item $j\in S^{\ast}$ whose demand $d_{j}\in{\cal D}_{2}$.
Let $z\triangleq\sum_{k\in S^{\ast}\setminus\\{j\\}}d_{k}$. Thus,
$d({S^{\ast}})=d_{j}+z$, i.e., the sum of demands of $S^{\ast}$ can be written
as the sum of a single demand $d_{j}$ and a subset sum $z$.444It is possible
that $S^{\ast}$ only consists of a single item $j$, in which case our
algorithm obviously produces the optimal answer. Note that $d_{j}\in{\cal
D}_{2}$ and $z\in{\cal D}_{1}$. Otherwise, the projection of
$d({S^{\ast}})=d_{j}+z$ on the $\frac{\pi}{4}$ line would exceed $2\cdot
C/\sqrt{2}>C$.
Moreover, we have $v({S^{\ast}\setminus\\{j\\}})\leq v({S_{1}^{\ast}})$,
because $S_{1}^{\ast}$ is an optimal solution for feasible region ${\cal
D}_{1}$. On the other hand, $v_{j}\leq v({S_{2}})$ since item $j$ with
$d_{j}\in{\cal D}_{2}$ is a candidate for $S_{2}$ in our algorithm. We obtain:
$v({S^{\ast}})=v_{j}+v({S^{\ast}\setminus\\{j\\}})\leq
v({S_{2}})+v({S_{1}^{\ast}})$
By the description of our algorithm, the total value of the output solution
$v(S)=\max(v({S_{1}}),v({S_{2}}))\geq\max(\rho\cdot
v({S_{1}^{*}}),v({S_{2}}))=\max(\rho\cdot v({S_{1}^{*}}),v({S_{2}^{*}}))$. Now
it remains to show that it is further
$\geq\frac{\rho}{1+\rho}(v({S_{2}})+v({S_{1}^{*}}))$.
If $\rho\cdot v({S_{1}^{*}})\geq v({S_{2}})$, we have that $v(S)$ is at least
$\rho\cdot v({S_{1}^{*}})=\frac{\rho}{1+\rho}(\rho\cdot
v({S_{1}^{*}})+v({S_{1}^{*}}))\geq\frac{\rho}{1+\rho}(v({S_{2}})+v({S_{1}^{*}}));$
otherwise, $v(S)$ is at least
$v({S_{2}})=\frac{\rho}{1+\rho}(v({S_{2}})+\frac{1}{\rho}v({S_{2}}))\geq\frac{\rho}{1+\rho}(v({S_{2}})+v({S_{1}^{*}})).$
Case (3): ($\frac{\rho}{2}$-approximation) We consider an optimal solution
$S^{\ast}$, such that $d({S^{\ast}})\in{\cal D}_{2}$, and $d_{k}\in{\cal
D}_{1}$ for every item $k\in S^{\ast}$.
First, we let $\tilde{d}(S)\triangleq\sum_{k\in S}\tilde{d}_{k}$. The
condition on $S^{\ast}$ is equivalent to the following condition on projected
demands on the $\frac{\pi}{4}$ line: $C/\sqrt{2}<\tilde{d}({S^{\ast}})\leq C$,
and $\tilde{d}_{k}\leq C/\sqrt{2}$ for every item $k\in S^{\ast}$.
We use Lemma 3.3 to show that $d({S^{\ast}})\in{\cal D}_{2}$ can be written as
the sum of two demand subset sums in ${\cal D}_{1}$. Lemma 3.3 is essentially
an equivalent statement of this on the projected demands, and will be proved
later in this subsection.
###### Lemma 3.3
For a set of $n$ positive real numbers $a_{1},...,a_{n}$ satisfying
$\sum_{i=1}^{n}a_{i}\leq C$, $a_{i}\leq C^{\prime}$ for all $i$ and
$C^{\prime}\geq C/\sqrt{2}$, there exists a subset $T\subseteq\\{1,...,n\\}$
such that
$\sum_{i\in T}a_{i}\leq C^{\prime}\mbox{\ \ and \ \
}\sum_{i\in\\{1,...,n\\}\backslash T}a_{i}\leq C^{\prime}.$
By Lemma 3.3, we have $\tilde{d}(T)$ and $\tilde{d}({S^{\ast}\setminus T})\leq
C/\sqrt{2}$ for some subset $T\subseteq S^{\ast}$. That is, $d(T)\in{\cal
D}_{1}$ and $d({S^{\ast}\setminus T})\in{\cal D}_{1}$.
Thus, $v(T)\leq v({S_{1}^{\ast}})$ and $v({S^{\ast}\setminus T})\leq
v({S_{1}^{\ast}})$. Moreover, since $v({S^{\ast}})=v(T)+v({S^{\ast}\setminus
T})$, we have $v({S^{\ast}})\leq 2v({S_{1}^{\ast}})$. Hence
$v(S)\geq v(S_{1})\geq\rho\cdot
v(S_{1}^{\ast})\geq\frac{\rho}{2}v({S^{\ast}}).$
Combining Cases (1)-(3): $\min\\{\rho,\rho/(1+\rho),\rho/2\\}=\rho/2$, we
complete the proof of the approximation ratio of ${\sf Alg}^{\rm a}$ as
$\rho/2$.
Finally, we prove Lemma 3.3:
###### Proof
The case $\sum_{i=1}^{n}a_{i}\leq C^{\prime}$ is trivial. Otherwise, let $j$
be the smallest index such that the partial sum exceeds $C^{\prime}$, i.e.,
$\sum_{i=1}^{j-1}a_{i}\leq C^{\prime}$ and $\sum_{i=1}^{j}a_{i}>C^{\prime}$.
Clearly $j\geq 2$ since all $a_{i}\leq C^{\prime}$.
Let $x=\sum_{i=1}^{j-1}a_{i}$, $z=a_{j}$ and $y=\sum_{i=j+1}^{n}a_{i}$.
Note that $\sum_{i=1}^{n}a_{i}=x+y+z$. We already have
$x\leq C^{\prime},\ \ z\leq C^{\prime},\ \ x+y+z>C^{\prime}\mbox{\ \ and\ \
}x+z>C^{\prime}$
The lemma holds if $y+z\leq C^{\prime}$, because we can set
$T=\\{1,...,j-1\\}$.
If $y+z>C^{\prime}$, then we obtain:
$\displaystyle x+y$ $\displaystyle=$ $\displaystyle 2(x+y+z)-(x+z)-(y+z)$
$\displaystyle<$ $\displaystyle
2C-2C^{\prime}\leq(2-\sqrt{2})C<\frac{C}{\sqrt{2}}\leq C^{\prime}$
because $x+y+z\leq C$. Hence, we can set $T=\\{1,...,j-1,j+1,...,n\\}$.
## 4 Monotone Approximation Algorithm for C-KP
As mentioned in Subsection 2.3, a monotone polynomial time algorithm for C-KP
implies an incentive compatible polynomial time mechanism. However, our
approximation algorithm ${\sf Alg}^{\rm a}$ presented in Section 3 does not
seem to have an easy proof for monotonicity. In this section, we give a slight
modification of ${\sf Alg}^{\rm a}$, for which monotonicity becomes immediate
and the approximation ratio is preserved.
### 4.1 Basic Idea
In ${\sf Alg}^{\rm a}$, monotonicity is not guaranteed due to the comparison
between $v(S_{1})$ and $v(S_{2})$, the total value of solution $S_{1}$ and
$S_{2}$. Although we assume ${\sf Alg}^{\rm 1d}$ for 1-KP is monotone,
$v(S_{1})$ can fluctuate since $S_{1}$ is an approximate solution. Our trick
here is to transform each solution candidate for $S_{2}$, a single item $k$
with demand $d_{k}\in{\cal D}_{2}$, to be a solution candidate for $S_{1}$: an
item of the same value whose demand is exactly the capacity limit $C/\sqrt{2}$
for ${\sf Alg}^{\rm 1d}$. These new items will not combine with each other or
with any original items to form new solution candidates for $S_{1}$. Then our
new algorithm ${\sf Alg}^{\rm b}$ only needs to run ${\sf Alg}^{\rm 1d}$ on
the modified set of items to produce a solution for C-KP.
### 4.2 Approximation Algorithm
Algorithm 2 ${\sf Alg}^{\rm b}[(d_{k},v_{k}:k\in K),C]$
1: for $k\in K$ do
2: Set $\tilde{d}_{k}=\min\\{\frac{d_{k}^{\rm R}+d_{k}^{\rm
I}}{\sqrt{2}},\frac{C}{\sqrt{2}}\\}$
3: end for
4: Set $S={\sf Alg}^{\rm 1d}[(\tilde{d}_{k},v_{k}:k\in K),\frac{C}{\sqrt{2}}]$
5: Output $S$
Recall that we assume every demand $d_{k}$ lies in ${\cal D}$ ($|d_{k}|\leq
C$). The preprocessing $\tilde{d}_{k}=\min\\{\frac{d_{k}^{\rm R}+d_{k}^{\rm
I}}{\sqrt{2}},\frac{C}{\sqrt{2}}\\}$ does exactly the transformation mentioned
above: For $d_{k}\in{\cal D}_{1}$, we simply do the projection onto the
$\frac{\pi}{4}$ line; otherwise, $d_{k}\in{\cal D}_{2}$, its projection is
larger than $C/\sqrt{2}$, and we cut it off to $C/\sqrt{2}$. Then we run ${\sf
Alg}^{\rm 1d}$ on the modified projected demands and outputs the answer.
The following theorem states that our modification of the algorithm does not
change the approximation ratio:
###### Theorem 4.1
If ${\sf Alg}^{\rm 1d}$ is a $\rho$-approximation algorithm for 1-KP, then
${\sf Alg}^{\rm b}$ is a $\frac{\rho}{2}$-approximation algorithm for C-KP.
The proof of Theorem 4.1 is essentially the same as that of Theorem 3.1. The
main difference is that, now instead of an explicit comparison between the
solutions $S_{1}$ and $S_{2}$ to the two subproblems on region ${\cal D}_{1}$
and ${\cal D}_{2}$ respectively, our algorithm make it implicit inside the
execution of ${\sf Alg}^{\rm 1d}$. Therefore, in the formal proof below, we
have to define the two subproblems explicitly and show that the total value of
our output $v(S)\geq\rho\cdot\max\\{v(S_{1}^{\ast}),v({S_{2}^{\ast}})\\}$.
The case analysis is easy given this inequality. Although ${\sf Alg}^{\rm a}$
has a better approximation guarantee in terms of the inequality
$v(S)\geq\max\\{\rho\cdot v(S_{1}^{\ast}),v({S_{2}^{\ast}})\\}$, overall, we
achieve the same approximation ratio of $\rho/2$. Just for case (2), we can
only prove an approximation ratio of $\rho/2$, instead of $\rho/(1+\rho)$ for
${\sf Alg}^{\rm a}$.
###### Proof
We partition $K$ into two disjoint sets $K_{1}$ and $K_{2}$, such that
$K_{1}\triangleq\\{k\in K:d_{k}\in{\cal D}_{1}\\}$ and $K_{2}\triangleq\\{k\in
K:d_{k}\in{\cal D}_{2}\\}$. Note that the projection of any demand in $K_{1}$
onto the $\frac{\pi}{4}$ line is at most $C/\sqrt{2}$, whereas that in $K_{2}$
is larger than $C/\sqrt{2}$.
Let $S_{1}$ be the output of ${\sf Alg}^{\rm b}$, when the input is $K_{1}$.
Let $S_{2}$ be the output of ${\sf Alg}^{\rm b}$, when the input is $K_{2}$.
Let $S_{1}^{\ast}$ and $S_{2}^{\ast}$ be their corresponding optimal
solutions. $S_{1}^{\ast}$ is an optimal solution to 1-KP on projected demands
within capacity $C/\sqrt{2}$, hence is an optimal solution to C-KP on feasible
region ${\cal D}_{1}$. On the other hand, since each demand in $K_{2}$ is
changed to one exactly equal to the capacity limit of 1-KP, only one of them
can be satisfied. Hence $S_{2}^{\ast}$ chooses the one with maximum value
$S_{2}^{*}=\\{\arg\max_{k\in K_{2}}v_{k}\\}$.
Since any demand in $K_{2}$ will not combine with any in $K_{1}$ to form new
feasible solutions to 1-KP, ${\sf Alg}^{\rm b}$ outputs either a solution
whose sum vector lies in ${\cal D}_{1}$ or a singleton set of a demand in
$K_{2}$, which is evidently a feasible solution to C-KP.
Optimally ${\sf Alg}^{\rm 1d}$ would output
$\arg\max\\{v(S_{1}^{\ast}),v(S_{2}^{\ast})\\}$. Since ${\sf Alg}^{\rm 1d}$ is
a $\rho$-approximation algorithm to 1-KP, we have
$v(S)\geq\rho\cdot\max\\{v(S_{1}^{\ast}),v({S_{2}^{\ast}})\\}$.
Based on this inequality, it is easy to go through the case analysis in the
proof of Theorem 3.1 (with slight modifications), hence we omit the rest of
the proof here.
On the other hand, our new algorithm is monotone according to Definition 2.1.
###### Theorem 4.2
If ${\sf Alg}^{\rm 1d}$ is a monotone algorithm for 1-KP, then ${\sf Alg}^{\rm
b}$ is a monotone algorithm for C-KP.
###### Proof
We need to show that, if item $k$ is selected by ${\sf Alg}^{\rm b}$ with
demand $d_{k}$ and value $v_{k}$, $k$ is also selected with demand
$d^{\prime}_{k}$ and value $v^{\prime}_{k}$, where $v^{\prime}_{k}\geq v_{k}$
and ${d^{\prime}}_{k}\leq d_{k}$ (i.e., ${d^{\prime}}_{k}^{\rm R}\leq
d_{k}^{\rm R}$ and ${d^{\prime}}_{k}^{\rm I}\leq d_{k}^{\rm I}$), while all
inputs of other agents do not change.
Item $k$ is selected by ${\sf Alg}^{\rm b}$ on $d^{\prime}_{k}$ and
$v^{\prime}_{k}$ if and only if it is selected by ${\sf Alg}^{\rm 1d}$ on
$\tilde{d}_{k}^{\prime}$ and $v^{\prime}_{k}$. Since
$\tilde{d}_{k}=\min\\{\frac{{d}_{k}^{\rm R}+{d}_{k}^{\rm
I}}{\sqrt{2}},\frac{C}{\sqrt{2}}\\}$, ${d^{\prime}}_{k}^{\rm R}\leq d_{k}^{\rm
R}$ and ${d^{\prime}}_{k}^{\rm I}\leq d_{k}^{\rm I}$ implies
$\tilde{d}_{k}^{\prime}\leq\tilde{d}_{k}$. Then from the monotonicity of ${\sf
Alg}^{\rm 1d}$, $k$ is selected by ${\sf Alg}^{\rm 1d}$, and hence by ${\sf
Alg}^{\rm b}$.
Combining Theorem 4.1, Theorem 4.2 with Theorem 2.2 gives:555Note that
algorithms are exact under this problem formulation where a solution is
specified as the selection of a subset of items.
###### Corollary 4.3
Since 1-KP has a monotone FPTAS BKV05KS , there is an incentive compatible
$(\frac{1}{2}-\epsilon)$-approximation algorithm for C-KP that runs in
polynomial-time in the size of input and $1/\epsilon$, for any $\epsilon>0$.
## 5 Inapproximability for C-KP
In this section, we complete the study of C-KP by providing an
inapproximability result. We show that C-KP does not admit an FPTAS, unless P
= NP.
We remark that it is known there is no FPTAS for 2-KP (see KPP10book ), which
does not have direct implications for C-KP. However, our proof is an extension
of the basic idea in the proof for 2-KP.
As in the reduction for 2-KP, we reduce the equipartition problem to C-KP:
###### Definition 5.1
(equipartition Problem): Given a set of positive integers $\\{w_{k}:k\in
K\\}$, with $|K|=n$ where $n$ is even, we determine if there is a subset of
items $S\subseteq K$ such that
$|S|=\frac{n}{2}\mbox{\ and\ }\sum_{k\in S}w_{k}=\sum_{k\notin S}w_{k}$
It is well-known that equipartition is NP-complete.
###### Theorem 5.2
There is no FPTAS for C-KP, unless P = NP.
###### Proof
We define a decision version of C-KP with a cardinality objective: given
$\\{w_{k}:k\in K\\}$, a capacity bound $C$ and a cardinality bound $M$, we
determine if there is a subset of items $S$ such that
$|S|\geq M,\mbox{\ and\ }\Big{|}\sum_{k\in S}d_{k}\Big{|}\leq C$
Now we map every instance of equipartition to an instance of the C-KP decision
problem that always yields the same answer.
Given $\\{w_{k}:k\in K\\}$ from equipartition, define
$M=n/2,\quad d_{k}^{\rm R}=w_{k},\quad d_{k}^{\rm I}=\beta(w_{\max}-w_{k}),$
$C=\sqrt{\Big{(}\frac{W}{2}\Big{)}^{2}+\beta^{2}\Big{(}\frac{nw_{\max}}{2}-\frac{W}{2}\Big{)}^{2}}$
where $W\triangleq\sum_{k=1}^{n}w_{k}$, $w_{\max}\triangleq\max\\{w_{k}:k\in
K\\}$. Note that in our reduction, $d_{k}^{\rm I}\geq 0$.
As shown in Fig. 2, the feasible region $\cal D$ for C-KP is the $\frac{1}{4}$
disk of radius $C$ in the first quadrant. Since for any subset $S\subseteq K$,
$\sum_{k\in S}d_{k}^{\rm I}=\beta(|S|\cdot w_{\max}-\sum_{k\in S}d_{k}^{\rm
R})$, the cardinality constraint $|S|\geq\frac{n}{2}$ imposes all solutions to
have its sum vector in the halfplane $H:$
$d^{I}\geq\beta(\frac{nw_{\max}}{2}-d^{R})$. The dividing line of $H$ goes
through point $P:$
$\Big{(}\frac{W}{2},\beta(\frac{nw_{\max}}{2}-\frac{W}{2})\Big{)}$. Our main
idea is to set $\beta>0$ such that the dividing line of $H$ coincides with the
tangent line at $P$. Thus we make the intersection of $H$ and ${\cal D}$
exactly $P$, which implies $|S|=\frac{n}{2}$ and $\sum_{k\in
S}w_{j}=\frac{W}{2}$ for any solution $S$ to our reduced C-KP decision problem
instance.
Figure 2: Reduction of inapproximability.
On the other hand, it is clear that each subset $S$ satisfying conditions of
equipartition also satisfies conditions of the reduced C-KP decision problem.
Therefore, the solution of the reduced C-KP decision problem is equivalent to
the solution of equipartition.
To determine a proper $\beta$, since the dividing line of halfplane $H$ goes
through $P$, it coincides with the tangent line at $P$ if and only if they
have the same slope, i.e.,
$-\frac{\frac{W}{2}}{\beta(\frac{nw_{\max}}{2}-\frac{W}{2})}=-{\beta}.$
Solving the above equation, we obtain
$\beta=\sqrt{\frac{W}{nw_{\max}-W}},$
which is $>0$ unless all weights are equal. In this case, we set $\beta=0$,
and it is trivially a ”yes” instance for both equipartition and our C-KP
decision problem.
So far we have shown the NP-hardness of the C-KP decision problem. So its
maximization version, where $|S|\geq M$ is replaced by $\max|S|$, is NP-hard.
We use the standard technique to prove the inapproximability of the
maximization version by FPTAS. Suppose that there exists an FPTAS for any
$\epsilon>0$ in time polynomial in $n$ and $1/\epsilon$. Then we choose
$\epsilon=\frac{1}{n+1}$. Let the optimal solution be $z^{\ast}>0$ and that of
the approximation solution produced by FPTAS be $z^{A}$. We obtain:
$z^{A}\geq(1-\epsilon)z^{\ast}>z^{\ast}-z^{\ast}/n\geq z^{\ast}-1$
because $z^{\ast}\leq n$. Moreover, since $z^{\ast}$ is an integer, this
implies that the FPTAS can solve the problem exactly in polynomial time,
contradicting the NP-hardness of the problem.
Finally, since the maximization version of C-KP decision problem is a special
case of the original C-KP with all $v_{k}=1$, there is no FPTAS to C-KP.
## 6 Approximation Algorithm for
GC-KP
We are also able to solve the generalized problem GC-KP, by changing our
approximation algorithm ${\sf Alg}^{\rm a}$ in Section 3. Now, instead of an
approximation algorithm ${\sf Alg}^{\rm 1d}$ for 1-KP as a subroutine, we rely
on an approximation algorithm for 3-KP (three-dimensional knapsack problem) as
a subroutine.
### 6.1 Basic Idea
Now a feasible solution of our problem is a subset of items whose sum of
demands lies in the intersection of halfplanes $d^{\rm R}\leq C^{\rm R}$,
$d^{\rm I}\leq C^{\rm I}$, and the $1/4$ disk of radius $C$ in the first
quadrant. In the most general case ($C^{\rm R},C^{\rm I}<C$), both halfplanes
cut the circle, which also cut the original regions ${\cal D}_{1}$ and ${\cal
D}_{2}$ defined in Section 3. Fig. 3 shows the new ${\cal D}_{1}$ (polygon
$PSTQO$) and ${\cal D}_{2}$. Clearly, the feasible region ${\cal D}$ is the
disjoint union of ${\cal D}_{1}$ and ${\cal D}_{2}$.
Figure 3: Graphical Picture for GC-KP.
Recall that $OR$ is perpendicular to $ST$ and the length of $OR$ is
$C/\sqrt{2}$. If we denote the projection of a demand $d_{k}$ onto line $OR$
by $\tilde{d}_{k}$, the region ${\cal D}_{1}$ corresponds to 3-dimensional
linear constraint $\sum_{k\in K}x_{k}\tilde{d}_{k}\leq C/\sqrt{2}$,
$\sum_{k\in K}x_{k}d_{k}^{\rm R}\leq C^{\rm R}$ and $\sum_{k\in
K}x_{k}d_{k}^{\rm I}\leq C^{\rm I}$. Thus the subproblem on feasible region
${\cal D}_{1}$ can be solved by a 3-dimensional knapsack algorithm.
On the other hand, the solutions in polygon ${\cal D}_{1}$ is almost the whole
story by the same reason as in Section 3. Again our algorithm takes the
maximum between an optimal solution for the subproblem on feasible region
${\cal D}_{1}$ and an optimal solution on input demands lying in ${\cal
D}_{2}$. This reduces the approximation ratio by at most a factor of 2.
The degenerate cases ($C^{\rm R}\geq C$ or $C^{\rm I}\geq C$ or both) can be
treated easily by setting $T,Q$ to be the intersection point of the circle and
the $d^{\rm R}$-axis, or setting $P,S$ to be the intersection point of the
circle and the $d^{\rm I}$-axis, or both.
### 6.2 Approximation Algorithm
Let ${\sf Alg}^{\rm c}[(d_{k},v_{k}:k\in K),C,C^{\rm R},C^{\rm I}]$ be our
approximation algorithm for GC-KP, where $C,C^{\rm R},C^{\rm I}$ are the
capacity on magnitude, real part and imaginary part of total satisfiable
demand respectively. Let ${\sf Alg}^{\rm
3d}[((d^{1}_{k},d^{2}_{k},d^{3}_{k}),v_{k}:k\in K),C^{1},C^{2},C^{3}]$ be an
approximation algorithm for 3-KP (e.g., from FC84alg or KPP10book ). We
describe our approximation algorithm to GC-KP as follows:
Algorithm 3 ${\sf Alg}^{\rm c}[(d_{k},v_{k}:k\in K),C,C^{\rm R},C^{\rm I}]$
1: for $k\in K$ do
2: Set $\tilde{d}_{k}=\frac{d_{k}^{\rm R}+d_{k}^{\rm I}}{\sqrt{2}}$
3: end for
4: Set $S_{1}={\sf Alg}^{\rm 3d}[((\tilde{d}_{k},d_{k}^{\rm R},d_{k}^{\rm
I}),v_{k}:k\in K),\frac{C}{\sqrt{2}},C^{\rm R},C^{\rm I}]$
5: Set $S_{2}=\\{\displaystyle{\arg\max}_{k\in K:d_{k}\in{\cal
D}_{2}}v_{k}\\}$
6: Set $S=\arg\max_{S_{1},S_{2}}\\{v(S_{1}),v(S_{2})\\}$
7: Output $S$
Our ${\sf Alg}^{\rm c}$ follows the same structure as ${\sf Alg}^{\rm a}$ for
C-KP. The difference is that, for GC-KP, the subproblem on feasible region
${\cal D}_{1}$ is equivalent to an instance of 3-KP, since ${\cal D}_{1}$ is
defined by three halfplanes. And to check if a demand $d_{k}$ lies in region
${\cal D}_{2}$, we need to check four inequalities: $|d_{k}|\leq C$,
$\tilde{d}_{k}>C/\sqrt{2}$, $d_{k}^{\rm R}\leq C^{\rm R}$ and $d_{k}^{\rm
I}\leq C^{\rm I}$.
###### Theorem 6.1
If ${\sf Alg}^{\rm 3d}$ is a $\rho$-approximation algorithm for 3-KP, ${\sf
Alg}^{\rm c}$ is a $\frac{\rho}{2}$-approximation algorithm for GC-KP.
###### Corollary 6.2
Since 3-KP has a PTAS FC84alg , there is a
$(\frac{1}{2}-\epsilon)$-approximation algorithm for GC-KP that runs in
polynomial-time in the size of input, for any $\epsilon>0$.
We omit the proof of Theorem 6.1 here since it is essentially the same as that
of Theorem 3.1 for C-KP.
## 7 Conclusions and Future Work
The knapsack problem has been one of the most popular algorithmic problems
since it is a simple abstraction that captures the tradeoff between limited
resource and value maximization in resource allocation. In this paper,
motivated by the need to model AC electrical systems, where power demands have
to be represented as complex numbers, we initiate the study of a new variation
called the complex-demand knapsack problem (C-KP).
By investigating its relationship with multi-dimensional knapsack problems
($m$-KP), we provide $(\frac{1}{2}-\epsilon)$-approximation algorithms for
C-KP and its generalization GC-KP; on the other hand, we also show its
inapproximability by FPTAS unless P = NP. Furthermore, our approximation
algorithm for C-KP can be monotonized, which implies the existence of a
mechanism/algorithm of the same approximation ratio that is incentive
compatible, individually rational, and computationally efficient.
Our results provide basic insights on efficient power allocation in AC
electrical systems, which is a fundamental problem in the design of multi-
agent systems for smart grid. Still, there are interesting directions to
continue in the future: First, we hope to find a PTAS for C-KP, closing the
gap between constant approximation and FPTAS, and a monotone algorithm for GC-
KP. Second, we will extend the problem to an electrical network setting, where
there is an underlying network connecting different devices with links of
different capacities on the magnitude of transmitted power. For mechanism
design, we may require an additional property called cancellability: the total
payment to be collected from the agents should always cover the cost to
generate the power supply, given a cost function of power generation. We are
not aware of any previous work related to this property in mechanism design,
and we expect new insights and techniques coming out of the study on it.
Acknowledgments We thank Mario Szegedy and anonymous referees for useful
suggestions. Lan Yu is supported by Singapore NRF Research Fellowship 2009-08.
Chi-Kin Chau is supported by MI-MIT Collaborative Research Project (11CAMA1).
## References
* [1] A. Archer and É. Tardos. Truthful mechanisms for one-parameter agents. In FOCS’01, pages 482–491, 2001.
* [2] P. Briest, P. Krysta, and B. Vocking. Approximation techniques for utilitarian mechanism design. In STOC’05, pages 39–48, 2005.
* [3] A. Frieze and M. Clarke. Approximation algorithm for the m-dimensional 0-1 knapsack problem. European Journal of Operational Research, 15:100–109, 1984.
* [4] E. Gerding, V. Robu, S. Stein, D. Parkes, A. Rogers, and N. Jennings. Online mechanism design for electric vehicle charging. In AAMAS’11, pages 811–818, 2011.
* [5] J. Grainger and W. Stevenson. Power System Analysis. McGraw-Hill, 1994.
* [6] H. Kellerer, U. Pferschy, and D. Pisinger. Knapsack Problems. Springer, 2010.
* [7] A. Kulik and H. Shachnai. There is no EPTAS for two-dimensional knapsack. Information Processing Letters, 110(16):707–710, 2010.
* [8] D. Lehmann, L. O’Callaghan, and Y. Shoham. Truth revelation in approximately efficient combinatorial auctions. In EC’99, pages 96–102, 1999.
* [9] N. Nisan. Introduction to mechanism design. In Algorithmic Game Theory, chapter 9, pages 209–242. 2007.
* [10] N. Nisan and A. Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35:166–196, 2001.
|
arxiv-papers
| 2012-05-10T14:59:58 |
2024-09-04T02:49:30.802974
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lan Yu and Chi-Kin Chau",
"submitter": "Chi-Kin Chau",
"url": "https://arxiv.org/abs/1205.2285"
}
|
1205.2322
|
# Metallicities, dust and molecular content of a QSO-Damped Lyman-$\alpha$
system reaching $\log N$(H i) = 22:
An analog to GRB-DLAs
R. Guimarães Programa de Modelagem Computacional - SENAI - Cimatec, 41650-010
Salvador, Bahia, Brasil rguimara@eso.org P. Noterdaeme and P. Petitjean
UPMC-CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis bd Arago, F-75014
Paris, France C. Ledoux European Southern Observatory, Alonso de Córdova
3107, Casilla 19001, Vitacura, Santiago 19, Chile R. Srianand Inter-
University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind,
Pune 411 007, India S. López Departamento de Astronomía, Universidad de
Chile, Casilla 36-D, Santiago, Chile H. Rahmani Inter-University Centre for
Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
###### Abstract
We present the elemental abundance and H2 content measurements of a Damped
Lyman-$\alpha$ (DLA) system with an extremely large H i column density, $\log
N$(H i) (cm-2) = 22.0$\pm$0.10, at $z_{\rm abs}=3.287$ towards the QSO SDSS J
081634$+$144612\. We measure column densities of H2, C i, C i⋆, Zn ii, Fe ii,
Cr ii, Ni ii and Si ii from a high signal-to-noise and high spectral
resolution VLT-UVES spectrum. The overall metallicity of the system is [Zn/H]
= $-1.10\pm 0.10$ relative to solar. Two molecular hydrogen absorption
components are seen at $z=3.28667$ and 3.28742 (a velocity separation of
$\approx 52$ km s-1) in rotational levels up to $J=3$. We derive a total H2
column density of log $N$(H2) (cm-2) = 18.66 and a mean molecular fraction of
$f$ = $2N($H${}_{2})/[2N$(H${}_{2})+N($H i$)]=10^{-3.04\pm 0.37}$, typical of
known H2-bearing DLA systems. From the observed abundance ratios we conclude
that dust is present in the Interstellar Medium (ISM) of this galaxy, with a
enhanced abundance in the H2-bearing clouds. However, the total amount of dust
along the line of sight is not large and does not produce any significant
reddening of the background QSO. The physical conditions in the H2-bearing
clouds are constrained directly from the column densities of H2 in different
rotational levels, C i and C i⋆. The kinetic temperature is found to be
$T\approx 75$ K and the particle density lies in the range $n_{\rm H}$ =
50$-$80 cm-3. The neutral hydrogen column density of this DLA is similar to
the mean H i column density of DLAs observed at the redshift of $\gamma$-ray
bursts (GRBs). We explore the relationship between GRB-DLAs and high column
density end of QSO-DLAs finding that the properties (metallicity and
depletion) of DLAs with $\log N$(H i) $>$ 21.5 in the two populations do not
appear to be significantly different.
quasars: general — quasars: absorption lines — ISM: molecules
## 1 Introduction
Despite accounting for only a small fraction of all the baryons in the
Universe (see, e.g., Petitjean et al. 1993), the physical state of the neutral
and molecular phases of the interstellar medium is a crucial ingredient of
galaxy formation. These gaseous phases are at any redshift the reservoir of
gas available for star formation. At high redshift, most of the neutral
hydrogen mass is revealed by the damped Lyman-$\alpha$ absorption system
(DLAs) detected in the spectra of background quasars (see e.g. Wolfe et al.
2005 for a review). Since DLAs are easy to identify in QSO spectra and the H i
column densities can be measured accurately, it is possible to derive the
cosmological mass density of the neutral gas at different redshifts,
independent of the exact nature of the absorbers, provided a sufficiently
large number of background quasars are observed (see Prochaska & Wolfe 2009,
Guimarães et al. 2009, Noterdaeme et al. 2009b).
Key results of DLA surveys include the indication that the $N$(H i) frequency
distribution deviates significantly from a single power-law with $f$($N$(H i))
sharply steepening at $\log N$(H i) $>$ 21.5. However, Zwaan & Prochaska
(2006) used CO emission maps in the nearby Universe to show that the H2 column
density distribution function is a continuous extension of the H i
distribution for high column densities. The transition happens at log $N$(H i)
$\sim$ 22 which is the approximate column density associated with the
conversion from H i into H2 (e.g., Schaye 2001).
The slope of the column density distribution at large $N$(H i),
$\alpha\sim−3.5$, implies that systems with very large column density are very
rare. Indeed, until very recently111In the last stages of this work, the
discovery of another DLA with log $N$(H i) $\geq$ 22 was reported by
Noterdaeme et al. (2012) and Kulkarni et al. (2012)., only one DLA system with
$\log N$(H i) $\sim$ 22 was reported in the literature: the system at $z_{\rm
abs}=3.287$ towards SDSS J 081634$+$144612 (from the SDSS-II DLA catalog,
Noterdaeme et al. 2009b). However, DLAs with such high column densities are
frequently detected at the redshift of $\gamma$-ray bursts (e.g. Savaglio et
al. 2003; Vreeswijk et al. 2004; Fynbo et al. 2006; Jakobsson et al. 2006;
Prochaska et al. 2007; Fynbo et al. 2009; Ledoux et al. 2009; Savaglio 2010).
Therefore, comparing the chemical and physical properties of the strongest
QSO-DLAs to that of GRB-DLAs may provide clues to understand the nature of
these absorbers.
We present here a detailed study of the gas-phase abundances of metals, dust
and molecules in the QSO-DLA towards SDSS J 081634$+$144612, based on high
spectral resolution data. The paper is organized as follows. In Sect. 2 we
provide details of observations and data reduction. The hydrogen column
density of the DLA, the metal, dust and molecular content are discussed in
Sect. 3. The physical state of the H2 bearing gas component is discussed in
Sect. 4. In Sect. 5, we finally investigate the connection between QSO-DLAs
and GRB-DLAs.
## 2 Observations
We observed quasar SDSS J 081634$+$144612 twice, in September 2008 and April
2009, with the high-resolution Ultraviolet and Visual Echelle spectrograph
(UVES, Ballester et al. 2000) mounted on the ESO Kueyen VLT-UT2 8.2 m
telescope at Cerro Paranal, Chile. Observations have been performed under
programs 081.A-0334(A), PI. S. López in visitor mode, and 282.A-5030(A), PI.
P. Noterdaeme in service mode. Ten exposures were taken for a total of 12.4
hours exposure time: nine exposures using Dichroic 2 with a setting 437+760 nm
plus one 5400 s exposure with the red arm centered at 550 nm that covers the
Lyman-$\alpha$ absorption. A slit width of 1 arcsec and 2x2 pixel binning were
used, resulting in a spectral resolution of 50,000.
The quasar spectrum was reduced using the UVES pipeline (see e.g. Ledoux et
al. 2003 for details). The main characteristics of the pipeline are to perform
a precise inter-order background subtraction, especially for master flat-
fields, and to allow for an optimal extraction of the object signal rejecting
cosmic rays and performing sky subtraction at the same time. The pipeline
products were checked step by step. The wavelength scale of each reduced
spectrum was then converted to vacuum-heliocentric values and the spectra
rebinned to a constant wavelength step. No further rebinning was performed
during the analysis of the whole spectrum. Individual 1-D exposures were
scaled, weighted and combined together.
In order to derive the physical parameters of the absorption features, we fit
the metal absorption profiles with multiple Voigt profiles using VPFIT
(Carswell et al. 1987). The continuum level was obtained locally in the
vicinity of each metal absorption feature. Molecular hydrogen features were
fitted altogether using fit/lyman and after normalizing the corresponding
region of the Lyman-$\alpha$ forest. Atomic data for metal species and H2 are,
respectively, from Morton (2003) and Bailly et al. (2010). In the following,
solar abundances are taken from Lodders (2003). The origin of the velocity
scale ($v$ = 0 km s-1) is set at the redshift of the single C i component,
$z=3.28746$.
## 3 Abundances
### 3.1 H i and metal content
The neutral hydrogen column density was measured from the fit of the damping
wings of the Lyman-$\alpha$ absorption at $z=3.287$. We find $\log N$(H i)
(cm-2) = 22.0$\pm$0.10. The observed Damped Lyman-$\alpha$ absorption together
with the best fitted Voigt profile is shown in Figure 1. The dashed lines
indicate the profiles corresponding to $\log N$(H i) = 21.9 and 22.1.
We detect absorption lines of C i, C i⋆, Zn ii, Fe ii, Cr ii, Ni ii and Si ii,
spread over about 150 km s-1. As can be seen in Fig. 2, the absorption
profiles are not strongly saturated except may be for Si ii$\lambda$1808\. We
are therefore confident that our column density determinations are robust. The
fit to the absorption lines are overplotted on the Figure as red solid lines.
The singly ionized species are expected to be the dominant contributors to the
abundances of the corresponding elements in H i clouds with such high column
densities.
Voigt profile fitting is performed simultaneously for all the absorption lines
keeping the same number of components having same redshifts and Doppler
parameters for all singly ionized species. Note that Zn ii$\lambda$2026 is
blended with Mg i$\lambda$2026\. The contributions of the latter is taken into
account in the fits and found to have negligible influence on the derived N(Zn
ii).
Eight velocity components are necessary to model the profiles (see Fig. 2). We
report in Table 1 the results of the fits, column density and Doppler
parameter, for each of the components. Because of saturation effects, the Si
ii column density should be considered a lower limit. However, given the shape
of the absorption, we are confident that the true value cannot be much larger.
The column density of singly ionized iron, despite their blending, could be
reliably measured.
Although the profile decomposition may not be unique, three distinct clumps
can be identified (#1, 2 and 3 from blue to red) at mean redshifts of 3.28661,
3.28735 and 3.28814, and made of respectively, 3, 2 and 3 components. H2
absorption is detected in #1, which is the strongest clump in metal species,
but most of the H2 is found associated to the C i component in clump #2.
Unfortunately, we cannot determine $N$(H i) in each clump. The mean
metallicity222Metallicities are given relative to solar: ${\rm[X/H]}={\rm
log}(N_{\rm X}/N_{\rm HI})_{\rm DLA}-{\rm log}({\rm X/H})_{\odot}$ in the
cloud, derived by adding the column densities of the eight components are:
[Zn/H] = $-1.10\pm 0.10$, [Si/H] $\geq$ $-1.23\pm 0.10$, [Cr/H] = $-1.58\pm
0.10$, [Ni/H] = $-1.68\pm 0.10$ and [Fe/H] = $-1.58\pm 0.10$.
### 3.2 Dust
In the ISM of the Galaxy, zinc is virtually undepleted onto dust grains when
Si, Cr and Ni are. We find in the present DLA mean relative abundances:
[Si/Zn] = $-0.13\pm 0.03$, [Cr/Zn] = $-0.48\pm 0.02$, [Ni/Zn] = $-0.58\pm
0.02$ and [Fe/Zn] = $-0.48\pm 0.02$ indicating that the overall depletion of
Si, Cr, Ni and Fe is similar to what is seen in the gas from the halo of our
Galaxy (Welty et al. 1999). This is also typical of DLAs where H2 is detected
(Noterdaeme et al. 2008). From Fig. 3, which presents the depletion patterns
relative to zinc observed component by component, it is clear however that the
depletion is enhanced in the main H2-bearing component. This situation is
similar to that of the H2-bearing component towards Q 0013$-$004 (Petitjean et
al. 2002), where higher depletion factors are seen in the H2 components.
From the flux-calibrated SDSS spectrum, it is also possible to estimate the
reddening induced by the presence of dust in the DLA to the background QSO
light, following the method described in Noterdaeme et al. (2009b). In Fig. 4,
we show that the SDSS spectrum of J 081634$+$144612 is well matched with the
SDSS composite spectrum from Vanden Berk et al. (2001), shifted to the same
emission redshift and reddened using a SMC extinction law (Gordon et al. 2003)
at $z=3.286$ with E(B-V) = 0.05$\pm$0.06. The associated uncertainty is
obtained from the dispersion measured for a control sample of 163 SDSS QSOs
from Schneider et al. (2010) with emission redshift within $\pm 0.02$ to that
of J 081634$+$144612\. This means that there is no significant reddening of
the quasar J 081634$+$144612\. Overall, the measured extinction-to-dust ratio,
A${}_{\rm v}/N$(H i) $<5\times 10^{-23}$ mag cm2 (2 $\sigma$) is typical of
that of the general DLA population (Vladilo et al. 2008). This, together with
the presence of H2 in a component with higher depletion factor suggests that
appreciable fraction of H i may be associated with components that do not have
H2.
### 3.3 H2 content
Molecular hydrogen is detected in two distinct sub-systems at $z_{\rm abs}$ =
3.28667 and 3.28742, separated by $\sim$52 km s-1 with absorption lines from
rotational levels up to J = 3 (see Figure 5). The results of the fits to the
numerous absorption lines are given in Table 2. The $z_{\rm abs}$ = 3.28742
component alone contains about 90% of the total H2 column density in the
absorber, log $N$(H2) = 18.62, and coincides with the C i component. This is
expected because the energy of the photons that ionize C i is close to that of
photons that dissociate H2. The total H2 column density integrated over the
two components and all rotational levels is log $N$(H2) = 18.66 $\pm$ 0.27
(cm-2) corresponding to a molecular fraction of log $f$ = log 2$\times$
$N$(H2)/(2$\times$ $N$(H2) + $N$(H i)) = $-3.04\pm 0.37$ if we assume that the
totality of neutral hydrogen is associated with the H2 components. This value
is amongst the lowest observed in H2 bearing DLAs with metallicities [Zn/H]
$>$ $-1.3$ (Petitjean et al. 2006, Noterdaeme et al. 2008). This again may
indicate that appreciable fraction of H i may be associated with components
that do not have H2 (see also Noterdaeme et al. 2010, Srianand et al. 2010,
2012).
## 4 Physical state of the gas
### 4.1 Excitation of H2
From the detection of H2 in different rotational levels (J=0 to J=3, see Table
2), it is possible to put constraints on the physical state of the gas. The
excitation temperature $T_{0J}$ between rotational levels 0 and J is defined
as
$\frac{N(\rm J)}{N(0)}=\frac{g(\rm J)}{g(0)}e^{-E(\rm 0J)/kT_{0J}}\noindent$
(1)
where $g(\rm J)$ is the statistical weight of the rotational level J: $g(\rm
J)=(2J+1)(2I+1)$ with nuclear spin I = 0 for even J (para-H2) and I = 1 for
odd J (ortho-H2), $k$ is the Boltzmann constant, and $E(\rm 0J)$ is the energy
difference between level J and the ground state (J=0). If the excitation
processes are dominated by collisions, then the populations of the rotational
levels follow a Boltzmann distribution described by a unique excitation
temperature for all rotational levels. This is generally the case for low
rotational levels which have a de-excitation time scale larger than the
collision time-scale. Indeed, $T_{01}$ is a good indicator of the kinetic
temperature (Roy et al. 2006; Le Petit et al. 2006) especially in clouds
similar to the one we study here (log $N$(H2) $>$ 18). However, because of the
small energy difference between the J=0 and J=1 levels, the value of $T_{01}$
is very sensitive to uncertainties on $N$(H2,J=0) and $N$(H2,J=1) and the use
of higher rotational levels may help derive a better constraint on $T_{\rm
K}$. From Fig. 6, it can be seen that the population of J=0 to J=2 levels can
be described by a unique excitation temperature $T_{\rm ex}=69_{-8}^{+10}$ K
and $T_{\rm ex}=79_{-10}^{+14}$ K for the first and second component,
respectively.
These temperatures are slightly smaller than what was found in previous
studies of H2-bearing DLAs (e.g. Ledoux et al. 2003, $T\sim$ 90 to 180 K;
Srianand et al. 2005, $T\sim$ 153$\pm$78 K), but similar to what is measured
in the ISM of our Galaxy (77$\pm$17 K; Rachford et al. 2002) and in the
Magellanic Clouds (82$\pm$21 K; Tumlinson et al. 2002), where H i column
densities are also large. Note that, temperatures observed through high
latitude Galactic sight lines are also larger (124$\pm$8 K; Gillmon et al.
2006, or ranging from 81 K at log $N$(H2) = 20 to 219 K at log $N$(H2) = 14;
Wakker 2006).
The population of J=3 rotational level is in turn enhanced compared to the
Boltzmann distribution, which indicates the presence of additional excitation
processes such as UV pumping (e.g. Noterdaeme et al. 2007a) and/or turbulent
dissipation (as possibly indicated by the larger $b$-parameters for higher-J
levels seen by Noterdaeme et al. 2007b).
### 4.2 Density
Absorption lines produced by neutral carbon are seen only in one component at
$z_{\rm abs}$ = 3.28746 (see Figure 2) and are associated with the strongest
H2 component.
We used the relative populations of the two first sub-levels of the C i ground
state to derive the excitation temperature of the C i fine-structure level,
according to the Boltzmann equation (see Eq. 1). We have adopted the energy
difference between the C i excited (C i⋆: 2s22p2 3P1) and true ground-state
(2s22p2 3P0) levels, $\Delta E_{\rm eg}$ = 23.6 K. The population ratio
$N$(J=1)/$N$(J=0) of the C i fine-structure level corresponds to an excitation
temperature of $T_{\rm ex}$ = 15.4$\pm$0.1 K. This is higher than the
temperature expected in the case the excitation is dominated by the cosmic
microwave background radiation ($T_{\rm CMBR}$ = 11.7 K at $z=3.287$)
indicating that excitation by collisions is important. Using the results shown
in fig. 12 of Srianand et al. (2005), we derive that the particle density,
$n_{\rm H}$, is in the range 50-80 cm-3.
## 5 Discussion
We have presented a detailed analysis of a QSO-DLA system with an extremely
large column density, $\log N($H i$)=22.0\pm 0.10$, at $z_{\rm abs}=3.287$
towards the quasar SDSS J 081634$+$144612\. The velocity structure of
associated metal absorption lines indicates the presence of 8 components
grouped into three sub-systems centered at $z=3.28661$, 3.28735 and 3.28814
respectively, spanning $\sim 115$ km s-1. C i is detected in the $z=3.28735$
sub-system whilst H2 is detected in both the $z=3.28661$ and 3.28735 sub-
systems. From the H2 excitation, we derive a kinetic temperature of $T_{\rm
K}\sim 75$ K and the observed column density ratio $N($C i$)^{\star}/N($C i$)$
yields a particle density in the range $n_{\rm H}\sim 50-80$ cm-3. The
depletion of metals onto dust grains measured in the strongest H2 component
located at $z_{\rm abs}=3.28735$ is similar to what is observed in the disc of
the Galaxy. All this shows that this system, apart from having unusually large
N(H i), has properties consistent with that of a typical H2-bearing DLA (see
Ledoux et al. 2003; Noterdaeme et al. 2008).
While log $N($H i$)\geq 22$ DLAs are very rarely seen in front of QSOs,
several have already been detected in the optical afterglow spectrum and at
the redshift of long-duration $\gamma$-ray bursts. Two of them (towards GRB
050401, GRB 080607 and GRB 060926) even have log $N($H i$)>22.5$ (Watson et
al. 2006; Prochaska et al. 2009; Jakobsson et al. 2006). It is not surprising
to observe a strong DLA at the redshift of GRBs since these objects are
expected to be associated with star-forming regions where the gas is likely to
be found in large quantities. Although QSO-DLAs are located close to regions
where stars form as indicated by the presence of metals, the detection of C
ii⋆ absorption and galaxy-like kinematics (Wolfe et al. 2003), their exact
nature is not completely elucidated. Some should be associated with the ISM of
galaxies especially when molecules are detected (Noterdaeme et al. 2008),
others are probably located in the outskirts of galactic haloes (Møller et al.
2002; Møller et al. 2004; Fox et al. 2007; Pontzen et al. 2008, Rauch et al.
2008; Rahmani et al. 2010; Fynbo et al. 2010, 2011).
The difference between the populations of GRB-DLAs and QSO-DLAs is apparent
because (i) the mean H i column density is higher in GRBs than in DLAs
reaching easily well beyond $10^{21.5}$ cm-2 in the former case (see, e.g.,
Jakobsson et al. 2006; Fynbo et al. 2009) whilst QSO-DLA with log $N($H
i$)\geq 22$ are very rare (Noterdaeme et al. 2009b), and (ii) the mean
metallicity is larger for GRB-DLAs (Savaglio et al. 2006; Fynbo et al. 2006;
Prochaska et al. 2007; Fynbo et al. 2008), about 0.1 solar at $z>2$ (to be
compared to $\sim 0.03$ solar for intervening DLAs). Note that molecules (H2,
CO) have been detected in a number of QSO-DLAs (Noterdaeme et al. 2008, 2011,
Srianand et al. 2008), whereas they are rarely seen in GRB-DLAs (Ledoux et al.
2009). This may be a consequence of small number statistics however (see Fynbo
et al. 2006 and Prochaska et al. 2009) and possibly of inadequate data as high
spectral resolution in the blue is usually needed (see Ledoux et al. 2009).
If these differences exist when comparing the overall populations, they may
not be that apparent if we restrict ourselves to high H i column density
systems. This is why it is interesting to compare the properties of SDSS J
081634$+$144612 with those of GRB-DLAs. In Fig. 7, we plot the logarithm of
the H i column density versus metallicity for QSO-DLAs where H2 is not
detected (open circles), QSO-DLAs where H2 is detected (filled circles), the
same for GRB-DLAs (squares) and SDSS J 081634$+$144612 (filled diamond). It
can be seen that there is a lack of systems with both a high metallicity and a
high column density. This is well known for QSO-DLAs and could be a
consequence of these DLAs being missed because of the high induced attenuation
which makes the QSO drop out of the sample (e.g. Boissé et al. 1998). Altough
GRB afterglows are for a little while much brighter than quasars, the presence
of a dust-bias could also affect their statistics (see e.g. Fynbo et al. 2009;
Ledoux et al. 2009; and Greiner et al. 2011). A possibility is that GRB-DLA
metallicities could be higher than measured as often only lower limits are
derived from intermediate resolution observations (see Prochaska 2006,
Petitjean & Vergani, 2011). This can however probably not explain this lack of
systems completely. SDSS J 081634$+$144612 is located at the limit of the
region where systems are missing and within the region where GRB-DLAs are
located. In Fig. 8, the logarithm of the H i column density is plotted for the
same systems versus the depletion of iron onto dust grains. We have scaled the
depletion of chromium in SDSS J 081634$+$144612\. It can be seen here again
that the DLA towards SDSS J 081634$+$144612 is well within the region where
GRB-DLAs are located. Therefore, our study indicates that extremely large H i
column density DLAs found toward QSOs have properties similar to those of GRB-
DLAs.
Following Schaye (2001), Krumholz et al (2009) proposed a radiation-
insensitive model –hence applying to both QSO and GRB-DLAs– where the lack of
high-$N$(H i), high-metallicity is explained by the conversion from atomic to
molecular gas. Here, the physical conditions in the $H_{2}$-bearing cloud
indicate that we are still observing diffuse gas. This is not in tension with
the above model since dense, cold and molecular gas is expected to have a
small cross section and is not easily intercepted by the line-of-sight.
Interestingly, a fully molecular cloud has been observed in the case of the
$\log N$(H i) = 22.7 DLA associated to GRB080607 (Prochaska et al. 2009),
which could be observed very quickly ($<$1 h) after the burst. Therefore,
while the absorption properties of high-$N$(H i) GRB and QSO-DLAs appear to be
similar, detecting fully molecular clouds will remain challenging in the case
of QSOs, because of the random distribution of the lines-of-sight and the high
induced extinction.
Very little is known about GRB host galaxies at high redshift ($z>2$, Savaglio
et al. 2009). Cosmological simulations show that GRB-DLAs are predominantly
associated with haloes of mass 10${}^{10}<M_{\rm vir}$/$M_{\odot}<10^{12}$, an
order of magnitude larger than the galaxies responsible for the bulk of QSO-
DLAs (Pontzen et al. 2010; see however Barnes & Haehnelt 2010). But what is
true for the overall QSO-DLA population does not hold for its high column
density end. We have shown here that DLAs with the highest H i column
densities, seem to have absorption properties similar to that of GRB-DLAs. It
would be most interesting to search for emission lines associated to these
DLAs which have probably small impact parameters (see e.g. Noterdaeme et al.
2012)333The galaxy responsible for the recently found $\log N($H i$)\sim 22$
DLA towards J 1135$-$0010 was detected at $b\approx 0.1$ arcsec from the
background QSO (Noterdaeme et al. 2012). In this particular case, Kukarni et
al. (2012) demonstrated that detecting the Lyman-$\alpha$ emission is even
possible from UVES data. Here, the DLA is covered at the edge of two UVES
echelle orders and possibly contaminated by scattered light in the red arm,
preventing us from putting any meaningful limit on the Ly-$\alpha$ flux
–within the slit. Follow-up observations specifically tuned to the search of
emission lines is thus desirable (see e.g. Fynbo et al. 2010, 2011, Péroux et
al. 2012).
Note that the above conclusion is probably not very surprising as high H i
column densities likely arise in gas located in the inner regions of star
forming galaxies of moderate attenuation. This idea should be investigated
with larger samples of both QSO and GRB-DLAs with high column densities. The
BOSS survey (Eisenstein et al. 2011) will soon increase the number of such
QSO-DLAs by an order of magnitude.
We thank the anonymous referee for helpful comments and suggestions which
improved this paper. PPJ and RS acknowledge the support of the Indo-French
Centre for the Promotion of Advanced Research (Centre Franco-Indien pour la
Promotion de la Recherche Avancée) under contract no. 4304-2. S. L. is
supported by FONDECYT grant No. 1100214.
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Figure 1: H i Lyman-$\alpha$ profile of the DLA at $z=3.286$ toward QSO SDSS J 081634$+$144612\. The overplotted solid line and accompanying dashed lines correspond to the best fit solution log $N$(H i) = 22.0$\pm$0.10. The origin of the velocity scale is taken at $z=3.287$. Figure 2: Observed absorption profiles of C i, C i*, Zn ii, Fe ii, Cr ii, Ni ii, Si ii together with fitted profiles. The dashed vertical lines indicate the positions of the velocity components. The origin of the velocity scale is taken at $z=3.287456$, the redshift of the C i component. Regions affected by blends (see Fe ii lines as well as the reddest clump of Cr ii$\lambda$2066) are not considered in the fitting process. Additional absorption seen at +150 km s-1 the C i⋆ $\lambda$1560.6 panel and in the red wing of C i$\lambda$1656 and C i⋆ $\lambda$1657.9 are due to C i⋆⋆. Figure 3: Depletion pattern of elements relative to zinc measured in individual components in the DLA toward SDSS J 081634$+$144612\. The points are slightly shifted along the x-axis according to their position in the system (for clarity purpose only). The filled circle indicates the component associated with the strongest H2 absorption. The dotted, dashed and continuous lines represent the typical relative abundances observed in, respectively, cold or warm gas in the Galactic disk and diffuse gas in the Galactic Halo, from Welty et al. (1999). Figure 4: Left: The SDSS spectrum of J 081634$+$144612 (black) is shown together with the unreddened SDSS composite (grey) and the same reddened using a SMC extinction law and E(B-V) = 0.05 (red). Right: Distribution of E(B-V) measured for a control sample of 163 QSOs within $\Delta z=\pm 0.02$ from SDSS J 081634$+$144612\. The arrow indicate the position of the latter. Figure 5: Portions of the SDSS J 081634$+$144612 normalized spectrum. The best fit to the H2 absorption lines in several Lyman bands (0-0 to 4-0) is superimposed onto the spectrum, with the corresponding uncertainty represented by the shaded area. The fit parameters are given in Table 2. The blue and green labels indicate the branch and the rotational level for the bluest and reddest component of H2, respectively. Figure 6: H2 excitation diagram for the 2 components observed toward SDSS J 081634$+$144612\. Triangles correspond to $z_{\rm abs}$ = 3.28667 and squares to $z_{\rm abs}$ = 3.28742. The column density $N_{\rm J}$ divided by the statistical weight, $g_{\rm J}$, is plotted for the J = 0 up to J = 3 H2 rotational levels on a logarithmic scale against the excitation energy, $E_{\rm J}$, in K. The J=0 to J=2 points have been fitted with straight lines using Eq. 1 giving the excitation temperatures $T_{\rm ex}^{012}$ indicated in Table 2. Figure 7: Logarithm of the total neutral hydrogen column density versus metallicity. Circles correspond to QSO-DLAs (Noterdaeme et al. 2008), squares to GRB-DLAs (Prochaska et al. 2009), triangles to GRB-DLAs(Ledoux et al. 2009), inverted triangle to the QSO-DLA towards SDSS J 113520$+$001053 (Noterdame et al. 2012) and the diamond indicates our measurement for the QSO-DLA towards SDSS J 081634$+$144612\. Filled symbols indicate systems in which H2 is detected, the filled square corresponds to the GRB080607 (Prochaska et al. 2009). Figure 8: Logarithm of the total neutral hydrogen column density versus depletion. Circles correspond to QSO-DLAs (Noterdaeme et al. 2008), squares to GRB-DLAs (Prochaska et al. 2009), triangles to GRB-DLAs (Ledoux et al. 2009), inverted triangle to the QSO-DLA towards SDSS J 113520$+$001053 (Noterdame et al. 2012) and the diamond indicates our measurement for the QSO-DLA towards SDSS J 081634$+$144612\. Filled symbols indicate systems in which H2 is detected, the filled square corresponds to the GRB080607 (Prochaska et al. 2009) . Table 1: Results of Voigt profile fitting analysis. $z$ | $v$ | $b$ | $\log N$ (cm-2)
---|---|---|---
| (km s-1) | (km s-1) | C i | C i⋆ | C i⋆⋆ | Si ii | Cr ii | Ni ii | Zn ii | Fe ii
3.286481 | -68 | 8.1$\pm$0.2 | | | | 15.73$\pm$0.03 | 13.43$\pm$0.02 | 13.87$\pm$0.01 | 13.04$\pm$0.02 | 15.09$\pm$0.04
3.286746 | -50 | 8.2$\pm$0.3 | | | | 15.99$\pm$0.04 | 13.71$\pm$0.02 | 14.15$\pm$0.01 | 13.12$\pm$0.02 | 15.48$\pm$0.03
3.286998 | -32 | 7.4$\pm$1.7 | | | | 14.90$\pm$0.10 | 12.68$\pm$0.11 | 13.18$\pm$0.10 | 12.19$\pm$0.10 | 14.79$\pm$0.04
3.287260 | -14 | 8.7$\pm$1.0 | | | | 14.99$\pm$0.05 | 12.83$\pm$0.06 | 13.44$\pm$0.04 | 12.40$\pm$0.05 | 14.85$\pm$0.04
3.287456 | 0 | 7.2$\pm$0.2 | 13.43$\pm$0.01 | 13.24$\pm$0.02 | 12.47$\pm$0.07 | 14.90$\pm$0.04 | 12.63$\pm$0.08 | 13.33$\pm$0.03 | 12.40$\pm$0.07 | 14.57$\pm$0.06
3.288098 | +45 | 20.6$\pm$0.8 | | | | 15.29$\pm$0.04 | 13.23$\pm$0.05 | 13.73$\pm$0.03 | 12.45$\pm$0.06 | 14.91$\pm$0.07
3.288169 | +50 | 4.4$\pm$0.6 | | | | 14.89$\pm$0.07 | 12.69$\pm$0.09 | 13.23$\pm$0.05 | 11.73$\pm$0.16 | 14.41$\pm$0.13
3.288345 | +62 | 6.1$\pm$3.7 | | | | 14.30$\pm$0.25 | | 12.30$\pm$0.48 | 11.34$\pm$0.38 | 14.82$\pm$0.05
Total | 13.43$\pm$0.01 | 13.24$\pm$0.02 | 12.47$\pm$0.07 | 16.31$\pm$0.01 | 14.07$\pm$0.02 | 14.54$\pm$0.01 | 13.53$\pm$0.01 | 15.89$\pm$0.02
Table 2: Molecular hydrogen column densities and excitation temperatures
$z_{\rm abs}$ | $v$ 1 | J | log $N($H2,J) 2 | $b$ | $T_{\rm ex}$
---|---|---|---|---|---
| (km s-1) | | (cm-2) | (km s-1) | (K)
3.28667 3 | 55.0 | 0.0 | 16.59$\pm$0.50 | $2_{-1}^{+2}$ | $T_{\rm ex}^{012}=69_{-8}^{+10}$
| | 1 | 17.55 $\pm$0.30 | $2_{-1}^{+2}$
| | 2 | 14.71 $\pm$0.20 | $8.3_{-0.4}^{+0.4}$
| | 3 | 15.13 $\pm$0.10 | $6.5_{-0.4}^{+0.4}$ | $T_{\rm ex}^{03}=158_{-28}^{+44}$
3.28742 4 | 2.5 | 0 | 18.19 $\pm$0.35 | $2_{-1}^{+2}$ | $T_{\rm ex}^{012}=79_{-10}^{+14}$
| | 1 | 18.41 $\pm$0.20 | $2_{-1}^{+2}$
| | 2 | 16.21 $\pm$0.25 | $8.3_{-0.4}^{+0.4}$
| | 3 | 15.75 $\pm$0.10 | $6.5_{-0.4}^{+0.4}$ | $T_{\rm ex}^{03}=117_{-12}^{+16}$
1 With respect to C i.
2 The errors on the H2 column densities correspond to the best fits using the
range of $b$-values.
3 Total log $N$(H2)= 17.60.
4 Total log $N$(H2)= 18.62.
|
arxiv-papers
| 2012-05-10T17:45:16 |
2024-09-04T02:49:30.813081
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Guimar\\~aes, P. Noterdaeme, P. Petitjean, C. Ledoux, R. Srianand,\n S. Lopez and H. Rahmani",
"submitter": "Rodney Guimar\\~aes",
"url": "https://arxiv.org/abs/1205.2322"
}
|
1205.2412
|
# Divergence form nonlinear nonsmooth elliptic equations with locally
arbitrary growth conditions and nonlinear maximal regularity
Qiao-fu Zhang
(Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, P. R. China)
###### Abstract
This is a simplification of our prior work on the existence theory for the
Rosseland-type equations. Inspired by the Rosseland equation in the
conduction-radiation coupled heat transfer, we use the locally arbitrary
growth conditions instead of the common global restricted growth conditions.
In the Lebesgue square integrable space, the solution to the linear elliptic
equation depends continuously on the coefficients matrix. This is a simple
version of the maximal regularity. There exists a fixed point for the
linearized map (compact and continuous) in a closed convex set.
Key words: growth conditions; nonlinear maximal regularity; nonlinear elliptic
equations; nonsmooth data; Rosseland equation;
## 1 Introduction
Consider the following elliptic problem: find $u$, $(u-u_{b})\in
H^{1}_{0}(\Omega)$, such that
$-\mbox{div}[A(u(x),x)\nabla u]=0,\quad\mbox{in\,}\,\Omega.$ (1.1)
For the Rosseland equation: $A(z,x)=K(x)+z^{3}B(x)$, where $K(x)$ and $B(x)$
are symmetric and positive definite.
(1) $K(x)+z^{3}B(x)$ is positive definite only in an interval for $z$.
(2) it doesn’t satisfy the common growth and smooth conditions and there may
be no $C^{2,\gamma}$ estimate (Theorem 15.11 [1]).
The problem of the existence theory for the Rosseland equation (also named
diffusion approximation) was proposed by Laitinen [6] in 2002. It may be
useful to keep this equation in mind while reading this paper.
It’s a little technical to prove the existence by the fixed point method in
$L^{\infty}(\Omega)$ [7, 8]. We will use $L^{2}(\Omega)$ in this paper.
Firstly, we make the following assumptions.
(A1) $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain.
(A2) $A=(a_{ij})$. $a_{ij}=a_{ji}$. $T_{min}\leq T_{max}$ are two constants.
$\lambda|\xi|^{2}\leq a_{ij}(z,x)\xi_{i}\xi_{j}\leq\Lambda|\xi|^{2},\quad
0<\lambda\leq\Lambda,$ (1.2)
$\forall\,\,(z,x,\xi)\in[T_{min},T_{max}]\times\Omega\times\mathbb{R}^{n}.$
(1.3)
(A3)
$u_{b}\in H^{1}(\Omega).\quad T_{min}\leq u_{b}(x)\leq
T_{max},\quad\mbox{a.\,e.\,\,in}\,\,\,\partial\Omega.$ (1.4)
(A4) $A(z,x)$ is uniformly continuous with respect to $z$ in $\mathfrak{C}$,
where
$\mathfrak{C}=\\{\varphi\in L^{2}(\Omega);\,\,T_{min}\leq\varphi(x)\leq
T_{max},\,\,\mbox{a.\,e.\,\, in}\,\,\Omega\\};$ (1.5)
In other words, if $z_{i},\,z\in\mathfrak{C}$, $\|z_{i}-z\|_{2}\to 0$,
$\sup_{1\leq p,q\leq n}\|a_{pq}(z_{i}(x),x)-a_{pq}(z(x),x)\|_{2}\to 0.$ (1.6)
###### Remark 1.1
In fact, we had considered a general case: parabolic equations with bounded
mixed boundary conditions and nonnegative bounded right-hand term $f(z,x)$ in
[8].
If $a_{pq}$ is uniformly Hölder continuous with respect to $z$, (A4) is
natural since
$\displaystyle\|a_{pq}(z_{i}(x),x)-a_{pq}(z(x),x)\|_{2}^{2}$
$\displaystyle\leq$ $\displaystyle\int_{\Omega}C|z_{i}(x)-z(x)|^{2\alpha}$
(1.7) $\displaystyle\leq$ $\displaystyle C\|z_{i}-z\|_{2}^{2}\to 0.$
## 2 Linearized map and fixed point
###### Theorem 2.1
$($Corollary 11.2 [1]$)$ Let $\mathfrak{C}$ be a closed convex set in a Banach
space $\mathfrak{B}$ and let $\mathcal{L}$ be a continuous mapping of
$\mathfrak{C}$ into itself such that the image $\mathcal{L}\mathfrak{C}$ is
precompact. Then $\mathcal{L}$ has a fixed point.
###### Lemma 2.1
The following set
$\mathfrak{C}=\\{\varphi\in L^{2}(\Omega);\,\,T_{min}\leq\varphi(x)\leq
T_{max},\,\,a.\,e.\,\,in\,\,\Omega\\}$ (2.8)
is a closed convex set in the Banach space $L^{2}(\Omega)$.
Proof Suppose $v_{i}\in\mathfrak{C}$, $v\in L^{2}(\Omega)$,
$\|v_{i}-v\|_{2}\to 0$. If $v\notin\mathfrak{C}$, there exist two constants
$\delta_{0}>0$, $\delta_{1}>0$, such that the Lebesgue measure of the set
$\Omega_{0}\equiv\\{x\in\Omega;\,v(x)\geq T_{max}+\delta_{0}\\}$ is bigger
than $\delta_{1}>0$. Then
$\|v_{i}-v\|_{2}^{2}=\int_{\Omega}|v_{i}-v|^{2}\geq\int_{\Omega_{0}}|v_{i}-v|^{2}\geq\delta_{0}^{2}\delta_{1}.$
(2.9)
It’s impossible since $\|v_{i}-v\|_{2}\to 0$. Similarly, $v\geq T_{min}$ and
$\mathfrak{C}$ is closed.
$\forall\,\theta\in[0,1],\quad\theta v_{1}+(1-\theta)v_{2}\leq\theta
T_{max}+(1-\theta)T_{max}=T_{max}.$ (2.10)
So $\mathfrak{C}$ is convex. $\square$
###### Theorem 2.2
If $(A1)-(A4)$ are satisfied, then
$(1)$ $\forall\,z\in\mathfrak{C}$, the following equation has a unique
solution $w\in\mathfrak{C}$, $(w-u_{b})\in H^{1}_{0}(\Omega)$.
$\int_{\Omega}A(z(x),x)\nabla w\cdot\nabla\varphi=0,\quad\forall\,\varphi\in
H^{1}_{0}(\Omega).$ (2.11)
$(2)$ Define a map $\mathcal{L}:\,\mathfrak{C}\to\mathfrak{C}$,
$\mathcal{L}z=w$, then $\mathcal{L}\mathfrak{C}$ is precompact in
$L^{2}(\Omega)$.
$(3)$ $\mathcal{L}$ is continuous in $L^{2}(\Omega)$. So $\mathcal{L}$ has a
fixed point in $\mathfrak{C}$.
Proof (1) Let $v=(w-u_{b})\in H^{1}_{0}(\Omega)$, then we have
$\int_{\Omega}A(z(x),x)\nabla
v\cdot\nabla\varphi=-\int_{\Omega}A(z(x),x)\nabla
u_{b}\cdot\nabla\varphi,\quad\forall\,\varphi\in H^{1}_{0}(\Omega).$ (2.12)
From (A2), if $z\in\mathfrak{C}$, $A(z(x),x)\in[\lambda,\Lambda]$. From (A3),
$|\nabla u_{b}|\leq C$. Let $\varphi=v$,
$\displaystyle\lambda|v|_{1}^{2}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}A\nabla v\cdot\nabla v=-\int_{\Omega}A\nabla
u_{b}\cdot\nabla v$ (2.13) $\displaystyle\leq$ $\displaystyle\|A\nabla
u_{b}\|_{2}\|\nabla v\|_{2}=(\int_{\Omega}A^{\top}A\nabla u_{b}\cdot\nabla
u_{b})^{1/2}\|\nabla v\|_{2}$ $\displaystyle\leq$
$\displaystyle\Lambda\|u_{b}\|_{H^{1}(\Omega)}|v|_{1}.$
Using the well-known Lax-Milgram Lemma, there exists a unique solution $v\in
H^{1}_{0}(\Omega)$ to this equation. Using the Poincaré inequality,
$\displaystyle\|w\|_{H^{1}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\|w-u_{b}\|_{H^{1}(\Omega)}+\|u_{b}\|_{H^{1}(\Omega)}$ (2.14)
$\displaystyle\leq$ $\displaystyle
C(\Omega)\frac{\Lambda\|u_{b}\|_{H^{1}(\Omega)}}{\lambda}+\|u_{b}\|_{H^{1}(\Omega)}.$
Using the maximum principle (Theorem 8.1 [1]), $u\in\mathfrak{C}$. In fact, we
can let $\varphi=(w-T_{max})_{+}\in H^{1}_{0}(\Omega)$,
$\displaystyle C(\Omega)\lambda\|(w-T_{max})_{+}\|_{1}^{2}$
$\displaystyle\leq$ $\displaystyle\lambda|(w-T_{max})_{+}|_{1}^{2}$ (2.15)
$\displaystyle\leq$
$\displaystyle\int_{\Omega}A(z(x),x)\nabla(w-T_{max})_{+}\cdot\nabla(w-T_{max})_{+}$
$\displaystyle=$ $\displaystyle\int_{\Omega}A(z(x),x)\nabla
w\cdot\nabla(w-T_{max})_{+}=0.$
So $(w-T_{max})_{+}=0$, $w\leq T_{max}$. In the same way, $w\leq T_{min}$.
(2) $\|w\|_{H^{1}(\Omega)}\leq C$. $\mathcal{L}\mathfrak{C}$ is bounded in
$H^{1}(\Omega)$. From the Rellich Theorem, $\mathcal{L}\mathfrak{C}$ is
precompact in $L^{2}(\Omega)$.
(3) Suppose
$z_{i},\,z\in\mathfrak{C},\quad\|z_{i}-z\|_{2}\to
0,\quad\mathcal{L}z_{i}=w_{i},\quad\mathcal{L}z=w.$ (2.16)
$H^{1}_{0}(\Omega)$ is a Hilbert and then a reflexive space, so there exists a
subsequence $\\{i_{k}\\}$ and $v_{0}=(w_{0}-u_{b})\in H^{1}_{0}(\Omega)$ such
that
$(w_{i_{k}}-u_{b})\to(w_{0}-u_{b}),\quad\mbox{weakly\,\,in\,\,}H^{1}_{0}(\Omega).$
(2.17) $H^{1}_{0}(\Omega)\subset
L^{2}(\Omega),\quad(L^{2}(\Omega))^{\prime}\subset(H^{1}_{0}(\Omega))^{\prime}.$
(2.18)
$(w_{i_{k}}-u_{b})\to(w_{0}-u_{b}),\quad\mbox{weakly\,\,in\,\,}L^{2}(\Omega).$
(2.19)
$\\{w_{i_{k}}-u_{b}\\}$ is bounded in $H^{1}(\Omega)$, so there exists a
subsequence $\\{i_{m}\\}\subset\\{i_{k}\\}$ and $v_{*}\in L^{2}(\Omega)$ such
that
$(w_{i_{m}}-u_{b})\to v_{*},\quad\mbox{strongly\,\,in\,\,}L^{2}(\Omega).$
(2.20) $(w_{i_{m}}-u_{b})\to v_{0},\quad\mbox{weakly\,\,in\,\,}L^{2}(\Omega).$
(2.21)
So $v_{*}=v_{0}$. Since each subsequence of
$\\{\|w_{i_{k}}-u_{b}-v_{0}\|_{2}\\}$ has a sub-subsequence which converges to
0, $\|w_{i_{k}}-u_{b}-v_{0}\|_{2}\to 0$, $\|w_{i_{k}}-w_{0}\|_{2}\to 0$.
Since
$(w_{i_{k}}-u_{b})\to(w_{0}-u_{b}),\quad\mbox{weakly\,\,in\,\,}H^{1}_{0}(\Omega).$
(2.22) $\forall\,\overrightarrow{\psi}\in
L^{2}(\Omega;\,\mathbb{R}^{n}),\quad\langle\overrightarrow{\psi},h\rangle_{H^{1}_{0}}\equiv\int_{\Omega}\nabla
h\cdot\overrightarrow{\psi},\quad\forall\,h\in H^{1}_{0}(\Omega),$ (2.23)
is a bounded linear functional.
$\int_{\Omega}\nabla(w_{i_{k}}-u_{b})\cdot\overrightarrow{\psi}\to\int_{\Omega}\nabla(w_{0}-u_{b})\cdot\overrightarrow{\psi}.$
(2.24) $\forall\,\overrightarrow{\psi}\in
L^{2}(\Omega;\,\mathbb{R}^{n}),\quad\int_{\Omega}\nabla
w_{i_{k}}\cdot\overrightarrow{\psi}\to\int_{\Omega}\nabla
w_{0}\cdot\overrightarrow{\psi}.$ (2.25)
From the Riesz representation theorem, the dual space
$(L^{2}(\Omega;\,\mathbb{R}^{n}))^{\prime}\simeq
L^{2}(\Omega;\,\mathbb{R}^{n}),$ (2.26) $\nabla w_{i_{k}}\to\nabla
w_{0},\quad\mbox{weakly\,\,in\,\,}L^{2}(\Omega;\,\mathbb{R}^{n}).$ (2.27)
From (A4) and $\|z_{i}-z\|_{2}\to 0$,
$\sup_{1\leq p,q\leq n}\|a_{pq}(z_{i_{k}}(x),x)-a_{pq}(z(x),x)\|_{2}\to 0.$
(2.28)
We can conclude that, $\forall\,\phi\in C^{\infty}_{0}(\Omega)$,
$\displaystyle|\int_{\Omega}[A(z_{i_{k}}(x),x)\nabla w_{i_{k}}-A(z(x),x)\nabla
w_{0}]\cdot\nabla\phi|$ (2.29) $\displaystyle\leq$
$\displaystyle|\int_{\Omega}[A(z_{i_{k}}(x),x)\nabla w_{i_{k}}-A(z(x),x)\nabla
w_{i_{k}}]\cdot\nabla\phi|$ $\displaystyle\,+\,|\int_{\Omega}[A(z(x),x)\nabla
w_{i_{k}}-A(z(x),x)\nabla w_{0}]\cdot\nabla\phi|$ $\displaystyle=$
$\displaystyle|\int_{\Omega}[A(z_{i_{k}}(x),x)-A(z(x),x)]\nabla
w_{i_{k}}\cdot\nabla\phi|$ $\displaystyle\,+\,|\int_{\Omega}[\nabla
w_{i_{k}}-\nabla w_{0}]\cdot A(z(x),x)^{\top}\nabla\phi|$ $\displaystyle\leq$
$\displaystyle C\sup_{1\leq p,q\leq
n}\|a_{pq}(z_{i_{k}}(x),x)-a_{pq}(z(x),x)\|_{2}+\epsilon(i_{k})\to 0.$
$\int_{\Omega}A(z_{i_{k}}(x),x)\nabla
w_{i_{k}}\cdot\nabla\phi=0,\quad\int_{\Omega}A(z(x),x)\nabla
w_{0}\cdot\nabla\phi=0.$ (2.30) $\int_{\Omega}A(z(x),x)\nabla
w_{0}\cdot\nabla\varphi=0,\quad\forall\,\varphi\in H^{1}_{0}(\Omega).$ (2.31)
Since the solution is unique from the step (1), $w_{0}=\mathcal{L}z=w$. So
$\|w_{i_{k}}-w\|_{2}\to 0$. Each subsequence of $\\{\|w_{i}-w\|_{2}\\}$ has a
sub-subsequence which converges to $0$, so $\|w_{i}-w\|_{2}\to 0$. We have
obtain the continuity of $\mathcal{L}$.
From Theorem 2.1, there exists a fixed point. $\square$
###### Remark 2.1
For the continuity of $\mathcal{L}$ in $C^{0}(\overline{\Omega})$, we can use
the well-known De Giorgi-Nash estimate: $\\{w_{i}\\}$ is bounded in
$C^{0,\alpha}(\overline{\Omega})$ if $u_{b}\in
C^{0,\alpha_{0}}(\partial\Omega)$ and $\Omega$ satifies a uniform exterior
cone condition (Theorem 8.29 [1]).
Then from the Arzel$\grave{\rm{a}}$-Ascoli Lemma,
$\|w_{i_{k}}-w_{0}\|_{C^{0}(\overline{\Omega})}\to 0$. By the same method,
$w_{0}=w$ and $\|w_{i}-w\|_{C^{0}(\overline{\Omega})}\to 0$.
From the linear maximal regularity [4], a natural conjecture is: $\mathcal{L}$
is continuous in $C^{0,\alpha}(\overline{\Omega})$ and $H^{1}(\Omega)$.
###### Definition 2.1
$\mathfrak{C}_{\infty}=\\{\varphi\in
L^{\infty}(\Omega);\,\,T_{min}\leq\varphi(x)\leq
T_{max},\,\,a.\,e.\,\,in\,\,\Omega\\}$ (2.32)
is a closed convex set in the Banach space $L^{\infty}(\Omega)$.
We replace $(A4)$ in $L^{2}$ with $(A4^{\prime})$ in $L^{\infty}:$
$\|A(z_{i}(x),x)-A(z(x),x)\|_{\infty}\to
0,\quad\mbox{if\,\,}\|z_{i}(x)-z(x)\|_{\infty}\to 0.$ (2.33)
###### Corollary 2.1
If $(A1)-(A3),\,\,(A4^{\prime})$ are satisfied, define a map
$\mathcal{L}:\,\mathfrak{C}_{\infty}\to\mathfrak{C}_{\infty}$,
$\mathcal{L}z=w:$ such that $w\in\mathfrak{C}_{\infty}$, $(w-u_{b})\in
H^{1}_{0}(\Omega)$ and
$\int_{\Omega}A(z(x),x)\nabla w\cdot\nabla\varphi=0,\quad\forall\,\varphi\in
H^{1}_{0}(\Omega).$ (2.34)
Then $\mathcal{L}$ is continuous in $H^{1}(\Omega)$.
Proof Suppose
$z_{i},\,z\in\mathfrak{C}_{\infty},\quad\|z_{i}-z\|_{\infty}\to
0,\quad\mathcal{L}z_{i}=w_{i},\quad\mathcal{L}z=w.$ (2.35)
(1) $\\{w_{i}\\}$ is bounded in $H^{1}(\Omega)$. For any
$\overrightarrow{\psi}\in L^{2}(\Omega;\,\mathbb{R}^{n})$, each subsequence of
$\int_{\Omega}\nabla(w_{i}-w)\cdot\overrightarrow{\psi}$ has a sub-subsequence
converges to 0. So $\nabla w_{i}\to\nabla w$ weakly in
$L^{2}(\Omega;\,\mathbb{R}^{n})$. $\|A(z_{i}(x),x)-A(z(x),x)\|_{\infty}\to 0$,
$\nabla u_{b}\in
L^{2}(\Omega;\,\mathbb{R}^{n}),\quad\int_{\Omega}A(z_{i})\nabla
w_{i}\cdot\nabla u_{b}\to\int_{\Omega}A(z)\nabla w\cdot\nabla u_{b}.$ (2.36)
$\int_{\Omega}A(z_{i})\nabla w_{i}\cdot\nabla
u_{b}=\int_{\Omega}A(z_{i})\nabla w_{i}\cdot\nabla w_{i},$ (2.37)
$\int_{\Omega}A(z)\nabla w\cdot\nabla u_{b}=\int_{\Omega}A(z)\nabla
w\cdot\nabla w.$ (2.38) $\int_{\Omega}A(z_{i})\nabla w_{i}\cdot\nabla
w_{i}\to\int_{\Omega}A(z)\nabla w\cdot\nabla w.$ (2.39)
$\int_{\Omega}[A(z_{i})-A(z)]\nabla w_{i}\cdot\nabla w_{i}\leq
C\|A(z_{i}(x),x)-A(z(x),x)\|_{\infty}\to 0.$ (2.40) $\int_{\Omega}A(z)[\nabla
w\cdot\nabla w-\nabla w_{i}\cdot\nabla w_{i}]\to 0.$ (2.41)
$\displaystyle\int_{\Omega}|\nabla w_{i}-\nabla w|^{2}$ (2.42)
$\displaystyle\leq$ $\displaystyle\frac{1}{\lambda}\int_{\Omega}A(z)[\nabla
w_{i}-\nabla w]\cdot[\nabla w_{i}-\nabla w]$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\int_{\Omega}A(z)[\nabla w\cdot\nabla w+\nabla
w_{i}\cdot\nabla w_{i}-2\nabla w\cdot\nabla w_{i}]$ $\displaystyle\to$
$\displaystyle 0.$
$(w_{i}-w)\in H^{1}_{0}(\Omega)$, then use the Poincaré inequality,
$\|w_{i}-w\|_{H^{1}}\to 0$. $\square$
## 3 Nonlinear maximal regularity
For the linear parabolic/ellptic equations with nonsmooth data, the theory of
maximal regularity has been established [2, 3, 4, 5]. In brief, maximal
regularity is about the smoothness of the data-to-solution-map [5]. This
smooth dependence has its physical meaning: many physical processes are stable
with respect to the parameters (except the chaos and critical theory). For the
mathematicians, ”the door is open to apply the powerful theorems of
differential calculus”([5], e.g. the Implicit Function Theorem).
In the following, we will discuss the continuous dependence (between the
solutions and the data) for the following kind of nonlinear equations: find
$u$, $(u-u_{b})\in H^{1}_{0}(\Omega)$, such that
$\int_{\Omega}A(u(x),x)\nabla u\cdot\nabla\varphi=0,\quad\forall\,\varphi\in
H^{1}_{0}(\Omega).$ (3.43)
In this section, we strengthen (A4) with the following equicontinuous
conditions.
(A4.1) Each element of $\mathfrak{A}(z,x)$ satisfies (A2):
$A(z,x)|_{\mathfrak{C}\times\Omega}\in[\lambda,\Lambda],\quad\forall\,A(z,x)\in\mathfrak{A}(z,x).$
(3.44) $\mathfrak{C}=\\{\varphi\in L^{2}(\Omega);\,\,T_{min}\leq\varphi(x)\leq
T_{max},\,\,\mbox{a.\,e.\,\, in}\,\,\Omega\\}.$ (3.45)
(A4.2) $\mathfrak{A}(z,x)$ is equicontinuous with respect to $z$ in
$\mathfrak{C}$ (uniformly for $x$). In other words, for any
$z_{i},\,z\in\mathfrak{C}$, if $\|z_{i}-z\|_{2}\to 0$,
$\sup_{A=(a_{pq})\in\mathfrak{A}(z,x)}\sup_{1\leq p,q\leq
n}\|a_{pq}(z_{i}(x),x)-a_{pq}(z(x),x)\|_{2}\leq\epsilon(i).$ (3.46)
$\epsilon(i)$ denotes a higher-order infinitesimal which depends only on $i$.
###### Theorem 3.1
If $(A1)-(A3),\,\,(A4.1)-(A4.2)$ are satisfied and the solution to the
nonlinear equation is unique, then
$(1)$ $u$ depends continuously $($in $L^{2}$ or $C^{0}(\overline{\Omega}))$ on
the coefficients matrix $A(\cdot,x)$ in $\mathfrak{A}(z,x)$.
$(2)$ $u$ depends continuously $($in $L^{2}$ or $C^{0}(\overline{\Omega}))$ on
the boundary value $u_{b}$.
$(3)$ $A(z,x)\nabla z$ is a continuous $($with respect to $z$ in
$\mathfrak{C}\cap H^{1}(\Omega))$ functional in
$(C^{0,1}(\overline{\Omega}))^{\prime}$.
Proof (1) Suppose $A_{i}(\cdot,x)\in\mathfrak{A}(\cdot,x)$, then there exists
a $u_{i}$, $(u_{i}-u_{b})\in H^{1}_{0}(\Omega)$, such that
$\int_{\Omega}A_{i}(u_{i}(x),x)\nabla
u_{i}\cdot\nabla\varphi=0,\quad\forall\,\varphi\in H^{1}_{0}(\Omega).$ (3.47)
We will prove that $\|u-u_{i}\|_{2}\to 0$ (or
$\|u-u_{i}\|_{C^{0}(\overline{\Omega})}\to 0$) if
$\|A(z(x),x)-A_{i}(z(x),x)\|_{2}\to 0$ for any $z(x)\in\mathfrak{C}$.
$\\{u_{i}\\}$ is bounded in $H^{1}(\Omega)$ (or in
$C^{0,\alpha}(\overline{\Omega})$). So $u_{i_{k}}\to u_{0}$, strongly in
$L^{2}(\Omega)$ (or in $C^{0}(\overline{\Omega})$); $\nabla u_{i_{k}}\to\nabla
u$, weakly in $L^{2}(\Omega;\,\mathbb{R}^{n})$.
$\displaystyle\|A_{i_{k}}(u_{i_{k}}(x),x)-A(u_{0}(x),x)\|_{2}$ (3.48)
$\displaystyle\leq$
$\displaystyle\|A_{i_{k}}(u_{i_{k}}(x),x)-A_{i_{k}}(u_{0}(x),x)\|_{2}+\|A_{i_{k}}(u_{0}(x),x)-A(u_{0}(x),x)\|_{2}$
$\displaystyle\to$ $\displaystyle 0.$
$\forall\,\phi\in C^{0,\infty}(\Omega)$,
$0=\int_{\Omega}A_{i_{k}}(u_{i_{k}}(x),x)\nabla
u_{i_{k}}\cdot\nabla\phi\to\int_{\Omega}A(u_{0}(x),x)\nabla
u_{0}\cdot\nabla\phi.$ (3.49)
Because of the density,
$\int_{\Omega}A(u_{0}(x),x)\nabla
u_{0}\cdot\nabla\varphi=0,\quad\forall\,\varphi\in H^{1}_{0}(\Omega).$ (3.50)
Since the solution is unique, so $w_{i}\to w_{0}$, strongly in $L^{2}(\Omega)$
(or in $C^{0}(\overline{\Omega})$).
(2) Let $u_{bi},\,u_{b0}\in H^{1}(\Omega)$,
$\|u_{bi}-u_{b0}\|_{H^{1}(\Omega)}\to 0$. $w_{i}=u_{i}-u_{bi}\in
H^{1}_{0}(\Omega)$ such that $\forall\,\varphi\in H^{1}_{0}(\Omega)$,
$\int_{\Omega}A(w_{i}+u_{bi})\nabla
w_{i}\cdot\nabla\varphi=-\int_{\Omega}A(w_{i}+u_{bi})\nabla
u_{bi}\cdot\nabla\varphi.$ (3.51)
$\\{w_{i}\\}$ is bounded in $H^{1}(\Omega)$ (or in
$C^{0,\alpha}(\overline{\Omega})$). So $w_{i_{k}}\to w_{0}$, strongly in
$L^{2}(\Omega)$ (or in $C^{0}(\overline{\Omega})$); $\nabla w_{i_{k}}\to\nabla
w_{0}$, weakly in $L^{2}(\Omega;\,\mathbb{R}^{n})$. By the same method in (1),
$\forall\,\phi\in C^{0,\infty}(\Omega)$,
$\int_{\Omega}A(w_{0}+u_{b0})\nabla
w_{0}\cdot\nabla\phi=-\int_{\Omega}A(w_{0}+u_{b0})\nabla
u_{b0}\cdot\nabla\phi.$ (3.52)
So $w_{i}\to w_{0}$, strongly in $L^{2}(\Omega)$.
(3) For any $\eta\in C^{0,1}(\overline{\Omega})$,
$\displaystyle|\langle A(z,x)\nabla z,\eta\rangle|$ $\displaystyle=$
$\displaystyle|\int_{\Omega}A(z,x)\nabla z\cdot\nabla\eta|$ (3.53)
$\displaystyle\leq$
$\displaystyle\|\eta\|_{C^{0,1}(\overline{\Omega})}\int_{\Omega}|A(z,x)\nabla
z|$ $\displaystyle\leq$ $\displaystyle
C\|\eta\|_{C^{0,1}(\overline{\Omega})}.$
For any $z$ in $\mathfrak{C}\cap H^{1}(\Omega)$, $\langle A(z,x)\nabla
z,\eta\rangle$ is a linear continuous functional in
$(C^{0,1}(\overline{\Omega}))^{\prime}$.
Suppose $z_{i}\to z$ in $H^{1}(\Omega)$,
$\displaystyle|\langle A(z_{i},x)\nabla z_{i}-A(z,x)\nabla z,\eta\rangle|$
(3.54) $\displaystyle\leq$
$\displaystyle\|\eta\|_{C^{0,1}(\overline{\Omega})}|\int_{\Omega}|A(z_{i},x)\nabla
z_{i}-A(z,x)\nabla z|\to 0.$
$\|A(z_{i},x)\nabla z_{i}-A(z,x)\nabla
z\|_{(C^{0,1}(\overline{\Omega}))^{\prime}}\to 0$. $\square$
###### Remark 3.1
We can consider the continuous dependence in $H^{1}(\Omega)$.
## 4 Existence for Garlerkin method
Let $h\in(0,1)$ be the step size, $\\{\phi_{i,h}\\}$ is a kind of finite
element basis in $H^{1}_{0}(\Omega)$.
$\forall\,z_{h}=(\sum_{i}z_{i,h}\phi_{i,h}+u_{b})\in\mathfrak{C}$, the
following equation has a unique solution $w_{h}=\sum_{i}w_{i,h}\phi_{i,h}$,
$(w_{h}+u_{b})\in\mathfrak{C}\cap H^{1}_{0}(\Omega)$.
$\displaystyle\int_{\Omega}A(\sum_{i}z_{i,h}\phi_{i,h}+u_{b})\nabla(\sum_{i}w_{i,h}\phi_{i,h})\cdot\nabla\phi_{j,h}$
(4.55) $\displaystyle=$
$\displaystyle-\int_{\Omega}A(\sum_{i}z_{i,h}\phi_{i,h}+u_{b})\nabla
u_{b}\cdot\nabla\phi_{j,h},\quad\forall\,\phi_{j,h}.$
Define $\mathcal{L}z_{h}=(w_{h}+u_{b})$, then
(1) $\mathcal{L}\mathfrak{C}\subset\mathfrak{C}$.
(2) $\mathcal{L}\mathfrak{C}$ is compact.
(3) $\mathcal{L}$ is continuous.
(4) There exists a fixed point $u_{h}=\sum_{i}w_{i,h}\phi_{i,h}$ and $u_{h}\to
u$.
## 5 Acknowledge
This work is supported by the National Nature Science Foundation of China (No.
90916027). This is a part of my PhD thesis [8] in AMSS, Chinese Academy of
Sciences, and a simplification of our prior paper [7]. So I will thank my
advisor Professor Jun-zhi Cui (he is also a member of the Chinese Academy of
Engineering) and the referees for their careful reading and helpful comments.
My E-mail is: zhangqf@lsec.cc.ac.cn.
## References
* [1] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order, volume 224. Springer Verlag, 2001.
* [2] J.A. Griepentrog. Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Advances in Differential Equations, 12(9):1031–1078, 2007\.
* [3] J.A. Griepentrog. Sobolev-Morrey spaces associated with evolution equations. Advances in Differential Equations, 12(7):781, 2007.
* [4] J.A. Griepentrog and L. Recke. Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces. Mathematische Nachrichten, 225:39–74, 2001.
* [5] J.A. Griepentrog and L. Recke. Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems. Journal of Evolution Equations, 10(2):341–375, 2010.
* [6] M.T. Laitinen. Asymptotic analysis of conductive-radiative heat transfer. Asymptotic Analysis, 29(4):323–342, 2002.
* [7] Q.F. Zhang and J.Z. Cui. Existence theory for Rosseland equation and its homogenized equation. Applied Mathematics and Mechanics, submitted.
* [8] Q.F. Zhang. Multi-scale analysis for Rosseland equation with small periodic oscillating coefficients $($in Chinese$)$. PhD thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 2012.
|
arxiv-papers
| 2012-05-11T01:36:16 |
2024-09-04T02:49:30.821927
|
{
"license": "Public Domain",
"authors": "Zhang Qiao-fu",
"submitter": "Qiaofu Zhang",
"url": "https://arxiv.org/abs/1205.2412"
}
|
1205.2719
|
# Can the exciton–polariton regime be defined by its quantum properties?
D. G. Suárez-Forero Universidad Nacional de Colombia - Bogotá, Facultad de
Ciencias, Departamento de Física, Grupo de Óptica e Información Cuántica,
Carrera 30 Calle 45-03, C.P. 111321, Bogotá, Colombia G. Cipagauta
Universidad Nacional de Colombia - Bogotá, Facultad de Ciencias, Departamento
de Física, Grupo de Óptica e Información Cuántica, Carrera 30 Calle 45-03,
C.P. 111321, Bogotá, Colombia H. Vinck-Posada hvinckp@unal.edu.co
Universidad Nacional de Colombia - Bogotá, Facultad de Ciencias, Departamento
de Física, Grupo de Óptica e Información Cuántica, Carrera 30 Calle 45-03,
C.P. 111321, Bogotá, Colombia K. M. Fonseca-Romero Universidad Nacional de
Colombia - Bogotá, Facultad de Ciencias, Departamento de Física, Grupo de
Óptica e Información Cuántica, Carrera 30 Calle 45-03, C.P. 111321, Bogotá,
Colombia B. A. Rodríguez Instituto de Física, Universidad de Antioquia,
Medellín, A.A. 1226, Medellín, Colombia
###### Abstract
Using a simple fully quantum model in an effective exciton scheme that takes
into account the system–environment interaction, we study the different
regimes arising in a microcavity–quantum dot system. Our numerical
calculations of the emission linewidth, emission energy, integrated intensity
and second- and third-order correlation functions are in good qualitative
agreement with reported experimental results. We show that the transition from
the polariton-laser to the photon-laser regime can be defined through the
critical points of both the negativity and the linear entropy of the steady
state.
###### pacs:
42.50.ct, 78.67.Hc, 42.55.sa, 03.75.Gg
The solid–state realization of Bose–Einstein Condensation (BEC) has been
achieved in exciton–polariton systems Kasprzak _et al._ (2006); Deng _et
al._ (2006). These quasi–bosons, arising from the strong coupling between
photons and electron–hole pairs in semiconductor microcavities ($\mu$-C), have
a high critical temperature due to their small effective mass (eight orders of
magnitude smaller than hydrogen atom mass). After more than two decades of
theoretical and experimental investigations, nowadays it is understood that
the large occupation of the polariton ground state cannot be identified with
usual thermodynamic equilibrium BEC states Deng _et al._ (2010); Kasprzak
_et al._ (2006); Snoke (2006). Instead, the corresponding experimentally
observed regime has been called polariton laser Kasprzak _et al._ (2006);
Bajoni _et al._ (2008); Azzini _et al._ (2011) because of its dynamical
nature and the gain in the light–emission intensity. The transition from this
regime to a second one, identified with the well-known photon laser, has also
been observed Bajoni _et al._ (2008); Azzini _et al._ (2011); Christopoulos
_et al._ (2007).
Cavity polariton systems have been studied from different theoretical
perspectives. Assuming thermal equilibrium, a trial wave function that takes
into account the coherence properties of both light and matter, has been able
to predict multiple phase transitions Littlewood _et al._ (2004); Kamide and
Ogawa (2010). On the other hand, when the matter–light state is obtained from
some equation of motion (mean field dynamics Vinck-Posada _et al._ (2005);
Lagoudakis _et al._ (2011); Liew _et al._ (2008), master equation in
multiexcitonic scheme Vera _et al._ (2009a), dissipative Jaynes–Cummings
model Perea _et al._ (2004); Laussy _et al._ (2009); del Valle _et al._
(2009); Laussy _et al._ (2011); Vera _et al._ (2009b); del Valle _et al._
(2007); Auffèves _et al._ (2009); Laucht _et al._ (2009)), the dynamical
character of the polariton laser regime is conspicuous, and the non-Gibbsian
character of the stationary state of the system is revealed. However, the
elucidation of the mechanisms behind the appearance of the different observed
regimes is still an open problem. Our work is a first step in this direction.
The aim of this Letter, the identification of the regimes observed in current
experiments from the quantum properties of the steady state, is possible due
to the simplicity of our model. Indeed, we consider a single pumped radiator
interacting with a leaky mode of the electromagnetic field, ignoring
collective effects. Our calculations, however, are in qualitative good
agreement with the experimental results. Moreover, we can correlate the
entanglement, mixedness and the coherence functions of the steady state not
only with the observed regimes but also with important physical parameters
like the pumping rate and the detuning. In addition, we are able to provide a
criterion to identify the “best” polariton that can be sustained by the
system.
We model a quantum dot (QD) embedded in a $\mu$-C, as a two-level system
(ground $\ket{G}$ and excited $\ket{X}$ states). Its interaction with a single
electromagnetic mode of frequency $\omega_{C}$, in the dipole and rotating
wave approximations, is described by the Hamiltonian ($\hbar=1$):
$\hat{H}=\omega_{C}\hat{a}^{\dagger}\hat{a}+(\omega_{C}-\Delta)\hat{\sigma}^{\dagger}\hat{\sigma}+g(\hat{a}\hat{\sigma}^{\dagger}+\hat{a}^{\dagger}\hat{\sigma}).$
(1)
The detuning $\Delta$ is the difference between cavity mode and exciton
energies, $g$ is the matter–light coupling constant,
$\hat{\sigma}=\ket{G}\bra{X}$ is the QD ladder operator, and
$\hat{a}^{\dagger}$ ($\hat{a}$) is the usual creation (annihilation) operator
of the cavity mode. The Hamiltonian (1) commutes with the excitation number
$\hat{N}=\hat{N}_{ph}+\hat{N}_{ex}=\hat{a}^{\dagger}\hat{a}+\hat{\sigma}^{\dagger}\hat{\sigma}$;
hence, it only causes transitions between matter–light states of the same
excitation manifold. Polaritons are defined as the energy eigenstates
$\hat{H}$, and are explicitly given by
$\displaystyle\ket{n,+}$ $\displaystyle=$
$\displaystyle\sin{\Phi_{n}}\ket{G,n}+\cos{\Phi_{n}}\ket{X,n-1}$
$\displaystyle\ket{n,-}$ $\displaystyle=$
$\displaystyle\cos{\Phi_{n}}\ket{G,n}-\sin{\Phi_{n}}\ket{X,n-1},$ (2)
where $\\{\ket{n}\\}$ denotes the Fock number states of the field and
$\tan{2\Phi_{n}}=2g\sqrt{n}/\Delta$. We include two non-conservative
processes, the loss of photons in the $\mu$-C ($\kappa$) and the continuous
pumping of excitons ($P$), in the master equation for the density operator
$\hat{\rho}$ of the system
$\displaystyle\frac{d\hat{\rho}}{dt}$ $\displaystyle=$ $\displaystyle
i[\hat{\rho},\hat{H}]+\tfrac{1}{2}P(2\hat{\sigma}^{\dagger}\hat{\rho}\hat{\sigma}-\hat{\sigma}\hat{\sigma}^{\dagger}\hat{\rho}-\hat{\rho}\hat{\sigma}\hat{\sigma}^{\dagger})$
(3) $\displaystyle+$
$\displaystyle\tfrac{1}{2}\kappa(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}),$
where we have made the Born-Markov approximation. We neglect other
system–environment interaction mechanisms (e.g., spontaneous emission,
dephasing, photon pumping, polariton pumping, etc.) because their effect is
either small or is already effectively contained in the master equation. If a
better adjustment with the experimental results is desired those mechanisms
can be included and fitted Laucht _et al._ (2009), but the qualitative
physical image remains essentially unchanged.
The basic assumption behind our approach, which focuses on the steady state
$\hat{\rho}_{\textrm{ss}}$ of the equation of motion (3), is that polariton
lifetime is much longer than the time required to reach the asymptotic
solution Deng _et al._ (2006). The steady state
$\hat{\rho}_{\textrm{ss}}=\hat{\rho}(\kappa,P,g,\Delta)$ of the system, is a
function of the dissipative rates $\kappa$ and $P$, the matter-light coupling
constant $g$ and the detuning $\Delta$. In the weak-coupling regime $g\ll
P,\kappa$, the pumping keeps the QD in its excited state while the dissipation
steers the electromagnetic field to its ground state. Other states are not
significantly populated because matter excitation cannot be converted into
photons. On the other extreme, the ultra-strong coupling regime $g\ggg
P,\kappa,\Delta$, the long-time density matrix becomes (almost) diagonal in
the basis of bare states $\ket{G/X,n}$. The larger the coupling $g$ the
smaller the difference of the populations of $\ket{G,n}$ and $\ket{X,n-1}$
($\propto 1/g^{2}$). The coherences $|\rho_{GnXn-1}|$, which decay as $1/g$,
also vanish as the coupling $g$ increases.
In this work we focus on the strong-coupling regime $g\gg P,\kappa$, in which
the coherences $\rho_{GnXn-1}$ are small, but different from zero. In
resonance they are purely imaginary. For small detunings they acquire a small
real part. If the detuning increases, $\Delta\gg g$, the matter-light
interaction becomes dispersive, i.e., the energies of the matter states depend
on the number of photons, and the mechanism which converts matter excitations
into photons is suppressed. Thereby, large detunings correspond to a weak
coupling regime. We conclude that in the regime $|\Delta|\sim g\gg P,\kappa$,
the steady state of the system is expected to exhibit a polaritonic behavior.
Unless stated otherwise, the steady state solution $\hat{\rho}_{ss}$ of (3) is
obtained for the initial condition $\hat{\rho}(0)=\ket{G0}\bra{G0}$, and
setting $\omega_{C}=1$ eV, $g=1$ meV and $\kappa=5\times 10^{-2}$ meV, while
$\Delta$ and $P$ are varied in ranges similar to those of current experiments
Bajoni _et al._ (2008); Azzini _et al._ (2011); Reithmaier _et al._ (2004).
Evidence of the spontaneous coherence build-up associated with polariton
states are currently detected through the photoluminescence properties in
quantum wells (QW) Bajoni _et al._ (2008); Azzini _et al._ (2011); Kasprzak
_et al._ (2006). We compare our theoretical predictions in QDs with the
experimental findings in QWs because: $i$) due to experimental difficulties no
analogous results for QDs have been reported and $ii$) it is reasonable to
assume that some of the physical mechanisms behind the exciton–polariton laser
regime are the same in both cases. Additionally, the present approach may shed
light on the separation of the collective effects in QWs from those of a QD
single emitter.
The calculated emission linewidth, emission energy, integrated intensity and
number of photons are shown in fig. 1 as a function of the pumping rate. This
numerical calculation used the quantum regression theorem Perea _et al._
(2004); Laussy _et al._ (2009); del Valle _et al._ (2009); Vera _et al._
(2009b) and the integrated intensity corresponds to the area under the curve
of the peak associated to the transition between the polaritons in the
manifolds $n=1$ and $n=2$. The linewidth (fig.1.a) exhibits the characteristic
reduction in the polariton regime, and the subsequent growth and decrease
Bajoni _et al._ (2008); Azzini _et al._ (2011); Kasprzak _et al._ (2006).
The emission energy blueshifts (fig.1.b) from the exciton transition frequency
to the cavity mode frequency. In the polaritonic region the blueshift is
smaller than observed in QWs Bajoni _et al._ (2008); Azzini _et al._ (2011).
In the intermediate region, $10^{-1}$ meV $\lesssim P\lesssim 1$ meV, the
calculated blueshift grows faster than the measured one. Despite the small
slope changes in the integrated intensity as a function of $P$ (fig.1.c) our
results are consistent with the previous prediction of absence of threshold in
a one–atom laser Rice and Carmichael (1994). However, the nonlinearity of the
model is evident in the curve of the average number of photons. Only for large
values of the pumping ($P\gtrsim 1$ meV), this curve is parallel to that of
the integrated emission intensity (whose slope is approximately one).
Figure 1: (color online). (a) Emission linewidth, (b) emission energy, (c)
integrated intensity (continuous line) and average number of photons (dashed
line) and (d) second– and third–order correlation functions versus the
incoherent exciton pumping $P$, for $\kappa=5\times 10^{-2}$ meV and
$\Delta=2.5$ meV. The marked regions correspond to the polariton–laser and
photon–laser regimes.
Statistics of the emitted light can be characterized by the normalized second–
$g^{(2)}(0)=\braket{\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}}/\braket{\hat{a}^{\dagger}\hat{a}}^{{}_{2}}$
and third–order
$g^{(3)}(0)=\braket{\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\hat{a}}/\braket{\hat{a}^{\dagger}\hat{a}}^{{}_{3}}$
coherence functions, plotted in fig.(1.d). For small pumping power ($P\lesssim
2\times 10^{-2}$ meV) $g^{(2)}(0),g^{(3)}(0)<1$, a footprint of quantum-like
light, i.e. the partial state of the field is nearly a Fock state with small
number of photons, as expected. Indeed, our calculations show that in this
regime
$\hat{\rho}_{ss}\approx\ket{G}\bra{G}\otimes\\{\sin{\varphi}\ket{0}\bra{0}+\cos{\varphi}\ket{1}\bra{1}\\}$,
with $\varphi\approx 0$. For intermediate values of the pumping ($10^{-2}$ meV
$\lesssim P\lesssim 10^{-1}$ meV) both correlations functions monotonically
grow beyond one (inset fig.1.d), as has been experimentally observed Horikiri
_et al._ (2010). We analyze this behavior below in the text. For larger
pumping rates ($P\gtrsim 2$ meV), the state of the field becomes coherent up
to third order (when $g^{(2)}(0)=g^{(3)}(0)=1$), and thus the linewidth falls
(fig.1.a). In this region, the state of the field obtained by the partial
trace over the excitonic degrees of freedom, has a fidelity of more than 0.99
with the random-phase coherent state
$\int_{0}^{2\pi}\tfrac{d\phi}{2\pi}\Ket{\alpha e^{i\phi}}\Bra{\alpha
e^{i\phi}}$, where $\Ket{\alpha e^{i\phi}}$ is an usual coherent state with
$|\alpha|^{2}$ average number of photons. This type of state has been proposed
to describe the features of a (true) photon-laser Mølmer (1997).
We stress that the regimes identified through the characteristics of the
emitted light, mirror quantum properties of the system state. We describe
those quantum properties by entanglement and mixedness of the steady state.
Linear entropy and negativity are employed to quantify mixedness and matter-
light entanglement, respectively. The former, defined as
$S_{L}(\hat{\rho})=1-\textrm{Tr}\hat{\rho}^{{}_{2}}$, vanishes for pure states
and is maximum for maximally mixed states. The latter is defined as
$\mathcal{N}(\hat{\rho})=2\sum_{\lambda<0}|\lambda|$, where $\lambda$ denotes
the eigenvalues of the partial transpose of $\hat{\rho}$ Życzkowski _et al._
(1998); Vidal and Werner (2002). Finally, searching for a relation between the
energy of the system and its quantum characteristics (such as entanglement),
we introduce the differential energy per excitation,
$\nu(\kappa,P)=\tfrac{1}{2}d\braket{\hat{H}}/d\braket{\hat{N}}$, as a
convenient measure of energy per particle. The factor of 2 in the definition
of $\nu$ has been chosen to satisfy the condition $\nu(P\ll
g,\kappa\ll\Delta)\approx\omega_{C}-\Delta$. The negativity, linear entropy
and the differential energy per excitation are depicted in fig. 2.
Figure 2: (color online). $\mathcal{N}(\hat{\rho}_{ss})$ (continuous red
line), $S_{L}(\hat{\rho}_{ss})$ (dashed black line) and $\nu$ (dashed-dotted
blue line) as a function of the incoherent exciton pumping $P$, for
$\kappa=5\times 10^{-2}$ meV and (a) $\Delta=0$, (b) $\Delta=1$ meV, (c)
$\Delta=2$ meV, (d) $\Delta=3$ meV, (e) $\Delta=7$ meV and (f) $\Delta=10$
meV. Note that for $\Delta\neq 0$ the maximum of
$\mathcal{N}(\hat{\rho}_{ss})$, the minimum of $S_{L}(\hat{\rho}_{ss})$ and
the inflection point of $\nu$ coincide.
Assuming the strong-coupling regime, the formation of the polariton is
hindered by two different mechanisms, which depend on the detuning. While the
mixedness of the state is large for small detuning (fig. 2.a and 2.b), matter
and light decouple for large detuning (fig. 2.f). In the intermediate region
$g\lesssim|\Delta|\lesssim 10g$, where the polariton is well defined, both
mechanisms compete. The matter-light entanglement is enhanced, and the entropy
of the asymptotic state of the system decreases with increasing detuning. The
negativity of the steady state $\mathcal{N}(\hat{\rho}_{ss})$ vanishes for
small and large values of the pumping power, and attains a maximum at the
point $P=\kappa$ –the mid-point of the polariton-laser region.
Now, we are able to give a possible explanation of the behavior of the
correlation functions in the polariton regime. The asymptotic state in the
polaritonic region is a mixed entangled state which satisfies
$g^{(3)}(0)>g^{(2)}(0)>1$. This behavior of the correlation functions is a
rather generic feature. As an example, we consider another mixed entangled
state $\hat{\rho}_{pol}(\bar{n})=\sum_{n}P_{n}(\bar{n})\ket{n,+}\bra{n,+}$.
Since the probabilities $P_{n}(\bar{n})=e^{-\bar{n}}\bar{n}^{n}/(n!)$ are
Poisson weights, this is a polariton coherent state. However, the reduced
photon state has super-poissonian statistics, i.e., the second- and third-
order coherence functions can not be expected to be unity. Hence, matter-light
entanglement is a viable alternative to the standard explanation of the
unexpected behavior of these functions, based on polariton-polariton and
polariton-phonon interactions.
The differential energy per excitation $\nu$ is plotted in fig. 2 as a
function of $P$ for several values of the detuning. For $\Delta=0$, $\nu$ is
independent of $P$ and equals to the cavity mode energy, since the number of
photons becomes much larger than the number of matter excitations. Our
numerical results show that for $|\Delta|\gtrsim g$, and small ($P<10^{-2}$
meV) or large ($10$ meV $>P>10^{-1}$ meV) pumping rates, $\nu$ is almost a
constant, equal to the exciton (photon) energy in the former (latter) case.
The same constant values would had been obtained with the definition
$\tilde{\nu}=H/N$. For intermediate values of $P$, $\nu$ displays an inflexion
point at $P=\kappa$, where $\nu=\omega_{C}-(\Delta/2)$ is halfway between the
exciton and photon energies. Moreover, since the mean number of excitations is
one, it is tempting to define $P=\kappa$ as the condition for the “optimum”
polariton. In order to quantify this idea we compare the steady state with the
polariton states $\ket{n,\pm}$ defined by (2), using the sequence of non-zero
fidelities $F_{n\pm}=\sqrt{\braket{n,\pm}{\hat{\rho}_{ss}}{n,\pm}}$. For small
values of $P$ (fig.3.a), when $\rho_{G0G0}$ is much larger than the other
populations, only $F_{1-}$ does not vanish –however it is relatively small–.
For $P=\kappa$ (fig.3.b) $F_{1-}$ increases up to more than 0.95, while the
remaining fidelities are still small. Hence, the steady state
$\hat{\rho}_{ss}$ is quite similar to the $\Lambda_{1}$–polariton $\ket{1-}$.
As $P$ increases, $F_{n\pm}$ is non-zero for larger excitation-numbers, but
their values are very small.
Figure 3: (color online). Sequence of non-zero fidelities $F_{n\pm}$ between
the steady state $\hat{\rho}_{ss}$ and the
$\Lambda_{n}$–lower(black)$/$upper(green) polaritons $\ket{n,\pm}$ for
$\kappa=5\times 10^{-2}$ meV, $\Delta=3$ meV and (a) $P=2\times 10^{-3}$ meV,
(b) $P=5\times 10^{-2}$ meV, (c) $P=1$ meV and (d) $P=5$ meV. $|\Lambda_{n}|$
denotes the excitation number of the polariton manifold $\Lambda_{n}$.
The excitation number ($\hat{N}$) symmetry associated with the Hamiltonian (1)
is broken in the time evolution provided by the master equation (3), in the
sense that the asymptotic state of the system cannot be labeled with a single
eigenvalue of $\hat{N}$. If polariton-like behavior is actually present,
restoration of the symmetry is expected. In order to account for this effect,
we introduce the participation ratio $PR=\sum_{n=0}^{\infty}P_{n}^{2}$, where
$P_{n}$ is the probability to have $n$ excitations in the asymptotic state.
This quantity, which varies from zero –all excitation numbers are
equiprobable– to one –only one occupied manifold–, displays a global maximum
at zero pumping rate and a local maximum at $P\approx\kappa$, in the strong
coupling regime, signaling a partial restoration of symmetry. This can be
understood as a combined effect of the decrease of the mixedness of the state
and the increase of its entanglement, occurring at $P\approx\kappa$, as
discussed above (again, $\Delta\sim g$).
As we have seen, the optimum polariton exhibits maximum negativity, minimum
linear entropy, a local maximum of the participation ratio and an inflection
point of the differential energy per excitation, provided that the system is
in strong coupling. To quantify our previous qualitative arguments –which show
that polaritons cannot be sustained neither for small nor for large
detunings–, it is worth examining the behavior of the linear entropy and the
negativity at $P=\kappa$, as a function of the detuning (fig.4). Three regions
can be identified. In the first region, $|\Delta|<0.68$ meV, the negativity of
the steady state is exactly zero and the linear entropy is larger than $0.72$.
In the second region, $0.68$ meV $<|\Delta|\lesssim 10$ meV, the linear
entropy and the negativity are still significative. In the third region the
dissipative polariton is very close to the Hamiltonian polariton
$\ket{1,\pm}$, for the corresponding $\Delta$. Nevertheless, the dressed
states (2) are virtually the bare states.
Figure 4: (color online). Linear entropy (dashed black line) and negativity
(continuous red line) of the steady state of the system $\hat{\rho}_{ss}$, and
negativity of polaritons of the excitation manifold $\Lambda_{1}$ (dashed-
dotted blue line) as a function of $\Delta$ for $P=\kappa=5\times 10^{-2}$
meV.
Our results might provide a guide to experiments, in the sense that, in the
strong-coupling regime, both the pumping rate and the detuning have to be
carefully adjusted. In our model, the best matter-light correlation properties
occur at $P=\kappa$ and $\Delta\approx 3g$, where $\mathcal{N}\approx 0.32$
and $S_{L}\approx 0.29$. From the theoretical point of view we propose the
following criterion: if the negativity rises above 0.25, close to its maximum
possible value, its inflection points, as a function of the pumping power, can
be used to define the polaritonic regime. When this condition is fulfilled all
the quantifiers that we have examined exhibit a characteristic change. The
emission energy presents a blueshift, the differential energy per excitation
has an inflection point, the emission line decreases, the second- and third-
order correlation functions increase beyond their value for photon coherent
states, the entropy decreases and the negativity increases. With the exception
of the first two, these changes can be understood as a coherence gain of the
asymptotic state of the system.
We are grateful with Prof. P.S.S. Guimarães from UFMG (Brazil), Prof. J.
Mahecha from UdeA (Colombia) and Dra. J. Restrepo from UAN (Colombia) for
critical reading of the manuscript. The authors acknowledge partial financial
support from Dirección de Investigación - Sede Bogotá, Universidad Nacional de
Colombia (DIB-UNAL) under project 12584, and technical and computational
support from Grupo de Óptica e Información Cuántica (GOIC-UNAL), and Grupo de
Física Atómica y Molecular (GFAM-UDEA).
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|
arxiv-papers
| 2012-05-11T20:46:13 |
2024-09-04T02:49:30.833573
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. G. Su\\'arez-Forero, G. Cipagauta, H. Vinck-Posada, K. M.\n Fonseca-Romero, B. A. Rodr\\'iguez",
"submitter": "Boris Anghelo Rodriguez Rey Prof.",
"url": "https://arxiv.org/abs/1205.2719"
}
|
1205.2742
|
Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index $3+\sqrt{5}$.
There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between $5$ and $3+\sqrt{5}$. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. We prove our uniqueness results by showing that there is a unique flat connection on each graph. The result on 1-supertransitive subfactors is proved by an argument using intermediate subfactors, running the `odometer' from the FusionAtlas` Mathematica package and paying careful attention to dimensions.
At this point, we have a complete classification of subfactor planar algebras with index less than 5 (and hence of amenable subfactors in that range). The work of <cit.> established this up to index 4. Haagerup's landmark work <cit.>, followed by the results in <cit.>, gave the classification up to index $3+\sqrt{3}$, and finally the classification up to index less than 5 appears in <cit.>. At index exactly 5 the classification is known (there are 5 group-subgroup subfactors) but has not yet appeared in the literature.
The great surprise from these classifications has been just how few small index subfactors there are: just 10 subfactors in the index range $(4,5)$, coming in 5 pairs which are either dual or conjugate to each other. We know that at higher indices there is an incredible profusion of subfactors, and even by index $6$ there are certain wild phenomena. As we have worked through low indices, on the other hand, we see a desert; the `little desert' of our title. In this paper, we give evidence, and a conjecture, that the desert continues further.
Beyond 5, the techniques used previously seem to lose traction. The combinatorial growth in possible principal graphs becomes too rapid, and we don't have effective obstructions at the level of graphs for principal graph which begin with quadruple or higher branches. Nevertheless, this paper gives some preliminary results on the range $(5, 3+\sqrt{5})$; we completely classify the 1-supertransitive case, and prove that the two known principal graphs are uniquely realized by quantum group subfactors.
There are two subfactors with index in the interval $(5,3+\sqrt{5})$ which are easy to construct from quantum groups.
Given a quantum group $U_q(\mathfrak{g})$, irreducible representation $V$ and root of unity $\zeta = \exp(2 \pi i/ \ell)$, there is a corresponding subfactor $\cQ(\mathfrak{g}, V, \ell)$, as long as a certain positivity condition is satisfied <cit.>. (This condition has been completely analysed in <cit.>.) We are interested in the subfactors $\cA = \cQ(\mathfrak{su}_2, V_{(2)}, 14)$ and $\cB = \cQ(\mathfrak{su}_3, V_{(1,0)}, 14)$. Here $V_{(2)}$ is the three dimensional adjoint representation of $\mathfrak{su}_2$, and $V_{(1,0)}$ is the three dimensional standard representation of $\mathfrak{su}_3$. Each has quantum dimension $q^{-2} + 1 + q^2$, and at $q = \exp(2 \pi i / 14)$ this is $d\approx2.24698$, the largest root of $x^3-2x^2-x+1$. These subfactors have principal graphs
\begin{align*}
\Gamma(\cA) & = \left(\bigraph{bwd1v1p1v1x0p1x1duals1v1x2}, \bigraph{bwd1v1p1v1x0p1x1duals1v1x2}\right) \\
\Gamma(\cB) & = \left(\bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}\right)
\end{align*}
and both have index $d^2 \approx 5.04892$. In the language of `levels', $\ell = 14$ corresponds to $SU(2)_5$ and $SU(3)_4$. The subfactor $\cA$ is the reduced subfactor construction for the third vertex in the $A_6$ subfactor. See the case $\ell=7$, $k=3$ of <cit.> for another realization of the subfactor $\cA$. The subfactor $\cB$ first appears in <cit.>, where its index (but not its principal graph) is calculated. Its principal graph was shortly known to the experts, but its first appearance in print is as a special case of the last section of <cit.>.
The object of this paper is to show that these are the only subfactors with these principal graphs, and moreover that $\cA$ is the only $1$-supertransitive subfactor with index in the interval $(5, 3+\sqrt{5})$. Our three main theorems are
*thm:only1STTheorem <ref>
\beginthm:only1ST
The only 1-supertransitive subfactors with index in the range $(5, 3+\sqrt{5})$ have principal graph $\Gamma(\cA)$.
\endthm:only1ST
*thm:uniquehatTheorem <ref>
\beginthm:uniquehat
There is a unique subfactor with principal graph $\Gamma(\cA)$.
\endthm:uniquehat
*thm:uniquehexTheorem <ref>
\beginthm:uniquehex
There is a unique subfactor with principal graph $\Gamma(\cB)$.
\endthm:uniquehex
We conjecture that these two subfactors are the only non-$A_\infty$ subfactors (of any supertransitivity) with index in that interval, although we cannot prove this at present. See Conjecture <ref> and Theorem <ref> for some evidence of this.
We begin with a section on graph planar algebras, introducing some new notions which are required for the rest of the paper. In <ref>, we prove Theorem <ref>, establishing that $\Gamma(\cA)$ is the only possible principal graph for a $1$-supertransitive subfactor with index in the interval $(5, 3+\sqrt{5})$. Next, in <ref> we find that
the eigenvalue condition from <cit.> ensures that there are at most two two gauge equivalence classes of bi-unitary connections on $\Gamma(\cA)$ which are flat. We show that there is exactly one bi-unitary connection on $\Gamma(\cB)$.
This shows there is at most one subfactor with principal graph $\Gamma(\cB)$; since the subfactor coming from the standard representation in $SU(3)_4$ has principal graph $\Gamma(\cB)$ in fact there is exactly one such subfactor (and the bi-unitary connection we found must in fact be flat).
We then turn to finding flat elements in the graph planar algebra, for each of the two connections on $\Gamma(\cA)$. By definition, for a bi-unitary connection on a $k$-supertransitive graph to be flat, there must be a low weight flat vector in the $(k+1)$-box space of the graph planar algebra. Conversely, the existence of such a low weight flat vector guarantees the existence of some $k$-supertransitive subfactor at the same index (although not necessary with the expected principal graph). In <ref>, we show that one of the two connections on $\Gamma(\cA)$ has such a flat low weight vector; the classification statement from <ref> ensures that the resulting $2$-supertransitive subfactor in fact has principal graph $\Gamma(\cA)$. We show that the other connection has no such flat low weight vectors, so can not be flat.
A word about our use of computers: Theorem <ref> uses the `odometer' of FusionAtlas in an essential way (see <cit.>, which also makes essential use of the odometer, for the details of this routine). However, with the addition of the assumption that the principal graph is finite depth, we can (and do) prove this theorem by hand. Theorems <ref> and <ref> are proved by explicitly providing connections and flat low-weight elements. Checking that these elements are flat under the given connection can be done either by hand or by computer, and we expect that you'll have more faith that we did this correctly when we say that we did it with a computer. (Also, the Mathematica code we used for this purpose is available with the arXiv sources for this article.)
§ GRAPH PLANAR ALGEBRAS
Planar algebras were first defined by Jones in <cit.>, which also explains their relation to subfactors. We do not reproduce the definition here.
§.§ Lopsided and spherical planar algebras
Starting with any shaded planar algebra, it is possible to rescale the pivotal structure in two different ways. This means introducing a scalar factor for each critical point in strings in the action of planar tangles.
The identity
\begin{tikzpicture}[baseline=0]
\clip (-1.1,-.9) rectangle (1.5,.9);
\filldraw[fill=white!50!gray] (-1,-1)--(-1,0) arc (180:0:5mm) arc (-180:0:5mm) -- (1,1)--(2,1)--(2,-1);
\end{tikzpicture}
\quad
\quad
\begin{tikzpicture}[baseline=0]
\clip (.5,-.9) rectangle (1.5,.9);
\filldraw[fill=white!50!gray] (1,-1) -- (1,1)--(2,1)--(2,-1);
\end{tikzpicture}
(and the corresponding identity in the opposite shading) contrains the possible rescalings to:
\begin{align*}
\begin{tikzpicture}[baseline=0]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-2,1)--(-2,-1) -- (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1)--(1,-1)--(1,1);
\end{tikzpicture}
\quad
\mapsto x \quad
\begin{tikzpicture}[baseline=0]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-2,1)--(-2,-1) -- (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1)--(1,-1)--(1,1);
\end{tikzpicture}
\begin{tikzpicture}[baseline=0,yscale=-1]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-2,1)--(-2,-1) -- (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1)--(1,-1)--(1,1);
\end{tikzpicture}
\quad
\mapsto y^{-1}
\begin{tikzpicture}[baseline=0, yscale=-1]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-2,1)--(-2,-1) -- (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1)--(1,-1)--(1,1);
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=0,yscale=-1]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1);
\end{tikzpicture}
\quad
\mapsto x^{-1}
\begin{tikzpicture}[baseline=0, yscale=-1]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1);
\end{tikzpicture}
\begin{tikzpicture}[baseline=0]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1);
\end{tikzpicture}
\quad
\mapsto y \quad
\begin{tikzpicture}[baseline=0]
\clip (-1.5,-.5) rectangle (.5,.6);
\filldraw[fill=white!50!gray] (-1,-1)--(-1,0) arc (180:0:5mm) --(0,-1);
\end{tikzpicture}
\end{align*}
If we want this rescaling to respect the star structure, we must have $y^{-1} = x^*$, since caps and cups are adjoint to each other. However, we will generally not do this; in fact throughout below we will take $y=1$.
We will call the planar algebra obtained from $\cP$ in this way $\cP^{\cap x,y}$. Even though $\cP$ and $\cP^{\cap x,y}$ are not equivalent, it is easy to describe the relationship between the actions of planar tangles. Say $\psi: \cP \to \cP^{\cap x,y}$ is the identity on the underlying vector spaces. If $T$ is a planar tangle and $z_i$ are elements in $\cP$, then
\begin{equation}
\label{eq:intertwiner}
\psi(T(z_i)) = x^{n} y^{m} T(\psi(z_i)),
\end{equation}
where $n$ is the signed count (minimums are positive, maximums are negative) of critical points which are shaded above in $T$ and $m$ is the signed count of critical points shaded below.
We call a shaded planar algebra spherical if the two circles (shaded or unshaded inside) have the same value (that is, they are the same multiple of the appropriate empty diagram), usually called $\delta$. We call a shaded planar algebra lopsided if the circle shaded inside has value $1$. We can always obtain a lopsided planar algebra from a spherical one, by choosing $x=\delta, y=1$ above. We can also go the other way.
For every rescaling, the product of the value of the two circles is constant; in particular in the lopsided planar algebra the unshaded circle has value $\delta^2$.
Generally, we have found that the lopsided pivotal structure is extremely helpful. Its essential importance is that it often allows us to work over a fixed number field (the values of loops must certainly lie in the scalars; only having to include the index, not the square root of the index, is a promising start). Once we are working in a fixed number field, a great many calculations become much easier, and it is possible to have a computer perform exact arithmetic very efficiently. The first use of the lopsided pivotal structure (although somewhat hidden) was in the construction of the extended Haagerup subfactor planar algebra in 6 of <cit.>, where we needed to compute the moments of some elements in the graph planar algebra. It has been used subsequently in <cit.> and <cit.>. This paper is the first time the lopsided pivotal structure has been used alongside the theory of connections; we are now able to explicitly check flatness in cases that would have been more difficult in the spherical pivotal structure.
§.§ The lopsided graph planar algebra
The planar algebra of a bipartite graph was first definied in <cit.>. We recall that definition, as well as the lopsided version of a graph planar algebra, which we explicitly describe. For comparison, we'll show the definition of both $\cG^{\text{spherical}}(\Gamma)$ and $\cG^{\text{lopsided}}(\Gamma)$. (Somewhat confusingly, it's not the case that $\cG^{\text{lopsided}}(\Gamma)$ is just $\cG^{\text{spherical}}(\Gamma)^{\cap \delta,1}$, the lopsided version of $\cG^{\text{spherical}}(\Gamma)$. It's also essential, to obtain the desired number-theoretic properties, to rescale the basis.)
In both, the underlying vector spaces $\cG^{\bullet}(\Gamma)_{n, \pm}$ are just functionals on loops of length $n$ on $\Gamma$, with the base point at either an even or odd depth vertex depending on $\pm$. We'll often abuse notation and think of a loop on $\Gamma$ in place of its indicator function. To define the action of a planar tangle $T$, we specify its values $T(\gamma_i)$, where the $\gamma_i$ are the indicator functions for loops corresponding to the input vector spaces for $T$. This element $T(\gamma_i) \in \cG^\bullet_{n}$ (here $n$ is the number of points on the outer boundary of $T$) is itself a functional on loops corresponding to the outside boundary of $T$, so we specify it by giving its values on loops $\gamma_0$:
\begin{equation}
\label{eq:action}
T(\gamma_i)(\gamma_0) = \sum_{b \in \cL} c(T, b) ,
\end{equation}
where the label set $\cL$ consists of all ways to compatibly color the strands of $T$ with edges of $\Gamma$ and the regions of $T$ with vertices of $\Gamma$, such that around each inner or outer boundary of $T$ the colors agree with the loops $\gamma_i$. The coefficients $c(T,b)$ are the so-called `critical point coefficients', which determine the difference between the spherical and lopsided versions of the graph planar algebra. In each case, $c(T,b)$ is a product over the critical points in the strings in $T$ of some function of the labels given by $b$ appearing above and below the critical point. In the spherical case, we have
$$c^{\text{spherical}}(T,b) = \prod_{\substack{\text{critical} \\ \text{points $x$}}} \sqrt{\frac{d_{\text{above }x}}{d_{\text{below }x}}}^{\operatorname{sign}(x)}$$
while in the lopsided case we have
$$c^{\text{lopsided}}(T,b) = \prod_{\substack{\text{critical} \\ \text{points $x$}}} \left(\frac{d_{\text{above }x}}{\delta^{\text{shading above }x}}\right)^{\operatorname{sign}(x)}.$$
In these formulas, $d_{\text{above }x}$ means the Frobenius-Perron dimension of the graph vertex appearing above the critical point in the labelling given by $b$, and similarly for $d_{\text{below }x}$. When we write $\delta^{\text{shading above }x}$ we just mean $1$ if the critical point is not shaded above, and $\delta$ if it is shaded above. (In the two-sided graph planar algebra, defined in <ref>, sometimes this quantity is $\delta^{-1}$.) Henceforth, we'll just call the ratio $d_a / \delta^{\operatorname{shading}(a)}$ the `lopsided dimension' of $a$, written $d^{\text{lopsided}}_a$. The quantity $\operatorname{sign}(x)$ is the sign of the second derivative at the critical point: $+1$ if the critical point is a local minimum, $-1$ if it is a local maximum.
The careful reader will note that in the above definition we have implicitly chosen a Morse function on our tangles $T$, so that we can talk about critical points and their signs. This is at first sight incompatible with the definition of a planar algebra, where the maps associated to tangles must be invariant under planar isotopies (even those which rotate the boundary discs). To resolve this, for each space $\cG(\Gamma)_{n,\pm}$ we introduce $n+1$ different bases, which we think of as having $k$ boundary points pointing upwards and $n-k$ boundary points pointing downwards. In Equation (<ref>) above, this removes the ambiguity in choosing the Morse function needed to classify boundary points. The basis corresponding to all boundary points upwards is just the basis of indicator functions on loops, as above, and the other bases are all defined by the transporting this first basis via the tangles
\newcommand{\turncontents}{
\draw (-0.5,0)--(-0.5,2);
\draw (0.5,0)--(0.5,2);
\draw (1.5,0)--(1.5,2);
\draw (2.5,0)--(2.5,1) arc (180:0:.5cm) -- (3.5,-2);
\draw (.2,0)--(.2,-2);
\draw (1.8,0)--(1.8,-2);
\filldraw[fill=white] (-1,-.5) rectangle (3,.5);
\begin{tikzpicture}[baseline=0,scale=0.5]
\turncontents
\end{tikzpicture},
\qquad
\begin{tikzpicture}[baseline=0,scale=0.5, xscale=-1]
\turncontents
\end{tikzpicture},
\qquad
\begin{tikzpicture}[baseline=0,scale=0.5, yscale=-1]
\turncontents
\end{tikzpicture}
\quad \text{ and } \quad
\begin{tikzpicture}[baseline=0,scale=0.5, xscale=-1, yscale=-1]
\turncontents
\end{tikzpicture}
These bases are coherent (that is, any two ways to modify the division into upper and lower boundary give equal maps) because in Equation (<ref>) the coefficient $c(\rho^{2\pi},b)$ for the $2\pi$ rotation is always $1$.
Consider now the linear map $\natural : \cG^{\text{spherical}} \to \cG^{\text{lopsided}}$ which rescales loops according to
$$\natural(\gamma) = \sqrt{\left(\prod_{{\text{$a$ above}}} d^{\text{lopsided}}_a \right)\left( \prod_{{\text{$b$ below}}} {d^{\text{lopsided}}_b}\right)^{-1}} \; \; \gamma.$$
Taking care here, the vertices of $\gamma$ appearing on the left and right sides (that is, at the changeovers between upper boundary points and the lower boundary points) do not count in either of the products.
One can readily check that this intertwines the actions of planar tangles on the spherical and lopsided graph planar algebras according to the formula
\begin{equation}
\label{eq:intertwiner2}
\natural(T(z_i)) = \sqrt{\delta}^{\; n} \sqrt{\delta}^{\; -m} T(\natural(z_i)),
\end{equation}
where again $n$ is the signed count (minimums are positive, maximums are negative) of critical points which are shaded above in $T$ and $m$ is the signed count of critical points shaded below.
Note that this intertwining condition means that we can locate the lowest weight spaces, or rotational eigenspaces, using the lopsided graph planar algebra, where arithmetic is easier. In particular, $\natural$ restricts to an isomorphism between the lowest weight spaces, and an isomorphism between each rotational eigenspace.
If one transported the action of planar tangles on $\cG^{\text{lopsided}}$ across to $\cG^{\text{spherical}}$ via the map $\natural$ and its inverse, this formula shows that the action is a rescaling (as described in the previous section) of the usual action of planar tangles on $\cG^{\text{lopsided}}$ with $x = \delta ^{1/2}$ and $y=\delta^{-1/2}$.
Nevertheless, for our purposes it wouldn't have been enough to simply rescale the original action. The map $\natural$, which rescales the basis, allows us to define the lopsided graph planar algebra over the field $\mathbb{Q}(d^{\text{lopsided}})$ generated by the lopsided dimensions. (Often, but not always, this is no bigger than the field $\mathbb{Q}(\delta^2)$ generated by the index of the graph.) The spherical graph planar algebra is defined instead over the field $\mathbb{Q}\left(\left\{\sqrt{d_a / d_b} \; \mid \text{$a$ and $b$ adjacent}\right\}\right)$ which is generally much much larger. Indeed, usually it's impossible to identify a single generator of this resulting number field.
Finally, we define the $*$ action on the lopsided graph planar algebra simply by transporting across the $*$ action form the spherical graph planar algebra, via $\natural$ and $\natural^{-1}$. Explicitly, this gives
$$\gamma^* = \left(\prod_{{\text{$a$ above}}} d^{\text{lopsided}}_a \right)\left( \prod_{{\text{$b$ below}}} {d^{\text{lopsided}}_b}\right)^{-1}\operatorname{reverse}(\gamma)$$
on loops, extending antilinearly to the entire space.
§.§ The two-sided graph planar algebra
Usually, the graph planar algebra is defined in terms of a single principal graph. We now introduce the `two-sided' graph planar algebra for a pair of principal graphs (with dual data) $(\Gamma, \Gamma')$. (The name comes from an interpretation of graph planar algebras, connections and flatness coming from Turaev-Viro theory; c.f. <cit.>). The two-sided graph planar algebra has region colours and strand types indexed by the square
\node(NN) [rectangle, fill=red-unshaded] at (0,0) {$N-N$};
\node(NM) [rectangle, fill=red-shaded] at (0,1) {$N-M$};
\node(MM) [rectangle, fill=blue-unshaded] at (2,1) {$M-M$};
\node(MN) [rectangle, fill=blue-shaded] at (2,0) {$M-N$};
\draw (NN)--(NM)--(MM)--(MN)--(NN);
\end{tikzpicture}
That is, there is a vector space $\cG_\pi$ for each closed loop $\pi$ on this square (namely, each sequence of `unshaded red', `shaded red', `unshaded blue' and `shaded blue', subject to the condition that the shadings alternate). This vector space has basis given by the loops in the 4-partite graph for $(\Gamma, \Gamma')$ which descend to the loop $\pi$ on the square. The even and odd vertices of $\Gamma$ lie over the unshaded red and shaded red vertices of the square, while the even and odd vertices of $\Gamma'$ lie over the unshaded blue and shaded blue vertices of the square.
As before we have two versions of the two-sided graph planar algebra, which we call spherical and lopsided.
The two actions of planar tangles are exactly as above for the one-sided graph planar algebra, with the obvious restriction that the labelings $b$ in Equation (<ref>) above respect the four different shadings in $T$, and a further interpretation of the quantity $\operatorname{shading}(a)$: this is $0$ if the shading is $N-N$ or $M-M$, $+1$ when the shading is $N-M$ and $-1$ when the shading is $M-M$.
The notion of a connection was originally formulated by Ocneanu in <cit.>. Notice that a connection, as usually defined, is exactly an element $K $ of the space $\cG^{\text{spherical}}_\zeta$, where $\zeta$ is the loop `unshaded red, shaded red, unshaded blue, shaded blue'.
The renormalization axiom is no longer an axiom; it is just a statement about the one-click rotation of a four-box.
The biunitarity condition is a pair of planar equations
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1,-1.8) rectangle (1,1.8);
\fill[fill=red-unshaded] (-1,-2)--(-.3,-2)--(-.3,2)--(-1,2);
\fill[fill=red-shaded] (-.3,-2)--(-.3,-1)--(.3,-1)--(.3,-2);
\fill[fill=red-shaded] (-.3,2)--(-.3,1)--(.3,1)--(.3,2);
\fill[blue-unshaded] (1,-2)--(.3,-2)--(.3,3)--(1,2);
\fill[blue-shaded] (-.3,-1)--(-.3,1)--(.3,1)--(.3,-1);
\draw (-.3,-2)--(-.3,2);
\draw (.3,-2)--(.3,2);
\node[conn, minimum size=8mm] at (0,1) {$K$};
\node[conn, minimum size=8mm] at (0,-1) {$K^*$};
\end{tikzpicture}
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1,-1.8) rectangle (1,1.8);
\fill[fill=red-unshaded] (-1,-2)--(-.3,-2)--(-.3,2)--(-1,2);
\fill[fill=red-shaded] (-.3,-2)--(-.3,2)--(.3,2)--(.3,-2);
\fill[blue-unshaded] (1,-2)--(.3,-2)--(.3,3)--(1,2);
\draw (-.3,-2)--(-.3,2);
\draw (.3,-2)--(.3,2);
\end{tikzpicture}
\quad
\text{and}
\quad
\begin{tikzpicture}[scale=.7,baseline=0]
\clip (-1.9,-1) rectangle (1.9,1);
\fill[red-shaded] (-2,1) rectangle (2,0);
\fill[blue-shaded] (-2,-1) rectangle (2,0);
\filldraw[fill=blue-unshaded] (-1,.3) .. controls (0,.6) .. (1,.3)--(1,-.3) .. controls (0,-.6) .. (-1,-.3) -- (-1,.3);
\filldraw[fill=red-unshaded] (-2,.6) -- (-1,.3) -- (-1,-.3) -- (-2,-.6);
\filldraw[fill=red-unshaded] (2,.6) -- (1,.3) -- (1,-.3) -- (2,-.6);
\node[conn, minimum size=8mm] at (-1,0) {$K$};
\node[conn, minimum size=8mm] at (1,0) {$K^*$};
\end{tikzpicture}
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1.9,-1) rectangle (1.9,1);
\fill[red-shaded] (-2,1) rectangle (2,0);
\fill[blue-shaded] (-2,-1) rectangle (2,0);
\filldraw[fill=red-unshaded] (-2,-.6) .. controls (0,0) .. (2,-.6)--(2,.6) .. controls (0,0) .. (-2,.6);
\end{tikzpicture} \; ,
where $K^*$ is defined using the usual $*$-structure (i.e., $K^*(e_1 e_2 e_3 e_4) = \overline{K(e_4 e_3 e_2 e_1)}$.)
Let $\cG_{\text{red}}$ (respectively $\cG_{\text{blue}}$) be the space indexed by paths which alternate between the two shades of red (respectively blue). Note that $\cG_{\text{red}}$ is a copy of the graph planar algebra of the principal graph $\Gamma$, and $\cG_{\text{ blue}}$ is the graph planar algebra for the dual graph $\Gamma'$.
We say that a pair of elements $(x, y) \in \cG_{\text{ blue}} \times \cG_{\text{ red}}$ is flat with respect to a connection $K$ if
\begin{equation*}
\xabove = \ybelow.
\end{equation*}
(This picture illustrates flatness of four-boxes. The $2n$-box generalization has $n$ copies of the connection or its star sitting above $x$ and below $y$.)
Using the map $\natural$ we can push a connection across to the lopsided analogue of the two-sided graph planar algebra. It is still a biunitary, although we have to be careful because the spherical $*$ structure transported across via $\natural$ is now more complicated. On loops, it is
$$ \gamma^* = \prod_{{\text{$a$ above}}} d^{\text{lopsided}}_a \cdot \prod_{{\text{$b$ below}}} {d^{\text{lopsided}}_b}^{-1} \cdot \operatorname{reverse}(\gamma).$$
It is worth noting at this point that a pair of elements $(x,y)$ is flat with respect to a connection $K$ in the spherical graph planar algebra exactly if $(\natural(x), \natural(y))$ is flat with respect to $\natural(K)$ in the lopsided graph planar algebra. This will be essential to our later calculations.
We abuse notation by saying that $x$ itself is flat if there exists a $y$ so that $(x,y)$ is flat. (The element $y$ is necessarily uniquely determined!) Notice that bi-unitarity immediately implies that every Temperley-Lieb diagram is flat. The flat elements form a sub planar algebra which we call the flat subalgebra
We will need the following results about flat elements:
If there are no flat elements besides Temperley-Lieb for a connection $K$ in each of the spaces $P_{j,+}$ for $j < n$, and a $k$-dimensional space of flat lowest weight vectors in $P_{n,+}$, then the flat subalgebra is $n-1$ supertransitive with excess $k$.
Any subfactor planar algebra $P$ defines a bi-unitary connection $K(P) \in \cG(\Gamma(P))_\zeta$, and $P$ is isomorphic to the flat subalgebra for $K(P)$ inside $\cG(\Gamma(P))$.
Although not stated in this language, this result is well known and appears in <cit.>. See also <cit.> for a proof via Turaev-Viro theory.
§.§ The gauge group
The gauge group for a given 4-partite graph $\Xi$ is a copy of the unit circle for every edge in $\Xi$, i.e.,
$$\text{Gauge}(\Xi) = \{g: \Xi \rightarrow \mathbb{T} \} \simeq \mathbb{T}^{|E(\Xi)|}.$$
Elements of the gauge group can be thought of as $2$-boxes in the graph planar algebra: If $g \in \text{Gauge}(\Xi)$ and $\gamma$ is the 2-loop going through edges $e_1$ and $e_2$, then $g (\gamma) = \delta_{e_1,e_2} g(e_1) $.
Thus, $\text{Gauge}(\Xi)$ acts on the graph planar algebra of $\Xi$, by
\begin{tikzpicture}[baseline=.5cm]
\draw (0,-.8)--(0,1.8);
\draw (1,-.8)--(1,1.8);
\draw (2,-.8)--(2,1.8);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,0) rectangle (2.5,1);
\node at (1,.5) {$X$};
\end{tikzpicture}
\mapsto
\begin{tikzpicture}[baseline=.5cm]
\draw (0,-.8)--(0,1.8);
\draw (1,-.8)--(1,1.8);
\draw (2,-.8)--(2,1.8);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,0) rectangle (2.5,1);
\node at (1,.5) {$X$};
\node[gauge] at (0, -.4) {$g$};
\node[gauge] at (0, 1.4) {$g$};
\node[gauge] at (1, -.4) {$g$};
\node[gauge] at (1, 1.4) {$g$};
\node[gauge] at (2, -.4) {$g$};
\node[gauge] at (2, 1.4) {$g$};
\end{tikzpicture}
If $K$ is a biunitary connection, then $gK$ is again a biunitary connection. Let a black bead represent $g$, and a white bead represent $g^*=g^{-1}$; then the first biunitarity equation for $gK$ holds:
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1,-2) rectangle (1,2);
\fill[fill=red-unshaded] (-1,-2)--(-.3,-2)--(-.3,2)--(-1,2);
\fill[fill=red-shaded] (-.3,-2)--(-.3,-1)--(.3,-1)--(.3,-2);
\fill[fill=red-shaded] (-.3,2)--(-.3,1)--(.3,1)--(.3,2);
\fill[blue-unshaded] (1,-2)--(.3,-2)--(.3,3)--(1,2);
\fill[blue-shaded] (-.3,-1)--(-.3,1)--(.3,1)--(.3,-1);
\draw (-.3,-2)--(-.3,2);
\draw (.3,-2)--(.3,2);
\node[conn, minimum size=8mm] at (0,1) {$K$};
\node[conn, minimum size=8mm] at (0,-1) {$K^*$};
\node[invgauge] at (-.3,1.8) {};
\node[invgauge] at (.3,1.8) {};
\node[invgauge] at (-.3,.2) {};
\node[invgauge] at (.3,.2) {};
\node[gauge] at (-.3,-1.8) {};
\node[gauge] at (.3,-1.8) {};
\node[gauge] at (-.3,-.2) {};
\node[gauge] at (.3,-.2) {};
\end{tikzpicture}
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1,-2) rectangle (1,2);
\fill[fill=red-unshaded] (-1,-2)--(-.3,-2)--(-.3,2)--(-1,2);
\fill[fill=red-shaded] (-.3,-2)--(-.3,-1)--(.3,-1)--(.3,-2);
\fill[fill=red-shaded] (-.3,2)--(-.3,1)--(.3,1)--(.3,2);
\fill[blue-unshaded] (1,-2)--(.3,-2)--(.3,3)--(1,2);
\fill[blue-shaded] (-.3,-1)--(-.3,1)--(.3,1)--(.3,-1);
\draw (-.3,-2)--(-.3,2);
\draw (.3,-2)--(.3,2);
\node[conn, minimum size=8mm] at (0,1) {$K$};
\node[conn, minimum size=8mm] at (0,-1) {$K^*$};
\node[invgauge] at (-.3,1.8) {};
\node[invgauge] at (.3,1.8) {};
\node[gauge] at (-.3,-1.8) {};
\node[gauge] at (.3,-1.8) {};
\end{tikzpicture}
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1,-2) rectangle (1,2);
\fill[fill=red-unshaded] (-1,-2)--(-.3,-2)--(-.3,2)--(-1,2);
\fill[fill=red-shaded] (-.3,-2)--(-.3,2)--(.3,2)--(.3,-2);
\fill[blue-unshaded] (1,-2)--(.3,-2)--(.3,3)--(1,2);
\draw (-.3,-2)--(-.3,2);
\draw (.3,-2)--(.3,2);
\node[invgauge] at (-.3,1.8) {};
\node[invgauge] at (.3,1.8) {};
\node[gauge] at (-.3,-1.8) {};
\node[gauge] at (.3,-1.8) {};
\end{tikzpicture}
\begin{tikzpicture}[scale=.7, baseline=0]
\clip (-1,-2) rectangle (1,2);
\fill[fill=red-unshaded] (-1,-2)--(-.3,-2)--(-.3,2)--(-1,2);
\fill[fill=red-shaded] (-.3,-2)--(-.3,2)--(.3,2)--(.3,-2);
\fill[blue-unshaded] (1,-2)--(.3,-2)--(.3,3)--(1,2);
\draw (-.3,-2)--(-.3,2);
\draw (.3,-2)--(.3,2);
\end{tikzpicture}
and the second biunitarity equation is verified similarly.
The complex gauge group for a given 4-partite graph $\Xi$ is a copy of the non-zero complex numbers for every edge in $\Xi$, i.e.,
$$\text{ComplexGauge}(\Xi)= \{g: \Xi \rightarrow (\mathbb{C}^{\times})\} \simeq (\mathbb{C}^{\times})^{|E(\Xi)|}.$$
Again, elements of the complex gauge group are $2$-boxes in the graph planar algebra, and act accordingly. For $g$ a complex gauge group element, $gK$ is no longer necessarily a biunitary connection, but it is bi-invertible and this is often sufficient.
We define $\text{Alt}(g)(X)$ to be the result of surrounding $X$ with alternating $g$'s and $g^{-1}$'s. It's easy to see that $\operatorname{Alt}(g)$ is an isomorphism of planar algebras (it fixes Temperley-Lieb diagrams, and commutes with disjoint union and applying caps). In fact, $\text{Alt}(g)$ acts trivially on any non-essential loop on $\Gamma$, and so for principal graphs without loops this action is always trivial.
Let $X$ be an element in the graph planar algebra of $\Xi$, $K$ be a connection on $\Xi$, and $g$ be an element of $\text{ComplexGauge}(\Xi)$.
$X$ is flat with respect to $K$ if and only if $\text{Alt}(g^{-1})(X)$ is flat with respect to $gK$.
Letting a black bead represent $g \in \text{ComplexGauge}(\Xi)$, and a white bead be $g^{-1}$, flatness of $X$ is the assertion that there is a $Y$ such that
\begin{tikzpicture}[baseline=1.5cm]
\draw (0,-1)--(0,3.5);
\draw (1.5,-1)--(1.5,3.5);
\draw (3,-1)--(3,3.5);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,0) rectangle (3.5,1);
\node at (1.5,.5) {$X$};
\draw (-1,3.5) .. controls (-1,2.9) .. (0,2.6) -- (3,2) .. controls (4,1.7) .. (4,0) -- (4,-1);
\node[conn, minimum size=7mm] at (0,2.6) {\tiny$K$};
\node[conn, minimum size=7mm] at (1.5,2.3) {\tiny$K^{{\hbox{-}\!}1}$};
\node[conn, minimum size=7mm] at (3,2) {\tiny$K$};
\end{tikzpicture}
\quad
\quad
\begin{tikzpicture}[baseline=3.5cm]
\draw (0,1)--(0,5.5);
\draw (1.5,1)--(1.5,5.5);
\draw (3,1)--(3,5.5);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,4) rectangle (3.5,5);
\node at (1.5,4.5) {$Y$};
\draw (-1,5.5) -- (-1,3.5) .. controls (-1,2.9) .. (0,2.6) -- (3,2) .. controls (4,1.7) .. (4,1);
\node[conn, minimum size=7mm] at (0,2.6) {\tiny$K$};
\node[conn, minimum size=7mm] at (1.5,2.3) {\tiny$K^{{\hbox{-}\!}1}$};
\node[conn, minimum size=7mm] at (3,2) {\tiny$K$};
\end{tikzpicture}
This is true if and only if
\begin{tikzpicture}[baseline=1.5cm]
\draw (0,-1)--(0,3.5);
\draw (1.5,-1)--(1.5,3.5);
\draw (3,-1)--(3,3.5);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,0) rectangle (3.5,1);
\node at (1.5,.5) {$X$};
\draw (-1,3.5) .. controls (-1,2.9) .. (0,2.6) -- (3,2) .. controls (4,1.7) .. (4,0) -- (4,-1);
\node[conn, minimum size=7mm] at (0,2.6) {\tiny$K$};
\node[conn, minimum size=7mm] at (1.5,2.3) {\tiny$K^{{\hbox{-}\!}1}$};
\node[conn, minimum size=7mm] at (3,2) {\tiny$K$};
\node[invgauge] at (4, -.5) {};
\node[invgauge] at (0, -.5) {};
\node[gauge] at (1.5, -.5) {};
\node[invgauge] at (3, -.5) {};
\node[invgauge] at (0, 3.2) {};
\node[gauge] at (1.5, 2.9) {};
\node[invgauge] at (3, 2.6) {};
\node[invgauge] at (-1, 3.2) {};
\end{tikzpicture}
\quad
\quad
\begin{tikzpicture}[baseline=3.5cm]
\draw (0,1)--(0,5.5);
\draw (1.5,1)--(1.5,5.5);
\draw (3,1)--(3,5.5);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,4) rectangle (3.5,5);
\node at (1.5,4.5) {$Y$};
\draw (-1,5.5) -- (-1,3.5) .. controls (-1,2.9) .. (0,2.6) -- (3,2) .. controls (4,1.7) .. (4,1);
\node[conn, minimum size=7mm] at (0,2.6) {\tiny$K$};
\node[conn, minimum size=7mm] at (1.5,2.3) {\tiny$K^{{\hbox{-}\!}1}$};
\node[conn, minimum size=7mm] at (3,2) {\tiny$K$};
\node[invgauge] at (-1, 5.3) {};
\node[invgauge] at (0, 5.3) {};
\node[gauge] at (1.5, 5.3) {};
\node[invgauge] at (3, 5.3) {};
\node[invgauge] at (0, 1.9) {};
\node[gauge] at (1.5, 1.6) {};
\node[invgauge] at (3, 1.3) {};
\node[invgauge] at (4, 1.3) {};
\end{tikzpicture}
And inserting lots of instances of the relation $1=g \cdot g^{-1}$ along strands, we see that this is equivalent to the equality
\begin{tikzpicture}[baseline=1.5cm]
\draw (0,-1)--(0,3.5);
\draw (1.5,-1)--(1.5,3.5);
\draw (3,-1)--(3,3.5);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,0) rectangle (3.5,1);
\node at (1.5,.5) {$X$};
\draw (-1,3.5) .. controls (-1,2.9) .. (0,2.6) -- (3,2) .. controls (4,1.7) .. (4,0) -- (4,-1);
\node[conn, minimum size=7mm] at (0,2.6) {\tiny$K$};
\node[conn, minimum size=7mm] at (1.5,2.3) {\tiny$K^{{\hbox{-}\!}1}$};
\node[conn, minimum size=7mm] at (3,2) {\tiny$K$};
\node[invgauge] at (4, -.5) {};
\node[invgauge] at (0, -.5) {};
\node[gauge] at (1.5, -.5) {};
\node[invgauge] at (3, -.5) {};
\node[invgauge] at (0, 3.2) {};
\node[gauge] at (1.5, 2.9) {};
\node[invgauge] at (3, 2.6) {};
\node[invgauge] at (-1, 3.2) {};
\node[invgauge] at (.6,2.48) {};
\node[gauge] at (.9,2.42) {};
\node[gauge] at (2.1,2.18) {};
\node[invgauge] at (2.4,2.12) {};
\node[gauge] at (0,1.2) {};
\node[invgauge] at (0,1.5) {};
\node[invgauge] at (1.5,1.2) {};
\node[gauge] at (1.5,1.5) {};
\node[gauge] at (3,1.2) {};
\node[invgauge] at (3,1.5) {};
\end{tikzpicture}
\quad
\quad
\begin{tikzpicture}[baseline=3.5cm]
\draw (0,1)--(0,5.5);
\draw (1.5,1)--(1.5,5.5);
\draw (3,1)--(3,5.5);
\filldraw[fill=white, thick, rounded corners = 4mm] (-.5,4) rectangle (3.5,5);
\node at (1.5,4.5) {$Y$};
\draw (-1,5.5) -- (-1,3.5) .. controls (-1,2.9) .. (0,2.6) -- (3,2) .. controls (4,1.7) .. (4,1);
\node[conn, minimum size=7mm] at (0,2.6) {\tiny$K$};
\node[conn, minimum size=7mm] at (1.5,2.3) {\tiny$K^{{\hbox{-}\!}1}$};
\node[conn, minimum size=7mm] at (3,2) {\tiny$K$};
\node[invgauge] at (-1, 5.3) {};
\node[invgauge] at (0, 5.3) {};
\node[gauge] at (1.5, 5.3) {};
\node[invgauge] at (3, 5.3) {};
\node[invgauge] at (0, 1.9) {};
\node[gauge] at (1.5, 1.6) {};
\node[invgauge] at (3, 1.3) {};
\node[invgauge] at (4, 1.3) {};
\node[invgauge] at (.6,2.48) {};
\node[gauge] at (.9,2.42) {};
\node[gauge] at (2.1,2.18) {};
\node[invgauge] at (2.4,2.12) {};
\node[gauge] at (0,3.8) {};
\node[invgauge] at (0,3.5) {};
\node[invgauge] at (1.5,3.8) {};
\node[gauge] at (1.5,3.5) {};
\node[gauge] at (3,3.8) {};
\node[invgauge] at (3,3.5) {};
\end{tikzpicture}
In fact, this argument easily shows that two gauge equivalent connections give isomorphic planar algebras; $\operatorname{Alt}(g)$ provides the isomorphism between the corresponding flat elements. Conversely, if two connections give the same planar algebra, they must be gauge equivalent, although we will not need this here.
§ CLASSIFICATION OF 1-SUPERTRANSITIVE SUBFACTORS WITH INDEX IN THE RANGE $(5,3+\SQRT{5})$.
Our main result in this section is
The only 1-supertransitive subfactors with index in the range $(5, 3+\sqrt{5})$ have principal graph $\left(\bigraph{bwd1v1p1v1x0p1x1duals1v1x2}, \bigraph{bwd1v1p1v1x0p1x1duals1v1x2}\right)$.
The proof uses a small amount of computer enumeration of possible principal graphs, using the FusionAtlas` software described in <cit.>, although this could easily be replaced by tedious hand calculations.
We separately prove the same statement with an additional hypothesis of finite depth as Proposition <ref>, because we think the proof is interesting.
In fact, we think a much stronger statement holds
Any subfactor with index in the range $(5, 3+\sqrt{5})$ has principal graph $A_\infty$, $\left(\bigraph{bwd1v1p1v1x0p1x1duals1v1x2}, \bigraph{bwd1v1p1v1x0p1x1duals1v1x2}\right)$ or $\left(\bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}\right)$.
Computer evidence from the FusionAtlas` enables us to prove the following theorem in support of this conjecture, but we won't give the details here.
Any subfactor with index in range $(5, 3+\sqrt{5})$ either appears in Conjecture <ref> or has rank at least 38.
To establish Theorem <ref> we need a few preliminary results.
The following lemma is stated in <cit.> and was known to experts well before then. It probably follows from <cit.>; we find the following proof more direct.
A subfactor $N\subset M$ in which some, but not all, depth-two projections have dimension 1 has a proper intermediate subfactor.
This proof is written in the language of algebra objects for tensor categories; a dictionary between subfactors and algebra objects is given in <cit.>.
$M$ is an algebra object in the category of $N-N$ bimodules; it is isomorphic to $ \sum_{p \in P_4} p$, where $P_4$ is the set of irreducible summands of $M$ as an $N-N$ bimodule (ie, vertices at depth zero and two in the principal graph).
Let $P'_4$ be the subset of $P_4$ consisting of objects of dimension 1. The objects in $P'_4$ form a group under tensor product, as the tensor product of two projections is again a projection, and dimension is multiplicative under tensor product.
Thus, $ \sum_{p \in P'_4} p$ is also an algebra object, and as it is intermediate between $N \simeq 1$ and $M \simeq \sum_{p \in P_4} p$, it corresponds to an intermediate subfactor for $N \subset M$.
A subfactor which has index in $(5,3+\sqrt{5})$ and an object at depth two with dimension in $(1,2)$ has principal graph $A=\bigraph{gbg1v1p1v1x1p0x1}$.
Let $X$ be the generating object for this subfactor, and $Z$ be the object at depth $2$ with $1 < \dim{Z} < 2$. Then $Z$ generates an ADET fusion category. Since $X \in X \tensor Z$, the principal graph for $- \tensor Z$ must contain a $T_n$, with $X$ the last vertex. If $X_0$ is the first vertex in the $T_n$, we have $\dim{X}/ \dim{X_0} = [n]_{q=\zeta_{4n+2}}$.
Case 2: If $n=2$ and if $\dim X_0 = 1$, then $\dim{X} < 2$, so we were below index 4. But if $\dim X_0 \geq \sqrt{2}$, then the index is at least $3+\sqrt{5}$.
Case 3: If $n=3$, then the only possible dimensions are $\dim{X_0}=1$, and $\dim{X} \approx 2.247$ is the largest root of $x^3 - 2 x^2 - x + 1$. We want to know what the $X\otimes -$ principal graph is.
The dimensions of the objects at depth two in the $X\otimes -$ principal graph are all at least $\sqrt{2}$, and because the index is $\dim{X}^2 \approx 5.049$, there are exactly two depth-two vertices in the principal graph. One is $Z$; call the other $Y$. We know $\dim{Z}$ is the graph norm of $T_3$, and then $\dim{Y}$ must be $\dim{X}$. We abbreviate these as
\begin{align*}
d & = \dim{X} = \dim{Y} \approx 2.247; \\
e & = \dim{Z} = d^2-d-1 \approx 1.802.
\end{align*}
In the following illustrations of the forms of principal graphs, filled-in circles are vertices whose neighbors are all known, while open circles are vertices which may connect to other vertices at the next depth.
The triple-point obstruction tells us that if our principal graph were of the first form below, then our dual principal graph would be of the second form:
\left(
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,-.5);
\draw (1,0)--(2,.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\draw[fill=black] (1,0) circle (1mm);
\draw[fill=black] (2,-.5) circle (1mm);
\draw[fill=black] (2,.5) circle (1mm);
\draw[fill=white] (3,-.5) circle (1mm);
\draw[fill=white] (3,.5) circle (1mm);
\end{tikzpicture}
\quad , \quad
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,-.5);
\draw (1,0)--(2,.5);
\draw (2,-.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\draw[fill=black] (1,0) circle (1mm);
\draw[fill=black] (2,-.5) circle (1mm);
\draw[fill=black] (2,.5) circle (1mm);
\draw[fill=white] (3,-.5) circle (1mm);
\draw[fill=white] (3,.5) circle (1mm);
\end{tikzpicture}.
\right)
But the univalent vertex at depth two has dimension 1, and we get a contradiction because this subfactor cannot have an intermediate subfactor (as the index does not factor into allowed index values).
Thus, our principal graph is either of the form
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,0);
\draw (1,0)--(2,.5)--(3,0);
\draw[fill=black] (0,0) circle (1mm);
\node at (0,0) [below left] {$1$};
\draw[fill=black] (1,0) circle (1mm);
\node at (1,0) [below left] {$X$};
\draw[fill=white] (2,-.5) circle (1mm);
\node at (2,-.5) [below left] {$Z$};
\draw[fill=white] (2,.5) circle (1mm);
\node at (2,.5) [above left] {$Y$};
\draw[fill=white] (3,0) circle (1mm);
\end{tikzpicture} \quad ,
\quad \text{or} \quad
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,-.5);
\draw (2,.5)--(3,0);
\draw (1,0)--(2,.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\node at (0,0) [below left] {$1$};
\draw[fill=black] (1,0) circle (1mm);
\node at (1,0) [below left] {$X$};
\draw[fill=white] (2,-.5) circle (1mm);
\node at (2,-.5) [below left] {$Z$};
\draw[fill=white] (2,.5) circle (1mm);
\node at (2,.5) [above left] {$Y$};
\draw[fill=white] (3,-.5) circle (1mm);
\draw[fill=white] (3,0) circle (1mm);
\draw[fill=white] (3,.5) circle (1mm);
\end{tikzpicture} \quad .
In both cases, note that $\dim{Z} \dim{X} - \dim{X} = de-d = e < 2$, so $Z$ must have degree two. In the first case, dimensions imply $Y$ also connects to a vertex of dimension 1. In the second case, $\dim{Y} \dim{X} - \dim{X} = d^2-d \approx 2.802$ can only be partitioned into a sum of allowable dimensions as $e +1$, whence $Y$ connects to three other vertices: $X$, a vertex of dimension $e$ and a vertex of dimension 1. These stronger restrictions say our graph is actually of the form
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,0);
\draw (1,0)--(2,.5)--(3,0);
\draw (2,.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\node at (0,0) [below left] {$1$};
\node at (0,0) [above left, blue] {$1$};
\draw[fill=black] (1,0) circle (1mm);
\node at (1,0) [below left] {$X$};
\node at (1,0) [above left, blue] {$d$};
\draw[fill=black] (2,-.5) circle (1mm);
\node at (2,-.5) [below ] {$Z$};
\node at (2,-.5) [above , blue] {$e$};
\draw[fill=black] (2,.5) circle (1mm);
\node at (2,.5) [below ] {$Y$};
\node at (2,.5) [above right, blue] {$d$};
\draw[fill=white] (3,0) circle (1mm);
\node at (3,0) [ right, blue] {$e$};
\draw[fill=black] (3,.5) circle (1mm);
\node at (3,.5) [ right, blue] {$1$};
\end{tikzpicture} \quad ,
\quad
\text{or}
\quad
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,-.5);
\draw (2,.5)--(3,0);
\draw (1,0)--(2,.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\node at (0,0) [below left] {$1$};
\node at (0,0) [above left, blue] {$1$};
\draw[fill=black] (1,0) circle (1mm);
\node at (1,0) [below left] {$X$};
\node at (1,0) [above left, blue] {$d$};
\draw[fill=black] (2,-.5) circle (1mm);
\node at (2,-.5) [below ] {$Z$};
\node at (2,-.5) [above right, blue] {$e$};
\draw[fill=black] (2,.5) circle (1mm);
\node at (2,.5) [below ] {$Y$};
\node at (2,.5) [above right, blue] {$d$};
\draw[fill=white] (3,-.5) circle (1mm);
\node at (3,-.5) [ right, blue] {$e$};
\draw[fill=white] (3,0) circle (1mm);
\node at (3,0) [ right, blue] {$e$};
\draw[fill=white] (3,.5) circle (1mm);
\node at (3,.5) [ right, blue] {$1$};
\end{tikzpicture} \quad .
In the first case, the graph
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,0);
\draw (1,0)--(2,.5)--(3,0);
\draw (2,.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\node at (0,0) [below left] {$1$};
\draw[fill=black] (1,0) circle (1mm);
\node at (1,0) [below left] {$X$};
\draw[fill=black] (2,-.5) circle (1mm);
\node at (2,-.5) [below left] {$Z$};
\draw[fill=black] (2,.5) circle (1mm);
\node at (2,.5) [above left] {$Y$};
\draw[fill=black] (3,0) circle (1mm);
\draw[fill=black] (3,.5) circle (1mm);
\end{tikzpicture}
has the correct norm, and is therefore is the only possible graph of this form.
It remains to rule out graphs of the first form. To do so, we need to move from considering bigraphs to considering pairs of bigraphs with dual data. First, name the vertices at depth three $A$, $B$ and $C$:
\begin{tikzpicture}[baseline=0]
\draw (0,0)--(1,0)--(2,-.5)--(3,-.5);
\draw (2,.5)--(3,0);
\draw (1,0)--(2,.5)--(3,.5);
\draw[fill=black] (0,0) circle (1mm);
\node at (0,0) [below left] {$1$};
\node at (0,0) [above left, blue] {$1$};
\draw[fill=black] (1,0) circle (1mm);
\node at (1,0) [below left] {$X$};
\node at (1,0) [above left, blue] {$d$};
\draw[fill=black] (2,-.5) circle (1mm);
\node at (2,-.5) [below ] {$Z$};
\node at (2,-.5) [above right, blue] {$e$};
\draw[fill=black] (2,.5) circle (1mm);
\node at (2,.5) [below ] {$Y$};
\node at (2,.5) [above right, blue] {$d$};
\draw[fill=white] (3,-.5) circle (1mm);
\node at (3,-.5) [ right, blue] {$e$};
\draw[fill=white] (3,0) circle (1mm);
\node at (3,0) [ right, blue] {$e$};
\draw[fill=white] (3,.5) circle (1mm);
\node at (3,.5) [ right, blue] {$1$};
\node at (3,-.5) [below] {$C$};
\node at (3,0) [left] {$B$};
\node at (3,.5) [above] {$A$};
\end{tikzpicture} \quad .
Standard path-counting arguments show that the `other graph' has the same form up to depth two; call its vertices at depth two $\hat{Y}$ and $\hat{Z}$. The depth-three vertices $A$, $B$ and $B$ must each connect to $\hat{Y}$ or $\hat{Z}$. Path counting implies that each of these three vertices connects to only one of $\hat{Y}$ or $\hat{Z}$; further neither $\hat{Y}$ nor $\hat{Z}$ can be univalent (or else there would be an intermediate subfactor). So one of the depth-two vertices has degree two, and the other has degree three. Without loss of generality say $\hat{Y}$ is the degree-three vertex. Then $A$ must connect to $\hat{Y}$ (because otherwise $\dim{Z} = \frac{d+1}{d}$ is not an allowed dimension). Thus, $Y$ and $Z$, and $\hat{Y}$ and $\hat{Z}$, are self-dual (since the two vertices in each pair have distinct norms).
We've just shown that the two possibilities for the 4-partite principal graph are
\begin{tikzpicture}[inner sep=.7mm, xscale=1.5]
\begin{scope}[xshift=-4mm]
\node at (-1,0) {$1$};
\node at (-1,2) {$\hat{1}$};
\node at (-1,4) {$1$};
\node at (0,1) {$\bar{X}$};
\node at (0,3) {$X$};
\node at (1,0) {$Z$};
\node at (1,2) {$\hat{Z}$};
\node at (1,4) {$Z$};
\node at (2,0) {$Y$};
\node at (2,2) {$\hat{Y}$};
\node at (2,4) {$Y$};
\node at (3,1) {$\bar{C}$};
\node at (3,3) {$C$};
\node at (4,1) {$\bar{B}$};
\node at (4,3) {$B$};
\node at (5,1) {$\bar{A}$};
\node at (5,3) {$A$};
\end{scope}
\node(1) at (-1,0) [circle,fill] {};
\node(h1) at (-1,2)[circle,fill] {} ;
\node(11) at (-1,4)[circle,fill] {};
\node(bX) at (0,1) [circle,fill]{};
\node(X) at (0,3) [circle,fill]{};
\node(Z) at (1,0) [circle,fill]{};
\node(hZ) at (1,2) [circle,fill]{};
\node(ZZ) at (1,4)[circle,fill] {};
\node(Y) at (2,0) [circle,fill]{};
\node(hY) at (2,2) [circle,fill]{};
\node(YY) at (2,4) [circle,fill]{};
\node(bC) at (3,1) [circle, draw, fill=white]{};
\node(C) at (3,3) [circle, draw, fill=white]{};
\node(bB) at (4,1)[circle,draw, fill=white] {};
\node(B) at (4,3)[circle,draw, fill=white] {};
\node(bA) at (5,1)[circle, draw, fill=white] {};
\node(A) at (5,3)[circle, draw, fill=white] {};
\draw (11)--(X)--(ZZ)--(C);
\draw (X)--(YY)--(B);
\draw (YY)--(A);
\draw (h1)--(X)--(hZ)--(C);
\draw (X)--(hY)--(B);
\draw (hY)--(A);
\draw (h1)--(bX)--(hZ)--(bC);
\draw (bX)--(hY)--(bB);
\draw (hY)--(bA);
\draw (1)--(bX)--(Z)--(bC);
\draw (bX)--(Y)--(bB);
\draw (Y)--(bA);
\end{tikzpicture}
\begin{tikzpicture}[inner sep=.7mm, xscale=1.5]
\begin{scope}[xshift=-4mm]
\node at (-1,0) {$1$};
\node at (-1,2) {$\hat{1}$};
\node at (-1,4) {$1$};
\node at (0,1) {$\bar{X}$};
\node at (0,3) {$X$};
\node at (1,0) {$Z$};
\node at (1,2) {$\hat{Z}$};
\node at (1,4) {$Z$};
\node at (2,0) {$Y$};
\node at (2,2) {$\hat{Y}$};
\node at (2,4) {$Y$};
\node at (3,1) {$\bar{C}$};
\node at (3,3) {$C$};
\node at (4,1) {$\bar{B}$};
\node at (4,3) {$B$};
\node at (5,1) {$\bar{A}$};
\node at (5,3) {$A$};
\end{scope}
\node(1) at (-1,0) [circle,fill] {};
\node(h1) at (-1,2)[circle,fill] {} ;
\node(11) at (-1,4)[circle,fill] {};
\node(bX) at (0,1) [circle,fill]{};
\node(X) at (0,3) [circle,fill]{};
\node(Z) at (1,0) [circle,fill]{};
\node(hZ) at (1,2) [circle,fill]{};
\node(ZZ) at (1,4)[circle,fill] {};
\node(Y) at (2,0) [circle,fill]{};
\node(hY) at (2,2) [circle,fill]{};
\node(YY) at (2,4) [circle,fill]{};
\node(bC) at (3,1) [circle, draw, fill=white]{};
\node(C) at (3,3) [circle, draw, fill=white]{};
\node(bB) at (4,1)[circle,draw, fill=white] {};
\node(B) at (4,3)[circle,draw, fill=white] {};
\node(bA) at (5,1)[circle, draw, fill=white] {};
\node(A) at (5,3)[circle, draw, fill=white] {};
\draw (11)--(X)--(ZZ)--(C);
\draw (X)--(YY)--(B);
\draw (YY)--(A);
\draw (h1)--(X)--(hZ)--(B);
\draw (X)--(hY)--(C);
\draw (hY)--(A);
\draw (h1)--(bX)--(hZ)--(bB);
\draw (bX)--(hY)--(bC);
\draw (hY)--(bA);
\draw (1)--(bX)--(Z)--(bC);
\draw (bX)--(Y)--(bB);
\draw (Y)--(bA);
\end{tikzpicture}
The first of these cases is ruled out by the triple point obstruction (as stated in <cit.>, where it is attributed to <cit.> and Ocneanu), applied to the vertices $Y$ and $\hat{Y}$. The second is ruled out by using dimension data to extend the 4-partite graph up to depth 4; it must have the form
\begin{tikzpicture}[inner sep=.7mm, xscale=1.5]
\begin{scope}[xshift=-4mm]
\node at (-1,0) {$1$};
\node at (-1,2) {$\hat{1}$};
\node at (-1,4) {$1$};
\node at (0,1) {$\bar{X}$};
\node at (0,3) {$X$};
\node at (1,0) {$Z$};
\node at (1,2) {$\hat{Z}$};
\node at (1,4) {$Z$};
\node at (2,0) {$Y$};
\node at (2,2) {$\hat{Y}$};
\node at (2,4) {$Y$};
\node at (3,1) {$\bar{C}$};
\node at (3,3) {$C$};
\node at (4,1) {$\bar{B}$};
\node at (4,3) {$B$};
\node at (5,1) {$\bar{A}$};
\node at (5,3) {$A$};
\node at (6,0) {$D$};
\node at (6,2) {$\hat{D}$};
\node at (6,4) {$D$};
\node at (7,0) {$E$};
\node at (7,2) {$\hat{E}$};
\node at (7,4) {$E$};
\end{scope}
\node(1) at (-1,0) [circle,fill] {};
\node(h1) at (-1,2)[circle,fill] {} ;
\node(11) at (-1,4)[circle,fill] {};
\node(bX) at (0,1) [circle,fill]{};
\node(X) at (0,3) [circle,fill]{};
\node(Z) at (1,0) [circle,fill]{};
\node(hZ) at (1,2) [circle,fill]{};
\node(ZZ) at (1,4)[circle,fill] {};
\node(Y) at (2,0) [circle,fill]{};
\node(hY) at (2,2) [circle,fill]{};
\node(YY) at (2,4) [circle,fill]{};
\node(bC) at (3,1) [circle,fill]{};
\node(C) at (3,3) [circle,fill]{};
\node(bB) at (4,1)[circle,fill] {};
\node(B) at (4,3)[circle,fill] {};
\node(bA) at (5,1)[circle,fill] {};
\node(A) at (5,3)[circle,fill] {};
\node(D) at (6,0) [circle, draw, fill=white]{};
\node(hD) at (6,2) [circle,draw, fill=white]{};
\node(DD) at (6,4) [circle,draw, fill=white]{};
\node(E) at (7,0) [circle,draw, fill=white]{};
\node(hE) at (7,2) [circle,draw, fill=white]{};
\node(EE) at (7,4) [circle,draw, fill=white]{};
\draw (11)--(X)--(ZZ)--(C);
\draw (X)--(YY)--(B);
\draw (YY)--(A);
\draw (h1)--(X)--(hZ)--(B);
\draw (X)--(hY)--(C);
\draw (hY)--(A);
\draw (h1)--(bX)--(hZ)--(bB);
\draw (bX)--(hY)--(bC);
\draw (hY)--(bA);
\draw (1)--(bX)--(Z)--(bC);
\draw (bX)--(Y)--(bB);
\draw (Y)--(bA);
\draw (DD)--(C)--(hE)--(bC)--(D);
\draw (EE)--(B)--(hD)--(bB)--(E);
\end{tikzpicture}
but then note that the number of paths from $A$ to $\bar{C}$ is different if you go down, versus up and around. Thus this cannot be a principal graph.
Case 4: If $n \geq 4$, the index is greater than $8.29$.
The only finite-depth $1$-supertransitive subfactor with index in $(5, 3+\sqrt{5})$ has principal graph $A=\bigraph{gbg1v1p1v1x1p0x1}$.
We analyze the possibilities in terms of the dimensions of their depth-two objects.
If there were an object of dimension 1 at depth two, then we would be in one of the following two cases. Either all objects at depth two would have dimension one (in which case, the principal graph is a star and has integer index), or there would be an intermediate subfactor by Lemma <ref> (requiring the index to factor into allowed index values, which does not happen in $(5,3+\sqrt{5})$).
An even object with dimension in $(2,\sqrt{5})$ would have dimension $\frac{\sqrt{3}+\sqrt{7}}{2}$, either by the analysis of possible small dimensions in fusion categories from <cit.>, or by the classification of subfactors with index less than $5$, from <cit.>.
If this held for all objects at depth 2, $\dim{X}^2 \geq 1+\sqrt{3} + \sqrt{7} > 3 + \sqrt{5}$.
Thus, some even object $Z$ at depth $2$ has $1 < \dim{Z} < 2$. Lemma <ref> shows that this subfactor has principal graph $A$.
What happens if we drop the finite depth condition?
The above argument fails — an object at depth $2$ could have dimension in the interval $(2, \sqrt{5})$, generating an infinite depth $A_\infty$ subcategory.
Happily, we can simply run the odometer, as in <cit.>.
We can ignore all the graphs in Figure <ref>, because they must have an intermediate subfactor, which isn't possible with index in $(5,3+\sqrt{5})$.
We can also ignore the graphs
$$\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1x2x3}\right)\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1x3x2}, \bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1x3x2}\right)
by Lemma <ref>; at least one of the three objects at depth 2 has dimension $<2$.
\begin{align*}
\left(\bigraph{bwd1v1p1v1x0p0x1duals1v1x2}, \bigraph{bwd1v1p1v1x0p1x0duals1v1x2}\right) & \qquad
\left(\bigraph{bwd1v1p1v1x0p1x0duals1v1x2}, \bigraph{bwd1v1p1v1x0p0x1duals1v1x2}\right)\displaybreak[1] \\ &
\left(\bigraph{bwd1v1p1v1x0p1x0p1x0duals1v1x2}, \bigraph{bwd1v1p1v1x0p1x0p0x1duals1v1x2}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v0x0x1p0x0x1duals1v2x1x3}, \bigraph{bwd1v1p1p1v0x0x1p0x0x1duals1v2x1x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v0x0x1p0x0x1duals1v2x1x3}, \bigraph{bwd1v1p1p1v1x0x0p0x1x0duals1v2x1x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v0x1x0p1x0x0duals1v1x2x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0duals1v1x2x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0duals1v2x1x3}, \bigraph{bwd1v1p1p1v0x0x1p0x0x1duals1v2x1x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0duals1v2x1x3}, \bigraph{bwd1v1p1p1v0x1x0p1x0x0duals1v2x1x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p1x0x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p0x1x0duals1v1x2x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p1x0x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0duals1v1x2x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v0x1x0p1x0x0p1x0x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1x2x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v0x1x0p1x0x0p1x0x0duals1v1x2x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0p0x0x1duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0p0x1x0duals1v1x2x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x1x0p0x0x1duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0p0x1x0duals1v1x2x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p0x0x1p0x1x0p0x0x1duals1v2x1x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0p0x1x0duals1v2x1x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x1x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x1x0duals1v1x2x3}\right)&\qquad
\left(\bigraph{bwd1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0p1x0x0duals1v1x2x3}\right)\displaybreak[1] \\&
\left(\bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0duals1v1x2x3}, \bigraph{bwd1v1p1p1v1x0x0p1x0x0p0x1x0duals1v1x2x3}\right)
\end{align*}
Principals graphs for which there must be an intermediate subfactor.
The result of running the odometer and ignoring the weeds mentioned above is the tree shown in Figure <ref> (in which the leaves, i.e. the red and blue graphs, are the remaining weeds).
\scalebox{0.65}{
\begin{tikzpicture}
\tikzset{grow=right,level distance=130pt}
\tikzset{every tree node/.style={draw,fill=white,rectangle,rounded corners,inner sep=2pt}}
\Tree
\end{tikzpicture}
Running the odometer.
The graph
$$(\tikz[baseline=-3pt]{\node[inner sep=0pt] {$\bigraph{bwd1v1p1v1x0p1x1duals1v1x2}$};}, \tikz[baseline=-3pt]{\node[inner sep=0pt] {$\bigraph{bwd1v1p1v1x0p1x1duals1v1x2}$};} )$$
has a depth 2 object with dimension less than $2$, and again Lemma <ref> shows that the only subfactors coming from this weed have principal graph $\Gamma(\cA)$.
For each red pair, some object has an impossible dimension. The hardest case is for the graphs of the form
The last two objects have dimensions
\begin{align*}
d_1 & = \frac{1}{2}(q^4-q^2-2-q^{-2}+q^{-4}), \\
d_2 & = \frac{1}{2}(q^5-q^3-3q-3q^{-1}-q^{-3}+q^{-5}).
\end{align*}
Now $d_2 < 1.145$, and $d_2 = 1$ only if $q=1.69068...^{\pm 1}$. But then $d_1 = 1.54231...$, which is not of the form $2 \cos(\pi/n)$.
Putting this together, we have the proof of Theorem <ref>.
§ CONNECTIONS
§.§ Bi-unitary connections on $\Gamma(\cA)$
The four-partite principal graph for $\cA$ is
\begin{tikzpicture}[inner sep=.7mm, xscale=1.5,yscale=1.2]
\begin{scope}[xshift=-4mm]
\node at (-1,0) {$1$};
\node at (-1,2) {$\hat{1}$};
\node at (-1,4) {$1$};
\node at (0,1) {$\bar{X}$};
\node at (0,3) {$X$};
\node at (1,0) {$Z$};
\node at (1,2) {$\hat{Z}$};
\node at (1,4) {$Z$};
\node at (2,0) {$Y$};
\node at (2,2) {$\hat{Y}$};
\node at (2,4) {$Y$};
\node at (3,1) {$\bar{W}$};
\node at (3,3) {$W$};
\node at (4,1) {$\bar{g}$};
\node at (4,3) {$g$};
\end{scope}
\node(1) at (-1,0) [circle,fill] {};
\node(h1) at (-1,2)[circle,fill] {} ;
\node(11) at (-1,4)[circle,fill] {};
\node(bX) at (0,1) [circle,fill]{};
\node(X) at (0,3) [circle,fill]{};
\node(Z) at (1,0) [circle,fill]{};
\node(hZ) at (1,2) [circle,fill]{};
\node(ZZ) at (1,4)[circle,fill] {};
\node(Y) at (2,0) [circle,fill]{};
\node(hY) at (2,2) [circle,fill]{};
\node(YY) at (2,4) [circle,fill]{};
\node(bW) at (3,1) [circle,fill]{};
\node(W) at (3,3) [circle,fill]{};
\node(bg) at (4,1)[circle,fill] {};
\node(g) at (4,3)[circle,fill] {};
\draw (11)--(X)--(YY)--(W)--(ZZ)--(X);
\draw (YY)--(g);
\draw (h1)--(X)--(hY)--(W)--(hZ)--(X);
\draw (hY)--(g);
\draw (h1)--(bX)--(hY)--(bW)--(hZ)--(bX);
\draw (hY)--(bg);
\draw (1)--(bX)--(Y)--(bW)--(Z)--(bX);
\draw (Y)--(bg);
\end{tikzpicture}
We determine which biunitary connections are flat, using a condition from <cit.> which follows from flatness.
Recall the notion of `diagrammatic gauge' from <cit.>. It is a subset of the full gauge group orbit of any bi-unitary connection, and characterized by having all connection entries in the 1-by-1 and 2-by-2 matrices corresponding to the edges of the principal graph before the first branch point real, and the connection matrix at the branch point having real first row and first column, with the top-left entry having sign $(-1)^{n+1}$ and the other entries in the first row and column being positive. It is called the diagrammatic gauge because it corresponds to using Temperley-Lieb diagrams as the basis for the appropriate 1-dimensional trivalent vertex spaces.
In particular, if a connection $K$ is in the diagrammatic gauge, and $U$ is the connection matrix at the first branch point of the principal graph, then the eigenvalues of $U U^t$ are $+1$ with multiplicity $2$, along with rotational eigenvalues of the planar algebra generators corresponding to the branches. For $\cA$, there is just one such generator (since the branch point is a triple point), and its rotational eigenvalue is $+1$, since the vertices past the branch point are self-dual. Thus all the eigenvalues of $U U^t$ are $+1$, so this matrix is the identity. We use this condition to drastically simplify the calculation of possible flat bi-unitary connections.
We in fact find there are two bi-unitary connections satisfying this eigenvalue condition, and subsequently, in <ref> discover that only one of them is actually flat, by explicitly looking for flat elements in the graph planar algebra.
\begin{equation*}
\begin{tikzpicture}
\principalmatrixA
\houses
\end{tikzpicture}
\begin{tikzpicture}
\dualmatrixA
\shuffledhouses
\end{tikzpicture}
\end{equation*}
Houses in Auvers, 2, by Vincent van Gogh, illustrating the bijection between nonzero connection entries on the principal and dual principal graphs.
The connection entries $1X\hat{1}\bar{X}, ZX\hat{1}\bar{X}, YX\hat{1}\bar{X}, 1X\hat{Z}\bar{X}$ and $1X\hat{Y}\bar{X}$ each lie in a 1-by-1 matrix, so have norm 1. The diagrammatic gauge choice means these are all exactly $+1$. We then use the renormalization axiom to transfer these across to the dual graph. Below $d=\dim{X}=\dim{Y} \approx 2.24698$ and $e=\dim{Z}=\dim{W}= d^2 - d - 1 \approx 1.80194$.
\begin{equation*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\stepAp
\end{tikzpicture}
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\stepAd
\end{tikzpicture}
\end{equation*}
We quickly see that the 3-by-3 $X \bar{X}$ matrix in the dual connection must be symmetric. Call this matrix $U$ and recall $U U^t = \bf{1}$. Now $U = U^t = U^{-1} = U^*$ tells us that $U$ is real, and this allows us to completely solve for $U$. We obtain two solutions,
$$U = \begin{pmatrix}
\frac{1}{d} & \frac{\sqrt{e}}{d} & \frac{\sqrt{d}}{d} \\
\frac{\sqrt{e}}{d} & \frac{e}{d} r^{(i)}_1 & \sqrt{\frac{e}{d}} r^{(i)}_2 \\
\frac{\sqrt{d}}{d} & \sqrt{\frac{e}{d}} r^{(i)}_2 & r^{(i)}_3
\end{pmatrix}
\begin{align*}
r^{(1)}_1 & = d^2 -4d+3 \\
r^{(1)}_2 & = d^2 -3d +2\\
r^{(1)}_3 & = d^2 -3d +1\\
\intertext{and}
r^{(2)}_1 & = -d^2+2d+1\\
r^{(2)}_2 & = -d^2+d+2\\
r^{(2)}_3 & = -d^2+d+3
\end{align*}
We next transfer these entries back to the principal graph.
\begin{equation*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\stepBp
\end{tikzpicture}
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\stepBd
\end{tikzpicture}
\end{equation*}
With $s_i = \sqrt{1-r_i^2}$, we next introduce six new variables $\xi_1, \xi_2, \xi_2', \alpha_1, \alpha_2$ and $\alpha_2' \in \bbT$ on the unit circle. Using these we complete the 2-by-2 matrices on the principal graph, and transfer all the new entries back over the to dual graph.
\begin{equation*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\stepCp
\end{tikzpicture}
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\stepCd
\end{tikzpicture}
\end{equation*}
We can now fill in the bottom right entries of these 2-by-2 matrices; we also introduce variables $\beta_1, \ldots \beta_5 \in \bbT$ on the unit circle for the remaining 1-by-1 matrices in the dual connection, and transfer everything back to the principal connection matrix:
\begin{align*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\stepDp
\end{tikzpicture}
% \displaybreak[1] \\
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\stepDd
\end{tikzpicture}
\end{align*}
We check our work up to this point by verifying that the rows of the 3-by-3 matrix in the principal connection have unit norm. Next, we eliminate some phases using orthogonality of the rows and of the columns of this matrix. All dot products of rows with each other, and columns with each other, have the form
$$m_1 \phi_1 + m_2 \phi_2 + m_3 \phi_3=0, \qquad m_i \in \mathbb{R}, \qquad \phi_i \in \mathbb{T}$$
with $m_1 + m_2+m_3=0$.
This implies (via the triangle inequality) that $\phi_1 = \phi_2 = \phi_3$, from which we deduce
\begin{align*}
\beta_2 &= \sigma^{(i)} \frac{\beta_1 \alpha_1 \xi_2 \xi_2'}{\xi_1 \alpha_2 \alpha_2'} \\
\beta_4 &= - \sigma^{(i)} \frac{\beta_3 \alpha_2 \alpha_2'}{ \alpha_1} \\
\beta_5 &= \beta_1 \beta_3
\end{align*}
where $\sigma^{(1)}=+1$ and $\sigma^{(2)}=-1$.
The connection matrices are thus:
\begin{align*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\stepEp
\end{tikzpicture}
%\displaybreak[1] \\
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\stepEd
\end{tikzpicture}
\end{align*}
One can readily verify that each block is a unitary matrix.
Now we act by the remaining gauge subgroup, namely the subgroup corresponding to edges $ZW, YW$ and $Yg$ (columns in the principal matrix), $\hat{Z}\bar{W}, \hat{Y}\bar{W}$ and $\hat{Y}\bar{g}$ (rows in the principal matrix), $\bar{W}{Z}, \bar{W}{Y}$ and $\bar{g} Y$ (columns in the dual matrix) and $W\hat{Z}$, $W\hat{Y}$ and $g\hat{Y}$ (rows in the dual matrix). In particular, we take the gauge element
\begin{align*}
\mu \left(W,\hat{Z}\right) & = -\frac{\alpha _2'}{\xi _2'} &
\mu \left(g,\hat{Y}\right) & = \frac{1}{\beta _3} &
\mu \left(\bar{W},Z\right) & = \frac{1}{\alpha _2} \\
\mu \left(\bar{g},Y\right) & = \frac{1}{\beta _1} &
\mu (Z,W) & = \frac{\alpha _1 \xi _2'}{\xi _1 \alpha _2'} &
\mu \left(\hat{Z},\bar{W}\right) & = \frac{\alpha _2}{\alpha _1} \\
\mu \left(\bar{W},Y\right) & = \frac{\alpha _1}{\alpha _2 \alpha _2'} &
\mu \left(W,\hat{Y}\right) & = -\frac{\alpha _2 \xi _1 \alpha _2'}{\alpha _1 \xi _2 \xi _2'} &
\mu (Y,g) & = 1 \\
\mu (Y,W) & = 1 &
\mu \left(\hat{Y},\bar{g}\right) & = 1 &
\mu \left(\hat{Y},\bar{W}\right) & = 1
\end{align*}
and see that any bi-unitary connection is gauge equivalent to $K^{(1)}$ or $K^{(2)}$, obtained from the matrices below by substituting in the values $r^{(i)}_j$ and $s^{(i)}_j$ given above for $r_j$ and $s_j$. (Equivalently, one can think of $K^{(i)}$ as obtained from the matrices immediately above by setting each $\alpha_i, \alpha'_i, \beta_i$ to $1$, and each $\xi_i, \xi'_i$ to $-1$.)
\begin{align*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\Gaugedp
\end{tikzpicture}
%\displaybreak[1] \\
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\Gaugedd
\end{tikzpicture}
\end{align*}
We next transfer these elements to the lopsided two-sided graph planar algebra, and make a gauge choice so that all of the coefficients lie in the field generated by the index $d^2 = \lambda$.
Thus, we define
$$K^{(i)}_{\text{lopsided},0} = \natural(K^{(i)})$$
for $i=1$ and $2$, and see that these are given by
\begin{align*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\cellA{r1X}{c1X}{}{}{}{$\frac{1}{d}$}
\cellA{r1X}{cZX}{}{}{}{$\frac{1}{d}$}
\cellA{r1X}{cYX}{}{}{}{$\frac{1}{d}$}
\cellA{rZW}{cZW}{}{}{}{$-\frac{r_1}{d}$}
\cellA{rZW}{cZX}{}{}{}{$\frac{s_1}{\sqrt{d} \sqrt{e}}$}
\cellA{rZW}{cYW}{}{}{}{$-\frac{r_2}{d}$}
\cellA{rZW}{cYX}{}{}{}{$\frac{s_2}{\sqrt{d} \sqrt{e}}$}
\cellA{rZX}{c1X}{}{}{}{$\frac{1}{d}$}
\cellA{rZX}{cZW}{}{}{}{$\frac{\sqrt{e} s_1}{d^{3/2}}$}
\cellA{rZX}{cZX}{}{}{}{$\frac{r_1}{d}$}
\cellA{rZX}{cYW}{}{}{}{$\frac{\sqrt{e} s_2}{d^{3/2}}$}
\cellA{rZX}{cYX}{}{}{}{$\frac{r_2}{d}$}
\cellA{rYX}{c1X}{}{}{}{$\frac{1}{d}$}
\cellA{rYX}{cZW}{}{}{}{$\frac{\sqrt{e} s_2}{d^{3/2}}$}
\cellA{rYX}{cZX}{}{}{}{$\frac{r_2}{d}$}
\cellA{rYX}{cYW}{}{}{}{$-\frac{e^{3/2} s_1}{d^{5/2}}$}
\cellA{rYX}{cYg}{}{}{}{$\frac{1}{d^2}$}
\cellA{rYX}{cYX}{}{}{}{$\frac{r_3}{d}$}
\cellA{rYg}{cYW}{}{}{}{$-\frac{e \sigma }{d^2}$}
\cellA{rYg}{cYg}{}{}{}{$\frac{1}{d^2}$}
\cellA{rYg}{cYX}{}{}{}{$\frac{1}{d}$}
\cellA{rYW}{cZW}{}{}{}{$-\frac{r_2}{d}$}
\cellA{rYW}{cZX}{}{}{}{$\frac{s_2}{\sqrt{d} \sqrt{e}}$}
\cellA{rYW}{cYW}{}{}{}{$\frac{e r_1}{d^2}$}
\cellA{rYW}{cYg}{}{}{}{$-\frac{\sigma }{d^2}$}
\cellA{rYW}{cYX}{}{}{}{$-\frac{\sqrt{e} s_1}{d^{3/2}}$}
\end{tikzpicture}
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\cellA{rXZ}{cXY}{}{}{}{$r_2$}
\cellA{rXZ}{cXZ}{}{}{}{$\frac{e r_1}{d}$}
\cellA{rXZ}{cX1}{}{}{}{$\frac{1}{d}$}
\cellA{rXZ}{cWZ}{}{}{}{$\frac{\sqrt{e} s_1}{\sqrt{d}}$}
\cellA{rXZ}{cWY}{}{}{}{$\frac{\sqrt{d} s_2}{\sqrt{e}}$}
\cellA{rX1}{cXY}{}{}{}{$1$}
\cellA{rX1}{cXZ}{}{}{}{$\frac{e}{d}$}
\cellA{rX1}{cX1}{}{}{}{$\frac{1}{d}$}
\cellA{rXY}{cXY}{}{}{}{$r_3$}
\cellA{rXY}{cXZ}{}{}{}{$\frac{e r_2}{d}$}
\cellA{rXY}{cX1}{}{}{}{$\frac{1}{d}$}
\cellA{rXY}{cWZ}{}{}{}{$\frac{\sqrt{e} s_2}{\sqrt{d}}$}
\cellA{rXY}{cWY}{}{}{}{$-\frac{\sqrt{e} s_1}{\sqrt{d}}$}
\cellA{rXY}{cgY}{}{}{}{$1$}
\cellA{rWZ}{cXY}{}{}{}{$\frac{\sqrt{d} s_2}{\sqrt{e}}$}
\cellA{rWZ}{cXZ}{}{}{}{$\frac{\sqrt{e} s_1}{\sqrt{d}}$}
\cellA{rWZ}{cWZ}{}{}{}{$-r_1$}
\cellA{rWZ}{cWY}{}{}{}{$-\frac{d r_2}{e}$}
\cellA{rWY}{cXY}{}{}{}{$-\frac{\sqrt{e} s_1}{\sqrt{d}}$}
\cellA{rWY}{cXZ}{}{}{}{$\frac{\sqrt{e} s_2}{\sqrt{d}}$}
\cellA{rWY}{cWZ}{}{}{}{$-r_2$}
\cellA{rWY}{cWY}{}{}{}{$r_1$}
\cellA{rWY}{cgY}{}{}{}{$-\sigma $}
\cellA{rgY}{cXY}{}{}{}{$1$}
\cellA{rgY}{cWY}{}{}{}{$-\sigma $}
\cellA{rgY}{cgY}{}{}{}{$1$}
\end{tikzpicture}
\end{align*}
Next, we choose elements of the complex gauge group $\mu^{(i)}$ given by the formulas (unspecified entries are all $1$)
\begin{align*}
\mu^{(1)}(1, X) & = \lambda_{1,-2,-1,1}^{(2.25)} &
\mu^{(1)}(Z, X) & = \lambda_{1,-2,-1,1}^{(2.25)} &
\mu^{(1)}(Z, W) & = \lambda_{13,0,-88,0,-17,0,1}^{(2.637)} \\
\mu^{(1)}(Y, X) & = \lambda_{13,-20,9,-1}^{(0.7645)} &
\mu^{(1)}(Y, W) & = \lambda_{169,-15,-16,1}^{(0.3244)} &
\mu^{(1)}(Y, g) & = \lambda_{13,-25,4,1}^{(1.718)} \\
\mu^{(1)}(W, \hat{Z}) & = \lambda_{1,0,-5,0,-22,0,13}^{(2.77)} &
\mu^{(1)}(\bar{X}, Y) & = \lambda_{1,1,-16,13}^{(2.94)} &
\mu^{(1)}(\bar{W}, Z) & = \lambda_{13,0,38,0,-45,0,1}^{(0.9413)} \\
\mu^{(1)}(\bar{g}, Y) & = \lambda_{1,1,-16,13}^{(2.94)} &
\mu^{(1)}(\hat{Z}, \bar{W}) & = \lambda_{1,0,-5,0,-22,0,13}^{(2.77)} & &
\end{align*}
\begin{align*}
\mu^{(2)}(1, X) & = \lambda_{1,-2,-1,1}^{(2.25)} &
\mu^{(2)}(Z, X) & = \lambda_{1,-2,-1,1}^{(2.25)} &
\mu^{(2)}(Z, W) & = \lambda_{1,0,-16,0,-29,0,1}^{(4.200)} \\
\mu^{(2)}(Y, X) & = \lambda_{1,-6,5,-1}^{(5.049)} &
\mu^{(2)}(Y, W) & = \lambda_{1,-15,12,1}^{(14.1)} &
\mu^{(2)}(Y, g) & = \lambda_{1,-11,-4,1}^{(11.34)} \\
\mu^{(2)}(W, \hat{Z}) & = \lambda_{1,0,-1,0,-2,0,1}^{(0.6671)} &
\mu^{(2)}(\bar{X}, Y) & = \lambda_{1,-1,-2,1}^{(0.445)} &
\mu^{(2)}(\bar{W}, Z) & = \lambda_{1,0,-2,0,-1,0,1}^{(1.499)} \\
\mu^{(2)}(\bar{g}, Y) & = \lambda_{1,-1,-2,1}^{(0.445)} &
\mu^{(2)}(\hat{Z}, \bar{W}) & = \lambda_{1,0,-1,0,-2,0,1}^{(0.6671)} & &
\end{align*}
(Here, $\lambda_p^{(x)}$ denotes the root of $\sum p_i x^{n-i}$ which is approximately equal to $x$.)
Applying these, we have $K^{(1)}_{\text{lopsided}} = \mu^{(1)} (K^{(1)}_{\text{lopsided},0})$ given by
\begin{tikzpicture}[xscale=2, yscale=.8]
\principalmatrixA
\longcellA{r1X}{c1X}{}{}{}{$1$}
\longcellA{r1X}{cZX}{}{}{}{$1$}
\longcellA{r1X}{cYX}{}{}{}{$1$}
\longcellA{rZX}{c1X}{}{}{}{$1$}
\longcellA{rZX}{cZW}{}{}{}{$1$}
\longcellA{rZX}{cZX}{}{}{}{$\{-4,21,-5\}$}
\longcellA{rZX}{cYW}{}{}{}{$1$}
\longcellA{rZX}{cYX}{}{}{}{$\{-3,16,-4\}$}
\longcellA{rZW}{cZW}{}{}{}{$\{4,-21,12\}$}
\longcellA{rZW}{cZX}{}{}{}{$1$}
\longcellA{rZW}{cYW}{}{}{}{$\left\{-\frac{11}{13},\frac{61}{13},-\frac{32}{13}\right\}$}
\longcellA{rZW}{cYX}{}{}{}{$1$}
\longcellA{rYg}{cYW}{}{}{}{$\left\{-\frac{11}{13},\frac{61}{13},-\frac{32}{13}\right\}$}
\longcellA{rYg}{cYg}{}{}{}{$1$}
\longcellA{rYg}{cYX}{}{}{}{$1$}
\longcellA{rYX}{c1X}{}{}{}{$1$}
\longcellA{rYX}{cZW}{}{}{}{$1$}
\longcellA{rYX}{cZX}{}{}{}{$\{-3,16,-4\}$}
\longcellA{rYX}{cYW}{}{}{}{$\left\{-\frac{15}{13},\frac{82}{13},-\frac{33}{13}\right\}$}
\longcellA{rYX}{cYg}{}{}{}{$1$}
\longcellA{rYX}{cYX}{}{}{}{$\{-3,16,-5\}$}
\longcellA{rYW}{cZW}{}{}{}{$\left\{-\frac{11}{13},\frac{61}{13},-\frac{32}{13}\right\}$}
\longcellA{rYW}{cZX}{}{}{}{$1$}
\longcellA{rYW}{cYW}{}{}{}{$\left\{\frac{105}{169},-\frac{613}{169},\frac{400}{169}\right\}$}
\longcellA{rYW}{cYg}{}{}{}{$\left\{-\frac{11}{13},\frac{61}{13},-\frac{32}{13}\right\}$}
\longcellA{rYW}{cYX}{}{}{}{$\left\{-\frac{15}{13},\frac{82}{13},-\frac{33}{13}\right\}$}
\end{tikzpicture}
\begin{tikzpicture}[xscale=2, yscale=.8]
\dualmatrixA
\longcellA{rXY}{cXY}{}{}{}{$\{2,-11,3\}$}
\longcellA{rXY}{cXZ}{}{}{}{$\{-2,11,-4\}$}
\longcellA{rXY}{cX1}{}{}{}{$1$}
\longcellA{rXY}{cWZ}{}{}{}{$\{-1,6,-3\}$}
\longcellA{rXY}{cWY}{}{}{}{$\left\{\frac{4}{13},-\frac{21}{13},\frac{1}{13}\right\}$}
\longcellA{rXY}{cgY}{}{}{}{$\{1,-5,2\}$}
\longcellA{rXZ}{cXY}{}{}{}{$\{3,-16,5\}$}
\longcellA{rXZ}{cXZ}{}{}{}{$\{-3,16,-6\}$}
\longcellA{rXZ}{cX1}{}{}{}{$1$}
\longcellA{rXZ}{cWZ}{}{}{}{$\{-1,6,-3\}$}
\longcellA{rXZ}{cWY}{}{}{}{$\{1,-5,2\}$}
\longcellA{rX1}{cXY}{}{}{}{$\{1,-5,2\}$}
\longcellA{rX1}{cXZ}{}{}{}{$\{-1,6,-3\}$}
\longcellA{rX1}{cX1}{}{}{}{$1$}
\longcellA{rWY}{cXY}{}{}{}{$\left\{-\frac{12}{13},\frac{63}{13},-\frac{16}{13}\right\}$}
\longcellA{rWY}{cXZ}{}{}{}{$\{1,-5,2\}$}
\longcellA{rWY}{cWZ}{}{}{}{$\left\{-\frac{4}{13},\frac{21}{13},-\frac{14}{13}\right\}$}
\longcellA{rWY}{cWY}{}{}{}{$\left\{-\frac{85}{169},\frac{456}{169},-\frac{187}{169}\right\}$}
\longcellA{rWY}{cgY}{}{}{}{$\left\{-\frac{1}{13},\frac{2}{13},\frac{3}{13}\right\}$}
\longcellA{rWZ}{cXY}{}{}{}{$\{-1,6,-2\}$}
\longcellA{rWZ}{cXZ}{}{}{}{$\{1,-5,2\}$}
\longcellA{rWZ}{cWZ}{}{}{}{$\{3,-13,7\}$}
\longcellA{rWZ}{cWY}{}{}{}{$\left\{-\frac{1}{13},\frac{2}{13},\frac{3}{13}\right\}$}
\longcellA{rgY}{cXY}{}{}{}{$\{0,1,0\}$}
\longcellA{rgY}{cWY}{}{}{}{$\left\{-\frac{5}{13},\frac{23}{13},-\frac{11}{13}\right\}$}
\longcellA{rgY}{cgY}{}{}{}{$\{0,1,0\}$}
\end{tikzpicture}
and $K^{(2)}_{\text{lopsided}} = \mu^{(2)} (K^{(2)}_{\text{lopsided},0})$ given by
\begin{align*}
\begin{tikzpicture}[yscale=.8]
\principalmatrixA
\cellA{r1X}{c1X}{}{}{}{$1$}
\cellA{r1X}{cZX}{}{}{}{$1$}
\cellA{r1X}{cYX}{}{}{}{$1$}
\cellA{rZX}{c1X}{}{}{}{$1$}
\cellA{rZX}{cZW}{}{}{}{$1$}
\cellA{rZX}{cZX}{}{}{}{$\{2,-11,5\}$}
\cellA{rZX}{cYW}{}{}{}{$1$}
\cellA{rZX}{cYX}{}{}{}{$\{1,-6,4\}$}
\cellA{rZW}{cZW}{}{}{}{$\{2,-11,4\}$}
\cellA{rZW}{cZX}{}{}{}{$1$}
\cellA{rZW}{cYW}{}{}{}{$\{1,-5,2\}$}
\cellA{rZW}{cYX}{}{}{}{$1$}
\cellA{rYW}{cZW}{}{}{}{$\{1,-5,2\}$}
\cellA{rYW}{cZX}{}{}{}{$1$}
\cellA{rYW}{cYW}{}{}{}{$\{1,-5,2\}$}
\cellA{rYW}{cYX}{}{}{}{$\{1,-6,3\}$}
\cellA{rYW}{cYg}{}{}{}{$\{1,-5,2\}$}
\cellA{rYg}{cYW}{}{}{}{$\{1,-5,2\}$}
\cellA{rYg}{cYX}{}{}{}{$1$}
\cellA{rYg}{cYg}{}{}{}{$1$}
\cellA{rYX}{c1X}{}{}{}{$1$}
\cellA{rYX}{cZW}{}{}{}{$1$}
\cellA{rYX}{cZX}{}{}{}{$\{1,-6,4\}$}
\cellA{rYX}{cYW}{}{}{}{$\{1,-6,3\}$}
\cellA{rYX}{cYX}{}{}{}{$\{1,-6,5\}$}
\cellA{rYX}{cYg}{}{}{}{$1$}
\end{tikzpicture}
\begin{tikzpicture}[yscale=.8]
\dualmatrixA
\cellA{rXY}{cX1}{}{}{}{$1$}
\cellA{rXY}{cXZ}{}{}{}{$\{-2,11,-6\}$}
\cellA{rXY}{cXY}{}{}{}{$\{2,-11,5\}$}
\cellA{rXY}{cWZ}{}{}{}{$\{-1,6,-3\}$}
\cellA{rXY}{cWY}{}{}{}{$\{0,-1,1\}$}
\cellA{rXY}{cgY}{}{}{}{$\{1,-5,2\}$}
\cellA{rXZ}{cX1}{}{}{}{$1$}
\cellA{rXZ}{cXZ}{}{}{}{$\{-1,6,-4\}$}
\cellA{rXZ}{cXY}{}{}{}{$\{1,-6,3\}$}
\cellA{rXZ}{cWZ}{}{}{}{$\{-1,6,-3\}$}
\cellA{rXZ}{cWY}{}{}{}{$\{1,-5,2\}$}
\cellA{rX1}{cX1}{}{}{}{$1$}
\cellA{rX1}{cXZ}{}{}{}{$\{-1,6,-3\}$}
\cellA{rX1}{cXY}{}{}{}{$\{1,-5,2\}$}
\cellA{rWY}{cXZ}{}{}{}{$\{1,-5,2\}$}
\cellA{rWY}{cXY}{}{}{}{$\{0,-1,0\}$}
\cellA{rWY}{cWZ}{}{}{}{$\{0,1,0\}$}
\cellA{rWY}{cWY}{}{}{}{$\{1,-4,1\}$}
\cellA{rWY}{cgY}{}{}{}{$\{1,-4,1\}$}
\cellA{rWZ}{cXZ}{}{}{}{$\{1,-5,2\}$}
\cellA{rWZ}{cXY}{}{}{}{$\{-1,6,-2\}$}
\cellA{rWZ}{cWZ}{}{}{}{$\{-1,5,-1\}$}
\cellA{rWZ}{cWY}{}{}{}{$\{1,-4,1\}$}
\cellA{rgY}{cXY}{}{}{}{$\{0,1,0\}$}
\cellA{rgY}{cWY}{}{}{}{$\{1,-3,1\}$}
\cellA{rgY}{cgY}{}{}{}{$\{0,1,0\}$}
\end{tikzpicture}
\end{align*}
We use the abbreviation $\{a,b,c\}$ as shorthand for $a d^4 + b d^2 + c$, where as usual $d^2 \approx 5.0489$ is the index.
These may look hideous, but secretly something wonderful has taken place. These explicit connections have all their entries in a fixed (and rather small) algebraic number field. This means that all subsequent calculations (looking for flat elements, particularly) can be efficiently performed by a computer, without the time consuming difficulties of exact arithmetic on arbitrary algebraic numbers.
§.§ Bi-unitary connections on $\Gamma(\cB)$
In this section, we show that there is a unique bi-unitary connection on $\Gamma(\cB)$, up to gauge equivalence. Moreover, we find a bi-invertible connection in the same complex gauge orbit whose entries lie in $\mathbb{Q}[\delta^2]$. The calculation is straightforward, and does not use flatness in any way.
\begin{tikzpicture}[inner sep=.7mm, yscale=1.7]
\begin{scope}[xshift=-4mm]
\node at (-1,0) {$1$};
\node at (-1,2) {$\hat{1}$};
\node at (-1,4) {$1$};
\node at (0,1) {$\bar{f}$};
\node at (0,3) {$f$};
\node at (1,0) {$A$};
\node at (1,2) {$\hat{H}$};
\node at (1,4) {$A$};
\node at (2,1) {$\bar{B}$};
\node at (2,3) {$B$};
\node at (3,1) {$\bar{F}$};
\node at (3,3) {$F$};
\node at (4,0) {$G$};
\node at (4,2) {$\hat{I}$};
\node at (4,4) {$G$};
\node at (5,0) {$C$};
\node at (5,2) {$\hat{J}$};
\node at (5,4) {$C$};
\node at (6,0) {$E$};
\node at (6,2) {$\hat{K}$};
\node at (6,4) {$E$};
\node at (7,3) {$z$};
\node at (7,1) {$\bar{z}$};
\node at (8,3) {$D$};
\node at (8,1) {$\bar{D}$};
\end{scope}
\node(1) at (-1,0) [circle,fill] {};
\node(h1) at (-1,2) [circle,fill] {};
\node(11) at (-1,4) [circle,fill] {};
\node(bf) at (0,1) [circle,fill] {};
\node(f) at (0,3) [circle,fill] {};
\node(A) at (1,0) [circle,fill] {};
\node(hH) at (1,2) [circle,fill] {};
\node(AA) at (1,4) [circle,fill] {};
\node(bB) at (2,1) [circle,fill] {};
\node(B) at (2,3) [circle,fill] {};
\node(bF) at (3,1) [circle,fill] {};
\node(F) at (3,3) [circle,fill] {};
\node(G) at (4,0) [circle,fill] {};
\node(hI) at (4,2) [circle,fill] {};
\node(GG) at (4,4) [circle,fill] {};
\node(C) at (5,0) [circle,fill] {};
\node(hJ) at (5,2) [circle,fill] {};
\node(CC) at (5,4) [circle,fill] {};
\node(E) at (6,0) [circle,fill] {};
\node(hK) at (6,2) [circle,fill] {};
\node(EE) at (6,4) [circle,fill] {};
\node(z) at (7,3) [circle,fill] {};
\node(bz) at (7,1) [circle,fill] {};
\node(D) at (8,3) [circle,fill] {};
\node(bD) at (8,1) [circle,fill] {};
\draw (11)--(f)--(AA)--(B)--(CC)--(D)--(EE)--(F)--(AA);
\draw (F)--(GG)--(z);
\draw (h1)--(f)--(hH)--(B)--(hJ)--(D)--(hK)--(F)--(hH);
\draw (F)--(hI)--(z);
\draw (h1)--(bf)--(hH)--(bB)--(hI)--(bD)--(hK)--(bF)--(hH);
\draw (bF)--(hJ)--(bz);
\draw (1)--(bf)--(A)--(bB)--(G)--(bD)--(E)--(bF)--(A);
\draw (bF)--(C)--(bz);
\end{tikzpicture}
Notice that the dual principal matrix has only 1-by-1 and 2-by-2 blocks in it. Further, each 2-by-2 block contains an entry which is in a 1-by-1 block in the principal matrix; hence, the norms of all the entries in the dual principal (and therefore also principal) matrix are easily determined. They are the following:
(Here $\lambda_1$, $\lambda_2$, and $\lambda_3$ are
the roots of $x^6-2x^4-x^2+1$, $x^6-x^4-2x^2+1$ and $x^6+9x^4-x^2-1$ which are approximately $-0.744955, 0.667115$ and $0.619712$, respectively.
As above, we use the abbreviation $\{a,b,c\}$ as shorthand for $a d^4 + b d^2 + c$.)
\begin{equation*}
% generated by the command 'PrincipalConnectionNormMatrix[H, connB, vNshort[H]]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\begin{tikzpicture}[yscale=.8]
\principalmatrixB
\normcellB{r1f}{c1f}{}{}{}{$1$}
\normcellB{r1f}{cAf}{}{}{}{$1$}
\normcellB{rHf}{c1f}{}{}{}{$1$}
\normcellB{rHf}{cAF}{}{}{}{$-\lambda _1$}
\normcellB{rHf}{cAf}{}{}{}{$\{1,-5,0\}$}
\normcellB{rHf}{cAB}{}{}{}{$\lambda _3$}
\normcellB{rHF}{cAF}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rHF}{cAf}{}{}{}{$-\lambda _1$}
\normcellB{rHF}{cAB}{}{}{}{$\lambda _2 \{-2,11,-4\}$}
\normcellB{rHF}{cCB}{}{}{}{$1$}
\normcellB{rHF}{cEF}{}{}{}{$1$}
\normcellB{rHB}{cAF}{}{}{}{$\lambda _2 \{-2,11,-4\}$}
\normcellB{rHB}{cAf}{}{}{}{$\lambda _3$}
\normcellB{rHB}{cAB}{}{}{}{$\{3,-16,5\}$}
\normcellB{rHB}{cGF}{}{}{}{$1$}
\normcellB{rID}{cGF}{}{}{}{$-\lambda _1$}
\normcellB{rID}{cGz}{}{}{}{$\lambda _2$}
\normcellB{rID}{cEF}{}{}{}{$1$}
\normcellB{rIB}{cAF}{}{}{}{$1$}
\normcellB{rIB}{cGF}{}{}{}{$\lambda _2$}
\normcellB{rIB}{cGz}{}{}{}{$-\lambda _1$}
\normcellB{rJF}{cAB}{}{}{}{$1$}
\normcellB{rJF}{cCB}{}{}{}{$\lambda _2$}
\normcellB{rJF}{cCD}{}{}{}{$-\lambda _1$}
\normcellB{rJF}{cED}{}{}{}{$1$}
\normcellB{rJz}{cCB}{}{}{}{$-\lambda _1$}
\normcellB{rJz}{cCD}{}{}{}{$\lambda _2$}
\normcellB{rKF}{cAF}{}{}{}{$1$}
\normcellB{rKF}{cCD}{}{}{}{$1$}
\normcellB{rKF}{cEF}{}{}{}{$\{-1,6,-4\}$}
\normcellB{rKF}{cED}{}{}{}{$\lambda _1 \{1,-6,4\}$}
\normcellB{rKD}{cGF}{}{}{}{$1$}
\normcellB{rKD}{cEF}{}{}{}{$\lambda _1 \{1,-6,4\}$}
\normcellB{rKD}{cED}{}{}{}{$\{-1,6,-4\}$}
\end{tikzpicture}
\end{equation*}
\begin{equation*}
% generated by the command 'DualConnectionNormMatrix[H, connB, vNshort[H]]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\begin{tikzpicture}[yscale=.8]
\dualmatrixB
\normcellB{rf1}{cfA}{}{}{}{$\lambda _3 \{2,-11,6\}$}
\normcellB{rf1}{cf1}{}{}{}{$\{2,-11,5\}$}
\normcellB{rfH}{cfA}{}{}{}{$\{2,-11,5\}$}
\normcellB{rfH}{cf1}{}{}{}{$\lambda _3 \{2,-11,6\}$}
\normcellB{rfH}{cBA}{}{}{}{$1$}
\normcellB{rfH}{cFA}{}{}{}{$1$}
\normcellB{rBH}{cfA}{}{}{}{$1$}
\normcellB{rBH}{cBA}{}{}{}{$1$}
\normcellB{rBH}{cFA}{}{}{}{$\{2,-11,5\}$}
\normcellB{rBH}{cFC}{}{}{}{$\lambda _3 \{2,-11,6\}$}
\normcellB{rBJ}{cFA}{}{}{}{$\lambda _3 \{2,-11,6\}$}
\normcellB{rBJ}{cFC}{}{}{}{$\{2,-11,5\}$}
\normcellB{rBJ}{czC}{}{}{}{$1$}
\normcellB{rFH}{cfA}{}{}{}{$1$}
\normcellB{rFH}{cBA}{}{}{}{$\{2,-11,5\}$}
\normcellB{rFH}{cBG}{}{}{}{$\lambda _3 \{2,-11,6\}$}
\normcellB{rFH}{cFA}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rFH}{cFE}{}{}{}{$\lambda _2 \{1,-5,1\}$}
\normcellB{rFK}{cFA}{}{}{}{$\lambda _2 \{1,-5,1\}$}
\normcellB{rFK}{cFE}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rFK}{cDE}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rFK}{cDG}{}{}{}{$\lambda _2 \{1,-5,1\}$}
\normcellB{rFI}{cBA}{}{}{}{$\lambda _3 \{2,-11,6\}$}
\normcellB{rFI}{cBG}{}{}{}{$\{2,-11,5\}$}
\normcellB{rFI}{cDE}{}{}{}{$\lambda _2 \{1,-5,1\}$}
\normcellB{rFI}{cDG}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rzI}{cBG}{}{}{}{$1$}
\normcellB{rzI}{cDG}{}{}{}{$1$}
\normcellB{rDK}{cFC}{}{}{}{$\lambda _2 \{1,-5,1\}$}
\normcellB{rDK}{cFE}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rDK}{cDE}{}{}{}{$1$}
\normcellB{rDJ}{cFC}{}{}{}{$\{-2,11,-4\}$}
\normcellB{rDJ}{cFE}{}{}{}{$\lambda _2 \{1,-5,1\}$}
\normcellB{rDJ}{czC}{}{}{}{$1$}
\end{tikzpicture}
\end{equation*}
From each 2-by-2 block, in either matrix, we get a single equation involving the four phases of that matrix. Solving these we obtain the following matrices. Here each $\alpha_i$ is an unknown unit complex number. (A computer has made arbitrary choices about which phases to write in terms of others; don't expect to see patterns here.)
\hspace{-1.2cm}
% generated by the command 'PrincipalConnectionMatrix[H, RootReduce[connB2 /. shortNames], vNshort[H], "medium"]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\begin{tikzpicture}[xscale=1.5,yscale=.8]
\principalmatrixB
\mediumcellB{r1f}{c1f}{}{}{}{$-\frac{\alpha _1 \alpha _2}{\alpha _3}$}
\mediumcellB{r1f}{cAf}{}{}{}{$\alpha _1$}
\mediumcellB{rHf}{c1f}{}{}{}{$\alpha _2$}
\mediumcellB{rHf}{cAf}{}{}{}{$\alpha _3 \{1,-5,0\}$}
\mediumcellB{rHf}{cAF}{}{}{}{$-\alpha _{11} \lambda _1$}
\mediumcellB{rHf}{cAB}{}{}{}{$\alpha _6 \lambda _3$}
\mediumcellB{rHB}{cAf}{}{}{}{$\alpha _4 \lambda _3$}
\mediumcellB{rHB}{cAF}{}{}{}{$\frac{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20} \lambda _2 \{2,-11,4\}}{\alpha _{15} \alpha _{17} \alpha _{19}}$}
\mediumcellB{rHB}{cAB}{}{}{}{$\alpha _7 \{3,-16,5\}$}
\mediumcellB{rHB}{cGF}{}{}{}{$\alpha _{12}$}
\mediumcellB{rHF}{cAf}{}{}{}{$-\alpha _5 \lambda _1$}
\mediumcellB{rHF}{cAF}{}{}{}{$\frac{\alpha _{13} \alpha _{16} \alpha _{25} \{-2,11,-4\}}{\alpha _{18} \alpha _{24}}$}
\mediumcellB{rHF}{cAB}{}{}{}{$\frac{\alpha _8 \alpha _9 \alpha _{22} \alpha _{24} \lambda _2 \{2,-11,4\}}{\alpha _{10} \alpha _{21} \alpha _{23}}$}
\mediumcellB{rHF}{cCB}{}{}{}{$\alpha _8$}
\mediumcellB{rHF}{cEF}{}{}{}{$\alpha _{13}$}
\mediumcellB{rID}{cGF}{}{}{}{$\frac{\alpha _{15} \alpha _{17} \lambda _1}{\alpha _{18}}$}
\mediumcellB{rID}{cGz}{}{}{}{$\alpha _{20} \lambda _2$}
\mediumcellB{rID}{cEF}{}{}{}{$\alpha _{15}$}
\mediumcellB{rIB}{cAF}{}{}{}{$\alpha _{14}$}
\mediumcellB{rIB}{cGF}{}{}{}{$\frac{\alpha _{15} \alpha _{17} \alpha _{19} \lambda _2}{\alpha _{18} \alpha _{20}}$}
\mediumcellB{rIB}{cGz}{}{}{}{$-\alpha _{19} \lambda _1$}
\mediumcellB{rJz}{cCB}{}{}{}{$-\alpha _{10} \lambda _1$}
\mediumcellB{rJz}{cCD}{}{}{}{$\alpha _{22} \lambda _2$}
\mediumcellB{rJF}{cAB}{}{}{}{$\alpha _9$}
\mediumcellB{rJF}{cCB}{}{}{}{$\frac{\alpha _{10} \alpha _{21} \alpha _{23} \lambda _2}{\alpha _{22} \alpha _{24}}$}
\mediumcellB{rJF}{cCD}{}{}{}{$\frac{\alpha _{21} \alpha _{23} \lambda _1}{\alpha _{24}}$}
\mediumcellB{rJF}{cED}{}{}{}{$\alpha _{21}$}
\mediumcellB{rKF}{cAF}{}{}{}{$\alpha _{16}$}
\mediumcellB{rKF}{cCD}{}{}{}{$\alpha _{23}$}
\mediumcellB{rKF}{cED}{}{}{}{$\alpha _{24} \lambda _1 \{1,-6,4\}$}
\mediumcellB{rKF}{cEF}{}{}{}{$\frac{\alpha _{18} \alpha _{24} \{1,-6,4\}}{\alpha _{25}}$}
\mediumcellB{rKD}{cGF}{}{}{}{$\alpha _{17}$}
\mediumcellB{rKD}{cED}{}{}{}{$\alpha _{25} \{-1,6,-4\}$}
\mediumcellB{rKD}{cEF}{}{}{}{$\alpha _{18} \lambda _1 \{1,-6,4\}$}
\end{tikzpicture}
\hspace{-1.2cm}
% generated by the command 'DualConnectionMatrix[H, RootReduce[connB2 /. shortNames], vNshort[H], "medium"]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\begin{tikzpicture}[xscale=1.5, yscale=.8]
\dualmatrixB
\mediumcellB{rf1}{cfA}{}{}{}{$\alpha _1 \lambda _3 \{2,-11,6\}$}
\mediumcellB{rf1}{cf1}{}{}{}{$\frac{\alpha _1 \alpha _2 \{-2,11,-5\}}{\alpha _3}$}
\mediumcellB{rfH}{cfA}{}{}{}{$\alpha _3 \{2,-11,5\}$}
\mediumcellB{rfH}{cf1}{}{}{}{$\alpha _2 \lambda _3 \{2,-11,6\}$}
\mediumcellB{rfH}{cBA}{}{}{}{$\alpha _4$}
\mediumcellB{rfH}{cFA}{}{}{}{$\alpha _5$}
\mediumcellB{rBJ}{cFA}{}{}{}{$\alpha _9 \lambda _3 \{2,-11,6\}$}
\mediumcellB{rBJ}{cFC}{}{}{}{$\frac{\alpha _{10} \alpha _{21} \alpha _{23} \{2,-11,5\}}{\alpha _{22} \alpha _{24}}$}
\mediumcellB{rBJ}{czC}{}{}{}{$\alpha _{10}$}
\mediumcellB{rBH}{cfA}{}{}{}{$\alpha _6$}
\mediumcellB{rBH}{cBA}{}{}{}{$\alpha _7$}
\mediumcellB{rBH}{cFA}{}{}{}{$\frac{\alpha _8 \alpha _9 \alpha _{22} \alpha _{24} \{-2,11,-5\}}{\alpha _{10} \alpha _{21} \alpha _{23}}$}
\mediumcellB{rBH}{cFC}{}{}{}{$\alpha _8 \lambda _3 \{2,-11,6\}$}
\mediumcellB{rFH}{cfA}{}{}{}{$\alpha _{11}$}
\mediumcellB{rFH}{cBA}{}{}{}{$\frac{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20} \{-2,11,-5\}}{\alpha _{15} \alpha _{17} \alpha _{19}}$}
\mediumcellB{rFH}{cBG}{}{}{}{$\alpha _{12} \lambda _3 \{2,-11,6\}$}
\mediumcellB{rFH}{cFA}{}{}{}{$\frac{\alpha _{13} \alpha _{16} \alpha _{25} \{-2,11,-4\}}{\alpha _{18} \alpha _{24}}$}
\mediumcellB{rFH}{cFE}{}{}{}{$\alpha _{13} \lambda _2 \{1,-5,1\}$}
\mediumcellB{rFK}{cFA}{}{}{}{$\alpha _{16} \lambda _2 \{1,-5,1\}$}
\mediumcellB{rFK}{cFE}{}{}{}{$\frac{\alpha _{18} \alpha _{24} \{2,-11,4\}}{\alpha _{25}}$}
\mediumcellB{rFK}{cDG}{}{}{}{$\alpha _{17} \lambda _2 \{1,-5,1\}$}
\mediumcellB{rFK}{cDE}{}{}{}{$\alpha _{18} \{-2,11,-4\}$}
\mediumcellB{rFI}{cBA}{}{}{}{$\alpha _{14} \lambda _3 \{2,-11,6\}$}
\mediumcellB{rFI}{cBG}{}{}{}{$\frac{\alpha _{15} \alpha _{17} \alpha _{19} \{2,-11,5\}}{\alpha _{18} \alpha _{20}}$}
\mediumcellB{rFI}{cDG}{}{}{}{$\frac{\alpha _{15} \alpha _{17} \{2,-11,4\}}{\alpha _{18}}$}
\mediumcellB{rFI}{cDE}{}{}{}{$\alpha _{15} \lambda _2 \{1,-5,1\}$}
\mediumcellB{rzI}{cBG}{}{}{}{$\alpha _{19}$}
\mediumcellB{rzI}{cDG}{}{}{}{$\alpha _{20}$}
\mediumcellB{rDK}{cFE}{}{}{}{$\alpha _{24} \{-2,11,-4\}$}
\mediumcellB{rDK}{cFC}{}{}{}{$\alpha _{23} \lambda _2 \{1,-5,1\}$}
\mediumcellB{rDK}{cDE}{}{}{}{$\alpha _{25}$}
\mediumcellB{rDJ}{cFE}{}{}{}{$\alpha _{21} \lambda _2 \{1,-5,1\}$}
\mediumcellB{rDJ}{cFC}{}{}{}{$\frac{\alpha _{21} \alpha _{23} \{2,-11,4\}}{\alpha _{24}}$}
\mediumcellB{rDJ}{czC}{}{}{}{$\alpha _{22}$}
\end{tikzpicture}$$
We still have further constraints: the phase equations coming from the three-by-three matrix in the principal graph connection. The equations coming from orthogonality of columns are all of the form
$$m_1 \phi_1 + m_2 \phi_2 + m_3 \phi_3=0, \qquad m_i \in \mathbb{R}_+, \qquad \phi_i \in \mathbb{T}.$$
In each of these equations, we find that two of the $m_i$ sum to the remaining third; thus the corresponding equations on phases are of the form $\phi_i = \phi_j = - \phi_k$.
So the orthogonality of the first and third columns let us determine $\alpha_3$ and $\alpha_5$ in terms of the phases of the remaining entries;
similarly the orthogonality of the second and third columns give us expressions for $\alpha_6$ and $\alpha_7$. The connection is then:
[xscale=2.3, yscale=.8]
r1fc1f$\frac{\alpha _1 \alpha _2 \alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20}}{\alpha _4 \alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19}}$
r1fcAf$\alpha _1$
rHfc1f$\alpha _2$
rHfcAB$\frac{\alpha _8 \alpha _9 \alpha _{11} \alpha _{18} \alpha _{22} \alpha _{24}^2 \lambda _3}{\alpha _{10} \alpha _{13} \alpha _{16} \alpha _{21} \alpha _{23} \alpha _{25}}$
rHfcAF$-\alpha _{11} \lambda _1$
rHfcAf$\frac{\alpha _4 \alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19} \{-1,5,0\}}{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20}}$
rHFcAB$\frac{\alpha _8 \alpha _9 \alpha _{22} \alpha _{24} \lambda _2 \{2,-11,4\}}{\alpha _{10} \alpha _{21} \alpha _{23}}$
rHFcAF$\frac{\alpha _{13} \alpha _{16} \alpha _{25} \{-2,11,-4\}}{\alpha _{18} \alpha _{24}}$
rHFcAf$-\frac{\alpha _4 \alpha _{13} \alpha _{15} \alpha _{16} \alpha _{17} \alpha _{19} \alpha _{25} \lambda _1}{\alpha _{12} \alpha _{14} \alpha _{18}^2 \alpha _{20} \alpha _{24}}$
rHFcCB$\alpha _8$
rHFcEF$\alpha _{13}$
rHBcAB$\frac{\alpha _8 \alpha _9 \alpha _{12} \alpha _{14} \alpha _{18}^2 \alpha _{20} \alpha _{22} \alpha _{24}^2 \{3,-16,5\}}{\alpha _{10} \alpha _{13} \alpha _{15} \alpha _{16} \alpha _{17} \alpha _{19} \alpha _{21} \alpha _{23} \alpha _{25}}$
rHBcAF$\frac{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20} \lambda _2 \{2,-11,4\}}{\alpha _{15} \alpha _{17} \alpha _{19}}$
rHBcAf$\alpha _4 \lambda _3$
rHBcGF$\alpha _{12}$
rIDcGz$\alpha _{20} \lambda _2$
rIDcGF$\frac{\alpha _{15} \alpha _{17} \lambda _1}{\alpha _{18}}$
rIDcEF$\alpha _{15}$
rIBcAF$\alpha _{14}$
rIBcGz$-\alpha _{19} \lambda _1$
rIBcGF$\frac{\alpha _{15} \alpha _{17} \alpha _{19} \lambda _2}{\alpha _{18} \alpha _{20}}$
rJzcCB$-\alpha _{10} \lambda _1$
rJzcCD$\alpha _{22} \lambda _2$
rJFcAB$\alpha _9$
rJFcCB$\frac{\alpha _{10} \alpha _{21} \alpha _{23} \lambda _2}{\alpha _{22} \alpha _{24}}$
rJFcCD$\frac{\alpha _{21} \alpha _{23} \lambda _1}{\alpha _{24}}$
rJFcED$\alpha _{21}$
rKFcAF$\alpha _{16}$
rKFcCD$\alpha _{23}$
rKFcED$\alpha _{24} \lambda _1 \{1,-6,4\}$
rKFcEF$\frac{\alpha _{18} \alpha _{24} \{1,-6,4\}}{\alpha _{25}}$
rKDcGF$\alpha _{17}$
rKDcED$\alpha _{25} \{-1,6,-4\}$
rKDcEF$\alpha _{18} \lambda _1 \{1,-6,4\}$
rf1cf1$\frac{\alpha _1 \alpha _2 \alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20} \{2,-11,5\}}{\alpha _4 \alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19}}$
rf1cfA$\alpha _1 \lambda _3 \{2,-11,6\}$
rfHcf1$\alpha _2 \lambda _3 \{2,-11,6\}$
rfHcfA$\frac{\alpha _4 \alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19} \{-2,11,-5\}}{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20}}$
rfHcBA$\alpha _4$
rfHcFA$\frac{\alpha _4 \alpha _{13} \alpha _{15} \alpha _{16} \alpha _{17} \alpha _{19} \alpha _{25}}{\alpha _{12} \alpha _{14} \alpha _{18}^2 \alpha _{20} \alpha _{24}}$
rBHcfA$\frac{\alpha _8 \alpha _9 \alpha _{11} \alpha _{18} \alpha _{22} \alpha _{24}^2}{\alpha _{10} \alpha _{13} \alpha _{16} \alpha _{21} \alpha _{23} \alpha _{25}}$
rBHcBA$\frac{\alpha _8 \alpha _9 \alpha _{12} \alpha _{14} \alpha _{18}^2 \alpha _{20} \alpha _{22} \alpha _{24}^2}{\alpha _{10} \alpha _{13} \alpha _{15} \alpha _{16} \alpha _{17} \alpha _{19} \alpha _{21} \alpha _{23} \alpha _{25}}$
rBHcFC$\alpha _8 \lambda _3 \{2,-11,6\}$
rBHcFA$\frac{\alpha _8 \alpha _9 \alpha _{22} \alpha _{24} \{-2,11,-5\}}{\alpha _{10} \alpha _{21} \alpha _{23}}$
rBJcFC$\frac{\alpha _{10} \alpha _{21} \alpha _{23} \{2,-11,5\}}{\alpha _{22} \alpha _{24}}$
rBJcFA$\alpha _9 \lambda _3 \{2,-11,6\}$
rBJczC$\alpha _{10}$
rFHcfA$\alpha _{11}$
rFHcBA$\frac{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20} \{-2,11,-5\}}{\alpha _{15} \alpha _{17} \alpha _{19}}$
rFHcBG$\alpha _{12} \lambda _3 \{2,-11,6\}$
rFHcFA$\frac{\alpha _{13} \alpha _{16} \alpha _{25} \{-2,11,-4\}}{\alpha _{18} \alpha _{24}}$
rFHcFE$\alpha _{13} \lambda _2 \{1,-5,1\}$
rFKcFA$\alpha _{16} \lambda _2 \{1,-5,1\}$
rFKcFE$\frac{\alpha _{18} \alpha _{24} \{2,-11,4\}}{\alpha _{25}}$
rFKcDG$\alpha _{17} \lambda _2 \{1,-5,1\}$
rFKcDE$\alpha _{18} \{-2,11,-4\}$
rFIcBA$\alpha _{14} \lambda _3 \{2,-11,6\}$
rFIcBG$\frac{\alpha _{15} \alpha _{17} \alpha _{19} \{2,-11,5\}}{\alpha _{18} \alpha _{20}}$
rFIcDG$\frac{\alpha _{15} \alpha _{17} \{2,-11,4\}}{\alpha _{18}}$
rFIcDE$\alpha _{15} \lambda _2 \{1,-5,1\}$
rzIcBG$\alpha _{19}$
rzIcDG$\alpha _{20}$
rDJcFC$\frac{\alpha _{21} \alpha _{23} \{2,-11,4\}}{\alpha _{24}}$
rDJcFE$\alpha _{21} \lambda _2 \{1,-5,1\}$
rDJczC$\alpha _{22}$
rDKcFC$\alpha _{23} \lambda _2 \{1,-5,1\}$
rDKcFE$\alpha _{24} \{-2,11,-4\}$
rDKcDE$\alpha _{25}$
We now pass to the lopsided planar algebra using the map $\natural$, and choose an element of the complex gauge group which simultaneously demonstrates that the connections shown above all form a single gauge orbit, and gives us a nice representative with all entries in the number field $\mathbb{Q}[d^2]$. Gauge entries that are not specified below are all equal to $1$.
\begin{align*}
\mu(1, f) & = \frac{\alpha _4 \alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19} \lambda_{1,-2,-1,1}^{(2.25)}}{\alpha _1 \alpha _2 \alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20}} &
\mu(A, f) & = \frac{\lambda_{1,-2,-1,1}^{(2.25)}}{\alpha _1} \\
\mu(A, B) & = \frac{\alpha _{10} \alpha _{13} \alpha _{16} \alpha _{21} \alpha _{23} \alpha _{25} \lambda_{1,-5,6,-1}^{(3.25)}}{\alpha _8 \alpha _9 \alpha _{11} \alpha _{18} \alpha _{22} \alpha _{24}^2} &
\mu(A, F) & = \frac{\lambda_{1,-2,-1,1}^{(2.25)}}{\alpha _{11}} \\
\mu(C, B) & = \frac{\alpha _{13} \lambda_{1,0,-4,0,3,0,1}^{(1.202)}}{\alpha _8} &
\mu(C, D) & = \frac{\alpha _{24} \lambda_{1,3,-4,1}^{(-4.049)}}{\alpha _{21} \alpha _{23}} \\
\mu(G, F) & = \frac{\lambda_{1,0,-1,0,-9,0,1}^{(1.869)}}{\alpha _{12}} &
\mu(G, z) & = \frac{\alpha _{17} \alpha _{25} \lambda_{1,0,-335,0,-44,0,-1}^{(-18.31)}}{\alpha _{12} \alpha _{18} \alpha _{20} \alpha _{24}} \\
\mu(E, D) & = \frac{\lambda_{1,0,-9,0,-1,0,1}^{(3.016)}}{\alpha _{21}} \
\mu(f, \hat{H}) & = \frac{\alpha _1 \alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20}}{\alpha _4 \alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19}} \\
\mu(B, \hat{J}) & = \frac{\alpha _8 \alpha _{22} \alpha _{24} \lambda_{1,0,12,0,-15,0,1}^{(1.037)}}{\alpha _{10} \alpha _{13} \alpha _{21} \alpha _{23}}
\mu(F, \hat{I}) & = \frac{\alpha _{25} \lambda_{1,0,-4,0,3,0,1}^{(-1.674)}}{\alpha _{15} \alpha _{24}} \\
\mu(F, \hat{K}) & = \frac{\alpha _{25} \lambda_{1,1,-2,-1}^{(1.25)}}{\alpha _{18} \alpha _{24}} &
\mu(D, \hat{K}) & = \frac{\alpha _{21} \lambda_{1,0,-2,0,-1,0,1}^{(-0.7450)}}{\alpha _{24}} \\
\mu(\bar{B}, A) & = \frac{\alpha _{11} \alpha _{15} \alpha _{17} \alpha _{19} \lambda_{1,-1,-2,1}^{(1.80)}}{\alpha _{12} \alpha _{14} \alpha _{18} \alpha _{20}} &
\mu(\bar{F}, A) & = \frac{\alpha _{11} \alpha _{18} \alpha _{24} \lambda_{1,2,-1,-1}^{(0.802)}}{\alpha _{16} \alpha _{25}} \\
\mu(\bar{z}, C) & = \frac{\alpha _{21} \alpha _{23} \lambda_{1,-4,3,1}^{(1.45)}}{\alpha _{22} \alpha _{24}} &
\mu(\bar{D}, G) & = \frac{\alpha _{12} \alpha _{18} \alpha _{24} \lambda_{1,0,11,0,-4,0,-1}^{(0.7181)}}{\alpha _{17} \alpha _{25}} \\
\mu(\bar{D}, E) & = -\frac{\alpha _{24}}{\alpha _{25}} &
\mu(\hat{H}, \bar{F}) & = \frac{\lambda_{1,-2,-1,1}^{(2.25)}}{\alpha _{13}} \\
\mu(\hat{I}, \bar{B}) & = \frac{\alpha _{12} \alpha _{18} \alpha _{20} \alpha _{24} \lambda_{1,0,15,0,12,0,-1}^{(-0.2758)}}{\alpha _{17} \alpha _{19} \alpha _{25}} &
\end{align*}
We call this bi-invertible connection $K^B_{\text{lopsided}}$.
% generated by the command 'PrincipalConnectionMatrix[H, connB4, vNshort[H]]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\begin{tikzpicture}[yscale=.8]
\principalmatrixB
\cellB{r1f}{c1f}{}{}{}{$1$}
\cellB{r1f}{cAf}{}{}{}{$1$}
\cellB{rHF}{cAB}{}{}{}{$\{1,-6,4\}$}
\cellB{rHF}{cAf}{}{}{}{$1$}
\cellB{rHF}{cAF}{}{}{}{$1$}
\cellB{rHF}{cCB}{}{}{}{$1$}
\cellB{rHF}{cEF}{}{}{}{$1$}
\cellB{rHB}{cAB}{}{}{}{$\{-1,6,-3\}$}
\cellB{rHB}{cAf}{}{}{}{$1$}
\cellB{rHB}{cAF}{}{}{}{$\{1,-6,4\}$}
\cellB{rHB}{cGF}{}{}{}{$1$}
\cellB{rHf}{c1f}{}{}{}{$1$}
\cellB{rHf}{cAB}{}{}{}{$1$}
\cellB{rHf}{cAf}{}{}{}{$\{-1,5,0\}$}
\cellB{rHf}{cAF}{}{}{}{$1$}
\cellB{rID}{cGz}{}{}{}{$\{2,-12,7\}$}
\cellB{rID}{cGF}{}{}{}{$1$}
\cellB{rID}{cEF}{}{}{}{$1$}
\cellB{rIB}{cAF}{}{}{}{$1$}
\cellB{rIB}{cGz}{}{}{}{$1$}
\cellB{rIB}{cGF}{}{}{}{$\{-3,16,-4\}$}
\cellB{rJz}{cCB}{}{}{}{$1$}
\cellB{rJz}{cCD}{}{}{}{$\{2,-12,7\}$}
\cellB{rJF}{cAB}{}{}{}{$1$}
\cellB{rJF}{cCB}{}{}{}{$\{-3,16,-4\}$}
\cellB{rJF}{cCD}{}{}{}{$1$}
\cellB{rJF}{cED}{}{}{}{$1$}
\cellB{rKF}{cAF}{}{}{}{$1$}
\cellB{rKF}{cCD}{}{}{}{$1$}
\cellB{rKF}{cED}{}{}{}{$\{-2,11,-5\}$}
\cellB{rKF}{cEF}{}{}{}{$\{-2,11,-5\}$}
\cellB{rKD}{cGF}{}{}{}{$1$}
\cellB{rKD}{cED}{}{}{}{$\{-1,6,-4\}$}
\cellB{rKD}{cEF}{}{}{}{$\{-2,11,-5\}$}
\end{tikzpicture}$$
% generated by the command 'DualConnectionMatrix[H, connB4, vNshort[H]]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\begin{tikzpicture}[yscale=.8]
\dualmatrixB
\cellB{rfH}{cf1}{}{}{}{$1$}
\cellB{rfH}{cfA}{}{}{}{$-1$}
\cellB{rfH}{cBA}{}{}{}{$\{0,1,-1\}$}
\cellB{rfH}{cFA}{}{}{}{$\{0,1,-1\}$}
\cellB{rf1}{cf1}{}{}{}{$1$}
\cellB{rf1}{cfA}{}{}{}{$\{0,1,-1\}$}
\cellB{rBJ}{cFA}{}{}{}{$\{1,-5,3\}$}
\cellB{rBJ}{cFC}{}{}{}{$\{-2,11,-4\}$}
\cellB{rBJ}{czC}{}{}{}{$\{-1,6,-3\}$}
\cellB{rBH}{cfA}{}{}{}{$\{1,-5,3\}$}
\cellB{rBH}{cBA}{}{}{}{$\{-1,7,-4\}$}
\cellB{rBH}{cFA}{}{}{}{$\{2,-12,7\}$}
\cellB{rBH}{cFC}{}{}{}{$\{-1,6,-3\}$}
\cellB{rFH}{cfA}{}{}{}{$\{1,-5,2\}$}
\cellB{rFH}{cBG}{}{}{}{$\{1,-5,1\}$}
\cellB{rFH}{cBA}{}{}{}{$\{1,-6,3\}$}
\cellB{rFH}{cFA}{}{}{}{$\{1,-5,2\}$}
\cellB{rFH}{cFE}{}{}{}{$\{-2,11,-3\}$}
\cellB{rFK}{cFA}{}{}{}{$\{1,-5,2\}$}
\cellB{rFK}{cFE}{}{}{}{$\{-3,16,-5\}$}
\cellB{rFK}{cDE}{}{}{}{$\{-3,16,-5\}$}
\cellB{rFK}{cDG}{}{}{}{$\{1,-5,1\}$}
\cellB{rFI}{cBG}{}{}{}{$\{6,-32,9\}$}
\cellB{rFI}{cBA}{}{}{}{$\{1,-5,2\}$}
\cellB{rFI}{cDE}{}{}{}{$\{-2,11,-3\}$}
\cellB{rFI}{cDG}{}{}{}{$\{1,-5,1\}$}
\cellB{rzI}{cBG}{}{}{}{$\{0,1,0\}$}
\cellB{rzI}{cDG}{}{}{}{$\{0,-3,2\}$}
\cellB{rDJ}{cFC}{}{}{}{$\{1,-5,2\}$}
\cellB{rDJ}{cFE}{}{}{}{$\{-1,6,-2\}$}
\cellB{rDJ}{czC}{}{}{}{$\{1,-7,4\}$}
\cellB{rDK}{cFC}{}{}{}{$\{1,-5,2\}$}
\cellB{rDK}{cFE}{}{}{}{$\{-1,5,-1\}$}
\cellB{rDK}{cDE}{}{}{}{$\{1,-5,2\}$}
\end{tikzpicture}$$
§ FLAT LOW WEIGHT VECTORS
This section contains the final ingredients necessary in the proofs of two of our main theorems:
There is a unique subfactor with principal graph $\Gamma(\cA)$.
There is a unique subfactor with principal graph $\Gamma(\cB)$.
In <ref> we showed that there are two biunitary connections on $\cA$ which pass the branch-point eigenvalue test.
Since the two vertices at depth 2 in $\Gamma(\cA)$ are self-dual, a flat low weigth $2$-box must have rotational eigenvalue $+1$.
Theorem <ref> shows that $K^{(1)}_{\text{lopsided}}$ has no flat, low-weight $2$-boxes with eigenvalue $+1$, hence by Theorem <ref> this connection is not flat.
Thus there is at most one gauge equivalence class of flat connections on the principal graph for $\cA$, and $K^{(2)}_{\text{lopsided}}$ provides a representative of its complex gauge group orbit. Since we know the subfactor exists (easily constructed from quantum groups), this connection must actually be flat by Theorem <ref>. Theorem <ref> explicitly describes the flat subalgebra.
In <ref> we showed that there is one biunitary connection on $\cB$ which passes the branch-point eigenvalue test. Since we know the subfactor exists (easily constructed from quantum groups), this connection must actually be flat by Theorem <ref>. Theorem <ref> explicitly describes the flat subalgebra.
The work in this section consists of calculating the flat low weight vectors in the graph planar algebra, for each of the three connections (two on $\Gamma(\cA)$ and one on $\Gamma(\cB)$) described above.
In order to do this, we will need to write down rather complicated elements of the graph planar algebra. We express these as a collection of matrices; the usual multiplication structure on a graph planar algebra space $\cG(\Gamma)_{n,+}$ breaks up as a direct sum of matrix algebras. Each matrix algebra is indexed by a pair of even vertices $a, b$ on $\Gamma$, and the rows and columns are indexed by paths of length $n$ from $a$ to $b$. Thus if $\pi$ and $\rho$ are paths on $\Gamma$ from $a$ to $b$, we denote by $ \bar{\rho}\pi$ the concatenation of $\pi$ with the reverse of $\rho$, a loop based at $a$, and the $(\rho,\pi)$ entry of the $(a,b)$ matrix gives the coefficient of the loop $\bar{\rho}\pi$.
We need to specify the ordering of paths from $a$ to $b$, in order to fix an ordering of the rows and columns of these matrices. For $\cA$, we use
\begin{align*}
% generated by the command 'typesetPathsA' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\paths^2_{\mathcal{A}}(1, 1) & = \left\{(1X1)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(1, Z) & = \left\{(1XZ)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(1, Y) & = \left\{(1XY)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(Z, 1) & = \left\{(ZX1)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(Z, Z) & = \left\{(ZXZ), (ZWZ)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(Z, Y) & = \left\{(ZXY), (ZWY)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(Y, 1) & = \left\{(YX1)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(Y, Z) & = \left\{(YXZ), (YWZ)\right\} \displaybreak[1]\\
\paths^2_{\mathcal{A}}(Y, Y) & = \left\{(YXY), (YWY), (YgY)\right\} \displaybreak[1]\\\end{align*}
and for $\cB$
\begin{align*}
% generated by the command 'typesetPathsB' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\paths^3_{\mathcal{B}}(1, f) & = \left\{(1f1f), (1fAf)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(1, B) & = \left\{(1fAB)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(1, F) & = \left\{(1fAF)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(A, f) & = \left\{(Af1f), (AfAf), (ABAf), (AFAf)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(A, B) & = \left\{(AfAB), (ABAB), (ABCB), (AFAB)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(A, F) & = \left\{(AfAF), (ABAF), (AFAF), (AFGF), (AFEF)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(A, z) & = \left\{(AFGz)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(A, D) & = \left\{(ABCD), (AFED)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(G, f) & = \left\{(GFAf)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(G, B) & = \left\{(GFAB)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(G, F) & = \left\{(GFAF), (GFGF), (GFEF), (GzGF)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(G, z) & = \left\{(GFGz), (GzGz)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(G, D) & = \left\{(GFED)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(C, f) & = \left\{(CBAf)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(C, B) & = \left\{(CBAB), (CBCB), (CDCB)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(C, F) & = \left\{(CBAF), (CDEF)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(C, D) & = \left\{(CBCD), (CDCD), (CDED)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(E, f) & = \left\{(EFAf)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(E, B) & = \left\{(EFAB), (EDCB)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(E, F) & = \left\{(EFAF), (EFGF), (EFEF), (EDEF)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(E, z) & = \left\{(EFGz)\right\} \displaybreak[1]\\
\paths^3_{\mathcal{B}}(E, D) & = \left\{(EFED), (EDCD), (EDED)\right\} \displaybreak[1]\\
\end{align*}
The lowest weight eigenspace with eigenvalue $+1$ in the graph planar algebra $\cG(\Gamma(\cA))_{2,+}$ is four dimensional, with basis $\{S_1, S_2, S_3, S_4\}$ given in <ref>.
Recall that the conditions for $S$ being a lowest weight eigenvector with eigenvalue $\mu$ are
\begin{align*}
\begin{tikzpicture}[baseline]
\draw (-.3,-1)--(-.3,0);
\draw (.3,-1)--(.3,0);
\draw (-.3,0) --(-.3,.5) .. controls (-.3,1) and (.3,1) .. (.3,.5)--(.3,0);
\node[minimum size=1cm, shape=rectangle, fill=white, draw] at (0,0) {$S$};
\end{tikzpicture} & = 0 &
\begin{tikzpicture}[baseline]
\draw (-.3,-1.5)--(-.3,1.5);
\draw (.3,.3)--(.3,.5) .. controls (.3,1) and (.9,1) .. (.9,.5)--(.9,-.5) .. controls (.9,-1) and (.3, -1) .. (.3,-.5) -- (.3,-.3);
\node[minimum size=1cm, shape=rectangle, fill=white, draw] at (0,0) {$S$};
\end{tikzpicture} & = 0
\end{align*}
\begin{align*}
\begin{tikzpicture}[baseline]
\draw (.3,.3)--(.3,.5) .. controls (.3, .8) and (.9,.8) .. (.9,.5) -- (.9,-.5) .. controls (.9,-1) and (-.3,-1) .. (-.3,-1.5);
\draw (-.3,-.3)--(-.3,-.5) .. controls (-.3, -.8) and (-.9,-.8) .. (-.9,-.5) -- (-.9,.5) .. controls (-.9,1) and (.3,1) .. (.3,1.5);
\draw (-.3,.3)--(-.3,.5) .. controls (-.3, 1.1) and (1.5,1.1) .. (1.5,.5) -- (1.5,-.5) .. controls (1.5,-1) and (.3,-1) .. (.3,-1.5);
\draw (.3,-.3)--(.3,-.5) .. controls (.3, -1.1) and (-1.5,-1.1) .. (-1.5,-.5) -- (-1.5,.5) .. controls (-1.5,1) and (-.3,1) .. (-.3,1.5);
\node[minimum size=1cm, shape=rectangle, fill=white, draw] at (0,0) {$S$};
\end{tikzpicture}\;
= \mu \;\;
\begin{tikzpicture}[baseline]
\draw (-.3,-1)--(-.3,1);
\draw (.3,-1)--(.3,1);
\node[minimum size=1cm, shape=rectangle, fill=white, draw] at (0,0) {$S$};
\end{tikzpicture}
\end{align*}
Setting $\mu=1$ and writing this explicitly in the lopsided graph planar algebra, we have
\begin{align*}
% generated by the command 'typesetEigenspaceEquations' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
0 & = \left(d^4-5 d^2+2\right) S(1,X,Y,X)+\left(-d^4+6 d^2-3\right) S(1,X,Z,X)+S(1,X,1,X) \displaybreak[1]\\
0 & = S(Z,X,1,X)+\left(d^4-5 d^2+2\right) S(Z,X,Y,X)+\left(-d^4+6 d^2-3\right) S(Z,X,Z,X) \displaybreak[1]\\
0 & = S(Y,X,1,X)+\left(-d^4+6 d^2-3\right) S(Y,X,Z,X)+\left(d^4-5 d^2+2\right) S(Y,X,Y,X) \displaybreak[1]\\
0 & = \left(2 d^4-10 d^2+3\right) S(Z,W,Y,W)+\left(-d^4+6 d^2-2\right) S(Z,W,Z,W) \displaybreak[1]\\
0 & = \left(-d^4+6 d^2-2\right) S(Y,W,Z,W)+\left(2 d^4-10 d^2+3\right) S(Y,W,Y,W) \displaybreak[1]\\
0 & = \left(d^4-3 d^2+1\right) S(Y,g,Y,g) \displaybreak[1]\\
0 & = S(1,X,1,X) \displaybreak[1]\\
0 & = \left(-2 d^4+11 d^2-4\right) S(Z,W,Z,X)+\left(-2 d^4+11 d^2-4\right) S(Z,X,Z,X) \displaybreak[1]\\
0 & = \left(-2 d^4+11 d^2-4\right) S(Z,X,Z,W)+\left(-2 d^4+11 d^2-4\right) S(Z,W,Z,W) \displaybreak[1]\\
0 & = \left(2 d^4-11 d^2+5\right) S(Y,g,Y,X)+\left(2 d^4-11 d^2+5\right) S(Y,W,Y,X)+\left(2 d^4-11 d^2+5\right) S(Y,X,Y,X) \displaybreak[1]\\
0 & = \left(2 d^4-11 d^2+5\right) S(Y,g,Y,W)+\left(2 d^4-11 d^2+5\right) S(Y,X,Y,W)+\left(2 d^4-11 d^2+5\right) S(Y,W,Y,W) \displaybreak[1]\\
0 & = \left(2 d^4-11 d^2+5\right) S(Y,W,Y,g)+\left(2 d^4-11 d^2+5\right) S(Y,X,Y,g)+\left(2 d^4-11 d^2+5\right) S(Y,g,Y,g) \displaybreak[1]\\
0 & = S(Z,X,1,X)-S(1,X,Z,X) \displaybreak[1]\\
0 & = S(Y,X,1,X)-S(1,X,Y,X) \displaybreak[1]\\
0 & = S(1,X,Z,X)-S(Z,X,1,X) \displaybreak[1]\\
0 & = \left(d^4-5 d^2+1\right) S(Z,W,Z,X)-S(Z,X,Z,W) \displaybreak[1]\\
0 & = \left(-d^4+6 d^2-4\right) S(Z,X,Z,W)-S(Z,W,Z,X) \displaybreak[1]\\
0 & = S(Y,X,Z,X)-S(Z,X,Y,X) \displaybreak[1]\\
0 & = \left(d^4-5 d^2+1\right) S(Y,W,Z,X)-S(Z,X,Y,W) \displaybreak[1]\\
0 & = \left(-d^4+6 d^2-4\right) S(Y,X,Z,W)-S(Z,W,Y,X) \displaybreak[1]\\
0 & = S(Y,W,Z,W)-S(Z,W,Y,W) \displaybreak[1]\\
0 & = S(1,X,Y,X)-S(Y,X,1,X) \displaybreak[1]\\
0 & = S(Z,X,Y,X)-S(Y,X,Z,X) \displaybreak[1]\\
0 & = \left(d^4-5 d^2+1\right) S(Z,W,Y,X)-S(Y,X,Z,W) \displaybreak[1]\\
0 & = \left(-d^4+6 d^2-4\right) S(Z,X,Y,W)-S(Y,W,Z,X) \displaybreak[1]\\
0 & = S(Z,W,Y,W)-S(Y,W,Z,W) \displaybreak[1]\\
0 & = \left(d^4-5 d^2+1\right) S(Y,W,Y,X)-S(Y,X,Y,W) \displaybreak[1]\\
0 & = \left(d^4-5 d^2+2\right) S(Y,g,Y,X)-S(Y,X,Y,g) \displaybreak[1]\\
0 & = \left(-d^4+6 d^2-4\right) S(Y,X,Y,W)-S(Y,W,Y,X) \displaybreak[1]\\
0 & = \left(-d^4+6 d^2-3\right) S(Y,g,Y,W)-S(Y,W,Y,g) \displaybreak[1]\\
0 & = \left(2 d^4-11 d^2+5\right) S(Y,X,Y,g)-S(Y,g,Y,X) \displaybreak[1]\\
0 & = \left(-2 d^4+11 d^2-4\right) S(Y,W,Y,g)-S(Y,g,Y,W) \displaybreak[1]\\
\end{align*}
Solving these linear equations gives the desired answer.
Inside this subspace, the flat elements with respect to the connection $K^{(2)}_{\text{lopsided}}$ are one-dimensional, spanned by the element $T$ given in <ref>.
An element $x \in \cG(\Gamma)_{n,+}$ in a graph planar algebra is flat with respect to a connection $K$ exactly if there exists an element $y \in \cG(\Gamma')_{n,-}$ satisfying
\begin{equation*}
\xabove = \ybelow.
\end{equation*}
It is easy to see that if $x$ is a lowest weight eigenvector, $y$ must be also, with the same eigenvalue. In our case, the principal and dual principal graphs are the same, so Theorem <ref> also suffices to describe the lowest weight eigenspace on $\Gamma'$. Thus we take $x = \sum_{i=1}^4 c_{0,i} S_i$ and $y = \sum_{i=1}^4 c_{1,i} \rho^{1/2}(S_i)$ (the `half-click' rotation $\rho^{1/2}$ is necessary here since $S_i$ itself lives in $\cG(\Gamma)_{n,+} \iso \cG(\Gamma')_{n,+}$, and we need a basis for the lowest weight eigenvectors in $\cG(\Gamma')_{n,-}$). The flatness condition reduces to the 129 equations listed in <ref>. Solving these linear equations, we find a one dimensional space of solutions spanned by
\begin{align*}
c_{0,1} & = 1\\
c_{0,2} & = -3 d^4+17 d^2-10\\
c_{0,3} & = 2 d^4-11 d^2+6\\
c_{0,4} & = -2 d^4+12 d^2-7\\
c_{1,1} & = d^4-5 d^2+2\\
c_{1,2} & = -2 d^4+11 d^2-6\\
c_{1,3} & = d^4-5 d^2+3\\
c_{1,4} & = -d^4+7 d^2-4
\end{align*}
Substituting these into the formula for $x$ above, we get the element $T$ claimed in the statement.
Remember that there is a whole gauge orbit of flat connections equivalent to $K^{(2)}_{\text{lopsided}}$. As we change the connection, the corresponding flat elements change as described in Lemma <ref>. This gauge group action gives rise to a 1-parameter family of embeddings of the $\cA$ planar algebra in its graph planar algebra; the formulas above give just one point on this curve. In previous cases where we've explicitly identified the embedding in the graph planar algebra <cit.> there's just been a discrete set of embeddings. As noted previously, the gauge group action on flat elements is always trivial for principal graphs with no loops. One sees a related phenomenon in solving the equation $S^2 = (1-r) S + r f^{(n)}$ (here $r$ is the ratio of dimensions past an initial branch point, $n$ the depth of those vertices) which must be satisfied by the lowest weight $n$-box in a $(n-1)$-supertransitive excess 1 planar algebra. For $\cA$, there is a one-parameter family of solutions, exactly agreeing with the gauge orbit of the flat element described above, while in previously studied examples there had been a discrete set of solutions.
Inside the subspace described in Theorem <ref>, there are no flat elements with respect to the connection $K^{(1)}_{\text{lopsided}}$.
This is essentially identical to the calculation described in Theorem <ref>
Thus, we've proved one of our main theorems:
The lowest weight $\mu$-eigenspaces in $\cG(\Gamma(\cB))_{3,+}$ are each 7 dimensional, for $\mu= 1, \omega$ or $\omega^{-1}$. There is a one-dimensional space of flat elements with respect to the connection $K^B_{\text{lopsided}}$ in the $1$-eigenspace, and no flat elements in the other eigenspaces. Those flat elements are spanned by the element $T$ given in <ref>.
Exactly analogous to the calculations above. We omit writing down the bases for the lowest weight eigenspaces, but they can be found explicitly calculated in the Mathematica notebook connections-and-flat-elements.nb available with the arXiv sources of this article.
§ LOW WEIGHT VECTORS AND FLAT ELEMENTS
§.§ Low weight vectors for $\cA$
The low weight vectors in $\cG(\Gamma(\cA))_{2,+}$ with eigenvalue 1 form a four-dimensional space, with basis given below.
\begin{align*}
% generated by the command 'LowestWeightLaTeX[G, "Lopsided", 1]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
(S_{1})_{1,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{1,Z} & = \left(
\begin{array}{c}
d^4-5 d^2+2
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{1,Y} & = \left(
\begin{array}{c}
d^4-6 d^2+3
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{Z,1} & = \left(
\begin{array}{c}
d^4-5 d^2+2
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{Z,Z} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{Z,Y} & = \left(
\begin{array}{cc}
-1 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{Y,1} & = \left(
\begin{array}{c}
d^4-6 d^2+3
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{Y,Z} & = \left(
\begin{array}{cc}
-1 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{1})_{Y,Y} & = \left(
\begin{array}{ccc}
-2 d^4+12 d^2-8 & -1 & d^4-6 d^2+3 \\
d^4-6 d^2+4 & 0 & -d^4+6 d^2-3 \\
d^4-6 d^2+4 & 1 & 0
\end{array}
\right) \displaybreak[1]\\
\\(S_{2})_{1,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{1,Z} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{1,Y} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{Z,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{Z,Z} & = \left(
\begin{array}{cc}
d^4-5 d^2+1 & 2 d^4-11 d^2+3 \\
-d^4+5 d^2-1 & -2 d^4+11 d^2-3
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{Z,Y} & = \left(
\begin{array}{cc}
-1 & 0 \\
0 & -d^4+5 d^2-1
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{Y,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{Y,Z} & = \left(
\begin{array}{cc}
-1 & 0 \\
0 & -d^4+5 d^2-1
\end{array}
\right) \displaybreak[1]\\
(S_{2})_{Y,Y} & = \left(
\begin{array}{ccc}
-d^4+6 d^2-4 & -1 & 0 \\
d^4-6 d^2+4 & 1 & 0 \\
0 & 0 & 0
\end{array}
\right) \displaybreak[1]\\
\\(S_{3})_{1,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{1,Z} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{1,Y} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{Z,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{Z,Z} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{Z,Y} & = \left(
\begin{array}{cc}
0 & 0 \\
-d^4+6 d^2-4 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{Y,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{Y,Z} & = \left(
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{3})_{Y,Y} & = \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}
\right) \displaybreak[1]\\
\\(S_{4})_{1,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{1,Z} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{1,Y} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{Z,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{Z,Z} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{Z,Y} & = \left(
\begin{array}{cc}
0 & d^4-5 d^2+1 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{Y,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{Y,Z} & = \left(
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}
\right) \displaybreak[1]\\
(S_{4})_{Y,Y} & = \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}
\right) \displaybreak[1]\\
\\
\end{align*}
§.§ Flat generators for $\cA$
Only the second connection on $\Gamma(\cA)$ has any flat lowest weight eigenvectors with eigenvalue 1, and it has a one dimensional space of such, spanned by the element $T$ specified below.
\begin{align*}
% generated by the command 'ToLaTeXString[GPA4Matrix[T], "T"]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
T_{1,1} & = \left(
\begin{array}{c}
\end{array}
\right) \displaybreak[1]\\
T_{1,Z} & = \left(
\begin{array}{c}
6 d^4-32 d^2+9
\end{array}
\right) \displaybreak[1]\\
T_{1,Y} & = \left(
\begin{array}{c}
3 d^4-16 d^2+4
\end{array}
\right) \displaybreak[1]\\
T_{Z,1} & = \left(
\begin{array}{c}
6 d^4-32 d^2+9
\end{array}
\right) \displaybreak[1]\\
T_{Z,Z} & = \left(
\begin{array}{cc}
-5 d^4+27 d^2-9 & -8 d^4+43 d^2-13 \\
5 d^4-27 d^2+9 & 8 d^4-43 d^2+13
\end{array}
\right) \displaybreak[1]\\
T_{Z,Y} & = \left(
\begin{array}{cc}
4 d^4-21 d^2+4 & -2 d^4+11 d^2-4 \\
d^4-6 d^2+5 & 5 d^4-27 d^2+9
\end{array}
\right) \displaybreak[1]\\
T_{Y,1} & = \left(
\begin{array}{c}
3 d^4-16 d^2+4
\end{array}
\right) \displaybreak[1]\\
T_{Y,Z} & = \left(
\begin{array}{cc}
4 d^4-21 d^2+4 & d^4-5 d^2 \\
2 d^4-11 d^2+5 & 5 d^4-27 d^2+9
\end{array}
\right) \displaybreak[1]\\
T_{Y,Y} & = \left(
\begin{array}{ccc}
5 d^4-26 d^2+4 & 4 d^4-21 d^2+4 & 3 d^4-16 d^2+4 \\
5-d^2 & 4 d^4-22 d^2+9 & -3 d^4+16 d^2-4 \\
-5 d^4+27 d^2-9 & -8 d^4+43 d^2-13 & 0
\end{array}
\right) \displaybreak[1]\\
\end{align*}
This element was found by solving the linear equations described below, equivalent to the flatness of a linear combination $\sum_{i=1}^4 c_{1,i} S_i$ of the lowest weight eigenvectors specified in the previous section.
We denote by $\cF$ the element of the two-sided graph planar algebra
\begin{equation*}
\xabove
\ybelow
\end{equation*}
(Here $x$ and $y$ are as described in Theorem <ref>.)
Each of its coefficients, which are indexed by a loop on the 4-partite principal graph (reading around the boundary clockwise from the left, first an $N-N$ bimodule, then an $M-N$, $M-M$, $M-N$, $N-N$ then $N-M$ bimodule), and are given below, must be zero.
\begin{align*}
% generated by the command 'typesetFlatnessEquations[flatness2]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
\cF(1\bar{X}\hat{1}\bar{X}1X) & = 0 \displaybreak[1] \\
\cF(1\bar{X}\hat{1}\bar{X}ZX) & = \left(d^4-5 d^2+2\right) c_{0,1}-c_{1,1} \displaybreak[1] \\
\cF(1\bar{X}\hat{1}\bar{X}YX) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(1\bar{X}\hat{Z}\bar{X}1X) & = 0 \displaybreak[1] \\
\cF(1\bar{X}\hat{Z}\bar{X}ZX) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(-2 d^4+11 d^2-6\right) c_{1,1}-c_{1,2} \displaybreak[1] \\
\cF(1\bar{X}\hat{Z}\bar{X}YX) & = \left(-2 d^4+12 d^2-7\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{1,1}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(1\bar{X}\hat{Z}\bar{W}ZX) & = \left(d^2-1\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{1,3}+c_{1,2} \displaybreak[1] \\
\cF(1\bar{X}\hat{Z}\bar{W}YX) & = \left(-d^4+5 d^2-3\right) c_{0,1}+\left(2 d^4-11 d^2+6\right) c_{1,3}+c_{1,2} \displaybreak[1] \\
\cF(1\bar{X}\hat{Y}\bar{X}1X) & = 0 \displaybreak[1] \\
\cF(1\bar{X}\hat{Y}\bar{X}ZX) & = \left(1-d^2\right) c_{0,1}+\left(2 d^4-11 d^2+7\right) c_{1,1}+c_{1,2} \displaybreak[1] \\
\cF(1\bar{X}\hat{Y}\bar{X}YX) & = \left(d^4-6 d^2+4\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{1,1}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(1\bar{X}\hat{Y}\bar{W}ZX) & = d^2 c_{0,1}+c_{1,1}+c_{1,2} - c_{1,4} \displaybreak[1] \\
\cF(1\bar{X}\hat{Y}\bar{W}YX) & = \left(d^4-4 d^2+2\right) c_{0,1}+\left(d^4-6 d^2+3\right) c_{1,1}+\left(d^4-6 d^2+3\right) c_{1,2} -c_{1,4} \displaybreak[1] \\
\cF(1\bar{X}\hat{Y}\bar{g}YX) & = \left(1-d^2\right) c_{0,1}+\left(-d^4+6 d^2-3\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{X}\hat{1}\bar{X}1X) & = \left(d^4-5 d^2+2\right) c_{0,1}-c_{1,1} \displaybreak[1] \\
\cF(Z\bar{X}\hat{1}\bar{X}ZX) & = \left(d^4-5 d^2+1\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{X}\hat{1}\bar{X}ZW) & = \left(2 d^4-11 d^2+3\right) c_{0,2}+\left(-2 d^4+11 d^2-5\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{X}\hat{1}\bar{X}YX) & = \left(-4 d^4+23 d^2-14\right) c_{1,1}-c_{0,1}-c_{0,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{1}\bar{X}YW) & = \left(d^4-5 d^2+1\right) c_{0,4}+\left(-2 d^4+11 d^2-6\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{X}1X) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(-2 d^4+11 d^2-6\right) c_{1,1}-c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{X}ZX) & = \left(d^4-6 d^2+3\right) c_{0,2}+\left(-6 d^4+34 d^2-19\right) c_{1,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{X}ZW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,1}+\left(-2 d^4+11 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{X}YX) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(2 d^4-11 d^2+6\right) c_{0,3}\\ & \quad +\left(-9 d^4+51 d^2-29\right) c_{1,1}+\left(-4 d^4+23 d^2-14\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{X}YW) & = \left(-d^4+5 d^2-2\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{0,4}+\left(-3 d^4+17 d^2-10\right) c_{1,1}\\ & \quad +\left(-2 d^4+11 d^2-6\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{W}ZX) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(2 d^4-11 d^2+5\right) c_{1,2}+\left(3 d^4-17 d^2+10\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{W}ZW) & = \left(d^4-6 d^2+2\right) c_{0,2}+\left(2 d^4-11 d^2+4\right) c_{1,2}+\left(d^4-6 d^2+4\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{W}YX) & = \left(d^4-6 d^2+4\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(d^4-5 d^2+3\right) c_{0,3}\\ & \quad +\left(2 d^4-11 d^2+5\right) c_{1,2}+\left(4 d^4-23 d^2+13\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Z}\bar{W}YW) & = d^2 \left(-c_{0,2}\right)+\left(d^4-5 d^2+2\right) c_{1,2}+c_{0,4}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{X}1X) & = \left(1-d^2\right) c_{0,1}+\left(2 d^4-11 d^2+7\right) c_{1,1}+c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{X}ZX) & = \left(-2 d^4+11 d^2-4\right) c_{0,2}+\left(9 d^4-51 d^2+28\right) c_{1,1}+\left(3 d^4-17 d^2+9\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{X}ZW) & = \left(-3 d^4+16 d^2-5\right) c_{0,2}+\left(5 d^4-28 d^2+14\right) c_{1,1}+\left(2 d^4-11 d^2+5\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{X}YX) & = \left(-3 d^4+17 d^2-9\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{0,2}+\left(-2 d^4+11 d^2-6\right) c_{0,3}\\ & \quad +\left(13 d^4-74 d^2+43\right) c_{1,1}+\left(4 d^4-23 d^2+14\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{X}YW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(-2 d^4+11 d^2-5\right) c_{0,4}+\left(5 d^4-28 d^2+16\right) c_{1,1}\\ & \quad +\left(2 d^4-11 d^2+6\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{W}ZX) & = d^2 \left(-c_{0,2}\right)+\left(d^4-6 d^2+4\right) c_{1,1}+\left(d^4-6 d^2+4\right) c_{1,2}\\ & \quad +\left(-2 d^4+11 d^2-5\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{W}ZW) & = \left(d^4-4 d^2+1\right) c_{0,2}+c_{1,1}+c_{1,2}\\ & \quad +\left(-2 d^4+11 d^2-4\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{W}YX) & = \left(-d^4+5 d^2-3\right) c_{0,1}+\left(-d^4+5 d^2-3\right) c_{0,2}+\left(-d^4+6 d^2-3\right) c_{0,3}\\ & \quad +\left(2 d^4-11 d^2+6\right) c_{1,1}+\left(2 d^4-11 d^2+6\right) c_{1,2}+\left(-2 d^4+11 d^2-5\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{W}YW) & = \left(d^2-1\right) c_{0,4}+\left(d^4-6 d^2+2\right) c_{0,2}+c_{1,1}\\ & \quad +\left(-d^4+5 d^2-2\right) c_{1,4}+c_{1,2} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{g}YX) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(-d^4+6 d^2-3\right) c_{0,2}+\left(-d^4+6 d^2-3\right) c_{0,3}\\ & \quad +\left(-2 d^4+11 d^2-6\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{X}\hat{Y}\bar{g}YW) & = \left(d^4-6 d^2+2\right) c_{0,2}+\left(-d^4+5 d^2-2\right) c_{0,4}+\left(-d^4+6 d^2-3\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{X}1X) & = \left(d^4-5 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,2}+\left(3 d^4-17 d^2+9\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{X}ZX) & = \left(-d^4+6 d^2-3\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,2}+\left(6 d^4-34 d^2+19\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{X}ZW) & = \left(-d^4+5 d^2-2\right) c_{0,2}+\left(-2 d^4+11 d^2-5\right) c_{1,2}+\left(3 d^4-17 d^2+9\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{X}YX) & = \left(-4 d^4+23 d^2-13\right) c_{0,1}+\left(-4 d^4+23 d^2-13\right) c_{0,2}+\left(-3 d^4+17 d^2-10\right) c_{0,3}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,2}+\left(9 d^4-51 d^2+28\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{X}YW) & = \left(-2 d^4+11 d^2-6\right) c_{0,4}+\left(-2 d^4+11 d^2-5\right) c_{1,2}+c_{0,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,4} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{W}ZX) & = \left(d^4-5 d^2+1\right) c_{0,2}+\left(2 d^4-11 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{W}ZW) & = \left(2 d^4-11 d^2+3\right) c_{0,2}+\left(3 d^4-16 d^2+5\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{W}YX) & = \left(-2 d^4+11 d^2-6\right) c_{0,1}+\left(-2 d^4+11 d^2-6\right) c_{0,2}+\left(-2 d^4+11 d^2-6\right) c_{0,3}\\ & \quad +\left(d^4-6 d^2+3\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Z}\bar{W}YW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(-d^4+6 d^2-3\right) c_{0,4}+\left(d^4-5 d^2+2\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}1X) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{1,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}ZX) & = \left(d^4-6 d^2+3\right) c_{0,2}+\left(-6 d^4+34 d^2-19\right) c_{1,1}+\left(-6 d^4+34 d^2-19\right) c_{1,2}\\ & \quad +\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}ZW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(-3 d^4+17 d^2-9\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}YX) & = \left(4 d^4-23 d^2+14\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{0,2}+\left(4 d^4-23 d^2+13\right) c_{0,3}\\ & \quad +\left(-9 d^4+51 d^2-28\right) c_{1,1}+\left(-9 d^4+51 d^2-28\right) c_{1,2}+\left(-7 d^4+40 d^2-23\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}YW) & = \left(-d^4+6 d^2-3\right) c_{0,2}+\left(d^4-6 d^2+5\right) c_{0,4}+\left(-3 d^4+17 d^2-10\right) c_{1,1}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,2}+\left(-3 d^4+17 d^2-9\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}ZX) & = \left(-d^4+5 d^2-1\right) c_{0,2}+\left(-2 d^4+11 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}ZW) & = \left(-2 d^4+11 d^2-3\right) c_{0,2}+\left(-3 d^4+16 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}YX) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(2 d^4-11 d^2+6\right) c_{0,2}+\left(2 d^4-11 d^2+6\right) c_{0,3}\\ & \quad +\left(-d^4+6 d^2-3\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}YW) & = \left(-d^4+5 d^2-2\right) c_{0,2}+\left(d^4-6 d^2+3\right) c_{0,4}+\left(-d^4+5 d^2-2\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{g}YX) & = \left(d^4-6 d^2+4\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(2 d^4-11 d^2+6\right) c_{0,3}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{g}YW) & = \left(-d^4+5 d^2-2\right) c_{0,2}+\left(d^4-6 d^2+3\right) c_{1,1}+c_{0,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}1X) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}ZX) & = \left(-4 d^4+23 d^2-14\right) c_{1,1}-c_{0,1}-c_{0,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}ZW) & = \left(-3 d^4+17 d^2-10\right) c_{1,1}+c_{0,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}YX) & = \left(-2 d^4+12 d^2-8\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(-7 d^4+40 d^2-24\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}YW) & = \left(-4 d^4+23 d^2-14\right) c_{1,1}-c_{0,1}-c_{0,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}Yg) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}1X) & = \left(-2 d^4+12 d^2-7\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{1,1}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}ZX) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(-d^4+6 d^2-4\right) c_{0,4}\\ & \quad +\left(-9 d^4+51 d^2-29\right) c_{1,1}+\left(-4 d^4+23 d^2-14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}ZW) & = \left(-3 d^4+17 d^2-10\right) c_{0,3}+\left(-6 d^4+34 d^2-19\right) c_{1,1}-c_{0,2}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}YX) & = \left(15 d^4-85 d^2+48\right) c_{0,1}+\left(6 d^4-34 d^2+19\right) c_{0,2}+\left(-13 d^4+74 d^2-43\right) c_{1,1}\\ & \quad +\left(-7 d^4+40 d^2-24\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}YW) & = \left(4 d^4-23 d^2+13\right) c_{0,1}+\left(3 d^4-17 d^2+9\right) c_{0,2}+\left(-7 d^4+40 d^2-23\right) c_{1,1}\\ & \quad +\left(-4 d^4+23 d^2-14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}Yg) & = \left(-3 d^4+17 d^2-10\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}ZX) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(2 d^4-11 d^2+6\right) c_{0,2}+\left(-2 d^4+11 d^2-5\right) c_{0,4}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2}+\left(4 d^4-23 d^2+14\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}ZW) & = \left(-2 d^4+11 d^2-4\right) c_{0,2}+\left(-2 d^4+11 d^2-6\right) c_{0,3}+\left(5 d^2-d^4\right) c_{1,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}YX) & = \left(6 d^4-35 d^2+20\right) c_{0,1}+\left(2 d^4-12 d^2+7\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{1,2}\\ & \quad +\left(6 d^4-34 d^2+19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}YW) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(d^4-5 d^2+3\right) c_{0,2}+c_{1,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}Yg) & = \left(-d^4+7 d^2-4\right) c_{0,1}+\left(d^4-6 d^2+3\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}1X) & = \left(d^4-6 d^2+4\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{1,1}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}ZX) & = \left(-3 d^4+17 d^2-9\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{0,4}\\ & \quad +\left(13 d^4-74 d^2+43\right) c_{1,1}+\left(4 d^4-23 d^2+14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}ZW) & = \left(3 d^4-17 d^2+9\right) c_{0,3}+\left(9 d^4-51 d^2+29\right) c_{1,1}+c_{0,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}YX) & = \left(-13 d^4+73 d^2-40\right) c_{0,1}+\left(-5 d^4+28 d^2-15\right) c_{0,2}+\left(20 d^4-114 d^2+67\right) c_{1,1}\\ & \quad +\left(7 d^4-40 d^2+24\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}YW) & = \left(-4 d^4+23 d^2-12\right) c_{0,1}+\left(-3 d^4+17 d^2-8\right) c_{0,2}+\left(11 d^4-63 d^2+37\right) c_{1,1}\\ & \quad +\left(4 d^4-23 d^2+14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}Yg) & = \left(2 d^4-11 d^2+7\right) c_{0,1}+\left(2 d^4-12 d^2+8\right) c_{1,1}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}ZX) & = \left(-2 d^4+11 d^2-5\right) c_{0,1}+\left(-2 d^4+11 d^2-5\right) c_{0,2}+\left(d^4-6 d^2+3\right) c_{0,4}\\ & \quad +\left(d^4-6 d^2+5\right) c_{1,1}+\left(d^4-6 d^2+5\right) c_{1,2}+\left(-d^4+6 d^2-4\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}ZW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(2 d^4-11 d^2+5\right) c_{0,3}+\left(d^4-6 d^2+4\right) c_{1,1}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2}+\left(d^4-5 d^2\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}YX) & = \left(-4 d^4+22 d^2-13\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(3 d^4-17 d^2+9\right) c_{1,1}\\ & \quad +\left(3 d^4-17 d^2+9\right) c_{1,2}+\left(-d^4+6 d^2-4\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}YW) & = \left(-d^4+7 d^2-4\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{1,1}-c_{0,2}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2}-c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}Yg) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(d^4-5 d^2+2\right) c_{1,1}+\left(d^4-5 d^2+2\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{g}YX) & = \left(-3 d^4+17 d^2-9\right) c_{0,1}+\left(-d^4+6 d^2-3\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{g}YW) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(-d^4+5 d^2-2\right) c_{0,2}+\left(-2 d^4+11 d^2-6\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{g}Yg) & = \left(1-d^2\right) c_{0,1}+\left(-d^4+6 d^2-3\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}1X) & = \left(4 d^4-23 d^2+13\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2}+\left(6 d^4-34 d^2+19\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}ZX) & = \left(-6 d^4+34 d^2-19\right) c_{0,1}+\left(-6 d^4+34 d^2-19\right) c_{0,2}+\left(3 d^4-17 d^2+10\right) c_{0,4}\\ & \quad +\left(-4 d^4+23 d^2-14\right) c_{1,2}+\left(13 d^4-74 d^2+42\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}ZW) & = \left(d^4-6 d^2+4\right) c_{0,2}+\left(6 d^4-34 d^2+19\right) c_{0,3}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(9 d^4-51 d^2+28\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}YX) & = \left(-25 d^4+142 d^2-81\right) c_{0,1}+\left(-9 d^4+51 d^2-29\right) c_{0,2}+\left(-6 d^4+34 d^2-19\right) c_{1,2}\\ & \quad +\left(19 d^4-108 d^2+61\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}YW) & = \left(-7 d^4+40 d^2-23\right) c_{0,1}+\left(-4 d^4+23 d^2-13\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(10 d^4-57 d^2+33\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}Yg) & = \left(5 d^4-28 d^2+16\right) c_{0,1}+\left(3 d^4-17 d^2+9\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}ZX) & = \left(-3 d^4+17 d^2-10\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{0,2}+\left(3 d^4-17 d^2+9\right) c_{0,4}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}ZW) & = \left(2 d^4-11 d^2+5\right) c_{0,2}+\left(3 d^4-17 d^2+10\right) c_{0,3}+\left(2 d^4-11 d^2+5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}YX) & = \left(-10 d^4+57 d^2-33\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{0,2}+\left(d^4-6 d^2+5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}YW) & = \left(-3 d^4+17 d^2-10\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}Yg) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}1X) & = \left(-4 d^4+23 d^2-13\right) c_{0,1}+\left(-6 d^4+34 d^2-19\right) c_{1,1}+\left(-6 d^4+34 d^2-19\right) c_{1,2}\\ & \quad +\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}ZX) & = \left(7 d^4-40 d^2+23\right) c_{0,1}+\left(7 d^4-40 d^2+23\right) c_{0,2}+\left(-4 d^4+23 d^2-14\right) c_{0,4}\\ & \quad +\left(-13 d^4+74 d^2-42\right) c_{1,1}+\left(-13 d^4+74 d^2-42\right) c_{1,2}+\left(-10 d^4+57 d^2-33\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}ZW) & = \left(-d^4+6 d^2-5\right) c_{0,2}+\left(-7 d^4+40 d^2-23\right) c_{0,3}+\left(-9 d^4+51 d^2-28\right) c_{1,1}\\ & \quad +\left(-9 d^4+51 d^2-28\right) c_{1,2}+\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}YX) & = \left(30 d^4-171 d^2+98\right) c_{0,1}+\left(10 d^4-57 d^2+33\right) c_{0,2}+\left(-19 d^4+108 d^2-61\right) c_{1,1}\\ & \quad +\left(-19 d^4+108 d^2-61\right) c_{1,2}+\left(-13 d^4+74 d^2-42\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}YW) & = \left(9 d^4-51 d^2+29\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{0,2}+\left(-10 d^4+57 d^2-33\right) c_{1,1}\\ & \quad +\left(-10 d^4+57 d^2-33\right) c_{1,2}+\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}Yg) & = \left(-6 d^4+34 d^2-19\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{1,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}ZX) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{0,4}\\ & \quad +\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}ZW) & = \left(-2 d^4+11 d^2-5\right) c_{0,2}+\left(-3 d^4+17 d^2-10\right) c_{0,3}+\left(-2 d^4+11 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}YX) & = \left(10 d^4-57 d^2+33\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(-d^4+6 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}YW) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}Yg) & = \left(-2 d^4+11 d^2-6\right) c_{0,1}-c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{g}YX) & = \left(5 d^4-29 d^2+17\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(3 d^4-17 d^2+10\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{g}YW) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(3 d^4-17 d^2+9\right) c_{1,1}+c_{0,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{g}Yg) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}1X) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}ZX) & = \left(-d^4+6 d^2-4\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(2 d^4-11 d^2+5\right) c_{0,4}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}ZW) & = \left(2 d^4-11 d^2+4\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{0,3}+\left(-3 d^4+17 d^2-9\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}YX) & = \left(-4 d^4+23 d^2-14\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(-4 d^4+23 d^2-14\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}YW) & = \left(-d^4+6 d^2-4\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{1,1}-c_{0,2} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}Yg) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}ZX) & = -c_{0,1}-c_{0,2}+c_{0,4} -c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}ZW) & = \left(-d^4+5 d^2-1\right) c_{0,2}+c_{0,3}-c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}YX) & = \left(-2 d^4+10 d^2-5\right) c_{0,1}+\left(-d^4+5 d^2-2\right) c_{0,2}+\left(-d^4+6 d^2-3\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}YW) & = \left(d^2-1\right) c_{0,1}+\left(-d^4+6 d^2-2\right) c_{0,2}-c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}Yg) & = d^2 c_{0,1}+\left(-d^4+5 d^2-2\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{g}YX) & = 0 \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{g}YW) & = 0 \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{g}Yg) & = 0 \displaybreak[1] \\
\end{align*}
§.§ Flat generators for $\cB$
Only the low weight eigenspace with eigenvalue $1$ in $\cG(\Gamma(\cB))_{3,+}$ contains any flat vectors, and these are spanned by the element $T$ below. (As usual, $d^2$ is the index of $\cA$ and $\cB$, the largest real root of $\lambda^3-6\lambda^2+5\lambda -1=0$, approximately $5.04892$.)
\begin{align*}
% generated by the command 'ToLaTeXString[GPA4Matrix[T], "T"]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
T_{1,f} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
T_{1,B} & = \left(
\begin{array}{c}
-5 d^4+34 d^2-20
\end{array}
\right) \displaybreak[1]\\
T_{1,F} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{A,f} & = \left(
\begin{array}{cccc}
0 & 0 & 9 d^4-50 d^2+29 & -5 d^4+27 d^2-13 \\
0 & 0 & -9 d^4+50 d^2-29 & 5 d^4-27 d^2+13 \\
4 d^4-16 d^2+9 & 3 d^4-19 d^2+12 & d^4-4 d^2+4 & -d^4+4 d^2+3 \\
-4 d^4+16 d^2-9 & -3 d^4+19 d^2-12 & 8 d^4-46 d^2+25 & -4 d^4+23 d^2-16
\end{array}
\right) \displaybreak[1]\\
T_{A,B} & = \left(
\begin{array}{cccc}
-9 d^4+50 d^2-29 & -3 d^4+19 d^2-12 & -6 d^4+31 d^2-17 & -4 d^4+23 d^2-16 \\
d^4-4 d^2+4 & 4 d^4-23 d^2+16 & 8 d^4-46 d^2+25 & -6 d^4+31 d^2-10 \\
-d^4+4 d^2-4 & -4 d^4+23 d^2-16 & -8 d^4+46 d^2-25 & 6 d^4-31 d^2+10 \\
8 d^4-46 d^2+25 & -d^4+4 d^2-4 & -2 d^4+15 d^2-8 & 10 d^4-54 d^2+26
\end{array}
\right) \displaybreak[1]\\
T_{A,F} & = \left(
\begin{array}{ccccc}
5 d^4-27 d^2+13 & -4 d^4+23 d^2-16 & 9 d^4-50 d^2+22 & 6 d^4-31 d^2+17 & -d^4+4 d^2-4 \\
-d^4+4 d^2+3 & -6 d^4+31 d^2-10 & -10 d^4+54 d^2-12 & -2 d^4+8 d^2-1 & -9 d^4+50 d^2-22 \\
-4 d^4+23 d^2-16 & 10 d^4-54 d^2+26 & d^4-4 d^2-10 & -4 d^4+23 d^2-16 & 10 d^4-54 d^2+26 \\
6 d^4-31 d^2+17 & -d^4+4 d^2-4 & 9 d^4-50 d^2+22 & 5 d^4-27 d^2+13 & -4 d^4+23 d^2-16 \\
-2 d^4+8 d^2-1 & -9 d^4+50 d^2-22 & -10 d^4+54 d^2-12 & -d^4+4 d^2+3 & -6 d^4+31 d^2-10
\end{array}
\right) \displaybreak[1]\\
T_{A,z} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{A,D} & = \left(
\begin{array}{cc}
-5 d^4+27 d^2-13 & 7 d^4-35 d^2+21 \\
7 d^4-35 d^2+14 & 2 d^4-8 d^2+1
\end{array}
\right) \displaybreak[1]\\
T_{C,f} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{C,B} & = \left(
\begin{array}{ccc}
8 d^4-46 d^2+25 & 9 d^4-50 d^2+29 & 3 d^4-19 d^2+12 \\
-8 d^4+46 d^2-25 & -9 d^4+50 d^2-29 & -3 d^4+19 d^2-12 \\
8 d^4-46 d^2+25 & 9 d^4-50 d^2+29 & 3 d^4-19 d^2+12
\end{array}
\right) \displaybreak[1]\\
T_{C,F} & = \left(
\begin{array}{cc}
2 d^4-8 d^2+1 & 7 d^4-35 d^2+14 \\
7 d^4-35 d^2+21 & -5 d^4+27 d^2-13
\end{array}
\right) \displaybreak[1]\\
T_{C,D} & = \left(
\begin{array}{ccc}
-3 d^4+19 d^2-12 & -d^4+4 d^2-4 & -3 d^4+19 d^2-12 \\
3 d^4-19 d^2+12 & d^4-4 d^2+4 & 3 d^4-19 d^2+12 \\
-3 d^4+19 d^2-12 & -d^4+4 d^2-4 & -3 d^4+19 d^2-12
\end{array}
\right) \displaybreak[1]\\
T_{G,f} & = \left(
\begin{array}{c}
-5 d^4+34 d^2-20
\end{array}
\right) \displaybreak[1]\\
T_{G,B} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{G,F} & = \left(
\begin{array}{cccc}
-4 d^4+23 d^2-16 & -3 d^4+19 d^2-12 & 8 d^4-46 d^2+25 & -4 d^4+16 d^2-9 \\
5 d^4-27 d^2+13 & 0 & -9 d^4+50 d^2-29 & 0 \\
-d^4+4 d^2+3 & 3 d^4-19 d^2+12 & d^4-4 d^2+4 & 4 d^4-16 d^2+9 \\
-5 d^4+27 d^2-13 & 0 & 9 d^4-50 d^2+29 & 0
\end{array}
\right) \displaybreak[1]\\
T_{G,z} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
T_{G,D} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{E,f} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{E,B} & = \left(
\begin{array}{cc}
-12 d^4+69 d^2-41 & -21 d^4+133 d^2-77 \\
7 & 9 d^4-50 d^2+29
\end{array}
\right) \displaybreak[1]\\
T_{E,F} & = \left(
\begin{array}{cccc}
10 d^4-54 d^2+26 & 8 d^4-46 d^2+25 & -d^4+4 d^2-4 & -2 d^4+15 d^2-8 \\
-4 d^4+23 d^2-16 & -9 d^4+50 d^2-29 & -3 d^4+19 d^2-12 & -6 d^4+31 d^2-17 \\
-6 d^4+31 d^2-10 & d^4-4 d^2+4 & 4 d^4-23 d^2+16 & 8 d^4-46 d^2+25 \\
6 d^4-31 d^2+10 & -d^4+4 d^2-4 & -4 d^4+23 d^2-16 & -8 d^4+46 d^2-25
\end{array}
\right) \displaybreak[1]\\
T_{E,z} & = \left(
\begin{array}{c}
-5 d^4+34 d^2-20
\end{array}
\right) \displaybreak[1]\\
T_{E,D} & = \left(
\begin{array}{ccc}
8 d^4-46 d^2+25 & 3 d^4-19 d^2+12 & 9 d^4-50 d^2+29 \\
8 d^4-46 d^2+25 & 3 d^4-19 d^2+12 & 9 d^4-50 d^2+29 \\
-8 d^4+46 d^2-25 & -3 d^4+19 d^2-12 & -9 d^4+50 d^2-29
\end{array}
\right) \displaybreak[1]\\
\end{align*}
|
arxiv-papers
| 2012-05-12T02:30:46 |
2024-09-04T02:49:30.845646
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Scott Morrison and Emily Peters",
"submitter": "Scott Morrison",
"url": "https://arxiv.org/abs/1205.2742"
}
|
1205.2826
|
# A New Extension of Serrin’s Lower Semicontinuity Theorem
Hu Xiaohong1,2 and Zhang Shiqing1 111Corresponding author: huxh@cqupt.edu.cn
and zhangshiqing@msn.com
1Department of Mathematics, Sichuan University,
Chengdu 610064, PR China
2 Department of Mathematics and Physics,
Chongqing University of Posts and Telecommunications,
Chongqing 400065, PR China
Supported partially by NSF of China.
Abstract: In this paper, we present a new extension of the famous Serrin’s
lower semicontinuity theorem for the variational functional
$\int_{\Omega}f(x,u,u^{\prime})dx$,we prove its lower semicontinuity in
$W_{loc}^{1,1}(\Omega)$ with respect to the strong $L_{loc}^{1}$ topology
assuming that the integrand $f(x,s,\xi)$ has the usual continuity on all the
three variables and the convexity property on the variable $\xi$ and the local
absolute continuity on the variable $x$.
Keyword: Lower semicontinuity, Serrin’s theorem, strong convergence in
$L^{1}$, convex function, local absolute continuity.
2002 Mathematics Subject Classification: Primary 49J45, Secondary 52A41.
## 1 Introduction and Main Results
The aim of this paper is to give some new sufficient conditions for lower
semicontinuity with respect to the strong convergence in $L_{loc}^{1}$ for
functionals of integral type
$F(u,\Omega)=\int_{\Omega}f(x,u(x),Du(x))dx,$ (1.1)
where $\Omega$ is an open set of $R^{n}$, $u$ is in the Sobolev Space[1]
$W_{loc}^{1,1}(\Omega)=\\{u:u\in L^{1}(K),Du\in L^{1}(K),\forall
K\subset\subset\Omega\\}$, $Du$ denotes the generalized gradient of $u$, and
the integrand $f(x,s,\xi):\Omega\times R\times R^{n}\rightarrow[0,\infty)$
satisfies the following conditions:
(H1) $f$ is continuous in $\Omega\times R\times R^{n}$ and $f(x,s,\xi)$ is
convex in $\xi\in R^{n}$ for all $(x,s)\in\Omega\times R$.
The integral functional $F$ is called lower semicontinuous in
$W_{loc}^{1,1}(\Omega)$ with respect to the strong convergence in
$L_{loc}^{1}$, if for every $u_{m},u\in W_{loc}^{1,1}(\Omega)$ such that
$u_{m}\rightarrow u$ in $L_{loc}^{1}$ (where $u_{m}\rightarrow u$ in
$L_{loc}^{1}$ means $\|u_{m}-u\|_{L^{1}(K)}\rightarrow 0\ as\
m\rightarrow+\infty,\forall K\subset\subset\Omega$), then
$\liminf_{m\rightarrow+\infty}F(u_{m},\Omega)\geq F(u,\Omega).$ (1.2)
It is well known that condition (H1) alone is not sufficient for strong lower
semicontinuity of the integral $F$ in (1.1) (see book [12]). In addition to
(H1),Serrin published in 1961 an article[13] proposing some sufficient
conditions for strong lower semicontinuity. One of the most known and
celebrated Serrin’s theorem on this subject is the following one.
Theorem 1.1[13] Let f satisfy, in addition to (H1), one of the following
conditions:
(a) $f(x,s,\xi)\rightarrow+\infty$ when $|\xi|\rightarrow+\infty$, for all
$(x,s)\in\Omega\times R$;
(b) $f(x,s,\xi)$ is strictly convex in $\xi\in R^{n}$ for all
$(x,s)\in\Omega\times R$;
(c) the derivatives $f_{x}(x,s,\xi)$, $f_{\xi}(x,s,\xi)$ and $f_{\xi
x}(x,s,\xi)$ exist and are continuous.
Then $F(u,\Omega)$ is lower semicontinuous in $W_{loc}^{1,1}(\Omega)$ with
respect to the strong convergence in $L_{loc}^{1}$.
The conditions (a), (b) and (c) quoted above are clearly independent, in the
sense that we can find a continuous function $f$ satisfying just one of them,
but none of the others . However, the proof of Theorem 1.1 is essentially the
same for every condition quoted above; indeed, the proof is based on an
approximation theorem for convex functions depending continuously on
parameters that can be applied, in particular, when f satisfies one of
conditions (a), (b) and (c). This fact suggests the possibility to find a
suitable condition weaker than one of conditions (a), (b) and (c). Many
attempts have been made to weaken the assumptions on the integrand $f$, such
as L. Ambrosio in paper [2], V. De Cicco in his paper [3] and I. Fonseca in
his papers [6] and [7] proposed several generalizations of Theorem 1.1. In the
papers [9] and [10], Gori prove the following theorems:
Theorem 1.2[9] Let us assume that $f$ satisfies (H1) and also assume that, for
every compact set $K\subset\Omega\times R\times R^{n}$, there exists a
constant $L=L(K)$ such that
$|f_{\xi}(x_{1},s,\xi)-f_{\xi}(x_{2},s,\xi)|\leq L|x_{1}-x_{2}|,\
\forall(x_{1},s,\xi),(x_{2},s,\xi)\in K,$ (1.3)
and, for every compact set $K_{1}\subset\Omega\times R$, there exists a
constant $L_{1}=L_{1}(K_{1})$ such that
$|f_{\xi}(x,s,\xi)|\leq L_{1},\ \forall(x,s)\in K_{1},\ \forall\xi\in R^{n},$
(1.4) $|f_{\xi}(x,s,\xi_{1})-f_{\xi}(x,s,\xi_{2})|\leq
L_{1}|\xi_{1}-\xi_{2}|,\ \forall(x,s)\in K_{1},\ \forall\xi_{1},\xi_{2}\in
R^{n}.$ (1.5)
Then $F(u,\Omega)$ is lower semicontinuous in $W_{loc}^{1,1}(\Omega)$ with
respect to the strong convergence in $L_{loc}^{1}$.
Theorem 1.3[9] Let f satisfy (H1) and such that, for every open set
$\Omega^{\prime}\times H\times K\subset\subset\Omega\times R\times R^{n}$,
there exists a constant $L=L_{\Omega^{\prime}\times H\times K}$ such that, for
every $x_{1},x_{2}\in\Omega^{\prime}$, $s\in H$ and $\xi\in K$ ,
$|f(x_{1},s,\xi)-f(x_{2},s,\xi)|\leq L|x_{1}-x_{2}|.$ (1.6)
Then the functional $F(u,\Omega)$ is lower semicontinuous on
$W_{loc}^{1,1}(\Omega)$ with respect to the $L_{loc}^{1}$ convergence.
Condition (1.6) means that $f$ is locally Lipschitz continuous with respect to
$x$, locally with respect to $(s,\xi)$ and not necessarily globally, that is,
the Lipschitz constant is not uniform for $(s,\xi)\in R\times R^{n}$. This is
an improvement of (c) of Serrin’s Theorem 1.1 since, when only the gradient
$\nabla_{x}f$ exists and is continuous, this implies the Lipschitz continuity
of $f$ with respect to $x$ on the compact subsets of $\Omega\times R\times
R^{n}$.
Then a question arises that whether there are weaker enough conditions more
than locally Lipschitz continuous condition? In this paper, we consider
absolutely continuous condition. Obviously, absolute continuity is weaker than
Lipschitz continuity. Is the local absolute continuity condition on the
integrand enough for the lower semicontinuity of the variational functional?
The following theorems give a confirmed answer.
Theorem 1.4 Let $\Omega\subset R$ be an open set, $f(x,s,\xi):\Omega\times
R\times R\longrightarrow[0,+\infty)$ satisfy the following conditions:
(H1) $f(x,s,\xi)$ is continuous on $\Omega\times R\times R$, $f(x,s,\xi)$ is
convex in $\xi\in R$ for all $(x,s)\in\Omega\times R$;
(H2) $f_{\xi}(x,s,\xi)$ is continuous on $\Omega\times R\times R$, and for
every compact set of $\Omega\times R\times R$, $f_{\xi}(x,s,\xi)$ is
absolutely continuous about $x$;
(H3) for every compact set $K_{1}\subseteq\Omega\times R$, there exists a
constant $L_{1}=L_{1}(K_{1})$, such that
$|f_{\xi}|\leq L_{1},\ \forall(x,s)\in K_{1},\ \ \forall\xi\in R,$ (1.7)
$|f_{\xi}(x,s,\xi_{1})-f_{\xi}(x,s,\xi_{2})|\leq L_{1}|\xi_{1}-\xi_{2}|,\
\forall(x,s)\in K_{1},\ \ \forall\xi_{1},\xi_{2}\in R.$ (1.8)
Then the functional $F(u,\Omega)=\int_{\Omega}f(x,u(x),u^{\prime}(x))dx$ is
lower semicontinuous on $W_{loc}^{1,1}(\Omega)$ with respect to the strong
convergence in $L_{loc}^{1}(\Omega)$.
Theorem 1.5 Let $\Omega\subset R$ be an open set, $f(x,s,\xi):\Omega\times
R\times R\longrightarrow[0,+\infty)$ satisfy (H1) and the following
conditions:
(H4) for every compact set $\Omega^{\prime}\times H\times
K\subseteq\Omega\times R\times R$, $f(x,s,\xi)$ is absolutely continuous about
$x$;
Then the functional $F(u,\Omega)$ is lower semicontinuous on
$W_{loc}^{1,1}(\Omega)$ with respect to the strong convergence in
$L_{loc}^{1}(\Omega)$.
## 2 Some Lemmas
In this section, we collect some preliminary definitions and lemmas (see
papers [1,4,8,11,14]) which will be used in the sequel.
Definition 2.1 Let $\Omega\subset R^{n}$ be on open set, we denote
$L^{p}_{loc}(\Omega)=\\{u:\Omega\rightarrow R|\
\int_{\Omega^{\prime}}|u|^{p}dx<+\infty,\forall\Omega^{\prime}\subset\subset\Omega\\},$
and
$W^{1,1}_{loc}(\Omega)=\\{u|\ u\in L^{1}_{loc}(\Omega),Du\in
L^{1}_{loc}(\Omega)\\}$
Remark 2.1 Notice that $u_{m}\rightarrow u$ in $L^{1}_{loc}(\Omega)$ implies
that $u_{m}$ converges to $u$ in measure.
Definition 2.2 Let $f:[a,b]\rightarrow R$ be a real function, if
$\forall\varepsilon>0,\exists\delta>0$, such that for any finite disjoint open
interval $\\{(a_{i},b_{i})\\}^{n}_{i=1}$ on $[a,b]$, when
$\sum\limits^{n}_{i=1}(b_{i}-a_{i})<\delta$, we have
$\sum\limits^{n}_{i=1}|f(b_{i})-f(a_{i})|<\varepsilon,$
then we call $f(x)$ is a absolutely continuous function on $[a,b]$.
Remark 2.2 If $f(x)$ is Lipschitz continuous on $[a,b]$, then $f(x)$ is
absolutely continuous on $[a,b]$.
Lemma 2.1 Let $f(x)$ is a absolutely continuous function on $[a,b]$, then
$f(x)$ is almost everywhere differentiable on $[a,b]$ and $f^{\prime}(x)$ is a
integrable function on $[a,b]$.
Lemma 2.2(Lebesgue Dominated convergence theorem) Let $(X,\mathcal{R},\mu)$ be
a measure space, $f$ and $\\{f_{n}\\}(n\geq 1)$ be measurable functions on
$E\in\mathcal{R}$, if
(1) $\\{f_{n}\\}$ is convergence in measure to $f$ on $E$;
(2) there exists a integrable function $h(x)$ on $E$, such that
$|f_{n}(x)|\leq h(x),\ \ \ \ a.e.x\in E,$
then $f(x),f_{n}(x)(n\geq 1)$ are integrable on $E$, and
$\lim\limits_{n\rightarrow+\infty}\int_{E}f_{n}(x)d\mu=\int_{E}f(x)d\mu.$
Lemma 2.3 Let $f(x)$ be a measurable function on $E$, the $f(x)$ is Lebesgue
integrable on $E$ if and only if $|f(x)|$ is Lebesgue integrable on $E$, and
$|\int_{E}f(x)dx|\leq\int_{E}|f(x)|dx$
Lemma 2.4 Let $f(x)$ be Lebesgue integrable on
$E,then\forall\varepsilon>0,\exists\delta>0$, for every measurable subset $A$
of $E$, when $m(A)<\delta$, we have
$|\int_{A}f(x)dx|\leq\int_{A}|f(x)|dx<\varepsilon.$
Definition 2.3 Let function $\eta(x):R^{n}\rightarrow[0,1]$ satisfies $\eta\in
C^{\infty}_{c}(R^{n}),{\rm supp}(\eta)\subseteq B(0,1),\eta(-x)=\eta(x)$ and
$\int_{R^{n}}\eta(x)dx=1$. Giving a function $v:\Omega\rightarrow R$ and
$\varepsilon>0$, we define the convolution of $v$ with step $\varepsilon$ as
$v_{\varepsilon}(x)=\eta_{\varepsilon}*v(x)=\int_{R^{n}}\eta_{\varepsilon}(x-y)v(y)dy=\int_{R^{n}}\eta_{\varepsilon}(y)v(x-y)dy,$
where
$\eta_{\varepsilon}(x)=\varepsilon^{-n}\eta\left(\frac{x}{\varepsilon}\right),{\rm
supp}(\eta)=\overline{\\{x\in R^{n},|\eta(x)\neq 0\\}}.$
We have the following approximations about the convolution of $v$:
Lemma 2.5[1] Let $\Omega\subset R^{n}$ be an open set, and a function
$v:\Omega\rightarrow R$, then
(i) If $v\in L^{1}_{loc}(\Omega)$ and ${\rm supp}(v)\subset\subset\Omega$,
then $v_{\varepsilon}\in C^{\infty}_{0}(\Omega)$ provided
$\varepsilon<dist({\rm supp}(v),\partial\Omega)$, and
$v_{\varepsilon}\rightarrow v\ {\rm in}\ L^{1}_{loc}(\Omega)\ {\rm as}\
\varepsilon\rightarrow 0^{+}.$
(ii) If $v\in L^{p}(\Omega)$ where $1\leq p<+\infty$, then $v_{\varepsilon}\in
L^{p}(\Omega)$. Also
$v_{\varepsilon}\rightarrow v\ {\rm in}\ L^{p}_{loc}(\Omega)\ {\rm as}\
\varepsilon\rightarrow 0^{+}.$
A function $f:R^{n}\rightarrow R$ is called convex if for every $x,y\in
R^{n},\lambda\in(0,1)$, we have
$f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)$
.
Now, we give some properties about convex functions:
Lemma 2.6[4]. Let $f_{i}:\Omega\rightarrow R,\\{f_{i}\\}_{i\in N}$ be a
sequence of functions ,
(i) if $f_{i}$ is convex, then $f=\sup_{i\in N}f_{i}$ is also convex;
(ii) if $f_{i}$ is lower semicontinuous, then $f=\sup_{i\in N}f_{i}$ is also
lower semicontinuous. The following approximation result was proved by De
Giorgi[8].
Lemma 2.7[8] Let $U\subseteq R^{d}$ be an open set and $f:U\times
R^{n}\rightarrow[0,+\infty)$ be a continuous function with compact support on
$U$, such that, for every $t\in U,f(t,\cdot)$ is convex on $R^{n}$. Then there
exists a sequence $\\{\alpha_{q}\\}^{\infty}_{q=1}\subseteq
C^{\infty}_{c}(R^{n}),\alpha_{q}\geq 0,\int_{R^{n}}\alpha_{q}dx=1,{\rm
supp}(\alpha_{q})\subseteq B(0,1)$. Let
$a_{q}(t)=\int_{R^{n}}f(t,\xi)\\{(n+1)\alpha_{q}(\xi)+<\nabla\alpha_{q}(\xi),\xi>\\}d\xi,$
and
$b_{q}(t)=-\int_{R^{n}}f(t,\xi)\nabla\alpha_{q}(\xi)d\xi.$
Then
$f_{j}(t,\xi)=\max\limits_{1\leq q\leq j}\\{0,a_{q}(t)+<b_{q}(t),\xi>\\},j\in
N,$
satisfy the following results:
(i) for every $j\in N,f_{j}:U\times R^{n}\rightarrow[0,+\infty)$ is a
continuous function with compact support on $U$ such that, $\forall t\in
U,f_{j}(t,\cdot)$ is convex on $R^{n}$. Moreover, $\forall(t,\xi)\in U\times
R^{n},f_{j}(t,\xi)\leq f_{j+1}(t,\xi)$ and
$f(t,\xi)=\sup\limits_{j\in N}f_{j}(t,\xi)$
(ii) for every $j\in N$, there exists a constant $M_{j}>0$, such that,
$\forall(t,\xi)\in U\times R^{n}$,
$|f_{j}(t,\xi)|\leq M_{j}(1+|\xi|),$
and, $\forall t\in U,\forall\xi_{1},\xi_{2}\in R^{n}.$
$|f_{j}(t,\xi_{1})-f_{j}(t,\xi_{2})|\leq M_{j}|\xi_{1}-\xi_{2}|.$
## 3 Proof of Theorem 1.4
We will divide into four steps to complete the proof of theorem 1.4.
Step 1: Let $\\{\beta_{i}(x,s)\\}_{i\in N}$ be a sequence of smooth functions
satisfying
(1) there exists a compact set $\Omega^{\prime}\times
H\subset\subset\Omega\times R$, such that
$\beta_{i}(x,s)=0,\forall(x,s)\in(\Omega\backslash\Omega^{\prime})\times(R\backslash
H),$
(2) for every $i\in
N,\beta_{i}(x,s)\leq\beta_{i+1}(x,s),\forall(x,s)\in\Omega^{\prime}\times H$,
(3)
$\lim\limits_{i\rightarrow+\infty}\beta_{i}(x,s)=1,\forall(x,s)\in\Omega^{\prime}\times
H$.
Let
$f_{i}(x,s,\xi)=\beta_{i}(x,s)f(x,s,\xi),\ \ \ \ \ i=1,2,\cdots.$
It is clear that, for each $i\in N,f_{i}$ satisfies all the hypothesis in
theorem 1.4 and also vanishes if $(x,s)$ is outside $\Omega^{\prime}\times H$.
Thus
$\lim\limits_{i\rightarrow+\infty}f_{i}(x,s,\xi)=f(x,s,\xi),\ \ \ \
\forall(x,s,\xi)\in\Omega^{\prime}\times H\times R,$
and
$f_{i}(x,s,\xi)\leq f_{i+1}(x,s,\xi)\leq f(x,s,\xi),\forall i\in
N,\forall(x,s,\xi)\in\Omega^{\prime}\times H\times R.$
By Levi lemma, we have
$\lim\limits_{i\rightarrow+\infty}\int_{\Omega^{\prime}}f_{i}(x,s,\xi)dx=\int_{\Omega^{\prime}}f(x,s,\xi)d\xi.$
Thus, without loss of generality, we can assume that there exists an open set
$\Omega^{\prime}\times H\subset\subset\Omega\times R$, such that
$f(x,s,\xi)=0,\ \ \ \
\forall(x,s,\xi)\in(\Omega\backslash\Omega^{\prime})\times(R\backslash
H)\times R.$ (3.1)
Let $u_{m},u\in W^{1,1}_{loc}(\Omega)$ such that $u_{m}\rightarrow u$ in
$L^{1}_{loc}(\Omega)$. We will prove that
$\liminf\limits_{m\rightarrow+\infty}F(u_{m},\Omega)\geq F(u,\Omega).$
Without loss of generality, we can assume that
$\liminf\limits_{m\rightarrow+\infty}F(u_{m},\Omega)=\lim\limits_{m\rightarrow+\infty}F(u_{m},\Omega)<+\infty.$
By (3.1), we have
$F(u_{m},\Omega)=F(u_{m},\Omega^{\prime}),F(u,\Omega)=F(u,\Omega^{\prime})$,
thus we will only prove the following inequality:
$\lim\limits_{m\rightarrow+\infty}F(u_{m},\Omega^{\prime})\geq
F(u,\Omega^{\prime}).$ (3.2)
Step 2: Let $\eta_{\varepsilon}\in C^{\infty}_{c}(R)$ be a mollifier and for
$\epsilon>0$,define
$v_{\varepsilon}(x)=\eta_{\varepsilon}*u(x)=\int_{\Omega}\eta_{\varepsilon}(x-y)u(y)dy,\
\ \ \ x\in[\Omega_{\varepsilon}],$ (3.3)
where $\left[\Omega_{\varepsilon}\right]\triangleq\\{x\in\Omega:{\rm
dist}(x,\partial\Omega)>\varepsilon\\}$. We have
$\displaystyle[u_{\varepsilon}(x)]^{\prime}$ $\displaystyle=$
$\displaystyle[\eta_{\varepsilon}*u(x)]_{x}=[\int_{\Omega}\eta_{\varepsilon}(x-y)u(y)dy]_{x}$
$\displaystyle=$
$\displaystyle\int_{\Omega}[\eta_{\varepsilon}(x-y)]_{x}u(y)dy=\int_{\Omega}-[\eta_{\varepsilon}(x-y)]_{y}u(y)dy$
$\displaystyle=$
$\displaystyle\int_{B(x,\varepsilon)}\eta_{\varepsilon}(x-y)[u(y)]_{y}dy=[u^{\prime}]_{\varepsilon}(x),\
\ \ \ x\in\Omega_{\varepsilon}.$
In the following, we denote the derivative of $u_{\varepsilon}$ as
$u^{\prime}_{\varepsilon}$. When $u\in W^{1,1}_{loc}(\Omega)$, we know
$u^{\prime}\in L^{1}_{loc}(\Omega)$. By Lemma 2.5, we know
$u^{\prime}_{\varepsilon}\in C^{\infty}_{0}(\Omega)$ and
$u^{\prime}_{\varepsilon}\rightarrow u^{\prime}\ {\rm in}\
L^{1}_{loc}(\Omega)\ {\rm as}\ \varepsilon\rightarrow 0^{+},$ (3.5)
i.e., $\forall\delta>0,\ \exists\epsilon>0$, such that
$\int_{\Omega^{\prime}}|u^{\prime}_{\varepsilon}-u^{\prime}|dx<\delta.$ (3.6)
New we estimate the difference for the integrand values on different points:
$\displaystyle f(x,u_{m},u^{\prime}_{m})-f(x,u,u^{\prime})$ $\displaystyle=$
$\displaystyle f(x,u_{m},u^{\prime}_{m})-f(x,u_{m},u^{\prime}_{\varepsilon})$
$\displaystyle+$ $\displaystyle
f(x,u_{m},u^{\prime}_{\varepsilon})-f(x,u,u^{\prime}_{\varepsilon})$
$\displaystyle+$ $\displaystyle
f(x,u,u^{\prime}_{\varepsilon})-f(x,u,u^{\prime}).$
By the convexity of $f(x,s,\xi)$ with respect to $\xi$, we have
$f(x,u_{m},u^{\prime}_{m})-f(x,u_{m},u^{\prime}_{\varepsilon})\geq
f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot(u^{\prime}_{m}-u^{\prime}_{\varepsilon}).$
(3.8)
By (3.8), we have
$\displaystyle f(x,u_{m},u^{\prime}_{m})-f(x,u_{m},u^{\prime}_{\varepsilon})$
$\displaystyle\geq$ $\displaystyle
f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{\varepsilon}$ $\displaystyle=$ $\displaystyle
f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot u^{\prime}$
$\displaystyle+$ $\displaystyle f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot
u^{\prime}-f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{\varepsilon}$ $\displaystyle=$ $\displaystyle
f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot u^{\prime}$
$\displaystyle+$ $\displaystyle
f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot(u^{\prime}-u^{\prime}_{\varepsilon})$
$\displaystyle+$
$\displaystyle[f_{\xi}(x,u,u^{\prime}_{\varepsilon})-f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})]\cdot
u^{\prime}_{\varepsilon}.$
By (3) and (3), we have
$\displaystyle f(x,u_{m},u^{\prime}_{m})-f(x,u,u^{\prime})$
$\displaystyle\geq$ $\displaystyle
f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot u^{\prime}$
$\displaystyle+$ $\displaystyle
f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot(u^{\prime}-u^{\prime}_{\varepsilon})$
$\displaystyle+$
$\displaystyle[f_{\xi}(x,u,u^{\prime}_{\varepsilon})-f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})]\cdot
u^{\prime}_{\varepsilon}$ $\displaystyle+$ $\displaystyle
f(x,u_{m},u^{\prime}_{\varepsilon})-f(x,u,u^{\prime}_{\varepsilon})$
$\displaystyle+$ $\displaystyle
f(x,u,u^{\prime}_{\varepsilon})-f(x,u,u^{\prime}).$
This implies
$\displaystyle\int_{\Omega^{\prime}}[f(x,u_{m},u^{\prime}_{m})-f(x,u,u^{\prime})]dx$
$\displaystyle\geq$
$\displaystyle\int_{\Omega^{\prime}}[f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\xi})\cdot u^{\prime}]dx$
$\displaystyle+$
$\displaystyle\int_{\Omega^{\prime}}[f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot(u^{\prime}-u^{\prime}_{\varepsilon})]dx$
$\displaystyle+$
$\displaystyle\int_{\Omega^{\prime}}[f_{\xi}(x,u,u^{\prime}_{\varepsilon})-f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})]\cdot
u^{\prime}_{\varepsilon}dx$ $\displaystyle+$
$\displaystyle\int_{\Omega^{\prime}}[f(x,u_{m},u^{\prime}_{\varepsilon})-f(x,u,u^{\prime}_{\varepsilon})]dx$
$\displaystyle+$
$\displaystyle\int_{\Omega^{\prime}}[f(x,u,u^{\prime}_{\varepsilon})-f(x,u,u^{\prime})]dx.$
Step 3: Now, we estimate the right side of inequality (3).
By (1.7) and (3.6), we have
$\int_{\Omega^{\prime}}[f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot(u^{\prime}-u^{\prime}_{\varepsilon})]dx\geq-
L_{1}\int_{\Omega^{\prime}}|u^{\prime}-u^{\prime}_{\varepsilon}|dx\geq-
L_{1}\delta.$ (3.12)
Thus
$\lim\limits_{\varepsilon\rightarrow
0}\int_{\Omega^{\prime}}[f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot(u^{\prime}-u^{\prime}_{\varepsilon})]dx\geq
0$ (3.13)
Since $f(x,s,\xi)$ and $f_{\xi}(x,s,\xi)$ are continuous functions , they are
bounded functions on compact subset. By remark 2.1 and Lebesgue dominated
convergence theorem, we obtain
$\lim\limits_{m\rightarrow+\infty}\int_{\Omega^{\prime}}[f_{\xi}(x,u,u^{\prime}_{\varepsilon})-f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})]\cdot
u^{\prime}_{\varepsilon}dx=0,$ (3.14)
and
$\lim\limits_{m\rightarrow+\infty}\int_{\Omega^{\prime}}[f(x,u_{m},u^{\prime}_{\varepsilon})-f(x,u,u^{\prime}_{\varepsilon})]dx=0.$
(3.15)
Now, we will prove
$\lim\limits_{\varepsilon\rightarrow
0}\int_{\Omega^{\prime}}[f(x,u,u^{\prime}_{\varepsilon})-f(x,u,u^{\prime})]dx\geq
0$ (3.16)
By lemma 2.7, there exists a sequence of non-negative continuous functions
$f_{j}(x,s,\xi)\ (j\in N)$, such that $f_{j}(x,s,\xi)$ is convex on $\xi$, and
$\forall(x,s,\xi)\in\Omega^{\prime}\times H\times R$,
$\displaystyle f_{j}(x,s,\xi)\leq f_{j+1}(x,s,\xi),$ (3.17) $\displaystyle
f(x,s,\xi)=\sup\limits_{j\in N}f_{j}(x,s,\xi),$ (3.18)
$\displaystyle|f_{j}(x,s,\xi_{1})-f_{j}(x,s,\xi_{2})|\leq
M_{j}|\xi_{1}-\xi_{2}|.$ (3.19)
By Levi lemma, we obtain
$\lim\limits_{j\rightarrow+\infty}\int_{\Omega^{\prime}}f_{j}(x,u,u^{\prime}_{\varepsilon})dx=\int_{\Omega^{\prime}}f(x,u,u^{\prime}_{\varepsilon})dx,$
(3.20)
and
$\lim\limits_{j\rightarrow+\infty}\int_{\Omega^{\prime}}f_{j}(x,u,u^{\prime})dx=\int_{\Omega^{\prime}}f(x,u,u^{\prime})dx.$
(3.21)
In order to prove (3.16), we only need to prove
$\lim\limits_{\varepsilon\rightarrow
0}\int_{\Omega^{\prime}}[f_{j}(x,u,u^{\prime}_{\varepsilon})-f_{j}(x,u,u^{\prime})]dx\geq
0,\ \forall j\in N.$ (3.22)
By (3.20) and (3.21), we have
$\int_{\Omega^{\prime}}[f_{j}(x,u,u^{\prime}_{\varepsilon})-f_{j}(x,u,u^{\prime})]dx\geq-
M_{j}\int_{\Omega^{\prime}}|u^{\prime}_{\varepsilon}-u^{\prime}|dx\geq-
M_{j}\delta.$
Thus (3.16) holds.
Step 4: Now, we need to prove
$\lim\limits_{m\rightarrow+\infty}\int_{\Omega^{\prime}}[f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot u^{\prime}]dx=0.$
(3.23)
Let
$\displaystyle g(x,s)\triangleq f_{\xi}(x,s,u^{\prime}_{\varepsilon}),$ (3.24)
$\displaystyle G_{m}(x)\triangleq\int^{u_{m}(x)}_{u(x)}g(x,s)ds.$ (3.25)
By condition $(H_{2})$, $g(x,s)$ is a absolutely continuous function on $x$.
By Lemma 2.1, $g(x,s)$ is almost everywhere differentiable, i.e.
$\frac{\partial g}{\partial x}$ exists almost everywhere. Derivating the both
sides of (3.25), we obtain
$G^{\prime}_{m}(x)=g(x,u_{m})\cdot u^{\prime}_{m}-g(x,u)\cdot
u^{\prime}+\int^{u_{m}(x)}_{u(x)}\frac{\partial g}{\partial x}dx,\ \ \ \
a.e.x\in\Omega^{\prime}.$ (3.26)
Because $G_{m}(x)$ vanishes outside $\Omega^{\prime}$, we obtain
$\int_{\Omega^{\prime}}G^{\prime}_{m}(x)dx=0.$ (3.27)
By (3.26), we have
$\displaystyle|\int_{\Omega^{\prime}}[f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot u^{\prime}]dx|$
$\displaystyle=$ $\displaystyle\left|\int_{\Omega^{\prime}}[g(x,u_{m})\cdot
u^{\prime}_{m}-g(x,u)\cdot u^{\prime}]dx\right|$ $\displaystyle=$
$\displaystyle|-\int_{\Omega^{\prime}}\int^{u_{m}(x)}_{u(x)}\frac{\partial
g}{\partial x}dsdx|\leq\int_{D_{m}}\left|\frac{\partial g}{\partial
x}\right|dxds,$
where
$D_{m}=\\{(x,s)\in\Omega^{\prime}\times H|\ \min\\{u_{m}(x),u(x)\\}\leq
s(x)\leq\max\\{u_{m}(x),u(x)\\}\\}.$
We note
$|D_{m}|=|\int_{\Omega^{\prime}}\int^{u_{m}}_{u}dsdx|\leq\int_{\Omega^{\prime}}|u_{m}-u|dx\rightarrow
0\ (m\rightarrow+\infty).$ (3.29)
By Fubini theorem, we have
$\int_{\Omega\times R}|\frac{\partial g}{\partial
x}|dxds=\int_{H}ds\int_{\Omega^{\prime}}\left|\frac{\partial g}{\partial
x}\right|dx$ (3.30)
Since $g(x,s)$ is absolutely continuous about $x$, $\frac{\partial g}{\partial
x}$ is integrable and absolutely integrable, i.e.
$\int_{\Omega^{\prime}}\left|\frac{\partial g}{\partial x}\right|dx<+\infty.$
(3.31)
Thus
$\int_{\Omega\times R}\left|\frac{\partial g}{\partial x}\right|dxds<+\infty$
(3.32)
Because of the absolute continuity of integral, we have
$\lim\limits_{m\rightarrow+\infty}\int_{D_{m}}\left|\frac{\partial g}{\partial
x}\right|dxds=0$ (3.33)
By (3), we obtain
$\lim\limits_{m\rightarrow+\infty}|\int_{\Omega^{\prime}}[f_{\xi}(x,u_{m},u^{\prime}_{\varepsilon})\cdot
u^{\prime}_{m}-f_{\xi}(x,u,u^{\prime}_{\varepsilon})\cdot u^{\prime}]dx|=0$
(3.34)
Thus we have proved (3.23). By (3.13)-(3.16) and (3.23), we have
$\lim\limits_{m\rightarrow+\infty}\int_{\Omega^{\prime}}[f(x,u_{m},u^{\prime}_{m})-f(x,u,u^{\prime})]dx\geq
0$ (3.35)
Thus we deduce that the functional $F(u,\Omega)$ is lower semicontinuous on
$W^{1,1}_{loc}(\Omega)$ with respect to the strong convergence in
$L^{1}_{loc}(\Omega)$, which does complete the proof.
## 4 Proof of Theorem 1.5
In order to prove Theorem 1.5, we will verify all the conditions in Theorem
1.4 from the assumptions in Theorem 1.5. Now we will divide into four steps to
complete the proof of Theorem 1.5:
Step 1: Similar to the first step of the proof in Theorem 1.4, without loss of
generality, we assume that the integrand $f(x,s,\xi)$ vanishes outside a
compact subset of $\Omega\times R$. Thus we assume that there exists an open
set $\Omega^{\prime}\times H\subset\subset\Omega\times R$, such that
$f(x,s,\xi)\equiv 0,\ \ \ \
\forall(x,s,\xi)\in(\Omega\backslash\Omega^{\prime})\times(R\backslash
H)\times R.$ (4.1)
Let $u_{m},u\in W^{1,1}_{loc}(\Omega)$, such that $u_{m}\rightarrow u$ in
$L^{1}_{loc}(\Omega)$, we need to prove
$\lim\limits_{m\rightarrow+\infty}F(u_{m},\Omega^{\prime})\geq
F(u,\Omega^{\prime}).$ (4.2)
By lemma 2.7, there exists a functions sequence $\\{f_{j}(x,s,\xi)\\}_{j\in
N}$, such that $\forall j\in N$, $f_{j}$ is a continuous function on
$\Omega^{\prime}\times H\subset\subset\Omega\times R$;
$\forall(x,s)\in\Omega^{\prime}\times H$, $f_{j}(x,s,\cdot)$ is convex on $R$,
and $\forall(x,s,\xi)\in\Omega^{\prime}\times H\times R$,
$\displaystyle f_{j}(x,s,\xi)\leq f_{j+1}(x,s,\xi)$ (4.3) $\displaystyle
f(x,s,\xi)=\sup\limits_{j\in N}f_{j}(x,s,\xi)$ (4.4)
$\displaystyle|f_{j}(x,s,\xi_{1})-f_{j}(x,s,\xi_{2})|\leq
M_{j}|\xi_{1}-\xi_{2}|,(x,s)\in\Omega^{\prime}\times H,\xi_{1},\xi_{2}\in R.$
(4.5)
Let $\eta_{\varepsilon}\in C^{\infty}_{c}(R)(0<\varepsilon<<1)$ be a mollifier
and define the $f_{j,\varepsilon}=f_{j}*\eta_{\varepsilon}$, i.e.
$f_{j,\varepsilon}(x,s,\xi)=\int_{R}f_{j}(x,s,\xi-z)\eta_{\varepsilon}(z)dz.$
(4.6)
By (4.5), we have
$\displaystyle|f_{j,\varepsilon}(x,s,\xi)-f_{j}(x,s,\xi)|$ $\displaystyle=$
$\displaystyle|\int_{R}f_{j}(x,s,\xi-z)\eta_{\varepsilon}(z)dz-\int_{R}f_{j}(x,s,\xi)\eta_{\varepsilon}(z)dz|$
$\displaystyle\leq$
$\displaystyle\int_{R}|f_{j}(x,s,\xi-z)-f_{j}(x,s,\xi)|\eta_{\varepsilon}(z)dz$
$\displaystyle\leq$ $\displaystyle\int_{{\rm
supp}\eta_{\varepsilon}}M_{j}|z|\cdot\eta_{\varepsilon}(z)dz\leq
M_{j}\cdot\varepsilon.$
Choosing $\varepsilon=\varepsilon_{j}=\frac{1}{jM_{j}}\rightarrow 0$. By (4),
we have
$|f_{j,\varepsilon_{j}}(x,s,\xi)-f_{j}(x,s,\xi)|\leq
M_{j}\varepsilon_{j}=\frac{1}{j}.$ (4.8)
So
$f_{j}(x,s,\xi)-\frac{2}{j}\leq f_{j,\varepsilon_{j}}(x,s,\xi)-\frac{1}{j}\leq
f_{j}(x,s,\xi)\leq f(x,s,\xi)$ (4.9)
By (4.3), (4.4) and Levi lemma, we have
$\lim\limits_{i\rightarrow+\infty}\int_{\Omega^{\prime}}f_{j}(x,u(x),u^{\prime}(x))dx=\int_{\Omega^{\prime}}f(x,u(x),u^{\prime}(x))dx.$
(4.10)
Denote
$F_{j}(u,\Omega^{\prime})=\int_{\Omega^{\prime}}[f_{j,\varepsilon_{j}}(x,u(x),u^{\prime}(x))-\frac{1}{j}]dx$
(4.11)
By (4.9)-(4.11), we have
$\lim\limits_{j\rightarrow+\infty}F_{j}(u,\Omega^{\prime})=F(u,\Omega^{\prime})=\int_{\Omega^{\prime}}f(x,u(x),u^{\prime}(x))dx.$
(4.12)
Obviously,
$F_{j}(u,\Omega^{\prime})\leq F(u,\Omega^{\prime}),\ \ \ \ \forall j\in N.$
Thus
$\sup\limits_{j\in N}F_{j}(u,\Omega^{\prime})=F(u,\Omega^{\prime}).$ (4.13)
By Lemma 2.6, in order to prove that $F(u,\Omega^{\prime})$ is lower
semicontinuous on $W^{1,1}_{loc}(\Omega)$ with respect to the strong
convergence in $L^{\prime}_{loc}(\Omega)$, it’s enough that we prove lower
semicontinuity with respect to the strong convergence in $L^{1}_{loc}(\Omega)$
for the functional sequence $\\{F_{j}(u,\Omega^{\prime})\\}_{j\in N}$ on
$W^{1,1}_{loc}(\Omega)$ .
Step 2: In order to prove that $\forall j\in N,F_{j}(u,\Omega^{\prime})$ is
lower semicontinuous on $W^{1,1}_{loc}(\Omega)$ with respect to the strong
convergence in $L^{1}_{loc}(\Omega)$, we will prove that $\forall j\in N$, the
integrand $f_{j,\varepsilon_{j}}(x,u(x),u^{\prime}(x))$ satisfy all conditions
of theorem 1.4.
$\forall\xi_{1},\xi_{2}\in R,0<\lambda<1$, by the convexity of
$f_{j}(x,s,\cdot)$ on $R$, we have
$f_{j}(x,s,\lambda\xi_{1}+(1-\lambda)\xi_{2})\leq\lambda
f_{j}(x,s,\xi_{1})+(1-\lambda)f_{j}(x,s,\xi_{2}).$ (4.14)
Thus
$\displaystyle f_{j,\varepsilon_{j}}(x,s,\lambda\xi_{1}+(1-\lambda)\xi_{2})$
$\displaystyle=$
$\displaystyle\int_{R}f_{j}(x,s,\lambda\xi_{1}+(1-\lambda)\xi_{2}-z)\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle=$
$\displaystyle\int_{R}f_{j}(x,s,\lambda(\xi_{1}-z)+(1-\lambda)(\xi_{2}-z))\cdot\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle\leq$
$\displaystyle\lambda\int_{R}f_{j}(x,s,(\xi_{1}-z))\cdot\eta_{\varepsilon_{j}}(z)dz+(1-\lambda)\int_{R}f_{j}(x,s,(\xi_{2}-z))\cdot\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle=$ $\displaystyle\lambda
f_{j,\varepsilon_{j}}(x,s,\xi_{1})+(1-\lambda)f_{j,\varepsilon_{j}}(x,s,\xi_{2}).$
Thus $f_{j,\varepsilon_{j}}$ satisfy $(H_{1})$.
Step 3: $\forall(x,s)\in\Omega^{\prime}\times H,\xi_{1},\xi_{2}\in R$, by
(4.5), we have
$\displaystyle|f_{j,\varepsilon_{j}}(x,s,\xi_{1})-f_{j,\varepsilon_{j}}(x,s,\xi_{2})|$
$\displaystyle=$
$\displaystyle|\int_{R}[f_{j}(x,s,\xi_{1}-z)\eta_{\varepsilon_{j}}(z)-f_{j}(x,s,\xi_{2}-z)\eta_{\varepsilon_{j}}(z)]dz|$
$\displaystyle\leq$
$\displaystyle\int_{R}|f_{j}(x,s,\xi_{1}-z)-f_{j}(x,s,\xi_{2}-z)|\cdot\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle\leq$ $\displaystyle\int_{{\rm
supp}\eta_{\varepsilon}}M_{j}|\xi_{1}-\xi_{2}|\eta_{\varepsilon_{j}}(z)dz\leq
M_{j}|\xi_{1}-\xi_{2}|.$
Thus
$\left|\frac{\partial f_{j,\varepsilon_{j}}}{\partial\xi}\right|\leq M_{j}.$
(4.17)
So $f_{j,\varepsilon_{j}}$ satisfies (1.7) in the condition (H3) of Theorem
1.4.
Now, we will prove $f_{j,\varepsilon_{j}}$ satisfies (1.8) in the condition
(H3) of Theorem 1.4. By ${\rm supp}(\eta_{\varepsilon_{j}})\subseteq
B(0,\varepsilon_{j})$,we have
$\displaystyle\frac{\partial f_{j,\varepsilon_{j}}}{\partial\xi}(x,s,\xi)$
$\displaystyle=$ $\displaystyle\int_{R}\frac{\partial
f_{j}(x,s,\xi-z)}{\partial\xi}\cdot\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle=$ $\displaystyle-\int_{R}\frac{\partial
f_{j}(x,s,\xi-z)}{\partial z}\cdot\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle=$
$\displaystyle\int_{R}f_{j}(x,s,\xi-z)\frac{\partial\eta_{\varepsilon_{j}}(z)}{\partial
z}dz.$
By (4.5) and (4), we have
$\displaystyle\left|\frac{\partial
f_{j,\varepsilon_{j}}}{\partial\xi}(x,s,\xi_{1})-\frac{\partial
f_{j,\varepsilon_{j}}}{\partial\xi}(x,s,\xi_{2})\right|$ $\displaystyle\leq$
$\displaystyle\int_{R}\left|f_{j}(x,s,\xi_{1}-z)-f_{j}(x,s,\xi_{2}-z)\right|\cdot\left|\frac{\partial\eta_{\varepsilon_{j}}(z)}{\partial
z}\right|dz$ $\displaystyle\leq$ $\displaystyle
M_{j}|\xi_{1}-\xi_{2}|\cdot\int_{R}\left|\frac{\partial\eta_{\varepsilon_{j}}(z)}{\partial
z}\right|dz=L_{j}M_{j}|\xi_{1}-\xi_{2}|,$
where
$L_{j}=\int_{R}\left|\frac{\partial\eta_{\varepsilon_{j}}(z)}{\partial
z}\right|dz$ (4.20)
is a constant depending on $\varepsilon_{j}$. Thus $f_{j,\varepsilon_{j}}$
satisfies (1.8). So we have proved that $f_{j,\varepsilon_{j}}$ satisfies
$(H_{3})$.
Step 4: Next we will prove that $f_{j,\varepsilon_{j}}$ satisfies condition
$(H2)$.
By the condition $(H4)$, for every compact subset $\Omega^{\prime}\times
H\times K$, $f(x,s,\xi)$ is absolutely continuous about $x$, i.e.
$\forall\varepsilon_{0}>0,\ \exists\delta>0$ such that for any finite disjoint
open interval $\\{(x_{i},y_{j})\\}^{n}_{i=1}$ in $\Omega^{\prime}$, when
$\Sigma^{n}_{i=1}(y_{i}-x_{i})<\delta$, we have
$\sum\limits^{n}_{i=1}|f(y_{i},s,\xi)-f(x_{i},s,\xi)|<\varepsilon_{0}$ (4.21)
By Lemma 2.7, there exists continuous functions sequence
$\\{f_{j}(x,s,\xi)\\}_{i\in N}$, $\forall j\in N$,
$\forall(x,s)\in\Omega^{\prime}\times H,\ f_{j}(x,s,\cdot)$ is convex on $R$,
and $\forall(x,s,\xi)\in\Omega^{\prime}\times H\times R$, we have
$f_{j}(x,s,\xi)=\max\limits_{1\leq q\leq j}\\{0,a_{q}(x,s)+b_{q}(x,s)\xi\\},\
j\in N.$ (4.22)
where
$\displaystyle
a_{q}(x,s)=\int_{R}f(x,s,\xi)\left[2\eta_{q}(\xi)+\xi\frac{\partial\eta_{q}(\xi)}{\partial\xi}\right]d\xi,$
(4.23) $\displaystyle
b_{q}(x,s)=-\int_{R}f(x,s,\xi)\frac{\partial\eta_{q}(\xi)}{\partial\xi}d\xi,$
(4.24)
and $\eta_{q}\in C^{\infty}_{c}(R)\ (q\in N)$ are mollifiers satisfying
$\eta_{q}\geq 0,\ \int_{R}\eta_{q}(\xi)d\xi=1$ and ${\rm
supp}(\eta_{q})\subseteq B(0,1)$, $\forall j\in N$. By (4.22), without of loss
generality, we assume that there exists $l\in\\{1,\cdots,j\\}$, such that
$f_{j}(x,s,\xi)=a_{l}(x,s)+b_{l}(x,s)\cdot\xi$ (4.25)
where $a_{l},b_{l}$ are given by (4.23)-(4.24). By (4.21), we obtain
$\displaystyle\sum\limits^{n}_{i=1}|a_{l}(y_{i},s)-a_{l}(x_{i},s)|$
$\displaystyle=$
$\displaystyle\sum\limits^{n}_{i=1}|\int_{R}[f(y_{i},s,\xi)-f(x_{i},s,\xi)]\cdot[2\eta_{l}(\xi)+\xi\frac{\partial\eta_{l}(\xi)}{\partial\xi}]d\xi|$
$\displaystyle\leq$
$\displaystyle\int_{R}\sum\limits^{n}_{i=1}|f(y_{i},s,\xi)-f(x_{i},s,\xi)|\cdot[2\eta_{l}(\xi)+|\xi\frac{\partial\eta_{l}(\xi)}{\partial\xi}|]d\xi$
$\displaystyle\leq$
$\displaystyle\varepsilon_{0}\int_{B(0,1)}[2\eta_{l}(\xi)+|\xi\frac{\partial\eta_{l}(\xi)}{\partial\xi}|]d\xi\leq(2+A_{l})\cdot\varepsilon_{0},$
where
$A_{l}=\int_{B(0,1)}|\frac{\partial\eta_{l}(\xi)}{\partial\xi}|d\xi$
is a constant. Similar to the above proof, we have
$\displaystyle\sum\limits^{n}_{i=1}|b_{l}(y_{i},s)-b_{l}(x_{i},s)|$
$\displaystyle=$
$\displaystyle\sum\limits^{n}_{i=1}|\int_{R}[f(y_{i},s,\xi)-f(x_{i},s,\xi)]\cdot\frac{\partial\eta_{l}(\xi)}{\partial\xi}d\xi|$
$\displaystyle\leq$
$\displaystyle\int_{R}\sum\limits^{n}_{i=1}|f(y_{i},s,\xi)-f(x_{i},s,\xi)|\cdot\left|\frac{\partial\eta_{l}(\xi)}{\partial\xi}\right|d\xi$
$\displaystyle\leq$
$\displaystyle\varepsilon_{0}\int_{B(0,1)}\left|\frac{\partial\eta_{l}(\xi)}{\partial\xi}\right|d\xi\leq
A_{l}\cdot\varepsilon_{0}.$
Thus
$\displaystyle\sum\limits^{n}_{i=1}|f_{j}(y_{i},s,\xi)-f_{j}(x_{i},s,\xi)|$
$\displaystyle=$
$\displaystyle\sum\limits^{n}_{i=1}|a_{l}(y_{i},s)-a_{l}(x_{i},s)+[b_{l}(y_{i},s)-b_{l}(x_{i},s)]\cdot\xi|$
$\displaystyle\leq$
$\displaystyle\sum\limits^{n}_{i=1}|a_{l}(y_{i},s)-a_{l}(x_{i},s)|+\sum\limits^{n}_{i=1}|b_{l}(y_{i},s)-b_{l}(x_{i},s)|\cdot|\xi|$
$\displaystyle\leq$
$\displaystyle(2+A_{l})\varepsilon_{0}+A_{l}\varepsilon_{0}K_{1}=(2+A_{l}+A_{l}K_{1})\varepsilon_{0}\triangleq\sigma.$
Since $\xi$ varies on a compact set, then
$K_{1}=\sup\limits_{\xi}\\{|\xi|\\}<+\infty$. Choosing $\varepsilon_{0}$
sufficient small so that $\sigma$ is enough small. Thus $f_{j}(x,s,\xi)$ is
absolutely continuous about $x$ on any compact subset of $\Omega\times R\times
R$. By (4.6) and (4), we have
$\displaystyle\sum\limits^{n}_{i=1}|f_{j,\varepsilon_{j}}(y_{i},s,\xi)-f_{j,\varepsilon_{j}}(x_{i},s,\xi)|$
$\displaystyle=$
$\displaystyle\sum\limits^{n}_{i=1}\left|\int_{R}[f_{j}(y_{i},s,\xi-z)\eta_{\varepsilon_{j}}(z)-f_{j}(x_{i},s,\xi-z)\eta_{\varepsilon_{j}}(z)]dz\right|$
$\displaystyle\leq$
$\displaystyle\int_{R}\sum\limits^{n}_{i=1}|f_{j}(y_{i},s,\xi-z)-f_{j}(x_{i},s,\xi-z)|\cdot\eta_{\varepsilon_{j}}(z)dz$
$\displaystyle\leq$
$\displaystyle\sigma\cdot\int_{B(0,\varepsilon_{j})}\eta_{\varepsilon_{j}}(z)dz=\sigma.$
By (4) and (4), we obtain
$\displaystyle\sum\limits^{n}_{i=1}\left|\frac{\partial
f_{j,\varepsilon_{j}}}{\partial\xi}(y_{i},s,\xi)-\frac{\partial
f_{j,\varepsilon_{j}}}{\partial\xi}(x_{i},s,\xi)\right|$ $\displaystyle\leq$
$\displaystyle\int_{R}\sum\limits^{n}_{i=1}|f_{j}(y_{i},s,\xi-z)-f_{j}(x_{i},s,\xi-z)|\cdot\left|\frac{\partial\eta_{\varepsilon_{j}}(z)}{\partial
z}\right|dz$ $\displaystyle\leq$
$\displaystyle\sigma\int_{R}\left|\frac{\partial\eta_{\varepsilon_{j}(z)}}{\partial
z}\right|dz=L_{j}\sigma$
where $L_{j}$ are constants depending on $\varepsilon_{j}$ and given by (4.20)
($\forall j\in N$). By (4), for every compact subset on $\Omega\times R\times
R,\frac{\partial f_{j,\varepsilon_{j}}}{\partial\xi}$ is absolutely continuous
about $x$. Thus $f_{j,\varepsilon_{j}}$ satisfy condition $(H_{2})$.
Now,we have proved $f_{j,\varepsilon_{j}}$ satisfies all conditions in Theorem
1.4, so $F_{j}(u,\Omega^{\prime})$ is lower semicontinuous in
$W^{1,1}_{loc}(\Omega)$ with respect to the strong convergence in
$L^{1}_{loc}(\Omega)$. Thus $F(u,\Omega)$ has the same lower semicontinuity.
This completes the proof of Theorem 1.5.
## References
* [1] Adams R.A.,Fournier J.F., Sobolev space. 2nd Ed,Academic press,2003.
* [2] Ambrosio,L., Fusco,N., Pallara,D., Functions of bounded variation and free discontinuity problems, Oxford University Press, Inc. New York, 2000.
* [3] Cicco,V.De, Leoni,G., A chain rule in $L^{1}(div;\Omega)$ and its applications to lower semicontinuity, Calculus of Variations, 2003,19: 23-51.
* [4] Ekeland,I., Témam,R., Convex analysis and variational problems, North-Holland, Amsterdam, 1976.
* [5] Evans,L.C., Partial Differential Equations, American Mathematical Society, Providence, 1998.
* [6] Fonseca,I., Leoni,G., Some remarks on lower semicontinuity, Indiana Univ. Math. J., 2000,49: 617-635.
* [7] Fonseca,I., Leoni,G., On lower semicontinuity and relaxation, Proc. Roy. Soc. Edinburgh, 2001,131A: 519-565.
* [8] Giorgi,E.De, Teoremi di semicontinuit’a nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968.
* [9] Gori,M., Marcellini,P., An extension of the Serrin s lower semicontinuity theorem, J. Convex Analysis 2002,9: 475-502.
* [10] Gori,M., Maggi,F., Marcellini,P., On some sharp conditions for lower semicontinuity in L1, Differential and Integral Equations, 2003,16(1): 51-76.
* [11] Jean B.,Hiriart U.,Claude, Fundametals of convex analysis.Springer,2001.
* [12] Pauc C.Y., La M thode mtrique en calcul des variations. Hermann, Paris,1941.
* [13] Serrin,J., On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 1961,101: 139-167.
* [14] Stein E. M., Shakarchi R., Real analysis. Princeton University Press,2005.
|
arxiv-papers
| 2012-05-13T03:50:18 |
2024-09-04T02:49:30.862579
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hu Xiaohong and Zhang Shiqing",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1205.2826"
}
|
1205.2851
|
# Reduction of dynamical biochemical reaction networks in computational
biology
O. Radulescu1, A.N. Gorban2, A. Zinovyev3 and V. Noel4
1 DIMNP UMR CNRS 5235, University of Montpellier 2, Montpellier, France.
2 Department of Mathematics, University of Leicester, LE1 7RH, UK.
3 Institut Curie, U900 INSERM/Curie/Mines ParisTech, 26 rue d’Ulm, F75248
Paris, France.
4IRMAR UMR 6625, University of Rennes 1, Rennes, France.
Abstract
Biochemical networks are used in computational biology, to model the static
and dynamical details of systems involved in cell signaling, metabolism, and
regulation of gene expression. Parametric and structural uncertainty, as well
as combinatorial explosion are strong obstacles against analyzing the dynamics
of large models of this type. Multi-scaleness is another property of these
networks, that can be used to get past some of these obstacles. Networks with
many well separated time scales, can be reduced to simpler networks, in a way
that depends only on the orders of magnitude and not on the exact values of
the kinetic parameters. The main idea used for such robust simplifications of
networks is the concept of dominance among model elements, allowing
hierarchical organization of these elements according to their effects on the
network dynamics. This concept finds a natural formulation in tropical
geometry. We revisit, in the light of these new ideas, the main approaches to
model reduction of reaction networks, such as quasi-steady state and quasi-
equilibrium approximations, and provide practical recipes for model reduction
of linear and nonlinear networks. We also discuss the application of model
reduction to backward pruning machine learning techniques.
## 1 Introduction
During the last decades, biologists have identified a wealth of molecular
components and regulatory mechanisms underlying the control of cell functions.
Cells integrate external signals through sophisticated signal transduction
pathways, ultimately affecting the regulation of gene expression, including
that of the signaling components. Metabolic functions are sustained and
controlled by complex machineries involving genes, enzymes and metabolites.
The genetic regulations result from the coordinate effect of many, mutually
interacting genes. These regulations involve many molecular actors, including
proteins and regulatory RNAs, which form large, intricate networks.
Current dynamical models of cellular molecular processes are small size
networks. These small scale models, that are subjective simplifications of
reality, can not take into account the specificities of regulatory mechanisms.
New methods are needed, allowing to reconcile small scale dynamical models and
large scale, but static, network architectures. The main obstacle to
increasing the size of dynamical networks is the incomplete information, on
the parameters and on the mechanistic details of the interactions. In vivo
values of the parameters depend on crowding and heterogeneity of the
intracellular medium, and can be orders of magnitude different from what is
measured in vitro. Furthermore, learning models from data suffer for non-
identifiability and over-fitting problems. Thus, model reduction is an
avoidable step in the study of large networks, allowing to extract the
essential features of the model, that can then be identified from data. Model
reduction in computational biology should have several particularities.
First of all, model reduction should cope with parametric incompleteness
and/or uncertainty.
A certain class of reduction methods are parameter independent and
automatically comply with this specificity. In biochemical networks, the
number of possible chemical species grows combinatorially due to numerous
possibilities of interactions between molecules with multiple interaction
sites. The exact lumping methods [12, 18] reduce the number of microstates and
avoid combinatorial explosion in the description and analysis of large models
of receptor and scaffold signalling. A similar technique [24] is used to
rationally organize supramolecular complexes in rule-based modeling [21] of
biochemical networks. Other, parameter independent, coarse-graining techniques
are graphical methods formalizing node deletion and merging operations in
biochemical networks [29], pooling of metabolites in large scale metabolic
networks [73, 54], or extensive searches in the set of all possible lumps
[22]. Finally, qualitative reduction methods were used to simplify large
logical regulatory graphs, adequately suppressing nodes and defining sub-
approximating dynamics [68, 69].
Secondly, biochemical processes governing network dynamics span over many
timescales. For example, changing gene expression programs can take hours and
even days while protein complex formation goes on the second scale and post-
translational protein modifications take minutes to happen. Protein life half-
times can vary from minutes to days. Model reduction should exploit
multiscaleness. Asymptotic dynamics of networks with slow and fast processes,
can be strongly simplified using various ideas such as inertial and invariant
manifolds (IM) and averaging approximations.
The iterative methods of IM aim to find a slow low dimensional IM, containing
the asymptotic dynamics [37, 38, 81]. The Computational Singular Perturbation
(CSP) [59, 15] aims to find even more, the slow IM and, in addition, the
geometry of fast foliation. Invariant manifolds can be calculated by various
other methods [39, 41, 81, 56, 58].
Very popular are the methods for computation of a “first approximations” to
the slow IM. The classical quasi steady-state approximation (QSS) was proposed
by [10] and was elaborated into an important tool for analysis of chemical
reaction mechanism and kinetics [87, 16, 50]. The classical QSS is based on
the relative smallness of concentrations of some of active reagents (radicals,
concentration of enzyme and substrate-enzyme complexes or amount of active
centers on the catalyst surface) [5, 86, 100]. The quasiequilibirium
approximation (QE) has two basic formulations: the thermodynamic approach,
based on conditional entropy maximum (or free energy conditional minimum), or
the kinetic formulation, based on equilibration of fast reversible reactions.
The very first use of the entropy maximum dates back to Gibbs [30].
Corrections to QE approximation with applications to physical and chemical
kinetics were developed by [40, 39]. An important, still unsolved, problem of
these two approximations is the detection of QSS species and QE reactions
without application of all machinery of the IM or CSP methods. Indeed, not all
reactions with large constants are at quasi-equilibrium, and there are no
simple rules to find QSS species if there is no such hints as a small amount
of a conserved quantity (like the total concentration of enzyme). The method
of Intrinsic Low Dimensional Manifolds (ILDM) [63, 14] provides an
approximation of a low dimensional invariant manifold and works as a first
step of CSP [55].
Another method allowing to simplify multiscale dynamics is averaging. This
idea can be tracked back to Poincaré’s perturbative treatment of the many body
problem in celestial mechanics [74], further developed in classical mechanics
by other authors [6, 62], and also known as adiabatic or Born-Oppenheimer
approximation in quantum mechanics [66]. Rather generally, averaging can be
applied when some fine scale variables of the system are rapidly oscillating.
Then, the dynamics of slow, coarse scale variables, can be obtained by time
averaging the system over a timescale much larger than the period of the fast
oscillations. The way to perform averaging, depends on the structure of the
system, namely on the definition of the coarse grained and fine variables [11,
7, 2, 85, 1, 34, 88].
Some of these ideas have been implemented in computational biology tools.
Systems biology markup language SBML [52] can allocate a ”fast” attribute to
reaction elements. Fast reaction specification can be taken into account by
computational biology softwares such as VirtualCell [90] that implements a QE
approximation algorithm [89]. Similarly, the simulation tool COPASI [51]
implements the ILDM method [93].
Finally, multiscaleness does not uniquely apply to timescales but equivalently
to abundances of various species in these networks. mRNA copy numbers can
change from some units to tens of thousands, and the dynamic concentration
range of biological proteins can reach up to five orders of magnitude.
Furthermore, the DNA molecule has only one or a few copies. Low copy numbers
lead, directly or indirectly (a species can be stochastic even if present in
large copy numbers), to stochastic gene expression. In computational biology,
model reduction should thus cope not only with deterministic, but also with
stochastic and hybrid models. The need to reduce large scale stochastic models
is acute. Indeed, stochastic simulation algorithm (SSA, [32, 31]) can be very
expensive in computer time when applied to large unreduced models, precluding
model analysis and identification. For this reason, extensive effort has been
dedicated to adapting the main ideas used for model reduction of deterministic
models, namely exact lumping, invariant manifolds, QSS, QE, and averaging, to
the case of stochastic models.
Reduction of stochastic rule-based models, based on a weakened version of the
exact lumpability criterion, has been proposed by [25] to define abstract
species or stochastic-fragments that can be further used in simplified
calculations. Multiscaleness of stochastic models is two-fold, it affects both
species and reaction rates. This has been exploited in hybrid stochastic
simulation schemes that are, for the most of them, based on a partition of the
biochemical reactions in fast and slow reactions [49, 13, 4, 48, 3, 82, 57,
47, 92, 83, 45, 9, 60, 36, 72]. Conversely, mixed partitions, using both
reactions and species can exploit both types of multiscaleness and more
appropriately unravel a rich variety of stochastic functioning regimes such as
piece-wise deterministic, switched diffusions, diffusions with jumps, as well
as averaged processes [80, 20, 19] only partially covered by some situations
discussed in [64].
Machine learning approaches to parameter identification [35] could profit from
Fokker-Planck approximations, also known as diffusion approximations or
Langevin approach, of the master equation describing dynamics of stochastic
networks. Traditional approaches such as central limit theorem [33, 65], the
$\Omega$ and the Kramers-Moyal expansions [80, 20] where used to derive
diffusion approximations. Alternatively, [23] propose diffusion approximations
for slow/fast stochastic networks, in which the drift and diffusion parameters
are obtained numerically. By the ergodic theorem, time averaging of multiscale
stochastic models boils down to a QE assumption for the fast variables. This
idea has been used in [20] to reduce stochastic networks. A few computational
biology tools implement stochastic approximations [83].
With the exception of the parameter independent methods, all the model
reduction methods described above need a full parametrization of the model.
This is a stringent requirement, and can not be easily bypassed. Indeed, the
reduction has a local validity. The elements defining a reduced model such as
IM, QSS species, QE species, depend on the model parameters and also on the
position on a trajectory of the dynamics. What one can expect is that model
reduction is robust, i.e. a given reduced model provides an accurate
approximation of the dynamics of the initial model for a wide range of
parameters and variables values. One can show that this property is satisfied
by biochemical networks with separated constants, because in this case the
simplified networks depend on the order relations among model parameters and
not on the precise values of these parameters [42, 76, 70].
The purpose of this review is not the exhaustive description of all the
reduction methods that we have delineated. We will revisit the fundamental
concepts of model reduction in the light of a new program, that should, in the
long term, lead to a new generation of reduction tools satisfying all the
specific requirements of computational biology. Due to space limitations, we
restrict ourselves to deterministic models.
## 2 Deterministic dynamical networks
To construct a dynamic reaction network we need the list of components,
$\mathcal{A}=\\{A_{1},...\,A_{n}\\}$ and the list of reactions (the reaction
mechanism):
$\sum_{i}\alpha_{ji}A_{i}\rightleftharpoons\sum_{k}\beta_{jk}A_{k},$ (1)
where $j\in[1,r]$ is the reaction number.
Dynamics of nonlinear networks in homogeneous isochoric systems (fixed volume)
is described by a system of differential equations:
$\frac{dc}{dt}=P(c)=\sum_{j=1}^{r}\nu_{j}(R^{+}_{j}(c)-R^{-}_{j}(c))$ (2)
$c\in{\mathbb{R}}^{n}$ is the concentration vector,
$\nu_{j}=\beta_{j}-\alpha_{j}$ is the global stoichiometric vector. The
reaction rates $R^{+/-}_{j}(c)$ are non-linear functions of the
concentrations. For instance, the mass action law reads
$R^{+}_{j}(c)=k_{j}^{+}\prod_{i}c_{i}^{\alpha_{ji}}$,
$R^{-}_{j}(c)=k_{j}^{-}\prod_{i}c_{i}^{\beta_{ji}}$, in which case $P_{i}(c)$
is a multivariate polynomial on the concentrations $c_{j}$.
## 3 Multi-scale reduction of monomolecular reaction networks
Monomolecular reaction networks are the simplest reaction networks. The
structure of these networks is completely defined by a digraph, in which
vertices correspond to chemical species $A_{i}$, edges correspond to reactions
$A_{i}\to A_{j}$ with kinetic constants $k_{ji}>0$.
The kinetic equation is
$\frac{dc_{i}}{dt}=\sum_{j}k_{ij}c_{j}-\left(\sum_{j}k_{ji}\right)c_{i},$ (3)
or in matrix form: $\dot{c}=Kc$.
The solutions of (3) can be expressed in terms of left and right eigenvectors
of the kinetic matrix $K$:
$c(t)=(l^{0},c(0))+\sum_{k=1}^{n-1}r^{k}(l^{k},c(0))\exp(-\lambda_{k}t)$ (4)
where $Kr^{k}=\lambda_{k}r^{k}$, and $l^{k}K=\lambda_{k}l^{k}$.
Each eigenvalue $\lambda_{k}$ is the inverse of a timescale of the network. A
reduced network having solutions of the type (4), with eigenvectors $r^{k}$,
$l^{k}$, and eigenvalues $\lambda_{k}$ approximating the eigenvectors and the
eigenvalues of the original network is called a multiscale approximation.
We say that the network constants are totally separated if for all
$(i,j)\neq(i^{\prime},j^{\prime})$ one of the relations
$k_{ji}<<k_{j^{\prime}i^{\prime}}$, or $k_{ji}>>k_{j^{\prime}i^{\prime}}$ is
satisfied.
It was shown in [42, 76, 43] that the multiscale approximations of arbitrary
monomolecular reaction networks with totally separated constants are acyclic
(have no cycles), and deterministic (have no nodes from which leave more than
one edge) digraphs.
In order to reduce a network with total separation, one needs only qualitative
information on the constants. More precisely, each edge of the reaction
digraph can be labeled by a positive integer representing the rank of the
reaction parameter in the ordered series of parameter values, the largest
parameter (the quickest reaction) having the lowest label. These integer
labels also indicate the timescales of the processes modeled by the network
reactions.
The reduced network is not always a subgraph of the initial graph. It is
obtained from this integer labeled digraph by graph re-writing operations,
that can be generically described as pruning and pooling. Two types of pruning
operations are of primary importance (see also Figure 1) :
Rule a)
If one has one node from which leave more than one edge, then all the edges
are pruned with the exception of the fastest one (lowest integer label). This
operation corresponds to keeping the dominant term among the terms
$c_{i}k_{ij}$ consuming a species $A_{i}$, and reduces the node outdegree to
one. The same principle can not be applied to reduce the indegree, because
which production term is dominant among $k_{ij}c_{j}$ depends not only on
$k_{ij}$ but also on the concentrations $c_{j}$.
Rule b)
Cycles with separated constants can be transformed into chains, by elimination
of the slowest step. This can be justified intuitively by topology, because
any two nodes of a cycle are connected by two paths, one containing the
slowest step and the other one not containing the slowest step. The latter
shortcuts the former.
However, a combination of rules a) and b) is not allowed to prune slow
reactions leaving cycles and further transform the cycles into chains. Indeed,
the total mass of such cycles is slowly decaying because of outgoing
reactions. Pruning the slow reactions that leave a cycle would keep the total
cycle mass constant and produce the wrong long time approximation. In this
case, pooling operations are needed:
Rule c)
Glue each cycle in the pruned system into a new vertex and transform the
network of all initial reactions into a new one. The concentration of this new
component is the sum of the concentration of the glued vertices. Reactions to
the cycles transform into reactions to the correspondent new vertices (with
the same constants). To transform the reactions from the cycles, we have to
calculate the normalized quasi-stationary distributions inside each cycle
(with unit sum of the concentrations in each cycle). Let for the vertex
$A_{i}$ from a cycle this concentration be $c_{i}^{\circ}$. Then the reaction
$A_{i}\to A_{j}$ with the constant $k_{ji}$ transforms into the reaction from
the new (“cycle”) vertex with the constant $k_{ji}c_{i}^{\circ}$. The
destination vertex of this reaction is $A_{j}$ if it does not belong to a
cycle of the pruned system, it is the correspondent glued cycle if it includes
$A_{j}$ and does not include $A_{i}$ and the reaction vanishes if both $A_{i}$
and $A_{j}$ belong to the same cycle of the pruned system.
After pooling we have to prune (Rule a) and so on, until we get an acyclic
pruned system. Then the way back follows: we have to restore cycles and cut
them (Rule b).
In more detail, the graph re-writing operations, are described in the Appendix
and illustrated in Figure 1. The dynamics of reduced acyclic deterministic
digraphs follows from their topology and from the timescale labels. First of
all, let us notice that the network has as many timescales as remaining edges
in the reduced digraph. The computation of eigenvectors of acyclic
deterministic digraphs is straightforward [42, 76, 43]. For networks with
total separation, these eigenvectors satisfy, in the first approximation, a
$0-1$ type property, the coordinates of $l^{k}$, $r^{k}$ belong to the sets
$\\{0,1\\}$, and $\\{0,1,-1\\}$ respectively. The $0-1$ property of
eigenvectors has a non-trivial consequence. On the timescale
$t_{k}=(\lambda_{k})^{-1}$, the reduced digraph behaves as an effective
reaction (single step approximation). The effective reaction receives (from
reactions acting on smaller timescales) the mass coming from the species with
coordinate $1$ in $l^{k}$ (pool) and transfers it (during a time $t_{k}$) to
the species with coordinate $1$ in $r^{k}$. The successive single step
approximations of an acyclic deterministic digraph are illustrated in Figure
2.
Monomolecular networks with separation represent instructive examples where
reduction and qualitative dynamics result from the network topology and from
the orders of magnitude of the kinetic constants. This type of models can be
used in computational biology to reduce linear subnetworks or even binary
reactions for which one reactant is present in much larger quantities than the
other (pseudo-monomolecular approximation).
As argued by a few authors, total separation could be a generic property of
biochemical networks [28]. This property can be checked empirically by
investigating the distribution of network timescales in logarithmic scale.
Whenever one finds distributions with large support in logarithmic scale (a
log-uniform distribution is equivalent to the Zipf law, i.e. a power law
distribution with exponent $-1$, well known in critical systems [28]) total
separation is valid and the above reduction method applies.
## 4 Separation, dominance, and tropical geometry
The previously presented algorithm is based on the idea of dominance, which
occurs at many levels. For instance, when several reactions compete for the
same pool, all can be pruned, excepting the dominant one (Rule a)). This
simple idea is widely spread, and corresponds to max-plus algebra: the sum of
positive, well separated terms, can be replaced by the maximum term. Max-plus
algebra, that found many applications to dynamical systems [17, 96, 8], belong
to the new mathematical field of tropical geometry [71]. Tropical geometry
offers convenient solutions to solve systems of polynomial equations with
separated monomials, to simplify and hybridize systems of polynomial or
rational ordinary differential equations with separated monomials. We can
conveniently use tropical geometry concepts to rationalize many model
reduction operations and find new ones.
The logarithmic transformation $u_{i}=logx_{i},\,1\leq i\leq n$, well known
for drawing graphs on logarithmic paper, plays a central role in tropical
geometry [97].
Let us consider multivariate monomials $M(\mathbold x)=a_{\alpha}\mathbold
x^{\alpha}$, where $\mathbold
x^{\alpha}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\ldots x_{n}^{\alpha_{n}}$.
Monomials with positive coefficients $a_{\alpha}>0$, become linear functions,
$logM=loga_{\alpha}+<\alpha,log(\mathbold x)>$, by this transformation.
There is a straightforward way to use the logarithmic transformation from
tropical geometry in order to obtain approximations of dynamical networks of
the type (2). Let us suppose that reaction rates are polynomial functions of
the concentrations (this is satisfied by mass action law and obviously, also
by monomolecular networks), such that
$\sum_{j=1}^{r}\nu_{j}(R^{+}_{j}(c)-R^{-}_{j}(c))=\sum_{\alpha\in
A}a_{\alpha}\mathbold c^{\alpha}$.
We call tropicalization of the smooth ODE system (2) the following piecewise-
smooth system:
$\frac{dc_{i}}{dt}=s_{i}exp[max_{\alpha\in
A_{i}}\\{log(|a_{i,\alpha}|)+<\mathbold c,\alpha>\\}],$ (5)
where $\mathbold u=(logc_{1},\ldots,logc_{n})$,
$s_{i}=sign(a_{i,\alpha_{max}})$ and $a_{i,\alpha_{max}},\,\alpha_{max}\in
A_{i}$ denotes the coefficient of a monomial for which the maximum occurring
in (5) is attained.
The tropicalization associates to a polynomial $\sum_{\alpha\in
A}a_{\alpha}\mathbold c^{\alpha}$, the max-plus polynomial
$P^{\tau}(\mathbold c)=exp[max_{\alpha\in
A}\\{log(|a_{\alpha}|)+<log(\mathbold c),\alpha>\\}].$
In other words, a polynomial is replaced by a piecewise smooth function, equal
to the largest, in absolute value, of its monomials. Thus, (5) is a piecewise
smooth model [95] because the dominating monomials in the max-plus polynomials
can change from one domain to another of the concentration space. The singular
set where at least two of the monomials are equal, and where the max-plus
polynomial $P^{\tau}(\mathbold c)$ is not smooth is called tropical variety
[67]. On logarithmic paper, the tropical varieties of various species define
polyhedral domains inside which the dynamics is defined by monomial
differential equations (Figure 3). Tropicalized systems remind of, but are not
equivalent to, Savageau’s S-systems [84] that have been used for modeling
metabolic networks. S-systems are smooth systems such that the production and
consumption terms of each species are multivariate monomials. Tropicalized
systems are S-systems locally, within the polyhedral domains defined by the
tropical varieties, and also along some parts of the tropical variety (that
carry sliding modes, see next section).
The tropicalization unravels an important property of multiscale systems, that
is to have different behavior on different timescales. We have seen that, on
every timescale, monomolecular networks with total separation behave like a
single reaction step. This is akin to considering only the dominant processes
in the network and implies that the tropicalization is a good approximation
for monomolecular networks with total separation. In the next section we
discuss other, more general situations, that include nonlinear networks, when
the tropicalization represents an useful approximation of the smooth dynamics.
## 5 Quasi-steady state and Quasi-equilibrium, revisited
Two simple methods for model reduction of nonlinear models with multiple
timescales: the quasi-equilibrium (QE) and the quasi-steady state (QSS)
approximations. As discussed in [43, 44], these two approximations are
physically and dynamically distinct. In order to understand these differences
let us refer to the simple example of the Michaelis-Menten mechanism,
$S+E\underset{k_{-1}}{\overset{k_{1}}{\rightleftharpoons}}ES\overset{k_{2}}{\rightarrow}P+E$
(6)
The QSS approximation, proposed for this system by Briggs and Haldane,
considers that the total concentration of enzyme, $[E]+[ES]$ is much lower
than the total concentration of substrate, therefore complex $ES$ is a low
concentration, fast species. Its concentration is driven by concentration of
$S$, hence, the simplified mechanism correspond to pooling the two reactions
of the mechanism into a unique irreversible reaction
$S\overset{R(S,E_{tot})}{\longrightarrow}P$, which means that
$\frac{d[P]}{dt}=-\frac{d[S]}{dt}=k_{2}[ES]_{QSS}$. The QSS value of the
complex concentration results from the equation
$k_{1}[S]([E]_{tot}-[ES]_{QSS})=(k_{-1}+k_{2})[ES]_{QSS}$. From this follows
that $R([S],[E]_{tot})=k_{2}[E]_{tot}[S]/(k_{m}+[S])$, where $[E]_{tot}$ is
the total enzyme concentration, and $k_{m}=(k_{-1}+k_{2})/k_{1}$.
The QE approximation considers that the first reaction of the mechanism is a
fast, reversible reaction. The simplified mechanism corresponds to a pooling
of species. Two pools $[S]_{tot}=[S]+[ES]$, and $[E]_{tot}=[E]+[ES]$ are
conserved by the fast reversible reaction, but only one, $[E]_{tot}$ is
conserved by the two reactions of the mechanism. The pool $[S]_{tot}$ is
slowly consumed by the second reaction and represents the slow variable of the
system. The single step approximation reads
$S_{tot}\overset{R([S]_{tot},[E]_{tot})}{\longrightarrow}P$, or equivalently
$\frac{d[P]}{dt}=-\frac{d[S]_{tot}}{dt}=k_{2}[ES]_{QE}$. The QE value of the
complex concentration is the unique positive solution of the quadratic
equation $k_{1}([S]_{tot}-[ES]_{QE})([E]_{tot}-[ES]_{QE})=k_{-1}[ES]_{QE}$.
From this it follows that
$R([S]_{tot},[E]_{tot})=2k_{2}[E]_{tot}[S]_{tot}([E]_{tot}+[S]_{tot}+k_{-1}/k_{1})^{-1}(1+\sqrt{1-4[E]_{tot}[S]_{tot}/([E]_{tot}+[S]_{tot}+k_{-1}/k_{1})^{2}})^{-1}$.
When the concentration of enzyme is small, $[E]_{tot}<<[S]_{tot}$, we obtain
the original equation of Michaelis and Menten, $R([S]_{tot},[E]_{tot})\approx
k_{2}\frac{[E]_{tot}[S]_{tot}}{k_{-1}/k_{1}+[S]_{tot}}$.
One of the main difficulties to applying QE or QSS reduction to computational
biology models is that QE reactions and QSS species should be specified a
priori. For some models, biological information can be used to rank reactions
according to their rates. For instance, one knows that metabolic processes and
post-transcriptional modifications are more rapid than gene expression.
However, this information is rather vague. In detailed gene expression models
some processes can be rapid, while others are much slower. Furthermore, the
relative order of these processes can be inverted from one functioning regime
to another, for instance the binding and unbinding rates of a repressor to
DNA, can be slow or fast depending on various conditions. Even if some
numerical approaches such as iterative IM, CSP and ILDM propose criteria for
detecting fast and slow processes, at present there is no general direct
method to identify QE reactions and QSS species.
Here we present two methods, based, the first one on singular perturbations,
and the second on tropical geometry ideas, allowing to detect QE reactions and
QSS species.
The first method uses simulation of the trajectories, therefore it can only be
applied to a fully parametrized model. However, in systems with separation,
the sets of QE reactions and QSS species are robust, ie remain the same for
broad ranges of the parameters. One can use imprecise parameters (resulting
for instance from crude estimates or fitting) to compute these sets. The
method starts by detecting slaved species. Given the trajectories $\mathbold
c(t)$ of all species, the imposed trajectory of the $i$-th species is a real,
positive solution $c_{i}^{*}(t)$ of the polynomial equation
$P_{i}(c_{1}(t),\ldots,c_{i-1}(t),c_{i}^{*}(t),c_{i+1}(t),\ldots,c_{n}(t))=0,$
(7)
where $P_{i}$ is the $i$-th component of the rhs of (2). We say that a species
$i$ is slaved if the distance between the trajectory $c_{i}(t)$ and some
imposed trajectory $c_{i}^{*}(t)$ is small for some time interval $I$,
$sup_{t\in I}|log(c_{i}(t))-log(c_{i}^{*}(t))|<\delta$, for some $\delta>0$
sufficiently small. The remaining species, that are not slaved, are called
slow species.
Slaved species are rapid and are constrained by the slow species. The minimum
number of variables that we expect for a reduced model is equal to the number
of slow species. The slow species can be obtained by direct comparison of the
imposed and actual trajectories. This method is illustrated for a model of
NF$\kappa$B canonical pathway in Figure 4.
There are two types of slaved species. Low concentration, slaved species
satisfy QSS conditions. Large concentration, slaved species are consumed and
produced by fast QE reactions and satisfy QE conditions. Because the reduction
schemes are different in the two situations, it is useful to have a method to
separate the two cases. Using the values of concentrations can work when
concentrations are well separated, but may fail for a continuum of values. A
better method is to identify which are the dominant terms in the Eq.(7). Using
again the example of Michaelis-Menten mechanism, the complex ES will be
detected as slaved in both QSS and QE conditions. Eq.(7) reads
$k_{1}[S][E]=(k_{-1}+k_{2})[ES]$. For QE condition, the term $k_{2}$ will be
dominated by $k_{-1}$. We call pruned version of Eq.(7) the equation obtained
after removing all the dominated monomials, in this case the equation
$k_{1}[S][E]-k_{-1}[ES]=0$. When the pruned version is a combination of
reversible reaction rates set to zero, then the slaved species satisfy QE
conditions. Again, the comparison of monomials is possible for a fully
parametrized model, however we expect this comparison to be robust for models
with separation.
The second method to identify QE and QSS conditions from the calculation of
the tropicalization (5). This can be done formally and do not require
simulation of trajectories and numerical knowledge of the parameters. Indeed,
is was shown in [95] that there is a relation between sliding modes of the
tropicalized system (5) and the QSS or QE conditions. Sliding modes are well
known for ordinary differential equations with discontinuous vector fields
[27]. In such systems, the dynamics can follow discontinuity hypersurfaces
where the vector field is not defined. When the discontinuity hypersurfaces
are smooth and $n-1$ dimensional ($n$ is the dimension of the vector field)
then the conditions for sliding modes read:
$<n_{+}(x),f_{+}(x)><0,\quad<n_{-}(x),f_{-}(x)><0,\quad x\in\Sigma,$ (8)
where $f_{+},f_{-}$ are the vector fields on the two sides of $\Sigma$ and
$n_{+}=-n_{-}$ are the interior normals.
In [95] we have shown the following. If the smooth dynamics obeys QE or QSS
conditions and if the pruned polynomial $\tilde{P}$ defining the fast dynamics
is a 2-nomial, $\tilde{P}_{i}(\mathbold c)=a_{1}\mathbold
c^{\alpha_{1}}+a_{2}\mathbold c^{\alpha_{2}}$, then the QE or QSS equations
define a hyperplane of the tropical variety of $\tilde{P}$, namely
$S=\\{<log(\mathbold c),\alpha_{1}-\alpha_{2}>=log(|a_{1}|/|a_{2}|)\\}$. The
stability of the QE of QSS manifold implies the existence of a sliding mode of
the tropicalization (5) along this hyperplane. This result suggests that
checking the sliding mode condition (8) on the tropical manifold, provides a
method of detecting QE reactions and QSS species.
To illustrate this method, let us use again the Michaelis-Menten example. In
this case, two conservation laws allow elimination of two variables $E$ and
$P$ and the dynamics can be described by two ODEs:
$\displaystyle\frac{d[S]}{dt}$ $\displaystyle=$ $\displaystyle-
k_{1}E_{tot}[S]+k_{1}[S][ES]+k_{-1}[ES]$ $\displaystyle\frac{d[ES]}{dt}$
$\displaystyle=$ $\displaystyle
k_{1}E_{tot}[S]-k_{1}[S][ES]-(k_{-1}+k_{2})[ES]$ (9)
The tropical manifolds of the two species $S$ and $ES$ are tripods with
parallel arms like in Figure 3. Indeed, the slopes of the arms of tropical
manifold are only given by the powers of different variables of the monomials,
and these are the same for the two species. Investigation of the flow field
close to the tripod arms identifies sliding modes on an unbounded subset $AOB$
of the tropical manifold of the species $ES$. This subset is a global
attractor of the tropicalized dynamics and represents a tropicalized version
of the invariant manifold of the smooth system. If the initial data is not in
this set, the tropicalized trajectory converges quickly to it and continues on
it as a sliding mode. When $k_{2}>>k_{-1}$, $ES$ satisfies QSS conditions
leading to the Michaelis-Menten equation. The arm $AO$ of the tropical
manifold of the species $ES$ carry a sliding mode, has the equation
$k_{1}E_{tot}[S]=(k_{-1}+k_{2})[ES]>>k_{1}[S][ES]$, and corresponds to the
linear regime of the Michaelis-Menten equation. Similarly, the arm $OB$ of the
tropical manifold of $ES$ has the equation
$k_{1}E_{tot}[S]=k_{1}[S][ES]>>(k_{-1}+k_{2})[ES]$ and corresponds to the
saturated regime of the Michaelis-Menten equation. When $k_{2}<<k_{-1}$, the
tropical manifolds of the two species $S$ and $ES$ practically coincide. Both
species are rapid and satisfy QE conditions, namely
$k_{1}E_{tot}[S]=k_{-1}[ES]>>k_{1}[S][ES]$ on the arm $AO$ and
$k_{1}E_{tot}[S]=k_{1}[S][ES]>>k_{-1}[ES]$ on the arm $OB$.
The tropicalization can thus be used to obtain global reductions of models.
Even when global reductions are not possible (sliding modes leave the tropical
manifold or simply do not exist), the tropicalization can be used to hybridize
smooth models, ie transform them into piecewise simpler models (modes) that
change from one time interval to another. These changes occur when the
piecewise smooth trajectory of the system meets a hyperplane of the tropical
manifold and continues as a sliding mode along this hyperplane or leaves
immediately the hyperplane. Hybridization is a particularly interesting
approach to modeling cell cycle. Indeed, progression of the cell cycle is a
succession of several different regimes (phases). This strategy is illustrated
in Figure 4 for a simple cell cycle model.
## 6 Graph rewriting for large nonlinear, deterministic, dynamical networks
We have seen that model reduction of monomolecular networks with total
separation is based on graph rewriting operations.
Similarly, QSS and QE approximations can be used to produce simpler networks
from large nonlinear networks. The classical implementation of these
approximations leads to differential-algebraic equations. It is however
possible to reformulate the simplified model as a new, simpler, reaction
network. We showed in the previous section how to do this for the Michaelis
Menten mechanism under different conditions. In general one has to solve the
algebraic equations corresponding to QE or QSS conditions, eliminate (prune)
QSS species and QE reactions, pool reactions (for QSS approximation) or
species (for QE approximation), and finally calculate the kinetic laws of the
new reactions.
By reaction pooling we understand here replacing a set of reactions by a
single reaction whose stoichiometry vector $\nu$ is the sum of the
stoichiometry vectors $\nu_{i}$ of the reactions in the pool,
$\nu=\sum_{i}\gamma_{i}\nu_{i}$. If the reactions are reversible then the
coefficients $\gamma_{i}$ can be arbitrary integers, otherwise they must be
positive integers. Reaction pools conserve certain species that where
previously consumed or produced by individual reactions in the pools. These
species were called intermediates in [76]. The species that are either
produced or consumed by the pools were called terminal in [76]. For example,
an irreversible chain of reactions $A_{1}\to A_{2}\to A_{3}$ can be pooled
onto a single reaction $A_{1}\to A_{3}$, which in terms of stoichiometry
vectors reads $\begin{bmatrix}-1\\\ 0\\\ 1\end{bmatrix}=\begin{bmatrix}-1\\\
1\\\ 0\end{bmatrix}+\begin{bmatrix}0\\\ -1\\\ 1\end{bmatrix}$. In this example
$A_{1}$, $A_{3}$ are terminal species and $A_{2}$ is an intermediate species.
Reaction pooling is used with QSS conditions, in which case the intermediates
are the QSS species.
By species pooling we understand replacing a set of species concentrations
$\\{c_{i}\\}$ by a linear combination with positive coefficients of species
concentrations, $\sum_{i}b_{i}c_{i}$. Species pooling is used with QE
conditions.
In general, the reaction and species pools result from linear algebra. Indeed,
let us consider the matrix $S^{f}$ that defines the stoichiometry of the rapid
subsystem. For the QSS approximation, the matrix $S^{f}$ has a number of lines
equal to the number of QSS species. The columns of this matrix are the
stoichiometries of the reactions in the model, restricted to the QSS species.
We exclude zero valued columns, i.e. reactions that do not act on QSS species.
For the QE approximation, the number of columns of the matrix $S^{f}$ is equal
to the number of QE reactions, and the lines of $S^{f}$ are the
stoichiometries of QE reactions. We exclude zero valued lines corresponding to
species that are not affected by QE reactions.
In QE conditions, species pools are defined by vectors in the left kernel of
$S^{f}$,
$b^{T}S^{f}=0$ (10)
The vectors $b$, that are conservation laws of the fast subsystem, define
linear combinations of species concentrations that are the new slow variables
of the system. Of course, one could eliminate from these combinations, the
conservation laws of the full reaction network, that will be constant (see
Appendix).
In QSS conditions, reaction pools (also called routes) are defined by vectors
in the right kernel of $S^{f}$,
$S^{f}\gamma=0$ (11)
According to the definition (11), a reaction pool does not consume or produce
QSS species (these are intermediates). One can impose, like in [76], a
minimality condition for choosing the reaction pools. A reaction pool is
minimal if there is no other reaction pool with less nonzero stoichiometry
coefficients. This is equivalent to choosing reaction pools as elementary
modes of the fast subsystem.
After pooling, QE and QSS algebraic conditions must be solved and the rates of
the new reactions calculated. The new rates should be chosen such that the
remaining species and pools of species satisfy the simplified ODEs. The choice
of the rates is not always unique (some uniqueness conditions are discussed in
[76], see also the Appendix). In order to compute the new rates, one has to
solve QE and QSS equations. For network with polynomial or rational rates,
this implies solving large systems of polynomial equations. The complexity of
this task is double exponential on the size of the system [70], therefore one
needs approximate solutions. Approximate solutions of polynomial equations can
be formally derived when the monomials of these equations are well separated.
Some simple recipes were given in [76] and could be improved by the methods of
tropical geometry.
These ideas were used in [76] to reduce several models of NF-$\kappa$B
signalling (Figure 6).
The NF-$\kappa$B activation pathway is complex at many levels. NF-$\kappa$B is
sequestered in the cytoplasm by inactivating proteins named I$\kappa$B. There
are five known members of the NF-$\kappa$B family in mammals, Rel (c-rel),
RelA (p65), RelB, NF-$\kappa$B1 (p50 and its precursor p105) and NF-$\kappa$B2
(p52 and its precursor p100). This generates a large combinatorial complexity
of dimers, affinities and transcriptional capabilities. I$\kappa$B family
comprises seven members in mammals (I$\kappa$B$\alpha$, I$\kappa$B$\beta$,
I$\kappa$B$\epsilon$, I$\kappa$B$\gamma$, Bcl-3). All these inhibitors display
different affinities for NF-$\kappa$B dimers, multiplying the combinatorial
complexity. The activation of NF-$\kappa$B upon signalling, occurs by
phosphorylation by a kinase complex, then ubiquitination, and finally
degradation of I$\kappa$B molecules. The activation signal is transmitted by
several possible pathways most of them activating the kinase IKK that modifies
I$\kappa$B. In the canonical pathway, one important determinant of IKK
dynamics is the protein A20 that inhibits IKK activation. A20 expression is
controlled by NF-$\kappa$B. In order to cope with this complexity a model
containing 39 species, 65 reactions and 90 parameters was proposed in [76]. Of
course, not all reactions and parameters of this complex model are important.
In order to determine, in a rational and systematic way, which of the model
features are critical, we have used model reduction.
Graph rewriting was performed in a modular way, by applying the pruning and
pooling rules to tightly connected submodels of the NF-$\kappa$B network. The
computation of the reaction pools was performed using Matlab and METATOOL
[98]. Using submodel decomposition reduces the complexity of computing
elementary modes and of solving large systems of algebraic equations needed
for recalculating the reaction rates.
To give an example of modular reduction, let us consider the set of reactions
involving six cytoplasmic located intermediates (IKK$|$active, IKK$|$inactive,
IKK, IKK$|$active:IkBa, IKK$|$active:IkBa:p50:p65, p50:p65@csl) and four
terminal species (A20, IkBa@csl, IkBa:p50:p65@csl, p50:p65@ncl). As can be
seen from Figure 5, the six intermediate species are slaved. The reactions of
this submodel form the cytoplasmic part of the signalling mechanism, including
11 kinase transformation reactions, a complex release reaction, a complex
formation reaction, and the NF-$\kappa$B translocation reaction. The
elementary modes of the submodel (computed using METATOOL [98]) are used to
define the reactions pools. For this submodel, we find two elementary modes,
that can be described as the modulated inhibitor degradation (IkBa@csl
$\rightarrow$ $\varnothing$), and a reaction summarizing the NF-$\kappa$B
release and translocation (IkBa:p50:p65@csl $\rightarrow$ p50:p65@ncl),
respectively. In order to compute the reaction rates of the two elementary
modes as functions of the concentrations of the terminal species, we find
approximate solutions of the QSS equations for the intermediate species and
equate, for the variation rates of each terminal species, the contributions of
elementary modes to the total known variation rate in the unreduced model (see
Appendix). The two rates are
$k_{21p1}[IkBa@csl][IkBa:p50:p65@csl]/((k_{21p2}+[IkBa@csl])(k_{21p3}+[A20]))$
for the modulated inhibitor degradation, and
$k_{15p1}[IkBa:p50:p65@csl]/((k_{15p2}+[IkBa@csl])(k_{15p3}+[A20]))$ for the
release and translocation reaction.
## 7 Model reduction and model identification
Computational biology models contain mechanistic details that can not all be
identified from available experimental data. Determining the parameters of
such complex models could lead to overfitting, describing noise, rather than
features of data, or can be simply meaningless, when model behavior is not
sensitive to the parameters. Furthermore, many model identification methods
[35] suffer from the ”curse of dimensionality” as it becomes increasingly
difficult to cover the parameter space when the number of parameters
increases. A rather efficient strategy to bypass these problems is to use
model reduction. This method is known in machine learning as backward pruning
or post-pruning [99]. It consists in finding a complex model that fits data
well and then prune it back to a simpler one that also fits the data well. Far
from being redundant, backward pruning can be successfully used in
computational biology. Rather often, one starts with a complex model coping
with mechanistic details of the network regulation. Then, over-fitting and
problems of identifiability of the parameters are avoided by model reduction.
By model reduction the mechanistic model is mapped onto a simpler,
phenomenological model. For instance, gene transcription and translation can
be represented as one step and one constant in a phenomenological model, but
can consist of several steps such as initiation, transcription of mRNA leading
region, ribosome binding, translation, folding, maturation, etc. in a complex
model. Not all of these steps are important for the network functioning and
not all parameters are identifiable from the observed quantities. Following
reduction, the inessential steps are pruned and several critical parameters
are compacted into a few effective parameters that are identifiable.
As discussed in [78, 76, 77, 26], model reduction unravels the important
features and the critical parameters of the model.
Using model reduction for determining critical features of the model has many
advantages relative to numerical sensitivity studies [75, 46, 53]: this
approach is less time consuming, brings more insight, and is based on
qualitative comparison of the order of the parameters and therefore does not
need exhaustive scans of parameter values. In the applications described in
[78, 76, 77, 26], the critical parameters of the pruned model are combinations
(most often monomials) of the parameters of the complex models. As only the
critical combinations can be fitted from data it is important to have
estimates of some individual parameters, allowing to determine the remaining
ones.
This methodology has been first proposed in [76]. The model reduction of the
NF-$\kappa$B model in [76] leads to new, effective parameters that are
monomials of the parameters of the complex model. The correspondence between
the initial parameters and the effective parameters is shown in Figure 7.
Although not fully exploited in the theoretical study [76], this mapping can
be used for model identification from experimental data. Parameters of the
reduced model have increased observability and could be obtained from
experimental data. The values of the effective parameters can be used to
constrain the parameters of the full model. Some of the parameters of the full
model, that are not critical or contribute to effective parameters together
with other parameters remain arbitrary and could be fixed to generic values.
## 8 Conclusion
The mathematical techniques described in this paper define strategies for the
study of large dynamical network models in computational biology. Large
networks are needed in order to understand context dependence, specialization,
and individuality of the cell behavior. Extensive pathway database
accumulation supports somehow the idea that biological cell is a puzzle of
networks and pathways, and that once these are put together in a tightly
bound, coherent map, the cell physiology should be unraveled by a computer
simulation. Actually, confronting biochemical networks with real life is not
an easy challenge. Model reduction techniques are needed to bring us one step
closer to this objective, as these methods can reveal critical features of
complex organizations.
We have proposed that the ideas of limitation and dominance are fundamental
for understanding computational biology dynamical models. The essential,
critical features of systems with many separated time scales, can be resumed
by a dominant, reduced, subsystem. This dominant subsystem depends on the
order relations between model parameters or combinations of model parameters.
We have shown how to calculate such a dominant subsystem for linear and
nonlinear networks. Geometrical interpretation of these concepts in terms of
tropicalization provides a powerful framework, allowing to identify invariant
manifolds, quasi-steady state species and quasi-equilibrium reactions. We have
also discussed how model reduction can be applied to backward pruning
parameter learning strategies.
Future efforts are needed to extend these mathematical ideas and model
reduction algorithms and implement them into computational biology tools.
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## Appendix : algorithms
### Algorithm 1 : reduction of monomolecular networks with separation
This algorithm consists of three procedures.
I. Constructing of an auxiliary reaction network: pruning.
For each $A_{i}$ branching node (substrate of several reactions) let us define
$\kappa_{i}$ as the maximal kinetic constant for reactions $A_{i}\to A_{j}$:
$\kappa_{i}=\max_{j}\\{k_{ji}\\}$. For correspondent $j$ we use the notation
$\phi(i)$: $\phi(i)={\rm arg\,max}_{j}\\{k_{ji}\\}$.
An auxiliary reaction network $\mathcal{V}$ is the set of reactions obtained
by keeping only $A_{i}\to A_{\phi(i)}$ with kinetic constants $\kappa_{i}$ and
discarding the other, slower reactions. Auxiliary networks have no branching,
but they can have cycles and confluences. The correspondent kinetic equation
is
$\dot{c}_{i}=-\kappa_{i}c_{i}+\sum_{\phi(j)=i}\kappa_{j}c_{j},$ (12)
If the auxiliary network contains no cycles, the algorithm stops here.
II gluing cycles and restoring cycle exit reactions
In general, the auxiliary network $\mathcal{V}$ has several cycles
$C_{1},C_{2},...$ with lengths $\tau_{1},\tau_{2},...>1$.
These cycles will be “glued” into points and all nodes in the cycle $C_{i}$,
will be replaced by a single vertex $A^{i}$. Also, some of the reactions that
were pruned in the first part of the algorithm are restored with renormalized
rate constants. Indeed, reaction exiting a cycle are needed to render the
correct dynamics: without them, the total mass accumulates in the cycle, with
them the mass can also slowly leave the cycle. Reactions $A\to B$ exiting from
cycles ($A\in C_{i}$, $B\notin C_{i}$) are changed into $A^{i}\to B$ with the
rate constant renormalization: let the cycle $C^{i}$ be the following sequence
of reactions $A_{1}\to A_{2}\to...A_{\tau_{i}}\to A_{1}$, and the reaction
rate constant for $A_{j}\to A_{j+1}$ is $k_{j}$ ($k_{\tau_{i}}$ for
$A_{\tau_{i}}\to A_{1}$). The quasi-stationary normalized distribution in the
cycle is:
$c_{j}^{\circ}=\frac{1}{k_{j}}\left(\sum_{j=1}^{\tau_{i}}\frac{1}{k_{j}}\right)^{-1}\,,\;j=1,\ldots,\tau_{i}\,.$
The reaction $A_{j}\to B$ ($A\in C_{i}$, $B\notin C_{i}$) with the rate
constant $k$ is changed into $A^{i}\to B$ with the rate constant
$c_{j}^{\circ}k$.
Let the cycle $C_{i}$ have the limiting steps that is much slower than other
reactions. For the limiting reaction of the cycle $C_{i}$ we use notation
$k_{\lim\,i}$. In this case, $c_{j}^{\circ}=k_{\lim\,i}/k_{j}$. If $A=A_{j}$
and $k$ is the rate constant for $A\to B$, then the new reaction $A^{i}\to B$
has the rate constant $kk_{\lim\,i}/k_{j}$. This rate is obtained using quasi-
stationary distribution for the cycle.
The new auxiliary network $\mathcal{V}^{1}$ is computed for the network with
glued cycles. Then we prune it, extract cycles, glue them, iterate until a
acyclic network is obtained $\mathcal{V}^{m}$, where $m$ is the number of
iterations.
III Restoring cycles
The previous procedure gives us the sequence of networks
$\mathcal{V}^{1},\ldots,\mathcal{V}^{m}$ with the set of vertices
$\mathcal{A}^{1},\ldots,\mathcal{A}^{m}$ and reaction rate constants defined
for each $\mathcal{V}^{i}$ in the processes of pruning and gluing.
The dynamics of species inside glued cycles is lost after their gluing. A full
multi-scale approximation (including relaxation inside cycles) can be obtained
by restoration of cycles. This is done starting from the acyclic auxiliary
network $\mathcal{V}^{m}$ back to $\mathcal{V}^{1}$ through the hierarchy of
cycles. Each cycle is restored according to the following procedure:
* •
We start the reverse process from the glued network $\mathcal{V}^{m}$ on
$\mathcal{A}^{m}$. On a step back, from the set $\mathcal{A}^{m}$ to
$\mathcal{A}^{m-1}$ and so on, some of glued cycles should be restored and
cut. On the $q$th step we build an acyclic reaction network on the set of
vertices $\mathcal{A}^{m-q}$, the final network is defined on the initial
vertex set and approximates relaxation of the initial networks.
* •
To make one step back from $\mathcal{V}^{m}$ let us select the vertices of
$\mathcal{A}^{m}$ that are glued cycles from $\mathcal{V}^{m-1}$. Let these
vertices be $A^{m}_{1},A^{m}_{2},...$. Each $A^{m}_{i}$ corresponds to a glued
cycle from $\mathcal{V}^{m-1}$, $A^{m-1}_{i1}\to
A^{m-1}_{i2}\to...A^{m-1}_{i\tau_{i}}\to A^{m-1}_{i1}$, of the length
$\tau_{i}$. We assume that the limiting steps in these cycles are
$A^{m-1}_{i\tau_{i}}\to A^{m-1}_{i1}$. Let us substitute each vertex
$A^{m}_{i}$ in $\mathcal{V}^{m}$ by $\tau_{i}$ vertices
$A^{m-1}_{i1},A^{m-1}_{i2},...A^{m-1}_{i\tau_{i}}$ and add to
$\mathcal{V}^{m}$ reactions $A^{m-1}_{i1}\to
A^{m-1}_{i2}\to...A^{m-1}_{i\tau_{i}}$ (that are the cycle reactions without
the limiting step) with corresponding constants from $\mathcal{V}^{m-1}$.
* •
If there exists an outgoing reaction $A^{m}_{i}\to B$ in $\mathcal{V}^{m}$
then we substitute it by the reaction $A^{m-1}_{i\tau_{i}}\to B$ with the same
constant, i.e. outgoing reactions $A^{m}_{i}\to...$ are reattached to the
heads of the limiting steps. Let us rearrange reactions from $\mathcal{V}^{m}$
of the form $B\to A^{m}_{i}$. These reactions have prototypes in
$\mathcal{V}^{m-1}$ (before the last gluing). We simply restore these
reactions. If there exists a reaction $A^{m}_{i}\to A^{m}_{j}$ then we find
the prototype in $\mathcal{V}^{m-1}$, $A\to B$, and substitute the reaction by
$A^{m-1}_{i\tau_{i}}\to B$ with the same constant, as for $A^{m}_{i}\to
A^{m}_{j}$.
* •
After the previous step is performed, the vertices set is $\mathcal{A}^{m-1}$,
but the reaction set differs from the reactions of the network
$\mathcal{V}^{m-1}$: the limiting steps of cycles are excluded and the
outgoing reactions of glued cycles are included (reattached to the heads of
the limiting steps). To make the next step, we select vertices of
$\mathcal{A}^{m-1}$ that are glued cycles from $\mathcal{V}^{m-2}$, substitute
these vertices by vertices of cycles, delete the limiting steps, attach
outgoing reactions to the heads of the limiting steps, and for incoming
reactions restore their prototypes from $\mathcal{V}^{m-2}$, and so on.
After all, we restore all the glued cycles, and construct an acyclic reaction
network on the set $\mathcal{A}$. This acyclic network approximates relaxation
of the initial network. We call this system the dominant system.
Note that the reduction algorithm does not need precise values of the
constants. It is enough to have an initial ordering of the constants. Then,
the auxiliary network is obtained only from this ordering. However, after a
first iteration, and if the initial network contains cycles, some of the exit
constant are renormalized and the new rate constants become monomials of the
old ones. In order to prune again, we need to compare these monomials.
Monomials of well separated constants are generically well separated [42].
However, a freedom remains on ordering these new monomials, leading to several
possible reduced acyclic digraphs, given an initial digraph with ordering of
the constants (Figure 1 of the main text).
### Algorithm 2 : reduction of nonlinear networks with separation
This algorithm consists of the following procedures.
I. Identification of QSS species and QE reactions.
There are two methods of identification, trajectory based, and tropicalization
based. Presently we are using the trajectory based method.
Detect slaved species.
After generating trajectories $c(t)$ for $t\in I$, for each species compute
the distances $\delta_{i}=sup_{t\in I}|log(c_{i}(t))-log(c_{i}^{*}(t))|$. Use
k-means clustering to separate species into two groups, slaved (small values
of $\delta$) and slow (large values of $\delta$) species.
Prune.
For each $P_{i}$ (polynomial rate) corresponding to slaved species, compute
the pruned version $\tilde{P}_{i}$ by eliminating all monomials that are
dominated by other monomials of $P_{i}$.
Identify QE reactions and QSS species.
Identify, in the structure of $\tilde{P}_{i}$ the forward and reverse rates of
QE reactions. This step can be performed by recipes presented in [91]. The
slaved species not involved in QE reactions are QSS.
II. Exploiting QSS conditions, pruning intermediate species, pooling reactions
Define subsets and matrices
Given the set of QSS (intermediate) species $I$, one defines the set
${\mathcal{R}}_{I}$ of reactions acting on them. The terminal species $T$, are
the the other species, different from $I$, on which act the reactions from
${\mathcal{R}}_{I}$. Define two stoichiometric matrices $S^{f}$ and $S^{T}$.
$S^{f}$ defines the fast subsystem and has a number of lines equal to the
number of QSS species, and a number of columns equal to the number of
reactions ${\mathcal{R}}_{I}$. $S^{T}$ contains the stoichiometries of the
terminal species for the same reactions ${\mathcal{R}}_{I}$. Species $I$ will
be pruned, and reactions ${\mathcal{R}}_{I}$ will be pooled.
Compute elementary modes (EMs)
Compute elementary modes of nonzero terminal stoichiometry as minimal
solutions of $S^{f}\gamma=0$, $S^{T}\gamma\neq 0$, the minimality being
defined with respect to the number of nonzero coefficients. $S^{T}\gamma\neq
0$ on the output of elementary modes packages such as METATOOL. If the
terminal stoichiometries of the EMs are dependent, restrict to a subset of
independent terminal stoichiometries.
Solve QSS equations
Find approximate formal solutions for systems of QSS algebraic equations. This
step is not yet automatic. It will be automatized in subsequent work by using
tropical geometry methods.
Find rates of EMs
To each elementary mode $\gamma_{i}$, associate a kinetic law giving the rate
of the EM as a fonction of the terminal species concentrations
$R_{i}^{*}(c_{T})$. Let $R(c_{T})$ be the vector of rates of terminal species
(the dependence on $c_{T}$ is direct, or indirect, via $c_{I}$ that can be now
expressed as function of $c_{T}$) of reactions in ${\mathcal{R}}_{I}$. Then
the EM rates $R_{i}^{*}(c_{T})$ must satisfy $S^{T}R(c_{T})=\sum
R_{i}^{*}(c_{T})S^{T}\gamma_{i}$. This equation has an unique solution if the
vectors $S^{T}\gamma_{i}$ are independent (this justifies the independence
condition for the terminal stoichiometries of EMs).
III. Exploiting QE conditions, pruning QE reactions, pooling species
Define subsets and matrices
Given the set of QE reactions $Q$, one defines the set $S$ of species that are
affected by them. The species $S$ are also affected by other reactions that we
call terminal, $Q_{T}$. Define two stoichiometric matrices $S^{f}$ and
$S^{T}$. $S^{f}$ defines the fast subsystem and has a number of lines equal to
the cardinal of $E$, and a number of columns equal to the cardinal of $Q$.
$S^{T}$ contains the stoichiometries of the reactions reactions $Q_{T}$ for
the same species $S$ (it has the same number of lines as $S^{f}$). Reactions
$Q$ will be pruned and species $E$ will be pooled.
Compute species pools
Species pools are computed as minimal solutions of $bS^{f}=0$, $bS^{T}\neq 0$
(the second condition stands for looking for conservation laws of the fast
subsystem that are not conserved by the entire network; the minimality
condition means that we compute elementary modes of the transpose matrix
$S^{f}$).
Solve QE equations
Same methods as for QSS conditions. Solve the QSS equations together with the
conservation of pools and express the concentrations of the species $E$ as
functions of the pools $c^{*}_{i}=b_{i}c$.
Find new rates
Re-express (by substitution) the rate of each reaction from $Q_{T}$ in terms
of pools $c^{*}_{i}=b_{i}c$.
Figure 1: A monomolecular network with total separation can be represented as
a digraph with integer labels (the quickest reaction has label 1). Two simple
rules allow to eliminate competition between reactions (rule a) and transform
cycles into chains (rule b). Rule b can not be applied to cycles with outgoing
slow reactions, in which case more complex, hierarchical rules should be
applied (rule c). In the rule c, first the cycle $A_{2}\to A_{3}\to A_{4}\to
A_{2}$ is “glued” to a new node (pool $A_{2}+A_{3}+A_{4}$) and the constant of
the slow outgoing reaction renormalized to a monomial $k_{5}k_{5}/k_{3}$. Rule
b is applied to the resulting network, which is a cycle with no outgoing
reactions. The comparison of the constants $k_{5}k_{5}/k_{3}$ and $k_{6}$
dictates where this cycle is cut. Finally, the glued cycle is restored, with
its slowest step removed. Figure 2: For a given timescale, monomolecular
networks with total separation behave as a single step: the concentrations of
some species (white) are practically constant, some species (yellow) are rapid
, low concentration, intermediates, one species (red) is gradually consumed
and another (pink) is gradually produced. We have represented the sequence of
one step approximations of a reduced, acyclic, deterministic digraph, from the
quickest time-scale $t_{1}=\lambda_{1}^{-1}$ to the slowest one
$t_{4}=\lambda_{4}^{-1}$. These one step approximations are activated when
mass is introduced at $t=0$ via the “boundary nodes” $A_{1}$ and $A_{6}$.
a) b)
Figure 3: a) The tropical manifold of the polynomial $ax+by+cxy$ on
“logarithmic paper” is a three lines tripod. b) The tropical manifolds for the
species ES (in red) and S (in blue) for the Michaelis-Menten mechanism. The
tropicalized flow is also represented on both sides of the tropical manifolds
(with arrows, red on one side, blue on the other side). Sliding modes
correspond to blue and red arrows pointing in opposite directions. Figure 4:
Model reduction and tropicalization of a 5 variables cell cycle model defined
by the differential equations
$y_{1}^{\prime}=k_{9}y_{2}-k_{8}y_{1}+k_{6}y_{3}$,
$y_{2}^{\prime}=k_{8}y_{1}-k_{9}y_{2}-k_{3}y_{2}y_{5}$,
$y_{3}^{\prime}=k_{4}^{\prime}y_{4}+k_{4}y_{4}y_{3}^{2}/C^{2}-k_{6}y_{3}$,
$y_{4}^{\prime}=-k_{4}^{\prime}y_{4}-k_{4}y_{4}y_{3}^{2}/C^{2}+k_{3}y_{2}y_{5}$,
$y_{5}^{\prime}=k_{1}-k_{3}y_{2}y_{5}$, proposed in [94]. (A) Comparison of
trajectories and imposed trajectories show that variables $y_{1}$, $y_{2}$,
$y_{5}$ are always slaved, meaning that the trajectories are close to the 2
dimensional hyperplane defined by the QE condition $k_{8}y_{1}=k_{9}y_{2}$,
the QSS condition $k_{1}=k_{3}y_{2}y_{5}$ and the conservation law
$y_{1}+y_{2}+y_{3}+y_{4}=C$. The variables $y_{3}$, $y_{4}$ are slaved and the
corresponding species are quasi-stationary on intervals. This means that the
dimensionality of the dynamics is further reduced to 1, on intervals. (B)
Tropicalization on logarithmic paper, in the plane of the variables $y_{3}$,
$y_{4}$. The tropical manifold consists of two tripods, represented in blue
and red, which divide the logarithmic paper into 6 polygonal sectors. Monomial
vector fields defining the tropicalized dynamics change from one polygonal
domain to another. The tropicalized (approximated) and the smooth (not
reduced) limit cycle dynamics stay within bounded distance one from another.
This distance is relatively small on intervals where the variables $y_{3}$ or
$y_{4}$ are quasi-stationary, which correspond to sliding modes of the
tropicalization. Figure 5: The ratio of the imposed and actual trajectories
has been calculated as a function of time for each species of the model of
$NF\kappa B$ canonical pathway (proposed in [61], model
${\mathcal{M}}(14,25,28)$ from [76]). If this ratio is close to one fold, the
species is slaved, otherwise the species is slow. Among the slaved species,
some have low concentrations and satisfy quasi-steady-state conditions,
whereas other have large concentrations and satisfy quasi-equilibrium
conditions. Figure 6: Model of NF-$\kappa$B signaling, proposing separate
production of the subunits p50, p65, the full combinatorics of their
interactions as well as with the inhibitor I$\kappa$B, the positive self-
regulation of p50, and in addition an A20 molecule whose production is
enhanced upon NF-$\kappa$B stimulation, and which negatively regulates the
activity of the stimulus responding kinase IKK [76]. This model, denoted
${\mathcal{M}}(39,65,90)$ contains 39 species, 65 reactions and 90 parameters.
We have reduced it to various levels of complexity. Among the reduced model we
obtained one, ${\mathcal{M}}(14,25,33)$ that has the same stoichiometry as a
model published elsewhere by another author [61] and denoted
${\mathcal{M}}(14,25,28)$. Incidently, this is also the simplest model in the
hierarchy related to ${\mathcal{M}}(39,65,90)$. The rate functions in the
reduced model are different, explaining the difference in number of
parameters. Comparison of the rate functions and of the trajectories of the
models ${\mathcal{M}}(14,25,33)$ and ${\mathcal{M}}(14,25,28)$ provided
insight into the consequences of various mechanistic modeling choices. Figure
7: The model ${\mathcal{M}}(14,25,28)$ from from [76] (first proposed in
[61]) was used to generate a hierarchy of simpler models, the simplest one
being ${\mathcal{M}}(5,8,15)$. We show the mapping between the parameters of
the models M(14, 25, 28) and M(5, 8, 15). Parameters of the first model are
gathered into monomials that are parameters of the reduced model. The integers
on the arrows connecting parameters represent the corresponding powers of the
parameters in the monomial. The innermost circle represents a dynamical
property of the model that is influenced positively, negatively, or negligibly
by the effective parameters (parameters of the reduced model). From [79].
|
arxiv-papers
| 2012-05-13T10:09:39 |
2024-09-04T02:49:30.870341
|
{
"license": "Public Domain",
"authors": "Ovidiu Radulescu, Alexander N. Gorban, Andrei Zinovyev, Vincent Noel",
"submitter": "Ovidiu Radulescu",
"url": "https://arxiv.org/abs/1205.2851"
}
|
1205.2854
|
# A new family of $q$-analogue of Genocchi numbers and polynomials of higher
order
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr and Jong Jin Seo
Department of Applied Mathematics, Pukyong National University, Busan 608-737,
Republic of KOREA seo2011@pknu.ac.kr
###### Abstract.
The new $q$-Euler polynomials was introduced by T. Kim in “$q$-Generalized
Euler numbers and polynomials, Russian Journal of Mathematical Physics, Vol.
13, No. 3, 2006, pp. 293-308” by means of the following generating function:
$\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}E_{j,q}\left(x\right)=\frac{\left[2\right]_{q}}{e_{q}\left(z\right)+1}e_{q}\left(xz\right)\text{.}$
In this work, we consider the generating function of Kim’s $q$-Euler
polynomials and introduce new generalization of $q$-Genocchi polynomials and
numbers of higher order. Also, we give surprising identities for studying in
Analytic Numbers Theory and especially in Mathematical Physics. Moreover, by
applying $q$-Mellin transformation to generating function of $q$-Genocchi
polynomials of higher order and so we define $q$-Hurwitz-Zeta type function
which interpolates of this polynomials at negative integers.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Genocchi numbers and polynomials, $q$-Genocchi numbers
and polynomials of higher order, $q$-Mellin transformation, $q$-Hurwitz-Zeta
function, $q$-Gamma function, $q$-Exponential function.
.$\sim$25th International Conference of the Jangjeon Mathematical
Society$\sim$
## 1\. Introduction
Throughout this work, we assume that $q\in\mathbb{C}$ with $\left|q\right|<1$.
The $q$-integer of $x$ is defined by $\left[x\right]_{q}=\frac{1-q^{x}}{1-q}$
and note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$. The $q$-derivative
is defined by F. H. Jackson as follows:
(1)
$D_{q}f\left(x\right)=\frac{d}{d_{q}x}f\left(x\right)=\frac{f\left(x\right)-f\left(qx\right)}{\left(1-q\right)x}\text{.}$
Taking $f\left(x\right)=x^{n}$ in (1), it becomes as follows:
$D_{q}x^{n}=\frac{x^{n}-\left(qx\right)^{n}}{\left(1-q\right)x}=\left[n\right]_{q}x^{n-1}\text{
and }\left(\frac{d}{d_{q}x}\right)^{n}f\left(x\right)=\left[n\right]_{q}!$
where
$\left[n\right]_{q}!=\left[n\right]_{q}\left[n-1\right]_{q}\cdots\left[1\right]_{q}$.
Now, we give definitions of two kinds of $q$-exponential functions as follows:
For any $z\in\mathbb{C}$ with $\left|z\right|<1$,
(2)
$e_{q}\left(z\right)=\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}\text{
and
}E_{q}\left(z\right)=\sum_{l=0}^{\infty}q^{\binom{l}{2}}\frac{z^{l}}{\left[l\right]_{q}!}.$
By (2), it is not difficult to show that
$\left[l\right]_{\frac{1}{q}}!=q^{-\binom{l}{2}}\left[l\right]_{q}!$. Then, we
have the following
(3) $e_{\frac{1}{q}}\left(z\right)=E_{q}\left(z\right)\text{.}$
For the $q$-commuting variables $x$ and $y$ such that $yx=qxy$, we know that
(4) $e_{q}\left(x+y\right)=e_{q}\left(x\right)e_{q}\left(y\right)\text{.}$
The $q$-integral was defined by Jackson as follows:
(5)
$\int_{0}^{x}f\left(\xi\right)d_{q}\xi=\left(1-q\right)x\sum_{l=0}^{\infty}f\left(q^{l}x\right)q^{l}$
provided that the series on the right hand side converges absolutely.
In particular, if $f(\xi)=\xi^{n}$, then we have
(6)
$\int_{0}^{x}\xi^{n}d_{q}\xi=\frac{1}{\left[n+1\right]_{q}}x^{n+1}\text{.}$
The definitions of $q$-integral and $q$-derivative imply the following
formula:
(7) $D_{q}\left(\int_{0}^{x}f\left(t\right)d_{q}t\right)=f\left(x\right)$
and
(8)
$D_{q}\left(f\left(x\right)g\left(x\right)\right)=f\left(x\right)D_{q}\left(g\left(x\right)\right)+g\left(qx\right)D_{q}\left(f\left(x\right)\right)\text{.}$
For more informations of Eqs. (1-8), you can refer to [19-23].
The ordinary Euler numbers and polynomials are defined via the following
generating function:
$e^{E\left(x\right)t}=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{2}{e^{t}+1}e^{xt}\text{,
}\left|t\right|<\pi\text{ }$
where the usual convention about replacing $E^{n}\left(x\right)$ by
$E_{n}\left(x\right)$ (see [1], [2], [3], [5], [6], [13]).
In [6], the new $q$-generalization of Euler polynomials are introduced by T.
Kim as follows:
$\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}E_{j,q}\left(x\right)=\frac{\left[2\right]_{q}}{e_{q}\left(z\right)+1}e_{q}\left(xz\right)\text{.}$
By using the above generating function, Kim gave some interesting and
fascinating properties for new $q$-generalization of Euler numbers and
polynomials. We note that these polynomials are used to study in Analytic
Numbers Theory. So, in the next section, we shall introduce generating
function of $q$-Genocchi numbers and polynomials of higher order.
Additionally, we shall give their applications.
## 2\. New $q$-Genocchi numbers and polynomials of higher order
In this section, we introduce generating function for $q$-Genocchi polynomials
of higher order by using Kim’s method in [6]. Thus, we now start as follows:
(9)
$S_{q}\left(t:\alpha\right)=\sum_{n=0}^{\infty}\frac{t^{n}}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\text{.}$
Here $G_{n,q}^{\left(\alpha\right)}$ is called as the $q$-Genocchi numbers of
higher order. By using $q$-derivative operator, we compute as follows:
(10)
$S_{q}\left(\frac{d}{d_{q}x}:\alpha\right)x^{k}=\sum_{n=0}^{\infty}\frac{1}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\left(\frac{d}{d_{q}x}\right)^{n}x^{k}=\sum_{n=0}^{k}\binom{k}{n}_{q}G_{n,q}^{\left(\alpha\right)}x^{k-n}\text{,}$
where
$\binom{k}{n}_{q}=\frac{\left[k\right]_{q}\left[k-1\right]_{q}\cdots\left[k-n+1\right]_{q}}{\left[n\right]_{q}!}\text{.}$
Similarly, by (10), we develop as follows:
$\displaystyle S_{q}\left(\frac{d}{d_{q}x}:\alpha\right)e_{q}\left(tx\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{G_{j,q}^{\left(\alpha\right)}}{\left[j\right]_{q}!}\left(\frac{d}{d_{q}x}\right)^{j}\sum_{k=0}^{\infty}\frac{x^{k}}{\left[k\right]_{q}!}t^{k}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{t^{j}}{\left[j\right]_{q}!}G_{j,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$ $\displaystyle S_{q}\left(x,t:\alpha\right)\text{.}$
From this point of view, we can also consider the $q$-Genocchi polynomials of
higher order in the form:
(11)
$\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\left(x\right)=\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}e_{q}\left(zx\right)\text{.}$
As $q\rightarrow 1$ and $\alpha=1$ in Eq. (11), we easily reach the following
$\lim_{q\rightarrow
1}G_{n,q}^{\left(1\right)}\left(x\right)=G_{n}\left(x\right)$
which $G_{n}\left(x\right)$ is known as ordinary Genocchi polynomials (for
details, see [5], [8], [14], [15], [17], [18]).
By (9) and (11), we readily see that
$\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}G_{j,q}^{\left(\alpha\right)}\left(x\right)=\sum_{j=0}^{\infty}\left(\sum_{n=0}^{j}\binom{j}{n}_{q}x^{j-n}G_{n,q}^{\left(\alpha\right)}\right)\frac{z^{j}}{\left[j\right]_{q}!}\text{.}$
By comparing the coefficients of $\frac{z^{j}}{\left[j\right]_{q}!}$ on both
sides of the above equation, then we obtain the following theorem.
###### Theorem 2.1.
For any $j\in\mathbb{N}$, we have
$G_{j,q}^{\left(\alpha\right)}\left(x\right)=\sum_{n=0}^{j}\binom{j}{n}_{q}x^{j-n}G_{n,q}^{\left(\alpha\right)}\text{.}$
By applying $q$-derivative operator to (11), then we see that
$\sum_{n=1}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}\left\\{\frac{d}{d_{q}x}G_{n,q}^{\left(\alpha\right)}\left(x\right)\right\\}=z\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\left(x\right)\text{.}$
By comparing the coefficients of $z^{n}$ on both sides of the above equation,
we arrive the following theorem.
###### Theorem 2.2.
For any $n\in\mathbb{N}^{\ast}=\left\\{0,1,2,3,\ldots\right\\}$, we have
$\frac{d}{d_{q}x}G_{n,q}^{\left(\alpha\right)}\left(x\right)=\left[n\right]_{q}G_{n-1,q}^{\left(\alpha\right)}\left(x\right)\text{.}$
For $q$-commuting variables $x$ and $y$ ($yx=qxy$), we note that
$\displaystyle\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}G_{l,q}^{\left(\alpha\right)}\left(x+y\right)$
$\displaystyle=$
$\displaystyle\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}e_{q}\left(z\left(x+y\right)\right)$
$\displaystyle=$ $\displaystyle
e_{q}\left(zy\right)\left(e_{q}\left(zx\right)\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}\right)$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\left(\sum_{j=0}^{l}\binom{l}{j}_{q}y^{l-j}G_{j,q}^{\left(\alpha\right)}\left(x\right)\right)\frac{z^{l}}{\left[l\right]_{q}!}\text{.}$
As a result, we procure the following theorem.
###### Theorem 2.3.
For any $n\in\mathbb{N}^{\ast}$, we have
$G_{n,q}^{\left(\alpha\right)}\left(x+y\right)=\sum_{j=0}^{n}\binom{n}{j}_{q}y^{n-j}G_{j,q}^{\left(\alpha\right)}\left(x\right)\text{.}$
By expression (11), we compute as follows:
$\displaystyle\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}G_{l,q}^{\left(\alpha+\beta\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}\right]\left[\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\beta}e_{q}\left(zx\right)\right]$
$\displaystyle=$
$\displaystyle\left[\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}G_{j,q}^{\left(\alpha\right)}\right]\left[\sum_{k=0}^{\infty}\frac{z^{k}}{\left[k\right]_{q}!}G_{k,q}^{\left(\beta\right)}\left(x\right)\right]$
by using Cauchy product on the above equation, we derive that
(13)
$\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}\left(\sum_{n=0}^{l}\binom{l}{n}_{q}G_{n,q}^{\left(\alpha\right)}G_{l-n,q}^{\left(\beta\right)}\left(x\right)\right)\text{.}$
Comparing the coefficients of Eqs. (2) and (13), then we present the following
theorem.
###### Theorem 2.4.
For $l\in\mathbb{N}^{\ast}$, then we have
$G_{l,q}^{\left(\alpha+\beta\right)}\left(x\right)=\sum_{n=0}^{l}\binom{l}{n}_{q}G_{n,q}^{\left(\alpha\right)}G_{l-n,q}^{\left(\beta\right)}\left(x\right)\text{.}$
Jackson are defined the $q$-analogue of the Gamma function by
(14)
$\Gamma_{q}\left(x\right)=\frac{\left(q;q\right)_{\infty}}{\left(q^{x};q\right)_{\infty}}\left(1-q\right)^{1-x}\text{,
}x\neq 0,-1,-2,\cdots.$
which have the following properties:
$\Gamma_{q}\left(x+1\right)=\left[x\right]_{q}\Gamma_{q}\left(x\right)\text{,
}\Gamma_{q}\left(1\right)=1\text{ and }\lim_{q\rightarrow
1^{-}}\Gamma_{q}\left(x\right)=\Gamma\left(x\right)\text{,
}\Re\left(x\right)>0\text{.}$
It has the $q$-integral representation as follows:
(15)
$\Gamma_{q}\left(s\right)=\int_{0}^{\frac{1}{1-q}}t^{s-1}E_{q}\left(-qt\right)d_{q}t\text{.}$
When $\frac{\log\left(1-q\right)}{\log q}\in\mathbb{Z}$, becomes
(16)
$\Gamma_{q}\left(s\right)=\int_{0}^{\infty}t^{s-1}E_{q}\left(-qt\right)d_{q}t\text{.}$
The $q$-Mellin transformation of a suitable function $f$ on $\mathbb{R}_{q,+}$
is defined by
(17)
$M_{q}\left(f\right)\left(s\right)=\int_{0}^{\infty}t^{s-1}f\left(t\right)d_{q}t$
(for details of Eqs. (14-17), see [10], [20], [21]).
In [23], the novel $q$-differential operator was defined by Rubin as follows:
(18)
$\partial_{q}\left(f\right)\left(x\right)=\frac{f\left(q^{-1}x\right)+f\left(-q^{-1}x\right)-f\left(qx\right)+f\left(-qx\right)-2f\left(-x\right)}{2\left(1-q\right)x}\text{.}$
By (18), we note that
(19) $\lim_{q\rightarrow
1}\partial_{q}\left(f\right)\left(x\right)=f{\acute{}}\left(x\right)\text{.}$
By applying Rubin’s $q$-differential operator to the generating function of
$q$-Genocchi numbers and polynomials of higher order, we compute as follows:
$\displaystyle\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\partial_{q}\left\\{\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}e_{q}\left(xz\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\frac{1}{2\left(1-q\right)}\sum_{l=0}^{n}\binom{n}{l}_{q}\left\\{\begin{array}[]{c}q^{-l}+\left(-1\right)^{l}q^{-l}-q^{l}\\\
+\left(-1\right)^{l}q^{l}+2\left(-1\right)^{l}\end{array}\right\\}x^{l-1}G_{n-l,q}^{\left(\alpha\right)}\right)\frac{z^{n}}{\left[n\right]_{q}!}\text{.}$
By comparing the coefficients of $\frac{z^{n}}{\left[n\right]_{q}!}$ on both
sides of the above equation. Then, we state the following theorem.
###### Theorem 2.5.
Let $T_{q}\left(l\right)=$
$q^{-l}+\left(-1\right)^{l}q^{-l}-q^{l}+\left(-1\right)^{l}q^{l}+2\left(-1\right)^{l}$,
then we get
(21)
$\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)=\frac{1}{2\left(1-q\right)}\sum_{l=0}^{n}\binom{n}{l}_{q}T_{q}\left(l\right)x^{l-1}G_{n-l,q}^{\left(\alpha\right)}\text{.}$
By (21), we readily derive the following
$\displaystyle\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q\right)}\sum_{l=0}^{\left[\frac{n}{2}\right]}\binom{n}{2l}_{q}\left\\{q^{-2l}+1\right\\}G_{n-2l,q}^{\left(\alpha\right)}$
$\displaystyle+\frac{1}{\left(q-1\right)}\sum_{l=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2l+1}_{q}\left\\{q^{2l+1}+1\right\\}G_{n-1-2l,q}^{\left(\alpha\right)}\text{.}$
Here $\left[.\right]$ is Gauss’ symbol. Consequently, we derive the following
theorem.
###### Theorem 2.6.
For any $n\in\mathbb{N}^{\ast}$, we have
$\displaystyle\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q\right)}\sum_{l=0}^{\left[\frac{n}{2}\right]}\binom{n}{2l}_{q}\left\\{q^{-2l}+1\right\\}G_{n-2l,q}^{\left(\alpha\right)}$
$\displaystyle+\frac{1}{\left(1-q\right)}\sum_{l=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2l+1}_{q}\left\\{q^{2l+1}+1\right\\}G_{n-1-2l,q}^{\left(\alpha\right)}\text{.}$
From (21) and (2.6), we conclude as follows:
###### Corollary 2.7.
The following identity
$\displaystyle\sum_{l=0}^{n}\binom{n}{l}_{q}T_{q}\left(l\right)x^{l-1}G_{n-l,q}^{\left(\alpha\right)}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\left[\frac{n}{2}\right]}\binom{n}{2l}_{q}\left\\{2q^{-2l}+2\right\\}G_{n-2l,q}^{\left(\alpha\right)}$
$\displaystyle+\sum_{l=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2l+1}_{q}\left\\{2q^{2l+1}+2\right\\}G_{n-1-2l,q}^{\left(\alpha\right)}$
is true.
By (21), we have the following corollary.
###### Corollary 2.8.
For any $n\in\mathbb{N}^{\ast}$, we get
$\lim_{q\rightarrow
1}\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)=nG_{n-1}^{\left(\alpha\right)}\left(x\right)$
where $G_{n}^{\left(\alpha\right)}\left(x\right)$ are called Genocchi numbers
and polynomials of higher order which are defined by the following generating
function:
$\sum_{n=0}^{\infty}G_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{n!}=\left(\frac{2t}{e^{t}+1}\right)^{\alpha}e^{xt}\text{
(see \cite[cite]{[\@@bibref{}{Jolany}{}{}]},
\cite[cite]{[\@@bibref{}{Rim}{}{}]}).}$
We now give a $q$-analogue of D. Miliĉic’s Lemma (see [22, page 1, Lemma
1.2.1]).
###### Lemma 2.9.
Let $a_{n,q}$, $n\in\mathbb{N}^{\ast}:=\left\\{0,1,2,3,...\right\\}$, be
complex numbers such that $\sum_{n=0}^{\infty}\left|a_{n,q}\right|$ converges.
Let
$\lambda=\left\\{-n\mid n\in\mathbb{N}^{\ast}\text{ and }a_{n,q}\neq
0\right\\}\text{.}$
Then,
$g_{q}\left(z\right)=\sum_{n=0}^{\infty}\frac{a_{n,q}}{\left[z+n\right]_{q}}$
converges absolutely for $z\in\mathbb{C}-\lambda$ and uniformly on bounded
subsets of $\mathbb{C}-\lambda$. The $q$-function is a meromorphic function on
complex plane with simple poles at the points in $\lambda$ and
Res$\left(g_{q},-n\right)=a_{n,q}$ for any $-n\in\lambda$.
###### Proof.
By using similar method in lecture notes of D. Miliĉic in [22], it is clear
that if $\left|\left[z\right]_{q}\right|<R$, we see
$\left|\left[z+n\right]_{q}\right|=\left|\left[z\right]_{q}+q^{zn}\left[n\right]_{q}\right|\geq\left|\left[n\right]_{q}-R\right|$
for all $\frac{1-q^{n}}{1-q}>R$. Then, we get
$\left|\frac{1}{\left[z+n\right]_{q}}\right|\leq\frac{1}{\left[n\right]_{q}-R}$
for $\left|\left[z\right]_{q}\right|<R$ and $\left[n\right]_{q}>R$. It follows
that $\left[n_{0}\right]_{q}>R$, we have
$\left|\sum_{n=n_{0}}\frac{a_{n,q}}{\left[z+n\right]_{q}}\right|\leq\sum_{n=0}^{\infty}\frac{\left|a_{n,q}\right|}{\left|\left[z+n\right]_{q}\right|}\leq\sum_{n=0}^{\infty}\frac{\left|a_{n,q}\right|}{\left[n\right]_{q}-R}\leq\frac{1}{\left[n_{0}\right]_{q}-R}\sum_{n=n_{0}}^{\infty}\left|a_{n,q}\right|$
Hence, the series $\sum_{n>R}\frac{a_{n,q}}{\left[z+n\right]_{q}}$ converges
absolutely and uniformly on the disk $\left\\{z\mid\left|z\right|<R\right\\}$
and defines there a meromorphic function. It follows that
$\sum_{n=0}^{\infty}\frac{a_{n,q}}{\left[z+n\right]_{q}}$
is a meromorphic function on that disk with the simple poles at the points of
$\lambda$ in $\left\\{z\mid\left|z\right|<R\right\\}$. Then, for any
$-n\in\lambda$, we have
$g_{q}\left(z\right)=\frac{a_{n,q}}{\left[z+n\right]_{q}}+\sum_{-j\in\lambda-\left\\{n\right\\}}^{\infty}\frac{a_{j,q}}{\left[z+j\right]_{q}}=\frac{a_{n,q}}{\left[z+n\right]_{q}}+\vartheta\left(z\right)$
where $\vartheta\left(z\right)$ is holomorphic at $-n$. This also shows that
$Res\left(g_{q},-n\right)=a_{n,q}.$
Thus, we successfully complete the proof of lemma.
We now want to indicate that $\Gamma$ extends to a meromorphic function by
using Lemma 2. 9. That is, we discover the following
$\Gamma_{q}\left(z\right)=\int_{0}^{\infty}t^{z-1}E_{q}\left(-qt\right)d_{q}t=\int_{0}^{1}t^{z-1}E_{q}\left(-qt\right)d_{q}t+\int_{1}^{\infty}t^{z-1}E_{q}\left(-qt\right)d_{q}t\text{.}$
Then, the second integral converges for any complex $z$ and represents an
entire function. On the other hand, the $q$-exponential function is entire,
and we have
$\displaystyle\int_{0}^{1}t^{z-1}E_{q}\left(-qt\right)d_{q}t$ $\displaystyle=$
$\displaystyle\int_{0}^{1}t^{z-1}\left\\{\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}t^{j}\right\\}d_{q}t$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\left\\{\int_{0}^{1}t^{z+j-1}d_{q}t\right\\}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\frac{1}{\left[z+j\right]_{q}}$
for any $z\in\mathbb{C}$. Now also, we can write as follows:
$\Gamma_{q}\left(z\right)=\int_{1}^{\infty}t^{z-1}E_{q}\left(-qt\right)d_{q}t+\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\frac{1}{\left[z+j\right]_{q}}$
for any $z$ in the right half plane. From Lemma 2. 9, the right hand-side of
the above identity defines a meromorphic function on the complex plane with
simple poles at $z=-j,$ $j\in\mathbb{N}^{\ast}$. Then, we have the following
theorem.
###### Theorem 2.10.
For any $j\in\mathbb{N}^{\ast}$, we derive the following
(23)
$\text{{Res}}\left(\Gamma_{q},-j\right)=\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\text{.}$
As $q\rightarrow 1$ into (23), we easily derive that
$\lim_{q\rightarrow
1}\text{{Res}}\left(\Gamma_{q},-j\right)=\frac{\left(-1\right)^{j}}{j!}$
which it is residue of Euler’s Gamma function (see [22]).
Now also, by applying $q$-Mellin Transformation to generating function of
$q$-Genocchi polynomials of higher order, then we compute as follows:
$\displaystyle\Im_{q}\left(z,x:\alpha\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma_{\frac{1}{q}}\left(z\right)}\int_{0}^{\infty}t^{z-\alpha-1}\left\\{\left(-1\right)^{\alpha}S_{q}\left(x,-t:\alpha\right)\right\\}d_{\frac{1}{q}}t$
$\displaystyle=$
$\displaystyle\sum_{l_{1},l_{2},...,l_{\alpha}=0}^{\infty}\left(-1\right)^{l_{1}+l_{2}+...+l_{\alpha}}\left\\{\frac{1}{\Gamma_{\frac{1}{q}}\left(z\right)}\int_{0}^{\infty}t^{z-1}E_{\frac{1}{q}}\left(-\frac{t}{q}\left(qx+q\sum_{k=1}^{\alpha}l_{k}\right)\right)d_{\frac{1}{q}}t\right\\}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}^{\alpha}\sum_{l_{1},l_{2},...,l_{\alpha}=0}^{\infty}\frac{q^{-z}\left(-1\right)^{l_{1}+l_{2}+...+l_{\alpha}}}{\left(l_{1}+l_{2}+...+l_{\alpha}+x\right)^{z}}\text{.}$
So, we now introduce definition of $q$-Hurwitz-Zeta type function as follows:
###### Definition 1.
For any $z\in\mathbb{C}$, then we define
$\Im_{q}\left(z,x:\alpha\right)=\left[2\right]_{q}^{\alpha}\sum_{l_{1},l_{2},...,l_{\alpha}=0}^{\infty}\frac{\left(-1\right)^{l_{1}+l_{2}+...+l_{\alpha}}}{\left(qx+q\sum_{k=1}^{\alpha}l_{k}\right)^{z}}\text{.}$
Via the above definition, we derive interpolation functions for $q$-Genocchi
numbers and polynomials of higher order at negative integers with the
following theorem.
###### Theorem 2.11.
The following equality holds:
$\Im_{q}\left(-n,x:\alpha\right)=\frac{q^{-n}G_{n+\alpha,q}^{\left(\alpha\right)}\left(x\right)}{\left[\alpha\right]_{q}!\binom{n+\alpha}{\alpha}_{q}}\text{.}$
## References
* [1] A. Bayad, Modular properties of elliptic Bernoulli and Euler function, Adv. Stud. Contemp. Math. 20 (2010), no. 3, 389-401
* [2] A. Bayad, T. Kim, Identities involving values of Bernstein, $q$-Bernoulli, and $q$-Euler polynomials, Russ. J. Math. Phys. 18(2011), No. 2, 133-143.
* [3] C. S. Ryoo, Some relations between twisted $q$-Euler numbers and Bernstein polynomials, Adv. Stud. Contemp. Math. 21 (2011), no. 2, 217-223.
* [4] C. S. Ryoo and T. Kim, An anologue of the zeta function and its applications, Applied Mathematics Letters 19 (2006), 1068-1072.
* [5] H. Jolany and H. Sharifi, Some results for Apostol-Genocchi Polynomials of higher order, In press in Bulletin of the Malaysian Mathematical Sciences Society vol 36, no 2, 2013
* [6] T. Kim, $q$-Generalized Euler Numbers and polynomials, Russian Journal of Mathematical Physics, Vol. 13, No. 3, 2006, pp. 293-308.
* [7] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russian J. Math. Phys. 16 (2009), 484–491.
* [8] T. Kim, On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458–1465.
* [9] T. Kim, On the analogs of Euler numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=1$, J. Math. Anal. Appl. 331 (2007) 779–792.
* [10] T. Kim, Some Identities on the integral representation of the product of several $q$-Bernstein-type polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 634675, 11 pages.
* [11] T. Kim, S. H. Lee, H. H. Han and C. S. Ryoo, On the values of the weighted $q$-Zeta and $L$-functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 476381, 7 pp.
* [12] Lee Chae Jang, The $q$-analogue of twisted Lerch type Euler Zeta functions, Bull. Korean Math. Soc. 47 (2010), No. 6, pp. 1181-1188.
* [13] H. Ozden, I. N. Cangul, Y. Simsek, Multivariate interpolation functions of higher order $q$-Euler numbers and their applications, Abstract and Applied Analysis 2008 (2008), Article ID 390857, 16 pages.
* [14] S-H. Rim, J-H. Jin, E-J. Moon and S-J. Lee, On multiple interpolation functions of the $q$-Genocchi polynomials, Journal of Inequalities and Applications, Volume 2010, Article ID 351419, 13 pages.
* [15] I. N. Cangul, H. Ozden, Y. Simsek, A new approach to $q$-Genocchi numbers and their interpolation functions, Nonlinear Analysis 71 (2009), pp. 793-799.
* [16] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, accepted in Bulletin of the Malaysian Mathematical Sciences and Society.
* [17] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, accepted in Bulletin of the Korean Mathematical Society.
* [18] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [19] F. H. Jackson, On $q$-definite integrals, The Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–2036, 1910.
* [20] K. Brahim, and R. Guanes, Some applications of the $q$-Mellin Transform, Tamsui Oxford Journal of Mathematical Sciences 26(3) (2010) 335-343.
* [21] V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.
* [22] D. Miliĉic, Notes on the Riemann’s Zeta function, http://www.math.utah.edu/~milicic/zeta.pdf.
* [23] Richard L. Rubin, A $q^{2}$-analogue operator for $q^{2}$-analogue Fourier Analysis, J. Math. Analys. App., 212 (1997), 571-582.
|
arxiv-papers
| 2012-05-13T12:47:42 |
2024-09-04T02:49:30.882614
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Jong Jin Seo",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1205.2854"
}
|
1205.2943
|
# Charm-strange baryon strong decays in a chiral quark model
Lei-Hua Liu, Li-Ye Xiao and Xian-Hui Zhong 111E-mail: zhongxh@hunnu.edu.cn
Department of Physics, Hunan Normal University, and Key Laboratory of Low-
Dimensional Quantum Structures and Quantum Control of Ministry of Education,
Changsha 410081, China
###### Abstract
The strong decays of charm-strange baryons up to $N=2$ shell are studied in a
chiral quark model. The theoretical predictions for the well determined charm-
strange baryons, $\Xi_{c}^{*}(2645)$, $\Xi_{c}(2790)$ and $\Xi_{c}(2815)$, are
in good agreement with the experimental data. This model is also extended to
analyze the strong decays of the other newly observed charm-strange baryons
$\Xi_{c}(2930)$, $\Xi_{c}(2980)$, $\Xi_{c}(3055)$, $\Xi_{c}(3080)$ and
$\Xi_{c}(3123)$. Our predictions are given as follows. (i) $\Xi_{c}(2930)$
might be the first $P$-wave excitation of $\Xi_{c}^{\prime}$ with
$J^{P}=1/2^{-}$, favors the $|\Xi_{c}^{\prime}\
{}^{2}P_{\lambda}1/2^{-}\rangle$ or $|\Xi_{c}^{\prime}\
{}^{4}P_{\lambda}1/2^{-}\rangle$ state. (ii) $\Xi_{c}(2980)$ might correspond
to two overlapping $P$-wave states $|\Xi_{c}^{\prime}\
{}^{2}P_{\rho}1/2^{-}\rangle$ and $|\Xi_{c}^{\prime}\
{}^{2}P_{\rho}3/2^{-}\rangle$, respectively. The $\Xi_{c}(2980)$ observed in
the $\Lambda_{c}^{+}\bar{K}\pi$ final state is most likely to be the
$|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}1/2^{-}\rangle$ state, while the narrower
resonance with a mass $m\simeq 2.97$ GeV observed in the
$\Xi_{c}^{*}(2645)\pi$ channel favors to be assigned to the
$|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}3/2^{-}\rangle$ state. (iii) $\Xi_{c}(3080)$
favors to be classified as the $|\Xi_{c}\ S_{\rho\rho}1/2^{+}\rangle$ state,
i.e., the first radial excitation ($2S$) of $\Xi_{c}$. (iv) $\Xi_{c}(3055)$ is
most likely to be the first $D$-wave excitation of $\Xi_{c}$ with
$J^{P}=3/2^{+}$, favors the $|\Xi_{c}\ ^{2}D_{\lambda\lambda}3/2^{+}\rangle$
state. (v) $\Xi_{c}(3123)$ might be assigned to the $|\Xi_{c}^{\prime}\
{}^{4}D_{\lambda\lambda}3/2^{+}\rangle$, $|\Xi_{c}^{\prime}\
{}^{4}D_{\lambda\lambda}5/2^{+}\rangle$, or $|\Xi_{c}\
^{2}D_{\rho\rho}5/2^{+}\rangle$ state. As a by-product, we calculate the
strong decays of the bottom baryons $\Sigma_{b}^{\pm}$, $\Sigma_{b}^{*\pm}$
and $\Xi_{b}^{*}$, which are in good agreement with the recent observations as
well.
###### pacs:
12.39.Jh, 13.30.-a, 14.20.Lq, 14.20.Mr
## I INTRODUCTION
In recent years, several new charm-strange baryons, $\Xi_{c}(2930)$,
$\Xi_{c}(2980)$, $\Xi_{c}(3055)$, $\Xi_{c}(3080)$ and $\Xi_{c}(3123)$, have
been observed. Their experimental information has been collected in Tab. 1.
$\Xi_{c}(2980)$ and $\Xi_{c}(3080)$ are relatively well-established in
experiments. Both of their isospin states were observed by Belle Collaboration
in the $\Lambda_{c}^{+}\bar{K}\pi$ channel Chistov:2006zj , and confirmed by
BaBar with high statistical significances Aubert:2007dt . Belle also observed
a resonance structure around $2.97$ GeV with a narrow width of $\sim 18$ MeV
in the $\Xi_{c}^{*}(2645)\pi$ decay channel in a separate study Lesiak:2008wz
, which is often considered as the same resonance, $\Xi_{c}(2980)$, observed
in the $\Lambda_{c}^{+}\bar{K}\pi$ channel. $\Xi_{c}(2930)$ was found by BaBar
in the $\Lambda_{c}^{+}K^{-}$ final state by analyzing the
$B^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}K^{-}$ process
Aubert:2007eb . However, this structure is not yet confirmed by Belle.
$\Xi_{c}(3055)^{+}$ and $\Xi_{c}(3123)^{+}$ were only observed by BaBar in the
$\Lambda_{c}^{+}K^{-}\pi^{+}$ final state with statistical significances of
6.4$\sigma$ and $3.0\sigma$, respectively Aubert:2007dt . No further evidences
of them were found when BaBar searched the inclusive $\Lambda_{c}^{+}\bar{K}$
and $\Lambda_{c}^{+}\bar{K}\pi^{+}\pi^{-}$ invariant mass spectra for new
narrow states. BaBar’s observations show that $\Xi_{c}(3055)^{+}$ and
$\Xi_{c}(3123)^{+}$ mostly decay though the intermediate resonant modes
$\Sigma_{c}(2455)^{++}K^{-}$ and $\Sigma_{c}(2520)^{++}K^{-}$, respectively. A
good review of the recent experimental results on charmed baryons can be found
in Charles:2009gr .
Charmed baryon mass spectroscopy has been investigated in various models
SilvestreBrac:1996bg ; Migura:2006ep ; Valcarce:2008dr ; Guo:2008he ;
Ebert:2011kk ; Roberts:2007ni ; Ebert:2007nw ; Patel:2008mv ; Romanets:2012hm
; Zhang:2008pm ; Garcilazo:2007eh . The masses of charm-strange baryons in the
$N\leq 2$ shell predicted within several quark models have been collected in
Tabs. 2 and 3. Comparing the experimental data with the quark model
predictions, one finds that $\Xi_{c}(2930)$ could be a candidate of the $2S$
excitation of $\Xi_{c}$ with $J^{P}=1/2^{+}$, or the $1P$ excitation of
$\Xi_{c}^{\prime}$ with $J^{P}=1/2^{-}$, $3/2^{-}$ or $5/2^{-}$.
$\Xi_{c}(2980)$ might be assigned to the $2S$ excitation of $\Xi_{c}$ or
$\Xi_{c}^{\prime}$ with $J^{P}=1/2^{+}$. $\Xi_{c}(3055)$ and $\Xi_{c}(3080)$
are most likely to be the $1D$ excitations of $\Xi_{c}$ with $J^{P}=3/2^{+}$
or $5/2^{+}$, or the $2S$ excitation of $\Xi_{c}^{\prime}$ with
$J^{P}=1/2^{+}$. $\Xi_{c}(3123)$ might be classified as $1D$ excitation of
$\Xi_{c}^{\prime}$ with $J^{P}=3/2^{+}$, $5/2^{+}$ or $7/2^{+}$. Obviously,
only depending on the mass analysis it is difficult to determine the quantum
numbers of these newly observed charm-strange baryons. On the other hand, the
strong decays of these newly observed charm-strange baryons have been studied
in the framework of heavy hadron chiral perturbation theory Cheng:2006dk and
${}^{3}P_{0}$ model Chen:2007xf ; Liu:2007ge , respectively. In Cheng:2006dk ,
Cheng and Chua advocated that the $J^{P}$ numbers of $\Xi_{c}(2980)$ and
$\Xi_{c}(3080)$ could be $1/2^{+}$ and $5/2^{+}$, respectively. They claimed
that under this $J^{P}$ assignment, it is easy to understand why
$\Xi_{c}(2980)$ is broader than $\Xi_{c}(3080)$. In Chen:2007xf ; Liu:2007ge ,
Chen _et al._ have analyzed the strong decays of the $N=2$ shell excited
charm-strange baryons in the ${}^{3}P_{0}$ model, they could only exclude some
assignments according to the present experimental information. As a whole,
although the new charm-strange baryons have been studied in several aspects,
such as mass spectroscopy and strong decays, their quantum numbers are not
clear so far. Thus, more investigations of these new heavy baryons are needed.
To further understand the nature of these newly observed charm-strange
baryons, in this work, we make a systematic study of their strong decays in a
chiral quark model, which has been developed and successfully used to deal
with the strong decays of charmed baryons and heavy-light mesons Zhong:2007gp
; Zhong:2010vq ; Zhong:2009sk ; Zhong:2008kd . It should be pointed out that
very recently, some important progresses in the observation of the bottom
baryons have been achieved in experiments as well: CDF Collaboration first
measured the natural widths of the bottom baryons $\Sigma_{b}^{\pm}$ and
$\Sigma_{b}^{*\pm}$, and improved the measurement masses CDF:2011ac , and CMS
Collaboration observed a new neutral excited bottom baryon with a mass
$m=5945.0\pm 0.7\pm 0.3\pm 2.7$ MeV, which is most likely to be the
$\Xi_{b}^{*0}$ CMS . As a by-product, in this work we also calculate the
strong decays of these bottom baryons according to the new measurements.
This work is organized as follows. In the subsequent section, the charm-
strange baryon in the quark model is outlined. Then a brief review of the
chiral quark model approach is given in Sec. III. The numerical results are
presented and discussed in Sec. IV. Finally, a summary is given in Sec. V.
Table 1: Summary of the experimental results of the newly observed charm-strange baryons. Resonance | Mass (MeV) | Width (MeV) | Observed decay channel | Collaboration | Status Nakamura:2010zzi
---|---|---|---|---|---
$\Xi_{c}(2930)^{0}$ | $2931\pm 3\pm 5$ | $36\pm 7\pm 11$ | $\Lambda_{c}^{+}K^{-}$ | BaBar Aubert:2007eb | *
$\Xi_{c}(2980)^{+}$ | $2978.5\pm 2.1\pm 2.0$ | $43.5\pm 7.5\pm 7.0$ | $\Lambda_{c}^{+}K^{-}\pi^{+}$ | Belle Chistov:2006zj | ***
| $2969.3\pm 2.2\pm 1.7$ | $27\pm 8\pm 2$ | $\Lambda_{c}^{+}K^{-}\pi^{+}$, $\Sigma_{c}(2455)^{++}K^{-}$ | BaBar Aubert:2007dt |
| $2967.7\pm 2.3^{+1.1}_{-1.2}$ | $18\pm 6\pm 3$ | $\Xi_{c}(2645)^{0}\pi^{+}$ | Belle Lesiak:2008wz |
$\Xi_{c}(2980)^{0}$ | $2977.1\pm 8.8\pm 3.5$ | $43.5$ fixed | $\Lambda_{c}^{+}\bar{K}^{0}\pi^{-}$ | Belle Chistov:2006zj | ***
| $2972.9\pm 4.4\pm 1.6$ | $31\pm 7\pm 8$ | $\Lambda_{c}^{+}\bar{K}^{0}\pi^{-}$ | BaBar Aubert:2007dt |
| $2965.7\pm 2.4^{+1.1}_{-1.2}$ | $15\pm 6\pm 3$ | $\Xi_{c}(2645)^{+}\pi^{-}$ | Belle Lesiak:2008wz |
$\Xi_{c}(3055)^{+}$ | $3054.2\pm 1.2\pm 0.5$ | $17\pm 6\pm 11$ | $\Sigma_{c}(2455)^{++}K^{-}$ | BaBar Aubert:2007dt | **
$\Xi_{c}(3080)^{+}$ | $3076.7\pm 0.9\pm 0.5$ | $6.2\pm 1.2\pm 0.8$ | $\Lambda_{c}^{+}K^{-}\pi^{+}$ | Belle Chistov:2006zj | ***
| $3077.0\pm 0.4\pm 0.2$ | $5.5\pm 1.3\pm 0.6$ | $\Lambda_{c}^{+}K^{-}\pi^{+}$, $\Sigma_{c}(2455,2520)^{++}K^{-}$ | BaBar Aubert:2007dt |
$\Xi_{c}(3080)^{0}$ | $3082.8\pm 1.8\pm 1.5$ | $5.2\pm 3.1\pm 1.8$ | $\Lambda_{c}^{+}\bar{K}^{0}\pi^{-}$ | Belle Chistov:2006zj | ***
| $3079.3\pm 1.1\pm 0.2$ | $5.9\pm 2.3\pm 1.5$ | $\Lambda_{c}^{+}\bar{K}^{0}\pi^{-}$, $\Sigma_{c}(2455,2520)^{0}K^{0}_{S}$ | BaBar Aubert:2007dt |
$\Xi_{c}(3123)^{+}$ | $3122.9\pm 1.3\pm 0.3$ | $4.4\pm 3.4\pm 1.7$ | $\Sigma_{c}(2520)^{++}K^{-}$ | BaBar Aubert:2007dt | *
Table 2: Masses (MeV) of the $\Xi_{c}$-type charm-strange baryons in the various quark models. N | $J^{P}$ | State | Valcarce:2008dr | Ebert:2011kk | SilvestreBrac:1996bg | Roberts:2007ni | Ebert:2007nw
---|---|---|---|---|---|---|---
0 | $\frac{1}{2}^{+}$ | 1S | 2471 | 2476 | 2467 | 2466 | 2481
1 | $\frac{1}{2}^{-}$ | 1P | 2799 | 2792 | 2792 | 2773 | 2801
1 | $\frac{3}{2}^{-}$ | 1P | … | 2819 | 2792 | 2783 | 2820
2 | $\frac{1}{2}^{+}$ | 2S | 3137 | 2959 | 2992 | … | 2923
2 | $\frac{3}{2}^{+}$ | 1D | 3071 | 3059 | 3057 | 3012 | 3030
2 | $\frac{5}{2}^{+}$ | 1D | 3049 | 3076 | 3057 | 3004 | 3042
Table 3: Masses (MeV) of the $\Xi_{c}^{\prime}$-type charm-strange baryons in the various quark models. N | $J^{P}$ | State | Valcarce:2008dr | Ebert:2011kk | SilvestreBrac:1996bg | Roberts:2007ni | Ebert:2007nw
---|---|---|---|---|---|---|---
0 | $\frac{1}{2}^{+}$ | 1S | 2574 | 2579 | 2567 | 2594 | 2578
0 | $\frac{3}{2}^{+}$ | 1S | 2642 | 2649 | 2647 | 2649 | 2654
1 | $\frac{1}{2}^{-}$ | 1P | 2902 | 2936 | 2897 | 2855 | 2934
1 | $\frac{3}{2}^{-}$ | 1P | … | 2935 | 2910 | 2866 | 2931
1 | $\frac{5}{2}^{-}$ | 1P | … | 2929 | 3050 | 2895 | 2921
2 | $\frac{1}{2}^{+}$ | 2S | 3212 | 2983 | 3087 | … | 2984
2 | $\frac{1}{2}^{+}$ | 1D | … | 3163 | … | … | 3132
2 | $\frac{3}{2}^{+}$ | 1D | … | 3160 | 3127 | … | 3127
2 | $\frac{5}{2}^{+}$ | 1D | 3132 | 3166 | 3167 | 3080 | 3123
2 | $\frac{7}{2}^{+}$ | 1D | … | 3147 | … | 3094 | 3136
## II charm-strange baryon in the quark model
The charmed baryon contains a heavy charm quark, which violates SU(4)
symmetry. However, the SU(3) symmetry between the other two light quarks ($u$,
$d$, or $s$) is approximately kept. According to the symmetry, the charmed
baryons can be classified two different SU(3) flavor representations: the
symmetric $\mathbf{6}$ and antisymmetric antitriplet $\bar{\mathbf{3}}$. For
the charm-strange baryon, the antisymmetric flavor wave function
($\Xi_{c}$-type) can be written as
$\displaystyle\phi_{\Xi_{c}}=\cases{\frac{1}{\sqrt{2}}(us-su)c&for
$\Xi^{+}_{c}$\cr\frac{1}{\sqrt{2}}(ds-sd)c&for $\Xi^{0}_{c}$},$ (1)
while the symmetric flavor wave function ($\Xi^{\prime}_{c}$-type) is given by
$\displaystyle\phi_{\Xi^{\prime}_{c}}=\cases{\frac{1}{\sqrt{2}}(us+su)c&for
$\Xi^{\prime+}_{c}$\cr\frac{1}{\sqrt{2}}(ds+sd)c&for $\Xi^{\prime 0}_{c}$}.$
(2)
In the quark model, the typical SU(2) spin wave functions for the charm-
strange baryons can be adopted Koniuk:1979vy ; Copley:1979wj , which are
$\displaystyle\chi^{s}_{3/2}$ $\displaystyle=$
$\displaystyle\uparrow\uparrow\uparrow,\ \
\chi^{s}_{-3/2}=\downarrow\downarrow\downarrow,$ $\displaystyle\chi^{s}_{1/2}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(\uparrow\uparrow\downarrow+\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow),$
$\displaystyle\chi^{s}_{-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(\uparrow\downarrow\downarrow+\downarrow\downarrow\uparrow+\downarrow\uparrow\downarrow),$
(3)
for the spin-3/2 states with a symmetric spin wave function,
$\displaystyle\chi^{\rho}_{1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\uparrow\downarrow\uparrow-\downarrow\uparrow\uparrow),$
$\displaystyle\chi^{\rho}_{-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\uparrow\downarrow\downarrow-\downarrow\uparrow\downarrow),$
(4)
for the spin-1/2 states with a mixed antisymmetric spin wave function, and
$\displaystyle\chi^{\lambda}_{1/2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}(\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow-2\uparrow\uparrow\downarrow),$
$\displaystyle\chi^{\lambda}_{-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(\uparrow\downarrow\downarrow+\downarrow\uparrow\downarrow-2\downarrow\downarrow\uparrow),$
(5)
for the spin-1/2 states with a mixed symmetric spin wave function.
The spatial wave function of a charm-strange baryon is adopted the harmonic
oscillator form in the constituent quark model. The details of the spatial
wave functions can be found in our previous work Zhong:2007gp .
The spin-flavor and spatial wave functions of baryons must be symmetric since
the color wave function is antisymmetric. The flavor wave functions of the
$\Xi_{c}$-type charm-strange baryons, $\phi_{\Xi_{c}}$, are antisymmetric
under the interchange of the $u$ ($d$) and $s$ quarks, thus, their spin-space
wave functions must be symmetric. In contrast, the spin-spatial wave functions
of $\Xi^{\prime}_{c}$-type charm-strange baryons are required to be
antisymmetric due to their symmetric flavor wave functions under the
interchange of the two light quarks. The notations, wave functions, and
quantum numbers of the $\Xi_{c}$-type and $\Xi^{\prime}_{c}$-type charm-
strange baryons up to $N=2$ shell classified in the quark model are listed in
Tabs. 4 and 5, respectively.
Table 4: The $\Xi_{c}$-type charm-strange baryons classified in the quark model and their possible two body strong decay channels. The notation of the $\Xi_{c}$-type charmed baryon is denoted by $|\Xi_{c}\ ^{2S+1}L_{\sigma}J^{P}\rangle$ as used in Ref. Copley:1979wj . The Clebsch-Gordan series for the spin and angular-momentum addition $|\Xi_{c}\ ^{2S+1}L_{\sigma}J^{P}\rangle=\sum_{L_{z}+S_{z}=J_{z}}\langle LL_{z},SS_{z}|JJ_{z}\rangle^{N}\Psi^{\sigma}_{LL_{z}}\chi_{S_{z}}\phi_{\Xi_{c}}$ has been omitted. Notation | N | $l_{\lambda}$ | $l_{\rho}$ | L | $S$ | $J^{P}$ | Wave function | Strong decay channel
---|---|---|---|---|---|---|---|---
$|\Xi_{c}\ ^{2}S\frac{1}{2}^{+}\rangle$ | 0 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{0}\Psi^{S}_{00}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}}$ | …
$|\Xi_{c}\ ^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{1}{2}$ | $\frac{1}{2}^{-}$ | ${}^{1}\Psi^{\lambda}_{1L_{z}}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}^{\prime}\pi$, $\Xi_{c}^{*}\pi$
$|\Xi_{c}\ ^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{1}{2}$ | $\frac{3}{2}^{-}$ | |
$|\Xi_{c}\ ^{2}P_{\rho}\frac{1}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{1}{2}$ | $\frac{1}{2}^{-}$ | ${}^{1}\Psi^{\rho}_{1L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}\pi$, $\Xi_{c}^{\prime}\pi$, $\Xi_{c}^{*}\pi$
$|\Xi_{c}\ ^{2}P_{\rho}\frac{3}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{1}{2}$ | $\frac{3}{2}^{-}$ | |
$|\Xi_{c}\ ^{4}P_{\rho}\frac{1}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{3}{2}$ | $\frac{1}{2}^{-}$ | |
$|\Xi_{c}\ ^{4}P_{\rho}\frac{3}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{3}{2}$ | $\frac{3}{2}^{-}$ | ${}^{1}\Psi^{\rho}_{1L_{z}}\chi^{S}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}\pi$, $\Xi_{c}^{\prime}\pi$, $\Xi_{c}^{*}\pi$
$|\Xi_{c}\ ^{4}P_{\rho}\frac{5}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{3}{2}$ | $\frac{5}{2}^{-}$ | |
$|\Xi_{c}\ ^{2}S_{A}\frac{1}{2}^{+}\rangle$ | 2 | 1 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{A}_{00}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$,$\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$, $\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{4}S_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 0 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{A}_{00}\chi^{s}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}\pi$, $\Xi_{c}\eta$,$\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$, $\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{2}P_{A}\frac{1}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{A}_{1L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$, $\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{2}P_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | |
$|\Xi_{c}\ ^{4}P_{A}\frac{1}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{3}{2}$ | $\frac{1}{2}^{+}$ | | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$, $\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{4}P_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | $\ \ {}^{2}\Psi^{A}_{1L_{z}}\chi^{S}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$,$\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$, $\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{4}P_{A}\frac{5}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{3}{2}$ | $\frac{5}{2}^{+}$ | | $\Xi_{c}\pi$, $\Xi_{c}\eta$,$\Lambda_{c}K$, $\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{2}D_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$,$\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{2}D_{A}\frac{5}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ | $\ \ {}^{2}\Psi^{A}_{2L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}}$ |
$|\Xi_{c}\ ^{4}D_{A}\frac{1}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{3}{2}$ | $\frac{1}{2}^{+}$ | | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$,$\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}\ ^{4}D_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | |
$|\Xi_{c}\ ^{4}D_{A}\frac{5}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{3}{2}$ | $\frac{5}{2}^{+}$ | $\ \ {}^{2}\Psi^{A}_{2L_{z}}\chi^{S}_{S_{z}}\phi_{\Xi_{c}}$ |
$|\Xi_{c}\ ^{4}D_{A}\frac{7}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{3}{2}$ | $\frac{7}{2}^{+}$ | |
$|\Xi_{c}\ ^{2}D_{\rho\rho}\frac{3}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | | $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$,$\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$
$|\Xi_{c}\ ^{2}D_{\rho\rho}\frac{5}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ | $\ {}^{2}\Psi^{\rho\rho}_{2L_{z}}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}}$ |
$|\Xi_{c}\ ^{2}D_{\lambda\lambda}\frac{3}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | | $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$,$\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$D\Lambda$
$|\Xi_{c}\ ^{2}D_{\lambda\lambda}\frac{5}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ | $\ \ {}^{2}\Psi^{\lambda\lambda}_{2L_{z}}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}}$ |
$|\Xi_{c}\ ^{2}S_{\rho\rho}\frac{1}{2}^{+}\rangle$ | 2 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{\rho\rho}_{00}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$,$\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$
$|\Xi_{c}\ ^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ | 2 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{\lambda\lambda}_{00}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}}$ | $\Xi_{c}^{\prime}\pi$,$\Sigma_{c}K$,$\Xi_{c}^{*}\pi$,$\Sigma_{c}^{*}K$,$D\Lambda$
Table 5: The $\Xi_{c}^{\prime}$-type charm-strange baryons classified in the quark model and their possible two body strong decay channels. The notation of the $\Xi_{c}^{\prime}$-type charmed baryon is denoted by $|\Xi_{c}^{\prime}\ {}^{2S+1}L_{\sigma}J^{P}\rangle$ as used in Ref. Copley:1979wj . The Clebsch-Gordan series for the spin and angular-momentum addition $|\Xi_{c}^{\prime}\ {}^{2S+1}L_{\sigma}J^{P}\rangle=\sum_{L_{z}+S_{z}=J_{z}}\langle LL_{z},SS_{z}|JJ_{z}\rangle^{N}\Psi^{\sigma}_{LL_{z}}\chi_{S_{z}}\phi_{\Xi_{c}^{\prime}}$ has been omitted. Notation | $N$ | $l_{\lambda}$ | $l_{\rho}$ | $L$ | $S$ | $J^{P}$ | Wave function | Strong decay channel
---|---|---|---|---|---|---|---|---
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}S\frac{1}{2}^{+}\rangle$ | 0 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{0}\Psi^{S}_{00}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | …
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}S\frac{3}{2}^{+}\rangle$ | 0 | 0 | 0 | 0 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | ${}^{0}\Psi^{S}_{00}\chi^{s}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{1}{2}$ | $\frac{1}{2}^{-}$ | ${}^{1}\Psi^{\lambda}_{1L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}\pi$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{1}{2}$ | $\frac{3}{2}^{-}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}P_{\lambda}\frac{1}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{3}{2}$ | $\frac{1}{2}^{-}$ | | $\Xi_{c}\pi$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{3}{2}$ | $\frac{3}{2}^{-}$ | ${}^{1}\Psi^{\lambda}_{1L_{z}}\chi^{s}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}P_{\lambda}\frac{5}{2}^{-}\rangle$ | 1 | 1 | 0 | 1 | $\frac{3}{2}$ | $\frac{5}{2}^{-}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}P_{\rho}\frac{1}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{1}{2}$ | $\frac{1}{2}^{-}$ | ${}^{1}\Psi^{\rho}_{1L_{z}}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}P_{\rho}\frac{3}{2}^{-}\rangle$ | 1 | 0 | 1 | 1 | $\frac{1}{2}$ | $\frac{3}{2}^{-}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}S_{A}\frac{1}{2}^{+}\rangle$ | 2 | 1 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{A}_{00}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}P_{A}\frac{1}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | $\ \ {}^{2}\Psi^{A}_{1L_{z}}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}P_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 1 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}D_{A}\frac{3}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | | $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}D_{A}\frac{5}{2}^{+}\rangle$ | 2 | 1 | 1 | 2 | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ | $\ \ {}^{2}\Psi^{A}_{2L_{z}}\chi^{\rho}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ |
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}D_{\rho\rho}\frac{3}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | | $\Xi_{c}\pi$, $\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}D_{\rho\rho}\frac{5}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ | $\ {}^{2}\Psi^{\rho\rho}_{2L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\rho\rho}\frac{1}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{3}{2}$ | $\frac{1}{2}^{+}$ | | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\rho\rho}\frac{3}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | $\ {}^{2}\Psi^{\rho\rho}_{2L_{z}}\chi^{s}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\rho\rho}\frac{5}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{3}{2}$ | $\frac{5}{2}^{+}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\rho\rho}\frac{7}{2}^{+}\rangle$ | 2 | 0 | 2 | 2 | $\frac{3}{2}$ | $\frac{7}{2}^{+}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}D_{\lambda\lambda}\frac{3}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$,$D\Lambda$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}D_{\lambda\lambda}\frac{5}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ | $\ \ {}^{2}\Psi^{\lambda\lambda}_{2L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{3}{2}$ | $\frac{1}{2}^{+}$ | | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$,$D\Lambda$
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\lambda\lambda}\frac{3}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | $\ {}^{2}\Psi^{\lambda\lambda}_{2L_{z}}\chi^{s}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\lambda\lambda}\frac{5}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{3}{2}$ | $\frac{5}{2}^{+}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}D_{\lambda\lambda}\frac{7}{2}^{+}\rangle$ | 2 | 2 | 0 | 2 | $\frac{3}{2}$ | $\frac{7}{2}^{+}$ | |
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}S_{\rho\rho}\frac{1}{2}^{+}\rangle$ | 2 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{\rho\rho}_{00}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}S_{\rho\rho}\frac{3}{2}^{+}\rangle$ | 2 | 0 | 0 | 0 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{\rho\rho}_{00}\chi^{s}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ | 2 | 0 | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{\lambda\lambda}_{00}\chi^{\lambda}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
$|\Xi_{c}^{{}^{\prime}}\ {}^{4}S_{\lambda\lambda}\frac{3}{2}^{+}\rangle$ | 2 | 0 | 0 | 0 | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{\lambda\lambda}_{00}\chi^{s}_{S_{z}}\phi_{\Xi_{c}^{{}^{\prime}}}$ | $\Xi_{c}\pi$,$\Xi_{c}\eta$, $\Lambda_{c}K$, $\Xi_{c}^{\prime}\pi$, $\Sigma_{c}K$,$\Xi_{c}^{*}\pi$, $\Sigma_{c}^{*}K$,$\Xi_{c}(2790,2815)\pi$
## III The chiral quark model
In the chiral quark model, the effective low energy quark-meson pseudoscalar
coupling at tree-level is given byLi:1995si ; Zhong:2007fx ; Li:1997gda ; qk3
; Zhong:2011ti
$\displaystyle
H_{m}=\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\vec{\tau}\cdot\partial^{\mu}\vec{\phi}_{m}.$
(6)
where $\psi_{j}$ represents the $j$-th quark field in a baryon and $f_{m}$ is
the meson’s decay constant. The pseudoscalar-meson octet $\phi_{m}$ is
expressed as
$\displaystyle\phi_{m}=\pmatrix{\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\cr\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\cr
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta}.$ (7)
To match non-relativistic harmonic oscillator spatial wave function
${}^{N}\Psi_{LL_{z}}$ in the quark model, we adopt the non-relativistic form
of Eq. (6) in the calculations, which is given by Li:1995si ; Zhong:2007fx ;
Li:1997gda ; qk3 ; Zhong:2011ti
$\displaystyle H^{nr}_{m}$ $\displaystyle=$
$\displaystyle\sum_{j}\Big{\\{}\frac{\omega_{m}}{E_{f}+M_{f}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{P}_{f}+\frac{\omega_{m}}{E_{i}+M_{i}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{P}_{i}$
(8)
$\displaystyle-\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+\frac{\omega_{m}}{2\mu_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}^{\prime}_{j}\Big{\\}}I_{j}\varphi_{m},$
where $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ and $\mu_{q}$ correspond to
the Pauli spin vector and the reduced mass of the $j$-th quark in the initial
and final baryons, respectively. For emitting a meson, we have
$\varphi_{m}=e^{-i\textbf{q}\cdot\textbf{r}_{j}}$, and for absorbing a meson
we have $\varphi_{m}=e^{i\textbf{q}\cdot\textbf{r}_{j}}$. In the above non-
relativistic expansions,
$\textbf{p}^{\prime}_{j}=\textbf{p}_{j}-m_{j}/M\textbf{P}_{c.m.}$ is the
internal coordinate for the $j$-th quark in the baryon rest frame.
$\omega_{m}$ and q are the energy and three-vector momentum of the meson,
respectively. $\textbf{P}_{i}$ and $\textbf{P}_{f}$ stand for the momenta of
the initial final baryons, respectively. The isospin operator $I_{j}$ in Eq.
(8) is expressed as
$\displaystyle I_{j}=\cases{a^{\dagger}_{j}(u)a_{j}(s)&for $K^{+}$,\cr
a^{\dagger}_{j}(s)a_{j}(u)&for $K^{-}$,\cr a^{\dagger}_{j}(d)a_{j}(s)&for
$K^{0}$,\cr a^{\dagger}_{j}(s)a_{j}(d)&for $\bar{K^{0}}$,\cr
a^{\dagger}_{j}(u)a_{j}(d)&for $\pi^{-}$,\cr a^{\dagger}_{j}(d)a_{j}(u)&for
$\pi^{+}$,\cr\frac{1}{\sqrt{2}}[a^{\dagger}_{j}(u)a_{j}(u)-a^{\dagger}_{j}(d)a_{j}(d)]&for
$\pi^{0}$,\cr\frac{1}{\sqrt{2}}[a^{\dagger}_{j}(u)a_{j}(u)+a^{\dagger}_{j}(d)a_{j}(d)]\cos\phi_{P}\cr-a^{\dagger}_{j}(s)a_{j}(s)\sin\phi_{P}&for
$\eta$,}$ (9)
where $a^{\dagger}_{j}(u,d,s)$ and $a_{j}(u,d,s)$ are the creation and
annihilation operators for the $u$, $d$ and $s$ quarks, and $\phi_{P}$ is the
mixing angle of $\eta$ meson in the flavor basis Nakamura:2010zzi ;
Zhong:2011ht .
For a light pseudoscalar meson emission in a baryon strong decays, the partial
decay amplitudes can be worked out according to the non-relativistic operator
of quark-meson coupling. The details of how to work out the decay amplitudes
can be seen in our previous work Zhong:2007gp . The quark model permitted two
body strong decay channels of each charm-strange baryon have been listed in
Tabs. 4 and 5 as well. With the partial decay amplitudes derived from the
chiral quark model, we can calculate the strong decay width by
$\Gamma=\left(\frac{\delta}{f_{m}}\right)^{2}\frac{(E_{f}+M_{f})|\textbf{q}|}{4\pi
M_{i}(2J_{i}+1)}\sum_{J_{iz},J_{fz}}|\mathcal{M}_{J_{iz},J_{fz}}|^{2},$ (10)
where $\mathcal{M}_{J_{iz},J_{fz}}$ is the transition amplitude, $J_{iz}$ and
$J_{fz}$ stand for the third components of the total angular momenta of the
initial and final baryons, respectively. $\delta$ as a global parameter
accounts for the strength of the quark-meson couplings. It has been determined
in our previous study of the strong decays of the charmed baryons and heavy-
light mesons Zhong:2007gp ; Zhong:2008kd . Here, we fix its value the same as
that in Refs. Zhong:2008kd ; Zhong:2007gp , i.e. $\delta=0.557$.
In the calculation, the standard quark model parameters are adopted. Namely,
we set $m_{u}=m_{d}=330$ MeV, $m_{s}=450$ MeV, $m_{c}=1700$ MeV and
$m_{b}=5000$ MeV for the constituent quark masses. The harmonic oscillator
parameter $\alpha$ in the wave function ${}^{N}\Psi_{LL_{z}}$ is taken as
$\alpha=0.40$ GeV. The decay constants for $\pi$, $K$ and $\eta$ mesons are
taken as $f_{\pi}=132$ MeV, $f_{K}=f_{\eta}=160$ MeV, respectively. The masses
of the mesons and baryons used in the calculations are adopted from the
Particle Data Group Nakamura:2010zzi . With these parameters, the strong decay
properties of the well known heavy-light mesons and charmed baryons have been
described reasonably Zhong:2007gp ; Zhong:2010vq ; Zhong:2009sk ; Zhong:2008kd
.
## IV RESULTS AND DISCUSSIONS
### IV.1 $\Xi_{c}^{*}(2645)$
$\Xi^{*}_{c}(2645)$ and $\Xi^{\prime}_{c}$ are the two lowest states in the
$\Xi^{\prime}_{c}$-type charm-strange baryons. They are assigned to the two
$S$-wave states, $|\Xi^{\prime}_{c}\ {}^{2}S\frac{1}{2}^{+}\rangle$ and
$|\Xi^{\prime}_{c}\ {}^{4}S\frac{3}{2}^{+}\rangle$, respectively. The decay
widths of $\Xi^{*}_{c}(2645)\to\Xi_{c}\pi$ are calculated. The results are
listed in Tab. 6, from which we find that our predictions are in good
agreement with the experimental data Nakamura:2010zzi , and compatible with
other theoretical predictions Chen:2007xf ; Ivanov:1999bk ; Aliev:2010ev ;
Albertus:2005zy ; Tawfiq:1998nk ; Cheng:2006dk .
Table 6: The decay widths (MeV) of the well-established charm-strange baryons $\Xi_{c}^{*}(2645)$, $\Xi_{c}(2790)$ and $\Xi_{c}(2815)$. | Notation | Channel | $\Gamma$(ours) | $\Gamma_{total}$(ours) | $\Gamma_{total}$Chen:2007xf | $\Gamma_{total}$Ivanov:1999bk | $\Gamma_{total}$Aliev:2010ev | $\Gamma_{total}$Albertus:2005zy | $\Gamma_{total}$Tawfiq:1998nk | $\Gamma_{total}$Cheng:2006dk | $\Gamma_{total}^{exp}$
---|---|---|---|---|---|---|---|---|---|---|---
$\Xi_{c}^{*}(2645)^{0}$ | $|\Xi_{c}^{\prime}\ {}^{4}S\frac{3}{2}^{+}\rangle$ | $\Xi_{c}^{+}\pi^{-}$ | 1.55 | 2.34 | 1.08 | $3.12\pm 0.44$ | $4.2\pm 1.3$ | $3.03\pm 0.1$ | 1.88 | $2.8\pm 0.2$ | $<5.5$
| | $\Xi_{c}^{0}\pi^{0}$ | 0.79 | | | | | | | |
$\Xi_{c}^{*}(2645)^{+}$ | $|\Xi_{c}^{\prime}\ {}^{4}S\frac{3}{2}^{+}\rangle$ | $\Xi_{c}^{+}\pi^{0}$ | 0.89 | 2.44 | 1.13 | $3.04\pm 0.50$ | $4.2\pm 1.3$ | $3.18\pm 0.1$ | 1.81 | $2.7\pm 0.2$ | $<3.1$
| | $\Xi_{c}^{0}\pi^{+}$ | 1.55 | | | | | | | |
$\Xi_{c}(2790)^{+}$ | $|\Xi_{c}\ ^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ | $\Xi_{c}^{{}^{\prime}+}\pi^{0}$ | 0.92 | 2.72 | 9.9 | | | | | $8.0^{+4.7}_{-3.3}$ | $<15$
| | $\Xi_{c}^{{}^{\prime}0}\pi^{+}$ | 1.80 | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{0}\pi^{+}$ | $4\times 10^{-5}$ | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{+}\pi^{0}$ | $2\times 10^{-5}$ | | | | | | | |
$\Xi_{c}(2790)^{0}$ | $|\Xi_{c}\ ^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ | $\Xi_{c}^{{}^{\prime}+}\pi^{-}$ | 1.84 | 2.77 | 10.3 | | | | | $8.5^{+5.0}_{-3.5}$ | $<12$
| | $\Xi_{c}^{{}^{\prime}0}\pi^{0}$ | 0.93 | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{+}\pi^{-}$ | $4\times 10^{-5}$ | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{0}\pi^{0}$ | $2\times 10^{-5}$ | | | | | | | |
$\Xi_{c}(2815)^{+}$ | $|\Xi_{c}\ ^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$ | $\Xi_{c}^{{}^{\prime}0}\pi^{+}$ | 0.15 | 1.50 | 5.3 | $1.26\pm 0.13$ | | | 7.67 | $3.4^{+2.0}_{-1.4}$ | $<3.5$
| | $\Xi_{c}^{{}^{\prime}+}\pi^{0}$ | 0.08 | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{0}\pi^{+}$ | 0.83 | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{+}\pi^{0}$ | 0.44 | | | | | | | |
$\Xi_{c}(2815)^{0}$ | $|\Xi_{c}\ ^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$ | $\Xi_{c}^{{}^{\prime}+}\pi^{-}$ | 0.17 | 1.64 | 5.5 | | | | | $3.6^{+2.1}_{-1.5}$ | $<6.5$
| | $\Xi_{c}^{{}^{\prime}0}\pi^{0}$ | 0.09 | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{+}\pi^{-}$ | 0.87 | | | | | | | |
| | $\Xi_{c}^{*}(2645)^{0}\pi^{0}$ | 0.51 | | | | | | | |
$\Sigma_{b}$ and $\Sigma_{b}^{*}$ ($\Xi_{b}^{\prime}$ and $\Xi_{b}^{*}$) are
counterparts of $\Xi^{\prime}_{c}$ and $\Xi^{*}_{c}(2645)$, respectively.
Recently, the improved measurements of the masses and first measurements of
natural widths of the bottom baryon states $\Sigma_{b}^{\pm}$ and
$\Sigma_{b}^{*\pm}$ were reported by CDF Collaboration CDF:2011ac , and a new
neutral excited bottom-strange baryon with a mass $m=5945.0\pm 0.7\pm 0.3\pm
2.7$ MeV was observed by CMS Collaboration CMS . Given the measured mass and
decay mode of the newly observed bottom-strange baryon, this state most likely
corresponds to $\Xi_{b}^{*0}$ with $J^{P}=3/2^{+}$. As a by-product, we have
calculated the strong decays of the bottom baryons $\Sigma_{b}^{\pm}$,
$\Sigma_{b}^{*\pm}$ and $\Xi_{b}^{*0}$. Our results together with other model
predictions and experimental data have been listed in Tab. 7. From the table,
it is seen that our predictions are in good agreement with the measurements
CDF:2011ac ; CMS and the other model predictions Chen:2007xf ;
Hernandez:2011tx ; Hwang:2006df ; Guo:2007qu ; Limphirat:2010zz . It should be
pointed out that the strong decay properties of $\Xi_{b}^{*}$ were studied in
Limphirat:2010zz ; Chen:2007xf , where a little large mass $m\simeq 5960$ MeV
was adopted. With the recent measured mass of $\Xi_{b}^{*0}$, the predicted
decay widths in Limphirat:2010zz ; Chen:2007xf should be a little smaller
than their previous predictions.
Table 7: The decay widths (MeV) of the ground $S$-wave bottom baryons $\Sigma_{b}^{\pm}$, $\Sigma_{b}^{*\pm}$ and newly observed $\Xi_{b}^{*}(5945)^{0}$. | Notation | Channel | $\Gamma$(ours) | $\Gamma$Chen:2007xf | $\Gamma$Hernandez:2011tx | $\Gamma$Hwang:2006df | $\Gamma$Guo:2007qu | $\Gamma$Limphirat:2010zz | $\Gamma_{exp}$
---|---|---|---|---|---|---|---|---|---
$\Sigma_{b}(5811)^{+}$ | $|\Sigma_{b}\ ^{2}S\frac{1}{2}^{+}\rangle$ | $\Lambda_{b}^{0}\pi^{+}$ | 5.9 | 3.5 | 6.0 | 4.35 | 6.73–13.45 | 4.82,4.94 | $9.7^{+5.0}_{-3.9}$ CDF:2011ac
$\Sigma_{b}(5816)^{-}$ | $|\Sigma_{b}\ ^{2}S\frac{1}{2}^{+}\rangle$ | $\Lambda_{b}^{0}\pi^{-}$ | 6.7 | 4.7 | 7.7 | 5.77 | 6.73–13.45 | 6.31,6.49 | $4.9^{+4.2}_{-3.2}$ CDF:2011ac
$\Sigma_{b}^{*}(5832)^{+}$ | $|\Sigma_{b}\ ^{4}S\frac{3}{2}^{+}\rangle$ | $\Lambda_{b}^{0}\pi^{+}$ | 10.2 | 7.5 | 11.0 | 8.50 | 10.00–17.74 | 9.68,10.06 | $11.5^{+3.7}_{-3.7}$ CDF:2011ac
$\Sigma_{b}^{*}(5835)^{-}$ | $|\Sigma_{b}\ ^{4}S\frac{3}{2}^{+}\rangle$ | $\Lambda_{b}^{0}\pi^{-}$ | 11.0 | 9.2 | 13.2 | 10.44 | 10.00–17.74 | 11.81,12.34 | $7.5^{+3.1}_{-3.2}$ CDF:2011ac
$\Xi_{b}^{*}(5945)^{0}$ | $|\Xi_{b}^{\prime}\ {}^{4}S\frac{3}{2}^{+}\rangle$ | $\Xi_{b}\pi$ | $0.6$ | 0.85 | … | … | … | 1.83,1.85 | $2.1\pm 1.7$ CMS
### IV.2 $\Xi_{c}(2790)$ and $\Xi_{c}(2815)$
$\Xi_{c}(2790)$ and $\Xi_{c}(2815)$ are two relatively well-determined
$P$-wave charm-strange baryons with quantum numbers $J^{P}=1/2^{-}$ and
$3/2^{-}$, respectively. They were observed in the $\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}\pi\pi$ channels, respectively. The Particle Dada Group suggests they
belong to the same SU(4) multiplet as $\Lambda_{c}(2593)$ and
$\Lambda_{c}(2625)$, respectively Nakamura:2010zzi . According to our previous
study, $\Lambda_{c}(2593)$ and $\Lambda_{c}(2625)$ can be well explained with
the $|\Lambda_{c}\ ^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ and $|\Lambda_{c}\
^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$ assignments Zhong:2007gp . Thus,
$\Xi_{c}(2790)$ and $\Xi_{c}(2815)$ should correspond to the $\Xi_{c}$-type
excited states $|\Xi_{c}\ ^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ and
$|\Xi_{c}\ ^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$, respectively. With these
assignments we have calculated the strong decay properties of $\Xi_{c}(2790)$
and $\Xi_{c}(2815)$, which are listed in Tab 6. Our predicted widths are in
the range of observations Nakamura:2010zzi and compatible with other
theoretical predictions Chen:2007xf ; Cheng:2006dk . On the other hand,
$\Xi_{c}(2790)$ as a dynamically generated resonance having $J^{P}=1/2^{-}$
was also discussed in JimenezTejero:2009vq .
Finally it should be pointed out that $\Xi_{c}(2790)$ and $\Xi_{c}(2815)$ can
not be $P_{\rho}$-mode excited states $|\Xi_{c}\
^{2}P_{\rho}\frac{1}{2}^{-}\rangle$ and $|\Xi_{c}\
^{2}P_{\rho}\frac{3}{2}^{-}\rangle$, because these excitations have large
partial decay widths into $\Xi_{c}\pi$ and $\Lambda_{c}^{+}\bar{K}$ channels
(see Fig. 1), which disagrees with the observations. The strong decay
properties of the $P_{\rho}$-mode excited states have been shown in Fig. 1. We
advise experimentalists to search these missing $P$-wave states in
$\Xi_{c}\pi$, $\Lambda_{c}^{+}\bar{K}$ and $\Xi_{c}^{*}(2645)\pi$ invariant
mass distributions around the energy region $(2.8\sim 2.9)$ GeV.
### IV.3 $\Xi_{c}(2930)$
$\Xi_{c}(2930)$ is not well-established. It was only seen by BaBar in the
$\Lambda_{c}^{+}\bar{K}$ invariant mass distribution in an analysis of
$B^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}K^{-}$. The mass analysis
of the charm-strange baryon spectrum indicates that $\Xi_{c}(2930)$ can be
assigned to the first orbital ($1P$) excitation of $\Xi_{c}^{\prime}$ or the
first radial ($2S$) excitation of $\Xi_{c}$ (see Tab. 3) Ebert:2011kk ;
Ebert:2007nw .
Figure 1: (Color online) The strong decay properties of the $P_{\rho}$-mode
excitations of $\Xi_{c}$.
We have analyze the strong decay properties of all the first $P$-wave
excitations of $\Xi_{c}^{\prime}$ and the first radial ($2S$) excitations of
$\Xi_{c}$, which have been shown in Fig. 2 and 3.
Firstly, we can exclude the first radial ($2S$) excitations of $\Xi_{c}$ as
assignments to $\Xi_{c}(2930)$ for the decay channel $\Lambda_{c}^{+}\bar{K}$
of these states is forbidden (see Fig. 3).
In the first $P$-wave excitations of $\Xi_{c}^{\prime}$, we have noted that
the decay modes $\Lambda_{c}^{+}\bar{K}$ and $\Xi_{c}\pi$ for the
$P_{\rho}$-mode excited states, ${}^{2}P_{\rho}(1/2^{-})$ and
${}^{2}P_{\rho}(3/2^{-})$, are forbidden, thus, these states as assignments to
$\Xi_{c}(2930)$ should be excluded. Furthermore, it is found that the strong
decays of ${}^{2}P_{\lambda}(3/2^{-})$ and ${}^{4}P_{\lambda}(5/2^{-})$ are
governed by the $\Xi_{c}\pi$ channel, and the $\Xi^{*}_{c}(2645)\pi$ decay
mode dominates the decay of ${}^{4}P_{\lambda}(3/2^{-})$. They might be hard
observed by BaBar for their small $\Lambda_{c}^{+}\bar{K}$ branching ratios.
Given the decay modes and decay widths, two $J^{P}=1/2^{-}$ states
${}^{4}P_{\lambda}(1/2^{-})$ and ${}^{2}P_{\lambda}(1/2^{-})$ seem to be the
possible assignments to $\Xi_{c}(2930)$. Considering $\Xi_{c}(2930)$ as the
${}^{2}P_{\lambda}(1/2^{-})$, from the figure we find that its decays are
dominated by $\Lambda_{c}^{+}\bar{K}$ and $\Xi^{\prime}_{c}\pi$, and the other
partial decay widths are negligibly small. Its total width and the partial
decay width ratio between $\Lambda_{c}^{+}\bar{K}$ and $\Xi^{\prime}_{c}\pi$
are
$\displaystyle\Gamma=10.6\mathrm{MeV},\ \
\frac{\Gamma(\Lambda_{c}^{+}\bar{K})}{\Gamma(\Xi^{\prime}_{c}\pi)}\approx
1.1.$ (11)
And considering $\Xi_{c}(2930)$ as the ${}^{4}P_{\lambda}(1/2^{-})$, we see
that the $\Lambda_{c}^{+}\bar{K}$ governs the decays of $\Xi_{c}(2930)$, and
the other two decay channels $\Xi_{c}\pi$ and $\Xi^{\prime}_{c}\pi$ have
sizeable widths. The calculated total width and partial decay width ratios are
$\displaystyle\Gamma=16.7\mathrm{MeV},\ \
\frac{\Gamma(\Lambda_{c}^{+}\bar{K})}{\Gamma(\Xi_{c}\pi)}\approx 2.8,\ \
\frac{\Gamma(\Lambda_{c}^{+}\bar{K})}{\Gamma(\Xi^{\prime}_{c}\pi)}\approx
4.6.$ (12)
Figure 2: (Color online) The strong decay properties of the first orbital
($1P$) excitations of $\Xi_{c}^{\prime}$.
As a whole, $\Xi_{c}(2930)$ is most likely to be the first orbital ($1P$)
excitation of $\Xi_{c}^{\prime}$ with $J^{P}=1/2^{-}$, favors
$|\Xi_{c}^{\prime}\ {}^{4}P_{\lambda}1/2^{-}\rangle$ or $|\Xi_{c}^{\prime}\
{}^{2}P_{\lambda}1/2^{-}\rangle$. To confirm $\Xi_{c}(2930)$ and finally
classify it, further observations in the $\Xi^{\prime}_{c}\pi$, $\Xi_{c}\pi$,
$\Lambda_{c}^{+}\bar{K}$ invariant mass distributions and measurements of
these partial decay ratios are very crucial in experiments.
### IV.4 $\Xi_{c}(2980)$
$\Xi_{c}(2980)$ with a width of $\sim 40$ MeV was first found by Belle
Collaboration in the $\Lambda_{c}^{+}\bar{K}\pi$ channel, and then confirmed
by BaBar with large significances in the intermediate-resonant
$\Sigma_{c}(2455)\bar{K}$ and nonresonant $\Lambda_{c}^{+}\bar{K}\pi$ decay
channels. Belle also observed a resonance structure around $2.97$ GeV with a
smaller width of $\sim 18$ MeV in the $\Xi_{c}^{*}(2645)\pi$ decay channel in
a separate study Lesiak:2008wz , which is often considered as the same state
of $\Xi_{c}(2980)$. It should be pointed out that BaBar and Belle had analyzed
the $\Lambda_{c}^{+}\bar{K}$ and $\Xi_{c}\pi$ invariant mass distributions,
respectively, but they did not find any structures around 2.98 GeV, which
indicates that these partial decay width are too small to be observed or these
decay modes are forbidden. Although $\Xi_{c}(2980)$ is well-established in
experiments, its quantum numbers are still unknown. Recently, Ebert _et al._
calculated the mass spectra of heavy baryons in the heavy-quark-light-diquark
picture in the framework of the QCD-motivated relativistic quark model, they
suggested $\Xi_{c}(2980)$ could be assigned to the first radial ($2S$)
excitation of $\Xi_{c}^{\prime}$ with $J^{P}=1/2^{+}$ Ebert:2011kk , which
also consists with their early mass analysis Ebert:2007nw . Cheng _et al._
also discussed the possible classification of $\Xi_{c}(2980)$. They considered
that $\Xi_{c}(2980)$ might be the first radial ($2S$) excitation of $\Xi_{c}$
with $J^{P}=1/2^{+}$ Cheng:2006dk .
Figure 3: (Color online) The strong decay properties of the $L=0$ excitations
of $\Xi_{c}$. Figure 4: (Color online) The strong decay properties of the
first radial ($2S$) excitations of $\Xi_{c}^{\prime}$.
We have analyzed the strong decay properties of the first radial ($2S$)
excitations of both $\Xi_{c}$ and $\Xi_{c}^{\prime}$, which have been shown in
Figs. 3 and 4, respectively. From the figures, it is seen that the $2S$
excitations of both $\Xi_{c}$ and $\Xi^{\prime}_{c}$ have narrow decay widths
($<2$ MeV), which are at least an order smaller than those of $\Xi_{c}(2980)$.
Furthermore, the decay modes of these first radial excitations are in
disagreement with the observations of $\Xi_{c}(2980)$. Thus, the $2S$
excitations of both $\Xi_{c}$ and $\Xi^{\prime}_{c}$ are excluded as
assignments to $\Xi_{c}(2980)$ in present work. Our conclusion is in agreement
with that of ${}^{3}P_{0}$ calculations Chen:2007xf .
Figure 5: (Color online) The strong decay properties of the $P_{A}$-mode
excitations of $\Xi_{c}$.
$\Xi_{c}(2980)$ might be one of the $N=2$ shell orbital excitations of
$\Xi_{c}$ since the masses of these states are close to 2980 MeV. Their
calculated partial decay widths and total widths have been shown in Figs. 5
and 6. It is seen that the $P_{A}(1/2^{+},3/2^{+},5/2^{+})$,
$D_{A}(5/2^{+},7/2^{+})$, $D_{\rho\rho}(3/2^{+},5/2^{+})$ and
$D_{\lambda\lambda}(3/2^{+},5/2^{+})$ states have too narrow decay widths to
compare with the observations of $\Xi_{c}(2980)$. Furthermore, although the
decay widths of $D_{A}(1/2^{+},3/2^{+})$ are compatible with the measurement,
their decay modes are dominated by $\Lambda_{c}^{+}\bar{K}$ and $\Xi_{c}\pi$,
which disagrees with the observations as well. As a whole, all the states
shown in Figs. 5 and 6 are not good assignments to $\Xi_{c}(2980)$ either
their decay widths are too narrow to compare with the observations or their
decay modes disagree with the observations.
The $P_{\rho}$-mode states, ${}^{2}P_{\rho}(1/2^{-})$ and
${}^{2}P_{\rho}(3/2^{-})$, in the first $P$-wave excitations of
$\Xi_{c}^{\prime}$ could be candidates of $\Xi_{c}(2980)$ (see Fig. 2). We
have noted that excitation of the $\lambda$ variable unlike excitation in
$\rho$ involves the excitation of the “odd” heavy quark. The $P_{\rho}$-mode
excitation of charm-strange baryon is $\sim 70$ MeV heavier than the
$P_{\lambda}$-mode Narodetskii:2008pn ; Grach:2008ij . According to our
analysis in IV.3, $\Xi_{c}(2930)$ might be assigned to a $P_{\lambda}$-mode
excitation of $\Xi_{c}^{\prime}$. Thus, the expected mass of the
$P_{\rho}$-mode excitation is $\sim 3.0$ GeV, which is comparable with that of
$\Xi_{c}(2980)$. As the ${}^{2}P_{\rho}(1/2^{-})$ and
${}^{2}P_{\rho}(3/2^{-})$ candidates, respectively, the partial decay widths
and total width of $\Xi_{c}(2980)$ have been listed in Tab. 8.
If the resonance structure around $2.97$ GeV in the $\Xi_{c}^{*}(2645)\pi$
decay channel is the same state, $\Xi_{c}(2980)$, observed in
$\Lambda_{c}^{+}\bar{K}\pi$ decay channel, $\Xi_{c}(2980)$ is most likely to
be the $J^{P}=1/2^{-}$ excited state ${}^{2}P_{\rho}(1/2^{-})$. The reasons
are as follows. (i) The decay modes of ${}^{2}P_{\rho}(1/2^{-})$ are in
agreement with the observations. From Tab. 8, we see that the strong decays of
${}^{2}P_{\rho}(1/2^{-})$ are dominated by $\Sigma_{c}\bar{K}$, and the
partial decay width of $\Xi^{*}_{c}(2645)\pi$ is sizeable as well. The
$\Lambda_{c}^{+}\bar{K}\pi$ final state mainly comes from a intermediate
process in
$\Xi_{c}(2980)\rightarrow\Sigma_{c}\bar{K}\rightarrow\Lambda_{c}^{+}\bar{K}\pi$.
(ii) The total decay width
$\displaystyle\Gamma\simeq 44\ \mathrm{MeV},$ (13)
is in good agreement with the data. (iii) The decay channels $\Xi_{c}\pi$,
$\Lambda_{c}^{+}\bar{K}$ and $\Sigma^{*}_{c}(2520)\bar{K}$ of
${}^{2}P_{\rho}(1/2^{-})$ are forbidden, which can naturally explain why these
decay channels were not observed by Belle and BaBar. It should be mentioned
that the same $J^{P}$ quantum number (i.e. $J^{P}$=$1/2^{-}$) for
$\Xi_{c}(2980)$ is also suggested in JimenezTejero:2009vq , where the
$\Xi_{c}(2980)$ is considered as a dynamically generated resonance.
We have noted that the total width of $\Xi_{c}(2980)$ measured by Belle and
BaBar in the $\Lambda_{c}^{+}\bar{K}\pi$ channel is about two times larger
than that measured by Belle in the $\Xi_{c}^{*}(2645)\pi$ decay channel in a
separate study. Thus, the resonance with a mass $m\simeq 2970$ MeV [denoted by
$\Xi_{c}(2970)$ in this work] observed in the $\Xi_{c}^{*}(2645)\pi$ decay
channel might be a different resonance from the $\Xi_{c}(2980)$ observed in
the $\Lambda_{c}^{+}\bar{K}\pi$ channel, although they have comparable masses.
According to our analysis, the $\Xi_{c}(2970)$ observed in the
$\Xi_{c}^{*}(2645)\pi$ channel and $\Xi_{c}(2980)$ observed in the
$\Lambda_{c}^{+}\bar{K}\pi$ channel might be assigned to the
${}^{2}P_{\rho}(1/2^{-})$ and ${}^{2}P_{\rho}(3/2^{-})$ excitations,
respectively. If the ${}^{2}P_{\rho}(3/2^{-})$ is considered as the
$\Xi_{c}(2970)$ observed in the $\Xi_{c}^{*}(2645)\pi$ channel, its total
decay width
$\displaystyle\Gamma\simeq 16\ \mathrm{MeV},$ (14)
and dominant decay channel $\Xi_{c}^{*}(2645)\pi$ are in good agreement with
the observations (see Tab. 8). Furthermore, it is interestedly found that when
the ${}^{2}P_{\rho}(1/2^{-})$ and ${}^{2}P_{\rho}(3/2^{-})$ excitations are
considered as the resonances observed in the $\Lambda_{c}^{+}\bar{K}\pi$ and
$\Xi_{c}^{*}(2645)\pi$, respectively, we can naturally explain why the width
measured in the $\Xi_{c}(2645)^{*}\pi$ channel is about a factor 2 smaller
than that measured in the $\Lambda_{c}^{+}\bar{K}\pi$ channel.
In brief, the $\Xi_{c}(2970)$ observed in the $\Xi_{c}^{*}(2645)\pi$ final
state is most likely a different state from the $\Xi_{c}(2980)$ observed in
the $\Lambda_{c}^{+}\bar{K}\pi$ final state. The $\Xi_{c}(2980)$ and
$\Xi_{c}(2970)$, as two largely overlapping resonances, favor to be classified
as the $|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}1/2^{-}\rangle$ and
$|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}3/2^{-}\rangle$, respectively. Of course,
for the uncertainties of the data we can not exclude the $\Xi_{c}(2970)$ and
$\Xi_{c}(2980)$ as the same resonance, which favors to be assigned to the
$|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}1/2^{-}\rangle$. To finally clarify whether
the $\Xi_{c}(2970)$ observed in $\Xi_{c}^{*}(2645)\pi$ is the same resonance
observed in the $\Lambda_{c}^{+}\bar{K}\pi$ channel or not, we expect to
measure the partial width ratio
$\Gamma[\Xi_{c}^{*}(2645)\pi]:\Gamma(\Sigma_{c}\bar{K})$ further. If there is
only one resonance assigned to $|\Xi_{c}^{\prime}\
{}^{2}P_{\rho}1/2^{-}\rangle$, the ratio
$\Gamma[\Xi_{c}^{*}(2645)\pi]:\Gamma(\Sigma_{c}\bar{K})$ might be $\sim 0.08$.
Otherwise, if the $\Xi_{c}(2980)$ and $\Xi_{c}(2970)$ corresponds two
overlapping resonances $|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}1/2^{-}\rangle$ and
$|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}3/2^{-}\rangle$, respectively, the ratio
might be $\Gamma[\Xi_{c}^{*}(2645)\pi]:\Gamma(\Sigma_{c}\bar{K})\simeq 0.41$.
Table 8: The partial decay widths and total width (MeV) for $\Xi_{c}(2980)$ as the ${}^{2}P_{\rho}(1/2^{-})$ and ${}^{2}P_{\rho}(3/2^{-})$ excitations of $\Xi_{c}^{\prime}$, respectively. | $\Sigma_{c}\bar{K}$ | $\Xi^{*}_{c}(2645)\pi$ | $\Xi^{\prime}_{c}\pi$ | total
---|---|---|---|---
${}^{2}P_{\rho}(1/2^{-})$ | 37 | 3 | 4 | 44
${}^{2}P_{\rho}(3/2^{-})$ | 0.1 | 12 | 4 | 16
### IV.5 $\Xi_{c}(3080)$
$\Xi_{c}(3080)^{+}$ and its isospin partner state $\Xi_{c}(3080)^{0}$ were
first observed by Belle in the $\Lambda_{c}^{+}K^{-}\pi^{+}$ and
$\Lambda_{c}^{+}K^{0}\pi^{-}$ final state, respectively. The existence of
$\Xi_{c}(3080)^{+,0}$ has been confirmed by BaBar Collaboration. Furthermore,
BaBar’s analysis shows that most of the decay of $\Xi_{c}(3080)^{+}$ proceeds
through the intermediate resonant modes $\Sigma_{c}(2455)^{++}K^{-}$ and
$\Sigma_{c}(2520)^{++}K^{-}$ with roughly equal branching fractions.
Although $\Xi_{c}(3080)$ has been established in experiments, its quantum is
still unclear. Recently, Ebert _et al._ suggested $\Xi_{c}(3080)$ might be
classified as the second orbital ($1D$) excitations of $\Xi_{c}$ with
$J^{P}=5/2^{+}$ according to their mass calculations in the QCD-motivated
relativistic quark model. Cheng _et al._ discussed the possible classification
of $\Xi_{c}(3080)$ as well. They suggested that $\Xi_{c}(3080)$ might be the
second orbital ($1D$) excitation of $\Xi_{c}$ with $J^{P}=3/2^{+}$ or
$J^{P}=5/2^{+}$. More possible assignments to the $\Xi_{c}(3080)$ were
suggested by Chen _et al._ in their ${}^{3}P_{0}$ strong decay analysis
Chen:2007xf .
BaBar’s observations provide us two very important constraints on the
assignments to $\Xi_{c}(3080)$: (i) the strong decay is governed by both
$\Sigma_{c}(2455)\bar{K}$ and $\Sigma_{c}(2520)\bar{K}$, (ii) and the partial
width ratio
$\Gamma(\Sigma_{c}(2455)\bar{K})/\Gamma(\Sigma_{c}(2520)\bar{K})\simeq 1$. We
analyzed the strong decay properties of all the $N=2$ shell excitations of
both $\Xi_{c}$ and $\Xi^{\prime}_{c}$, which were shown in Figs. 3–8. From the
figures we find that only the $|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$
(i.e., the first radial ($2S$) excitation of $\Xi_{c}$) satisfies the two
constraints of BaBar’s observations at the same time: (i) at $m\simeq 3.08$
GeV the strong decays of $|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$ are
dominated by $\Sigma_{c}(2455)\bar{K}$ and $\Sigma_{c}(2520)\bar{K}$, the
partial other two decay modes $\Xi_{c}^{*}(2645)\pi$ and $\Xi^{\prime}_{c}\pi$
only contribute a very small partial width to the decay, (ii) and the
predicted partial width ratio between $\Sigma_{c}(2455)\bar{K}$ and
$\Sigma_{c}(2520)\bar{K}$ is
$\displaystyle\frac{\Gamma[\Sigma_{c}(2455)\bar{K}]}{\Gamma[\Sigma_{c}(2520)\bar{K}]}\simeq
0.8.$ (15)
Furthermore, if the $|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$ is considered
as an assignment to $\Xi_{c}(3080)$, the predicted total width
$\displaystyle\Gamma\simeq 4\ \mathrm{MeV}$ (16)
is also in good agreement with the measurements.
Finally, it should be point out that as a candidate of $\Xi_{c}(3080)$, the
mass of $|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$ consists with the quark
model expectations as well. According to our analysis in Sec. 8, the
$\Xi_{c}(2980)$ (observed in the $\Lambda_{c}^{+}\bar{K}\pi$ final state) and
$\Xi_{c}(2930)$ could be assigned to $P_{\rho}$ and $P_{\lambda}$-mode
excitations of $\Xi_{c}^{\prime}$, respectively. The estimated mass splitting
between $P_{\rho}$ and $P_{\lambda}$-mode excitation in the $N=1$ shell is
$\displaystyle\Delta
M\simeq\hbar\omega_{\rho}-\hbar\omega_{\lambda}\simeq(2980-2930)\
\mathrm{MeV}=50\ \mathrm{MeV}.$ (17)
With the above relation, we can estimate the mass splitting between
$S_{\rho\rho}$ and $S_{\lambda\lambda}$ excitations in the $N=2$ shell, which
is
$\displaystyle M(S_{\rho\rho})-M(S_{\lambda\lambda})\simeq
2\hbar\omega_{\rho}-2\hbar\omega_{\lambda}\simeq 100\ \mathrm{MeV}.$ (18)
In most of the quark models, the predicted masses for the $S_{\lambda\lambda}$
excitation of $\Xi_{c}$ are in the range of $(2.92\sim 2.99)$ GeV (see Tab.
2), thus, the mass of $\Xi_{c}S_{\rho\rho}$ excitation should be in the range
of $(3.02\sim 3.09)$ GeV, which is comparable with the mass of
$\Xi_{c}(3080)$.
As a whole, the mass, decay modes, partial width ratio
$\Gamma(\Sigma_{c}(2455)\bar{K}):\Gamma(\Sigma_{c}(2520)\bar{K})$ and total
decay width of $|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$ strongly support it
is assigned to $\Xi_{c}(3080)$.
### IV.6 $\Xi_{c}(3055)^{+}$
The $\Xi_{c}(3055)^{+}$ as a new structure was found by BaBar in the
$\Lambda_{c}^{+}\bar{K}\pi$ mass distribution with a statistical significance
of 6.4$\sigma$. It decays through the intermediate resonant mode
$\Sigma_{c}(2455)^{++}K^{-}$. BaBar also searched the inclusive
$\Lambda_{c}^{+}\bar{K}$ and $\Lambda_{c}^{+}\bar{K}\pi\pi$ invariant mass
spectra for evidence of $\Xi_{c}(3055)^{+}$, but no significant structure was
found. This state has not yet been confirmed by Belle. According to the
calculations of the charm-strange baryon spectrum in various quark models,
$\Xi_{c}(3055)$ might be assigned to the second orbital ($1D$) excitation of
$\Xi_{c}$ (see Tab. 2).
We have analyzed the strong decay properties of the second orbital excitations
of $\Xi_{c}$, which have been shown in Figs. 5 and 6. From Fig. 5, we find
that the $P_{A}(1/2^{+},3/2,5/2^{+})$ excitations can be firstly excluded as
the candidates of $\Xi_{c}(3055)^{+}$ for neither their decay modes nor their
decay widths consist with the observations. Furthermore, from Fig. 6 it is
seen that the $\Lambda_{c}^{+}\bar{K}$ is one of the main decay modes of
${}^{4}D_{A}(1/2^{+},3/2,7/2^{+})$ and ${}^{2}D_{A}(3/2^{+},5/2^{+})$, if the
$\Sigma_{c}(2455)^{++}K^{-}$ decay mode for these states is observed in
experiments, the $\Lambda_{c}^{+}\bar{K}$ decay mode should be observed as
well, which disagrees with the observations of BaBar. Thus, these states as
assignments to $\Xi_{c}(3055)^{+}$ should be excluded. The strong decays of
${}^{4}D_{A}(5/2^{+})$, ${}^{2}D_{\lambda\lambda}(5/2^{+})$ and
${}^{2}D_{\rho\rho}(5/2^{+})$ are dominated by $\Xi_{c}^{*}(2645)\pi$ and
$\Sigma_{c}(2520)\bar{K}$, the partial width of $\Sigma_{c}(2455)\bar{K}$ is
negligibly small, thus, these states can not be considered as candidates of
$\Xi_{c}(3055)^{+}$ as well.
Figure 6: (Color online) The strong decay properties of the second orbital
($1D$) excitations of $\Xi_{c}$. Some decay channels, such as $\Xi_{c}\eta$,
$\Xi_{c}(2790,2815)\pi$ are not shown in the figure for their small partial
decay widths. Figure 7: (Color online) The strong decay properties of the
$l_{\lambda}=l_{\rho}=1$ excitations of $\Xi_{c}^{\prime}$.
Finally, we find that only two $J^{P}=3/2^{+}$ states $|\Xi_{c}\
^{2}D_{\lambda\lambda}3/2^{+}\rangle$ and $|\Xi_{c}\
^{2}D_{\rho\rho}3/2^{+}\rangle$, might be candidates of the $\Xi_{c}(3055)$.
The partial decay widths and total width of $\Xi_{c}(3055)$ as the $|\Xi_{c}\
^{2}D_{\lambda\lambda}3/2^{+}\rangle$ and $|\Xi_{c}\
^{2}D_{\rho\rho}3/2^{+}\rangle$ candidates have been listed in Tab. 9,
respectively. From the table it is seen that the total widths of both states
are compatible with the observations of $\Xi_{c}(3055)$ within its
uncertainties. The strong decays of both states are dominated by
$\Sigma_{c}(2455)\bar{K}$ and the partial width of $\Sigma_{c}(2520)\bar{K}$
is negligibly small, which can explain why BaBar only observed the
intermediate resonant decay mode $\Sigma_{c}(2455)^{++}K^{-}$ for
$\Xi_{c}(3055)$. The $\Lambda_{c}^{+}\bar{K}$ decay mode is forbidden for both
$|\Xi_{c}\ ^{2}D_{\lambda\lambda}3/2^{+}\rangle$ and $|\Xi_{c}\
^{2}D_{\rho\rho}3/2^{+}\rangle$, which agrees with the observation that no
structures were found around $M(\Lambda_{c}^{+}\bar{K})\simeq 3.05$ GeV. As a
whole, $\Xi_{c}(3055)$ could be assigned to the second orbital ($1D$)
excitations of $\Xi_{c}$ with $J^{P}=3/2^{+}$, our conclusion is in agreement
with that of Ebert _et al._ according to their mass analysis. However, it is
difficult to determine which one can be assigned to $\Xi_{c}(3055)^{+}$ in the
$|\Xi_{c}\ ^{2}D_{\lambda\lambda}3/2^{+}\rangle$ and $|\Xi_{c}\
^{2}D_{\rho\rho}3/2^{+}\rangle$ candidates only according to the strong decay
properties. We have noted that $\Xi_{c}(3080)$ is most likely to be the
$\Xi_{c}S_{\rho\rho}$ assignment. According to various quark model
predictions, the mass of the second orbital excitation $\Xi_{c}D_{\rho\rho}$
should be larger than that of the first radial excitation
$\Xi_{c}S_{\rho\rho}$, which indicates that the mass of $\Xi_{c}D_{\rho\rho}$
might be larger than 3.08 GeV. From this point of view, the $|\Xi_{c}\
^{2}D_{\rho\rho}3/2^{+}\rangle$ as an assignments to $\Xi_{c}(3055)^{+}$
should be excluded. Thus, the $\Xi_{c}(3055)$ is most likely to be classified
as the $|\Xi_{c}\ ^{2}D_{\lambda\lambda}3/2^{+}\rangle$ excitation.
Table 9: The partial decay widths and total width (MeV) for $\Xi_{c}(3055)$ as the ${}^{2}D_{\lambda\lambda}(3/2^{+})$ and ${}^{2}D_{\rho\rho}(3/2^{+})$ excitations of $\Xi_{c}$, respectively. | $\Sigma_{c}\bar{K}$ | $\Xi^{*}_{c}(2645)\pi$ | $\Xi^{\prime}_{c}\pi$ | $\Sigma_{c}^{*}\bar{K}$ | $D\Lambda$ | total
---|---|---|---|---|---|---
${}^{2}D_{\lambda\lambda}(3/2^{+})$ | 2.3 | 0.5 | 1.0 | 0.1 | 0.1 | 4.0
${}^{2}D_{\rho\rho}(3/2^{+})$ | 5.6 | 0.8 | 3.3 | 0.3 | – | 10.0
### IV.7 $\Xi_{c}(3123)^{+}$
$\Xi_{c}(3123)^{+}$ is another new narrow structure was observed by BaBar in
the $\Lambda_{c}^{+}\bar{K}\pi$ final state only with week statistical
significance $3.0\sigma$. It decays through a intermediate resonant process in
$\Xi_{c}(3123)^{+}\rightarrow\Sigma_{c}(2520)^{++}K^{-}\rightarrow\Lambda_{c}^{+}K^{-}\pi^{+}$.
BaBar also searched $\Xi_{c}(3123)^{+}$ in the $\Lambda_{c}^{+}\bar{K}$ and
$\Lambda_{c}^{+}\bar{K}\pi\pi$ final states further, however, they did not
find any evidence in these channels. $\Xi_{c}(3123)^{+}$ has not yet been
confirmed by Belle.
From Tab. 2, it is seen that the predicted masses of the second orbital ($1D$)
excitations of $\Xi_{c}^{\prime}$ in various quark models are $(3.12\sim
3.17)$ GeV. Thus, the $1D$ excitations of $\Xi_{c}^{\prime}$ might be
candidates of $\Xi_{c}(3123)^{+}$. We have analyzed the strong decay
properties of these excitations, which have been shown in Fig. 8.
In these $D$-wave states, we can first excluded the
${}^{2}D_{\rho\rho}(3/2^{+})$, ${}^{4}D_{\rho\rho}(1/2^{+})$,
${}^{2}D_{\lambda\lambda}(3/2^{+})$, ${}^{2}D_{\lambda\lambda}(5/2^{+})$,
${}^{4}D_{\lambda\lambda}(1/2^{+})$ and ${}^{4}D_{\lambda\lambda}(7/2^{+})$ as
assignments to $\Xi_{c}(3123)^{+}$ for their partial width of
$\Sigma_{c}(2520)\bar{K}$ is negligibly small compared with that of
$\Sigma_{c}(2455)\bar{K}$ or $\Lambda_{c}^{+}\bar{K}$. Furthermore, we do not
consider the ${}^{2}D_{\rho\rho}(5/2^{+})$, ${}^{4}D_{\rho\rho}(3/2^{+})$ and
${}^{4}D_{\rho\rho}(7/2^{+})$ as good candidates of $\Xi_{c}(3123)^{+}$
although the $\Sigma_{c}(2520)\bar{K}$ is their dominant decay channel. The
reason is that the $\Lambda_{c}^{+}\bar{K}$ has a large partial width which
should be observed by BaBar, however, this decay mode was not observed yet.
Finally, only three excitations ${}^{4}D_{\lambda\lambda}(3/2^{+})$,
${}^{4}D_{\lambda\lambda}(5/2^{+})$ and ${}^{4}D_{\rho\rho}(5/2^{+})$ might be
candidates of $\Xi_{c}(3123)^{+}$. They decay mainly through
$\Sigma_{c}(2520)\bar{K}$ with a narrow decay width, which is consistent with
the observations of $\Xi_{c}(3123)^{+}$. To clearly see the decay properties
of ${}^{4}D_{\lambda\lambda}(3/2^{+})$, ${}^{4}D_{\lambda\lambda}(5/2^{+})$
and ${}^{4}D_{\rho\rho}(5/2^{+})$, as candidates of $\Xi_{c}(3123)^{+}$ their
partial decay widths and total width have been listed in Tab. 10.
According to our analysis in Sec. 9, $\Xi_{c}(3055)$ is most likely to be the
$|\Xi_{c}\ ^{2}D_{\lambda\lambda}3/2^{+}\rangle$ excitation. We have noted
that the quark model predicted mass of $\Xi_{c}^{\prime}D_{\lambda\lambda}$ is
typically $\sim 100$ MeV heavier than that of $\Xi_{c}D_{\lambda\lambda}$.
Thus, when the $|\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}3/2^{+}\rangle$ or
$|\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}5/2^{+}\rangle$ excitation is
assigned to the $\Xi_{c}(3123)$, the quark model predicted mass $\sim 3.15$
GeV is compatible with the observation. With the relation of
$(\hbar\omega_{\rho}-\hbar\omega_{\lambda})\simeq 50$ MeV in Eq.(17), we can
further estimate the mass splitting between $D_{\rho\rho}$ and
$D_{\lambda\lambda}$ excitations in the $N=2$ shell, which is
$\displaystyle M(D_{\rho\rho})-M(D_{\lambda\lambda})\simeq
2\hbar\omega_{\rho}-2\hbar\omega_{\lambda}\simeq 100\ \mathrm{MeV}.$ (19)
Thus, the estimated mass of $|\Xi_{c}^{\prime}\
{}^{4}D_{\rho\rho}5/2^{+}\rangle$ is $\sim 3.25$ GeV. Obviously, the
$|\Xi_{c}^{\prime}\ {}^{4}D_{\rho\rho}5/2^{+}\rangle$ could not be considered
as a good assignment to $\Xi_{c}(3123)$ for its mass is too heavy to compare
with the measurement.
Figure 8: (Color online) The strong decay properties of the second orbital
($1D$) excitations of $\Xi_{c}^{\prime}$. Some decay channels, such as
$\Xi_{c}\eta$, $\Xi_{c}(2790,2815)\pi$ are not shown in the figure for their
too narrow partial decay widths to compare with the others’.
It should be pointed out that in second orbital ($1D$) excitations of
$\Xi_{c}$, the $D_{\rho\rho}$ excitation $|\Xi_{c}\
^{2}D_{\rho\rho}5/2^{+}\rangle$ is also a good assignment to $\Xi_{c}(3123)$.
According to Eq. (19) the mass of the $\rho$ variable excitation
$D_{\rho\rho}$ is $100$ MeV heavier than the $D_{\lambda\lambda}$ excitation.
In Sec. 9, we predicted that $\Xi_{c}(3055)$ is most likely to be the
$|\Xi_{c}\ ^{2}D_{\lambda\lambda}3/2^{+}\rangle$ excitation, thus, the
estimated masses for $|\Xi_{c}\ ^{2}D_{\rho\rho}5/2^{+}\rangle$ might be $\sim
3.15$ GeV, which are close to the mass of $\Xi_{c}(3123)$. Its partial decay
widths and total width have been listed in Tab. 10. From the table it is seen
that both the decay modes and total width of $|\Xi_{c}\
^{2}D_{\rho\rho}5/2^{+}\rangle$ are compatible with the observations of
$\Xi_{c}(3123)$.
As a conclusion, for the scare experimental information, we can not determine
the $J^{P}$ of $\Xi_{c}(3123)$. Given the mass, decay mode and total width
observed in experiment, $\Xi_{c}(3123)$ could be assigned to the excitation
$|\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}3/2^{+}\rangle$,
$|\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}5/2^{+}\rangle$ or $|\Xi_{c}\
^{2}D_{\rho\rho}5/2^{+}\rangle$. Since the $|\Xi_{c}^{\prime}\
{}^{4}D_{\lambda\lambda}3/2^{+}\rangle$, $|\Xi_{c}^{\prime}\
{}^{4}D_{\lambda\lambda}5/2^{+}\rangle$, and $|\Xi_{c}\
^{2}D_{\rho\rho}5/2^{+}\rangle$ have a comparable mass, the $\Xi_{c}(3123)$
structure might correspond to several highly overlapping states around $3.1$
GeV. From Tab. 10, it is seen that the partial decay width ratios
$\Gamma(\Sigma_{c}\bar{K}):\Gamma(\Sigma_{c}^{*}\bar{K})$,
$\Gamma(\Xi_{c}^{*}2645\pi):\Gamma(\Sigma_{c}^{*}\bar{K})$ and
$\Gamma(\Xi_{c}\pi):\Gamma(\Sigma_{c}^{*}\bar{K})$ for these possible
assignments to $\Xi_{c}(3123)$ are very different, thus, the measurements of
these ratios are important to understand the nature of $\Xi_{c}(3123)$.
Table 10: The partial decay widths and total width (MeV) for $\Xi_{c}(3123)$ as the $\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}(3/2^{+})$, $\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}(5/2^{+})$, $\Xi_{c}^{\prime}\ {}^{4}D_{\rho\rho}(5/2^{+})$ and $\Xi_{c}\ ^{2}D_{\rho\rho}(5/2^{+})$ excitations, respectively. | $\Sigma_{c}\bar{K}$ | $\Xi^{*}_{c}(2645)\pi$ | $\Xi_{c}\pi$ | $\Sigma_{c}^{*}\bar{K}$ | $\Lambda_{c}\bar{K}$ | $\Xi^{\prime}_{c}\pi$ | $\Xi_{c}(2815)\pi$ | $\Xi_{c}(2790)\pi$ | $D\Lambda$ | total | $\frac{\Gamma(\Sigma_{c}\bar{K})}{\Gamma(\Sigma_{c}^{*}\bar{K})}$ | $\frac{\Gamma(\Xi^{*}_{c}(2645)\pi)}{\Gamma(\Sigma_{c}^{*}\bar{K})}$ | $\frac{\Gamma(\Xi_{c}\pi)}{\Gamma(\Sigma_{c}^{*}\bar{K})}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}(3/2^{+})$ | 1.2 | 1.0 | 0.9 | 2.6 | 0.9 | 0.4 | 0.9 | 0.4 | 1.2 | 10.5 | 0.46 | 0.38 | 0.35
$\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}(5/2^{+})$ | 0.07 | 2.6 | 1.1 | 2.9 | 0.6 | 0.1 | 0.1 | 0.05 | 0.09 | 7.8 | 0.02 | 0.90 | 0.38
$\Xi_{c}^{\prime}\ {}^{4}D_{\rho\rho}(5/2^{+})$ | 0.1 | 4.3 | 1.5 | 6.3 | 0.7 | 0.2 | 0 | 0 | 0 | 13.0 | 0.01 | 0.68 | 0.24
$\Xi_{c}\ ^{2}D_{\rho\rho}(5/2^{+})$ | 0.8 | 4.5 | 0 | 4.8 | 0 | 1.5 | 0 | 0 | 0 | 11.6 | 0.17 | 0.94 | 0
## V summary
In the chiral quark model framework, the strong decays of charm-strange
baryons are studied. As a by-product we also calculate the strong decays of
the $S$-wave bottom baryons $\Sigma_{b}^{\pm}$, $\Sigma_{b}^{*\pm}$,
$\Xi_{b}^{\prime}$ and $\Xi_{b}^{*}$. We obtain good descriptions of the
strong decay properties of the well-determined charm-strange baryons
$\Xi^{*}(2645)$, $\Xi(2790)$ and $\Xi(2815)$. Furthermore, the calculated
strong decay widths of $\Sigma_{b}^{\pm}$, $\Sigma_{b}^{*\pm}$, and
$\Xi_{b}^{*}$ are in good agreement with the recent measurements.
$\Xi_{c}(2930)$, if it could be confirmed in experiments, might be the first
$P$-wave excitations of $\Xi_{c}^{\prime}$ with $J^{P}=1/2^{-}$.
$|\Xi_{c}^{\prime}\ {}^{2}P_{\lambda}1/2^{-}\rangle$ and $|\Xi_{c}^{\prime}\
{}^{4}P_{\lambda}1/2^{-}\rangle$ could be candidates of $\Xi_{c}(2930)$
according to the present data. Further observations in the
$\Xi^{\prime}_{c}\pi$, $\Xi_{c}\pi$, $\Lambda_{c}^{+}\bar{K}$ invariant mass
distributions and measurements of these partial decay ratios are very crucial
to confirm $\Xi_{c}(2930)$ and classify it finally.
$\Xi_{c}(2980)$ might correspond to two different $P_{\rho}$-mode excitations
of $\Xi_{c}^{\prime}$: one resonance is the broader ($\Gamma\simeq 44$ MeV)
excitation $|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}1/2^{-}\rangle$, which was
observed in the $\Lambda_{c}^{+}\bar{K}\pi$ final state by BaBar and Belle,
and the other resonance is the narrower ($\Gamma\simeq 16$ MeV) excitation
$|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}3/2^{-}\rangle$, which was observed in the
$\Xi_{c}^{*}(2645)\pi$ channel by Belle in a separate study. If the structures
were observed in the $\Lambda_{c}^{+}\bar{K}\pi$ and $\Xi_{c}^{*}(2645)\pi$
final states correspond to the same state $\Xi_{c}(2980)$, which could only be
assigned to the $|\Xi_{c}^{\prime}\ {}^{2}P_{\rho}1/2^{-}\rangle$ excitation.
To finally clarify whether the $\Xi_{c}(2970)$ observed in
$\Xi_{c}^{*}(2645)\pi$ is the same state observed in the
$\Lambda_{c}^{+}\bar{K}\pi$ channel or not, we expect to measure the partial
width ratio $\Gamma[\Xi_{c}^{*}(2645)\pi]:\Gamma(\Sigma_{c}\bar{K})$ further.
Figure 9: (Color online) The charm-strange baryon spectrum up to $N=2$ shell
according to our predictions. In $1P$, $2S$ and $1D$ excitations, there are
two lines for each $J^{P}$ value, which correspond to the masses of the
excitations of $\rho$ variable (upper line) and $\lambda$ variable (lower
line), respectively. The mass gap between the $\lambda$ variable excitation
and the $\rho$ variable excitation is assumed to be $50$ MeV for the $1P$
states, and $100$ MeV for the $2S$ and $1D$ states. The thin lines stand for
the states unobserved in experiments. In the $1P$ ($1D$) excitations, the fist
two $J^{P}$ values are for the excitations of $\Xi_{c}$, while the last two
$J^{P}$ values are for the excitations of $\Xi_{c}^{\prime}$. In $2S$
excitations, the fist $J^{P}$ value is for the excitations of $\Xi_{c}$, while
the second $J^{P}$ value is for the excitations of $\Xi_{c}^{\prime}$.
$\Xi_{c}(3080)$ favors to be identified as the first radial excitation
$|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$. The width, decay modes and ratio
$\Gamma(\Sigma_{c}(2455)\bar{K})/\Gamma(\Sigma_{c}(2520)\bar{K})\simeq 0.8$
are in good agreement with the observations. As a assignment to
$\Xi_{c}(3080)$, the mass of $|\Xi_{c}\ ^{2}S_{\rho\rho}1/2^{+}\rangle$ is
also consistent with the quark model expectations.
Given the mass, decay modes and decay width, $\Xi_{c}(3055)$ is most likely to
be classified as the second orbital $\Xi_{c}$ excitation $|\Xi_{c}\
^{2}D_{\lambda\lambda}3/2^{+}\rangle$. To confirm it in experiments, more
observations in the $\Sigma_{c}\bar{K}$, $\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}^{*}(2645)\pi$ channels are needed.
$\Xi_{c}(3123)$ is most likely to be the second orbital ($1D$) excitations of
the charm-strange baryon with $J^{P}=3/2^{+}$ or $5/2^{+}$. It could be
assigned to the $\Xi_{c}^{\prime}$ excitation $|\Xi_{c}^{\prime}\
{}^{4}D_{\lambda\lambda}3/2^{+}\rangle$ or $|\Xi_{c}^{\prime}\
{}^{4}D_{\lambda\lambda}5/2^{+}\rangle$. For the scare experiment information
about $\Xi_{c}(3123)$, we can not exclude it as the assignment to the
$\Xi_{c}$ excitation $|\Xi_{c}\ ^{2}D_{\rho\rho}5/2^{+}\rangle$. Since the
$|\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}3/2^{+}\rangle$,
$|\Xi_{c}^{\prime}\ {}^{4}D_{\lambda\lambda}5/2^{+}\rangle$ and $|\Xi_{c}\
^{2}D_{\rho\rho}5/2^{+}\rangle$ have a comparable mass, the $\Xi_{c}(3123)$
structure might correspond to several largely overlapping resonances. To good
understand $\Xi_{c}(3123)$ structure, further observations in the
$\Xi^{*}_{c}(2645)\pi$, $\Sigma_{c}^{*}\bar{K}$ and $\Xi_{c}\pi$ channels are
expected.
Finally, according to our predictions we establish a spectroscopy for the
observed charm-strange baryons, which is shown in Fig. 9. We also estimate the
masses of the charm-strange baryons with different variable ($\lambda$ or
$\rho$) excitation from these newly observed states in experiments, which are
given in Fig. 9. These missing states might be found in future experiments. To
provide helpful information for search for the missing charm-strange baryons,
in Figs. 1–8 our predictions of their strong decay properties have been shown
as well.
## Acknowledgements
This work is supported, in part, by the National Natural Science Foundation of
China (11075051), Program for Changjiang Scholars and Innovative Research Team
in University (IRT0964), the Program Excellent Talent Hunan Normal University,
and the Hunan Provincial Natural Science Foundation (11JJ7001).
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|
arxiv-papers
| 2012-05-14T06:01:17 |
2024-09-04T02:49:30.890726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei-Hua Liu, Li-Ye Xiao and Xian-Hui Zhong",
"submitter": "Xianhui Zhong",
"url": "https://arxiv.org/abs/1205.2943"
}
|
1205.3153
|
# Etch Induced Microwave Losses in Titanium Nitride Superconducting Resonators
Martin Sandberg Electronic addreass: martin.sandberg@nist.gov Michael R.
Vissers Jeffrey S. Kline Martin Weides Current address: Karlsruhe Institute
of Technology, 76131 Karlsruhe, Germany Jiansong Gao National Institute of
Standards and Technology, Boulder, Colorado 80305, USA David S. Wisbey Saint
Louis University, St. Louis, Missouri 63103, USA David P. Pappas Electronic
address: david.pappas@nist.gov National Institute of Standards and Technology,
Boulder, Colorado 80305, USA
###### Abstract
We have investigated the correlation between the microwave loss and patterning
method for coplanar waveguide titanium nitride resonators fabricated on Si
wafers. Three different methods were investigated: fluorine- and chlorine-
based reactive ion etches and an argon-ion mill. At high microwave probe
powers the reactive etched resonators showed low internal loss, whereas the
ion-milled samples showed dramatically higher loss. At single-photon powers we
found that the fluorine-etched resonators exhibited substantially lower loss
than the chlorine-etched ones. We interpret the results by use of numerically
calculated filling factors and find that the silicon surface exhibits a higher
loss when chlorine-etched than when fluorine-etched. We also find from
microscopy that re-deposition of silicon onto the photoresist and side walls
is the probable cause for the high loss observed for the ion-milled
resonators.
Superconducting resonators are essential building blocks of quantum electrical
circuits. They are used for dispersive readout and coupling of superconducting
quantum bits, quantum information storage and for detector detector
applications Wallraff2004 ; Leek2010 ; DiCarlo2010 ; Mariantoni2011 ; Day2003
. This has lead to extensive efforts to understand and minimizing the loss of
these devicesBarends2010 ; Sage2011 ; Wang2009 ; Khalil2011 ; Gao2008a ;
Wenner2011 . One very promising material for building low-loss superconducting
resonators is titanium nitride (TiN), which has been shown to have very low
loss at high as well as at low drive powersLeduc2010 ; Vissers2010 .
Here we present experimental results relating the observed microwave loss in
thin-film TiN resonators to the method of etching used for patterning the
devices. The resonators used were frequency-multiplexed quarter-wavelength
coplanar waveguides (CPW) on a silicon substrate. On each chip, ten resonators
with varying coupling strength and resonance frequencies were coupled to a
common feedlinePappas2011 . The coupling was designed to give an external
quality factor ranging from 0.5 million to 5 million, and a resonance
frequency between 4 GHz and 7 GHz, depending on kinetic inductance.
In this work, three different etches were investigated: a fluorine (F)-based
reactive ion etch (RIE), a chlorine (Cl)-based RIE and an argon ion mill. The
etches were chosen due to their wide use and fundamentally different natures.
The F etch has a relatively low etch rate of TiN compared to its Si etch rate,
the Cl etch has almost identical etch rates for Si and TiN, and the argon-ion
mill is a completely physical etch.
All devices were fabricated on highly resistive intrinsic Si(100) 3” wafers.
The wafers were exposed to a hydrofluoric (HF) vapor etch to remove the native
oxide prior to the film growth. The HF etch also hydrogen-terminates the Si
surface, which has been shown to be crucial for achieving low loss in
resonators on SiWisbey2010 . Within minutes after the HF etch, the wafers were
transferred into a high-vacuum sputtering system. A TiN film (40 nm thick) was
deposited by reactive sputtering at 500 $\tccentigrade$, 250 W DC power, 4 mT
chamber pressure, 15 sccm argon flow, 10 sccm nitrogen flow, and an RF-induced
100 V substrate DC bias. The TiN films were patterned into CPW resonators by
the use of optical lithography and etched by the use of one of the three
different methods. Parameters for the etches are summarized in Table 1.
The wafers were diced into chips and wire-bonded into an aluminium sample box
equipped with microwave launchers. To reduce the risk of trapping magnetic
flux during cool-down, the sample box was placed inside niobium and cryoperm
shields. The devices were then cooled to the base temperature of an adiabatic
demagnetization refrigerator ($\approx$ 50 mK). The microwave line used to
drive the resonators was attenuated by 80 dB and filtered by the use of low-
pass filters with a cut-off frequency of 12 GHz. A high-electron-mobility
transistor (HEMT) placed at the 3 K stage was used to amplify the signal from
the sample. To reduce thermal radiation from the HEMT, a 3.5 GHz high- and a
12 GHz low-pass filter as well as a two stage isolator with an isolation of
$\approx$ 40 dB were placed between the sample and the HEMT. The microwave
response of each device was measured through the scattering parameter $S_{21}$
of the feedline by the use of a vector network analyzer. The internal and
external quality factors ($Q_{int}$ and $Q_{c}$ respectively) were extracted
through a circular fitting procedure Gao2008b of the real and imaginary parts
of the $S_{21}$ response. To maximize the fit accuracy, resonators with
similar $Q_{int}$ and $Q_{c}$ were chosen.
Measurements of six resonators, labelled F1, and F2 for the F-etched, Cl1,
Cl2, and Cl3 for the Cl-etched, and IM for the ion-milled, respectively, are
presented here. The parameters of the resonators are summarized in Table 2.
The extracted internal loss ($\tan(\delta_{int})=1/Q_{int}$) is plotted in
Figure 1 as a function of the internal resonator voltage. It is clear that the
ion-milled resonator, IM, shows substantially higher internal loss compared to
the RIE etched resonators. Among the RIE treated resonators it is also clear
that F etched resonators have the lowest internal loss.
The power dependence of the internal loss results from two level systems
(TLSs). There are three different surfaces where the TLSs are likely to be
located Gao2008a : at the substrate-vacuum (S-V) interface in the CPW gap, the
conductor-vacuum (C-V) interface, and the conductor-substrate (C-S) interface.
The different surfaces are depicted in the inset of Fig. 1 (a). The loss
contribution $\delta_{V}$, from a volume $V$ with dielectric constant
$\epsilon_{V}$, is obtained asKhalil2011 :
$\tan(\delta_{V})=\tanh(\frac{\hbar\omega_{r}}{2k_{B}T})\frac{\tan(\delta_{V}^{0})\int_{V}\epsilon_{V}|E(\vec{r})|^{2}\left(1+\left(\frac{E(\vec{r})}{E_{c}}\right)^{2}\right)^{-1/2}\mathrm{d}v}{\int_{V_{tot}}\epsilon(\vec{r})|E(\vec{r})|^{2}\mathrm{d}v}$
(1)
where $\vec{r}$ is the position inside the volume under integration and
$V_{tot}$ is the total volume. Here $E_{c}$ is the saturation electrical field
of the TLSs and $E(\vec{r})$ is the local electric field strength,
$\delta_{V}^{0}$ is the loss at small electric field and low temperature, and
$\omega_{r}=2\pi f_{r}$ is the (angular) resonance frequency. The device
temperature $T$ was found to be low enough that thermal excitation of the TLSs
was negligible, i.e., $T<\hbar\omega_{r}/2k_{B}$.
To fit Eqn. 1 to the measured loss, the electric field $E(\vec{r})$ was
numerically calculated on a cross-section of the resonators by the use of a
finite-element (FEM) solver. The actual CPW profile was obtained through
microscopy on neighboring devices cross sections, see Figure 2. The sinusoidal
voltage dependence along the length of the resonator is also considered when
fitting Eqn. 1.
In the calculations we used the following assumptions: fist, that the
conductor-substrate (C-S) interface consists of a 2 nm thick SiNx layer
(confirmed from pre-sputtering ellipsometer measurements(Vissers2010, )) with
a relative dielectric constant $\epsilon_{r}=$7.6; second, that the substrate-
vacuum (S-V) interface is a 3 nm thick layer with $\epsilon_{r}=$ 3.9
(dielectric constant of SiO2); third, that the conductor-vacuum (C-V)
interface has a $\epsilon_{r}=$ 10 (value of many metal oxides) and that it is
3 nm thick. We do not know the actual dielectric constant of the C-S
interface, but as will be shown later, this is not important as long as
$\epsilon_{r}\gg 1$.
We find that Eqn. 1 fits the power dependence of the loss for the F- and Cl-
etched resonators if we include a constant loss term in the expression. The
origin of this power-independent loss is, as of yet, not determined. One
possible reason for the loss could be, despite the double-layer magnetic
shielding, trapped magnetic flux in the vicinity of the resonatorsSong2009 .
We find that it varies by two orders of magnitude between the resonators
(ranging from $4\times 10^{-8}$ for resonator Fl2 to $1.8\times 10^{-6}$ for
resonator Cl1). However, since this loss is much less than the TLS loss at low
powers for a given resonator it can be extracted as a fitting parameter.
The power dependence of the loss is well fitted by the use of the calculated
electric field at both the S-V and the C-S interface by changing the critical
electric field. For resonator F1 and F2 we find $E_{c}$ = 8 to 10 V$\cdot$m-1
for the S-V and $E_{c}=12$ V$\cdot$m-1 for the C-S interface. For resonator
Cl1, Cl2 and Cl3 we find $E_{c}=$25 to 40 V$\cdot$m-1 for both interfaces. The
fact that the F-etched and Cl-etched resonator loss data do not fit to the
same $E_{c}$ suggests that the loss is dominated by TLSs in different
environments.
The much greater loss observed for the ion-milled resonator IM can, most
likely, be attributed to the fence-like structure found on the CPW edges. The
fences are formed due to re-deposition of Si onto the edges of the photoresist
during the ion-mill process. After the photoresist is stripped off, the fences
remains on the edges of the CPW and hence causing the higher loss, see in
Figure 2 (d).
To analyze the low-power loss, we calculate the filling factors for the S-V,
C-V and C-S interfaces. The filling factor, $F_{V}$, of region $V$ is the
ratio of the electric energy stored in region $V$ to the total electric energy
stored:
$F_{V}=\frac{\int_{V}\epsilon_{V}|E(\vec{r})|^{2}\mathrm{d}v}{\int_{V_{tot}}\epsilon(\vec{r})|E(\vec{r})|^{2}\mathrm{d}v}.$
(2)
The filling factors depend on the geometry of the devices. It has previously
been shown that the loss is well explained through filling factor arguments as
the resonator trench is changed Vissers2012 . Assuming that all TLSs are
located at the interfaces, the total TLS loss becomes:
$\delta_{TLS}=F_{\mathrm{S\mbox{-}V}}\delta_{\mathrm{S\mbox{-}V}}+F_{\mathrm{C\mbox{-}V}}\delta_{\mathrm{C\mbox{-}V}}+F_{\mathrm{C\mbox{-}S}}\delta_{\mathrm{C\mbox{-}S}}.$
(3)
The filling factors for the different resonators are shown in Figure 3. We
find that the filling factor of the C-V interface is about one order of
magnitude smaller than that of the S-V and C-S interfaces. This agrees with
the result of Wenner et al. Wenner2011 , indicating that the loss at the
conductor surface would have to be one order of magnitude higher than the loss
at the other interfaces in order to dominate. This is interesting, since the
electric field in the Si substrate and the vacuum are nominally the same,
thus, by looking only at Eqn. 2, one can easily be led to be believe the
participation ratios of the top and bottom conductor interfaces should also be
nominally the same. The much lower filling factor of the top surface comes
from the fact that it is the perpendicular displacement field
($D_{\perp}=\epsilon E_{\perp}$), and not the electric field that has to be
continuous at the interface. The imposed boundary condition on $D_{\perp}$
causes the electric field to be either enhanced or suppressed when going to a
region with higher $\epsilon_{r}$ or lower $\epsilon_{r}$, respectively.
Therefore, relatively high $\epsilon_{r}$ of the conductor interfaces compared
to the substrate is desirable to reduce the loss.
In the calculation of the filling factor of the C-V interface, we assumed
$\epsilon_{r}$ = 10. If we instead assume that the top surface is TiOx
($\epsilon_{r}\gtrsim$ 40), the filling factor would be suppressed even
further.
To compare the TLS losses of the resonators we first subtract the power-
independent background loss. We then compare resonators Cl1 and F1, which are
fabricated on the same Si wafer. Using Eqn. 3 and assuming that the loss of
the C-S interface $\delta_{\mathrm{C\mbox{-}S}}$ is equal for the two
resonators, we find that $\delta_{\mathrm{C\mbox{-}S}}\leq 0.4\times 10^{-3}$.
The loss of the F-etched trench is $\delta_{\mathrm{S\mbox{-}V}}\leq 0.9\times
10^{-3}$. Finally, upper and lower bounds for the Cl-etched trench, $1.8\times
10^{-3}\leq\delta_{\mathrm{S\mbox{-}V}}\leq 3.16\times 10^{-3}$ are obtained.
If we compare resonators F2 and Cl2 that are co-fabricated on a different
wafer, we find that $\delta_{\mathrm{S\mbox{-}V}}\leq 1.8\times 10^{-3}$,
$\delta_{\mathrm{C\mbox{-}S}}\leq 0.7\times 10^{-3}$ and $3.45\times
10^{-3}\leq\delta_{\mathrm{S\mbox{-}V}}\leq 5.3\times 10^{-3}$.
In both cases the loss of the F-etched trench is lower by at least a factor of
two than the loss of the Cl-etched trench. Why the loss varies so much between
wafers is not clear. Possible explanations include that the removal of
photoresist was done under different conditions (different removers and
temperatures) or that the assumptions made for the calculations of the filling
factors are not correct. The loss of resonator Cl3 agrees well with the loss
of resonator Cl1, considering the filling factors of the S-V and C-S
interfaces; see figure 3. Since the changes in filling factors are almost
identical for the two interfaces, we cannot quantify the loss contribution
from each region.
The higher loss of the Cl-etched trench is also accompanied by a very high
phase noise of the resonator. We found that the phase noise at 1 kHz of the
Cl-etched trench is two orders of magnitude higher than what would been found
for F-etched niobium resonators of identical geometry.
There are several possible reasons why the Cl-etched surfaces have a higher
loss. One is that the surface layer of the F-etched trench could have a lower
dielectric constant, due to deposition of fluorocarbon polymers during the
etch. This would decrease the filling factor of the S-V region and hence
decrease the contribution to the total loss. Another possible reason for the
higher loss is radiation damage. We have investigated two methods of
decreasing the loss due to the Cl etch. First, we decreased the DC bias during
the etch to 0 V to reduce potential ion damage. This did not notability
decrease the loss of the resonator. Secondly we performed a short (11 seconds)
F etch. This decreased the measured loss of the Cl-etched resonator by more
than a factor of two. These results lead us to believe that the higher loss is
most likely due to the result of a lower etch rate and potentially also the
presence of boron in the etch gas, which could be implanted into the Si
substrate and act as a dopant. The higher etch rate of the F-process is also
preferred, because any induced defects get removed at a higher rate, leaving
fewer defects at the substrate surfaceFonash1990 .
In conclusion, we have investigated how different etch processes affect the
loss of TiN CPW resonators on Si substrates. We found the highest loss for
resonators patterned by an Ar-ion mill. We attribute the high loss to a fence-
like structure found on the edges of the CPW. The fence structures are formed
due to re-deposition of Si onto the photoresist during processing.
The lowest loss was observed for a F-based RIE process. From calculated
filling factors we conclude that the loss of the F-processed Si surface is
lowered by at least a factor of two than that of the Cl-processed surface. We
found that it is the loss originating from the CPW trench that dominates for
the Cl-etched resonators. The trench loss is related to the etch chemistry and
not to the DC bias or the amount of trenching.
These results suggest that even higher quality factors could be achieve by
optimizing the etch as well as by post-etch processing of the resonators.
However, it is also possible that the remaining loss for the F-etched
resonator is dominated by the conductor-substrate interface or even the bulk
loss of the substrate. In this case, the loss could be lower by going to
larger geometries.
This work was supported by the NIST Quantum Information initiative. The views
and conclusions contained in this document are those of the authors and should
not be interpreted as representing the official policies, either expressly or
implied, of the U.S. government.
Table 1: Parameters for the different etches used in the experiment. An 8” ion gun was used for the ion mill. Etch | Pressure [mT] | Gas | Flow [sccm] | Power [W] | DC bias [V] | Etch rate [nm/s]
---|---|---|---|---|---|---
F | 100 | SF6 | 50 | 80 | -68 | TiN/Si: 1/20
Cl | 30 | Cl | 10 | 200 | -200 | TiN/Si: 3/3
| | BCl3 | 30 | | |
Etch | Pressure [mT] | Gas | Flow [sccm] | Beam current [mA] | Beam voltage [V] | Etch rate [nm/s]
IM | 100 | Ar | 40 | 40 | 300 | TiN/Si: 0.033/0.16
Table 2: Extracted parameters for the different resonators (see Fig. 1 (a)). The parameters of the different etches are given in Table 1. The geometric inductance Lg and kinetic inductance Lk are calculated through the method of SheenSheen1991 et al. The best fit is obtained assuming a penetration depth of 245 nm, close to that previously obtained for TiN films(Vissers2010, ). The capacitance is obtained from FEM calculations. Resonator | Etch | Depth | Gap | Width | Undercut | C | Lg | Lk | $\ell$ | fr | Qc
---|---|---|---|---|---|---|---|---|---|---|---
| | [nm] | [$\mu$m] | [$\mu$m] | [nm] | [pF/m] | [$\mu$H/m] | [$\mu$H/m] | [mm] | [GHz] |
IM | Ar | 650 | 2 | 3 | 0 | - | - | - | - | 6.612 | 168k
Cl1 | Cl | 270 | 1.9 | 3.0 | 0 | 176 | 0.42 | 0.52 | 3.318 | 5.58 | 140k
Cl2 | Cl | 200 | 2 | 3.0 | 0 | 187 | 0.42 | 0.42 | 3.318 | 6.02 | 140k
Cl3 | Cl | 40 | 2.1 | 2.7 | 0 | 189 | 0.45 | 1.02 | 3.318 | 4.32 | 300k
Fl1 | F | 1200 | 2.3 | 2.4 | 150 | 124 | 0.47 | 0.71 | 3.114 | 6.29 | 1360k
Fl2 | F | 200 | 2 | 3 | 10 | 183 | 0.42 | 0.44 | 3.05 | 6.53 | 602k
Figure 1: (a) Sketch of a coplanar waveguide structure. Here G denotes the
gap between the ground plane and the centerstrip, W the width of the center
strip, D the depth of the trench, and T the thickness of the TiN film. The
inset shows the top corner of the CPW center strip and illustrates the
position of substrate-vacuum (S-V), conductor-vacuum (C-V), and conductor-
substrate (C-S) interfaces in the cross-section of the CPW. (b) Extracted
internal loss as a function of internal voltage of the resonators described in
Table 2. The different markers represent the different resonators:
($\ocircle$) Fl1, ($\Box$) Fl2,($\rhd$) Cl1, ($\triangle$) Cl2, ($\Diamond$)
Cl3 and ($\times$) IM. The lines are fits of Eqn.1 by the use of the
calculated electric field in region S-V (solid) and region C-S (dashed).
Figure 2: SEM images of different etched samples. (a) Trenched F-etch Fl1. (b)
Trenched Cl-etch Cl1. (c) Non trenched Cl-etch Cl3. (d) Trenched ion milled
IM. From the cross sections it is clear that the F-etched resonator has an
undercut profile that is not observed for the Cl-etched resonators. It can
also be seen that the ion-milled profile has re-deposited material on the top
and sides of the TiN film.
Figure 3: Calculated filling factors of the substrate-vacuum (S-V),
conductor-vacuum (C-V), and conductor-substrate (C-S) interfaces for
resonators F1, F2, Cl1, Cl2 and Cl3. Only a minor part of the total electric
energy is stored at the interfaces with $\sim$ 90 % of the total electric
energy is stored in the bulk of the Si substrate and only $\sim$ 10 % is
stored in the vacuum. The difference in measured loss between the two Cl-
etched resonator Cl1 and Cl3 is well explained by the difference in filling
factors of the S-V and C-S interfaces .
## References
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|
arxiv-papers
| 2012-05-14T19:56:05 |
2024-09-04T02:49:30.905165
|
{
"license": "Public Domain",
"authors": "Martin Sandberg, Michael R. Vissers, Jeffrey S. Kline, Martin Weides,\n Jiansong Gao, David S. Wisbey, and David P. Pappas",
"submitter": "Martin Sandberg O",
"url": "https://arxiv.org/abs/1205.3153"
}
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